yastrebov.fryastrebov.fr/DOCs/ARTICLES/boyce2016ijf.pdfInt J Fract (2016) 198:5–100 DOI...

Int J Fract (2016) 198:5–100DOI 10.1007/s10704-016-0089-7

SANDIA FRACTURE CHALLENGE 2014

The second Sandia Fracture Challenge: predictions ofductile failure under quasi-static and moderate-ratedynamic loading

B. L. Boyce · S. L. B. Kramer · T. R. Bosiljevac · E. Corona · J. A. Moore · K. Elkhodary ·C. H. M. Simha · B. W. Williams · A. R. Cerrone · A. Nonn · J. D. Hochhalter · G. F. Bomarito ·J. E. Warner · B. J. Carter · D. H. Warner · A. R. Ingraffea · T. Zhang · X. Fang · J. Lua ·V. Chiaruttini · M. Mazière · S. Feld-Payet · V. A. Yastrebov · J. Besson · J.-L. Chaboche ·J. Lian · Y. Di · B. Wu · D. Novokshanov · N. Vajragupta · P. Kucharczyk · V. Brinnel ·B. Döbereiner · S. Münstermann · M. K. Neilsen · K. Dion · K. N. Karlson · J. W. Foulk III ·A. A. Brown · M. G. Veilleux · J. L. Bignell · S. E. Sanborn · C. A. Jones · P. D. Mattie ·K. Pack · T. Wierzbicki · S.-W. Chi · S.-P. Lin · A. Mahdavi · J. Predan · J. Zadravec ·A. J. Gross · K. Ravi-Chandar · L. Xue

Received: 2 November 2015 / Accepted: 5 February 2016 / Published online: 14 March 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Ductile failure of structural metals is rele-vant to a wide range of engineering scenarios. Com-putational methods are employed to anticipate the crit-ical conditions of failure, yet they sometimes provideinaccurate and misleading predictions. Challenge sce-narios, such as the one presented in the current work,provide an opportunity to assess the blind, quantita-tive predictive ability of simulation methods against apreviously unseen failure problem. Rather than eval-uate the predictions of a single simulation approach,the Sandia Fracture Challenge relies on numerous vol-unteer teams with expertise in computational mechan-ics to apply a broad range of computational methods,numerical algorithms, and constitutive models to the

B. L. Boyce (B) · S. L. B. Kramer · T. R. Bosiljevac ·E. Corona · M. K. Neilsen · J. L. Bignell · S. E. Sanborn ·C. A. Jones · P. D. MattieSandia National Laboratories, Albuquerque, NM, USAe-mail: [email protected]

S. L. B. Kramere-mail: [email protected]

T. R. Bosiljevace-mail: [email protected]

E. Coronae-mail: [email protected]

M. K. Neilsene-mail: [email protected]

challenge. This exercise is intended to evaluate thestate of health of technologies available for failure pre-diction. In the first Sandia Fracture Challenge, a widerange of issues were raised in ductile failure modeling,including a lack of consistency in failure models, theimportance of shear calibration data, and difficulties inquantifying the uncertainty of prediction [see Boyceet al. (Int J Fract 186:5–68, 2014) for details of theseobservations]. This second Sandia Fracture Challengeinvestigated the ductile rupture of a Ti–6Al–4V sheetunder both quasi-static and modest-rate dynamic load-ing (failure in ∼0.1 s). Like the previous challenge, thesheet had an unusual arrangement of notches and holesthat added geometric complexity and fostered a com-petition between tensile- and shear-dominated failure

J. L. Bignelle-mail: [email protected]

S. E. Sanborne-mail: [email protected]

C. A. Jonese-mail: [email protected]

P. D. Mattiee-mail: [email protected]

J. A. MooreNorthwestern University, Evanston, IL, USAe-mail: [email protected]

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6 B. L. Boyce et al.

modes. The teams were asked to predict the fracturepath and quantitative far-field failure metrics such asthe peak force and displacement to cause crack ini-tiation. Fourteen teams contributed blind predictions,and the experimental outcomes were quantified in threeindependent test labs. Additional shortcomings wererevealed in this second challenge such as inconsistencyin the application of appropriate boundary conditions,need for a thermomechanical treatment of the heat gen-eration in the dynamic loading condition, and furtherdifficulties in model calibration based on limited real-world engineering data. As with the prior challenge,this work not only documents the ‘state-of-the-art’ incomputational failure prediction of ductile tearing sce-narios, but also provides a detailed dataset for non-blindassessment of alternative methods.

Keywords Fracture · Rupture · Tearing · Defor-mation · Plasticity · Metal · Alloy · Simulation ·Prediction · Modeling

K. ElkhodaryThe American University in Cairo, New Cairo, Egypte-mail: [email protected]

C. H. M. Simha · B. W. WilliamsCanmetMATERIALS, Natural Resources Canada,Hamilton, ON, Canadae-mail: [email protected]

B. W. Williamse-mail: [email protected]

A. R. CerroneGE Global Research Center, Niskayuna, NY, USAe-mail: [email protected]

A. NonnOstbayerische Technische Hochschule (OTH) Regensburg,Regensburg, Germanye-mail: [email protected]

J. D. Hochhalter · G. F. Bomarito · J. E. WarnerNASA Langley Research Center, Hampton, VA, USAe-mail: [email protected]

G. F. Bomaritoe-mail: [email protected]

J. E. Warnere-mail: [email protected]

B. J. Carter · D. H. Warner · A. R. IngraffeaCornell University, Ithaca, NY, USAe-mail: [email protected]

D. H. Warnere-mail: [email protected]

1 Introduction

Computational simulations are often called upon to ren-der predictions for a wide range of failure scenarios inmechanical, structural, aerospace, and civil engineer-ing, where full-scale field tests are usually difficult,expensive, and time-consuming. Fracture simulation isdeeply rooted in computational solid mechanics thatcan date back to the 1970s, e.g. Norris et al. (1977).Since that time, there have been ongoing efforts todevelop realistic physical models and efficient compu-tational implementation that improve reliability, speed,robustness and above all, accuracy. However, generat-ing predictions that have adequate confidence levelsstill pose significant difficulties to the simulation com-munity. To elucidate the accuracy of these predictions,it is necessary to evaluate existing models in validationscenarios that approximate the conditions seen in prac-tical applications. For this purpose, Sandia NationalLaboratories has organized a series of fracture chal-

A. R. Ingraffeae-mail: [email protected]

T. Zhang · X. Fang · J. LuaGlobal Engineering and Materials Inc., Princeton, NJ, USAe-mail: [email protected]

X. Fange-mail: [email protected]

J. Luae-mail: [email protected]

V. Chiaruttini · S. Feld-Payet · J.-L. ChabocheOnera, Université Paris-Saclay, Châtillon, Francee-mail: [email protected]

S. Feld-Payete-mail: [email protected]

J.-L. Chabochee-mail: [email protected]

M. Mazière · V. A. Yastrebov · J. BessonMINES ParisTech, Centre des Matériaux, CNRS UMR7633, PSL Research University, Evry, Francee-mail: [email protected]

V. A. Yastrebove-mail: [email protected]

J. Bessone-mail: [email protected]

J. Lian · Y. Di · B. Wu · D. Novokshanov · N. Vajragupta ·P. Kucharczyk · V. Brinnel · B. Döbereiner ·S. MünstermannRWTH Aachen University, Aachen, Germanye-mail: [email protected]

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The second Sandia Fracture Challenge 7

lenges where participants are asked to predict quan-tities of interest (QoIs) in a given fracture scenario.The participants have never seen the challenge sce-nario previously, so the predictions are “blind” as isoften the case in real engineering predictions. More-over, while the scenarios are geometrically simple, theypresent mechanical complexities that are impossibleto predict with intuition or simple calculations alone.After the blind predictions are reported, they are subse-quently compared against experimental results to deter-mine how closely they replicated the fracture behaviorobserved in the laboratory. The current paper describesthe second Sandia Fracture Challenge (SFC2) issuedin 2014 to exercise capabilities in predicting fracture atboth quasi-static and modest dynamic loading rates ina Ti–6Al–4V sheet.

The Sandia Fracture Challenge series has three mainpurposes. Firstly, the Challenge is an assessment ofstate-of-the-art techniques to predict problems involv-ing ductile fracture accurately. These methods covermany models and approaches pursued by academia andindustry to deal with ductile fracture analysis. Sec-

Y. Die-mail: [email protected]

B. Wue-mail: [email protected]

D. Novokshanove-mail: [email protected]

N. Vajraguptae-mail: [email protected]

P. Kucharczyke-mail: [email protected]

V. Brinnele-mail: [email protected]

B. Döbereinere-mail: [email protected]

S. Münstermanne-mail: [email protected]

K. Dion · K. N. Karlson · J. W. Foulk III · A. A. Brown ·M. G. VeilleuxSandia National Laboratories, Livermore, CA, USAe-mail: [email protected]

K. N. Karlsone-mail: [email protected]

J. W. Foulk IIIe-mail: [email protected]

A. A. Browne-mail: [email protected]

ondly, the blind prediction environment offers indi-vidual participating teams an environment of “trueblindness” to evaluate the strengths and weaknessesof their methodology. This unique setup is a preciousopportunity to refine their methods and tools. Thirdly,the Sandia Fracture Challenge has brought togethera group of teams that have been actively working inthe ductile fracture area for many years. Each teamworked independently on the same fracture problem.The collective wisdom obtained by attacking a sin-gle specific task with a variety of strategies strength-ens our understanding of ductile fracture. This Chal-lenge process facilitates identifying current difficul-ties, acquiring experience to avoid certain pitfalls infuture efforts, and fosters collaboration between uni-versities, national laboratories, and industries aroundthe world. It also contributes to a cumulative learningprocess.

In 2012, the first Sandia Fracture Challenge (SFC1)was designed to assess the mechanics community’scapability to predict the failure of a ductile struc-

M. G. Veilleuxe-mail: [email protected]

K. Pack · T. WierzbickiMassachusetts Institute of Technology, Cambridge, MA,USAe-mail: [email protected]

T. Wierzbickie-mail: [email protected]

S.-W. Chi · S.-P. Lin · A. MahdaviUniversity of Illinois at Chicago, Chicago, IL, USAe-mail: [email protected]

S.-P. Line-mail: [email protected]

A. Mahdavie-mail: [email protected]

J. Predan · J. ZadravecUniversity of Maribor, Maribor, Sloveniae-mail: [email protected]

J. Zadravece-mail: [email protected]

A. J. Gross · K. Ravi-ChandarUniversity of Texas at Austin, Austin, TX, USAe-mail: [email protected]

K. Ravi-Chandare-mail: [email protected]

L. XueThinkviewer LLC, Sugar Land, TX, USAe-mail: [email protected]

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tural metal in an unfamiliar test geometry. The resultswere documented in a special issue of the Interna-tional Journal of Fracture (Boyce et al. 2014). Withno direct funding provided to the participants, thir-teen teams with over 50 participants from over 20different institutions responded to the challenge. Thetask was to predict the initiation and propagation of acrack in a ductile structural stainless steel (15-5PH)under quasi-static room temperature test conditions.The test geometry was a flat panel that contained around root pre-cut slot and multiple nearby holes thatcould influence the notch-tip stress state. The place-ment of the holes created a competition between atensile-dominated and shear-dominated failure mode.To calibrate their models, the participants were pro-vided with tensile test data, sharp crack mode-I frac-ture data, and some limited microstructural informa-tion.

The first Sandia Fracture Challenge was designedbased on lessons learned in earlier double-blind assess-ments, where it became obvious that the double-blindevaluation methodology should be governed by somecommon constraints (Boyce et al. 2011). These com-mon constraints also apply to the second Sandia Frac-ture Challenge presented here. They include: (1) Thesample geometry should be readily manufactured witheasily measured geometric features. (2) The manu-facturing process should avoid unintentional compli-cations such as significant residual stresses or non-negligible surface damage. (3) The QoIs, such asforces and displacements, should be readily measur-able with common instrumentation so that the tests canbe repeated in multiple labs in a cost effective man-ner. (4) The experiment should involve simple, uni-axial loading conditions that are readily tested withcommon lab-scale load frames and common grips.(5) The sample and loading conditions should avoidunwanted modes of deformation such as buckling.Since the challenge scenario involves a novel test geom-etry, the repeatability of the behavior may not be appar-ent until after significant experimental effort. In thepresent work and similar, prior efforts at Sandia, theexperiments were not performed until after the com-putational challenge had been issued. This approachensured that all participants, including the experimen-talists, were not biased by any prior knowledge of theoutcome.

The outcome of the first Challenge motivated thissecond Challenge, which explores several new facets:

(1) in addition to quasi-static loading, the scenario alsoinvolves modest dynamic loading spanning three ordersof magnitude in strain-rate, (2) the sharp-crack fracturetoughness data is replaced with V-notch shear failuredata, (3) rather than providing only machining toler-ances based on engineering drawings, the participantswere provided with actual specimen dimension mea-surements, (4) the alloy was changed to a titaniumalloy with low work hardening behavior. Ti–6Al–4Vis a common lightweight, heat treatable, alloy that pro-vides an excellent combination of mechanical prop-erties with high specific strength, corrosion resistance,and weldability, widely used in aircraft, spacecraft, andmedical devices.

As with the first challenge, the second challenge pro-vided all participants with experimental data, for modelcalibration, on the same lot of material used to producethe blind challenge geometries. Two types of mater-ial testing were performed and provided to all partic-ipants. One is the commonly used simple tension testof a dog-bone coupon. Because of possible anisotropy,the simple tension tests were performed in the rolling(longitudinal) direction and the transverse direction. Ashear-dominated test was also performed on a doubleV-notch plate for the rolling and transverse materialdirections. These tests may not be sufficient to cali-brate all parameters for the material constitutive mod-els used by the participants since some of these modelsare complicated and can have several dozen parameters.These materials tests were chosen to mimic a real engi-neering scenario where limited material testing data isavailable, but should be sufficient to characterize theessential parts of the mechanical response for use in anumerical simulation. These tests are all performed atthe two different nominal loading rates that are threeorders of magnitude apart and that are the same as theloading rates used for loading the Challenge sample.

Similar to the first Sandia Fracture Challenge, theSecond Challenge specimen was also a flat plate withmultiple geometrical features that facilitates fractureinitiation and growth. Two anti-symmetric slots per-pendicular to the loading direction were cut. The rootsof these initial slots were rounded to obscure the frac-ture initiation site. Three holes of different sizes weredrilled in the vicinity of the two slots. The spatialarrangement of the holes was chosen to create a com-petition between tensile and shear-dominated failure,so that the failure path was not simply intuitive. Inthe Second Sandia Fracture Challenge, the loading rate

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creates additional complexity. This includes harden-ing, strain rate dependence and the influence of thermaldissipation of plastic work on deformation. These rateeffects were not accounted for in the computationalapproaches in the first Sandia Fracture Challenge, sothese phenomena represented an uncharted area forblind predictions.

As with the first challenge, this second challenge wasperformed under conditions that are generally com-mensurate with typical engineering tasks: (a) the out-come is unknown during the simulation process (theprediction is blind), (b) the time for the project islimited, and (c) the available calibration data is lim-ited. After prediction results are submitted, the adoptedmethodologies are evaluated in their ability to pre-dict macroscopic scalar QoIs such as the peak allow-able force before failure and the amount of compo-nent deflection at various force levels. The experimen-tal results were only shared with the simulation teamsafter their blind predictions had been reported. Afterthe blind predictions and experimental outcomes hadbeen disseminated to all participants, a meeting washeld at the University of Texas at Austin on March 2–3, 2015 to synthesize commonalities that contributedto accurate predictions or systematic errors. The work-shop laid the groundwork for this paper that presentsthe Challenge and the comparison of the predictions.

This paper is organized as follows: Sect. 2 posesthe challenge scenario and describes the experimen-tal testing and results for the material characterization.Section 3 provides details of the geometry, test setups,and the experimental results of the Challenge prob-lem specimen. Section 4 is a synopsis of the modelsand methods used by the participating teams, while thecomparison of the prediction results from each team isgiven in Sect. 5. This is followed by Sect. 6, with dis-cussions on the results and the strength and weakness ofadopted methods. Section 7 provides a brief summary.The detailed contributions of each team’s procedureand their engineering judgment are given in “Appen-dix 1” and detailed measurements of test sample dimen-sions are provided in “Appendix 2” for potential futureassessment.

2 The Challenge

A meaningful, efficient, and ‘fair’ Challenge shouldbe governed by a set of common constraints (Boyce

et al. 2011, 2014). First, this ‘toy problem’ or ‘puz-zle’ should have no obvious or closed-form solution.It should be sufficiently distinct from other standard orknown test geometries so that the outcome of the exer-cise is unknown to the participants. The scenario shouldbe readily confirmed through experiments. It may bedesirable for the challenge scenario to result in a singleunambiguous repeatable experimental outcome, or aswas the case for the first Sandia Fracture Challenge, thescenario could be near a juncture of two competing out-comes. Since the challenge scenario involves a noveltest geometry, the repeatability of the behavior may notbe apparent until after significant experimental effort.In the present work and similar, prior efforts at Sandia,the experiments were not performed until after the com-putational challenge had been issued. This approachensured that all participants (including the experimen-talists) were not biased by any prior knowledge of theoutcome.

2.1 The 2014 Sandia Fracture Challenge Scenario

The fracture challenge was advertised and issuedto potentially interested parties through a mechanicsweblog site, imechanica.org, and through an e-mailsolicitation to many known researchers in the fracturecommunity. The fracture challenge was issued on May30, 2014 and final predictions from participants weredue on November 1, 2014, approximately five monthsafter the issuance of the challenge. The initial packetof information contained material processing and ten-sile test data on mechanical properties, the test spec-imen geometry, the loading conditions, and instruc-tions on how to report the predictions. Supplementalshear test data was released on August 13, 2014. Thedegree of detail provided was intended to be commen-surate with the information that is typically availablein real engineering scenarios in industry. These detailsregarding the material, test geometry/loading condi-tions, and QoIs are described in the following threesubsections.

2.1.1 Material

The alloy of interest was a commercial sheet stockof mill-annealed Ti–6Al–4V. This alloy was chosenbecause it is a commonly used alloy in aerospaceapplications—it is moderately rate sensitive and its

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very shallow work hardening rate can be challeng-ing for computational models. All test specimens wereextracted from a single plate purchased from RTIInternational Metals, Inc. with a specified thicknessof 3.124 ± 0.050 mm. The original material certifi-cation was provided to the participants, and includedthe following chemical analysis (in wt%): C 0.010,N 0.004, Fe 0.19, Al 6.02, V 3.94, O 0.16, and Ti89.676 (Y was also present with less than 50 ppm).The sheet was annealed by the mill first at 381.3 ◦C(718.3 ◦F) for 20 min and then at 419.9 ◦C (787.8 ◦F)for 15 min.

Rockwell C hardness measurements were performedon the Ti–6Al–4V plate. The average of 6 measure-ments was 36.1 HRC (Rockwell C) which is consistentwith mill-annealed Ti–6Al–4V.

2.1.2 Tensile calibration tests

Eighteen tensile coupons were tested, in two orienta-tions and at two loading rates, fast (25.4 mm/s) andslow (0.0254 mm/s), in the Structural Mechanics Lab-oratory at Sandia National Laboratories. Eight tensiletests were oriented along the rolling direction, withthree at the fast rate and five at the slow rate, and tenwere oriented along the transverse direction, with fiveat the fast rate and five at the slow rate. All tests wereconducted according to ASTM E8/E8M-13 using thenominal geometry shown in Fig. 1. The as-measuredthickness and width dimensions of all eighteen tensilecoupons were provided to the participants. The ten-sile specimen tests were conducted using a uniaxialMTS servo-hydraulic load frame with a 100-kN loadcell (±1 % uncertainty of the measured value) at ambi-ent temperature, controlled by an MTS FlexTest Con-troller. The actuator has an internal calibrated linearvariable displacement transformer (LVDT) (±0.5 %uncertainty of the measured value) that measured the

actuator stroke displacement, which was the controlvariable. The specimens were gripped using standardmanual wedge grips. The velocity of the fast-rate testas measured by the actuator LVDT in the tension testswas also confirmed using digital image correlation totrack fiducial markings on the grips imaged by a high-speed Phantom 611 camera. Strain was measured usingan extensometer with a 25.4 mm gage length. Engi-neering stress–strain curves were provided, as well asthe raw force-displacement data for each tensile test.Figure 2 shows the resulting engineering stress–straincurves for all eighteen tests. The yield strength, ulti-mate strength, and elongation values are presented inTable 1.

The test data indicates that there is a dependenceof the tensile behavior on the loading rate and to alesser extent on loading direction. Optical and electronmicroscope images of the fracture surface morphol-ogy (Fig. 3) as well as side-view macro images of thenecking profile and fracture plane (Fig. 4) were alsoprovided to the participants.

2.1.3 Shear calibration tests

Eight shear specimens were tested in the StructuralMechanics Laboratory in the same load frame as thetensile samples. Four were loaded in shear in the rollingdirection (denoted as VA for the remainder of this sec-tion), with two at the fast loading rate and two at theslow loading rate. Four were loaded in shear transverseto the rolling direction (denoted as VP for the remain-der of this section), with two at the fast loading rate andtwo at the slow loading rate. The geometry of the shearspecimens were based on ASTM D7078/D7078M-05 with the V-notched rail shear geometry modified(deeper notch than the standard) to reduce the stressarea by more than half and allow induced failureat lower forces to minimize grip rotation and mini-

Fig. 1 Tensile bargeometry used to providestress–strain data for modelcalibration. Dimensions arein millimeters. Meanthickness of tensilespecimens was 3.150 mm

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0

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TD1TD5TD8TD11TD12

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Fig. 2 Engineering stress–strain curves for eighteen tensile coupons. “RD” and “TD” refer respectively to those oriented along therolling direction and the transverse-to-rolling direction

mize grip slippage. The test specimen geometry forthe shear test is shown in Fig. 5. The as-measureddimension of the shear specimens were provided to theparticipants.

As shown in Fig. 6, stacked rosette strain gages,with a gauge length of 3.18 mm, were affixed to thecenter of the V-notch shear test specimens and to the

left (relative to the front face) of the center. The rosettestrain gages to the left of the front face were stackedfor four of the test specimens and unstacked for theremaining three test specimens. The rosette gages onthe front were paired with gages on the back. Theelastic shear modulus was calculated using the cen-tral stacked strain gage rosette measurements for the

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Table 1 Average tensileyield strength, ultimatestrength, and strain atfailure at two loading ratesand in two orthogonalloading directions

Direction Loading rate 0.2 % offsetyield

Ultimate tensilestrength

Strain at failure

mm/s MPa MPa mm/mm (%)

Rolling 0.0254 1016 1065 17.0

Transverse 0.0254 1045 1056 17.9

Rolling 25.4 1121 1140 15.9

Transverse 25.4 1150 1151 15.8

shear strain and an assumption that the shear stress wasthe measured load divided by the V-notch gage area.The elastic shear modulus from these calculations was44 GPa, which is consistent with literature values forTi–6Al–4V.

The shear test fixture was a commercial off the shelf“Adjustable Combined Loading Shear (CLS) Fixture”from Wyoming Test Fixtures. In this fixture, each testspecimen was held in place by a grip on each side. Eachgrip had two face grip inserts pressed against the frontand back face of the test specimen and one horizon-tal insert. The specimen was held in place by tighten-ing 5/8-18 UNF stainless bolts on the face grip insertsto 67.8 N m torque and hand tightening 1/2-20 UNFstainless bolts to secure the horizontal insert for eachgrip. The grip fixtures were made of 17-4PH stainlesssteel and were rigidly attached to the load frame. Fig-ure 7 shows a close up view of the shear test setup,including the grip inserts on the fixed side. An axialLVDT mounted in the back of the fixture betweenthe two grip halves provided the overall displace-ment measurement and the load cell provided the loadmeasurement.

Two issues arose when performing the shear tests.The first was that the shear specimen slipped withinthe grips. The second was that the fixture itself had anon-negligible compliance. Additional tests (includingtests on non-notched rectangular specimens to quantifythe fixture compliance and cyclic load tests to quan-tify the slip) were performed, and those detailed mea-surements were provided to the participants. As dis-cussed later, most participants made use of the axialLVDT displacement vs. load data provided either inthe direct form as shown in Fig. 8, or with a slipcorrection that was provided. The non-ideal behav-ior of this shear test method highlights the need todevelop better shear testing standards for metal failurecharacterization.

Figure 9 shows the post-test failure surfaces ofselected shear specimens. As seen from the figure, thefailure surfaces are at a slight angle with respect to theloading direction. The VA specimens showed a largerangle of the failure surface than the VP specimens.Additionally, the VA slow specimens visually showeda rougher failure surface than the other specimens.

2.1.4 Fracture challenge geometry and loadingcondition

The Challenge geometry consisted of an S-shaped sheetspecimen with two notches and three holes, as shown inFig. 10 with detailed dimensions. The notch locationswere labeled “A” and “B”, in Fig. 11, and consist of a6.35 mm wide notch of overall length 28.575 mm. Thesize of the three holes for “C”, “D”, and “E” were nomi-nally 3.175, 3.988, and 6.35 mm, respectively. In addi-tion, ‘knife-edge’ features were added to each notchedge for the purpose of mounting Crack Opening Dis-placement (COD) gages. Three holes were introducedinto the S-shaped geometry to provide multiple com-peting crack initiation sites and propagation paths ateach notch location.

Pin holes were machined away from each notch tipfor the insertion of an 18-mm diameter loading pin.These pin holes provided for standard clevis grip load-ing in a hydraulic uniaxial load frame. Clevis grips con-forming to ASTM E 399 were used; these grips haveD-shaped holes with flats to provide a rolling contactthat minimizes friction effects. It is important to notethat each computational team and experimental testinglab was provided no additional constraints regardingthe decision of how to apply boundary conditions. Thislimited definition of the boundary conditions bears sim-ilarity to real world engineering problems, where thedetailed boundary conditions are rarely well defined.This limited definition of the boundary conditions for

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Fig. 3 Images of fracture morphology for the tensile spec-imens with different loading rates and material orientations:a–d are optical images from a Keyence microscope and e–fare SEM images. a RD4—25.4 mm/s loading rate. b TD11–

25.4 mm/s loading rate. c RD7—0.0254 mm/s loading rate. dTD4—0.0254 mm/s loading rate. e RD8—0.0254 mm/s loadingrate. f RD9—25.4 mm/s loading rate

the Challenge geometry would prove to be a source ofdiscrepancy among the participants.

The participants were instructed that the FractureChallenge would evaluate two different displacement

rates of 25.4 and 0.0254 mm/s, spanning 3 orders ofmagnitude in strain-rate. Any undeclared aspects ofloading that were salient to the outcome were consid-ered as sources of potential uncertainty.

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Fig. 4 Images of thegeometry of the necking forthe tensile specimens withdifferent loading rates andmaterial orientations. Note:The grid lines in each imagehave a spacing of 5.08 mm,and each sample had a gripvertical height of 12.6 mm. aRD7—0.0254 mm/s loadingrate. b TD4—0.0254 mm/sloading rate. cRD4—25.4 mm/s loadingrate. d TD12—25.4 mm/sloading rate

Fig. 5 Specimen geometry for shear tests. Dimensions are inmillimeters. The average plate thickness for these specimens was3.100 mm. VA specimens have the sheet rolling direction parallel

to the 55.88 mm length, and the rolling direction in VP specimensis perpendicular to the rolling direction in VA specimens

2.1.5 Quantities of interest

A set of quantitative questions were posed to the par-ticipants to facilitate comparing the analyses to theexperimental results. These questions were meant toevaluate the robustness of the analysis technique inpredicting deformation, necking conditions, crack ini-tiation, and crack propagation. All challenge partic-ipants were requested to provide the QoIs embed-

ded in the following six questions to facilitate quan-titative evaluation and analysis of the blind predic-tions:

For each of the two loading rates, please predict thefollowing outcomes:

Question 1: Report the force at the followingCrack-Opening-Displacements (COD):

• COD1=1-, 2-, and 3-mm.

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Fig. 6 Strain gageconfigurations for the shearspecimens. a Sample VA1with stacked rosettes in thecenter and to the left of thecenter. b Sample VP4 withstacked rosettes in thecenter and adjacent gages tothe left of the center

(a)

(b)

Actuator Mo�on

Load Cell

Axial LVDT 1

Specimen Touching Edge of Opening

Horizontal Grip Insert

Fig. 7 Close view of the shear test facility showing the axialdisplacement measurement cell LVDT 1

Question 2: Report the peak force of the test.Question 3: Report the COD1 and COD2 valuesafter peak force when the force has dropped by 10 %(to 90 % of the peak value).

Fig. 8 Load versus displacement response of shear failure cali-bration tests

Question 4: Report the COD1 and COD2 valuesafter peak force when the force has dropped by 70 %(to 30 % of the peak value).Question 5: Report the crack path (use the featurelabels in Fig. 11 to report crack path).

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Fig. 9 Post-test images ofshear specimens at slow andfast loading rates in bothdirections (ruler scale:smallest division =0.254 mm)

Question 6: Report the expected force-COD1 andforce-COD2 curves as two separate ASCII datafiles.

A COD gage was used to monitor load-line displace-ment at the point of the ‘knife-edge’ features, akin tofracture toughness testing. Figure 11 shows the loca-tions of COD1 and COD2. The COD measurement

was defined for the participants in the following way:“COD1 and COD2= 0 at the start of the test; the CODvalues refer to the change in length from the beginningof the test”. All participants were also asked to reporttheir entire predicted force-COD displacement curve.These force-COD curves provided further insight intothe efficacy of the methods.

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Fig. 10 Dimensions of the second Sandia Fracture Challenge S-shape specimen geometry in millimeters

Fig. 11 Plate orientation,actuation direction, CrackOpening Displacement(COD) gauges and legendof features labeled A–G

3 Experimental method and results

The behavior of the challenge specimens were evalu-ated by three different experimental teams. The primaryresults are those from the Sandia Structural Mechan-ics Laboratory, with additional experiments performedat the Sandia Material Mechanics Laboratory and the

laboratory of Prof. Ravi-Chandar at the University ofTexas at Austin. All three laboratories tested specimensthat had been fabricated at the same time from the sameplate of material. All three sets of experimental datawere generally consistent with one another with regardto the force-displacement behavior and observed fail-ure path. The primary differences in the tests from each

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laboratory can be summarized as follows: the SandiaStructural Mechanics Lab performed in situ imagingfor both loading rates, the Sandia Material Mechan-ics Lab provided independent confirmation tests withadditional local thermal measurements in the high-deformation zones, and the University of Texas Labperformed optical measurements to capture local andglobal kinematic fields for the slow loading rate. Thefollowing subsections cover the pretest characteriza-tion of the challenge specimens, the details of the exper-imental methods used in each laboratory to test them,and post-test characterization of the specimens.

3.1 Observations from the Sandia StructuralMechanics Laboratory

3.1.1 Test setup and methodology

Pre-test geometry measurements All of the challengespecimens were fabricated by the same machine shop,where the specimens were inspected and measuredto ensure compliance with the fabrication drawing.Detailed specimen dimensions were provided to theparticipants and are tabulated in “Appendix 2”.

Test setup in structural mechanics laboratory In theStructural Mechanics Lab, eight specimens were testedat a loading rate of 0.0254 mm/s, and seven specimenswere tested at a loading rate of 25.4 mm/s. The testsetup is shown in Fig. 12. These tests were conductedin displacement control using the same uniaxial MTSservo-hydraulic load frame and 100-kN load cell as wasused for the tensile and shear tests. The actuator LVDTwas the control variable. Clevis grips from MaterialsTesting Technology (model number ASTM.E0399.08)were made of 17-4PH stainless steel with a pin diam-eter of 17.93 mm. These clevis grips conformed to theASTM E 399 testing standard, as prescribed by theChallenge. Each grip was securely mounted in the loadframe using spiral washers to allow for individual rota-tional alignment of each clevis grip relative to the uni-axial load-train. Precision-cut spacers were used oneither side of the sample at each clevis to center thesample in each clevis. Prior to testing, the clevises werealigned using a strain gaged dummy sample, loaded in

Fig. 12 Experimental test setup in the Structural MechanicsLaboratory

its elastic regime. The relative displacement betweenthe two clevis grips was measured during each testusing an LVDT mounted on each side of the grips. Theclevis grip LVDTs were calibrated at the time of usewith a Boeckeler Digital Micrometer, with ±0.508 μmresolution and repeatability within ±0.508 μm. TwoCOD gages from Epsilon Tech Corp. (SNs E93896,S93897) were used to measure the opening of the twonotches in the challenge specimens. The Epsilon CODgages were calibrated at the time of use with a StarretMicrometer, with ±0.508 μm resolution and repeata-bility within ±2.54 μm.

In the fast loading rate tests, a vibrational responsewas induced in the COD gages, thus causing oscilla-tions in the recorded output. The frequency of the oscil-lation was approximately 440 Hz. The frequency of the

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output signal matched what was recorded in the tests.To remove the oscillations, the COD data from all fast-rate tests were filtered using a low-pass, second-orderButterworth filter with cutoff frequency of 220 Hz.

A sequence of images of the deforming samples wascollected for both the slow and fast tests. The slow-ratetest imaging was controlled by Correlated SolutionsVicSnap software with NI-DAQ synced data collectionfrom the MTS FlexTest Controller. The imaging setupincluded two Point Grey Research 5MP Grasshoppermonochromatic CCDs (2448 × 2048 pixels) with 50-mm Schneider lenses, viewing the front and back sur-faces of the samples, with a frame rate of 2 fps. Thefast-rate test imaging included a Phantom V7 high-speed camera (800×600 pixels) with a 24–85 mm lens,viewing the front of the samples, and controlled byPhantom camera software based on a trigger sent bythe MTS FlexTest controller at the start of the test,with a frame rate of 3200 fps and a 0.3-ms exposure.Black and white fiducial markings applied to the sam-ple close to the COD gage knife edges enabled a sec-ondary COD measurement via a digital image corre-lation (DIC) algorithm (Chu et al. 1985) based on acustom image tracking script in a software packagecalled ImageJ (Schneider et al. 2012). Since the mark-ings were not precisely at the COD knife edges, thesewere not used to determine the COD measurements

that served as the basis for the Challenge questions. Inhindsight, these markings were useful for capturing theCOD measurement after the Epsilon COD gages hadfallen off the samples at the onset of unstable crackgrowth. The displacements from the COD gages andthe DIC tracking of the markings will be compared inthe next section.

3.1.2 Test results and observations

Load versus COD profiles Seven of the eight samplestested at 0.0254 mm/s and all seven of the samplestested at 25.4 mm/s in the Structural Mechanics Labora-tory failed along the B–D–E–A path defined in Fig. 11,while Sample 30 tested at 0.0254 mm/s failed along theA–C–F path. Figure 13 contains the load-COD1 pro-files for all of the samples. For the samples that failedvia the B–D–E–A path, the COD1 gage fell off the sam-ple at the onset of unstable crack growth, so the load-COD1 profiles are all truncated to the crack initiationpoint. The load-COD1 profile of Sample 30 has beentruncated to when the crack first initiated (in the A–Cligament). At each loading rate, the load-COD profilesfor the B–D–E–A samples show that these sampleshave repeatable behavior through peak load, with vari-ation in the COD1 value at crack initiation. In compar-ing the responses from the two loading rates, the faster

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COD1 values when crack propagated

unstablyB-D-E

{

COD1 value when crack propagated

unstablyA-C

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Sample 09Sample 15Sample 18Sample 22Sample 24Sample 25Sample 27

For

ce (

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COD1 values whencrack propagated unstably

{(a) (b)

Fig. 13 Load versus COD1 measurement for all samples tested in the Structural Mechanics Laboratory: a samples tested at 0.0254 mm/s;b samples tested at 25.4 mm/s

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loading rate led to higher peak loads, smaller COD1values at unstable crack initiation, and less spread inCOD1 values at unstable crack initiation.

The failures of the B–D and D–E ligaments occurrednearly simultaneously and were indistinguishable in theload-displacement data. The sequential still images col-lected during testing did not provide sufficient temporalresolution to clearly identify which ligament failed first.The high speed camera used during the high rate test-ing lacked the needed spatial resolution to definitivelyidentify the first ligament failure. Later post-challengeobservations at the University of Texas indicated thatligament D–E likely failed first for the slower rate loadcase (Gross and Ravi-Chandar 2016), as described indetail in “Post-Challenge experimental observationsfrom the Ravi-Chandar Lab at University of Texas atAustin” section of “Appendix 2”. The load-COD1 pro-file of Sample 30 falls in line with all the other sam-ples up to failure, with the A–C ligament failing at least0.5 mm earlier than any B–D–E failure in the other sam-ples. Figure 14 compares the load-displacement pro-files for Samples 11 and 30. The load-COD1 profilesare remarkably similar up to the A–C failure of Sample30, but the load-COD2 profiles deviate earlier at around1.4 mm, well before peak load. The load-clevis LVDTbehavior is nearly identical up to the crack initiationin notch A of Sample 30. The early failure of Sample

30 is thought to be an outlier based on fractography, aswill be discussed later in Sects. 3.1 and 3.3.

Figures 15 and 16 show the load-COD measure-ments for both COD gages with six associated in situimages for Samples 11 and 27, representing typicalsample behavior for each displacement rate. The force-COD1 data was used extensively for evaluation of pre-dictions in Sect. 5, and the COD2 curves and corre-sponding images are included here for completeness.For Sample 11 at the 0.0254 mm/s displacement rate,the COD1 gage, tracking the opening of notch B wherea crack formed, lags behind COD2 for images 1 and2, catching up by image 3, and then increasing morerapidly until failure. Alternatively, for Sample 27 at the25.4 mm/s displacement rate, the COD1 gage laggedfor most of the high-loading regime, except immedi-ately before failure. For both loading rates, consider-able strain localization is evident in the B–D, D–E andA–C ligaments as the tests progressed. Videos of thetests for Samples 11 and 27 are available in the Sup-plementary Information.

After the conclusion of the challenge, Team E per-formed follow-up DIC analysis of the fiducial mark-ing motion for Samples 11 and 27 using their cus-tom feature tracking algorithm in ImageJ. The corepurpose of this analysis was to evaluate the changein COD with force after unstable fracture when the

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Sample 11 Clevis LVDT

Sample 30 Clevis LVDT

For

ce (

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Clevis LVDT Displacement (mm)

Initiation from Notch A

Initiation from Hole C

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Comparison of Samples 11 and 30

Sample11 COD1Sample 11 COD2Sample 30 COD 1Sample30 COD 2

For

ce (

N)

COD (mm)

Initiation from Notch A

Initiation from Hole C

(a) (b)

Fig. 14 Load versus displacement for the Samples 11 and 30 with two different crack paths: a displacement is measured by CODgages; b displacement is measure by the clevis LVDT

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Sample 11 - Slow Rate

COD1COD2

For

ce (

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Displacement (mm)

1

23 4

5

0

0 1 2

3 4 5

Fig. 15 Load versus COD measurements for Sample 11 with associated in situ images

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Sample 27 - Fast Rate

COD1COD2

For

ce (

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Displacement (mm)

1

23

4

5

0

0 1 2

3 4 5

Fig. 16 Load versus COD measurements for Sample 27 with associated in situ images

clip gage extensometers detached. The fiducial markingdisplacements were slightly less than the COD gagessince the fiducial markings were inboard of the knifeedges in each notch. Figure 17 includes the load ver-sus COD1/COD1-DIC (the fiducial marking displace-ments across the notch with the COD1 gage) curvesfor Samples 11 and 27. COD1-DIC measurements aresimilar in profile to the COD1 measurements. For Sam-ple 27, the frame rate was fast enough to capture afew measurements during the load drop; the load ver-sus COD1-DIC curve at the first failure shows that

the load immediately drops with very little change inCOD1. This near-vertical unloading associated withunstable cracking is used later to compare to modelpredictions.

3.2 Confirmation observations from the SandiaMaterial Mechanics Laboratory

The Sandia Material Mechanics Laboratory providedfour tests to confirm the results independently from

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COD1COD1-DIC

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Displacement (mm)

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Sample 27 - Fast Rate

COD1COD1-DIC

For

ce (

N)

Displacement (mm)

Fig. 17 Load versus displacement for COD1 and DIC Fiducial Markings for Samples 11 and 27

the Sandia Structural Mechanics Laboratory: three atthe slow displacement rate (Samples 13, 23, and 29)and one at the fast displacement rate (Sample 20).The tests were performed on a MTS 311.21 four-postload frame under LVDT displacement control usinga FlexTest 40 digital controller. The LVDT had arange of ±100 mm (SN 645LS) with a maximum cali-brated error of ±0.35 % across the entire range. TheInterface model 1020AF load cell (SN 3069) had acapacity of 55,600 N with a maximum calibrated errorof ±0.45 % in tension. The same COD gages wereused by both labs, although the gages were indepen-dently calibrated at the time of testing in the Mate-rial Mechanics Laboratory with a MTS Calibratormodel 650.03 with a maximum calibration error of±1 %. The clevis grips were the same model used bythe Sandia Structural Mechanics Laboratory. Angularand concentric misalignments between the two gripswere compensated using a MTS model 609 align-ment fixture. Three-dimensional (3-D) printed plas-tic spacers ensured that the test coupon was locatedin the center of the wide clevis grips. Type K ther-mocouples with 0.25 mm wide junction tips were spotwelded in the center of the B–D ligament (Position 1)and in the A–C ligament (Position 2) to monitor thethermal history during the tests for Sample 20 and29.

All four samples tested in the Material Mechan-ics Laboratory failed along the B–D–E–A path. Theload-COD1 profiles for these samples are plotted withthe data from the Structural Mechanics Laboratory inFig. 18. Good agreement between the data demon-strates repeatability of the experimental results at twoindependent labs.

Figure 19 contains the thermal histories of ligamentsB–D (Position 1) and A–C (Position 2) of Samples 20and 29 during the tests measured via the thermocouplesinstalled on the test samples. Generally, the temperaturerise on the surface in the B–D ligament during the slowrate test was less than 10 ◦C, whereas the temperaturerise during the fast rate test was greater than 45 ◦C.For both samples, the temperature rise in the B–Dligament was greater than that in the A–C ligament.These thermal measurements are only approximate val-ues; there was no diagnosing the thermal resolution ofthe thermocouple readout. The thermocouple size was1-mm; the position may not have been precisely at themidpoint in the ligament between the notch and hole.The instantaneous spikes at the end of the slow rate dataand during the deformation process in the fast rate testare artifacts likely caused by the motion of the thermo-couple wire and not representative of the actual tem-perature. Beyond these details, it is important to see themajor trends of greater temperature rise in the fast ratetest and for the B–D ligament where the samples failed.

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Material MechanicsSample 29



Structural MechanicsSamples in Gray

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Experiments - Fast

For

ce (

N)




(a) (b)

Fig. 18 Comparison of force-displacement curves measured by the two Sandia mechanical testing labs

Fig. 19 Thermal histories of ligaments B–D and A–C: a Sam-ples 29 at the slow displacement rate; and b Sample 29 at the fastdisplacement rate. Note that scales for the axes are not the same.

The thermal rise in the fast test is delayed after the mechani-cal event presumably due to the rate of thermal conduction totransfer to the thermocouple

3.3 Collation of experimental values for the challengeassessment based on both Sandia Laboratories’testing

Prior to collating the experimental values for the QoIsof the Challenge, two aspects of the experimental

results require additional discussion. The first aspectis that one of the specimens followed the path A–C–F,while all the rest of the specimens followed the pathB–D–E. While it may be tempting to consider the oneexperiment indicating path A–C–F to be an outlier,and not representative of the response of the specimen,

123


any such decision must be based on proper evaluationof the underlying reasons for the observed response.We recall that in the 2012 Challenge, one specimenexhibited a different path from all the rest, but a care-ful analysis indicated that this response was the nomi-nal response to the challenge problem, while the otherdominant response observed in the experiments wasdue to systematic deviations in the specimen geome-try. In the present Challenge, Sample 30 exhibited pathA–C–F, while all others followed the path B–D–E–A;geometrical measurements identified this specimen tohave a greater out-of-plane warpage than all other spec-imens. However, no causal link has been establishedto connect the lack of flatness to the observed failure.Fractographic evaluation of the failed surface A–C ofSample 30 indicated that the failure was not consistentwith typical tensile failures observed previously (seesection “Fractographic observations” of Appendix 2).Based on these factors, it is considered that the fail-ure along A–C is not the expected or nominal responseof the specimen and the correct failure path identifiedfrom the experiments is along path B–D–E–A. The sec-ond issue is that since the COD gages fell off the sam-ples at the initiation of unstable crack growth at a forcegreater than 90 % of the peak load, two of the QoIsposed in the original challenge—the COD1 and COD2values when the load had dropped by 10 and 70 % ofthe peak load—were not directly measured; only thequestions related to the force at COD1 of 1, 2, and3 mm, the peak load and the crack path could be usedfor comparison to the computational predictions. Addi-tional QoIs, not listed in the original Challenge that helpcharacterize the failure, become necessary in order tocompare the experiments and simulations. Such addi-tional QoIs must be identified carefully so as to main-tain the original intent of the challenge. While thereare many options for this selection, it is clear that theexperimental results themselves point to key featuresof the response that should be predictable: all exper-iments indicated unstable crack initiation and growththrough the ligaments B-E and D-E. Therefore, threeadditional QoIs are chosen from the experiments: thebinary quantity on the nature of crack initiation (sta-ble/unstable), and the force and COD1 values at theinitiation of unstable crack growth. The specific valuesof the QoIs obtained from the challenge experimentsare summarized in Tables 2 and 3, including the forceand COD1 values when unstable crack propagationbegan. As can be seen from the results, the forces

obtained at the different COD levels, at the peak andat onset of unstable fracture, do not exhibit a largescatter: about ±3.2 % of the average for the slow-rate tests and ±1.8 % of the average for the fast-ratetests. In contrast, the COD value at onset of unsta-ble crack propagation had significantly greater varia-tion, with the range being ±19 % of the average forthe slow-rate tests and ±5.4 % of the average for thefast-rate tests. This scatter in the load and COD valuesis intrinsic and points to an important issue in under-standing the nature of the problem and generating mod-els: fracture is a much more stochastic process thandeformation.

4 Brief synopsis of modeling approach

In this section, a brief synopsis that compares and con-trasts the varied approaches employed for predictionis provided. Each approach illustrated in subsequentappendices has been partitioned into methods and mod-els in this section. The methods component highlightsthe character and solution of the partial differentialequations while the modeling component focuses onthe constitutive models requisite for localization, crackinitiation, and unstable propagation. The selected par-tition and array of preferred terminology only seeks tobe helpful. One must be careful to note that disparatesolutions can stem from the application of the samemethods and model.

The finite element method was employed by allteams except Team K who instead employed the Repro-ducing Kernel Particle Method (RKPM). Althoughmost teams employed dynamics to characterize theunstable process, some did adopt a quasi-static approachthat neglected inertial effects. Both implicit and explicitschemes for time integration were used. These char-acteristics are grouped into Solver in Table 4. Higherrates of loading can lead to temperature effects whichcan be modeled through the inclusion of energy bal-ance. The Coupling label differentiates between theinclusion of thermo-mechanical coupling (segregated)and approaches that employ variations of the adia-batic assumption. Teams that did not utilize a cou-pled thermo-mechanical model attempted to capturethe effects of temperature through the material fittingprocess. The application of the loading is categorizedas Boundary conditions. An important boundary con-dition in the model is the method used to define the

123


Table 2 Summary of the load and COD measurements for slow rate testing from the two Sandia Labs

Testing Lab Sample Crack path Force atCOD1=1 mm

Force atCOD1=2 mm

Force atCOD1=3 mm

Peakforce

Force atunstablecrack growth

COD1 atunstablecrack growth

N N N N N mm

0.0254mm/s rate testing

StructuralMechanics Lab

4 B–D–E–A 14,950 19,460 19,860 19,870 19,330 3.358

6 B–D–E–A 14,990 19,230 – 19,410 19,230 2.819

7 B–D–E–A 15,010 19,360 19,630 19,660 19,200 3.149

11 B–D–E–A 15,180 19,430 19,670 19,710 19,300 3.128

14 B–D–E–A 15,070 19,420 19,650 19,690 18,730 3.208

19 B–D–E–A 14,970 19,450 19,630 19,710 19,150 3.181

28 B–D–E–A 15,170 19,450 19,630 19,710 19,380 3.088

30a A–C–F 15,200 19,460 – 19,610 19,370 2.252

MaterialMechanics Lab

13 B–D–E–A 14,240 19,010 – 19,443 19,150 2.824

23 B–D–E–A 14,980 19,280 19,470 19,560 18,970 3.096

29 B–D–E–A 14,700 19,460 – 19,704 19,470 2.479

Minimum of B–D–E–A – 14,240 19,010 19,470 19,410 18,730 2.479

Average of B–D–E–A B–D–E–A 14,926 19,355 – 19,647 19,191 3.033

Maximum of B–D–E–A – 15,180 19,460 19,860 19,870 19,470 3.358

a While Sample 30 is included in the table for completeness, that result is considered to be an anomaly and is not included in theMin/Max/Avg values or subsequent comparison to predictions

Table 3 Summary of the load and COD measurements for fast rate testing from the two Sandia Labs

Testing Lab Sample Crack path Force atCOD1=1 mm

Force atCOD1=2 mm

Force atCOD1=3 mm

Peakforce

Force atunstablecrack growth

COD1 atunstablecrack growth

N N N N N mm



9 B–D–E–A 15,640 20,310 – 20,320 19,480 2.387

15 B–D–E–A 16,200 20,650 – 20,660 19,800 2.453

18 B–D–E–A 16,130 20,360 – 20,390 19,560 2.338

22 B–D–E–A 15,770 20,440 20,440 19,220 2.481

24 B–D–E–A 16,030 20,230 – 20,320 19,390 2.275

25 B–D–E–A 15,860 20,560 – 20,560 19,570 2.537

27 B–D–E–A 15,790 20,480 – 20,480 19,550 2.512


20 B–D–E–A 15,780 20,320 – 20,317 19,210 2.502

Minimum – 15,640 20,230 – 20,317 19,210 2.275

Average B–D–E–A 15,900 20,419 – 20,436 19,473 2.436

Maximum – 16,200 20,650 – 20,660 19,800 2.537

123


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ion

Bou

ndar

yco

nditi

ons,

mat

eria

lmod

el,

mat

eria

lpar

amet

ers

HIm

plic

itdy

nam

ics

The

rmo-

mec

hani

cal

Con

tiguo

us,d

efor

mab

le,

rota

ting

Hex

8Fu

lly-i

nteg

rate

dIP

:0.1

75m

m,T

T:

0.17

5m

mD

OF:

600

KE

lem

entd

elet

ion,

surf

ace

elem

ents

Mat

eria

lpar

amet

ers

IIm

plic

itdy

nam

ics

No

Surf

ace

node

s,fix

ed+

tran

slat

ion

Hex

8,R

educ

edin

tegr

atio

nIP

:0.2

5m

mT

T:0

.50

mm

DO

F:13

5KE

lem

entd

elet

ion

&cr

ack

band

Bou

ndar

yco

nditi

ons,

mat

eria

lpar

amet

ers

JE

xplic

itdy

nam

ics

Mix

edis

othe

rmal

/adi

abat

icC

onta

ct,f

rict

ionl

ess,

rigi

dH

ex8,

redu

ced

inte

grat

ion

IP:0

.1m

mT

T:0

.1m

mD

OF:

2500

KE

lem

entd

elet

ion

Mat

eria

lpar

amet

ers

KE

xplic

itdy

nam

ics

No

Surf

ace

node

s,fix

edR

KPM

w/s

tabi

lized

conf

orm

ing

noda

lin

tegr

atio

n

IP:0

.44

mm

,TT

:1.5

6m

mD

OF:

160

KPa

rtic

levi

sibi

lity

Det

erm

inis

tic

LE

xplic

itdy

nam

ics

No

Con

tact

,fri

ctio

n,de

form

able

Qua

d4,f

ully

inte

grat

edIP

:0.2

5m

mT

T:(

2DE

lem

ents

)D

OF:

200

KX

FEM

+co

hesi

vesu

rfac

eM

ater

ialP

aram

eter

s

ME

xplic

itdy

nam

ics

The

rmo-

mec

hani

cal

(hig

hra

te)

Con

tact

,fri

ctio

nles

s,ri

gid

Hex

8,re

duce

din

tegr

atio

nIP

:0.0

45m

mT

T:

0.14

2m

mD

OF:

2100

KE

lem

entd

elet

ion

Mat

eria

lpar

amet

ers

and

geom

etry

NE

xplic

itdy

nam

ics

No

Con

tact

,fri

ctio

n,ri

gid

Hex

8,re

duce

din

tegr

atio

nIP

:0.2

5m

mT

T:0

.24

mm

DO

F:21

60K

Ele

men

tdel

etio

nB

ound

son

mat

eria

lm

odel

123


pin loading. Methods included specimen-pin contact,a contiguously meshed pin, and the direct applicationof essential boundary conditions to the surface of thespecimen. For approaches that employ contact, the pinis specified as deformable or rigid and the specimen-pin interface is classified as having friction or beingfrictionless. In Table 4, a label describing the friction-less contact of a rigid pin would be contact, frictionless,rigid. Less rigorous approaches avoid the complexityof contact through a contiguously meshed pin. Thisassumption can also employ a deformable or rigid pin.In an attempt to approximate specimen rotation relativeto the pin, the contiguously meshed pin can both trans-late and rotate (about its centerline). The label rotat-ing communicates that the contiguously meshed pinis both rotating and translating while the label fixedimplies a contiguously meshed pin that can translatebut not rotate. Thus, the label contiguous, rotating,deformable would highlight a contiguously meshed,deformable pin that can rotate about its centerline. Thelast simplification involves the direct application of dis-placements to surface nodes. Although this approachcan accommodate translation, the direct applicationof displacements on a patch of nodes will constrainspecimen rotation and is thus labeled surface nodes,fixed.

In addition to specifying the balance laws andboundary conditions, the element type, discretization,and numerical approach for crack initiation and prop-agation is highlighted. Regarding element technology,approaches that employ full integration and reducedintegration via Element type are distinguished. In thisbrief summary, fully integrated refers to the devia-toric response. The pressure is almost always on alower-order basis to avoid volumetric locking. Reducedintegration implies element formulations that seek toreduce computational time through a single integra-tion point and require hourglass stiffness and/or vis-cosity to suppress zero-energy modes (Reese 2005).The level of refinement is noted in Discretization.The in-plane element size, through-thickness elementsize, and degrees of freedom are labeled IP, TT, andDOF, respectively. If a single element was employed inthe through-thickness direction, 2D is appended. Themethod employed for free surface creation is labeledFracture methodology. The most common methodol-ogy employed was element deletion. Elements maybe loaded and unloaded (via damage evolution) priorto deletion. Teams employing criteria (of loaded ele-

ments) may have also employed a crack-band modelfor unloading. Crack band models attempt to main-tain a common measure of surface energy (Bazant andPijaudier-Cabot 1988). Teams also employed surfaceelement approaches which regularize the effective sur-face energy through a characteristic length scale. Thosepaths were seeded or adaptively simulated throughX-FEM (Dolbow and Belytschko 1999). The frac-ture process originates from smooth notch having abounded stress concentration. After crack initiation,the process is over-driven. This particular problemmay exhibit less sensitivity to non-regularized meth-ods (de Borst 2004) because solutions will be less sen-sitive to the modeled fracture resistance. The final cat-egory, Uncertainty, summarizes each team’s effort tocharacterize uncertainty in the methods and/or models.For this effort, all teams employed bounds to charac-terize uncertainty. Those bounds are partitioned intogeometry, boundary conditions, and material parame-ters.

The solution of the balance laws requires the spec-ification of a constitutive model. Both the literatureand the provided data characterize Ti–6Al–4V sheetas anisotropic and having rate and temperature depen-dence for the applicable loading rates. The flow charac-teristics of the models used by the different teams arenoted under Plasticity in Table 5. The yield functionand the hardening, labeled Yield function and Hard-ening, respectively, may have rate and temperaturedependence. Table 5 contrasts yield functions whichemploy a single invariant (von Mises, J2), or multi-ple invariants (J2/J3) with anisotropic yield surfaces[Hill (Hill 1948), Cazacu–Plunkett–Barlat (Cazacuet al. 2006)]. The labels Hill and CPB06 representthe Hill and Cazacu–Plunkett–Barlat yield surfacesrespectively. The hardening is characterized through afunctional form. There is no distinguishing betweenapproaches that employ only the equivalent plasticstrain or additional internal state variables for hard-ening. Models for fracture/failure are partitioned intoCriteria and Damage evolution. Criteria refers to thefailure metric for a material point. The post-failureresponse may be abrupt or be governed by an additionallaw for unloading (such as the crack-band model).Damage evolution attempts to communicate if mod-els include the micromechanics of the failure processthrough a softening response termed Damage accu-mulation. That response may depend on the evolutionof an additional internal state variable (such as void

123


Tabl

e5

Mod

els

empl

oyed

byte

ams

part

icip

atin

gin

the

Sand

iaFr

actu

reC

halle

nge

Team

Plas

ticity

Frac

ture

/fai

lure

Cal

ibra

tion

Yie

ldfu

nctio

nH

arde

ning

Rat

e/Te

mp

Cri

teri

aD

amag

eev

olut

ion

AJ 2

Pow

er-l

awY

es/Y

esIn

com

patib

ility

Dam

age

accu

mul

atio

nTe

nsio

n

BJ 2

/CPB

06Sw

ift

Yes

/No

Xue

-Wie

rzbi

cki,

stra

inD

amag

eac

cum

ulat

ion

Tens

ion

&sh

ear

CJ 2

Piec

ewis

elin

ear

No/

No

Tri

axia

lity-

depe

nden

tfa

ilure

locu

scu

rve

Exp

onen

tialu

nloa

ding

(cra

ckba

nd)

Tens

ion

&sh

ear

DJ 2

Piec

ewis

elin

ear

Yes

/No

Plas

ticst

rain

Tra

ctio

n-di

spla

cem

entl

awTe

nsio

n

EHill

Mul

tiple

exp

Yes

/No

Nor

mal

,she

arst

reng

thT

ract

ion-

disp

lace

men

tlaw

Tens

ion

&sh

ear

FJ 2

/J 3

Pow

er-l

awY

es/Y

esB

ai-W

ierz

bick

i,st

rain

Dam

age

accu

mul

atio

nTe

nsio

n&

shea

r

GJ 2

Piec

ewis

elin

ear

Yes

/Yes

Cri

tical

plas

ticst

rain

Dam

age

accu

mul

atio

nw

/5st

eps

for

unlo

adin

gTe

nsio

n,re

ject

edsh

ear

HJ 2

Lin

ear-

exp

Yes

/Yes

Cri

tical

void

volu

me

frac

tion

Voi

dnu

clea

tion,

grow

th,

coal

esce

nce

Tens

ion

&sh

ear

IH

illPo

wer

-law

Yes

/No

Bai

-Wie

rzbi

ckis

trai

n(t

abul

ar)

Lin

ear

unlo

adin

g(c

rack

band

)Te

nsio

n&

shea

r

JH

illSw

ift+

Voc

eY

es/Y

esH

osfo

rd/C

oulo

mb

stra

inD

amag

eac

cum

ulat

ion

Tens

ion

(pla

s,fr

ac),

shea

r(pl

as)

KJ 2

Bi-

linea

rY

es/N

oPr

inci

pals

trai

nN

oTe

nsio

n

LJ 2

Pow

er-L

awY

es/N

oN

orm

al,s

hear

stre

ngth

Tra

ctio

n-di

spla

cem

entl

awTe

nsio

n&

shea

r

MH

illSp

line

Yes

/Yes

Tri

axia

lity

depe

nden

tstr

ain

No

Cou

pled

tens

ion+s

hear

(pla

st,f

rac)

NJ 2

Swif

tY

es/N

oX

ue(2

009)

Dam

age

accu

mul

atio

nTe

nsio

n

123


volume fraction) or a post-processed quantity. Con-tinuum damage mechanics approaches are contrastedwith methods that lump damage evolution into an effec-tive traction-displacement law. The Calibration labelin Table 5 communicates if the team employed theprovided tension and shear data for model calibra-tion.

5 Comparison of predictions and experiments

5.1 Comparison of scalar quantities of interest andcrack path

In real-world engineering scenarios, modeling is oftenused to predict scalar performance metrics such as themaximum allowable service load that a component cansupport or how far the component can be deformedbefore it will fail. Motivated by this, the challenge sce-nario specified certain scalar QoIs to be reported, aspreviously described in Sect. 2. Table 6 compares thescalar QoI predictions of all 14 teams to the exper-imentally measured range. In this table, several keyQoIs are tabulated: the applied load when COD1 wasequal to 1 and 2 mm, the peak applied load, and thecrack path for both the slow and fast loading rates. Inthe original challenge, the teams were asked to pre-dict the COD1 value when the load had dropped by 10and 70 % below peak load. However, as discussed ear-lier, these QoI’s were not experimentally measurablequantities due to the dynamic nature of the unstablecrack initiation and growth. This was the reason for theformulation of alternate QoIs described in Sect. 3.2.Therefore, Table 6 includes two additional columnsfor these additional QoIs: COD1 and force values atthe point of unstable crack initiation. These QoIs werereadily extracted from the force-COD curves reportedby each team. A sudden drop in the load at fixed COD1level was taken as indication of prediction of unsta-ble fracture; the force and COD1 corresponding to thistransition was taken to be the prediction of failure loadand COD1. Some teams predicted a gradual drop inthe load, with either an increasing or decreasing COD1level, indicative of stable fracture. In these cases, it wasnot possible to evaluate the alternative QoI.

The expected value column for the scalar QoI met-rics reported in Table 6 are color coded to indicate ifthe predicted values were consistent with experimen-tal observation. Black values were within the range

of experimental scatter, with an additional buffer of±10 % added to the experimental range. This some-what arbitrary buffer is intended to allow for the pos-sibility of ‘valid’ scenarios outside of the limited num-ber of experimental observations. Numbers reportedin the expected value column that are color codedred or blue indicate predictions that are high or low,respectively, compared to the buffered experimentalrange.

Figures 20 and 21 provide a graphical representa-tion of the tabulated data from Table 6, comparingeach team’s predictions (points) to the upper and lowerbounds for the experimental range (horizontal dashedlines). Figure 20 assesses the QoI metrics for earlydeformation up to the point of necking whereas Fig. 21assesses the QoI metrics for the first unstable crackevent. In addition to the numerical data, an indicationis made if the team did not predict the B–D–E–A crackpath, or if the team predicted a stable crack growthprocess (e.g. slowly evolving damage) rather than anunstable crack event. Four teams predicted crack pathsthat deviated from the B–D–E–A. One team (Team N)predicted a failure path for the slow rate test of A–C–F, and for the high rate test of B–D–E–A. Two otherteams predicted an A–C–F failure path for both theslow and fast rate tests (Team C and G). One team(Team F) predicted a mixed failure path (for the slowrate test, A–C failure, followed by partial failure alongthe B–D–E path, and then finally through the remain-ing ligament, C–F; and for the high rate test B-D-Efailure, followed by failure along the A–C ligament,and then finally failure through the remaining E–A lig-ament).

5.2 Comparison of force-COD curves

Force-COD1 curves can provide additional insightsinto the efficacy of the various modeling approaches.Figure 22 provides a comparison between the experi-mentally measured force-COD1 curves and those pre-dicted by the teams. While this figure provides anoverview of the extent of prediction scatter, it is dif-ficult to assess any specific team’s direct comparisonto experimental observations. For that purpose, Figs. 23and 24 provide a team-by-team comparison of the sameforce-COD1 curves for the slow and fast loading rates,respectively. A more detailed discussion comparing the

123


Table 6 Comparison of blind predictions to experimental values for both the slow and fast loading rates

(a) Slow (0.0254 mm/s) Loading Rate.

(b) Fast (25.4 mm/s) Loading Rate.

Quantities of Interest (QoI's) Prescribed in the Initial Challenge Additional Post-Challenge QoIs

Force at Unstable Crack (N)

COD1 at Unstable Crack (mm)

Minimum Average Maximum Minimum Average Maximum Minimum Average Maximum Min/Avg/Max Min/Avg/MaxTest Data

14240 14951 15200 19010 19365 19460 19410 19643 19870 18730/19191/19470 2.479/3.033/3.358

TeamLower Bound

AverageUpper Bound

Lower Bound

Expected Value

Upper Bound

Lower Bound

Expected Value

Upper Bound

Lower Bound

Expected Value

Upper Bound

Expected Value Expected Value

A 15000 16300 17700 20100 21500 23000 23500 23900 24300 23300 5.1

B 16655 20982 21156 19763 20411 23741 20137 22940 23753 22500 2.1C B-D-E-A A-C-F A-C-F 16300 17800 17800 20200 23000 23000 21600 24000 24000 22100 not path BDEA 3.15 not path BDEAD --- B-D-E-A B-D-E-A --- 18796 19255 --- 21748 23688 --- 21831 24495 17000 3.6E --- B-D-E-A B-D-E-A --- 14830 15250 --- 19150 19300 --- 19320 19450 192005 not unstable 2.935 not unstableF 15988 16150 16311 20579 20787 20994 21779 21999 22219 21900 not path BDEA 3.42 not path BDEA

GA-C-F

orB-D-E-A

A-C-FA-C-F

orB-D-E-A

16364 16364 18579 21120 21120 22604 20938 21516 22935 21100 not path BDEA 2.65 not path BDEA

H 14530 14998 15167 18420 19399 19832 19182 20244 20884 19300 4.4I B-D-E-A --- 15250 --- --- 19314 --- --- 20388 --- 20000 3.45J3 B-D-E-A B-D-E-A A-C-F 15625 15875 16125 19350 19630 19910 20085 20615 21145 19800 4.4K --- B-D-E-A --- --- 17087 --- --- 22584 --- --- 24231 --- 22700 4.0L4 992 1055 1118 1623 1727 1830 1854 1973 2091 19336 not unstable 3.66 not unstableM 15430 15530 15630 19460 19860 20260 20950 21570 22180 21500 4.6N 14000 15300 17000 19000 21159 23000 20000 21651 23000 20800 not path BDEA 2.8 not path BDEA

B-D-E-AA-C-F

B-D-E-A

I.D. Crack PathForce at COD1 = 1 mm

(N)Force at COD1 = 2 mm

(N)Peak Force of Test

(N)

B-D-E-A(10 of 11 Samples)1

B-D-E-A

B-D-E-A

A-C-(B-D-E)-F2

B-D-E-A

Quantities of Interest (QoI's) Prescribed in the Initial Challenge Additional Post-Challenge QoIs

Force at Unstable Crack (N)

COD1 at Unstable Crack (mm)

Minimum Average Maximum Minimum Average Maximum Minimum Average Maximum Min/Avg/Max Min/Avg/Max

Test Data

15640 15900 16200 20230 20418 20650 20317 20436 20660 19210/19473/19800 2.275/2.436/2.537

TeamLower Bound

Expected Value

Upper Bound

Lower Bound

Expected Value

Upper Bound

Lower Bound

Expected Value

Upper Bound

Expected Value Expected Value

A --- 19300 --- --- 29800 --- --- 33600 --- 35600 5.1B 17230 22163 22435 10058 20202 24983 20019 24268 25232 23700 1.9C B-D-E-A A-C-F A-C-F 15200 17200 17200 17900 21700 21700 20300 22400 22400 24000 not path BDEA 3.07 not path BDEAD --- B-D-E-A B-D-E-A --- 19734 21381 --- 23565 25612 --- 23816 27284 18500 3.5E A-C-F B-D-E-A B-D-E-A 15400 15590 15900 20230 20350 20450 20230 20440 20550 197005 not unstable 2.95 not unstableF 16397 16732 17066 21329 21764 22200 23081 23552 24023 21600 not path BDEA 2.78 not path BDEA

G A-C-F A-C-FA-C-F

orB-D-E-A

16897 16956 19433 22070 22070 23431 21366 22208 23456 21100 not path BDEA 2.15 not path BDEA

H 15194 15648 15816 19221 20260 20925 19251 20310 21179 18400 3.0I --- B-D-E-A --- --- 15955 --- --- 20664 --- --- 20960 --- 20500 3.0J3 A-C-F B-D-E-A B-D-E-A 16440 16680 16920 20255 20645 20000 20230 20670 22500 19800 2.6K --- B-D-E-A --- --- 16599 --- --- 21304 --- --- 22801 --- 24200 4.0L4 971 1079 1187 1549 1721 1893 1953 2088 2298 20436 not unstable 3.76 not unstableM 15800 15900 16000 20240 20650 21060 20460 21090 21720 20300 4.1N 14300 15857 17400 20000 21935 24000 20500 22722 25000 21500 2.7B-D-E-A

B-D-E-A

Crack Path

B-D-E-AB-D-E-A

A-C-(B-D-E)-F2

B-D-E-A

Force at COD1 = 1 mm(N)

Force at COD1 = 2 mm(N)

Peak Force of Test(N)

B-D-E-AB-D-E-A

I.D Crack Path

Blue colored numbers highlight expected value predictions that are more than 10 % below the average measured experimental value,whereas red colored numbers highlight expected value predictions that are more than 10 % above the average measured experimentalvalue.1 Minimum, average, and maximum values based only on samples that failed by the B–D–E–A path.2 Parenthesis indicate that a second crack path developed during the fracture process (along the path indicated in brackets) but that thefinal fracture occurred on the initial fracture path.3 Expected values reported for Team J are an average of the “Max” and “Min” expected values reported by the team.4 Team L realized after the submittal deadline that their initial submittal values were in error due to a mistake they made in summingreaction forces to get their applied load values.5 For Team E, values for Force and COD1 at unstable cracking were based on inflections in the force-COD1 curve, even though unstablecracking was not predicted6 For Team L, the unstable fracture condition was not apparent, so instead the failure condition was estimated here as the values whenthe force had dropped to 98 % of its peak value

123


Fig. 20 Evaluation ofplasticity predictions up tothe onset of necking:comparison of blindpredictions to the range ofexperimental data

predictions to the experimentally measured values iscontained in Sect. 6.

6 Discussion

The goal of the present study was to assess the pre-dictive capability of modeling and simulation method-ology when applied to a model engineering problemof ductile deformation and fracture. The success ofthe predictive tools relies on five successful elements:

(1) realistic physical constitutive models for defor-mation and failure, (2) accurate calibration of modelparameters based on available data, (3) proper numer-ical implementation in a simulation code, (4) repre-sentative boundary conditions, and (5) correct post-processing to extract desired quantities. The chosenproblem explored several phenomenological aspects ofmaterial behavior such as elasticity, anisotropic plas-ticity with coupled heat generation, localization, dam-age initiation, coalescence, crack propagation, and finalfailure. Furthermore, the damage and failure process

123


Fig. 21 Evaluation ofpredictions of the firstunstable fracture event:comparison of blindpredictions to the range ofexperimental data. Note thatthere were no upper orlower prediction bounds,since these quantities wereassessed after the resultshad been reported

considered a competition between a tensile mode fail-ure process and a shear mode failure process. The chal-lenge was open to the public and agnostic with regardto the computational methods employed. The challengemerely postulated a specific set of QoI’s to be predictedbased on the provided geometry, material properties,and loading conditions. This report combined with theoutcome of the previous challenge provide an assess-ment of the state-of-the art in failure modelling andareas for improvement.

6.1 Assessing agreement and discrepancy betweenpredictions and experiments

6.1.1 Generic categorization of potential sources ofdiscrepancy

In the context of the present challenge, multiple rep-etitions of data were provided on the material cali-bration tests to enable treatment of material propertyuncertainty. The as-machined specimen dimensions of

the challenge geometry were also provided to enablebounding based on dimensional variability. No guid-ance was provided on how to use this uncertainty infor-mation and different teams chose differing approachesdepending on engineering judgement and availabletime/resources. Most teams only utilized what was con-sidered as the most representative features of both thematerial response and the geometry in calibration andsimulation. Only one team, Team M, reported that theyhad intentionally explored geometric variations acrossthe range of reported specimen dimensions to helpbound their blind predictions. The representation of thecompliant boundary condition for the shear calibrationtest and the pin-loaded contact boundary condition forthe challenge geometry were addressed differently byeach team, resulting in different responses, even in theelastic regime. Some teams accounted for anisotropy ofthe plastic response of the material through the use ofthe Hill48 or equivalent models, whereas other teamsconsidered only the isotropic yielding model. The deci-sion was based on engineering judgment and/or modelavailability. Similarly, some teams ignored the heat

123


Fig. 22 Combinedcomparisons of force-CODpredictions (colored lines)to experimentalobservations (gray circles).a Slow (0.0254 mm/s)loading rate. b Fast(25.4 mm/s) loading rate

generation during high-rate deformation while otherschose adiabatic or coupled-thermomechanical condi-tions. Some teams incorporated strain-based damageinduced softening of the plastic response, while oth-ers used a strain-to-failure criterion based on a dam-age indicator function, in the spirit of the Johnson-Cook model. This lack of consensus on the modelingassumptions and techniques, as compared in Tables 4and 5, results in significant heterogeneities in the pre-dictions. Nevertheless, overall predictions were moreconsistent and in line with the experimental resultsthan in previous challenges. The general improve-ment in predictions is likely due, at least in part, from

learning and model improvement as a result of priorchallenges.

Elastic stiffness Perhaps one of the most surprisingoutcomes of the challenge was that several teams pre-dicted a stiffness in the elastic regime that deviated>10 % beyond the experimental slope. When predic-tions were in error, they were all consistently too stiffrelative to the experimental stiffness. For most if not allof the teams in error, the stiffness was overestimateddue to an unrealistic representation of the pin loadingboundary condition (e.g. no free rotation at the pin). A

123


Fig. 23 Individual teamcomparisons of force-COD1predictions (black lines) toexperimental observations(colored lines) for the slow(0.0254 mm/s) loading rate

comparison of the results for all of the teams suggeststhat if similar, realistic pin-loading boundary condi-tions had been adopted by the teams all of the resultswould be more consistent with the experimental elasticslope. It is also interesting to speculate that the elastic

slopes were better modelled in the previous challengebecause data from a compact tension specimen withpin loading was provided to calibrate models for theblind prediction.

123


Fig. 24 Individual teamcomparisons of force-COD1predictions (colored lines)to experimentalobservations (gray circles)for the fast (25.4 mm/s)loading rate

Yield surface The yield behavior and all post-yielddeformation/failure processes can depend on extrin-sic factors of the environment and loading condi-tions, including strain-rate, temperature, triaxiality,

Lode angle, and load-path history. In addition, yieldbehavior can depend on intrinsic aspects of the materialitself such as grain size, grain shape, crystal structure,phase distribution, etc. Rarely are all of these extrin-

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sic and intrinsic factors described in a single plastic-ity model, due in part to the limited calibration datathat is available, and also to uncertainty on the propermathematical descriptions. In the absence of a compre-hensive, universally-accepted plasticity law, and all thenecessary data to calibrate its parameters, each teamchooses to represent some subset of the dependencies.Two characteristics of the challenge material responsethat were included by some teams in their constitutivemodel description of the material are the dependence ofthe yield locus on Lode angle and material anisotropy.Wrought titanium alloys are known to display plas-tic anisotropy. The test data provided, be it the uni-axial tension test data or the shear test data, indicatedsome anisotropy in the material’s response. Compari-son of the tension test data against the shear test dataalso indicated dependence of the yield surface on Lodeangle. Several teams made note of this behavior, point-ing out that use of a von Mises yield locus (which hasno dependence on Lode angle) resulted in a poor fitbetween the model response and the shear test datawhen the tensile test data was utilized to determinethe model parameters. This is because the challengematerial exhibited yielding under shear loading at astress that was approximately 0.88 times lower thanthat predicted by a von Mises model (indicating thatthe yield surface for this alloy is closer to a Tresca likeyield surface). To account for these response charac-teristics (anisotropy and Lode angle dependence) sev-eral teams (B, E, F, I, J, and M) made use of mater-ial models that are capable of capturing these depen-dencies. (Teams C and H accounted for some of thesedependencies in an approximate way, using differentsets of material properties/models in different regionsto account for the expected material response charac-teristics.) Most of these teams (all, except B and C)made use of the Hill plasticity model, with its abilityto capture anisotropy and/or Lode angle dependence ofthe yield locus. While teams that accounted for LodeAngle (J3) dependence, and/or sheet anisotropy gener-ally produced better elastoplastic predictions, the Chal-lenge may not have been able to sufficiently discrimi-nate between relative importance of these two differentcontributions. Team I noted that the Hill yield func-tion could be utilized to account for the Lode angledependence of the yield locus without incorporationof material anisotropy, with acceptable results. Analy-ses by Team H and Team J concluded that an isotropicmodel would be more likely to predict failure along the

incorrect path A–C–F, whereas the models with lodeangle dependent yield loci were necessary to drive fail-ure to the correct B–D–E–A path. The results suggestthat models incorporating Lode angle dependence ofthe yield locus were able to predict the elasto-plasticresponse more accurately (perhaps prior to the onsetof localization). Predictions of loads and CODs byteams using the von Mises yield criterion (which hasno lode angle dependence) differed by about 10 %.Several teams also incorporated anisotropy in additionto Lode angle dependence of the yield locus. Basedon the QoIs requested, it is difficult to ascertain ifthe inclusion of anisotropy for the challenge problemwas truly necessary; however, post-challenge assess-ments carried out by one team suggest that alternate,more intrusive QoIs (such as inter-ligament strains)are more sensitive to the incorporation of anisotropyand if used may have resulted in a more conclu-sive determination of the importance of anisotropy(Gross and Ravi-Chandar 2016).

Work-hardening Calibration and extrapolation of thehardening behavior was typically handled either byfitting specific functional forms or by a spline/multi-linear approach. Models that represent the stress–straincurve through splines or other piecewise functions pro-vide greater flexibility. On the other hand, functionalforms of work hardening such as the Swift and Voceforms are perhaps cheaper computationally becausethey reduce the number of simulations required for cal-ibration and uncertainty quantification. While the addi-tional flexibility associated with the added parametersin a spline or piecewise-linear approach may at firstseem more accommodating, the alternative functionalforms were generally more successful in the elasto-plastic regime. In fact, among the teams that had themost accurate quantitative predictions for the elasto-plastic regime (Teams E, H, I, J), none of them used aspline or piecewise-linear approach. In discussion, oneteam noted that the flexibility of the extra degrees offreedom in a spline or piecewise linear approach canbe challenging to calibrate given limited data, can bemore difficult to assess in terms of uncertainty quan-tification, and can also be difficult to incorporate intocoupled physics models such as a thermomechanicalcoupling.

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Thermal effects This was crucial for this class of mate-rials, especially under the moderate rate loading con-ditions, where the conversion of plastic work to heat,as well as heat conduction play a role in the localresponse. Load versus COD curves in the fast-rate caseindicated a rounded response at the peak in the force-displacement curve rather than being flat-topped as inthe slow-rate case. Some teams that incorporated thethermo-mechanical coupling (e.g. Teams G, H, and M)or adiabatic conditions (Team J) were able to better cap-ture this feature at the faster rate. Several teams reportedambiguity in determining the appropriate thermal workcoupling parameter, i.e. Taylor–Quinney coefficient.

Localization The experiments indicated a clear peak-load and geometric softening associated with localiza-tion of the structural response. The localization wassubtle in the slow-rate force-COD curves and morepronounced at the fast-rate. Most teams predicted somedegree of localization at least in a qualitative sense. Thisis not too surprising, because the onset of localizationis essentially a geometric effect due to inter-ligamentnecking and a sufficiently fine mesh should capture thisfeature. The fidelity of models for work hardening andthermal softening will impact the accurate predictionof the onset of necking, and this is a potential source ofdiscrepancy. It bears emphasis that the Sandia tests pro-vide information about the onset and process of local-ization in both the tensile and shear testing that wascarried out, but the lack of full-field measurements ofstrain in the challenge sample prevent further valida-tion of the model with regard to onset and progressof localization. Follow-up experiments at the Univer-sity of Texas at Austin provided this detailed full-fieldligament deformation data (Gross and Ravi-Chandar2016).

Crack initiation and propagation The prediction QoImetrics for this challenge focused on elastoplasticityand the conditions for initial unstable fracture froma smooth notch. When the crack did initiate, it wasoverdriven and propagated catastrophically to the nextarrest feature. The elements of this challenge could besuccessfully navigated without wrestling with resis-tance curve behavior or incorporating a crack lengthscale and the related mesh dependence of sharp crackbehavior. For this reason, the current challenge does notdelve into the relative merits of different crack propa-

gation modeling approaches. Most teams had markedsuccess in predicting the correct propagation path andin predicting the unstable rupture. Incorrect predic-tions may be attributed to inadequate yield surface,work hardening response or boundary conditions. Theadopted crack initiation models fall into three cate-gories: a threshold strain value, voids, or damage. Inthe two instances of the cohesive zone and the XFEM-based XSHELL model, cracks and their location areassumed at the onset of the computation so initiationis not a consideration. Crack propagation was mod-eled using element deletion, XFEM, and the cohesivezone approaches. Based on the load-COD predictions,there are discrepancies in the onset and propagation ofcracks. A key source of the discrepancy is the trans-lation of model parameters from the tension and sheartests. For most mesh-based methods there is difficultyscaling the mesh size commensurately for the calibra-tion and challenge geometries so that the calibrationparameters translate appropriately. Cohesive zone andthe XFEM approaches seek to bypass these difficul-ties. At the post-challenge workshop, there was con-siderable discussion on the tradeoffs between relativelysimple models for failure that may lack fidelity or uni-versality versus more sophisticated models that may bemore challenging to calibrate or are less mature.

Numerical methods The finite element method was themethod used by all of the teams, except Team K whoused the Reproducing Kernel Particle Method. Mesheswith fine zones in regions of potential crack propaga-tion in the challenge sample and meshes with similardensities to model the calibration tests were employedwith the expectation that mesh-dependence would bemitigated. Only two teams that adopted the XFEM anda non-local approach tried to mitigate the dependenceof results on the mesh characteristics that is expectedin failure modelling. This remains a source of discrep-ancy and is suggested as an item for improvement forblind predictions in future challenges. A second aspectis the time integration scheme. With the exception offour teams, all other teams used explicit time integra-tion; mass-scaling was used to reduce computationaltime. The degree of discretization varied widely, withthe in-plane element sizes spanning an order of mag-nitude from 0.05–0.5 mm, and the number of degreesof freedom varying by more than two orders of mag-nitude from 30 K to >3 M. While a detailed sensitiv-ity study would be needed to fully address the dis-

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cretization tradeoff for a given method, there was notan immediately clear general trend that more elementsand higher spatial discretization consistently improvedaccuracy. The tradeoff between computational cost andprediction accuracy is specific to the individual methodemployed. It is not trivial to achieve Einstein’s com-plexity balance that “everything should be made as sim-ple as possible, but not simpler”.

Post processing An often undiscussed source of dis-crepancy is the process of extracting the correct quan-tities, translating them into desired units, and commu-nicating them correctly. This can be particularly prob-lematic with a looming deadline, as the post-processingoccurs in haste and rudimentary mistakes are made. Ina previous challenge for example, one team reportedQoIs in incorrect units. In the current challenge, TeamL accidentally accounted for reaction forces at onlyone boundary node rather than summing the forces onall boundary nodes. This mistake was not discovereduntil after the comparison to the experimental predic-tions. While the team requested to report the correctedsummed values, we have also included their originalmistake for the purpose of illustrating this point. Eventhe most accurate model can be rendered useless ifthese mistakes are not painstakingly avoided. In criticalapplications where the model has implications such asloss of life, it is recommended to utilize at least twoindependent prediction teams to provide a peer reviewor cross-check for glaring discrepancies.

6.1.2 Overview of agreement between predictions andexperiments

It is possible to assess the efficacy of each team’s pre-diction methodology comparison to the experimentaloutcome. The prescribed QoI’s that were readily mea-sured are the metrics associated with elasticity, plas-ticity, work hardening, and the onset of necking (peakforce). With respect to the metrics associated with plas-tic deformation up to peak force, half of the teams(E, G, H, I, J, M, N) were able to successfully pre-dict all prescribed quantitative metrics within the 10 %buffered experimental range. This result reinforces thenotion that the elastoplastic behavior is not trivial tocorrectly capture. In fact, all seven teams that did notcapture the peak force, were also outside the bufferedexperimental range in their early deformation predic-tion at COD1= 1 mm. Six of those seven teams over-

predicted the early deformation and subsequently over-predicted the peak force of the test. As discussed pre-viously, one of the most common culprits for thisoverprediction was inadequate pin-loading boundaryconditions.

Of the seven teams that predicted the elastoplasticresponse, five (E, H, I, J, M) were able to also pre-dict the B–D–E–A crack path for both loading rates.Based on these prescribed QoI’s that were experimen-tally measured, all five teams could claim success inthe fracture challenge. However, there were additionalpost-Challenge QoI’s that shed additional light on theability to predict the onset of cracking. Of the fiveteams that successfully predicted all the deformationand crack path metrics, four of the teams (E, H, I, J)also predicted the forces associated with crack advance.All four of these teams overpredicted the COD1 valuefor cracking for at least one of the two loading rates.This highlights a general tendency for overpredictingthe COD1 value for cracking: of the eight teams thatpredicted unstable B–D–E–A cracking, seven overpre-dicted the COD1 value of cracking for at least one ofthe loading rates. This outcome suggests the need for amore comprehensive, accurate suite of failure calibra-tion tests.

6.1.3 Uncertainty bounds

To this point, the assessment has focused largely ona comparison of the predicted ‘expected’ values forthe quantities of interest. However, each team was alsoallowed to report uncertainty bounds for their predic-tion. The use of uncertainty bounds is an importantengineering tool to represent the potential sources oferror in the prediction so that they do not result in mis-leading engineering interpretation.

The uncertainty bound also plays a psychologicalrole in conveying the degree of confidence in a predic-tion. For example, reporting an uncertainty bound thatvaries by a factor of 10 from the lower to upper boundhelps the user of the data understand that the result isonly an order of magnitude estimate, whereas reportinga uncertainty bound that only differs by 1 % from thelower to upper bound suggests a much higher degreeof confidence in the prediction. When teams reporteduncertainty bounds, they typically ranged from ∼5–20 % variation on the expected value, however therewere cases where the uncertainty bounds were ∼1 %

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and other cases where the uncertainty bounds werequite large. Team B, for example, predicted an expectedvalue for force at COD1= 2 mm (fast rate) of 20,202 N,remarkably consistent with the experimental rangeof 20,230–20,650 N. However, for that same quan-tity, Team B bounded their prediction from 10,058 to24,983 N, more than a factor of 2 different from thelower bound to the upper bound. From a practical per-spective, such a broad uncertainty window may leadengineers to overcompensate for the low (underpre-dicted) allowable minimum force.

It is possible to assess the efficacy of the pre-dicted uncertainty bounds in their ability to bracketthe experimental outcomes. It is only possible to makethis assessment on the pre-fracture QoIs associatedwith plasticity and the onset of necking (COD1=1 mm, COD1= 2 mm, and Peak Force), since the pre-scribed post-fracture QoIs were not measured exper-imentally. There were 14 teams that reported predic-tion on these 3 QoIs at each of the 2 different load-ing rates, resulting in a product of 84 predicted out-comes. Of these 84 predictions, the predicted ‘expectedvalue’ fell within the experimentally observed range inonly 17 (20 %) of the 84. Of these 84 predictions, 69were reported with uncertainty bounds and the remain-ing 15 predictions were reported as expected valueswith no declaration of an uncertainty bound, presum-ably because the team ran out of time to assess thebounds. Of these 69 reported values with uncertaintybounds, there were 16 cases where the expected valuewas outside the range of experimental values, but thebounds successfully overlapped, at least partially, withthe experimental range. To put this in perspective,while 20 % of the 84 predictions had expected val-ues that fell within the experimental range; another19 % (16/84) of the predictions had employed boundsthat included some portion of the experimental range.The positive conclusion is that the use of uncertaintybounds roughly doubles the chances of capturing theoutcome. Conversely, the negative conclusion is that77 %((69−16)/69) of the time that uncertainty boundswere employed, they still did not bracket the experi-mentaloutcome.

This rather weak performance for uncertaintybounds is likely attributed to the time consuming natureof a formal detailed Uncertainty Quantification (UQ)analysis. Most teams instead bound their predictions byidentifying one or two key parameter(s) that were dif-

ficult to calibrate, and varying them over some range.Based on the outcome of the previous challenge, a fewteams intentionally varied the dimensions of the geom-etry across the range of values in the manufacturingtolerances. Even this degree of rudimentary paramet-ric analysis can be much more time consuming thanthe baseline prediction, since it requires running sev-eral instantiations. There is also a tradeoff between thefidelity of the baseline prediction, and the feasibilityof detailed UQ analysis. For example, the fine-zonedmeshes used in some cases to model the challengesample preclude implementation of UQ in a timelymanner. As with the previous challenge, integrationbetween solid mechanics modeling and UQ has notmatured to the point where it is readily accessed fortime-sensitive, resource-limited predictions. Beyondthe time-consuming nature of UQ, there is a lack ofstandardized processes for the assessment.

6.1.4 Computational efficiency

There is generally a trade-off between fidelity andspeed to solution, known in cognitive psychology as thespeed-accuracy tradeoff (Wickelgren 1977). In engi-neering practice, it is often viewed that a reasonableapproximation within a short timeframe is more valu-able than a highly accurate estimate obtained in a muchlonger timeframe. The two Sandia Fracture Challengesboth provided only the most basic of constraints in thisregard: all teams had the same number of months toarrive at their prediction. Yet the number of man-hourseach team spent in arriving at their predictions likelyvaried widely, although that number may be difficult toestimate.

While it may be difficult to evaluate the overallefficiency of the team’s entire prediction stream, oneclear first-order metric is the number of computa-tional degrees of freedom (DOF) in the team’s analy-sis method. Even this parameter varied over a surpris-ingly wide range: nearly 2 orders of magnitude from30 K to 2.5 M DOF. The two teams with the lowestDOF (<100 K, Teams D and E) both noted the rel-ative speed of their approach as a pragmatic path tosolution. To achieve the lowest DOF of 30 K, Team Demployed an intriguing shell element approach. Team Ewas notable in that they were able to predict the scalarQoI’s with only 90 K DOF. Team I was also notablyefficient in this regard, with only 135 K DOF. In anengineering environment these pragmatic approaches

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may be more attractive. Of the other teams that faredwell in the QoI predictions, Teams H, J, and M utilized600 K, 2.5 M, and 2.1 M DOF respectively. The overallteam efficiency is a metric that will be valuable to trackin future Challenges.

6.2 Future needs for improving predictive ability ofcomputational models in the area of ductilefracture

The style of this challenge does not isolate a sin-gle modeling phenomenology (e.g. crack initiation)to render a detailed comparison between differingmethods while holding all other factors constant.Indeed, such a pure assessment is difficult to makebecause often each phenomenological model is con-strained to certain numerical implementations, cali-bration requirements, etc. For this reason, while theeffort is dubbed the “Sandia Fracture Challenge”,it is not simply an assessment of fracture model-ing paradigms but rather a holistic assessment ofapproaches that can render predictions of fracture sce-narios.

6.2.1 Constitutive modeling

While the quantities of interest (QoIs) were selectedto reflect the quantities that would be called upon in atypical engineering design scenario in industry, theseglobal QoIs such a far-field force and displacementare not the most sensitive discriminating parametersfor assessing the relative efficacy of different model-ing paradigms. As shown in Figs. 20 and 21, evenwhen teams predicted an incorrect crack path or astable fracture event, they still could obtain reason-able values for the far-field QoIs. In pursuit of a morerigorous assessment of constitutive modeling of boththe deformation and fracture events, it will be nec-essary to employ more invasive QoIs that probe thelocal state in the vicinity of extensive deformation andfailure. For example, the QoIs probed in the currentchallenge did not fully probe the impact of anisotropyof strength on predictions—an isotropic model wouldhave given all of the load and COD predictions towithin 10 %. As demonstrated by the DIC measure-ments of Gross and Ravi-Chandar, if more invasiveQoIs such as the inter-ligament strains are measured,existing plasticity models may need improvement to

capture these details. Of course, improvements to con-stitutive descriptions will entail varied experiments formodel calibration or alternatively, full-field measure-ments in standard tension tests to calibrate anisotropyof strength description, e.g. Gross and Ravi-Chandar(2016).

6.2.2 Failure modeling

There appears to be consensus on the importance ofincorporating triaxiality and Lode-angle dependenceon the failure strain. Because the test data provided inthe challenge spans limited triaxialities and shear paths,only a limited calibration of damage models was pos-sible. Some of the teams used published empirical datafor the Ti–6Al–4V failure locus of failure strain as afunction of stress triaxiality, also known as the “garlandcurve”, e.g. Giglio et al. (2013, 2012), to calibrate thedependence of the failure strain on triaxiality and Lodeangle. Furthermore, the effect of orientation and ratedependence on this type of failure curve is not knownand techniques to obtain such data are still to be devel-oped. There appears to be an emerging consensus on theimportance of incorporating triaxiality and Lode-angledependence on the failure strain. However, the fact thatteams that used simple triaxiality-dependent strain-to-failure criterion without damage were also able to pre-dict equally well the failure path, instability and mostof the scalar QoIs makes it difficult to indicate thatany one model is better than another; for example fourteams were able to predict all QoIs except the criticalCOD. A different kind of challenge, where the stressstate is systematically altered to introduce failure underdifferent triaxiality and Lode angle could be triggered,may have to be developed in order to sort out the suit-ability of different types of models. The precise natureof such a model and methods of obtaining a calibrationof these models remain topics of active research andfurther development is necessary.

While the previous challenge provided crystallo-graphic texture and microstructural information for thestainless steel that was tested, the current challengedid not. Instead, the current challenge assumed that theeffects of microstructure would be effectively capturedin the provided macroscale mechanical tests. Tensileand V-notch shear properties were only measured in thetwo orthogonal in-plane directions. However, given therecent community-wide emphasis on multiscale mod-eling, a future challenge may focus instead on provid-

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ing detailed microstructural information and predictingthe ensuing mechanical response.

A pervasive disconnect between the models andexperimental observations lies in the details of the frac-ture morphology. As shown in Fig. 3e, the slow loadingrate tensile test resulted in a well-defined cup-and-conemorphology with a flat fibrous zone in the central regionof high triaxial stress and a clear transition to shear lipsaround the perimeter of the fracture surface. There areclearly two distinct failure modes with an abrupt tran-sition, yet failure models generally lack this detail (e.g.compare to Figs. 54b, 63b, or 68b). Moreover, in theexperimental tensile tests there was a transition fromthis cup-and-cone morphology which dominated at theslow loading rate to a slant fracture more reminiscentof sheet failure that occurred at the higher loading rate,as seen in the comparison between Fig. 3e, f (also seeFig. 4a, b compared to Fig. 4c, d). It should be possibleto predict these feature transitions with a sufficientlyfine 3D mesh and a failure model that can accuratelydiscriminate between tensile and shear failure modes.As the community strives to add fidelity to failure pre-diction and more physically-realistic micromechanicalmodels for failure, this apparent disconnect may be avaluable area for future detailed assessment.

6.2.3 Computational methods

In the present challenge, only Team K used a meshlessmethod, in this case the Reproducing Kernel ParticleMethod. The absence of these other methods such asPeridynamics may be happenstance as the voluntarynature of this effort may not have captured the interestof a team with other methods. The relative lack of mesh-less methods may also be due to their rarity or the matu-rity of these techniques for engineering fracture scenar-ios compared to conventional approaches. Enhance-ments to conventional finite elements using cohesivezones and XFEM were used by some teams to modelcrack propagation in the former case, and initiation andpropagation in the latter. Team E, who employed acohesive zone approach to describe the damage evo-lution, was not able to capture the abrupt unstablecrack advance and instead predicted a steady softeningbehavior. The majority of teams instead used elementdeletion to advance the crack and with a sufficientlydense mesh. This approach appeared to yield reason-able results, although admittedly, the metrics or QoIs

of the current study focused largely on the deformationand first crack initiation rather than crack advance.

Most teams bound uncertainty in their predictionsbased on variations in material parameters and/orboundary conditions. Only two of the fourteen teamsconsidered geometric variation, even though geometricvariation was revealed to be a critical source of outcomediscrepancy in the previous challenge. This is likelybecause of the time consuming nature of a geomet-ric variation study. While many simulation codes nowoffer the possibility to vary geometry, the automaticmeshing that is needed may not be sufficiently robust.

6.3 Recommendations for future challenge scenarios

6.3.1 Specific topical areas in deformation andfracture where blind assessment is needed

The present challenge provided insight into the mod-eling of ductile failure at quasi-static and moderatelyhigh rates. In contrast to the SFC1 predictions, theSFC2 predictions were more in line with the exper-imental results; and as suggested this may likely bethe targeted nature of the calibration data provided.However, the current challenge and the post challengesummit held at the University of Texas at Austin high-lighted the need for measurements of local behaviorusing optical techniques. In the current study, clip-ongauges used to measure the COD were easy to imple-ment but unreliable. Optical methods like DIC pro-vide more detailed measurements, which highlightedthe short comings of the constitutive description. Full-field calibration data would allow for more fidelity inmodel calibration, while at the same time more invasiveQoIs could be interrogated in the challenge scenario.Aside from local DIC in the vicinity of failure, DICmay also prove useful in other locations such as nearthe loading pins to characterize the degree of slidingand validate boundary conditions. Beyond DIC, otherlocal QoIs such as could be provided by an infraredtemperature map could also prove useful for dynamicloading scenarios. An important caveat is that the DICand infrared measurements described here will onlyprovide surface information. This surface informationis unfortunately removed from the high triaxial stressessubsurface where cracks nucleate. In fact, these surfacemetrics may better reflect the formation of shear lipsin failure, a feature that most models fail to capture.For subsurface information, x-ray computed tomogra-

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phy may prove useful to indicate the state of subsurfacedamage evolution.

In addition to full-field data for calibration and val-idation, there was continued interest in adding morematerial calibration data. Some examples include: (1)temperature-dependent mechanical properties wouldaid the thermomechanical coupling analysis, (2) ther-mocouple temperature measurements in the gauge sec-tion during tensile tests to estimate the thermal workcoupling parameter, (3) in-plane 45-degree orienta-tion tensile tests to more completely evaluate theanisotropy, (4) additional tests to estimate the out-of-plane anisotropy, (5) additional strain-to-failure testsunder a broader range of stress states. While accessto this breadth of data should enhance the predictiveability of some models, there is a countervailing moti-vation to mimic the breadth of data that is typicallyavailable for engineering assessment. Clearly, this andthe previous Sandia Fracture Challenge exercise havedemonstrated the need for extensive material calibra-tion data to render accurate predictions.

A round-table discussion at the summit and subse-quent post-summit discussions yielded several poten-tial focus areas for follow-up challenges in ductile fail-ure:

1. Elevated temperature Subject the candidate mate-rial to loading under high temperature. High-temperature constitutive behavior will be crucial.

2. Complex load history and load path This couldinclude either multi-axial loads or alternatively,interrupted loading with change in direction. Notethat this will bring kinematic hardening effects intoplay.

3. Assembled systems, welds, connected systems withbolts, welds or fasteners It is interesting to note thatthe stiffness predictions of the current challenge area consequence of the interaction of the machineand the sample which functions as an assembly.Modelling welds will pose problems with a lack ofinformation regarding the weld constitutive behav-ior and the need for residual stress fields.

4. Puncture and out-of-plane deformation Usinglaboratory-scale tests to calibrate models and makeblind predictions of tests which involve perfora-tion and out-of-plane deformation. This exercisecould also explore even higher rate deformation, asis often found in puncture scenarios.

5. Effect of microstructure/multiscale modeling Someof the teams are engaged in developing multi-scale models for deformation and fracture. Whilethese models may lack the maturity of continuummacroscale failure modeling, it could be instruc-tive to evaluate the efficacy of various modelingapproaches to predict the effects of a modifiedmicrostructure on the outcome of a failure scenario.

6.3.2 Guidance for execution of a future challenge

The ‘lessons learned’ in the previous challenge hadbeen assimilated in this second challenge; most impor-tantly, sample manufacturing tolerances were main-tained within sufficient bounds so as to eliminate (savethe one test) crack path ambiguities. With regard tothe key lessons learned in the current challenge, it isthe deployment of full-field optical-based instrumenta-tion, in addition to the conventional clip gauges. Withthe full-field measurements in place a more invasive setof QoIs could be demanded (principal strains in the lig-ament region prior to localization for instance). As dis-cussed in the last section of the article on the previouschallenge, specification of a conditional set of QoIs isrecommended. These conditional QoIs mimic the dis-tinct phases of elastic, elastic-plastic response followedby localization and failure. Conditional QoIs whichserve as ‘go/no-go’ decision points could be postulated.If elastic response is correctly predicted, proceed toevaluate the elastic–plastic response and so on. See sec-tion 6.3.2 of Boyce et al. (2014) for further elaboration.

The discussion from the post-prediction summitsuggested that the next challenge might employ thesame material on a more challenging problem (see listin previous section): on the one hand, this is an eco-nomical option as fewer tests need to performed, lesstime will be spent for returning teams on model calibra-tion, and predictions should improve accordingly; onthe other hand if the challenge seeks to replicate real-world demands on engineering predictions, this ‘com-fort’ of working with the same material may be lessrealistic. An additional recommendation was raised toemploy more localized quantities of interest such asdisplacements in the ligament region between the holesand notches. While these more local measures wouldlikely be more discriminating, it will continue to beimportant to also assess the far-field metrics that mimicthe needs of “real world” scenarios.

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In the conclusion of the previous article on the chal-lenge, there was a call to apply modern uncertaintyquantification techniques to the challenge problem.Likewise, we conclude by suggesting that future chal-lenges could call for confidence intervals, or probabil-ities on the predictions. While it may be difficult for‘volunteer’ teams to allocate sufficient resources for anintensive uncertainty assessment, one of the benefits ofconducting a follow-up challenge on the same materialis that more effort could be spent exploring the uncer-tainty landscape. A focus on uncertainty assessmentwill likely require adoption of efficient iterative meth-ods. Calibrated material uncertainties must be deconvo-luted from the applied boundary conditions and speci-men geometry. Stochastic distributions must then beefficiently propagated through blind predictions andresponse surfaces for the QoIs to be adequately sam-pled. Unless the community can become more effi-cient in this iterative process, intuition-based uncer-tainty bounds will continue to be the norm.

Finally, in addition it will be useful in future scenar-ios to track the efficiency of the team solution. Whilethe computational degrees of freedom provide one clearmetric, it may also be beneficial for teams to track man-hours spent on the solution.

7 Summary and conclusions

The current study documented the capability of 14independent computational mechanics teams to pre-dict blindly the quasi-static and moderate-rate dynamicrupture of a geometrically complex specimen manufac-tured from a common titanium alloy, Ti–6Al–4V. This‘challenge’ scenario is the second such challenge, fol-lowing on a similar challenge two years prior evaluat-ing predictions of quasi-static rupture of a precipitationhardened stainless steel. On average, the predictionswere generally more consistent with the experimentaloutcome compared to the first challenge. The improve-ments are thought to be due to two factors (1) the mate-rial test data provided was more extensive than the firstchallenge (or typical engineering scenarios), includingthe provision of a shear failure test to complement themore common tensile tests, (2) while this second chal-lenge scenario was markedly different from the firstscenario in material, geometry, and loading conditions,the outcome of the first challenge still helped guideteams away from immature or difficult modeling para-digms towards robust approaches.

It was evident that the modeling of ductile rup-ture relies first and foremost on an accurately cal-ibrated model of orientation-dependent, stress-statedependent, rate-dependent elastoplastic deformationand localization. Under modest dynamic loading rates,the heat generated by plastic work can also lead toa non-negligible coupled thermal contribution. Whenteams erred in their prediction of the elastoplasticbehavior there was a consistent trend towards over-prediction of the stiffness and the peak force duringdeformation. This over-prediction of the stiffness andeffective strength appears to be correlated to inaccu-rate boundary conditions associated with the pin load-ing condition. Without getting these ‘basic’ boundaryconditions right, efforts to implement more advancedmechanics models for deformation and failure are invain. There were significant differences from team toteam in the approach to model the failure process, rang-ing from cohesive zone damage evolution to elementdeletion based on a critical strain criterion; and therewas significant quantitative disagreement across themodels regarding the conditions associated with theonset of failure. It is clear that the mechanics commu-nity will benefit from the development, standardization,and adoption of test methods to explore ductile ruptureloading modes beyond simple tension.

Detailed uncertainty analyses continue to be chal-lenging under real-world time constraints. While infor-mation was provided on variation of material proper-ties and variation of geometry, only a few teams had anopportunity to explore these variations, and even thenby the simple-but-often-effective method of samplingthe extremes. Under real-world engineering scenarioswith firm deadlines, detailed sensitivity analyses anduncertainty studies are typically underutilized. Finally,there is a clear benefit from comparing predictionsfrom multiple teams. In critical engineering scenarios,peer review and/or multiple independent predictionscan provide a broader perspective on true uncertainty,mitigate ‘tunnel vision’, and catch common pitfalls.

Acknowledgments BLB and HEF would like to thank Dr.James Redmond for managing Sandia’s role in this work throughthe DOE Advanced Scientific Computing program. SLBK andTRB would like to thank Dr. Dennis Croessmann and Dr. DavidEpp for their management role supporting the experimentalefforts at Sandia for this work through the NNSA Weapon Sys-tem Engineering and Assessment Technology Engineering Cam-paign. JLB, SES, and CAJ would like to thank DOE/NE andRyan Bechtel for partially supporting their participation in thischallenge. The Sandia authors would like to thank the follow-

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ing individuals for providing laboratory support of the experi-ments: Thomas Crenshaw, John Laing, Jhana Gearhart, MathewIngraham, Artis Jackson, Darren Pendley, Jack Heister, and AliceKilgo. Sandia National Laboratories is a multi-program labo-ratory managed and operated by Sandia Corporation, a whollyowned subsidiary of Lockheed Martin Corporation, for the U.S.Department of Energy’s National Nuclear Security Administra-tion under contract DE-AC04-94AL85000. The work of AJGand KRC at the University of Texas was performed during thecourse of an investigation into ductile failure under two relatedresearch programs funded by the Office of Naval Research:MURI Project N00014-06-1-0505-A00001 and FNC Project:N00014-08-1-0189; this support is gratefully acknowledged. Theauthors from GEM are grateful for the support provided by theOffice of Naval Research (N00014-11-C-0487) for which Dr.Paul Hess and Dr. Ken Nahshon serve as the technical monitors.KP and TW are grateful to Dr. Borja Erice at Ecole Polytechniquefor the development of the user material subroutine; thanks arealso due to Dr. Christian C. Roth at MIT for a valuable discussion.The authors gratefully acknowledge financial support from theNational Science Foundation (Grant Number CMMI-1532528,“Summit on Predictive Modeling of Ductile Failure”) towardsholding a Summit to discuss and distill the results reported inthis article.

Open Access This article is distributed under the terms ofthe Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

Appendix 1: Additional details on modelingapproaches and results

Team A

Team members:J.A. Moore, [email protected], North-

western University, Evanston IL, USAK. Elkhodary, [email protected], The Ameri-

can University in Cairo, New Cairo, Egypt

Approach

Simulations were performed using the finite element(FE) software ABAQUS 6.12. A dynamic analysis,with explicit time stepping was used. Failure surfaceswere generated by the element removal method. Anelement is removed when a proposed damage variableδ∗ ≥ δ∗Crit . Domain decomposition parallelizationwas also applied for computational expediency.

Material Law Isotropic elasticity was assumed. Forinelastic behavior, the von Mises stress r(σ ) is tested

against an isotropically hardening yield surface, whoseradius r(σy) is defined via a proposed simple modifica-tion to the Johnson-Cook flow rule. The modificationherein proposed aims to account for the evolving dam-age with a single variable and a corresponding parame-ter. As such,

r(σ ) =√

1

2

[(σ1 − σ2)2 + (σ1 − σ3)2 + (σ2 − σ3)2

]≤ r(σy), (1)

with a modified Johnson-Cook (JC) rule defined as,

r(σy) =(A + Bεnp

) (1 + CIn ε∗)

(1 + T ∗m) (1 − Dδ∗2

). (2)

Here, A, B, C , n and m are the ordinary JC mater-ial parameters, εp is the effective plastic strain, ε∗ is anormalized strain rate, and T ∗ is the homologous tem-perature. The first term on the right hand side of (2)accounts for hardening, the second accounts for rateeffects, and the third for thermal softening. Tempera-ture rise is computed herein from adiabatic heating dueto plastic dissipation, which ignores heat transfer. Toavoid over-heating at slow (quasi-static) rates, a smallfraction for the rate of conversion from plastic dissipa-tion to heat is taken (i.e. 0.3 instead of 0.9).

In addition, a fourth term in (2) is proposed, andaccounts for damage nucleation and evolution at amaterial point. D is a single parameter introduced topre-multiply the newly defined damage variable δ∗ out-lined below. Of importance, δ∗ does not require anyadditional parameters in its definition. This model wasimplemented as a user material subroutine (VUMAT),using a radial return algorithm.

Damage Law In nonlinear solids, the necessary andsufficient condition for compatible deformation maybe expressed by the first-order differential law (Kroner1981), Curl(FeFp)= 0, having assumed a multiplica-tive decomposition of the deformation gradient F intoa plastic distortion Fp followed by an elastic distortionFe. Application of Stokes’ theorem to the differentiallaw yields an incompatibility vector δ (Kroner 1981),

δ ≡∮c

FdX =∮c

FeFpdX (3)

Any non-zero value of δ indicates the initiation of atopological defect at a material point, which we hereininterpret as crack nucleation in the continuum.

In finite elements, the deformation gradient F iscomputed from unique nodal displacements and the

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assumption of differentiable displacement fields, irre-spective of the constitutive behavior that evolves atthe integration points. We will designate such a defor-mation gradient by FFE (i.e. ∇Xx ≡ FFE ). FFE iscompatible throughout an FE simulation by defini-tion (i.e. contour integral (3) vanishes identically withF = FFE ). It follows that (3) can be re-hashed as,

− δ = 0 − δ =∮c(FFE − FeFp)dX. (4)

As such, the integrand in (4) becomes non-zero onlywhen the product FeFp is not equal to FFE , whichmeans that the evolving physical mechanisms in thesolid cannot accommodate the deformation imposedin the FE simulation at that given time. Thus Eq. (4)serves as a condition for crack nucleation. We hereinfurther simplify (4), and define a scalar measure ofincompatibility δs by Elkhodary and Zikry (2011),

δs

∣∣∣∣∣∣FFE − ReUP

∣∣∣∣∣∣Le

0 (5)

where Le0 is the initial element characteristic length.

In (5) we have ignored elastic stretching; hence wetook Fe ∼= Re, where Re is the rotation from the polardecomposition of Fe. We also assumed plasticity is irro-tational, so that Fp = Up, where Up is the right plasticstretch tensor. Finally, in (2), δ∗ = δs/Le, where Le isthe current element characteristic length.

Selection of parameters for material model and failure

This modified Johnson-Cook rate, temperature anddamage dependent flow rule was calibrated againsttensile data using a finite element model of a dog-bone sample. The linear-elastic, density, and heat trans-fer properties were taken from MMPDS (Rice 2003);

whereas, failure and Johnson-Cook parameters were fitto the provided tensile stress strain data. The dog-bonemesh and resulting stress strain curves are comparedto experimental tensile test data in Fig. 25. Note, sheartest-data was not used for calibration.

Modeling details for the challenge specimen

The nominal dimensions provided were used for mod-eling the test specimen. The region where FE bound-ary conditions where applied to the pin holes is shownin Fig. 26. This region was fixed for the upper holeand loaded with a prescribed velocity for the lowerhole. For the slow load rate (0.0254 mm/s), a constantvelocity was applied to the bottom hole and a fixedmass-scaling factor of 1000 was used, having removedthe rate dependence from the flow rule. For the fastload rate (25 mm/s), two velocity profiles were tested.A smooth “actuator” type profile was applied to thebottom hole for the first 0.05 s; a constant velocity of25 mm/s was applied after this time. For the fast simu-lation, no mass scaling was used. Lateral velocity wasset to zero over this region for both holes. Out of planevelocity was also set to zero over the entire surface ofboth holes.

Element removal, based on the above defined frac-ture criterion, was used to model failure surface genera-tion. No initial crack or preferential crack direction wasapplied. To allow for unbiased crack growth a uniformfinite element mesh density was used throughout themodel. The element type was an 8-node reduced inte-gration linear hexahedral element with coupled thermalanalysis capabilities (C3D8RT in ABAQUS\Explicit).No formal mesh convergence study was performed on

Fig. 25 Comparison oftensile testing stress–straincurves (for transverse androlling direction) to finiteelement prediction. Theequivalent plastic strain inthe dog-bone mesh is shownfor reference. The analysisshowed failure at the pointmarked “x”

0.0 0.1 0.1 0.2 0.20

500

1000

1500

Engineering Strain

Eng

inee

ring

Str

ess,

MPa

25.4mm/sec

TransverseRollingFEM

0.0 0.1 0.1 0.2 0.20

500

1000

1500

Engineering Strain

Eng

inee

ring

Str

ess,

MPa

0.0254 mm/sec

TransverseRollingFEM

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Fig. 26 The loaded surface for the upper hole

Fig. 27 The failed specimen shapes from both the 0.0254 and25.4 mm/s analyses. On the left, the deformed and undeformed0.0254 mm/s results are superimposed and the crack path isshown in both configurations

Table 7 Force displacement results

Load rate,mm/s

Maximumload, kN

COD at first crack initiation,mm

COD1 COD2

0.0254 23.9 5.05 4.66

25.4 36.6 5.06 4.98

the sample; however, the mesh density was based onvalues from the previous fracture challenge. A meshdensity of 12 elements through the thickness was thusused. This resulted in 36 elements around the radii ofthe large notch tips, and a total of 1,280,112 elements.A total of 48 processors were used in the parallelizedanalysis.

Blind predictions

The failed specimen shape for both the 0.0254 and25.4 mm/s analyses are shown in Fig. 27. From theseresults the predicted crack path was B–D–E–A for bothloading rates. The 25.4 mm/s analysis was not run tocomplete failure; however, based on the plastic strainand failure criteria profile, we were confident that thecrack would terminate in the upper notch.

The maximum load and CODs at first crack initiationfor both load rates are given in Table 7. The reactionforce and COD are shown in Fig. 28.

Sources of discrepancy

We have identified three main sources of discrep-ancy with experimental results caused by the bound-ary conditions, post-processing method and failurecriterion calibration method. We found that bound-ary conditions were very influential, particularly inthe dynamic case. We tried two other sets of bound-ary conditions. Prior to blind predictions we did notconstrain lateral motion at grips, and after the blindpredictions we used rigid cylinders (Fig. 29) to loadthe specimen. For both these boundary conditions weobserved fracture initiation in the top hole; when weconstrained the lateral direction (which is what wereported) we predicted the experimentally observedfailure sequence.

The post processing method we used for the reportedblind predictions extracted forces only at bound-ary nodes and lead to an erroneous over-estimateof the force curve, in the dynamic model in par-ticular, due to highly localized deformation at thegrips. Therefore, this node averaging does not repre-sent the experimentally observed values. For our cor-rected curves (Fig. 30), the forces at the referencenodes that drive the motion of the rigid cylinders wasextracted. These forces give a better representation ofthe experimental results (less sensitive to grip localizedbehavior).

Finally, we believe the COD at failure predictionswould be more accurate if we used a more system-atic approach for determining the δ∗

critical (for elementremoval) as a function of mesh and sample size. Forthe analysis presented here, the same δ∗

critical was usedfor calibration and specimen failure predictions.

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Fig. 28 Force and CODfrom the 0.0254 and25.4 mm/s analyses

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

COD2, mm

Forc

e, N

25.4 mm/s0.0254 mm/s

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

COD1, mm

Forc

e, N

25.4 mm/s0.0254 mm/s

Rigid Cylinder

Fig. 29 Modified rigid cylinder boundary conditions

Team B

Team members:C.H.M. Simha, [email protected],

CanmetMATERIALS, Natural Resources Canada,Hamilton, Ontario, Canada

B.W. Williams, [email protected] CanmetMATERIALS, Natural ResourcesCanada, Hamilton, Ontario, Canada

Finite element models, meshed with 8-noded brickelements, were used to simulate loading of the ten-sile, shear, and fracture specimens. The approximateelement size near localization and failure in each ofthe specimens ranged from about 0.2 to 0.4 mm transi-tioning to larger elements away from the failure zone.Simulations were performed using the explicit dynamicsolvers in ABAQUS and DYNA3D. For both solvers, a

Fig. 30 Corrected 25 mm/s force displacement curves with orig-inal blind prediction

user-defined subroutine was implemented to describethe material behavior. The subroutine implemented inABAQUS was based on von Mises yielding, with ahardening rule of the form,

σ − σy(1 + K ε p)n (6)

with the Xue–Weirzbicki damage model used to des-cribe the failure of the material (Simha et al. 2014;Xue 2007). The DYNA3D subroutine also utilized thehardening rule given by Eq. 6 and the Xue-Wierzbickidamage model, but the deformation was based onthe Cazacu–Plunkett–Barlat 2006 (CPB06) asymmet-

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ric/anisotropic yield function (Cazacu et al. 2006). Bothsubroutines used the Bazant-Pijaudier-Cabot non-localapproach to mitigate the mesh dependence of finiteelement simulations (Simha et al. 2014). Though thetwo subroutines were very similar, there were smalldifferences in the implementations of the two mod-els, such as the tolerances utilized for convergence,which led to two slightly different predictions. TheCowpers-Symonds form, ( ˙ε/ ˙εre f )m , was used to cap-ture the effect of strain-rate in the DYNA3D simu-lations, whereas two different sets of hardening lawcoefficients were used for the slow and fast cases inthe ABAQUS simulations. The hardening law coeffi-cients at the slow and fast rates are provided in Table 8and the hardening law is compared to the experimentalstress versus strain curves in Fig. 31. Three simulationresults are presented; ‘Mean’, ‘LB’, and ‘UB’ whichcorrespond to the curves used in the mean, lower, andupper bound predictions of the fracture specimen.

In the determination of the yield function coeffi-cients required for the CPB06 model, it was deemedreasonable to assume that the material behavior in theRD and TD directions of the sheet was equal. Oden-berger et al. (2013) showed that the balanced biaxialflow stress state at room temperature for Ti–6Al–4Vsheet was about 1.1 times the flow stress in the rolling

direction. Hammer (Hammer 2012) showed that thecompressive stress is about 1.1 times the tensile stressfor Ti–6Al–4V sheet. These data were used to cali-brate the in-plane coefficients for the CPB06 modelwith L11 = 1.0, L12 = 0.0, L13 = 0.0, L22 = 1.0,L23 = 0.0, L33 = 0.82, and k = −0.12. The in-planeyield surface obtained from the CPB06 yield functionis compared to the isotropic von Mises yield function(Fig. 32). The data from the shear tests conducted in thecurrent work were used to determine the shear coeffi-cient, L44(= L55 = L66) = 1.15. The predicted shearbehavior is compared with the experimental data inFig. 33. Differences in predictions between ABAQUSand DYNA3D simulations were attributed to differentboundary conditions employed to model the shear tests.

In addition to the tensile and shears tests, addi-tional fracture data for Ti–6Al–4V provided by Ham-mer (2012) and Giglio et al. (2012) were used to deter-mine the coefficients for the damage model. To accountfor the influence of mesh sensitivity on failure, a non-local implementation of the damage model was used inthe simulations (Simha et al. 2014), in which the effec-tive plastic strain used in the failure model was basedon an averaged value between elements within a spec-ified radius. In the current work, the non-local radiuswas about 1.5 mm in the ABAQUS simulations and

Table 8 Hardening lawcoefficients for determinedfor Ti–6Al–4V from tensiletests in the rolling andtransverse orientations

σy (MPa) K n ˙ε ˙εre f m

ABAQUS—Slow 1010 20 0.15 – – –

ABAQUS—Fast 1100 21 0.10 – – –

DYNA3D—Slow 1030.0 37.3 0.087 0.0006667 0.0006667 0.009

DYNA3D—Fast 1030.0 37.3 0.087 0.6667 0.0006667 0.009

Fig. 31 Left results of tensile test simulations for slow loading. Right results of tensile test simulations for fast loading

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Fig. 32 Comparison of yield surfaces at zero plastic strain

0.25 mm in the DYNA3D simulations. As discussed byXue (2007), the damage model parameters can be esti-mated based on calculating an upper and lower bound toexperimental fracture data. The upper and lower boundestimated for Ti–6Al–4V for the slow rate case is com-pared to experimental data in Fig. 34 for the DYNA3Dsets of parameters. Though the upper and lower boundsdo not bound the experimental data, they provide a rea-sonable estimate to capture the response of the Ti–6Al–4V alloy. The coefficients determined for Ti–6Al–4Vbased on the tensile and shear tests are provided inTable 9.

Fig. 34 Comparison of experimental fracture data of Ti–6Al–4V with upper and lower bounds predicted from damage model

The predicted force versus COD at Location 1(COD1) are shown in Figs. 35 and 36 for the ABAQUSand DYNA3D simulations of the fracture specimen andcompared with the experimental data. A contour plot ofthe effective stress predicted from the slow DYNA3Dsimulation, after initial crack propagation, is shown inFig. 37.


The results show that the ABAQUS simulations overpredicted the force response and under predicted COD1at failure in both the slow and fast cases. The DYNA3Dsimulations better captured the force response, but mar-ginally under predicted COD1 at crack propagation inboth the slow and fast cases. After crack propagation,

Fig. 33 Left results of shear test simulations for slow loading. Right results of shear test simulations for fast loading

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Table 9 Damage modelcoefficients utilized inABAQUS and DYNA3Dsimulations for Ti–6Al–4V

εo plim (MPa) q γ m β

ABAQUS—Slow 1.75 1250 1.25 0.3 2.0 2.0

ABAQUS—Fast 1.75 1250 1.25 0.2 2.0 2.0

DYNA3D—Slow 1.75 1250 1.2 0.32 2.0 2.0

DYNA3D—Fast 1.75 1250 1.2 0.37 1.0 2.0

Fig. 35 Results for slow loading of challenge sample

Fig. 36 Results for fast loading of challenge sample

there were oscillations predicted in the DYNA3D simu-lations, which were not predicted in the ABAQUS sim-ulations. The cause of these oscillations was unknown,but could be due differences applied boundary con-ditions of the two finite element models. Both theABAQUS and DYNA3D simulations were able to pre-dict the crack propagation path correctly (B–D–E).Results from the DYNA3D were used as the lowerbound, and results from ABAQUS served to provide

mean and upper bounds. For a discussion of the sourcesof the discrepancies between predictions and experi-ments and a description of Team B’s attempt to reducethe discrepancies, this team’s separate paper in this spe-cial issue can be consulted.

The difference in the size of the local radius isattributed to different implementations in the two codesand the mesh density; higher mesh density in DYNA3Dand lower in the ABAQUS model. In the ABAQUSimplementation during the stress-return procedure thestress, strain, and damage are calculated during eachincrement of the stress return. In the DYNA3D imple-mentation, only the stresses and strain are computedduring the stress-return procedure. The damage is com-puted after the stress return algorithm has completed.The different implementation might account for someof the discrepancy. Also, the radius of 1.5 mm is simi-lar to what has been reported in the literature (see, forinstance, Belnoue et al. 2010) and it is smaller than thediameter of the hole so that non-interacting boundariesdo not impact the non-local calculation. Furthermore,it is our experience with the current non-local model inabaqus, that the non-local radius is usually a fractionof the smallest feature in the sample.

Team C

Team members:A.R. Cerrone, [email protected], GE Global

Research Center, Niskayuna, NY, USAA. Nonn, [email protected], Ostbay-

erische Technische Hochschule (OTH) Regensburg,Germany

J.D. Hochhalter, [email protected], NASALangley Research Center, Hampton, VA, USA

G.F. Bomarito, [email protected], NASALangley Research Center, Hampton, VA, USA

J.E. Warner, [email protected], NASA Lan-gley Research Center, Hampton, VA, USA

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Fig. 37 Left contour plot ofpredicted damage afterfracture of the Ti–6Al–4Vspecimen for the slowsimulation performed withDYNA3D. Right typicalmesh density used in model

B.J. Carter, [email protected], Cornell University,Ithaca, NY, USA

D.H. Warner, [email protected], Cornell Univer-sity, Ithaca, NY, USA

A.R. Ingraffea, [email protected], Cornell Univer-sity, Ithaca, NY, USA

Approach

An over-the-counter methodology was used to pre-dict fracture initiation and propagation in the chal-lenge specimen. Specifically, the finite element soft-ware Abaqus/Explicit was used to simulate deforma-tion and damage in the challenge specimen geometrywith nominal dimensions. The continuum (Ti–6Al–4V) was modeled as linear elastic isotropic with thevon Mises yield criterion. Hardening was defined bya tabular function of plastic strain. Damage initiation,in turn, was modeled with a ductile damage initiationcriterion wherein the equivalent plastic strain at failurewas a function of stress triaxiality. Damage propaga-tion, finally, was modeled with an energy-based lawwith exponential softening. To discriminate betweenthe fast (dynamic) and slow (static) actuation rates, adifferent set of plasticity and damage parameters wascalibrated for each individually.

Material models

Linear elastic isotropic material parameters The lin-ear elastic isotropic (LEI) properties of room tempera-ture Ti–6Al–4V used in this study are given in Table 10.

Calibration of hardening laws For both loading rates,yielding in shear started at approximately a 12 % lowervon-Mises stress than in tension. Clearly, the mater-ial exhibited anisotropic hardening. Hardening curves

Table 10 LEI properties of room temperature Ti–6Al–4V

E η

114 GPa 0.34

Fig. 38 Calibrated flow curves

from both the tensile and shear tests were fit for boththe static and dynamic cases. These curves are plottedin Fig. 38.

The anisotropic Drucker–Prager yield criterion inAbaqus/Explicit could have been employed to addressthe issue of anisotropic hardening; however, due to lackof time for calibration, it was passed over in favor of asimpler approach. Sensitivity studies showed that whenassigned to the entire discretization, both the tensileand shear flow curves consistently yielded the A–C–F failure path; however, based on engineering goodjudgment, the B–D–E–A path was expected. To estab-lish upper bound and best (“expected”) predictions,the tensile flow curve was assigned to the entire dis-cretization. To establish lower bound predictions, the

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model was split into two regions. The shear and tensileflow curves were assigned to the shear-stress controlledregion (beyond the lower notch including the B–D–E–A path) and axial-stress controlled region (beyond theupper notch including the A–C–F path), respectively,Fig. 39a.

Calibration of damage initiation and propagation lawsTwo damage initiation laws were considered, Fig. 40.The first is a failure locus calibrated against the slowtensile (stress triaxiality= 0.8) and shear (stress triaxi-ality= 0) test data. However, as evident from Fig. 39b,the relevant triaxiality levels for the challenge specimenare in the range between 0.2 and 0.6. Consequently, the

1.00.850.700.550.400.250.10-0.05-0.20-0.35-0.50-0.65-0.80-1.0

Stress Triaxiality

(a) (b)

Fig. 39 a Flow curve assignment to establish lower bound pre-dictions: red demarcates shear–stress controlled region and graydemarcates axial-stress controlled region. b Stress triaxiality atpredicted time of initiation (A–C–F) for slow actuation rate

Fig. 40 Fracture loci. Note that each faint dashed-dotted linerepresents the evolution of strain as a function of stress triaxialityat the presumed location of failure

remainder of the locus had to be estimated with goodengineering judgment. Based on results from the sheartest wherein failure strain for the fast actuation rate wasapproximately 20 % lower than for the slower one, thislocus was decreased by 20 % to give its dynamic coun-terpart. Denoted “estimated” in forthcoming sections,it was employed to establish all blind predictions. Thesecond is another failure locus given by Giglio et al.(2013) for room-temperature Ti–6Al–4V under quasi-static loading. This locus was also reduced by 20 % togive a locus appropriate for dynamic loading. Due tolack of available calibration data, damage was assumedto evolve based on a critical fracture energy (10 N/mm)criterion with exponential softening.

Modeling details

Geometry and boundary conditions The nominaldimensions of the tensile, shear, and challenge spec-imens were considered in the calibration and predic-tion phases. With regards to boundary conditions, rigidbodies were used for both the shear and challenge spec-imens. For the shear specimen specifically, all contactnodes were tied to a rigid body. The challenge speci-men, in turn, showed considerable sensitivity to how itspins were modeled—in general, frictionless rigid bodypins favored the B–D–E–A failure path whereas rigidbody pins with friction favored the A–C–F path. It isnoteworthy that Pack et al. (2014) explored this issueduring the First Sandia Fracture Challenge (Boyce et al.2014) and noted that boundary condition selection hadno influence on the crack path and minimal influence onthe specimen’s response. Clearly, this was not the casefor the challenge geometry. To resolve this ambiguity,kinematic coupling constraints were considered whichyielded the A–C–F path. Additionally, based on goodengineering judgment, pins with friction were deemedto be more representative of the actual loading condi-tions. Consequently, rigid body pins with friction (withcoefficient 0.10) were adopted for this study.

Crack propagation Elements began to accumulatedamage once a critical failure strain was reached. Dam-age evolved according to an energy-based criterion. Ifa given element’s stiffness degraded beyond accept-able limits, it was removed from the discretization (noremeshing or element-state mapping was required).This removal introduced new free surface into the dis-

123


cretization, and with subsequent removals, a faceted“crack” began to form in the discretization.

Mesh refinement A mesh refinement study on the chal-lenge specimen was conducted. It was observed thatdoubling the number of elements from four to eight inthe thickness direction beyond the notches resulted insteeper load drops. A mesh size of 0.2 mm beyond thenotches was adopted based on the authors’ past experi-ences with high-strength materials and plasticity-baseddamage models (Cerrone et al. 2014). 98,460 eight-noded brick elements with reduced integration dis-cretized the geometry.

Blind predictions

The predictions and experimental results are givenin Fig. 41. The A–C–F path was predicted for bothactuation rates when no anisotropy was considered(the “expected” predictions). In the case of the lowerbound predictions, the B–D–E–A failure path was pre-dicted. In experiments, all eight samples failed B–D–E–A in the fast test while ten of eleven failedB–D–E–A in the slow test. It is noteworthy thatthe lower bound predictions captured maximum loadaccurately; however, the predicted COD1 at fail-ure was off by well over 50 %. Moreover, the pre-dicted maximum load and COD1 at failure in theexpected predictions were off by approximately 15 %.Additionally, the predictions were too stiff duringloading. In hindsight, the simplifications made toaddress the issue of anisotropic hardening were inade-quate. Furthermore, it seems that the estimated failure

locus over-predicted damage initiation for low stresstriaxialities.

These two issues were addressed shortly after sub-mission of the blind predictions to Sandia. First, ear-lier onset of yielding due to shear stress was enforcedby leveraging the tensile flow curve in conjunctionwith the “*POTENTIAL” option in Abaqus/Explicit.As noted earlier, yielding in shear starts at approx-imately a 12 % lower von-Mises stress than in ten-sion in this titanium alloy. Consequently, the R12parameter was set to 0.88 to obtain the reductionof the yield stress by 12 % under shear. Addition-ally, the failure locus given by Giglio et al. (2013)was used as it gave more realistic failure strains forlow stress triaxialities. As was done for the blindprediction, the static failure locus was decreased by20 % to give the dynamic locus. These two minoradjustments resulted in a marked improvement inthe predictions, Fig. 41—not only was the B–D–E–A crack path now predicted for both actuation rates,but the maximum loads and COD1 values at fail-ure were well within the scatter of the experimentaldata.


Failure to account for anisotropic hardening/yieldingfavored the A–C–F failure path while discriminat-ing between tensile and shear-dominance favoredthe experimentally-consistent B–D–E–A path. Underthe current scheme, this implies that accounting foranisotropy in the nonlinear regime is necessary to pre-dict the correct failure path. The reader is directed

Fig. 41 Force versus COD profiles from experimental data and original and improved predictions for slow (a) and fast (b) actuationrates

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to Cerrone et al. (2016) for a detailed discussion onthis topic. Additionally, applying a failure locus curvethat gave lower failure strains for lower stress triaxi-alities resulted in a more accurate prediction of dam-age initiation. While these modifications improved thequality of the predictions considerably, the discrep-ancy of stiffness during early loading was left unad-dressed. The predictions did not account for some com-pliance in the challenge specimen. This is an indica-tion that perhaps the proposed boundary conditionswere incorrect; for example, the 0.10 friction coef-ficient assigned between the loading pin and spec-imen might have been too high. Additionally, dif-ferences between the as-tested geometries and spec-imen with nominal dimensions and variation in theelastic properties could have also contributed to thediscrepancy.

Team D

Team members:T. Zhang, [email protected], Global

Engineering and Materials Inc., Princeton, NJ, USAX. Fang, [email protected], Global Engi-

neering and Materials Inc., Princeton, NJ, USAJ. Lua, [email protected], Global Engi-

neering and Materials Inc., Princeton, NJ, USA

Summary of XSHELL methodology

The XSHELL toolkit developed in house is employedto predict the fracture patterns and its associated load-deflection curves at two different loading rates for the2014 Sandia Challenge problems. Plane strain coremodel in XSHELL is used to capture the stress tri-axiality effect in the vicinity of the crack tip. Crackinitiation and propagation is accomplished through anelement-wise crack insertion with cohesive injectiononce its accumulative plastic strain reaches a criticalvalue. Maximum plastic strain direction is employedas the crack growth direction law.

XSHELL is an extended finite element based toolkitfor Abaqus, which is developed for dynamic failureprediction of thin walled shell structures (Zhang et al.2014). The use of the extended finite element method-ology (XFEM) allows a mesh topology independentof any arbitrary crack surface and is proven to havegreat potential in automating the process as the crack

grows with time with fixed mesh. The XSHELL toolkitfeatures the kinematic representation of a crackedshell via its phantom paired elements, crack initia-tion prediction using an accumulative plastic straincriterion, mesh independent crack insertion through acracked shell along the direction determined by thecrack growth law, cohesive injection for characteri-zation of the energy dissipation during crack growth,and display of the fractured pattern and the load dis-placement curve using a customized Abaqus CAEinterface.

Calibration of material properties and failure parame-ters

Based on the uniaxial tensile experimental testing datafrom Sandia National Lab, the true stress strain curvesat two different loading rates have been iteratively cal-ibrated (Fig. 42). Abaqus 3D FEA model with datalook rate dependent plasticity is employed to deter-mine the material constitutive properties by matchingthe numerically predicted load displacement curve withthe experimentally measured load displacement curves.An additional stress strain curve at strain rate 400/s isselected by referring to the paper by Lee and Lin (1998)with a small modification to ensure that it is alwayshigher than the other two stress stain curves at lowerstrain rates (Fig. 42). This curve serves as the upperbound for the numerical analysis. The failure strains attwo testing rates can be also calibrated using uniaxialtensile tests, which are the same as around 0.6.

With Sandia Challenge 3D Abaqus model and ratedependent data look model, two analyses have beenperformed: one with the displacement loading rate0.0254 mm/s; and another with the displacement load-ing rate 25.4 mm/s. Average strain rates at critical areacan be estimated for these two loading cases: 0.015/s

Fig. 42 Stress strain curves at different strain rate

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for slow loading and 40/s for fast loading. Based onthese two average strain rates, the stress strain curvesfor Sandia Challenge problem at low loading rate andhigh loading rate can be interpreted from the materialstress strain curves at different strain rates as in Fig. 42.The resulting stress strain curves for these two loadingcases and the elastic material properties are listed inTable 11.

Mesh sensitivity study has also been performedusing Abaqus shell FEA model with S4R element toidentify a rational mesh size. The load displacementcurves obtained from a dense mesh and a coarse meshagree with each other, which indicates that the coarsemesh is good enough for the numerical simulation ofthe Sandia Challenge problem.

Blind prediction for Sandia challenge problem withXSHELL

Sandia Challenge FEA model for XSHELL plane straincore application is shown in Fig. 43. Given the locationof the notch and holes in the Sandia challenge problem,a user defined XSHELL zone denoted by the red boxin Fig. 43 is used with plane strain core option. Dueto the nature of this Sandia Challenge problem, theplane strain core model will be invoked only at thecrack tip. The edges of the loading holes are linked withtwo Reference Points by using kinematic coupling. Aconstant velocity loading condition is applied at theupper Reference Point while the lower Reference Pointis fixed. Abaqus explicit solver is used to perform thenumerical analysis.

The Sandia Challenge Problem has been analyzedwith the calibrated failure strain as 0.6 and the approx-imate plane strain core coefficient α as 0.001 basedon experience. As shown in Fig. 44c, the predictedcrack path for 0.0254 mm/s loading rate is B–D–E, which agrees with experimental tested crack path

Fig. 43 XSHELL model for the Sandia challenge problem

shown in Fig. 44d. The predicted Load-CODs curve for0.0254 mm/s loading rate is compared with the exper-imental testing curve in Fig. 44a. The solid red line inthis figure represents GEM’s prediction, while the dot-ted blue line represents the experimental testing curve.The predicted COD1 value at the instant of the firstcrack initiation is 2.21 mm, and the peak load fromthe numerical simulation is 23,830.9 N which is higherthan the experimental tested average peak force value19,643.36 N.

The predicted crack path for 25.4 mm/s loading rateis shown in Fig. 45c with crack initiation at Hole D fol-lowed by its growth to Hole B and the second crack ini-tiation at Hole D followed by its growth to Hole E. The

Table 11 Summary of material properties for Sandia challenge problem

E(MPa) v

113,800 0.342

0.0254 mm/s ε p 0.0 0.012 0.03 0.07 0.09 0.11 0.20 0.60

σ (MPa) 1002 1056.54 1101.23 1160.96 1185.86 1205.84 1241.24 1352.15

25.4 mm/s ε p 0.0 0.01 0.035 0.05 0.08 0.1 0.3 0.60

σ (MPa) 1157 1176.44 1207.51 1224.9 1252.36 1270.52 1426.29 1538.14

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Fig. 44 a Load versusCOD1 curves betweenXSHELL and experimentfor 0.0254 mm/s loadingcase; b numerical predicteddeformation shape beforecrack initiation; c numericalpredicted crack path and dexperimental tested crackpath

predicted Load-CODs curves are compared with exper-imental result in Fig. 45a. The predicted COD1 valueat first crack initiation is 2.4 mm, and the peak loadfrom the XSHELL prediction is 23,815.8 N, which ishigher than the experimental tested average peak value20,435.88 N.


The initial stiffness of Load-COD1 curves predicted bythe XSHELL toolkit has a discrepancy in comparisonwith the test data. This is mainly due to the use of kine-matic coupling boundary condition with all degrees offreedom fixed for the loading hole. The high peak loadprediction may be related to the use of a plasticity modelwithout J3 dependence.

Team E

Team members:V. Chiaruttini*, [email protected], Onera,

Université Paris-Saclay, Châtillon, France.M. Mazière, [email protected],

MINES ParisTech, PSL Research University, Centredes Matériaux, CNRS UMR 7633, Evry, France.

S. Feld-Payet, [email protected], Onera,Université Paris-Saclay, Châtillon, France.

V.A. Yastrebov, [email protected], MINES ParisTech, PSL Research University,Centre des Matériaux, CNRS UMR 7633, Evry, France.

J. Besson, [email protected],MINES ParisTech, PSL Research University, Centredes Matériaux, CNRS UMR 7633, Evry, France.

J.-L. Chaboche, [email protected],Onera, Université Paris-Saclay, Châtillon, France.

*corresponding author: [email protected]

For the blind round robin prediction of ductile frac-ture, we used finite element simulation with a visco-elasto-plastic material model in finite strain. This modelhas been calibrated for both tensile and shear tests.Afterwards some numerical analyses were performedto identify a suitable approach to obtain a satisfactorycrack path and force-displacement curve for differentloading rates.

Visco-elasto-plastic constitutive model

To identify the model, a finite element analysis hasbeen carried out on uni-axial tests on samples orientedmainly in the rolling direction, which is the most rep-

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Fig. 45 a Load versusCOD1 curves betweenXSHELL and experimentfor 25.4 mm/s loading case;b numerical predicteddeformation shape beforecrack initiation; c numericalpredicted crack path and dexperimental tested crackpath

resentative loading condition for the Sandia FractureChallenge. The strain rate has been estimated close tothe gauge for both loading rates (0.66 10−3 s−1 and0.66 s−1). At the higher rate, the hardening effectsshould be lower than expected due to a considerablethermal heat produced. The provided test curves weretransformed to represent the effective strain and stressbut they do not take into account the necking effects.

The chosen constitutive model is initially basedsolely on a modified Norton flow with an anisotropicHill criterion. To insure an accurate description of hard-ening at both low and high rates, a softening evolutionis added using a negative value for K :

p =( 3

2 σ : MHill : σ)− R

K0 + K(1 − e−bp

)n

To allow a finer tuning of the hardening in the studiedplastic strain domain up to four terms were tested incombination with nonlinear isotropic terms in the finalmodel:

R = R0 +∑N

i=1Qi

(1 − e−βi p

)This model was calibrated using implicit 3D finite

element computation within the Z-set finite elementsoftware (Besson and Foerch 1997). Such an approachallows us to accurately fit uni-axial tests using a

finite strain implicit quasi-static finite element solutionprocess, the mesh size was chosen to ensure the correctdeformation level for the appearance of the neckingeffect (Fig. 46).

To deal with material anisotropy, the provided sheartest result was studied. The first stage was to correctly fitthe initial stiffness of the numerical model with the test-ing data, dealing with modification on the prescribedboundary conditions. To satisfactory reproduce thoseexperiments, an anisotropic Hill criterion was finallyused, whose calibration was done using results fromGilles et al. (2012, 2011).

Failure modeling

Our driving idea was to identify the simplest modelthat could produce sufficiently accurate results on theSandia challenge’s specimen. Due to the specific spec-imen geometry, only two different kind of failure canbe observed: dominant shear failure under low triaxial-ity for crack path B–D–E–A, or high triaxiality failurealong crack path A–C–F. Thus, it is not strictly neces-sary to apply a Gurson-type model, which can predicta failure path whose geometry would highly dependon triaxiality. A pragmatic approach could be adopted,involving a cohesive zone model to insure the soften-

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Fig. 46 Comparisonbetween finite elementsimulations and uni-axialtests in rolling direction

ing process due to the material damaging. The chosenmodel (Alfano and Crisfield 2001) (allowing to decou-ple the material toughness for traction and shear fail-ure modes), was calibrated using both tests (uni-axialand shear, see Fig. 47), with a special attention paid tothe region of the negative slope observed in the shearexperiments, the mesh size was chosen according tothe necking effect.

Numerical simulation and discussion

To predict the crack path, complete 3D finite elementcomputations were performed with only visco-elastic-plastic constitutive model up to the maximal forcebefore the necking appears. Only after the peak forcecohesive elements were introduced in the zones withhigh accumulated plastic strain, which were observedin certain ligaments when reaching the peak load.

Both fast and slow loading cases were simulatedusing a 3D mesh in finite strain, with a quasi-staticimplicit solution and quadratic elements producingabout 90,000 degrees of freedom. Approximately 250incremental loading steps were used to reach the E–Aligament failure (with a computational time limited toless than 2 h).

The obtained result confirmed that, as expected, theshear dominant failure occurs through the B–D–E–Acrack path with the first B–D–E failure stage followed

a second crack initiation process on the E–A final liga-ment. As the test COD gauges were not able to deliverglobal Force-COD plots after the first failure, a sim-ple image analysis from the organizers movie (on thefast case) has been carried out to extend the curve afterthe unstable crack propagation (Fig. 48). This analysisshows quite a good agreement with our blind predic-tion and the experimental data during the stable failurephases (see Fig. 49).


The pragmatic approach that was applied for our pre-dictions has some evident drawbacks and limitations:the quasi-static simulation, combined with excessivelydissipative cohesive zone, failed to start the damageprocess sufficiently early (especially at high speed) andhence produced the softening slope which is not suffi-ciently steep. Furthermore, for a much more complexcase (where triaxiality has a significant impact on thecrack path), such a method cannot be successful. How-ever, this rather simple approach (from computationalpoint of view) offers the possibility to properly capturethe peak force and to correctly fit the testing curve dur-ing the ultimate crack initiation process on this specificSandia Fracture Challenge conditions.

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Fig. 47 Calibrated cohesive model compared with traction (a) and shear (b) test experiment

Fig. 48 Comparison between experimental results and our blind prediction: deformed mesh (showing accumulated plastic strainisolines) and fast testing at the synchronized time step, after the unstable crack propagation stage

Team F

Team members:J. Lian, [email protected], RWTH

Aachen University, Aachen, GermanyY. Di, [email protected], RWTH

Aachen University, Aachen, GermanyB. Wu, [email protected], RWTH

Aachen University, Aachen, GermanyD. Novokshanov, [email protected]

aachen.de, RWTH Aachen University, Aachen, Ger-many

N. Vajragupta, [email protected], RWTH Aachen University, Aachen, Germany

P. Kucharczyk, [email protected], RWTH Aachen University, Aachen, Germany

V. Brinnel, [email protected],RWTH Aachen University, Aachen, Germany

B. Döbereiner, [email protected], RWTH Aachen University, Aachen, Germany

S. Münstermann, [email protected], RWTH Aachen University, Aachen,Germany

Approach

The material model used in the team is based on ahybrid damage mechanics model (Lian et al. 2013)

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Fig. 49 Comparison between blind prediction (solid curve) andrebuilt experimental data (from image analysis and force sensor,dashed curve), for the Force-COD1 (N vs. mm) plot

which was further extended to incorporate the effect ofstrain rate and temperature for Charpy test and machin-ing under adiabatic condition (Buchkremer et al. 2014;Münstermann et al. 2012). The distinction of the modelis that it differentiates the damage initiation and frac-ture and it is developed in a hybrid way by combin-ing the uncoupled model to act as the damage thresh-old and coupled model to represent the microstruc-ture degradation till final fracture. The model isimplemented into the FE code Abaqus/Explicit bymeans of a user material subroutine (VUMAT) and allthe simulations presented here are conducted in thisenvironment.

The yield function of the model is given in Eq. 7. Itis noted that the coupling effect of the damage into theyield function is only valid once the damage initiationcriterion isfulfilled.

Φ = σ − (1 − D) σy(ε p, η, θ, ˙ε p, T

) ≤ 0 (7)

Generally, the isotropic yielding and hardening areemployed based on the negligible difference betweenthe flow responses from tensile tests along rollingand transverse direction for the investigated material.However, a more general plasticity model (Bai andWierzbicki 2008) to account for the stress state effecton yielding is employed, as defined in Eq. 8. In addi-

tion, the influence of strain rate and temperature on theyielding is also defined (Eq. 8).

σy(ε p, ε p, T, η, θ

)=[σy(ε p) ·

(c1˙εpln ˙εp + c ˙pε2

)+ c3˙εp · ˙ε p

]

×[c1

Texp(c2

TT)

+ c3T

] [1 − cη · (η − η0)

]

×[csθ + (

caxθ − cs

θ

) ·(

λ − λm+1

m + 1

)](8)

In the equation, σy(εp) stands for the flow curve

under the reference condition, in the context, i.e. qua-sistatic tensile test at room temperature; c1˙εp −c3˙εp are the

material parameters for the strain rate effect; c1T−c3

T arethe material parameters for temperature effect; cη andη0 are the material parameters for the effect of stresstriaxiality; cs

θ, caxθ and m are the material parameters

for the effect of the Lode angle. For details, readersare referred to Bai and Wierzbicki (2008) and Mün-stermann et al. (2013).

It is also noted that under the adiabatic condi-tion, the temperature evolution is defined according toEq. 9.

T = δ · σ · ˙εp

ρ · cp(9)

where δ, ρ and cp are the specific heat fraction, materialdensity and heat capacity, respectively.

For damage modelling, a discontinuous damage evo-lution law is assumed, i.e. the initiation of damage isnot associated with the plastic deformation but a char-acteristic strain-based criterion which depends on stresstriaxiality and Lode angle. Afterwards, a simple linearincrease is assumed with respect to the equivalent plas-tic strain, and the rate of damage evolution is governedby the energy dissipation between damage initiationand the complete fracture of the material point. Thecritical damage to fracture is assumed to be a functionof Lode angle. Element deletion technique is used forthe crack propagation.

D =

⎧⎪⎪⎨⎪⎪⎩

0; εp ≤ εpi∫ εp

εpi

σy02Gf

d εp; εpi < εp < ε

pf

Dcr; εpf ≤ εp

(10)

where Gf is the material parameter and σy0 is the stressat damage initiation. The damage initiation strain, ε

pi ,

and the critical damage accumulation for fracture are

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defined in Eqs. 11 and 12, respectively.

εpi =

[C1e

−C2η − C3e−C4η

]θ2 + C3e

−C4η (11)

Dcr = c1cr · θ2 + c2

cr (12)

where C1 −C4 are material parameters for damage ini-tiation locus and c1

cr and c2cr are the material parameters

for the fracture locus.

Material parameter calibration

The model involves a large number of material parame-ters for accurate characterization of material behavior.For a complete material parameter calibration proce-dure, several types of tests and specimens are requiredas described by Lian et al. (2013, 2015) and Buchkre-mer et al. (2014). In the SFC2, as only limited testswere provided, the calibration of the plasticity parame-ters, including the effect of stress state, strain rate andtemperature, and damage initiation locus was assistedby material database of the Steel Institute. The ref-erence flow curve was derived from the tensile testsalong the rolling direction (specimen RD5) providedby Sandia National Laboratory. The Ludwik equa-tion was used to extrapolate the flow curve to largestrains. The most important parameters for fractureprediction, the damage evolution and critical damageaccumulation parameters, were calibrated by an iter-ative fitting procedure such that force–displacementresponses of tensile and shear tests from simulationscan meet the experiments. The 8-node linear brickelement with reduced integration (C3D8R) was used

for both models and a mesh size of 0.1 mm wasapplied to the critical region to represent the highorder of strain gradient. The comparison of the force–displacement response between the experiment andsimulation with the final calibrated material parame-ters are shown in Figs. 50 and 51 for both tensileand shear tests, respectively. It is noted that only thetests under slow loading rate were used for the cali-bration and it is assumed that the damage parameterskeep the same for fast loading rate for simplicity asthe simulations of tensile and shear tests with the samedamage parameters under fast loading rate also givereasonably good results compared to the experiments.All the calibrated material parameters are listed inTable 12.

Blind prediction

For the simulation of the blind prediction challenge, thenumerical model was constructed based on the receivedideal geometry and the dimension scattering as a resultfrom the manufacturing process was neglected. Thesame element type (C3D8R) was applied to the model.To account for the high strain gradient in the criticaldeformation region, a finer mesh was implemented inthis area whereas the coarser mesh was applied on therest of the model and the model finally consisted ofabout 800,000 elements. As the calibration models, themesh size of 0.1 mm was chosen for the fine mesh. Theloading roll and the fixing roll were modeled as therigid body while the contact interaction between thespecimen and the rolls were assumed to be frictionless.

Fig. 50 Experimental and numerical force–displacement responses for tensile tests under both slow (a) and fast (b) loading rates

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Fig. 51 Experimental and numerical force–displacement responses for shear tests under both slow (a) and fast (b) loading rates

Table 12 Calibrated material parameters for the model

Plasticity Stress state effect Temperature effect Strain rate effect

cη csθ cc

θ ctθ m c1

T c2T c3

T c1˙εp c2

˙εp c3˙εp

0 0.95 1.0 1.0 5 1.6164 0.0063 0.7535 0.0123 1.089 0

Damage Damage initiation Damage evolution Fracture

C1 C2 C3 C4 Gf c1cr c2

cr

0.18 0.4421 0.06604 0.4982 20,000 mJ/mm3 0.040 0.008

Regarding the boundary condition assignment, the fix-ing roll was constrained in all degrees of freedom andonly displacement along the vertical axis was appliedto the loading roll. The overview of the model beforedeformation and the critical region after loading forboth slow and fast loading rates are shown in Fig. 52.To balance the predictive quality and computationalefficiency, time and mass scaling were applied to thesimulations for slow and fast loading rates, respectively.A convergence of the force–displacement response wasreached for the time scaling and a minor effect of thekinetic energy over the internal energy was also met forthe mass scaling.

As illustrated in Fig. 52, the crack path predictionof the slow loading rate is A–C–(B–D–E)–F. In gen-eral, the crack path A–C–F is in a competition withthe crack path B–D–E, and these two are roughly cor-responding to the fracture patterns under tensile andshear loadings, respectively. Based on the calibratedparameters, the model gives very sensitive and similarresponses of these two modes as these two crack paths

are concurrent. This observation is also confirmed bythe experiments. Although most of the fracture path forslow loading rate is B–D–E, A–C–F was also observedfor one specimen. For the fast loading rate, the pre-diction agrees with the experiments. Despite the smallcrack from A to C during the deformation owning tothe sensitivity of the model, the dominant crack pathis B–D–E–A. The force versus COD1 for both loadingrates are plotted in Fig. 53. For both rates, the maxi-mum force is overestimated 10–15 %, but the predictedfracture displacements are in a reasonable agreementwith the experiments consistently for both slow and fastloading rates.


Although overall accepted prediction was achievedwith several assumptions and literature data, there isstill certain degree of space to improve the blind pre-diction, such as an optimized strain hardening extrap-olation, mesh regularization or non-local formulationto decrease the scaling factors. The overestimation of

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Fig. 52 The numericalmodel for the blindprediction simulation alongwith the schematic sketch ofthe implemented boundarycondition and the crack pathsequence comparison

Fig. 53 Experimental and numerical force–displacement responses for tensile tests under both slow (a) and fast (b) loading rates

the force level for both slow and fast condition isclearly related to a higher description of the stress–strain response of the material. Both the strain harden-ing extrapolation and the damage evolution parametersare responsible for this, as the material owns a quiteearly damage initiation from microstructural point ofview. Another critical point for the challenge simula-

tion is the expensive computational cost. As the meshsize is pre-defined in the calibration test, to maintainconsistent softening response of the material, a largenumber of elements resulted in the challenge simu-lation. To balance the computational time and pre-dictive accuracy, high time and/or mass scaling fac-tor is employed, which also results in the oscillation

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of the force response under fast loading condition. Inthis regard, the implementation of mesh regulariza-tion or non-local formulation is of interest. The furtherimprovement together with the details for the modelingis reported in a full length paper in the same SpecialIssue as this article.

Team G

Team members:M.K. Neilsen, [email protected], Sandia National

Laboratories, Albuquerque, NM, USAK. Dion, [email protected], Sandia National Lab-

oratories, Livermore, CA, USA

Approach

Predictions for the SFC2 challenge were generatedusing a quasi-static finite element code in SIERRAsolid mechanics (2011). A unified creep plasticitymodel was used to capture temperature and strain rateeffects. For this model, the inelastic (creep + plastic)strain rate is given by

εin = 1

3γ n = 3

2e f sinhp

[τ

αD(1 − cwd)

]n (13)

where γ is the equivalent plastic strain rate, n is theassociated flow direction, τ is the vonMises effectivestress, D is a user-prescribed function of equivalentplastic strain to define isotropic strain hardening, f , p,and α, are temperature-dependent material parameters,and w is a scalar measure of damage. Material parame-ters c and d define the reduction in isotropic strengthdue to damage. Damage evolution is given by Wilkinset al. (1980).

w =∫ (

1

1 + pp

)a

(2 − A)β dγ (14)

where

s1 ≥ s2 ≥ s3 A = Max

(s2

s1,s2

s3

)p = −1

3σ : i

(15)

si are the eigenvalues of the stress deviator, p is pres-sure. Damage evolution depends on both pressure (firstinvariant of total stress) and third invariant of deviatoricstress. The pressure-dependent, first term in Wilkins etal. damage evolution equation is similar to the dam-age evolution equation proposed by Wellman (2012)

and used by us in the initial Sandia Fracture Chal-lenge (Boyce et al. 2014; Neilsen et al. 2014). Notethat the stress dependence can be removed by settingthe material parameters α and β equal to zero in Eq. 14,then damage is simply accumulated equivalent plasticstrain. When damage has reached a critical level, theelement is not instantaneously removed nor is the stressinstantaneously reduced to zero; instead the constitu-tive response is changed in five solutions steps to be thatof a very flexible elastic material with moduli equal to0.0001 times the original elastic moduli. This approachis used to make the acquisition of post failure equilib-rium solutions possible for most problems.

The effects of heating due to plastic work were cap-tured with fully coupled thermal stress simulations inwhich the volumetric heating rate, Q, was given by

Q = ηW p = ησ : εin (16)

where η, the Taylor-Quinney coefficient prescribes thefraction of plastic work that is converted to heat, W p

is the plastic work rate, σ is the Cauchy stress, and εin

the inelastic strain rate. A survey of literature yieldeda wide range of values, 0.1 to 0.9, for the fraction ofplastic work converted to heat. A value of 0.5 was usedfor η in these simulations.

Material parameters for the model were obtainedfrom simulations of the uniaxial tension tests at dif-ferent rates. Values for α in Eq. 13 which define theeffects of temperature on the isotropic strength werebased on data from (Rice 2003). Strain hardening wasbased on a fit to the slow rate experiment (black curvein Fig. 54a). The high rate test was then simulated (bluecurve in Fig. 54a). Finally, a fully coupled thermalstress simulation predicted that the apparent ductilityof the material would be dramatically reduced whenheating due to plastic work was included (red curvein Fig. 54a). This occurs because heating causes theapparent hardening of the material to decrease whichleads to the initiation of necking and subsequent crack-ing earlier in the simulation. The predicted deformedshape at failure is in reasonably good agreement withexperiments (Fig. 54b). In this figure, LIFE is simplythe current damage divided by the critical damage, sowhen LIFE obtains a value greater than one the elementis cracked and turns white. The model predictions wereinsensitive to changes in the damage parameters; how-ever, these parameters were highly correlated with thefailure strain.

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Fig. 54 Comparison ofuniaxial tension simulationswith experimentalmeasurements andobservations. a Engineeringstress–strain. b Deformedshape after failure with lifecontours (white elementshave failed)

Next, the shear test was simulated with y-displace-ments applied directly to all surface nodes on the spec-imen where the loading blocks contacted the specimen(Fig. 55a). These simulations predicted a crack simi-lar to the experiment (Fig. 55b) but load-displacementcurves that had a much higher initial slope and peakvalue than experiments (Fig. 55d). To try and under-stand this discrepancy, loading blocks were added tothe shear test simulation (Fig. 55c). The loading blockswere preloaded by preventing normal displacement ofthe back surface of the back blocks and clamping thesample by displacing nodes on the front surface ofthe front blocks to generate a total clamping force of170.8 kN which is equal to the expected clamping forceof the eight bolts torqued to 5.65 N-m. This simulationmatched the experimental load-displacement curve bet-ter (green curve in Fig. 55d) but did not fail because thesample was just rotating between the loading blocks. Inthe tests, the horizontal grip inserts (Fig. 7) which werehand-tightened would eventually prevent rotation of thesample and cause the observed shear failure. Due to thisshear test discrepancy, we decided to go ahead and gen-

erate SFC2 challenge geometry predictions with para-meters we had obtained from the uniaxial tension tests.

The SFC2 challenge geometry was then simulatedusing a model with 451,536 elements and a typical ele-ment edge length of 0.254 mm. Contact between theloading pins and sample was not included and insteadhalf of each loading pin was modeled. The nodes atthe centerline of the bottom pin were given zero dis-placement in all three directions while the nodes at thecenterline of the top pin were given vertical displace-ment at the prescribed rate. Sliding was not allowedbetween the pin and the sample but the center of thepin could freely rotate about the pin axis for ‘free rota-tion’ simulations. Pin rotation was not allowed in ‘norotation’ simulations. The free rotation and no rotationsimulations were performed to bound expected behav-ior. Free rotation simulations were expected to be closerto the experiments. Free rotation simulations predicteda crack path of A–C–F (Fig. 56a) and no pin rotationsimulations a crack path of B–D–E–A (Fig. 56b). Themodels bound the experimental displacements to fail-ure but predict too high of a failure load.

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(a)

(c) (d)

(b)

Fig. 55 Comparison of experimental measurements with modelpredictions for shear test. (a) Model with displacement pre-scribed. b Experimental and predicted crack on front and back

surfaces of sample. c Model with loading blocks, friction coeff.=0.36. d Load-displacement curves


The experimentally measured peak load for the chal-lenge geometry was 86 to 91 percent of the predictedpeak load. This discrepancy indicates that the materialis likely weaker in shear than this model predicted witha von Mises yield. Simulations of the shear test withloading blocks indicated that the sample may be rotat-ing more than expected which would contribute to thedisplacement discrepancy in the experiment. However,even with this discrepancy the model should have stillpredicted close to the correct loads. The discrepancy in

load at yield and peak load in the shear test is similarin magnitude to load discrepancy with the challengegeometry again indicating that this material is likelyweaker in shear than our model predicted. The mostsignificant weakness in these simulations was that theydid not account for this reduced strength in shear.

Team H

Team members:K.N. Karlson, [email protected], Sandia National

Laboratories, Livermore, CA, USA

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Fig. 56 Comparison of experimental measurements with blindmodel predictions—challenge geometry. a Path A–C–F pre-dicted w/free pin rotation. b Path B–D–E–A predicted w/no pin

rotation. c Force-COD1 for 0.0254 mm/s loading. d Force-COD1for 25.4 mm/sec loading

J.W. Foulk III, [email protected], Sandia NationalLaboratories, Livermore, CA, USA

A.A. Brown, [email protected], Sandia NationalLaboratories, Livermore, CA, USA

M.G. Veilleux, [email protected], Sandia NationalLaboratories, Livermore, CA, USA

Approach

The material, time scale, and mode of loading dic-tated our path forward. Provided experimental data

and literature advocate models that incorporaterate dependence, temperature dependence, andanisotropy in both the yield stress and the harden-ing. Void evolution must include multi-axial nucle-ation, growth, and coalescence. The low thermal con-ductivity of titanium and the time scales for char-acterization and testing requires thermomechanicalcoupling and implicit time integration. Local mate-rial softening requires regularized methods forsolution.

123


Team Sandia California (Team H) used SIERRASolid Mechanics (SIERRA SM) to capture the requiredphysics and numerics for solution. SIERRA SM is aLagrangian, three-dimensional, implicit code for theanalysis of solids and structures. It contains a versa-tile library of continuum and structural elements, andan extensive library of material models. For all SFC2related simulations, our team used Q1P0, 8 node hexa-hedral elements with element side lengths on the order0.175 mm in failure regions. To model crack initiationand failure, element death removed elements from thesimulation according to a continuum damage model.Exploratory studies were also conducted with regu-larized methodologies (nonlocality, surface elements).Unstable modes of fracture were resolved with implicitdynamics [HHT time integration with numerical damp-ing (Hilber et al. 1977)].

Thermo-visco-poro-plasticity. We chose SIERRA SM’sisotropic Elasto Viscoplastic (EV) material model forour simulations because it contains the most relevantphysics to accurately predict the SFC2 challenge prob-lem such as the flexibility to include temperature andrate dependence for a material. However, since the EVmodel does not support anisotropic plastic behavior, theanisotropy evident in the provided data was includedthrough other means described in detail in the followingsection.

The EV plasticity model is an internal state variablemodel for describing the finite deformation behavior ofmetals. The model incorporates strain rate and temper-ature sensitivity, as well as damage, and tracks historydependence through the use of internal state variables.In its full form, the model has considerable complex-ity, but most of the material parameters and resultingbehavior are optional. The form of the material modelspecific to our use for SFC2 will now be outlined for thesimplified case of uniaxial tension. For this simplifiedcase, the stress evolves according to

σ = E(ε − εp) (17)

where ε is the total strain and εp is the plastic strain.The flow rule is defined by

εp = f sinhn(

σy − k

Y− 1

)(18)

where σy is the equivalent stress; Y is a material para-meter representing the rate independent, initial yieldstress; f and n are material parameters that govern thematerial rate dependence; and κ is the isotropic harden-ing variable for the material, which evolves according

to a hardening minus dynamic recovery model origi-nally proposed by Kocks and Mecking (1980):

κ = κμ

μ+ (H − Rdκ)εp. (19)

The temperature dependence for all material para-meters (Y, f, n, H, Rd ) can be specified explicitly withuser specified scaling functions or using functionalforms built into the model. Heat generation due to plas-tic work is calculated with

q = βσ εp (20)

where the material parameter β is the fraction of plasticwork dissipated as heat.

The EV model contains a void growth model and avoid nucleation model to account for isotropic materialdamage. For void growth, damage evolves accordingto the model proposed by Cocks and Ashby (1980):

φ =√

2

3εp

1 − (1 − φ)m+1

(1 − φ)msinh

[1(2m − 1)

2m + 1

p

σvm

]

(21)

where σvm is the von Mises stress, p is the hydrostaticstress, φ is the void volume fraction of the materialand the damage exponent m is a material parameter.With this void growth model, damage will only increasewhen p/σvm > 0. To account for damage resultingfrom other stress states, a void nucleation model basedon J3 (Horstemeyer and Gokhale 1999; Nahshon andHutchinson 2008) was also included in the materialmodel:

η = ηεpN1

[4

27− J 2

3

J 23

](22)

where N1 is a material parameter, η is the numberof nucleated voids, and Ji are the deviatoric stressinvariants. These two damage models can be usedindependently or concurrently to model void nucle-ation and growth. Including damage evolution throughthese models reduces the material’s elastic modulus andshear modulus by a factor of 1 − φ, and the flow rulebecomes

εp = f sinhn[σy − κ(1 − φ)

Y (1 − φ)− 1

]. (23)

The damage models require the definition of the ini-tial void volume fraction φ0, the initial size of nucleatedvoids φ

η0 , and the initial void count per volume η0. Void

coalescence is modeled through φcoal . The materialpoint is unloaded for φ > φcoal . In contrast to surface

123


Fig. 57 The parameter setcalibrated to the tension testdata accurately captures theplasticity and failurebehavior of the data

elements, we remove elements (element death) whenany integration points satisfy the coalescence criteria.Stabilization of fully integrated formulations is prob-lematic with loaded and unloaded integration points.

Material parameter calibration

We populated the EV material parameters for Ti–6Al–4V sheet using a combination of the data provided in thechallenge announcements and data from literature. Ini-tially, the yield (Y, f , and n) and hardening (H and Rd )parameters were calibrated to the provided tensile datausing a non-linear, least squares algorithm where theobjective function consisted of the error between theprovided data and model data. Since the rate depen-dence for the initial yield stress is not uniquely con-strained by two data points, we used rate dependencedata from Follansbee and Gray (1989) to supplementthe data at two rates provided for the challenge. Tem-perature dependence was added to the initial yield stressY and the elastic material properties according to dataavailable in MMPDS-08 (Rice 2003). Various litera-ture sources were employed to inform our choice of β.Accurately modeling the temperature rise in the cali-bration specimens and the resulting softening requireda coupled thermo-mechanical simulation with ther-mal expansion, specific heat, thermal conductivity andemissivity determined from MMPDS-08. Void growthdamage parameters were chosen based on prior experi-ence with the material model and a sensitivity study ofthe model to the damage exponent m. Figure 57 con-tains initial tension simulation results and parametervalues.

After calibrating the model to the tension data, theshear data was incorporated into the model. Usingmaterial parameters calibrated to the tension data, amodel of the shear test did not accurately predict theyield behavior of the specimen thus indicating thatthe material exhibits an anisotropic yield surface. Byreducing the initial yield parameter Y by ∼ 83 %, theshear simulation results improved and compared wellto the test data.

Since the triaxiality driven void growth model can-not evolve damage in pure shear, sensitivity studies forvoid nucleation lead to the selection of the appropriateN1, φ

η0 and η0 parameters to capture shear failure. Fig-

ure 58 contains the calibrated shear simulation resultsand the corresponding parameters.

Challenge specimen modeling details

Model development for the challenge specimen includedspecifying the appropriate boundary conditions andincorporating anisotropy. The solid mechanics bound-ary conditions consisted of a symmetry boundary con-dition along the half-thickness plane of the specimenand approximations of the pin boundary conditionsin the test. A half-pin contiguously meshed into thespecimen with the center node line having prescribeddisplacements approximated frictionless pins. The toppin’s centerline was fixed and the bottom pin’s cen-terline was displaced downward with a rate corre-sponding to the test rates. As stated previously, accu-rately modeling the calibration specimens required acoupled thermo-mechanical simulation. The thermalboundary conditions included radiation from the spec-imen surface to the room temperature surroundings

123


Fig. 58 A separateparameter set with loweryield and void nucleationparameters accuratelymodels the yield behavior ofthe shear test for both ratesand failure for the slow rate

and a symmetry boundary condition along the half-thickness plane of the specimen. Since the EV modelcannot accommodate an anisotropic yield surface, themodel of the specimen was split into two elementblocks: Block 1 with a yield corresponding to the ten-sion initial yield Y t

RT and Block 2 with a lower yieldY s∗RT = 441 MPa since that region is initially predom-

inantly in shear. Figure 59 depicts Block 2 outlined inred with the remaining elements belonging to Block1. Since the stress state in Block 2 does not directlycorrespond to that of the failure region in the shearmodel, a simulation of the challenge specimen at theslow rate using a rate and temperature independent Hillplasticity model influenced the selection of Y s∗

RT = 441MPa. All simulations consisted of models constructedat the nominal dimensions according to the specimendrawings.

Blind predictions

Using the material model parameters and boundaryconditions specified in the previous sections, the chal-lenge specimen model predicted failure through crackpath B–D–E–A for both rates. For both rates, thecrack propagated unstably through B–D–E, as shownin Fig. 59, while the remaining ligament carried loaduntil tensile failure occurred much further into the sim-ulation (∼375 s for the slow rate and ∼ .36 secondsfor the fast rate). Table 13 lists the maximum loadsand CODs at crack initiation for each rate and Fig. 60displays the predicted load versus COD1 plot for bothrates.

Fig. 59 On the left, Block 2 is outlined in red on the undeformedmodel geometry. On the right, the deformed geometry is shownafter the crack has propagated into the upper hole in an unstablemanner


Several sources of error were present in the challengespecimen model. For example, an isotropic materialmodel was used to simulate the anisotropic materialthrough the use of separate element blocks and mate-rial parameters. Ideally, an anisotropic material modelwith rate and temperature dependence similar to EV

123


would have been used. Additionally, material para-meter uncertainties were large (e.g. β) and sensitivitystudies show these uncertain parameters had significanteffects on the simulation results. Numerical modelingissues also introduced error. SIERRA SM’s implicitcontact algorithm would not converge. Consequently,we employed a contiguously meshed and rotating half-pin. Our inability to quickly resolve the evolution oflocal damage did not permit resources for regularizedsolutions. We were able to nicely resolve both crack ini-tiation and propagation with surface elements (withoutthermomechanical coupling). Nonlocal studies will bethe subject of future work.

Team I

Team members:J.L. Bignell, [email protected], Sandia National

Laboratories, Albuquerque, NM, USAS.E. Sanborn, [email protected], Sandia National

Laboratories, Albuquerque, NM, USAC.A. Jones, [email protected], Sandia National

Laboratories, Albuquerque, NM, USAP.D. Mattie, [email protected], Sandia National

Laboratories, Albuquerque, NM, USATeam I approached the problem using the commer-

cially available general purpose finite element softwareAbaqus (Abaqus Standard Versions 6.13 and 6.14).Team I chose to use a “typical” finite element approach

to this problem because in many real world applica-tions (given time and budget constraints) this is theonly viable approach. An implicit solver was usedwith numerical stabilization to overcome global insta-bilities during the fracture event. To reduce the runtime, reduced integration 8-node hexahedral elementswere used rather than fully integrated elements. Abaqusallows tabular input of yield, potential functions, andstrain-to-failure curves. This feature was used to assignmultiple yield functions and multiple strain-to-failurecurves corresponding to different strain rates.

The Hill plasticity material model was used withrate dependent yield curves and isotropic hardening.In the Hill model, the strain rate is decomposed intothe sum of an elastic strain rate and an inelastic strainrate. In the Abaqus implementation of the model, theuser inputs six stress ratios Ri j to define the Hill yieldsurface. If they are all unity, the von Mises yield surfaceis recovered. The parameters of the Hill yield surfacewere determined for different strain rates during thecalibration procedure.

To capture material degradation and failure, strain-to-failure curves, that are dependent on both the stressstate (in terms of stress triaxiality η) and the strain rate,were employed:

εplD

(η, ˙ε pl

D

),

where η = − pq , p is the pressure stress, q is the von

Mises equivalent stress, and ˙ε plD is the equivalent plas-

tic strain rate. Because the stress and strain rate are

Table 13 Results predictedusing the challengespecimen model

Displacementrate (mm/s)

Peak load (N) COD1 @ Crackinitiation (mm)

COD2 @ Crackinitiation (mm)

25.4 20,310 2.966 2.644

0.0254 20,244 4.359 3.451

Fig. 60 The load versusCOD1 predictions for fastrate (left) and the slow rate(right) correspond to theexperimental data; however,both models over predictedCOD1 at crack initiation

123


Fig. 61 Tension test mesh and model details (left); close-up view of shear test mesh and model details (right). a Tension test model. bShear test model

changing throughout the simulation, degradation initi-ates when the state variable ωD , given by the followingintegral:

ωD =∫

dεplD

εplD

(η, ε

plD

) = 1,

∫dε

plD

εplD

(η, ε

plD

) = 1

reaches unity. For the time discretized problem, theintegral above is calculated by incrementing ωD at eachtime step, for each integration point, in the followingway (note ωD can never decrease):

ΔωD = �εplD

εplD

(η, ε

plD

) ≥ 0

Once ωD reaches unity, the material stress isdegraded in the following way,:

σ = (1 − D) σ ,

using a continuum damage variable D whose valueranges from zero to unity.

As the plastic strain increases the damage variableevolves as follows:

D = L ˙ε pl

˙u plf

= ˙u pl

˙u plf

, where u plf = 2G f

σy0

Here G f is the material fracture energy, L is thecharacteristic element length, and σy0 is the value of theyield stress at the time of failure initiation. This methodattempts to ensure that the energy dissipated during thedamage evolution process equals the fracture energy forthe material. While this method attempts to remove thedependence of failure on the size of the elements used,

the same element size used in the calibration processwas also used for the prediction. Degraded elementsare removed from the model when D attains a valuenear unity (full degradation).

Finite element models representing the uniaxial ten-sion and shear tests were constructed (see the meshrepresentations in Fig. 61). Two mesh sizes were ini-tially investigated for this calibration, 0.5 and 0.25 mm,with 0.25 mm being adopted for the challenge predic-tions. Symmetry was utilized when possible. Using thetwo models, along with the given uniaxial tension andshear test data, the following steps were employed inthe calibration process:

• Use the uniaxial tension test data and model todetermine the material hardening curves.

• Use the shear test data and model to define the Hillplasticity potential ratios

• Use the uniaxial tension test data and model todetermine the failure initiation parameters in thetension regime (η ≥ 0.33)

• Use the shear test data and model to determinethe failure initiation parameters in the pure shearregime (η = 0.0)

• Rerun the uniaxial tension and shear test models toverify all the inputs

• Using the calibrated model inputs, make predic-tions for the two double-notch tension tests

For the calibration of the material hardening parame-ters the following strain rate relationship was assumed

123


Fig. 62 a Yield stress versus plastic strain curves at differentstrain rates input into the model. Curves at strain rates of 0.001and 1.0 were determined by fitting the A, B, and n parameters

to the test data. b Plastic failure strain versus stress triaxialitycurves at different strain rates input into the model

for the yield stress loaded at an arbitrary strain rate:

σy

( ˙ε pl)

=⎛⎝1 + C

(ε pl

)ln

⎛⎝ ˙ε pl

˙ε plre f

⎞⎠⎞⎠ σy0.001, where,

C(ε pl

)=((

σy1.0(ε pl

)/σy0.001

(ε pl

))− 1)

ln (1.0/0.001),

along with the following assumed hardening relation-ships:

σy0.001

(ε pl

)= A0.001 + B0.001ε

pln0.001 and

σy1.0

(ε pl

)= A1.0 + B1.0ε

pln1.0.

The parameters A0.001, B0.001, n0.001, A1.0, B1.0,n1.0 were determined by fitting the model response tothe available tension test data. Figure 62a shows theyield stress vs. plastic strain curves at the 0.001 and1.0 strain rates determined from the calibration process,along with the curves at other rates determined usingthe assumed relationships above. Since Abaqus lin-early interpolates between the hardening curves that aredefined in the input, it is necessary to enter a sufficientnumber of curves to maintain the assumed log-linearrelationship above.

Using the rate dependent yield stress curves deter-mined, the shear model was run with Hill stress ratiosequal to unity. Results showed that the model over pre-dicted the onset of yielding when compared with theavailable shear test data. Hill stress ratios were adjustedto R12 = R23 = R13 = 0.88 to bring the model’s shearresponse in line with the test data.

The strain to failure curves were determined startingwith data found in the literature for Ti–6Al–4V (Giglio

et al. 2012). This is referred to as the reference curveεD−re f

pl (η) in the following discussion. The follow-ing rate dependent relationship was assumed:

εplD

(η, εD

pl)

=(

1 + E (η) ln

( ˙ε pl

˙ε plre f

))

εD−re fpl (η) Q (η) .

The scaling factors Q (η) and rate multiplier con-stants E (η) were determined through a fitting processusing the shear and tensile test models and test data.Figure 62b shows the final fitted plastic strain to failurevs. stress triaxiality curves for the strain rates input intothe model.

Figure 63 illustrates the response of the calibratedtension test model and Fig. 64 illustrates the response ofthe calibrated shear model. As mentioned above, onlytwo mesh sizes were studied during the calibration andprediction efforts. Ultimately, the mesh size of 0.25 mmwas used for the final calibration and prediction. Due totime constraints and Abaqus license restrictions, othermesh refinements were not considered. As discussedabove, the material degradation and failure model usedremoves the dependence on the element size, no meshsensitivity studies were performed to assess the degreeto which this holds true.

Having fitted the parameters for the material model,the double notch coupon (challenge problem) modelwas run (Fig. 65 illustrates the mesh representation uti-lized). A half symmetry model was created with an ele-ment size of 0.25 mm in the potential failure regions.The loading pins were not explicitly modeled. Instead,velocities were prescribed to the top and bottom halves

123


Fig. 63 Calibrated tension test model response. a Model response versus test data. b Example response

of the pin holes. It was recognized that this choiceof boundary conditions artificially restricts rotation ofthe two ends of the coupon while the pins utilized inthe testing likely allow for rotation. It was demon-strated later by additional analyses, that this choiceof boundary conditions introduces only a small errorin the predictions up to the first fracture in the spec-imens, whereas the behavior following the first frac-ture event (along ligament B–D–E) is significantlyaffected. Figure 67 illustrates the differences. Thenominal dimensions provided were used to constructthe model. Implicit dynamic simulations were perfor-med with numerical dissipation added to stabilize thesolution during the fracture event. Blind predictionswere made for both the slow and fast loading rates.Table 6 summarizes the results of these analyses.Figure 66 shows the predicted location 1 load vs. COD1curves, along with illustrations of the predicted frac-tures in the specimens for the two loading rates. Theresults show that the model over predicts both the peakforce and COD1 at failure for both loading rates, with

the discrepancy being more pronounced for the fastloading case. In addition, the model fails to capture thesoftening in the response corresponding to localizationof the strain immediately preceding the first fractureevent for both loading rates, again with the discrepancybeing more pronounced for the fast loading case. It isthought that these discrepancies are directly attribut-able to the model’s neglect of the plastic strain inducedheating and thermal softening of the material.


There are a number of potential sources of discrepancybetween the blind prediction and the actual challengeproblem results. The most significant source being thelack of inclusion of thermal effects in the materialresponse and the accounting of temperature changesinduced by plastic straining of the material (coupledthermal-mechanical response modeling). Results of thechallenge problem tests indicated a significant temper-ature rise in the material in the failure regions, par-

123


Fig. 64 Calibrated shear test model response. a Slow loading rate. b Fast loading rate. c Example response

Fig. 65 Double notchcoupon (challenge problem)mesh and boundaryconditions

ticularly for the fast loading cases. Another potentialsignificant source of discrepancy is the boundary con-ditions used. After the results of the challenge werereleased, Team I reran the challenge predictions with“pin-like” boundary conditions achieved using multi-

point constraints. It was found that there was not muchdifference in the response up to and including the peakload; however, beyond the peak load the response ofthe specimen was significantly different with a muchlarger COD achieved before fracture of the final liga-

123


Fig. 66 a Predicted loadversus COD1 curves for thechallenge specimen. bClose-up view ofdeformation and fracturepaths for the slow and fastloading rates

ment (Fig. 67). This is expected as with the updatedboundary conditions, rotation of the specimen ends isallowed as fracture progresses.

Team J

Team members:K. Pack, [email protected], Massachusetts Institute of

Technology, Cambridge, MA, USAT. Wierzbicki, [email protected], Massachusetts Insti-

tute of Technology, Cambridge, MA, USA

Plasticity modeling

The MIT team modeled the plasticity of the Ti–6Al–4V sheet using conventional description of continuummechanics for metallic materials, namely a yield func-tion, a hardening law, and a flow rule. The observa-tion of negligible difference in the engineering stress–strain curve before necking between the rolling direc-

tion (RD) and the transverse direction (TD) at eachloading speed led to the assumption of the identicalflow stress under uniaxial tension along three per-pendicular axes of RD, TD, and the thickness direc-tion. However, additional information from a shear testrequired to make use of the Hill’48 yield function (Hill1948) given in Eq. (24), which has the parameter Ncontrolling the plastic flow under in-plane pure shearstress.

σHill

[F(σ22 − σ33)2 + G(σ33 − σ11)2

+H(σ11 − σ22)2 + 2Lσ 223 + 2M2

13 + 2Nσ 212

] 12

(24)

Parameters other than N are reduced to those of thevon-Mises yield function, and N was calibrated so asfor numerical simulation to predict a correct level offorce in the V-notched rail shear test as demonstratedin Fig. 68a. The value of N larger than 1.5 implies therelative weakness in deformation resistance under pureshear loading.

123


Fig. 67 Comparison of the response for the original predictionmodel (with pin boundary conditions that restricted coupon rota-tion) and a model with modified (more representative) boundary

conditions. a Slow loading rate. b Fast loading rate. c Modelresponse comparison with test

Fig. 68 Comparisonbetween experiments andsimulations for: a V-notchedrail shear test forquasi-static loading; buniaxial tension test forquasi-static and dynamicloadings

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

25

30

35

Forc

e [k

N]

Displacement [mm]

Exp_VP2_Slow Sim_N=1.5 Sim_N=1.99

0.00 0.05 0.10 0.150

200

400

600

800

1000

1200

1400

Engi

neer

ing

stre

ss [M

Pa]

Engineering strain [-]

Exp_RD6_Slow Sim_Force_Slow Sim_PEEQ_Slow Exp_RD4_Fast Sim_Force_Fast Sim_PEEQ_Fast

0.0

0.4

0.8

1.2

1.6

2.0

Equi

vale

nt p

last

ic s

trai

n

(a) (b)

Accurate prediction of crack initiation in ductilemetals depends highly on the hardening curve in thepost-necking regime. Moreover, dynamic loading con-sidered in the 2nd Sandia Fracture Challenge is inex-tricably linked with the thermal softening as well asthe strain-rate effect. The MIT team employed a mod-ified Johnson–Cook law, proposed by Roth and Mohr(2014), which uses a weight-average of the Swift and

the Voce law for the strain hardening in the form ofEq. (25).

K [εp, ˙εp, T ]=[α · A(ε0+εp)

n+(1−α) · (k0 + Q(1 − eβε p ))]

[1 + CIn

( ˙εpε0

)][1 −

(T − TrTm − Tr

)m](25)

123


Table 14 Plasticity parameters of the Ti–6Al–4V sheet for the lower bound case

F G H L M N A (MPa) ε0

0.5 0.5 0.5 1.5 1.5 1.95 1393 0.01763

n k0 (MPa) Q (MPa) β α ε0 (/s) εi t (/s) εa (/s)

0.07955 1011 147.2 24.94 1.2 0.0006 0.0006 1.0

C T0 (K) Tr (K) Tm (K) m ηk ρ (kg/m3) Cp (J/kg K)

0.01605 293 293 1900 0.74 0.9 4430 5.263E2

In order to avoid high computational costs and ambi-guity of boundary conditions, fully coupled thermo-mechanical analysis was simplified by purely mechan-ical analysis in which a fraction of incremental plasticwork was converted into heat causing temperature risebased on Eq. (26).

dT = w[ ˙εp] ηk

ρCpσd εp (26)

ηk is the Taylor–Quinney coefficient assumed to beconstant. The weighting factor changes smoothly fromzero to unity by Eq. (27), which implies transition fromisothermal to adiabatic condition.

w[ ˙εp] =

⎧⎪⎨⎪⎩

0 for ˙εp < εi t( ˙εp−εi t )

2(3εa−2 ˙εp εi t )εa−ε3

i tfor εi t ≤ ˙εp ≤ εa

1 for ˙εa < ˙εp(27)

The optimized parameters for the lower bound caseare provided in Table 14 together with six parameters ofthe yield function. The recommended calibration pro-cedure through inverse analysis is explained in Rothand Mohr (2014). The only distinction was made inthat a dog-bone specimen was used to find hardeningparameters instead of a notched tensile specimen withcircular cutouts, which was found to be more appropri-ate for the hardening curve optimization (see Pack et al.2014). Careful attention was paid to the velocity profileapplied to the boundary of the specimen for fast loadingbecause it is not constant due to the compliance of thecross-head of a testing machine. It was confirmed thatthe engineering stress–strain curve predicted by finiteelement simulation showed a good agreement all theway to fracture with test results that showed the loweststress after necking (regarded as lower bound cases) asdepicted in Fig. 68b. Very fine mesh of 0.1 mm was usedin the necked region, following the recommendation byDunand and Mohr (2010).

Fracture modeling

Crack is assumed to initiate when the indicator D,calculated by a linear damage accumulation rule inEq. (28) reaches unity. The corresponding finite ele-ment is then eliminated from a whole model.

D =∫ ε f

0

d εp

εprf (η, θ)

(28)

The function εprf defines the strain to fracture under

proportional loading as a function of two stress-statedependent variables: the stress triaxiality η and theLode angle θ , whose combination specifies a load-ing path. Following Roth and Mohr (2014), the rate-dependent Hosford–Coulomb fracture model was takenfor ε

prf .

εprf (η, θ) = b0

(1 + γ In

( ˙εpε0

))

(1 + c)1n

[{1

2(( f1 − f2)

a(( f2 − f3)a

+ ( f3 − f1)a)

1a

}+ c(sη + f1 + f3)

] 1n

(29)

f1(θ) = 2

3cos

[π

6(1 − θ )

](30)

f2(θ) = 2

3cos

[π

6(1 − θ )

](31)

f3(θ) = 2

3cos

[π

6(1 − θ )

](32)

Mohr and Marcadet (2015) formulated the originalrate-independent Hosford–Coulomb fracture model,inspired by the onset of microscopic shear localization.Even though shear tests were carried out by Sandia, andthe test data were provided, the fact that slip occurredduring the tests and the axial displacement was mea-sured with an LVDT attached to fixtures whose com-pliance was not negligible is attributed to not including

123


A

C

D

E

B

Fig. 69 Sequence of damage accumulation and crack development for fast loading

shear tests for the present fracture calibration. As a con-sequence, the identification of five fracture parametershad to rely solely on two uniaxial tensile tests. TheMIT team was in possession of the fracture parametersfor a similar alloy, so a, c, and n were taken from ourdatabase. This basically assumes that the current alloyof interest has the same dependence on η and θ as thesimilar alloy in our database. The remaining parame-ters of b0 and γ control the height of a fracture enve-lope and its strain-rate sensitivity, respectively. Thesetwo values were found such that the engineering strain(equivalently displacement) to fracture of a dog-bonespecimen in both slow and fast loading condition isaccurately captured as noted in Fig. 68b.

Blind prediction

In light of the symmetry through thickness, only ahalf of the specimen geometry with print dimensionswas discretized by approximately 750,000 reduced-integration eight-node three-dimensional tri-linear solidelements (C3D8R of the Abaqus element library) withthe smallest ones of 0.1 × 0.1 × 0.1 mm3 around twonotches and three holes. This size of elements was cho-sen to be the same as the one used for the dog-bone andthe shear specimen to minimize a possible mesh sizeeffect. Abaqus/Explicit was used in simulation with thematerial models implemented through the user mate-rial subroutine (VUMAT). The upper and the lowerpin were modeled as analytical rigid bodies, and thepenalty contact with no friction was defined betweenthe pins and the specimen. The lower pin was pulled

down at 0.0254 mm/s for slow and 25.4 mm/s for fastloading with the first one tenth of the total simulationtime reserved for acceleration. Uniform mass scalingwas applied to reduce computational time in so far asit guarantees the negligible ratio of kinetic energy tointernal energy before the first crack initiation. Crackpropagation was considered to be nothing more than theproblem of consecutive crack re-initiation, thus beingmodeled by continuing element deletion.

Figure 69 visualizes the sequence of damage accu-mulation and crack path for fast loading. Slow load-ing exhibited a similar non-uniform distribution of thedamage indicator and ended up with the same crackpath of B–D–E–A. In the early stage, plasticity comesinto play in the ligament between A and C by ten-sion and in two ligaments between B and D and Dand E by combined shear and tension. Because of therelative weakness in shear resistance characterized byN = 1.95, the specimen prefers shear localization tonecking, which results in rapid crack propagation. Thetwo ligaments between B and D and D and E fracturealmost at the same time, by which a huge amount ofelastic energy is released all of a sudden. This causesthe vibration of the whole system. Finally, the ligamentbetween A and E deforms mostly due to bending andreaches complete failure.


The comparison of the force-COD1 curve betweenexperiments and simulations is made in Fig. 70. Elasticregion as well as the maximum force was very accu-

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0 1 2 3 4 50

5

10

15

20

25

Exp_Slow Sim_Slow_Lower_bound Exp_fast Sim_fast_Lower_bound

Forc

e [k

N]

COD1 [mm]

Fig. 70 Comparison of the force-COD1 curve between experi-ments and simulations in slow and fast tensile tests of the secondSandia Fracture Challenge specimen

rately predicted for both slow and fast loading with3.6 % overestimation of the maximum force for slowloading. This error might have been caused partiallyby dimensional discrepancy between actual specimensand the finite element model. The COD1 at the firstcrack (abrupt load drop) for fast loading was predictedwith great accuracy while it was 37 % overshot for slowloading. This is mainly due to the over-adjusted hard-ening curve after necking as can be deduced from thedelayed COD1 at the maximum load. As noted by Packet al. (2014), the flow stress after necking optimized bya dog-bone specimen deteriorates numerical predictionin other specimen geometries, and a notched tensilespecimen with circular cutouts serves an improvementin the prediction.

Concluding remarks

Plasticity and fracture modeling on the basis of uniaxialtension and shear tests subjected to two different load-ing speeds made a satisfying prediction both in the slowand the fast challenge problem. The present plasticitymodel accounts for relatively low shear resistance ofthe Ti–6Al–4V sheet, which was crucial to capture thetype of instability that led to a correct prediction of thecrack path. More detailed explanation about the cali-bration procedure and the improvement in numericalprediction based on an exhaustive testing program per-

formed on the leftover material can be found in Packand Roth (2016).

Team K

Team members:S.-W. Chi, [email protected], University of Illinois at

Chicago, USAS.-P. Lin, [email protected], University of Illinois at

Chicago, USAA. Mahdavi, [email protected], University of Illi-

nois at Chicago, USA

Approach

The simulation methodology in this work is basedon an enriched Reproducing Kernel Particle Method(RKPM) (Chen et al. 1996; Dolbow and Belytschko1999; Krysl and Belytschko 1997; Liu et al. 1995).The crack surface/tip discontinuity is embedded in thedisplacement approximation as follows.

uh =∑

I∈N−Ncut−Ntip

�I (x) dI

+∑

J∈Ncut

∑i=1,2

Si (x) �J (x) a j

+∑

K∈Ntip

∑j=1,2

fi (x) bK ≡∑I

�I (x)dI (33)

Here �I is the reproducing kernel (RK) shape func-tion with the Cubic B-spline (C2 continuous) as thekernel function centered at node xI ; Ncut and Ntip arenode sets, in which the support of node contains thecrack surface and crack tip, respectively; N is the totalnode set; dI , aJ , and bK are nodal coefficients. Si isintroduced to represent the continuity across the cracksurface and expressed as:

SI (x) ={

1 ζ+ > 00 ζ− < 0

and S2 = 1 − S1. (34)

where ζ+ and ζ+ denote the above and below crackregions, respectively. The crack tip enrichment func-tion, f j , is formulated based on the visibility criterion(Krysl and Belytschko 1997; Lin 2013) and has thefollowing form:

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For θ0 > 0, f1(x) =

⎧⎪⎨⎪⎩

1 θ0 − π/2 ≤ θ ≤ π

sin(θ − θ0 + π) −(π − θ0) ≤ θ < θ0 − π/2

0 −π ≤ θ < −(π − θ0)

For θ0 ≤ 0, f1(x) =

⎧⎪⎨⎪⎩

0 π + θ0 ≤ θ < π

sin(θ0 − θ + π) −θ0 + π/2 < θ ≤ π + θ0

1 −π ≤ θ ≤ θ0 + π/2

and f2 = 1 − f1 (35)

where θ0 is the angle from the crack direction to theenriched node and θ is the angle from the crack direc-tion to the evaluation point. Note that the enrichmentshape functions in (33), �I , satisfy the partition of unitycondition,

∑�I = 1. Therefore, the mass lumping in

the explicit time integration in the Galerkin formulationis straightforward (Lin 2013).

Material model and calibration of material parameters

The material in simulations is modeled by the Johnson–Cook rate-dependent model with a bi-linear hardeningJ2 elastoplasticity:

H(e p) = 1 (36)

K (e p, ˙ep){Ks(e p)[1 + C ln( ˙ep)/ ˙e0)], if ˙ep < ˙epcrit

Ks(e p)[1 + C ln( ˙epcrit)/˙e0)], esle

(37)

Ks(ep)

={ [Y0 + α0e p], if e p < e pcrit

Y1 + α1(e p − e pcrit), if e p < e pcrit, Y1 = Y0 + α0epcrit

(38)

where H and K are the kinematic and isotropic hard-ening parameters, respectively; e p is the effectiveplastic strain. The initial yield stress, y0 = 1.0474GPa, the hardening parameters α0 = 1.157 GPa, andα1 = 0.45 GPa, and the critical equivalent plastic straine pcrit = 0.12 were calibrated from slow-loading-ratetensile test data (Fig. 71). For rate effect, the referenceplastic strain rate ˙e0 was chosen to be the unity; theconstant C = 0.015 and the critical plastic strain ratee pcrit = 28 were obtained from the numerical tensiletests as shown in Fig. 71. Considering the maximumprinciple tensile strain as the crack initiation criterion

and assuming that the tensile failure is the predominantfailure mode, the crack opening strain in the initiationcriterion can be obtained according to the rupture pointin the tensile test. The principal strain at the rupturepoint, 0.946, was assumed as the crack initiation strainfor both slow- and fastloading-rate cases.

Numerical simulations

An explicit updated Lagrangian reproducing kernel for-mulation with the enriched displacement approxima-tion in (33) was employed to model the challenge prob-lem. The crack propagation speed in the dynamic sim-ulations was assumed to be 0.02 % of Rayleigh wavespeed, based on analytical and empirical studies fornon-branching crack propagation (Freund 1972, 1979).A RKPM discretization containing a total of 55,520nodes, with 2 nodes in the thickness direction, was usedfor simulations. The mean in-plane nodal distance nearthe notched area in the discretization is 0.44 mm, which

0

200000000

400000000

600000000

800000000

1000000000

1200000000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Experimental - loading rate = 25.4 mm/sNumerical - loading rate = 25.4 mm/sExperimental - loading rate = 0.0254 mm/sNumerical - loading rate = 0.0254 mm/s

P=0.946

Nominal Stress (Pa)

Nominal Strain

P: Principal strain

Nominal Stress-Strain Curves

Fig. 71 Nominal stress–strain of the tensile failure test. Thenumerical tensile failure test was conducted to obtain the hard-ening and rate parameters in the plasticity model. The crack open-ing strain in the initiation criterion was also calibrated from therupture point in the simulation

123


is consistent with the nodal density used for materialparameter calibration.

Figure 72 shows the predicted deformations and vonMises stress contours for both slow- and fast-loading-rate cases. The predicted crack paths for both casesfollow a similar pattern, “B–D–E–A”, which agreeswith experimental results. Figure 73 compares numer-ical load-COD curves to experimental ones for bothloading cases. The numerical simulations over-predictthe slopes of load-COD before yielding and peak load-ing values. This is likely mainly due to inappropriateboundary conditions. Unlike loaded at both bolt-holesthrough low-friction pins in the experiments, the spec-imen in the simulations was loaded through prescribedleftward displacement with fixed vertical direction onthe left hole (Fig. 72). The extra constraint limits theCOD development and therefore leads to an increase inforce response. Furthermore, for the case with fast load-ing rate, the numerical model predicts much larger rup-ture COD than the experimental data. The main causeof the discrepancy may be attributed to an inaccuratecrack initiation threshold and an inaccurate constantcrack propagation speed for fast loading rate.


The crack paths obtained from simulations for bothslow- and fast-loading-rate cases are in good agreementwith experimental data; however, the force responsesand the crack initiation COD are over-predicted. Theover-prediction is mainly attributed to inappropriateboundary conditions, and inappropriate material para-meters, including, but not limited to, rate-dependenthardening parameters and crack initiation thresholds.In the dynamic simulation, the crack tip speed alsoplays an important role and needs to be properlyestimated based on a fracture energy release ratealgorithm.

Team L

Team members:J. Predan, [email protected], University of Mari-

bor, SloveniaJ. Zadravec, [email protected], University

of Maribor, Slovenia

Fig. 72 Deformation, vonMises stress, and failurepattern

Slow loading rate Fast loading rate

Fig. 73 Comparison offorce-COD curves

Force (N)

COD1 (mm) COD1 (mm)

Force (N)Slow loading rate Fast loading rate

NumericalNumerical

Experimental Experimental

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Approach

The 2D plane strain simulation was performed withAbaqus CAE 6.14 software using an elasto-plasticmodel (continuum) and a damage model (damageinitiation and damage evolution). The Abaqus Stan-dard/Static solver and isotropic elastic plastic materialbehaviour including isotropic deformation hardeningwas employed. The elastic, plastic, damage initiationand evolution parameters were fit to experimental data.The parameter fitting utilized both tensile and sheardata at both loading rates. The Extended Finite ElementMethod (XFEM) was used to simulate damage in thespecimens. The finite elements mesh was refined in theregions where predicted crack propagation, as shownin Fig. 74. The damage initiation criterion was based onthe quadratic traction-interaction law and the damageevolution criterion was energy dissipation with linearsoftening. Damage stabilization was used to reach con-vergence in the static solver.

Procedure for selection of parameters for materialmodel and failure For parameter calibration, CAEmodels of the tensile and shear specimens were con-structed. The initial calibration utilized the elastoplastictensile data. The cross section of a broken test specimenwas used to calculate true stress and calibrate the plasticparameters. Further refinement of the parameters wasachieved by first explicitly simulating the tensile testand subsequently simulating the shear test to achieveresults that matched the experimentally reported out-comes.

Cohesive zone elements were used and the fail-ure criteria chosen was a nominal stress based on aquadratic combination of all three ratios for crack initi-ation and maximal deformation energy for crack prop-agation. Results of both loading rates simulations werefit to the reported experimental calibration data.

Modeling details A 2D assembly model was used forcrack growth path prediction. This model consistedof the challenge specimen with exact nominal dimen-sions that were defined in challenge documentation andtwo pins. Between the pins and specimen the follow-ing contact properties were defined: for normal behav-iour hard contact and for tangential behaviour the fric-tion coefficient of 0.05. The center of the bottom pinwas a fixed constraint in directions X and Y with freerotations about the Z axis. At the centre of upper pin,

Fig. 74 FE model configuration

the following motions were prescribed: displacementof 4 mm in direction Y, and fixed in direction X, freerotation around axis Z. The model was quasi-static. Asurface region was defined to allow possible cracking.The crack was formed from notch geometry accordingbased on the cohesive zone damage criteria. For crackgrowth, XFEM technology was employed. The XFEMregion was prescribed around all holes and notches toallow for possible crack initiation and propagation.

The Quads criterion was used for cohesive zone ele-ment placement, which is based on a quadratic combi-

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Fig. 75 Mesh with refinement edges

nation of nominal stress ratios between a given stressvalue and the peak nominal stress value in each of threedirections.

Mesh refinement In the model, 61,315 linear quadri-lateral plane strain elements were used of type CPE4.The maximum global size of any element was 1 mm or0.25 mm in the XFEM crack prediction region. Refine-ment edges are shown in purple in Fig. 75. Because ofthe small, unstructured elements there was no influenceof element orientation.

Blind predictions

Based on results of simulation shown on Figs. 76 and77, Team L predicted, that the crack will follow pathB–D–E–A. The predicted Force-COD1 curve for boththe slow and fast loading rates are shown in Fig. 78.

Sources of discrepancies

The primary source of discrepancy likely was related toinadequate shear calibration due to convergence prob-lems. Further, Team L also had made a post-processingmistake by exporting the reaction force for only 1 fix-ing point, whereas the pin had been constrained at 8points and for this reason, our initial blind predictionsreported forces that were too low.

Team M

Team members:A.J. Gross, [email protected], Univer-

sity of Texas at Austin, Austin, TX, USAK. Ravi-Chandar, [email protected], University of

Texas at Austin, Austin, TX, USAIn some recent work we have identified that, for a

class of ductile materials, plastic deformation proceeds

without intervening damage until very large strain lev-els; this is confirmed through observations and mea-surements of deformation at multiple scales, from themacroscopic to the level of the grains (Ghahrema-ninezhad and Ravi-Chandar 2012, 2013; Haltom et al.2013). The upshot of these investigations is twofold:first, it is essential that the plastic response of the mate-rial be calibrated to much larger strain levels than isusually achieved in a standard tensile test. Second, themechanisms of final failure—void nucleation, growthand coalescence—occur within a highly localized zonein the plastically deformed material, and only at thevery end of the material’s ability to withstand deforma-tion. Thus, a model for the final failure of the materialmay be implemented numerically by a simple damagecriterion such as element deletion. However, it is neces-sary to perform a careful evaluation of the plastic strainlevels at which damage may initiate under multiaxialloading. We have adopted this approach in formulatingthe simulation of the challenge problem.

The plastic constitutive properties of Ti–6Al–4V aremodeled by the flow theory of plasticity with isotropichardening. Due to both the limited time to implement anappropriate yield criterion for HCP metals in ABAQUS(the FEM package that was used for this work) andthe sparsity of stress paths used the calibration exper-iments, Hill’s 1948 anisotropic yield criterion (Hill1948) was selected as the governing model for plas-ticity. The two parameters affecting the yield stress forout of plane shear conditions are assumed to be equal totheir isotropic values since no data are available to cali-brate them; this is considered to be appropriate becausethe corresponding stresses will be negligible in both thecalibration experiments and challenge geometry. Theremaining four parameters are subject to calibration.It is evident that uniaxial tensile test results cannot beused to determine the stress–strain behavior beyonda logarithmic strain of ∼4 % because of the inhomo-geneous deformation that occurs beyond the Consid-ère strain. After this point, some model for the stress–strain relation must be considered. In this work, thebehavior is represented by a monotonically increasingspline with seven segments beyond the Considère strain(Gross and Ravi-Chandar 2015). This form is chosenbecause it provides much more flexibility in the shapeof the stress–strain curve than typical power law behav-ior, yet does not have an undue number of parameters.Temperature and rate effects were included by a multi-plicative modification to the stress–strain curve defined

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Fig. 76 Results of von Mises stress (left loading rate 0.0254 mm/s, right 25.4 mm/s)

by three parameters, following the form first used byJohnson and Cook (1983).

The parameters for the spline and the yield criterionare then found by an inverse procedure, where itera-tive finite element simulations of the slow rate calibra-tion tests are performed with different trial parametersfor the constitutive model. A nonlinear optimizationscheme is used to minimize the sum of the errors for therolling direction (RD) tensile, transverse direction (TD)tensile, and shear (VA; shear against grain) simulations.Error for each simulation is measured as the sum of therelative error between the net load in the experimentand simulation at 100 levels of global deformation.The resulting stress–strain curve and simulated load-elongation behavior for the calibration experiments areshown in Fig. 79. The VP shear data (shear parallel tograins) was not used, as the chosen constitutive modelis not influenced by the orientation difference between

VP and VA shear. Thus, only one shear test, or anaverage of the two could be used for calibration. VAshear was used exclusively for model calibration as itcorresponds to the dominant orientation of shear load-ing in the challenge geometry. After calibration of theanisotropy and stress–strain curve from the slow ratetests was completed, the parameters for temperatureand rate sensitivity were found using the high rate RDtensile and VA shear tests with the same inverse pro-cedure. TD tension was not used because temperatureand rate sensitivity are expected to be isotropic. Thetwo tests used for calibration were chosen because theyare dominated by different strain rates and temperatureranges.

Material failure is modeled by a strain to failuremodel where the failure strain, εf , is dependent exclu-sively on stress triaxiality. When an element in the FEMsimulations accumulates a damage parameter equal to

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Fig. 77 Results of plastification stress (left loading rate 0.0254 mm/s, right 25.4 mm/s)

Fig. 78 Force versus COD plot

unity according to the rule,∫ dε p

ε f , its stiffness is setto zero, where dε p is the plastic strain increment. Thefailure strain was calibrated by using the optimized RDtensile and VA shear simulations. For tension, the cen-

tral element in the neck has both the highest triaxial-ity and strain. Since rupture of the specimen occursrapidly, it corresponds to failure of this central ele-ment. By matching the experimental elongation at rup-ture in the simulation, the central element in the neckprovides a strain to failure estimate under moderatelevels of triaxiality. Strain to failure estimation in theshear specimen is based on past experimental expe-rience indicating that the peak load in the test cor-responds to the formation of a crack at one of thenotch tips. Then the grip displacement at peak loadin the experiment corresponds to global deformationstate where the element at the current notch tip inthe simulation must fail. This provides an estimate onthe strain to failure under negative triaxiality condi-tions. After crack initiation, stable growth occurs inthe experiment and could be used to perform a moredetailed failure calibration. Due to time constraintsthis data was not used. It was found that the strains

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Fig. 79 a Calibratedtrue-stress–strain behaviorfound for Ti–6Al–4V bysimultaneously minimizingthe load-displacement errorin b RD tension, c TDtension, and d VA shear

to failure over the large range of triaxiality spanned bythese two tests were nearly identical (0.79 from tensionand 0.82 from shear), so the strain to failure betweenthem was simply interpolated linearly. For triaxialitiesin excess of those in the tensile test, a conservativestrain to failure curve that is motivated by the exponen-tial behavior first suggested by McClintock (1968) wasadopted.

The challenge geometry is simulated with anABAQUS/Explicit FEM model for both the slow andfast rate scenarios. Mass scaling is used to increase thestable time step to for both the slow and fast rate simu-lations. A highly refined mesh is used in the ligamentsbetween the holes and notches, where strain localiza-tion is most likely to occur. Typical elements in thisregion were about 40 x 45 micron with 22 elementsthrough the thickness. A total of about 700,000 eight-noded linear elements with reduced integration andhourglass control were used. 1.26 million and 0.93 mil-lion time steps were used in the slow and fast rate testsrespectively, each simulation requiring several hundredhours of CPU time. Loading is applied on rigid, fric-tionless pins. The bottom pin was held fixed and aquadratic displacement rate was applied at the top pinfor the slow loading scenario. For the fast loading alinear displacement rate at the top pin was imposed,

consistent with the loading rate in the experiment. Theresults are summarized in Figs. 80 and 81 where thestrain field at selected levels of deformation from theslow rate prediction and the load-COD1 variations forthe slow and high rate predictions are shown.

For the slow rate test, early strain accumulation is thelargest in ligament A–C, and remains so until a COD1value of about 1.75 mm. After this point, both liga-ments B–D and D–E have comparable levels of strainthat exceed those in ligamentA–C. The prediction indi-cates an increase in load carrying capability of the spec-imen until a COD1 of about 4.5 mm, where, on the cuspof localization, failure initiates, triggering fast fracturein ligament D–E, almost immediately followed by thefast fracture of ligament B–D. At initiation of failure,the load is predicted to drop abruptly to nearly zero,thus ending the simulation. It is projected that liga-ment EA will fail under continued loading, but this wasnot examined. The main details of the high rate test aresimilar to the slow rate.


The largest difference between predictions and experi-ments is that failure was predicted to occur at a COD1∼4.1 mm while it was observed at a COD1 ∼3.03 mm

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Fig. 80 The equivalent plastic strain field at COD1 values of a 1 mm, b 1.75 mm, c 3 mm, and d after failure

Fig. 81 Load-COD1 curveswith the predicted outcomeand experimental results fora slow rate and b high rateloading

in the slow-rate tests, with a similar difference in thehigh-rate tests. All other aspects of the response werepredicted within the desired tolerance. After a careful

examination of the predictions in comparison to addi-tional experiments that were instrumented to measurelocal deformation fields (see Gross and Ravi-Chandar

123


2015, 2016), it was determined that the elastic–plasticconstitutive model did not satisfactorily represent theyield surface along the stress paths followed in the criti-cal ligaments of the challenge geometry beyond a strainof ∼0.2; this deficiency in identifying and calibratinga proper constitutive model was the main source of thediscrepancy between the predictions and the experi-ments. However, this deficiency did not manifest itselfin the calibration exercise; this points to the need fora better approach to designing, performing and imple-menting experiments that are used in the calibrationprocedure.

Team N

Team Members:L. Xue, [email protected], Thinkviewer LLC, Sugar

Land, TX, USA

Approach

The second Sandia Fracture Challenge gives an excel-lent opportunity to evaluate how different loading rateconditions can influence fracture behavior of a givenconfiguration of ductile metals. In general, the effectsof loading rate on mechanical behavior of metals cancome in from several aspects, for instance (1) manymetals display higher resistances under high strainrates; (2) the post-yield plastic hardening exhibits lowerhardening capability at high strain rates; (3) the plas-tic work in the deformation zone converts to thermalenergy at high strain rates, which heat up the mater-ial locally and hence reduces the material resistance;(4) high strain rates can activate different plastic flowmechanism and thus results in higher or lower fracturestrain depending on the micro structures of the material.In the present study, the effects (1) and (2) are modeledby calibrating the material yielding and strain harden-ing behaviors separately for each loading rates, whilethe effect (3) and (4) are not considered explicitly, butrather they are inherently embedded in the calibrationprocedure because they are not singled out in the mod-eling.

Two pulling rates at the grips are used for the coupontests of the material properties and the S-shaped speci-men for fracture prediction. Several fracture initiationsites can be foreseen for the S-shaped flat plate withthree holes in the vicinity of the roots of rounded slots.

At first glance, this S-shaped structure is in severalways like the first Sandia Fracture Challenge, exceptthat a second slot was added to the configuration. Pre-vious studies have shown that, in this type of duc-tile fracture simulations, the constitutive relationshipof the material plays a central role in the accuracy ofthe prediction results (Boyce et al. 2014). There aremany choices of constitutive models that are capableof discerning between a mode I dominant and a modeII dominant ductile fracture (Xue 2007, 2008, 2009).The Xue (2009) damage plasticity model requires onlyone parameter to be calibrated for the simplest case.Yet, this model is enabled by a full 3D non-linear dam-age coupled yield function and has been shown to beable to capture different fracture modes through a seriesof numerical simulations. In the first Sandia FractureChallenge, the Xue (2009) model was used and showedextraordinary capability in predicting fracture behaviorof the given structure using limited simple material test-ing data. In this second Sandia Fracture Challenge, thesame procedure was used to calibrate necessary mater-ial parameters and to obtain finite element predictions.

From the material testing results at the two load-ing rates, it is clear that the selected material Ti–6Al–4V exhibits some rate sensitivity. The nominal loadingrates at the pins of the slow and fast condition are 0.0254and 25.4 mm/s respectively. The strain rate in the defor-mation region of the S-shaped structure is further higherthan that of the high rate in the tensile material testing,because the deformation zone in the material testingis greater than the deformation zone in the S-shapedstructure while the end separation velocity are the same.This difference in strain rate should be about an orderor so given the size of the holed zone in the S-shapedstructure, which converts to about 3 % increase in themagnitude of yield stresses for the S-shaped structureconsidering a logarithmic relationship of the rate hard-ening coefficient. Note, this difference should be aboutthe same in the slow rate scenario. However, it is con-sidered relatively a small error and is not factored inthe present study, i.e. the calibrated matrix stress–strainrelationship from curve fitting of the tensile coupon inthe rolling direction was used directly for the S-shapedspecimen at the same nominal loading rate.

A parallel finite element simulation was used to cal-ibrate the material parameters. In both calibration andprediction simulations, a one-point reduced explicittime integration scheme was adopted. In this type ofductile fracture analyses, it is well-known that the sim-

123


ulation results are subjected to mesh size of the finiteelement model. In order to minimize mesh size depen-dence, the element sizes were carefully chosen suchthat the elements in the material tests and the centralregion of the S-shaped structure used for predictionwere about the same. A total of 16 elements throughthe thickness were used in order to capture the throughthickness fracture pattern and this through thicknesselement length was used in meshing the in-plane ele-ments. Thus, the element size was about 0.2 mm in alldirections. Same mesh was used for both slow and fastloading rate.

Material parameters calibration

The material tension test data in the rolling directionsfor the two rates were used to calibrate the mate-rial stress–strain curve using Swift relationship, seeEq. (39) σM = σy0(1 + εp

ε0)n where σM is the mater-

ial matrix resistance at given plastic strain εp, σy0 isthe initial yield stress, ε0 is a reference strain and n isthe hardening exponent. A set of initial fitting parame-ters for low loading rate were used to run a detailedfinite element analysis of the tensile tests and are thenadjusted by matching the simulation load-displacementcurve with the experimental one [Xue EFM 2009]. Dueto the coupled nature of the stress–strain relationshipwith damage associated weakening in the damage plas-ticity model, an iterative process is needed to calibrateall the material parameters in Eqs. (39–42), where σf0

is a reference fracture stress, kp is a pressure sensitiv-ity parameter, and m and β are damage and weakeningexponents. In the present study, both m and β were setto 2.0 according to previous studies on various met-als. A reference fracture strain εf0 is substituted for thereference fracture stress σf0, which is related to εf0 by

εf0 = ε0

(σf0σy0

− 1)1/n

. After several iterations, the final

fitted material parameters that give good match of theload-displacement curves for simple tension coupontests are listed in Table 15 for the two loading rates.Then, the above process was repeated for the high load-

ing rate.

σM = σy0

(1 + εp

ε0

)n

(39)

εf = ε0

⎧⎨⎩(

σf0

σy0

)1/n[(

1 + kpp) √

3

2 cos θL

]1/n

− 1

⎫⎬⎭(40)

D = m( εp

ε f

)m−1 εp

εf(41)

σeq =(

1 − Dβ)

σM (42)

From the given tensile coupon tests, the materialstress–strain curves show that the fracture strains undersimple tension at the two loading rates are not too differ-ent from each other; however, the hardening exponentappears to be lowered significantly when the loadingrate is high. It is also noticed that the material yieldstress displays some strain rate hardening. At the highloading rate, the yield stress is about 10% higher whenthe strain rate is 1000 times higher.

Experiments in literature show that at higher loadingrates, the fracture pattern often favors shear mode, e.g.it can change from a mode I to a mixed model I/III fora flat panel when a compact tension specimen loadeddynamically (Rivalin et al. 2001; Xue and Wierzbicki2009). In the present study, path A–C–F represents amode I fracture and path B–D–E–A represents an in-plane shear mode. These two modes should be the dom-inant fracture modes of the present structure for mostmetallic materials.

In addition to the tensile tests, a double V-notchedspecimen was also used for material shear testing. Thefracture plane of the double V-notched specimen undersimple shear is a little skewed and is not exactly alongthe straight line connecting the roots of the V-notches.The shear test shows the material yield stress undernominal simple shear is lower than that under simpletension. It was estimated that the initial yield stress wasabout 13 % lower under nominal simple shear condi-tion. There are two possibilities to model this differ-ence: (1) adopt a Tresca yield condition or similar J3-

Table 15 Material parameters for Ti–6Al–4V for damage plasticity model

Nominal loading rate σy0 ε0 n σ f 0 kp m β

Slow rate 0.0254 mm/s 1011 MPa 0.1432 0.05608 2.2 6.44e−5 MPa−1 2.0 2.0

Fast rate 25.4 mm/s 1124 MPa 0.07475 0.04540 3.5 1.90e−5 MPa−1 2.0 2.0

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Fig. 82 Predicted fracturepaths using the damageplasticity model: A–C–F forslow (left) loading rate andB–D–E–A for fast (right)loading rate. Plottedcontours are theaccumulated plastic strains

dependent yield condition or (2) assuming the mater-ial is anisotropic (likely due to rolling process, but wewere not sure because of limited experimental data) andchoose an anisotropic yield function such as Hill1948.In either way, it should be more accurate in describingthe yield condition, at least in the initial yielding phaseof the material. However, this J3-dependence yieldingis ignored in the present prediction using the simpledamage plasticity model (Xue 2009).

Results and discussions

Damage plasticity model for the material was usedthroughout the numerical simulation. Using the abovedescribed method and the calibrated material parame-ters, the pulling of the S-shaped structures at the twoloading rates were simulated. The nominal geometry ofthe specimen was used. No variation in the actual sizewas considered for the geometrical tolerance. Two setsof hole-pin contacts used to pull the specimen apartwere modeled explicitly, such that the rotation at thepin-hole contact was allowed. Fracture is simulated byelement removal, i.e. when damage index D is accumu-

lated to unity for an element, that element loses all itsload carrying capacity. This can be seen from Eq. (42).

In the low loading rate case, a tensile fracture path(A–C–F) was predicted; in the high loading rate case, anin-plane shear fracture path (B–D–E–A) was predicted.The after fracture paths are shown in Fig. 82.

The predicted COD1-loading curves are shown inFig. 83. There are some minor oscillations in theCOD1-loading curve due to the exaggerated pullingrate used in the explicit numerical simulation. The loadoscillation in the fast loading rate scenario after fractureoccurred in the B–D–E segments is due to the suddenbreak of the structure.

The post-mortem experiment examination revealsthat all S-shaped specimens fractured in the in-planeshear mode (B–D–E–A) under remote extension, exceptone that was believed to be not entirely flat that failedin A–C–F path. This differs from the numerical pre-dictions in that the slow rate experiments result in aB–D–E–A fracture path as well as under the fast load-ing rate. The simulation of the fast loading rate casematches the experimental results well. In both cases,the predicted peak loads is about 10 % higher than thatobtained experimentally.

123


Fig. 83 Predicted load-COD1 curves for the slow (left) and fast (right) loading rates using the damage plasticity model


In retrospect, neglecting the J3-dependence of the yieldcondition of the material appears to be the single mostsignificant source of discrepancy in these predictions.For instance, it was shown by the MIT team that theHill 1948 yield condition are capable of predictionthe in-plane shear mode (path B–D–E–A) for bothslow and fast loading rate. When the Hill 1948 yieldcondition degenerates to von Mises yield condition,the fracture path changes to A–C–F. Furthermore, thehigher predicted peak loads were also resulted fromthe tensile stress–strain curve used in simulations. Inboth slow and fast loading the regions along the shearpath were subjected to the most severe deformation,where the resistance was over-predicted. Meanwhile,the predicted COD1 values at fracture of both slowand fast loadings have a good match with the exper-imental data. This suggests the calibration procedureare fairly accurate under the assumption of the mater-ial obeys a von Mises yield criterion. It also suggeststhe adoption of a J3-dependent yield condition or ananisotropic yield function such as Hill 1948 will fixthese observed errors in the current prediction. In eithercase, a re-calibration of the material parameters fordamage accumulation and fracture initiation may benecessary.

Appendix 2: Further experimental details

Dimensional measurements

A calibrated coordinate measurement machine was uti-lized for in-plane dimensional measurements, and acalibrated micrometer was utilized for thickness mea-surements. All tested specimens had dimensions withinthe drawing tolerances. In addition to the in-planedimensional measurements, thickness measurementswere performed at Sandia with a calibrated MitutoyoIP65 micrometer with 0.001-mm resolution in eightlocations (TL, TC, TR, Pt. O, 5, BL, BC, and BR)shown in Fig. 84a; Tables 16 and 17 in “Appendix2” provide the details of these measurements. Despitebeing within the thickness tolerance of the drawing, thechallenge specimens had out-of-plane distortions thatwere caused by unbalanced milling of the plate thick-ness. The relative surface height measurements weretaken using a Brown & Sharpe BestTest dial indica-tor with a 0.0127-mm precision attached to a heightgage resting on a flat precision ground granite sur-face plate. The height was measured in 12 locationsrelative to the reference height “PT. O” as shown inFig. 84a. Tables 18 and 19 provide the relative surfaceheight measurements for each test sample. Sample 30had the largest relative curvature out of all the sam-

123


Fig. 84 Thickness and surface height measurements of the Challenge geometry samples: a measurement locations; b relative surfaceheight measurement setup with a dial indicator, surface plate, and jackscrews

ples, which may have contributed to this one samplefailing by a different crack path than all other sam-ples.

Fractographic observations

Images of the fracture surfaces and crack paths for Sam-ples 11 (‘slow’ loading rate) and 27 (‘fast’ loading rate)are shown in Figs. 85 and 86 respectively. Both loadingrates had remarkably similar fracture surfaces, so theywill be described together. The ligament B-D was pre-dominated by small ductile dimples, roughly 1μm indiameter. In addition, there were relatively flat patches∼50–100μm in diameter on the fracture surface. Inthese flat patches, there were very few dimples, and thematerial had a smeared appearance somewhat reminis-cent of a wear surface presumably associated with shearfailure. This smeared zone has an appearance some-what reminiscent of “slickenlines” associated with theshear fracture of geologic structures. Ligament D-Ehad similar features, although there was also a singlerather large smeared flat patch 300–500μm in diameterwith a cluster of very large 10–20μm diameter dim-ples on one side of the smear. These large dimples arethought to be associated with stable microvoid coales-

cence and the finer dimples are thought to be associatedwith fast fracture. While the large dimples and associ-ated large smear patch was most evident in ligamentD–E, it was also possible to find similar, albeit lesspronounced, features in ligament B–D. The final rup-ture ligament, E-A, also had very pronounced clusterof large dimples as well as an even larger smear patch.In all cases, the dimples were not perfectly equiaxedas would be expected in a tensile failure, but hadsome degree of directionality consistent with a shearfailure.

Sample 30 was the only exception that failed by pathA–C–F. The fracture surface of the A–C ligament thatfailed apparently prematurely (at a lower COD1 thanany other sample) is shown in Fig. 87. This ligament,A–C, should have nominally been loaded in pure ten-sion. Therefore, it is reasonable to compare this liga-ment to the tensile fractures observed in the tensile barsprovided for material calibration, specifically Fig. 3e.There are important differences that suggest that Sam-ple 30 could have exhibited anomalously low ductility.Firstly, unlike the tensile bar the outer shape of theligament A–C shows very little curvature suggestiveof limited necking. Secondly, the center of the frac-ture surface shows marked differences. This centralzone is typically called the ‘fibrous zone’ in tensile

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Table 16 Thickness measurements for all samples tested at the 0.0254 mm/s displacement rate

Testing Lab Sample Thickness measurement (mm)

TL TC TR Pt. 0 5 BL BC BR



4 3.099 3.084 3.099 3.117 3.119 3.117 3.114 3.122

6 3.147 3.124 3.139 3.132 3.124 3.124 3.129 3.129

7 3.139 3.124 3.145 3.138 3.131 3.147 3.129 3.134

11 3.142 3.124 3.132 3.136 3.118 3.142 3.124 3.127

14 3.129 3.124 3.139 3.125 3.124 3.129 3.127 3.124

19 3.139 3.119 3.132 3.127 3.124 3.129 3.124 3.109

28 3.099 3.101 3.127 3.113 3.103 3.119 3.117 3.119

30 3.139 3.129 3.139 3.123 3.120 3.139 3.129 3.129


13 3.120 3.122 3.125 3.118 3.108 3.127 3.119 3.128

23 3.129 3.120 3.127 3.115 3.123 3.123 3.115 3.099

29 3.120 3.117 3.119 3.115 3.112 3.131 3.125 3.117

UT Austin 2 3.103 3.103 3.120 3.096 3.110 3.119 3.115 3.112

5 3.128 3.115 3.125 3.125 3.119 3.136 3.128 3.118

31 3.087 3.090 3.098 3.103 3.098 3.096 3.096 3.103

Minimum 3.0874 3.0836 3.0975 3.0963 3.0975 3.0963 3.0963 3.0988

Average 3.1230 3.1140 3.1262 3.1202 3.1166 3.1270 3.1209 3.1192

Maximum 3.1471 3.1293 3.1445 3.1382 3.1306 3.1471 3.1293 3.1344

Table 17 Thickness measurements for all samples tested at the 25.4 mm/s displacement rate

Testing Lab Sample Thickness measurement (mm)

TL TC TR Pt. 0 1 2 3 4



9 3.142 3.131 3.139 3.136 3.125 3.131 3.124 3.125

15 3.117 3.117 3.106 3.114 3.119 3.112 3.112 3.104

18 3.138 3.134 3.145 3.136 3.131 3.139 3.134 3.125

22 3.117 3.103 3.120 3.127 3.113 3.117 3.113 3.115

24 3.136 3.129 3.136 3.131 3.127 3.123 3.125 3.124

25 3.125 3.117 3.127 3.122 3.119 3.134 3.122 3.106

27 3.120 3.115 3.136 3.122 3.125 3.136 3.127 3.131


20 3.141 3.129 3.136 3.139 3.131 3.124 3.125 3.119

Minimum 3.1166 3.1026 3.1064 3.1140 3.1128 3.1115 3.1115 3.1039

Average 3.1294 3.1218 3.1306 3.1282 3.1237 3.1269 3.1228 3.1188

Maximum 3.1420 3.1344 3.1445 3.1394 3.1306 3.1394 3.1344 3.1306

cup-and-cone failure, and it is surrounded by ‘shearlips’. The fibrous zone where fracture originates is typ-ically flat, with only microscale perturbations from the

mode-I crack path. In the tensile test fracture, Fig. 3e,this central fibrous zone is indeed reasonably flat withonly minor ridges running horizontally in the image

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Table 18 Surface height measurements for all samples tested at the 0.0254 mm/s displacement rate

Testing Lab Sample Surface height measurement (mm)

TL TC TR Pt 0 1 2 3 4 5 BL BC BR



4 −0.1016 −0.0762−0.0254 0.0000 0.0381 0.0381 −0.0254−0.0127 0.0127−0.0381−0.0127 0.0000

6 −0.0762 −0.0508−0.0254 0.0000 0.0000 0.0127 0.0127 0.0127 0.0127−0.0381−0.0381−0.0381

7 −0.1016 −0.0889−0.0508 0.0000 0.0254 0.0381 0.0000 0.0000 0.0127−0.0635−0.0508−0.0508

11 −0.1143 −0.0635 0.0127 0.0000 0.0508 0.0508 0.0000 0.0127 0.0254−0.0254−0.0254 0.0000

14 −0.1143 −0.0889−0.0381 0.0000 0.0381 0.0381 0.1270 0.0127 0.0254−0.0635−0.0508−0.0254

19 −0.0762 −0.0762−0.0635 0.0000 0.0000 0.0000 0.0000 0.0000 0.0127−0.0508−0.0508−0.0508

28 −0.1524 −0.1524−0.1143 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000−0.0889−0.0889−0.0889

30 −0.1778 −0.1651−0.1270 0.0000 0.0381 0.0381 −0.0381−0.0254 0.0000−0.1270−0.1270−0.1016


13 0.0000 −0.0127−0.0127 0.0000 0.0000 0.0000 −0.0127 0.0000 0.0000 0.0127 0.0000−0.0127

23 −0.0254 −0.0381−0.0254 0.0000 0.0000 0.0000 0.0000 0.0000 0.0127 0.0000−0.0254−0.0635

29 −0.0889 −0.0889−0.1016 0.0000−0.0127 0.0000 0.0254 0.0254 0.0000−0.0381−0.0635−0.0762

UT Austin 2 −0.1016 −0.1143−0.1143 0.0000−0.0127 −0.0127 0.0127 0.0254 0.0000−0.0508−0.0635−0.0889

5 −0.0254 −0.0508−0.0762 0.0000−0.0254 −0.0254 0.0381 0.0381 0.0000 0.0000−0.0381−0.0508

31 −0.0127 −0.0381−0.0635 0.0000−0.0254 N/A 0.0127 0.0127 0.0000−0.0127−0.0127−0.0254

Minimum −0.1778 −0.1651−0.1270 0.0000−0.0254 −0.0254−0.0381−0.0254 0.0000−0.1270−0.1270−0.1016

Average −0.0835 −0.0789−0.0590 0.0000 0.0082 0.0137 0.0109 0.0073 0.0082−0.0417−0.0463−0.0481

Maximum 0.0000 −0.0127 0.0127 0.0000 0.0508 0.0508 0.1270 0.0381 0.0254 0.0127 0.0000 0.0000

Table 19 Surface height measurements for all samples tested at the 25.4 mm/s displacement rate

Testing Lab Sample Surface height measurement (mm)

TL TC TR Pt 0 1 2 3 4 5 BL BC BR



9 −0.0762 −0.0635 −0.0127 0.0000 0.0254 0.0381 0.0000 0.0127 0.0127 −0.0508 −0.0381 −0.0254

15 −0.0762 −0.0635 −0.0508 0.0000 0.0254 0.0254−0.0254−0.0254 0.0000 −0.0635 −0.0508 −0.0381

18 −0.1143 −0.1016 −0.0635 0.0000 0.0127 0.0254−0.0127 0.0000 0.0127 −0.0762 −0.0762 −0.0635

22 −0.0889 −0.0889 −0.0762 0.0000 0.0254 0.0254−0.0127−0.0127 0.0000 −0.0635 −0.0381 −0.0254

24 −0.0381 −0.0381 −0.0127 0.0000 0.0254 0.0127−0.0127 0.0000 0.0000 −0.0508 −0.0254 −0.0127

25 −0.1016 −0.1016 −0.0762 0.0000 0.0127 0.0254 0.0000 0.0127 0.0254 −0.0635 −0.0635 −0.0762

27 −0.1270 −0.1016 −0.0508 0.0000 0.0381 0.0381−0.0127 0.0000 0.1270 −0.0635 −0.0508 −0.0381


20 0 −0.0381 −0.0635 0 −0.0254−0.0254 0.0127 0.0127 0 −0.0254 −0.0381 −0.0635

Minimum −0.1270 −0.1016 −0.0762 0.0000−0.0254−0.0254−0.0254−0.0254 0.0000 −0.0762 −0.0762 −0.0762

Average −0.0778 −0.0746 −0.0508 0.0000 0.0175 0.0206−0.0079 0.0000 0.0222 −0.0572 −0.0476 −0.0429

Maximum 0.0000 −0.0381 −0.0127 0.0000 0.0381 0.0381 0.0127 0.0127 0.1270 −0.0254 −0.0254 −0.0127

indicative of the lamellar microstructure developed dur-ing rolling. However, in Sample 30’s A–C ligamentfracture surface, these ridges are markedly more pro-

nounced and faceted indicating a much coarser andperhaps less uniform underlying crystallographic tex-ture. In fact in Sample 30 there is no nominal mode-I

123


Sample 11 – fractography example of slow-rate loading, ‘bo�om’ half of specimen

B D D EE A

Fig. 85 Sample 11: example fractography of the slow rate loading condition. Images taken from ‘bottom’ half of broken specimen

B D D E E A

Sample 27 – fractography example of fast-rate loading, ‘top’ half of specimen

Fig. 86 Sample 27: example fractography of the fast-rate loading condition. Images taken from ‘top’ half of broken specimen

crack plane and the entire central fibrous zone appearsmore like alternating shear lips. Inspection of the sec-ondary crack ligament, C–F also revealed a pronouncedridge that deviated from the expected mode-I crack

plane. This unusual fracture morphology raises sus-picion that Sample 30 was not a nominal failure. Thisapparently anomalous behavior is discussed further inSect. 3.3

123


500 µm

Fig. 87 Fracture surface of ligament A–C from Sample 30

Fig. 88 The experimental setup used in the University of Texaslab. Specimen is shown mounted in the grips and being viewedwith a DSLR camera for DIC based COD measurements, twoangled cameras for 3D-DIC, and a high speed camera to resolvethe cracking sequence

Post-challenge experimental observations from theRavi-Chandar Lab at University of Texas at Austin

The University of Texas volunteered to perform addi-tional tests after the predictions had been made in orderto produce a complimentary set of data to those alreadycompiled by the two Sandia laboratories. Experimentswere performed on three samples; 2, 5, and 31. Thesesamples were obtained from the same manufacturinglot as the samples tested at Sandia National Laborato-ries. Due to the limited number of samples, only exper-iments with the slow loading rate were performed.

The experimental setup used in the University ofTexas laboratory can be seen in Fig. 88. The experi-

0

5x103

1x104

1.5x104

2x104

2.5x104

0 0.5 1 1.5 2 2.5 3 3.5 4

Experiments - Slow

Forc

e (N

)


UT AustinSample 05

UT Austin Sample 31

UT AustinSample 02


Fig. 89 Comparison of the Load-COD curves measured in theUniversity of Texas tests with the data obtained from the SandiaStructural Mechanics Laboratory tests (grey lines). The COD1in the UT tests was obtained from 2D DIC measurements ratherthan clip gages. Specimen S02 had coarse temporal sampling forCOD measurements and is estimated to have failed at a COD1of 2.9 mm. S31 was not loaded to failure; the test was interruptedand preserved for microscopic characterization

ments utilized a 100-kN Instron electromechanical loadframe, with a 100-kN load cell (±0.25 % uncertainty ofthe measured value) at ambient temperature. The levelof noise in the load signal was measured to be 2 N.The crosshead rate was maintained at 0.0254 mm/s, asprescribed in the challenge. Two universal joints wereplaced, one each at the upper and lower grips in order tominimize the effect of loading misalignments. In addi-tion, the same clevises used by the Sandia laboratorieswere used. Instead of using COD gages to measure thedisplacements at the notch mouths, a digital single-lensreflex (DSLR) camera was used to view the entire spec-imen to allow the COD measurement to be made usingDIC. Two additional cameras focused on the regionbetween the notches were used to perform 3D-DICand to obtain the kinematic fields in the regions ofhighest deformation and eventual failure. A high-speedvideo camera, with high frame rate capability, was posi-tioned to view the ligaments B-D and D-E and resolvethe sequence of failure; the camera was post-triggeredwith the falling signal from the load drop associatedwith specimen failure. Further details of the experi-

123


Fig. 90 High speed images of sample 2 during failure of the firsttwo ligaments with overlaid vertical displacement fields fromDIC (positive displacement is down): a Both ligaments are intact;

b ligament DE has broken but BD remains intact; and c ligamentBD has also failed

mental methods, sensitivity, resolution, and results aredescribed by Gross and Ravi-Chandar (2016).

Confirmation of the load-COD1 results produced atboth of the Sandia laboratories is shown in Fig. 89.All three samples indicated the same crack path (B–D–E–A); failure occurred in ligaments B–D and D–E, although the loading on sample 31 was haltedjust after localization occurred in these ligaments, butbefore they failed. Follow up microscopy on this sam-ple is presented in Gross and Ravi-Chandar (2016).For the two samples loaded until failure, fast fractureoccurred in ligaments B-D and D-E nearly simultane-ously. To resolve which ligament failed first, sample2 was imaged at 20,000 fps and sample 5 was imagedat 40,000 fps. An image sequence showing three sub-sequent frames captured at 50-μs intervals by the highspeed camera at the time of failure for sample 2 is shownin Fig. 90. The overlaid color contours are of the ver-tical displacement field (relative to the trigger point)calculated with DIC from the high speed images. Thefirst image shows the state of the sample just beforecracking of any ligament, the second image shows lig-ament BD intact with ligament DE completely severedby a crack and the final image shows both ligamentsfully cracked. Identification of cracking in the secondimage is made by observing a displacement field con-sistent with the elastic recovery expected after externalloading is released from the ligament by the presenceof a crack. A high speed video with more frames forthis sample and without DIC processing is includedas Supplementary Information for this article. Despiteincreasing the framing rate for sample 5, only three sub-

sequent frames capture the same behavior observed forsample 2. Sufficient temporal resolution to determinethe location of crack initiation in each ligament wasnot pursued. Thus, the greatest specificity that can begiven for the cracking sequence of these two speci-mens is that fracture initiates in the ligament D–E, theassociated unloading and the subsequent fracture of theligament B–D occur within the next 100μs, suggest-ing a very dynamic process of fracture. After continuedloading ligament E–A is expected to fail as observed inthe experiments performed by the Sandia laboratories.

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