ductile fracture analysis

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A ductile failure model applied to the determinationof the fracture toughness of welded joints. Numerical

simulation and experimental validation

I. Penuelas *, C. Betegon, C. Rodrguez

Departamento de Construccion e Ingeniera de Fabricacion, Universidad de Oviedo, Edificio Departamental Oeste,

Desp. 23, Bloque 5, 33203 Gijon, Spain

Received 11 November 2005; received in revised form 28 April 2006; accepted 5 May 2006Available online 19 June 2006

Abstract

A ductile-failure model for analysing the fracture behaviour of welded joints has been implemented. Finite elementanalyses of mismatched welded joints have been performed using the computational cell methodology applied to SE(B)specimens. Different crack lengths, material mismatching, and widths of weld metal have been considered. Ductile para-meters have been experimentally and numerically obtained. The influence of geometry and material mismatching on thefracture behaviour of cracked welded joints has been validated by means of a testing program. In addition, the experimen-tal results have been explained through the crack tip constraint, which has been numerically determined. 2006 Elsevier Ltd. All rights reserved.

Keywords: Ductile fracture; Constraint parameter; Welded joints

1. Introduction

Ductile fracture of metallic materials involves micro-void nucleation and growth, and final coalescence ofneighbouring voids to create new surfaces of a macro-crack. The ductile failure process for porous materials isoften modelled by means of the Gurson model [1], which is one of the most widely known micro-mechanical

models for ductile fracture, and describes the progressive degradation of material stress capacity. In thismodel, which is a modification of the von Mises one, an elasticplastic matrix material is considered and anew internal variable, the void volume fraction, f, is introduced.

Although the original Gurson model was later modified by many authors, particularly by Tvergaard andNeedleman [24], the resultant model is not intrinsically able to predict coalescence, and is only capable ofsimulating micro-void nucleation and growth. This deficiency is solved by introducing an empirical void coa-lescence criterion: coalescence occurs when a critical void volume fraction, fc, is reached. The value of fc is

0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2006.05.007

* Corresponding author. Tel.: +34 985181980; fax: +34 985181945.E-mail address: [email protected] (I. Penuelas).

Engineering Fracture Mechanics 73 (2006) 27562773

www.elsevier.com/locate/engfracmech
mailto:[email protected]:[email protected]


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Nomenclature

a crack lengtha0 initial crack lengthDa crack length incrementA strain controlled nucleation rateB specimen thicknessBL slope of the blunting lineCMOD crack mouth opening displacementCTOD crack tip opening displacemente total elongationE Youngs modulusf current void volume fractionf* modified void volume fractionf0 initial void volume fractionfc critical void volume fraction

fF void volume fraction at final failurefN void volume fraction of nucleating particles in the Gaussian distribution of the nucleation ratefu ultimate void volume fraction_fgrowth void volume fraction growth rate_fnucleation; _fn void volume fraction nucleation rateh weld semi-widthH weld widthJ J-integralKI stress intensity factor in mode I of loadl0 initial element size at the fracture process zonem mismatch ratio (level of mismatching)

n strain hardening exponentp macroscopic hydrostatic stressP applied loadq macroscopic von Mises effective stressq1, q2, q3 fitting parameters introduced by Tvergaard and NeedlemanQ constraint Q-parameter (triaxility parameter)r, h polar coordinates with origin at the crack tiprvoids void space ratioR R-curve (resistance curve)S Distance between the support cylindersSij deviatoric components of the Cauchy stress tensorSN standard deviation in the Gaussian distribution of the nucleation rateT T-stressTg T-stress due to geometryTm T-stress due to material mismatching (ficticious)W specimen widthZ reduction of area

Greek symbols

a proportionality parameter in the RambergOsgood lawbg geometrical constraint parameter (biaxiality parameter)bm constraint parameter due to material mismatchingbT total constraint parameter

I. Penuelas et al. / Engineering Fracture Mechanics 73 (2006) 27562773 2757


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selected beforehand or is numerically fitted from tension tests [5], and is introduced in the model as a constantvalue which depends on the material. This criterion was initially validated by several numerical studies [3,6].However, later theoretical, numerical and experimental studies [7,8] suggested that the coalescence criterionshould include some micro-structural information related to the void/ligament rate, the stress state and thegeometry of the specimen, since different values of fc can be obtained from the same tension tests [914].

The plastic limit load proposed by Thomason [9] for void coalescence improves the prediction of ductile frac-ture. Accordingly to Thomason, on a surface of ductile fracture, micro-void coalescence is the result of the fail-ure by plastic limit load (microscopic internal necking) of the matrix between voids. Thus, the localiseddeformation state of void coalescence is completely different to the homogeneous deformation state during void

nucleation and growth. It is necessary to take into account both types of deformation in the analysis of ductilefracture, coalescence depending on the competition between these two types of deformation. As deformationbegins, voids are small and the stress needed to follow a homogeneous deformation is lower than that requiredto follow a localised deformation. As the plastic deformation grows and the void volume fraction increases, thestress required for the localised deformation mode decreases. When this stress equals that needed for the homo-geneous mode, a fork point is reached and coalescence takes place. Assuming that voids are initially sphericaland that they evolve in a spherical way, a ductile failure model is formulated, based on the so-called completeGurson model [13]. This complete model combines the GursonTvergaardNeedleman model and the coales-cence criterion proposed by Thomason, and it has been shown to give accurate predictions for any level of stresstriaxiality, for both strain non-hardening and strain hardening materials.

The implementation of such model requires the integration of the constitutive equations. This integrationrequires the calculation of the stress values and the different plastic variables in a finite strain increment, anddifferent integration procedures have been developed, using explicit and implicit methods. In the former, theyield function and the plastic variables are evaluated at known stress states, and although it is not necessary touse an iterative procedure, it is common to restore the final stress and the plastic variables to the yield surfaceby means of an iterative correction, since this condition is not forced by integration [15]. In the implicit meth-ods [16], stresses are unknown and so a system of non-linear equations should be solved, with iterative meth-ods usually being employed for this purpose. Since the yield criterion is used for defining the system ofequations to be solved, the obtained stresses directly satisfy the yield criterion.

In this work, an implicit method for integrating the constitutive equations that describe the ductile failureprocess of a metallic material in a welded joint, is implemented. The model is based on the complete Gursonmodel and employs RungeKutta based methods, which have been shown to provide good results in elasto-plasticity [17], to solve the resulting system of differential equations. The model has been implemented in

FORTRAN and it has been introduced in the ABAQUS finite element commercial code by means of the

_epij components of the plastic strain rate tensor

ep equivalent plastic strain_ep equivalent plastic strain ratee0 strain at yield stresse1, e2, e3 principal strainseN mean strain in the Gaussian distribution of the nucleation ratem Poissons ratior0 yield stressr0b yield stress of the base metalr0w yield stress of the weld metalr1 current maximum principal stressrm macroscopic mean stressrR tensile strengthr flow stress of the matrix materialU yield function of GursonTvergaardNeedlemans normalised T-stress

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UMAT user subroutine. Variable update algorithm is based on the return mapping algorithms [18], and arobust iteration scheme is used.

To simulate the extension of the crack, the computational cell methodology proposed by Xia and Shih [1922] is used. This methodology includes a realistic void growth mechanism and a micro-structural length scalephysically coupled to the size of the fracture process zone. Thus, according to these authors, under Mode I

ductile fracture occurs within a thin layer of material symmetrically located about the crack plane. This layerconsists of cell elements of height l0, where l0 is associated with the mean spacing of the larger void-initiatinginclusions. In this paper, progressive damage and subsequent macroscopic material softening in each cell ismodelled using the complete Gurson model. Finally, when the void volume fraction reaches a specific valueof final failure, the crack extends in a discrete manner by a distance of one cell.

Within this context, finite element analyses of SE(B) specimens with mismatched welded joints have beenperformed. Different crack lengths, widths of the weld material, and mechanical material mismatching havebeen considered. Ductile parameters have been experimentally and numerically obtained. The influence ofgeometry and material mismatching on the fracture behaviour of cracked welded joints, which has been ana-lysed in previous papers [23], has been here validated by means of a testing program, and the results have beenexplained through the crack tip constraint, which has been numerically determined.

2. Constitutive model: the complete Gurson model

To describe the evolution of void growth and subsequent macroscopic material softening in computationalcells, the yield function of Gurson [1], modified by Tvergaard [2,3] and Tvergaard and Needleman [4,24], isused in this work. This modified yield function is defined by an expression in the form:

Uq;p; r;f qr

2 2 q1 f cosh

3 q2 p2 r

1 q3 f2 0 1

where p = rm with rm the macroscopic mean stress, r is the flow stress of the matrix material (microscopic),which relates to the equivalent plastic strain by means of an uniaxial equation in the form r Hep, f iscurrent void volume fraction, f* is the modified void volume fraction (related to f), and q is the macroscopic

von Mises effective stress given by

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3

2 Sij Sij

r2

where Sij denotes the deviatoric components of the Cauchy stress.The constants q1, q2 and q3 are fitting parameters introduced by Tvergaard [2,3] to provide better agreement

with the results of detailed unit cell calculations. For metallic materials the usual values of these constants areq1 = 1.5, q2 = 1, q3 q21. The modified void volume fraction, f*, was introduced by Tvergaard and Needleman[4] to predict the rapid loss in strength that accompanies void coalescence, and is given by

f f if f 6 fc

fc fu fcfFfc ffc if f > fc( 3

where fc is the critical void volume fraction, fF is the void volume fraction at final failure, which is usuallyfF = 0.15, and f

u 1=q1 is the ultimate void volume fraction.

The internal variables of the constitutive model are r and f. Thus the evolution law for the void volumefraction is given in the model by an expression in the form:

_f _fgrowth _fnucleation 4The void nucleation law implemented in the current model takes into account the nucleation of both small andlarge inclusions. The nucleation originated at large inclusions is stress controlled, and it is assumed that largeinclusions are nucleated at the beginning of the plastic deformation, and are considered as the initial void vol-

ume fraction.



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The nucleation of smaller inclusions is strain controlled and, according to Chu and Needleman [25], thenucleation rate is assumed to follow a Gaussian distribution, which is

_fnucleationsmall particles A _ep 5where _ep is the equivalent plastic strain rate, and

A fNSN

ffiffiffiffiffiffiffiffiffi2 pp exp 12

ep

eNSN

2 6where eN is mean strain, SN is the standard deviation and fN is the void volume fraction of nucleating particles.

The growth rate of the existing voids can be expressed as a function of the plastic strain rate in the form:

_fgrowth 1 f _epkk 7where _epij is the plastic strain rate tensor.

As was stated in the introduction, in the modified GursonTvergaardNeedleman model the critical voidvolume fraction, fc, is a material constant. However, in the plastic limit load criterion proposed by Thomason[9] for void coalescence, fc is not a material constant, but the material response to coalescence. Thus, in thecomplete Gurson model [13,26,27] the coalescence criterion can be written in the form:

r1r< acoalescence 1rvoids 1

2 bcoalescenceffiffiffiffiffiffiffi

rvoidsp

! 1 p r2voids ) no coalesc: occurs

r1r

acoalescence 1rvoids 1 2

bcoalescenceffiffiffiffiffiffiffirvoids

p !

1 p r2voids ) coalesc: occurs

8>>>>>:

8

where the first term is related to the homogeneous deformation state, and the second one to the plasticlimit load needed for coalescence. In the previous expression, r1 is the current maximum principal stress,acoalescence = 0.1, bcoalescence = 1.2, and rvoids is the void space ratio, which is given by

rvoids ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3f4p expe1 e2 e33q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpe2e3p

2

9

with f, the actual void volume fraction; and e1, e2 and e3, the principal strains.The complete Gurson model has been verified by Zhang et al. [13] for non-hardening materials

(acoalescence = 0.1). In addition, Pardoen and Hutchinson [26] have shown that in the case of a RambergOsgood material with a strain hardening exponent n, and considering that voids remain always spherical,better predictions are obtained with

acoalescence 0:12 1:68n

; bcoalescence 1:2 10

These are the values used in this paper.

3. Ductile fracture in mismatched welded joints

3.1. Crack tip stress fields in welded joints

When analysing welded joints, the heterogeneity of the joint is a key factor for understanding the fracturebehaviour of any welded structure containing cracks. In an ideal weld, with a crack contained within theweld material and running along the materials centre-line, parallel to the weld-base material interface, andwhere the effect of the heat affected zone is negligible [2830], a two material idealisation of the weld structurecan be taken into account. This consists of weld material with yield stress r0w and width 2h, which contains thecrack, and base material with yield stress r0b. Fig. 1 shows this idealisation for a short-crack bend specimen. Inthis case, the level of material mismatching can be defined by the mismatch ratio m, in the form:

m

r0w

r0b 11



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with m > 1 referring to material strength overmatching, m < 1 to undermatching and m = 1 corresponding to ahomogeneous specimen of weld metal.

Using the finite element method and the slip-line field theory, Hao et al. [31], Joch et al. [32] and Burstow

and Ainsworth [33] have demonstrated that material strength mismatching significantly affects both the stressfields and the crack resistance curves of tension and bend specimens [29,34,35]. That is, constraint is not only afunction of geometry, but also of material mismatching. Thus, in addition to any geometrical constraint, con-straint due to material mismatching should be taken into account [36].

Geometrical constraint can be quantified by means of both the elastic T-stress [3740], which directly char-acterises the geometrical constraint effect, and the non-linear Q-parameter [41,42], which is a direct measure ofthe elasticplastic stress fields that can be related with the HRR field [43,44], and it usually describes the devi-ation of the stress field from a reference stress state at a specified position ahead of the crack tip, thus

Q rhh rRefhh

r0at

r r0J

2; h 0 12

where r0 is the yield stress of the material and rRefhh is the stress distribution for which T= 0. On the basis of a

dimensional argument ODowd and Shih [41] demonstrated that, at least under small scale yielding (SSY), foreach material Q and T are univocally related by expressions of the form:

Q A s B s2 C s3 13where s = T/r0 is the normalised T-stress. The evolution of T with the load can be characterised by the so-called biaxiality parameter, or geometrical constraint parameter, bg; so that, for each geometry and at eachinstant, the value of the T-stress is defined by an expression in the form:

T bg KI

ffiffiffiffiffiffiffiffiffip ap 14

Geometries with positive bg values lead to positive T-stresses and raise the stress fields slightly. Conversely,geometries with negative bg values lead to negative T-stresses and lower the stress fields significantly.

Constraint due to material mismatching has been analysed in depth by Burstow et al. [34], who used bound-ary layer formulations to investigate the crack tip stress fields in different mismatched cases. They also foundthat, even in the case of a null T-stress, T= 0, the development of the stress distribution around the crack tipdepends not only on the applied load, but also on the level of mismatching, m, and the weld material width, 2h.In the case of overmatching (m > 1) the crack tip stress fields are lowered relative to those obtained in homo-geneous weld material; conversely, in the case of undermatching (m < 1) the stress fields are raised relative tothe homogeneous reference situation. In all cases, the higher the level of mismatching and the thinner the weldmetal strip are, the more severe the former effects.

In order to quantify constraint due to material mismatching, different methodologies, based on that utilisedfor the quantification of geometrical constraint, have been defined. Thaulow and co-workers [4547] defined a

procedure similar to the JQ one. Thus, they established a JQM formulation for the crack tip stress fields.

W

S = 4W

B = W

a = W/2

P

2h

a b0

w0

Fig. 1. Two material idealisation of a three-point bend mismatched specimen.



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In this three-parameter formulation, J is related with the load, Q with the geometrical constraint and M withthe material constraint. Later, Burstow et al. [34] defined a normalised load parameter h r0w/Jthat scales theplastic zone with the width of the weld region. Thus, for a given mismatch ratio, the crack tip stress fieldsdepend only on this normalised load parameter. By means of this parameter, they also established a JQMformulation in order to examine the crack tip stress fields, and related Q and m for the overmatching case.

Thaulow and co-workers [47,48], also extended the model to interface cracks and three-material problemswhere the HAZ was taken into account. Detailed analyses of the crack tip stress fields have also been per-formed by Burstow et al. [34,35], who explored how both brittle and ductile fracture can be influenced bymaterial mismatching, observing that fracture toughness for this kind of idealised overmatched welds is, ingeneral, higher than that for undermatched specimens.

On the other hand, Betegon and Penuelas [23] defined a procedure similar to the JTone. Thus, they devel-oped a JTgTm formulation where J is related with the load, Tg with the geometrical constraint and Tm withthe material constraint. By means of finite element analyses of plane strain crack tip stress fields from homo-geneous and heterogeneous modified boundary layer formulation, as well as homogeneous and mismatchedfull field solutions, they established a new constraint parameter, bm, for overmatched welded joints, that quan-tifies the material mismatching effect on the crack tip stress fields. In the case of complete specimens, bothgeometry and material mismatching affect the crack tip stress fields and they also defined a total constraint

parameter bT, obtained by addition of bg and bm. In the case of undermatching, the constraint parameterbm cannot be defined. Although this three-parameter formulation of the stress fields in welded joints is definedfor small scale yielding conditions, it can be extended to large scale yielding conditions for some configurationsand enlarged the validity of other approaches.

3.2. Ductile fracture in welded joints

Due to the nature of the welding process, the welded zone contains, in general, more flaws and defects thanthe surrounding base material, and it is a critical place for fracture. Although fracture can develop by mech-anisms of cleavage and ductile tearing, in steels at room temperature under plain strain conditions, the crackusually grows by ductile tearing, and the fracture behaviour can be characterised by the ductile crack resis-

tance curve. In addition, any typical steel welded joint contains both small and large inclusions. Thus, the frac-ture behaviour of welded joints can be analysed by means of the complete Gurson model described in theprevious sections.

Within the context of the computational cell methodology proposed by Xia and Shih [1922], finite ele-ment analyses of three-point bend specimens with single edge cracks of different lengths (a/W= 0.1, 0.2,0.5), and values of the weld semi-width h ranging from 1 mm to 10 mm, have been performed. The ABAQUS[49] finite element commercial code has been used. Simulations have been carried out under plane strain con-ditions, and they have accounted for large strains around the crack tip, and contacts. Four-node bilinear,hybrid elements (CPE4H) have been used. Table 1 gives details of the yield stress of the materials and therange of material mismatch conditions analysed. Fig. 2(a) shows two finite element meshes used to modelthe problem; also shown are the support cylinder and the cylinder where the load is applied, which are mod-elled as rigid bodies. The mesh represents one-half of the specimen, since symmetry has been applied.Fig. 2(b) shows details of the mesh at the crack tip region; layers of uniformly sized void-containing elementsare also shown.

Table 1Material mismatch conditions analysed

Configuration m r0w (MPa) r0b (MPa)

Homogeneous 1 625 62520% overmatched 1.2 625 520.840% overmatched 1.4 625 446.460% overmatched 1.6 625 390.620% undermatched 0.8 625 781.3

40% undermatched 0.6 625 1041.7



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The selected material parameters correspond to structural steels with moderate strength, hardening andtoughness. All the materials are assumed to follow a RambergOsgood law written in the form:

e

e0 r

r0 a r

r0

n15

with e0 = r0/E, Youngs modulus E= 200 000 MPa, strain hardening exponent n = 10 and a = 1.56. The Pois-

sons ratio is assumed m = 0.3.

Fig. 2. Meshes for the three-point bend models: (a) general meshes and (b) the crack tip region and ductile cells (in grey).

Fig. 3. Resistance curves for short crack (a/W= 0.2) specimens with l0 = 0.1 mm, weld semi-width h = 1 mm and different levels ofmaterial mismatching, m.



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In the fracture process zone the micro-structural model parameters are: initial porosity f0 = 5.0 104,

which includes the void volume fraction inherent to the welding process and the large inclusions volume frac-tion; Gaussian distribution of the nucleation rate of small inclusions with void volume fraction of nucleatingparticles fN = 0.002, mean strain eN = 0.3 and standard deviation SN = 0.1; fitting parameters of the yield

Fig. 4. Resistance curves for short crack (a/W= 0.2) specimens with l0 = 0.1 mm, material mismatching m = 1.6 and different weld semi-widths, h.

Fig. 5. Resistance curves for short crack (a/W= 0.2) specimens with l0 = 0.1 mm, material mismatching m = 0.6 and different weld semi-

widths, h.



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function of Gurson, Tvergaard and Needleman q1 = 1.5, q2 = 1.0 and q3 q21; void volume fraction at finalfailure fF = 0.15; and different initial element dimensions l0 = 0.05 mm, l0 = 0.1 mm and l0 = 0.2 mm.

First of all, the results obtained from three-point bend specimen solutions for different mismatchingand geometrical configurations are shown. Fig. 3 shows resistance curves for different levels of material

Fig. 6. Fracture toughness at two different crack growth values as a function of the constraint parameter.

Fig. 7. Resistance curves for long crack (a/W= 0.5) specimens with l0 = 0.1 mm, different levels of material mismatching, m, and different

weld semi-widths, h.



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mismatching, m, a fixed weld semi-width, h = 1 mm, a short crack ofa/W= 0.2, and l0 = 0.1 mm. Also shownare the values of the total constraint parameter, bT, for the different configurations. In the case of materialstrength overmatching, the reduction in constraint results in a significant increment in weld resistance tofracture. Both, fracture initiation and fracture propagation (R-curve gradient) are increased. In the case ofmaterial strength undermatching, the increment in constraint results in a significant reduction in weld resis-

tance to fracture. Both, fracture initiation and fracture propagation (R

-curve gradient) are reduced. In allcases, the higher the level of mismatching and the thinner the weld width are, the more severe the formereffects. Figs. 4 and 5 show the resistance curves and the bT values for different weld semi-widths for the over-matched case m = 1.6 and the undermatched case m = 0.6, respectively, for l0 = 0.1 mm. For the overmatchedconfigurations, the bT values are also shown. In addition, the smaller the ratio a/Wis, and in consequence thelower the constraint is, the higher the fracture resistance. This crack size effect can be explained in terms of thegeometrical constraint that affects the crack tip stress fields [50].

Fig. 6 shows, for the overmatched configurations, the J values, at crack extensions of 0.2 mm and 1.0 mm,respectively, as a function of the total constraint parameter. The effect of bT on the crack resistance curves isthe same as on the crack tip stress fields. Thus, the more negative the constraint parameter is, the higher the Jvalue at fracture initiation. At a crack extension of 1.0 mm this constraint effect is higher than at a crack exten-sion of 0.2 mm, since bT also affects the gradient of the resistance curves.

In geometries with long cracks, that is to say with positive or higher values of the constraint parameter, theinfluence of varying both material mismatching and the weld width is lighter than in the case of geometrieswith quite negative values of the constraint parameter. Fig. 7 shows the resistance curves for a three-pointbend specimen with a long crack ofa/W= 0.5, different mismatched situations and different weld semi-widths.Although the effect on the resistance curve of a 60% overmatched welded joint with h = 1 mm is quite notice-able, bigger weld semi-widths (h = 5 mm) lead to resistance curves close to the one of the homogeneous allweld case, and for h = 10 mm the resistance curve is close enough to a homogeneous curve to accept thatthe welded joint behaves as homogeneous all weld material. What is more, for a 40% undermatched welded

joint, the effect is negligible for big semi-widths (h = 5 mm and h = 10 mm), but it is important for smallersemi-widths (h = 2 mm), where a profound reduction in fracture resistance is observed.

As was stated in the introduction, the size of elements in the fracture process zone, l0, is associated with the

mean spacing of the larger void-initiating inclusions. Thus, the fracture resistance curves obtained with differ-ent meshes, where the initial element size varies, have to be affected by this geometrical dimension l0. Fig. 8

Fig. 8. Resistance curves for short crack (a/W= 0.2) homogeneous specimens obtained with different cell element sizes.



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shows the resistance curves for a homogeneous specimen obtained with cell elements of l0 = 0.05 mm,l0 = 0.1 mm and l0 = 0.2 mm, other micro-structural parameters remaining the same. The bigger the elementsize is, the higher the fracture resistance to initiation and propagation (higher gradient of the resistance curve),the differences between each curve and the closest one of being nearly 40%. This figure reveals the importanceof a correct material characterisation in order to obtain realistic predictions of the ductile fracture behaviour

of any material under analysis. Different normalisations, applied to modified boundary layer formulations,have been found in literature [13]. However, although they have been taken into account in this paper, no pos-sible normalisation has been found for full field solutions.

4. Experimental procedure and validation

In order to analyse the transferability of the numerical results and the validation of the ductile failure modelimplemented, it is necessary to study the real fracture behaviour of different pre-cracked welded joints bymeans of experimental techniques. Thus, specimens extracted from overmatched and undermatched welded

joints, obtained with one weld material and two base materials (MB1 and MB2), with short and long cracksand different widths of the weld strip, have been tested within a complete experimental procedure.

In previous sections, computational studies have been carried out of different welded joints, with geomet-

rical and material configurations, as well as micro-structural fracture parameters, suitable for understandingthe influence of both geometry and material mismatching in the fracture behaviour of welded joints. Thus,welds of very small semi-widths have been simulated. However, in practice it is not possible to obtain rightwelds when welding the thick plates necessary to assure plain strain conditions (B% 20 mm) if a very smallroot gap (2 h = 2 mm) is required. On this basis, joints with mean widths of h = 5 mm and h = 10 mm (thatis, H= 2 h = 10 mm and H= 20 mm) have been obtained, although the effect of material mismatching is, inthese situations, less noticeable.

The welding procedure has been developed in order to obtain welded joints with no mechanical or geomet-rical discontinuities, and a heat affected zone small enough to consider a two-material (base and weld material)idealisation of the welded joint. An appropriate joint geometry is also required for obtaining specimens wherethe crack grows within the weld material. Thus, in the case of welded joints with big widths, a multi-run open

square butt weld, with full penetration welded from both sides, was utilised. However, in the case of smallwidths, due to accessibility problems, a multi-run single V-butt weld with backing strip and full penetrationwas used. The optimal operational and metallurgical conditions have been reached using a semiautomaticmechanically operated welding process with active shielding gas (MAG 88% Ar + 12% CO2), with weldingwire type E 70 S6, welding current 140200 A, welding voltage 2025 V, welding speed 2030 cm/min, heatinputs 10 kJ/cm and inter-run temperature lower than 250 C.

Coupons have been inspected with non-destructive tests including visual inspection of macrographs, X-raysand ultrasonics, in order to reject the coupons with high levels of defects or with no negligible heat affectedzones. Fig. 9(a) shows a macrograph of an undermatched H= 20 mm coupon; Fig. 9(b) shows a macrographof an overmatched H= 10 mm coupon. Destructive tests have also been conducted on samples of the couponsfor qualification purposes. Thus, the verification of the mechanical properties of the welded joints by means oftensile, hardness and bend tests has been carried out. The inclusion content of the weld material has also been

Fig. 9. (a) Macrograph of an undermatched H= 20 mm coupon and (b) macrograph of an overmatched H= 10 mm coupon.



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determined using quantitative microscopic optical techniques. Two different populations of inclusions, smalland large, have been observed. In addition, the initial void volume fraction inherent to the welding process isalso determined. Thus, an initial porosity value f0 = 1.0 10

4, and a volume fraction of void-nucleating par-ticles fN = 0.004, were considered.

The tensile properties of the weld and base metals have been determined, according to the ASTM E8M-04

standard [51], using round specimens extracted from the weld and base regions parallel to the weld direction.Table 2 gives details of the tensile properties at room temperature of the materials used. The mechanicalbehaviour of these materials within the plastic strain hardening zone has been fitted by a RambergOsgoodlaw (see Eq. (15)).

Single edge bend specimens (SE(B)) extracted normal to the weld direction, have been used for estimatingboth the crack growth resistance (JR) curves and the fracture toughness of the welded joints. The specimenshave been notched, fatigue pre-cracked, and sidegrooved. The notch was located in the centre line of the weldstrip with the crack growing in the thickness direction. Fig. 1 shows the bend test specimen geometry.

Table 3 gives details of the different specimen configuration tested, which consist of four overmatched, m =1.4, and three undermatched, m = 0.8, welded joints with weld mean widths H= 10 mm and H= 20 mm. Twocrack lengths corresponding to a short crack, a/W= 0.24, and a long crack, a/W= 0.5, have been tested. In allthe configurations the specimen width and thickness were W= 18 mm and B= W, respectively, and the span

was S= 4

W. For overmatched configurations, also shown are the values of the total constraint parameter,bT, defined by Betegon and Penuelas [23].

Fracture tests were carried out at room temperature, in accordance with the ASTM E1280-05a standard[52], and the single specimen method for the JR curve determination has been used. This method allows

Table 2Tensile properties at room temperature: r0 yield stress, rR tensile strength, e total elongation, Z reduction of area

Material r0 (MPa) rR (MPa) e (%) Z (%)

Weld 450 530 26 70MB1 320 427 31 69MB2 576 607 13 61

Table 3Specimen configurations tested, values of the total constraint parameter (bT), and slope of the blunting line (J= BL CTOD)

Configuration r0w (MPa) r0b (MPa) a/W H (mm) bT BL

1_Weld-MB1 450 320 0.24 10 0.255 9062_Weld-MB1 450 320 0.24 20 0.237 11303_Weld-MB1 450 320 0.5 10 +0.035 11604_Weld-MB1 450 320 0.5 20 +0.061 12405_Weld-MB2 450 576 0.24 10 12906_Weld-MB2 450 576 0.24 20 14107_Weld-MB2 450 576 0.5 10 1500

Initial notch

Fatigue pre-crackedInitial crack tip

Final crack tip

Spray paintedStable crack growth

Side groove

Fatigue

Final fracture

Fig. 10. Fracture surface of a short crack and H= 10 mm overmatched specimen.



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the crack length during test to be indirectly obtained from the specimen compliance variation. However, sincethis method leads to big inaccuracies when testing specimens of very ductile materials and short-cracked spec-imens, a colour marking technique of the crack front has been applied. This technique allows additional infor-mation to be obtained from a single test. Thus, two physical measurements of the crack growth are determinedduring each test: an intermediate painting sprayed crack length, and the final crack length. Fig. 10 shows a

detailed fracture surface of a short crack and thin width (H

= 10 mm) overmatched specimen.The amount of stable crack growth has been obtained by means of post-test examinations of the fracturesurface, and crack length has been obtained, according to ASTM standards [52], by a weighted mean of ninethrough-thickness crack length values measured on the fracture surface.

In a post-analysis of the testing results, the crack growth values, Da, indirectly obtained using the compli-ance method [41], have first been corrected to match the values physically measured on the fracture surface,and then interpolated between these values. This procedure leads to lower (JR) curve dispersion. It is impor-tant to note that, whereas in the experimental curves, where the crack growth values are obtained using thecompliance method, is included the effect of blunting at the crack tip, in physical measurements of the crackgrowth this effect is not included. Thus, and in order to compare both types of measurements, blunting at thecrack tip has been added to the later ones. After correction, the resistance curves obtained for each configu-ration from the individual pairs (J,Da), have been numerically fitted by expressions in the form [52,53]:

J C1 DaC2 16Fig. 11 shows the crack growth resistance curves obtained for the different configurations detailed in Table

3. In these curves, two different zones are represented: an initial zone that corresponds to the blunting behav-iour of each configuration, where the experimental results have been corrected in accordance with the stan-dards; and a second zone, where the experimental results have been fitted with expression (16).

In order to apply the proposed ductile model to simulate ductile tearing in SE(B) specimens, various modelparameters must be determined. The first set of parameters corresponds to the stressstrain law of the matrix

Fig. 11. Fitted experimental resistance curves for the tested specimens.



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material. The law has been obtained from the tensile test already described, and has been fitted by a RambergOsgood law with n ffi 7 and a = 1 for MB1 and weld metals, and n ffi 20 and a = 1 for MB2 metal.

The second set of parameters consists on q1, q2 and q3, which are related to the hardening of the matrixmaterial. In this work, q1 and q2 are fixed based on cell calculations by Kopling and Needleman [6], so thatq1 = 1.5, q2 = 1 and q3 q21.

The third set of parameters, eN,SN,

fN and

f0, is related to the void nucleation and the initial porosity value.Both, the void volume fraction of nucleating particles, fN, and the initial void volume fraction, f0, have been

determined using quantitative microscopic optical techniques. Thus, they have been obtained from the totalvolume of inclusions due to debonding particles. Two different populations of inclusions have been observed.The first one consists on particles of small mean diameter (about 0.6 lm) uniformly distributed. The secondone consists on bigger globular inclusions (mean diameter 7 lm) randomly distributed in the sample material.In this paper, the uniformly distributed small particles are considered to be nucleating particles, so fN has beenchosen to correspond to the volume fraction of these particles, fN = 0.004. The remaining nucleation para-meters have been chosen eN = 0.3 and SN = 0.1, in accordance to Chu and Needleman [25]. The initial voidvolume fraction, f0, is obtained from the large particles population, since, as it was pointed out in Section 2,the debonding between the matrix and the inclusions occurs as soon as load begins, so that the large inclusionsvolume fraction equals the initial void volume fraction and f0 = 1.0 10

4.

The porosity value that determines the final failure, fF, has been set to fF = 0.15. Finally, as fc is not a mate-rial parameter, but determined from Eq. (8), the only unknown parameter is the critical length, l0, which cor-responds to the size of the cell elements near the crack tip, that is, to the crack tip mesh size. This critical lengthis related to the material micro-structure. However, in this paper, as all the other parameters have been fixed,the mesh size has been determined by adjusting the numerical predictions to match the experimental fractureresistance curve of certain configurations, in this case a/W= 0.5, H= 10 mm, overmatched. And it has beenheld fixed for the other configurations. An alternative procedure would consists on fixing this critical length asthe mean inter-large-particle spacing and then determine the values of q1, q2 and q3 by fitting the numericaland experimental data. It is necessary to point out that, because of the parameter selection procedure, it isthe complete set of values which describe the ductile behaviour of the material. Thus, if any parameter is chan-ged, the numerically fitted parameters must be recalculated.

Once the mechanical properties and the ductile parameters were known, the geometries and mismatchedconfigurations tested have been numerically simulated and their resistance curves have been obtained.

Finally, in order to analyse the transferability of the numerical results and the validation of the ductile fail-ure model implemented, the numerical and experimental results have to be compared. To do this, the numer-ical resistance curves need to be corrected in order to include the blunting effect of each mismatchconfiguration and geometry. Thus, to the crack growth value determined numerically, one-half of the CTODvalue is added (Dacorrected = Da + CTOD/2), following a similar procedure to that followed for the experimen-tal physical values. The CTOD values are obtained from: (a) the CMOD calculated in simulations, ductile fail-ure being taken into account, and (b) the CTOD obtained from numerical simulations where, in order tosimulate blunting at the crack tip, finite element meshes with an inner radius at the crack tip have been used,ductile failure not being taken into account. For each configuration, these two numerical CTOD values havebeen found to be the same, and they agree with those obtained experimentally. Table 3 shows the slopes of theblunting lines for the different configurations analysed. From this table it can be observed that, in the case ofovermatching, the lower the constraint is, the lower the slope of the blunting line. Besides, in the case of under-matching, the slope of the blunting line grows with the relative size of the crack and the width of the weld strip.

Once corrected, numerical and experimental curves can finally be compared. Fig. 12 shows the resistancecurves obtained numerically and experimentally. The former are represented by isolated points, and the latterby continuum and dot lines. It has to be noted that all configurations with a similar resistance curve in Fig. 11(experimentally determined), have been represented in Fig. 12 as a unique curve.

From Fig. 12, it can be observed that constraint, measured through the total constraint parameter bT, qual-itatively determines the crack growth resistance curve, even for loads out of the range of application of bT.Thus, the more negative the constraint parameter is, see Table 3, the higher the fracture initiation, the steeperthe fracture propagation, and the lower the slope of the blunting line are. That is to say, the lowest constraint

configuration tested, which corresponds to a short-crack overmatched specimen with thin weld width, leads to



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the highest fracture resistance values, followed by the short-crack overmatched specimen with thick weldwidth. The remaining configurations, which correspond to positive or close to zero total constraint values,have very similar constraint values, and both fracture initiation and propagation are close enough to each

other to be considered the same for all of them, as was pointed out earlier. Thus, in the case of short-crackundermatched specimens, whatever the weld width, as well as in the case of long-crack specimens, whateverthe weld width and the mismatch nature, a single resistance curve is obtained. Such behaviours are shown byboth numerical and experimental results. In addition, very good agreement has been found between bothkinds of results for all the configurations analysed.

These results confirm the high dependence on constraint of both fracture toughness and ductile propaga-tion, and show the ability of numerical models for quantifying fracture behaviour.

5. Conclusions

An implicit method for integrating the constitutive equations for materials which behave according to thecomplete Gurson model has been implemented. A RungeKutta based method for solving differential equa-tions has been used within an error controlled sub-stepping procedure.

The implemented model has been applied to mismatched welded joints, and the key role of the constrainteffects on ductile fracture has been analysed. These effects have been quantified by means of a total constraintparameter, bT. In addition, a validating experimental programme has been carried out, and good agreementbetween numerically predicted and testing-obtained values is shown. Thus, the ability of numerical models forpredicting the fracture behaviour of complete geometries with welded joints not only qualitatively but alsoquantitatively, has been shown.

Acknowledgements

The authors acknowledge the financial support of the Spanish Ministry of Science and Technology, project

MAT2000-0602, and the access to ABAQUS under academic license from Hibbitt, Karlsson and Sorensen.

Fig. 12. Comparison of the experimental and numerical resistance curves (corrected with the blunting line) for the tested specimens.



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