Simulation of periodical shearband propagation in aluminum

alloys at the mesoscale

Ruslan R. Balokhonov a, Varvara A. Romanova b , Siegfried Schmauder C , Pavel V. Makarov d

a Institute ofStrength Physics and Materials Science SB RAS pr.Academicheskii 2/1, 634021 Tomsk

RUSSIA

e-mail: rusy@ms.tsc.ru

b Institute of Strength Physics and Materials Science SB RAS pr.Academicheskii 2/1, 634021 Tomsk

RUSSIA

e-mail: varvara@ms.tsc.ru

C Institutfiir Materialprüfung, WerkstojJkunde und Festigkeitslehre, University ofStuttgart

Pfaffenwaldring 32, D-70569 Stuttgart, Germany

d Institute ofStrength Physics and Materials Science SB RAS pr.Academicheskii 2/1, 634021 Tomsk RUS­

SIA

Abstract

Presented in this paper is a simulation of periodicalshear band propagation at the mesoscale. The phe­nomenon is experimentally observed in aluminumand cooper alloys, wbile Portevm-LeChatelier effectoccurs. An earlier developed macro-meso· yieldcriterion is modified to take into consideration in­

termittent yielding. Calculations of tension of theAl6061 alloy test piece as an example are performedto investigate the serrated character of the stress­strain curve and dynamies of stress and strain pat­terns atthe mesolevel.

Kevwords: Mesomechanics, numerical simulation,elasto~plastic behavior, intermittent yielding.

1. INTRODUCTION

The phenomenon of intermittent yielding in alumi­num and copper alloys is weIl studied experimen­tally, for instance in [1-5], and attributed toconsecutive periodical formation of shear bands atthe meso scale. As a special case of such ananomalous behavior, Lüders band propagation ischaracterized by single displacements ofmacroscopic localization zones along the test piece.

Earlier Lüders band origination and propagationwas numerically investigated using 20MnMoNi55steel as an example [6]. In order to describe local­ized plastic yielding, we formulated a specific yieldcriterion, assuming for any local region to be plasti­cally deformed,provided the equivalent plasticstrain reaches its critical value in one of the nearest

local regions. Such an approach combines numeri­cal techniques of continuum and discrete mechanics,particularly the finite-difference approximation [7]and the method of cellular automata [8].

In this paper tbis approach is extended to the case ofmultiple shear band generation and propagation. Forso doing, we modify the yield criterion earlier de­veloped in [6], with taking into account periodicalorigination of shear bands in the vicinity of a clampand their step-by-step propagation throughout thespecimen as loading continues. Mathematical de­scription of the 2D-problem in its dynamic formula­tion and the yield criterion are given in Seetion 2.

Using the model developed, two-dimensionalcalculations were carried out for the aluminum alloyAl6061 as an example. Model parameters for thealloy were derived from the experiment on tensileloading [I].Computational results and their analysis based on amesomechanical point of view are presented inSection 3. Physical mesomechanics [8,9] considersasolid under loading as a multilevel self-organizedsystem of different scale levels. Within this concept,plastic deformation is treated as shear stability lossthat occurs in a self-consistent way at the micro-,meso-, and macroseale levels. Therefore, analyzingthe computational results, we have paid specialattention to the examination of the interconnectionbetween elasto-plastic patterns on the meso andmacroseale levels. The computational results showgood agreement with those observed experimenfullyin [I].

46 Ruslan R. Balokhonov et aL/ Simulation of periodical shear band propagation in aluminium alloys at the mesoscale

2. Mathematical formulation

The physical problem of periodical shear bandpropagation along the test piece can be reduced to atwo-dimensional statement. In this paper the phe­nomenon of intermittent· yielding in aluminum al­loys was simulated within the plane strain formula­tion. Numerical solutions were performed in termsof Lagrangian variables using the finite-differencemethod [7].

*

In this paper we modify an ex~ression eeq = eo,

where Co is the modal mesoscale parameter [6], to

* * /ceq = C (~,co). Here ~ = f(ceq,O"o) 0"0' 0"0 isthe yield point.(1)

the density given in [10] as weIl as the critical valueof plastic strain derived from the computations forthe aluminum alloy are presente~ in Table 1.

PropenyBulk modulusShear modulusDensiCritical strain eModel constant AModel constant BModel constant C

Table 1: Material and model parameters of A16061.

(O'eq ), MPa

Figure 1: Calculated stress-strain curve for A16061(a) and itsfragment (b) presented in more details.

Calculation results presented in figs. 1, 2 demon­strate essentially irregular stress and strain behavior.Fig. 1 shows the integral stress-strain diagram (a)and its part plotted in details (b). The stress wascalculated as an average value of equivalent stressover the mesovolume:

a)

0,08 e

0,0052

f)

0,06

0,00510,0050

(crea ), MPa

o

0,00

50

Shear bands generate periodically near the c1amps, 150

provided ~C:qin = e(~,co). Here ~c:qin - mini- 100

mal increment of equivalent plastic strain resultedfrom previous shear band propagation through the

region under study. By analogywith f(ceq,O"o),*

functions C (~,eo) and e(~,co) are of a sense of

limiting surfaces in space of strains, Simple ratios

e* (~, co) = co eXP(-~-J and1-~c(~, co) = co (~-1)were derived during computa­

tional tension tests for the aluminum alloy A1606l.

3. COMPUTATIONAL RESULTS: ANALYSISAND DISCUSSION

Calculations were performed for the aluminum alloyA16061 which demonstrates intermittent yield be­havior [1]. In this work we investigate the deforma­tion of a rectangular region. According to the planestrain formulation the region corresponds to thelateral face of a flat test piece experimentally inves­tigated under tension,Boundary conditions on the right and left surfacesof the region under calculation simulate c1amp dis­placement with constant speed, while on the top andbottom surfaces they correspond to the free surfaceand symmetry conditions, respectively:

To describe strain hardening use was made of theexponentialfUnction:

f(ceq)=A-Bexp(-eeq/C) MPa, where0"0 =f(O)=A-BThe values of the parameters A, B and C chosenfrom the experimental data [1], elastic moduli and

w~ere n is the number of computational mesh points,Vi is the local volume.

G.C. Sill, Th. Kermanidis & Sp Pantelakis (Eds.) / Meso2004 - Multiscaling and Applied Science (2004) 47

stress and strain distribution, a shear band formsnear one of the grips and then starts to propagatetowards the opposite end of the specimen.Due to intensive deformation in loealized plastieflow, strain hardening develops behind its front.Sinee plastic deformation and~ therefore, strainhardening evolve throughout the speeimen, newshear bands originate in the vicinity of one or theother grip, änd, sometimes, near both ends simulta­neously. The periodical generation and propagationof plastic fronts on the mesoseale level result inserration of the maeroseopic stress-strain eurve.Comparing the maeroseopic stress-strain eurve withplastie deformation patterns on the mesoscale level,we have made. the following eonc1usion. Each jumpeorresponds to shear band formation and its propa­gation towards the opposite side of the region understudy. As tension eontinues, the shape, amplitudeand periodof the serrated jumps change due to thedevelopment of strain hardening.

Let us eonsider a fragment of deformation on themesoscale level in more details (fig. 2). lnitiallyplastie deformation originates in the vieinity of theright grip. At the very beginning, the plastic frontpropagates perpendicular to the direetion of tension(fig.2 e). Moving away from the grip, the frontexhibits a tendeney to deviate from its initial oriem­tation, inclining towards the axis of tension. Fromthe mathematical point of view, this phenomenon isc1early explained by a difference in stress-straineonditions near the free surface and inside thespecimen. Indeed, under the plane strain state thecomponent of stress tensor normal to the free sur­face is equal to zero on the surfaee and non-zero inthe bulk of the material that leads to a higher levelof equivalent stresses near the surfaee then that inthe volume. Thus, the yield eriterion in the surfaeeareas fulfils at lower external forces then that in

corresponding volume points.

In this paper we presented a mesomechanical modelwhieh describes' the phenomenon of intermittent)rielding in aluminum alloy A16061. The yield erite­rion developed in our earlier work has been modi­fied and applied in the simulation of multiple gen­erations of plastic flows in the vicinity of stresseoncentrators and their step-by-step propagationthroughout the specimen. Dynamies of propagationof the plastic fronts on the meso-seale level and itsinfluence on the macroscopie behavior is investi­gated. The interrelation between material elasto­plastic responses on the meso- and macro-scalelevels is studied in details.Acknowledgement

The .strain represents relative elongation cf the re­

gion along the Xl direction. First, the specimen . Summaryundergoes uniform elastie deformation, that corre­sponds to the linear monotonous part of the stress­strain curve. As the plastic yield criterion ineludingEq. (1) is fulfi1led in the vicinity of either one oranother grip, plastic flow originates there and, 'asloading continues, starts to propagate towards theopposite grip as a front of localized deformation.Since the first shear band appears, essentially non­uniform stress and strain patterns are observed onboth meso and macro seale levels. Aeeording to the*model fonnulation the values of C (~, co) andc(~,co) are rather small at the initial stage of plas­tic deformation. That is why low amplitudes of theserrations ean be found from the figure.As a general tendency, plastie yielding on the mesoscale level is obeyed to the following. Depending on

Figure 2: A sequence of strain rate patternscorresponding to the stress - strain curvepresented in Fig. 2b

48 Ruslan R. Balokhonov et al./ Simulation of periodical shear band propagation in aluminium alloys at the mesoscale

This work is supported by INTAS (YSF2002-159),Russian Foundation for Basic Research (N!! 02-01­

01195), and Russian Science Support Foundation.

REFERENCES

[1] Deryugin, E.E.,Panin, V.E., Schmauder, S.,Storozhenko, LV., (2001), Phys. Mesomech., 4, No.3, pp. 35.[2] Casarotto, 1., Tutsch, R., Ritter, R., Weidenmül­ler, J., Ziegenbein, A., Klose, F., Neuhäuser, H.,(2003), Comput. Mater. Sei., 26, pp. 210-218.[3] Nagornih, S.N., Sarafanov, G.F., Kulikova, G.A.,at al. (1993), Russian Physics Journal, 36, No.2, pp.112-117.

[4] Toyooka, S., Madjarova, V., Zhang, Q., andSuprapedi, (2001), Phys. Mesomech., 4, No. 3, pp.23-27 ..

[5] Klose, F.B., Ziegenbein, A., Weidenmüller, J.,Neuhäuser, H., Hähner, P., (2003), Comput. Mater.Sei., 26, pp. 80-86.

[6] Balokhonov, R.R., Romanova, V.A., Schmauder,S., Makarov, P.V., (2003), Comput. Mater. Sei., 28,No. 3-4, pp. 505-511.[7] Wilkins, M.L., (1964) In:.Methods in computa­tional physics, pp. 211-263,. Alder, S., Fernbach,Rotenberg, M. (Eds.) Academic Press 3, New YorkB.

[8] Physical Mesomechanies of HeterogeneousMedia and Computer-Aided Design of Materials,(1998), Ed. by V. E. Panin, Cambridge InternationalScience Publishing.[9] Panin, V.E., (2000), Phys. Mesomech., 6, pp. 5­36.

[10] Babichev, A.P., Babushkina, N.A., Bratkovskii,A.M., etc., Physical values: Reference book, Grigo­rieva, I.S., Meilihova, E.Z., Moscow, Energoa­tomizdat, (1991), ISBN 5-283-04013-5, (in Rus­sian).

MULTISCALING IN APPLIED SCIENCE AND

EMERGING TECHNOLOGIY

FundameJitals and Applications in Mesomechanics

Proceedings of the Sixth International Conference for Mesomechaniesheld in Patras, Greece, May 31 - June 4, 2004

Edited by

G. C. Sih

Department of Mechanical Engineering and MechanicsLehigh University, Bethlehem PA 18015, USA

and

SchoolofMechanicalEngineeringEast China University of Science and Technology

Shanghai 200237, China

Th. B. Kermanidis and Sp. G. PantelakisLaboratory ofTechnology and Strength ofMaterials

Department of Mechanical Engineering and AeronauticsUniversity of Patras

Patras 26500, Greece

Published by: Laboratory ofTechnology and Strength ofMaterialsDepartment ofMechanical EngiIieering and AeronauticsUniversity ofPatrasPatras 26500, Greece

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