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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 66 (2014) 133–153

http://d0022-50

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Effect of Lode parameter on plastic flow localizationafter proportional loading at low stress triaxialities

Matthieu Dunand a,b, Dirk Mohr a,b,n

a Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology,Cambridge, MA, USAb Solid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, École Polytechnique, Palaiseau, France

a r t i c l e i n f o

Article history:Received 12 March 2013Received in revised form3 January 2014Accepted 31 January 2014Available online 28 February 2014

Keywords:Ductile fractureLocalizationLode angleStress triaxialityUnit cell analysis

x.doi.org/10.1016/j.jmps.2014.01.00896 & 2014 Elsevier Ltd. All rights reserved.

esponding author. Phone +33 1 6933 5801.ail address: mohr@mit.edu (D. Mohr).

a b s t r a c t

The effect of the stress state on the localization of plastic flow in a Levy–von Misesmaterial is investigated numerically. A unit cell model is built with a spherical central voidthat acts as a defect triggering the onset of flow localization along a narrow band. Periodicboundary conditions are defined along all boundaries of the unit cell. Shear and normalloading is applied such that the macroscopic stress triaxiality and Lode parameter remainconstant throughout the entire loading history. Due to the initially orthogonal symmetryof the unit cell model the deformation-induced anisotropy associated with void shapechanges, both co-rotational and radial loading paths are considered. The simulationresults demonstrate that the macroscopic equivalent plastic strain at the onset oflocalization after monotonic proportional loading decreases in stress triaxiality and is aconvex, non-symmetric function of the Lode parameter. In addition to predicting the onsetof localization through unit cell analysis, an analytical criterion is proposed for monotonicproportional loading which defines an open convex envelope in terms of the shear andnormal stresses acting on the plane of localization.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The localization of plastic deformation within a narrow band is an important precursor to ductile fracture. Following theworks of Marciniak and Kuczynski (1967) and Rice (1977), it is common practice to predict the onset of localization based onmacroscopic constitutive theories through infinite band localization analysis (e.g. Mear and Hutchinson, 1985, Duszek andPerzyna, 1991). As noted by Rice (1977), the identified onset of localization corresponds to the loss of ellipticity of thegoverning equilibrium equations. Consequently, onset of localization maps can also be directly computed by assessing theloss of ellipticity of incremental moduli (Michel et al., 2007, Danas and Ponte Castañeda, 2012).

For most metals, the strains at the onset of localization are very large. As a consequence, the effect of voids on the elasto-plastic moduli needs to be taken into account when computing the instant of the onset of localization. This requiresadvanced constitutive theories such as the Gurson model (Gurson, 1977) and its extensions accounting for void nucleation(e.g. Chu and Needleman, 1980), for the loss of load-carrying capacity associated with void coalescence (e.g. Tvergaard andNeedleman, 1984), for void shape effects (e.g., Gologanu et al., 1993, 1994; Garajeu et al., 2000; Pardoen and Hutchinson,2000) and for plastic anisotropy (e.g., Benzerga et al., 2004). As shown by Nahshon and Hutchinson (2008), additional

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153134

modifications representing shear softening are necessary to obtain reasonable predictions of strain localization at low stresstriaxialities.

Unit cell models provide a computationally-expensive alternative to macroscopic constitutive theories to describe thelarge deformation response of metals of low porosity. The early analysis with unit cell models was mostly limited to two-dimensional models, e.g. axisymmetric mechanical systems with spheroidal voids (e.g. Koplik and Needleman, 1988, Brockset al., 1995, Pardoen and Hutchinson, 2000) or plane strain models with cylindrical voids (e.g. Tvergaard, 1981). Fully three-dimensional models have only been employed rather recently for plane strain conditions (e.g. Scheyvaerts et al., 2011,Nielsen et al., 2012, Rahman et al., 2012) and selected three-dimensional stress states (e.g. Barsoum and Faleskog, 2007,2011, Tekoglu et al., 2012). Aside from the macroscopic response, unit cell models provide valuable insight in the localdeformation fields and allow for the detailed analysis of the void growth and coalescence process (e.g. Scheyvaerts et al.,2011). As discussed by Pardoen and Hutchinson (2000), it is useful to define void growth as the phase prior to the localizationof deformation inside the intervoid ligament, while void coalescence describes the deformation process thereafter. Normallocalization may be seen as diffuse necking of the ligament, while shear localization is characterized by the development ofa shear band at the microscale. Due to the inherent periodicity of microstructures defined through unit cell models, theonset of coalescence corresponds to the onset of normal and/or shear localization of plastic flow within a narrow band at thescale of the void.

Substantial efforts have been devoted to the development of micromechanics-based coalescence criteria (see review ofBenzerga and Leblond, 2010). The first generation of coalescence criteria (Brown and Embury, 1973, Thomason, 1985,Benzerga, 2002) is primarily concerned with the prediction of internal necking as a function of the void shape, relativespacing and the applied normal stress. The effect of shear in addition to normal loads on the coalescence has been addressedrecently by Tekoglu et al. (2012). They demonstrate that the introduction of non-linear parameter functions into theBenzerga model leads to an excellent agreement with their unit cell simulations for combined shear and tension.Furthermore, Tekoglu et al. (2012) present a micromechanical analysis to come up with an analytical coalescence modelfor general loading conditions.

Reliable experimental results on ductile fracture at low stress triaxialities are still difficult to obtain because of significantexperimental challenges associated with the proper introduction of loading, the localization of deformation at the specimenlevel (necking), and the detection of the onset of fracture (e.g. Bao and Wierzbicki, 2004, Mohr and Henn, 2007, Brünig et al.,2008, Fagerholt et al., 2010, Gao et al., 2011, Dunand and Mohr, 2011a, Haltom et al., 2014). Numerical results on localizationare therefore of particular value for the development of ductile fracture modes at low stress triaxialities. Tvergaard (2008,2009) analyzed the behavior of a row of circular cylindrical holes under shear loading. He reports the formation of rotatingmicro-cracks as the result of void closure at low stress triaxialities. Furthermore, he points out that a maximum in themacroscopic shear stress accompanies the onset of localization of plastic flow. Nielsen et al. (2012) confirmed theseobservations using a three-dimensional unit cell model. In his most recent work, Tvergaard (2012) considered a square unitcell with a cylindrical void and fully periodic boundary conditions. By varying the normal stress during shearing, he foundthat increasing the stress triaxiality facilitates failure through shear localization.

Barsoum and Faleskog (2007) performed a micromechanical analysis on three-dimensional unit cells with sphericalvoids for combined tension and shear loading. Their model represents a layer of preexisting voids in a Levy–von Misesmaterial; it features a height-to-width ratio of 2:1 along with periodic boundary conditions on all three pairs of parallelboundaries. Using a kinematic condition comparing the deformation gradient rate inside and outside a band of localization(as proposed by Needleman and Tvergaard, 1992), they define the onset of shear localization and report the correspondingmacroscopic von Mises equivalent strain as strain to failure (due to localization). Their simulation results for a constantstress triaxiality of 1.0 elucidate the effect of the Lode parameter on shear localization for stress states between generalizedshear and axisymmetric tension. Their computational results also agree well with the observations from experiments wherecoalescence occurred by internal necking (triaxiality above 0.7). However, for low stress triaxialities and stress states closerto generalized shear, the macroscopic strains to failure predicted by the unit cell model are significantly higher than thosefound experimentally.

Gao et al. (2010) applied macroscopic normal stresses along the symmetry axes of a cubic unit cell with a spherical voidand boundaries that remain flat and perpendicular throughout deformation. Assuming that void coalescence occurs whenthe macroscopic strain state shifts to a uniaxial strain state, they computed the corresponding macroscopic effective strainas a function of the stress triaxiality (ranging from 0.33 to 2) and of the Lode angle. Their results indicate that themacroscopic strain to coalescence increases monotonically as a function of the Lode angle from axisymmetric tension toaxisymmetric compression. Furthermore, their simulations indicate that this Lode angle effect on coalescence becomesmore pronounced at low stress triaxialities. They also show that the effective strain to coalescence decreases whenassuming a Gurson instead of a Levy–von Mises matrix.

More recently, Barsoum and Faleskog (2011) made use of their unit cell model to investigate the localization ofdeformation into a narrow planar band for a wider range of stress states. Irrespective of the stress state, they observe thelowest macroscopic effective strain to localization for bands oriented at an angle of about 451with respect to the direction ofthe minimum principal macroscopic stress. The computed localization loci for stress triaxialities ranging from 0.75 to 2show the lowest strains to localization for generalized shear. The loci are approximately symmetric with respect to the Lodeparameter, showing slightly higher localization strains for axisymmetric compression than axisymmetric tension. Tekogluet al. (2012) considered an elastic perfectly plastic matrix material in their unit cell simulations and performed a limit load

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 135

analysis in a small strain setting. Two of the three macroscopic normal strain components (along the lattice axes) werealways identical in their simulations. They considered about 25 different shear-to-normal strain ratios with macroscopicstress triaxialities ranging from about �3 to 8. Nielsen et al. (2012) also used a 3D unit cell model with a power lawhardening matrix. They investigated ductile failure for six different far field shear-to-normal stress ratios, but limit theiranalysis to plane strain conditions.

The present work is an extension of the three-dimensional unit cell analysis of Barsoum and Faleskog (2011).The simulations are performed for finite strain kinematics and for an isotropic saturation hardening matrix material.The particular focus is on the effect of the Lode parameter on the localization of plastic flow in void containing solids at lowstress triaxialities (from 0 to 1). More than 180 different monotonic proportional loading paths are analyzed in that range.After a brief characterization of the stress state, the underlying assumptions of the unit cell analysis are discussed in detail.The boundary conditions are chosen such that the macroscopic stress state remains constant throughout each simulation.Furthermore, the macroscopic principal stress directions are assumed to follow the rotation of the unit cell. The simulationresults reveal that for monotonic proportional loading the macroscopic equivalent plastic strain at the onset of localization isa monotonically decreasing function of the stress triaxiality and a convex non-symmetric function of the Lode parameter. Inaddition to predicting the macroscopic strain at the onset of localization through unit cell analysis, a localization criterionfor monotonic proportional loading is formulated in terms of the normal and shear stress acting on the plane of localization.

2. Characterization of the stress state

Given the Cauchy stress tensor r, and its deviatoric part s, the first, second and third stress invariants read

I1 ¼ trðrÞ ð1Þ

J2 ¼12s : s ð2Þ

J3 ¼ detðsÞ: ð3ÞNormalizing the stress space by the equivalent von Mises stress,

s¼ffiffiffiffiffiffiffi3J2

pð4Þ

allows us to introduce two dimensionless parameters that characterize the stress state:

�

Firstly, we make use of the stress triaxiality,

η¼ I13s

ð5Þ

to characterize the spherical part of the stress tensor.

� Secondly, we introduce the Lode angle parameter θ to characterize the deviatoric part of the stress tensor,

θ¼ 1� 2πarccos

3ffiffiffi3

p

2J3

J2 3=2

!ð6Þ

Note that θ is close to the Lode parameter (Lode, 1926) multiplied by minus one,

θffi�L¼ � 2sII�sIII

sI�sIII�1

� �ð7Þ

with sIZsIIZsIII denoting the ordered principal stresses. The inverse relationships read

sI ¼ sðηþ f IðθÞÞ ð8Þ

sII ¼ sðηþ f IIðθÞÞ ð9Þ

sIII ¼ sðηþ f IIIðθÞÞ ð10Þwith the Lode angle parameter dependent functions

f IðθÞ ¼23

cosπ

6ð1�θÞ

� �; ð11Þ

f IIðθÞ ¼23

cosπ

6ð3þθÞ

� �; ð12Þ

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153136

f IIIðθÞ ¼ � 23

cosπ

6ð1þθÞ

� �: ð13Þ

Throughout this manuscript, the term stress state is employed to make reference to the pair of parameters fη; θg.Examples for stress states that we refer to frequently are:

�

Uniaxial tension ðη¼ 0:33; θ¼ 1Þ � Pure shear ðη¼ 0; θ¼ 0Þ � Generalized shear ð θ¼ 0Þ � Generalized axisymmetric tension ð θ¼ 1Þ � Generalized axisymmetric compression ð θ¼ �1Þ

3. Micromechanical model

A unit cell model with an initial spherical void is built and used to obtain a relationship between the macroscopicequivalent plastic strain to plastic localization after monotonic proportional loading and the stress state. The underlyingmotivation is the hypothesis that the localization of plastic deformation in a band indicates that the onset of fracture isimminent (Rice, 1977). Even though the overall elasto-plastic response of many metals can be described accurately up tovery large strains without considering the nucleation and growth of voids, a low porosity is needed as imperfectiontriggering the localization of plastic deformation.

3.1. Matrix material

The matrix material is modeled as a rate-independent isotropic elastic-plastic Levy–von Mises solid. Only isotropic strainhardening is considered by a saturation hardening law that describes the relationship between the von Mises equivalentplastic strain λ and the flow resistance k¼ kðλÞ. In differential form, the saturation law reads

kð0Þ ¼ k0dkdλ ¼H0 1� k

k1

� �r8<: ð14Þ

with the initial flow resistance k0 and the saturation value k1. The strain hardening parameters fk0; k1; H0; rg and theelastic constants used in the calculations (Table 1) correspond to a TRIP-assisted advanced high strength steel (TRIP780).

3.2. Unit cell geometry and kinematic boundary conditions

The undeformed unit cell consists of a rectangular cuboid of matrix material containing a spherical void at its center. Weintroduce the fixed Cartesian frame ½e1; e2; e3� corresponding to the normals to the cell's outer surfaces in the initial(undeformed) configuration. We limit our attention to a unit cell with the initial edge lengths L1 ¼ L2 ¼ L and L3 ¼ 2L(Fig. 1a); the height-to-width ratio of L3 =L1 ¼ 2 is chosen to facilitate the detection of localization using a purely kinematiccriterion. The cell features a central spherical void of radius R¼ 0:15L which corresponds to an initial porosity of f 0ffi0:7%.

Defining the macroscopic deformation gradient F as the spatial average of the local deformation gradient over the unitcell volume, the boundary conditions are chosen such that a gradient of the form

F ¼ F11e1 � e1þF22e2 � e2þF33e3 � e3þF13e1 � e3 ð15Þcan be applied to the unit cell. In other words, the unit cell is subject to normal loading along all its boundaries and shearloading in the e1–e3-plane. Denoting the average normal displacement along a boundary of normal 7ei as 7Ui, and theaverage tangential displacements along the direction 7e1 on the boundaries of normal 7e3 as 7Ut , the macroscopicdeformation gradient reads

F ¼1þ 2U1

L 0 UtL

0 1þ 2U2L 0

0 0 1þ U3L

2664

3775: ð16Þ

For the isotopic matrix material, the associated mechanical problem is symmetric with respect to the ðe1; e3Þ plane, andantisymmetric with respect to the ðe1; e2Þ plane. Thus, only one quarter of the cell (�L=2rX1rL=2, 0rX2rL=2 and0rX3rL) is considered for finite element analysis (Fig. 1b).

Symmetry conditions are imposed to the boundaries of the quarter model with the normal vectors 7e2,

u2ðX1;0; X3Þ ¼ 0u2ðX1; L=2; X3Þ ¼U2

(ð17Þ

Table 1Matrix material model parameters.

Elastic constants Strain hardening constants

E [MPa] ν [dimensionless] k0 [MPa] k1 [MPa] H0 [MPa] r [dimensionless]

185,000 0.3 450 1200 20,000 2.0

Fig. 1. (a) Initial geometry of the unit cell; the displacements U1, U2, U3 and Ut imposed to the unit cell boundaries are depicted in red. (b) Finite elementmesh of the unit cell. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 137

while anti-symmetry conditions (with respect to the ðe1; e2Þ-plane) plane are imposed to the boundary of normal vector �e3,

u1ðX1;X2;0Þþu1ð�X1;X2;0Þ ¼ 0u2ðX1;X2;0Þ�u2ð�X1;X2;0Þ ¼ 0u3ðX1;X2;0Þþu3ð�X1;X2;0Þ ¼ 0

8><>: ð18Þ

Both tangential and normal displacements are applied to the boundary of normal vector þe3, i.e.

u1ðX1;X2; LÞþu1ð�X1;X2; LÞ ¼ 2UT

u2ðX1;X2; LÞ�u2ð�X1;X2; LÞ ¼ 0u3ðX1;X2; LÞþu3ð�X1;X2; LÞ ¼ 2U3

8><>: ð19Þ

The periodicity conditions for the boundaries of normal vectors 7e1 read

u1ðL=2;X2;X3Þ�u1ð�L=2;X2;X3Þ ¼ 2U1

u2ðL=2;X2;X3Þ�u2ð�L=2;X2;X3Þ ¼ 0u3ðL=2;X2;X3Þ�u3ð�L=2;X2;X3Þ ¼ 0

8><>: ð20Þ

Note that the periodic boundary conditions impose a strong kinematic restriction on the possible formation of a planarband of localized plastic deformation. It can only take the vector e3 as normal vector in order to be kinematically admissible.

3.3. Macroscopic rate of deformation

According to the above boundary conditions, the four displacement degrees of freedom,

uðtÞ ¼ fU1; U2; U3; UT g ð21Þ

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153138

control the average deformation gradient in the unit cell. Introducing the velocity vector _uðtÞ, the macroscopic rate ofdeformation tensor can also be expressed as a function of these four degrees of freedom,

D¼ 2 _U1

Lþ2U1e1 � e1þ

2 _U2

Lþ2U2e2 � e2þ

_U3

LþU3e3 � e3

þ 12ðLþU3Þ

_Ut�2 _U1Ut

Lþ2U1

� �ðe1 � e3þe3 � e1Þ ð22Þ

This symmetric tensor features only four non-zero components which are summarized in a rate of deformation vector

D!¼ fD11; D22; D33; 2D13gT ð23Þ

The linear relationship (22) between the rate of deformation tensor and the current macroscopic velocities, can thus beconveniently rewritten as

D!¼Q _u ð24Þ

with the time-dependent linear transformation

Q ðtÞ ¼

2Lþ2U1

0 0 0

0 2Lþ2U2

0 0

0 0 1LþU3

0�1

LþU3

2UtLþ2U1

0 0 1LþU3

26666664

37777775

ð25Þ

3.4. Control of the loading path in macroscopic stress space

The macroscopic stress tensor rðtÞ is defined as the spatial average of the local Cauchy stress field over the currentvolume VðtÞ of the unit cell. However, without explicitly calculating the macroscopic stress tensor, the stress state and thedirections of the principal stresses are controlled throughout the simulations through a time-dependent kinematicconstraint on the applied macroscopic velocities _uðtÞ.

Consider first the rate of mechanical work for the unit cell at time t. It may be written as

_φ¼ Vr : D¼ V r!U D!¼ f U _u ð26Þ

In (26), f denotes the vector of the work conjugate forces to the macroscopic translational degrees of freedom, while thevector r! is work conjugate to D

!. The vector r! summarizes four of the six independent components of the symmetric

macroscopic Cauchy stress tensor,

r!¼ fs11; s22; s33; s13gT: ð27ÞIt is assumed that the shear components s12 and s23 (which may become non-zero due to the deformation-induced

anisotropy) remain always small as compared to the other stress components and may be neglected when computing thestress triaxiality and/or the Lode angle parameter.

The stress vector may be written as

r!¼ sIðtÞ ~aðtÞ ¼ sIðtÞ‖ ~a‖aðtÞ ð28Þwith sIðtÞ‖ ~a‖Z0 denoting the amplitude and the unit vector aAℝ4 denoting the direction of the loading path in thereduced stress space. As will be shown below, the latter is a function of the orientation of the principal stresses and thestress state (i.e. the principal stress ratios).

Instead of specifying four non-homogeneous boundary conditions on the components of f , a new set of velocity degreesof freedom _v is created through a linear transformation of _u,

_v¼ A _u: ð29ÞBased on the rate of work,

~f U _v¼ f U _u ð30Þwe define the work-conjugate force vector

~f ¼ A�T f ð31ÞThe time-dependent transformation AðtÞ is then chosen such that (28) is readily satisfied by specifying homogeneous

boundary conditions on all but one component of the force vector ~f . In other words, ~f takes the simple form

~f ¼ f ðtÞei ð32Þ

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 139

with ei denoting one of the four orthogonal basis vectors of ℝ4. Rewriting the rate of work, we have

~f U _v¼ V r!UQ _u¼ V r!UQA�1 _v¼ VðQA�1ÞT r!U _v: ð33ÞIn the weak formulation of the unit cell problem, Eq. (33) must be fulfilled for any _v and hence

~f ¼ V QA�1� �T

r!: ð34Þ

Satisfaction of the boundary condition ~f ¼ f ðtÞei implies r¼ sIðtÞ ~a if the transformation matrix A satisfies the identity

VsI‖ ~a‖ QA�1� �T

a¼ f ei: ð35Þ

Among all invertible transformation matrices that satisfy (35), we chose A such that LQA�1 becomes a rotation matrix,

LQA�1 ¼ a � e1þb � e2þc � e3þd � e4: ð36Þwith the unit vectors a, b, c and d forming an orthogonal basis of ℝ4. Note that we also randomly chose the first componentof ~f as the only non-zero force component, while a zero force boundary condition is imposed on the remaining threedegrees of freedom.

By updating the coordinate transformation at each time step, we can control the loading path in stress space. Note thatinstead of prescribing the time history of the first component of ~f , the work-conjugate velocity (first component of _v) isapplied.

3.5. Loading scenarios

In the above control scheme, linear and non-linear loading paths can be prescribed by specifying ~aðtÞ. The correspondingmacroscopic Cauchy stress tensor in 3D reads

rðtÞ ¼ sIðtÞ ~a1e1 � e1þ ~a2e2 � e2þ ~a3e3 � e3þ ~a4ðe1 � e3þe3 � e1Þ½ � ð37ÞIn terms of the principal stresses sIZsIIZsIII , the same stress tensor may be written as

rðtÞ ¼ ∑i ¼ I; II; III

siðtÞpiðtÞ � piðtÞ ¼ sIðpI � pIþψ IIpII � pIIþψ IIIpIII � pIIIÞ ð38Þ

with the unit vectors piðtÞAℝ3 denoting the principal stress directions. Note that the stress state fη; θg is a function of theprincipal stress ratios ψ II ¼ sII=sI and ψ III ¼ sIII=sI only,

η¼ffiffiffi2

p

31þψ IIþψ IIIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1�ψ IIÞ2þð1�ψ IIIÞ2þðψ II�ψ IIIÞ2q ð39Þ

θ¼ 1� 2πarccos

12ð2�ψ II�ψ IIIÞð1�2ψ IIþψ IIIÞð1þψ II�2ψ IIIÞ

ð1þψ2IIþψ2

III�ψ II�ψ III�ψ IIψ IIIÞ3

!: ð40Þ

The directions of the principal stresses can thus be chosen independently from the stress state. Recall that

(a)

According to the periodic boundary conditions, the ðe1; e2Þ-plane is the only possible plane of localization. (b) According to Rudnicki and Rice (1975), the direction of the second principal stress is always parallel to the plane of

localization.

For computational convenience, we impose

pII ¼ e2 ð41Þin all simulations which is consistent with (a) and (b). The other two principal stress directions can then be expressed as afunction of the angle αðtÞ between the first principal stress and the normal n¼ e3 to the plane of localization,

pI ¼sin α

0cos α

8><>:

9>=>; and pIII ¼

� cos α

0sin α

8><>:

9>=>;: ð42Þ

Using (41) and (42) in (38) allows us to express the loading path direction ~a as a function of the stress statefη; θg ¼ f ðψ II ; ψ IIIÞ and the orientation α of the principal stresses,

~a ¼

sin 2αþψ III cos 2α

ψ II

cos 2αþψ III sin2α

ð1�ψ IIIÞ cos α sin α

8>>>><>>>>:

9>>>>=>>>>;: ð43Þ

Fig. 2. (a) Rotation of the material coordinate system (blue); (b) simultaneous rotation of the material coordinate system (blue) and the principal stressdirections; and (c) rotation of the material coordinate system (blue) with respect to the principal stress directions. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153140

Since the planar band of localization can only form with its normal in the e3-direction, it is important to perform the unit cellanalysis for all possible principal stress orientations to determine the lowest estimate of the strain to failure for a given stress state.

When specifying the loading path, there are also two sources of anisotropy which are worth considering:

1.

Topological anisotropy associated with the spatial distribution of voids: according to the periodic boundary conditions,the determined effective behavior corresponds to a porous solid with voids positioned at the vertices of an orthorhombiclattice. The microstructure therefore features three orthogonal planes of symmetry which results in an orthotropiceffective mechanical response. Here, the orientation of the band of localization is fixed with respect to the initial latticeorientation. A thorough investigation of the effect of topological anisotropy is omitted within the scope of the presentwork due to the extremely high computational costs associated with the unit cell simulations. Instead, it is assumed thatthe effect of initial topological anisotropy on the simulations results is weak as compared to the effect of stress state.

2.

Morphological anisotropy due to deformation-induced void shape changes: the initially spherical void changes into anellipsoidal-like void. Even in the hypothetical case of an isotropic spatial void distribution (topological isotropy), theeffective behavior would become anisotropic due to the evolution of the void shape. In the special case of rotation freeloading, the void shapes are expected to remain symmetric with respect to the directions of the principal stress.

The rotation R of the macroscopic material coordinate frame (Fig. 2a) is defined through the decomposition

F ¼ VR¼UR ð44Þwith V ¼VT and U ¼UT denoting the left and right stretch tensors, respectively. Due to the two sources of anisotropy,

material rotation may affect the results of the localization analysis. In particular, since the orientation of the possible band oflocalization is fixed in space, the directions of loading must be rotated (i.e. UT a0) to simulate different band orientations.

To shed some light on the effect of material rotation due to the (undesired, but unavoidable) anisotropy of the unit cell model onthe predicted macroscopic strain to localization, we consider two scenarios for the evolution of the principal stress directions:

�

Co-rotational loading: the evolution of the principal stress directions is coupled with the rotation of the material,

piðtÞ ¼ RðtÞp0i : ð45Þ

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 141

As illustrated in Fig. 2b, the angle between the principal stress directions and the directions of the orthogonal coordinateframe attached to the material remains constant throughout loading.

�

Radial loading: the principal stress directions are kept constant throughout loading, i.e.

piðtÞ ¼ p0i : ð46Þ

In this case, the material rotates with respect to the loading, i.e. the angle between the principal stresses and the materialchanges as illustrated in Fig. 2c.

In both cases, the computations are performed over a wide range of initial principal stress orientations α0. While α is keptconstant throughout radial loading, the angle α evolves throughout co-rotational loading. After expressing R as a rotation bythe angle Δα around the e2-axis (Fig. 2a),

R¼cos ðΔαÞ 0 sin ðΔαÞ

0 1 0� sin ðΔαÞ 0 cos ðΔαÞ

264

375; ð47Þ

and decomposing the deformation gradient according to (44), the evolution of the principal stress directions (with respectto the stationary e3-axis) for co-rotational loading reads

αðtÞ ¼ α0þΔα ð48Þwith

tan ðΔαÞ ¼ F13F11þF33

: ð49Þ

3.6. Definition of the strain to failure

The definition of the “strain to failure” requires the definition of the instant of failure as well as that of a suitablemacroscopic strain measure. Here, the onset of failure corresponds to the onset of localization of plastic flow. After the onsetof localization, all additional plastic deformation is expected to accumulate within a narrow band of width w and of normalvector e3 in the vicinity of X3 ¼ 0. At the same time, the matrix material outside this band ( X3j j4w=2) should experiencepartial elastic unloading. Let FU be the volume average of the deformation gradient of the upper part of the unit celldelimited by L=2rX3rL, in which the deformation is expected to be approximately homogeneous (since w⪡L). Knowingthe displacements uM

1 and uM3 of a point M located at ðX1;X2;X3Þ ¼ ðL=2; L=2; L=2Þ (Fig. 1b), FU can then be estimated as

FUC

F11 0 Ut �ðuM1 �U1ÞL=2

0 F22 0

0 0 1þ U3 �uM3

L=2

26664

37775: ð50Þ

Following Needleman and Tvergaard (1992), the localization of deformation into a narrow planar band is then assumedto occur when

ξ� ‖ _F‖‖ _FU‖

⪢1 ð51Þ

where the tensorial norm ‖U‖ is defined as ‖X‖¼ffiffiffiffiffiffiffiffiffiffiffiX : X

p. In all our computations, the instant of onset of failure, tf , is defined

as the instant when ξ¼ 5.The macroscopic equivalent strain εp is defined based on the integral of the plastic work over the entire unit cell, φ, using

the stress-strain relationship for the matrix material,Z εp

0kðλÞdλ¼ φ

V: ð52Þ

This definition is identical to defining the macroscopic equivalent plastic strain as work-conjugate to the macroscopicvon Mises equivalent stress s,

sdεp ¼dφV

ð53Þ

provided that the deformation within the unit cell is uniform (i.e. approximately homogeneous local deformation field).We evaluated both definitions for selected computations and found almost identical results. However, definition (52) turnedout to be more convenient from a computational point of view (no need to compute s).

With the definitions of the instant of failure and the equivalent plastic strain at hand, the macroscopic strain to failure εffor a given stress state is then defined as the minimum of the macroscopic equivalent plastic strain at the instant of failure

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153142

over all possible angles α0,

εf ðη; θÞ : ¼minα0

εpðη; θ; tf ; α0Þ: ð54Þ

Note that due to the strictly monotonic relationship between the equivalent plastic strain and the plastic work, thecritical angle α0 can also directly determined from the minimization of the total plastic work.

3.7. Computational aspects

The implicit solver of the commercial Finite Element package Abaqus is used. The unit cell is discretized by 23,338 fullyintegrated first-order solid elements (element C3D8 of the Abaqus/Standard library), as shown in Fig. 1b. The boundaryconditions described by Eqs. (17) to (20) are then enforced by imposing the corresponding kinematic constrains to the nodeslocated on the cell boundaries. The non-linear loading path control is achieved through a user defined subroutine (timedependent multi-point constraint). For low triaxiality loadings, initially spherical voids tend to collapse into penny-shapedcracks (Tvergaard, 2008). Here the kinematic self-contact formulation of Abaqus/Standard is used to prevent interpenetra-tion of the void walls. The contact is modeled as frictionless (μ¼ 0), therefore transmitting only normal forces at theinterface (see Dahl et al. (2012) for more details on the effect of friction). For each stress state, at least ten different possibleshear band orientations have been considered. With an average computational time of about three hours per simulation run(with parallel processing on 12 CPUs), it took about eight months of elapsed time on a high performance workstation toobtain the simulation results presented below.

4. Results

Simulations with monotonic proportional loading are performed for a dense grid of stress states within the range

0rηr1 and �1rθr1: ð55ÞGiven the high computational costs, we performed the complete analysis for co-rotational loading only. Our discussion

therefore makes reference to results for co-rotational loading. The only exception is Section 4.3, where results for radial

Fig. 3. Influence of the stress state on the onset of localization: (a) Evolution of the localization indicator, and (b) macroscopic equivalent stress–straincurves for uniaxial tension ðη¼ 0:33; θ¼ 1Þ, pure shear ðη¼ 0; θ¼ 0Þ and a triaxial state of stress ðη¼ 0:5; θ¼ 0Þ. (c) Pre- and post-localization responseobserved at the microscale. The three contour plots correspond to the instants highlighted by dark dots in (b).

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 143

loading are discussed and compared with those for co-rotational loading. Before showing the results for all stress states inSection 4.2, selected examples are discussed in detail in Section 4.1 showing the evolution of the localization indicator, theeffect of the shear band orientation and the captured deformation mechanisms at the unit cell level.

4.1. Demonstration of the analysis procedure

4.1.1. Evolution of the localization indicatorFig. 3a shows the evolution of the localization indicator as a function of the macroscopic effective plastic strain for three

different stress states. The corresponding macroscopic equivalent stress–strain curves are depicted in Fig. 3b.

�

The black curve (η¼ 0:5, θ¼ 0) is representative for simulations in which localization occurred: the localization indicatorinitially remains close to 1 followed by a moderate increase, before diverging to infinity. The divergence of thelocalization indicator is usually associated with a maximum of the macroscopic effective stress-strain curve. With theonset of localization (i.e. ξ45), the effective load carrying capacity drops. At this stage, most of the cell matrix materialexperiences elastic unloading, while increases in plastic deformation are limited to the material within the band oflocalization (gray area in Fig. 3c).

�

The red curve (η¼ 0:33; θ¼ 1Þ shows an example where no localization occurred. The localization indicator remainsmore or less constant (ξffi1Þ and the effective stress–strain curve continues to follow the solution for a homogeneousdeformation field inside the unit cell.

�

The blue curve (η¼ 0; θ¼ 0Þ depicts another example with no localization. However, in this case, the localizationindicator evolution shows the signs of some pronounced non-uniform deformation at a strain of 0.6, but the unit cellremains stable and continues to deform in a uniform manner after an isolated peak in the localization indicator history.

Choosing a rather large critical value of the localization indicator (ξ¼ 5) increased the reliability of the localizationdetections. Note that changes in the effective equivalent plastic strain are almost insignificant when the localizationindicator increases from 2 to 5. In other words, the uncertainty in the reported strain to failure due to the choice of thecritical value of the localization indicator is small.

4.1.2. Orientation of the localization bandFor each stress state fη; θg, the localization analysis is repeated for different band orientations. In our model, the band

orientation is indirectly varied by changing the initial angle α0 of the maximum principal stress. Fig. 4 presents simulationresults for a stress state of η¼ 0:5 and θ¼ 0, and three distinct initial orientations: α0 ¼ 201 (black curves), α0 ¼ 26:21 (redcurves) and α0 ¼ 29:51 (blue curves). The computed macroscopic equivalent stress-strain curves (Fig. 4a) lie on top of eachother prior to the onset of localization, but the evolution of the localization indicator ξ clearly depends on α0 (Fig. 4b).

The macroscopic plastic strain at which localization occurs for α0 ¼ 26:21 is about 20% lower than that for α0 ¼ 201 andα0 ¼ 29:51. Fig. 4c depicts the dependency of εðη; θ; tf Þ – in black – and αðη; θ; tf Þ – in blue – on α0 over the interval201rα0r301. Each solid dot in Fig. 4c is the outcome of a unit cell analysis. A minimum of strain to localization is reachedfor α0 ¼ 26:21 (red dot in Fig. 4c), which therefore corresponds to the initial principal stress orientation which is mostfavorable for localization. We typically performed computations for ten initial orientations varying from 101 to 501. In mostcases, the plot of the dissipation φðtf Þ as a function of α0 showed a local minimum which is identified through cubicinterpolation with a relative precision of 3%. Otherwise, the range of initial orientations is increased further until a minimumis found.

4.1.3. Captured failure mechanismsIt is noted that the localization criterion captures both inter-void ligament shear bands and internal necking. The cross-

sectional cuts shown in Fig. 5 provide some insight into the post-localization behavior of the cell for an intermediate stresstriaxiality loading (η¼ 0:5, θ¼ 0) and at high stress triaxiality loading (η¼ 1, θ¼ 0). The superposed contour plots show thelocal equivalent plastic strain distribution for a macroscopic equivalent strain of εp ¼ 0:39 and εp ¼ 0:27, respectively. Themacroscopic equivalent stress-strain curves and the evolution of void volume fraction during deformation are also includedin Fig. 5 (in black for η¼ 0:5, in blue for η¼ 1). It can be seen that

�

At the intermediate stress triaxiality (η¼ 0:5), the overall shearing of the void is dominant and the void volume fractionremains approximately constant throughout loading.

�

At the high stress triaxiality (η¼ 1:0), the volume change associated with the hydrostatic pressure is more significant andthe void volume fraction increases from its initial value of 0.01 to about 0.04 at the onset of localization; the inter-voidligaments are therefore also being stretched in addition to shearing which introduces a neck into the sheared ligaments.

A detailed analysis of the deformation response at the unit cell level is omitted as the observed responses are inagreement with those reported in the literature. Readers interested in a comprehensive discussion of the failure mechanismunder shear loading are referred to the literature (e.g. Tvergaard, 2008, 2012).

Fig. 4. Effect of the principal stress orientation: (a) macroscopic equivalent stress-strain curves, (b) evolution of the localization indicator for differentinitial orientations, and (c) predicted failure strain (in black) and orientation of the plane of localization α (in blue) as a function of the initial orientation α0.Solid dots depict the onset of localization in (a); in (c), each dot corresponds to a cell calculation. The most favorable initial orientation αn

0 is shown in red.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153144

4.2. Strain to failure as function of stress state

The localization analysis has been executed for 178 different stress states within the domain specified in (55). Summaryplots showing the strain to failure as a function of the Lode angle parameter and the stress triaxiality are presented in Fig. 6.Solid dots correspond to simulation results, while the solid lines correspond to cubic spline interpolations. Dashed linescorrespond to spline extrapolations in a range where no localization was found in the calculations.

Fig. 5. Deformed unit cells at the instant of failure: (a) shear-dominated hole ligament deformation ðη¼ 0:5; θ¼ 0; εp ¼ 0:39Þ and (b) necking of the holeligaments (η¼ 1; θ¼ 0; εp ¼ 0:27Þ. The contour plots show the local plastic strain after the onset of localization (ξ45); and (c) evolution of the macroscopicequivalent stresses and void volume fractions as a function of the macroscopic plastic strain. Solid dots depict the onset of localization (ξ¼ 5), while thetriangles indicate the instants at which the contour plots are extracted. (For interpretation of the references to color in this figure, the reader is referred tothe web version of this article.)

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 145

The failure strain εf exhibits a strong dependence on both the stress triaxiality η and the Lode parameter θ. In particular, itis observed that:

�

The strain to failure is a monotonically decreasing function of the stress triaxiality. This can be clearly seen from Fig. 6aand b, where the strain to fracture is plotted as a function of the stress triaxiality for constant Lode angle parameters;

�

The strain to fracture exhibits an asymptotic behavior at low stress triaxiality: there appears to exist a “cut-off value” oftriaxiality ~ηc below which no localization occurs. This can be seen from Fig. 6a and b and is elucidated further in Fig. 6d.The cut-off stress triaxiality ηc depends on the Lode parameter, and increases as the stress state departs from generalizedshear (θ¼ 0) towards axisymmetric states of stress (θ¼ 71): ηcðθ¼ 0ÞC0:15, while ηcðθ¼ 1ÞC0:4 and ηcðθ¼ �1ÞC0:55.

�

It is worth noting that no localization occurs for pure shear (η¼ 0, θ¼ 0) and uniaxial tension (η¼ 0:33, θ¼ 1). For stresstriaxialities greater than ηZ0:6, localization occurs over the complete range of Lode parameters.

�

The strain to failure is a non-symmetric convex function of the Lode angle parameter θ, exhibiting its minimum in theinterior of its domain of definition (Fig. 6c). The minimum of the strain to failure does not exactly correspond togeneralized shear loadings (θ¼ 0): for a triaxiality of η¼ 1, the minimum is at θ¼ 0:3; for η¼ 0:2, the minimum is atθ¼ þ0:05.

�

In the specific case of axisymmetric loadings, the failure strain εf is almost twice higher for θ¼ �1 than for θ¼ þ1 at thesame triaxiality η.

Fig. 6. Failure strain as function of the stress triaxiality for (a) θr0, and (b) θZ0; (c) representation of the same data as a function of the Lode angleparameter with the stress triaxiality as curve parameter. (d) Projection on the ðη; θ)-plane showing the boundary of the domain of stress states withlocalization. Each solid dot corresponds to a unit cell analysis in which the onset of localization was reached.

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153146

4.3. Effect of loading path

4.3.1. Void shape evolution under co-rotational loadingAs detailed in Section 3.5, the principal macroscopic stress directions pI and pIII evolve during the cell deformation, when

a transverse displacement Uta0 is imposed (i.e. Ra1Þ. Fig. 7 depicts the evolution of the void shape in the ðe1; e3Þ- cross-section (solid lines) along with the major principal macroscopic stress direction pI (dashed lines) at different levels ofdeformation for the stress states considered under 4.1.1: (1) uniaxial tension (Fig. 7a), (2) pure shear (Fig. 7b) and (3) triaxialloading (Fig. 7c). In all three cases, a transverse displacement is imposed to the cell boundaries (as α0a0), thereby inevitablyshearing the unit cell.

During loading, the void evolves from an initially spherical shape to an elongated ellipsoidal-like shape. At low triaxialityloadings, the void eventually reduces to a penny-shaped crack (Fig. 7b). In addition to the evolution of its shape, the voidalso rotates with respect to the stationary unit cell frame ðe1; e2; e3Þ. The major principal macroscopic stress direction pI alsorotates about the e2 axis (co-rotational loading), and remains approximately aligned with the major void axis throughoutloading. Note that the matrix material model does not account for matrix failure. In a reality, matrix failure might also comeinto play and precede the onset of localization.

4.3.2. Co-rotational versus radial loadingThe unit cell analysis is performed under radial loading for generalized shear (θ¼ 0), axisymmetric tension (θ¼ 1) and

axisymmetric compression (θ¼ �1). As for co-rotational loading, the simulations for radial loading predict a monotonicdecrease of the strain to failure as a function of the stress triaxiality (Fig. 8). Furthermore, the simulations for radial loadingconfirm that the microstructure is more prone to localization under axisymmetric tension than under axisymmetriccompression. However, the quantitative comparison reveals lower cut-off stress triaxialities for radial loading (blue curves)as compared to co-rational loading (black curves). In other words, the rotation of the orthotropy axes with respect to the

Fig. 7. Computed void shape evolution (a) under uniaxial tension ðη¼ 0:33; θ¼ 1Þ, (b) pure shear ðη¼ 0; θ¼ 0Þ and (c) triaxial loading ðη¼ 0:5; θ¼ 0Þ.The solid lines depict the void contour in the ðe1; e3Þ-plane at different macroscopic equivalent plastic strain. The dashed lines indicate the currentorientation of the first principal macroscopic stress. Note that the picture boundaries do not correspond to the boundaries of the unit cell.

Fig. 8. Comparison of the strain to failure for radial loading (blue curves) with that for co-rotational loading (black curves) for generalized shear (θ¼ 0), foraxisymmetric tension (θ¼ 1) and for axisymmetric compression (θ¼ �1). (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 147

directions of the principal stresses (radial loading) makes the unit cell more prone to localization at low stress triaxialities.This effect vanishes at higher stress triaxialities which is anticipated intuitively as the deformation-induced morphologicalanisotropy is expected to be less pronounced.

Fig. 9. Effect of the initial porosity on the predicted strain to failure for generalized shear (θ¼ 0), for axisymmetric tension (θ¼ 1) and for axisymmetriccompression (θ¼ �1). The results are shown for f 0 ¼ 0:21% (blue), f 0 ¼ 0:71% (black) and f 0 ¼ 1:68% (red). (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153148

4.3.3. Effect of initial porosityThe central void acts as a defect that triggers the localization of the plastic deformation in the cell matrix. The computed

failure strains are therefore expected to depend on the initial porosity. All results presented hereinbefore have beenobtained for an initial void volume fraction of f 0 ¼ 0:71%. To shed more light on the dependence of the cell behavior on theinitial porosity, additional simulations have been carried out on unit cells of lower porosity (f 0 ¼ 0:21%) and of higherporosity (f 0 ¼ 1:68%), respectively.

Fig. 9 summarizes the simulation results for all three porosities (f 0 ¼ 0:21%¼blue, f 0 ¼ 0:71%¼black, f 0 ¼ 1:68%¼red).It is observed that the initial volume fraction has two major effects on the onset of localization:

1.

For a given stress state, the failure strain decreases as a function of the porosity. For example, for η¼ 1 and θ¼ �1, thestrain to failure for the lowest porosity is 1.5 times higher than that for f 0 ¼ 0:71%, and 2.4 times higher than that forf 0 ¼ 1:68%.

2.

The cut-off stress triaxiality below which no localization occurs decreases as the porosity increases. In case of ageneralized shear stress state, the cut-off trixiality is ηc ¼ 0:35 for f 0 ¼ 0:21%, and ηc ¼ 0:15 for f 0 ¼ 1:68%. For f 0 ¼ 1:68%,we have ηco0, i.e. localization occurs for all stress triaxialities considered in this study.

5. Macroscopic localization criterion for monotonic proportional loading

In view of developing ductile fracture models, we propose a macroscopic criterion to describe the strain to failure as afunction of the stress state. Starting point is the formulation of a criterion in stress space, followed by a transformation intothe mixed stress-strain space fη; θ; εf g. The criterion is expected to be valid for monotonic proportional loading only, i.e.loading at constant stress triaxiality and Lode angle parameter all the way to the point of failure. As already noted byMcClintock (1968), incorporated into most phenomenological fracture initiation models (e.g. Dunand and Mohr, 2011b), andexplicitly shown by Benzerga et al. (2012), the fracture locus describing the onset of fracture initiation in ductile solids ispath dependent. For arbitrary loading, the strain to fracture is not only a function of the current stress state, but also of theplastic loading history.

5.1. Criterion in terms of the normal and shear stress on the plane of localization

An attempt is made to find a relationship between the shear and normal stresses on the plane of localization. Accordingto our coordinate definitions, the vector normal to the plane of localization reads

n¼ e3 ¼ cos ðαÞpIþ sin ðαÞpIII : ð56Þ

Recall that the direction of the second principal stress is always parallel to the plane of localization. As a consequence, thenormal and shear stress acting on that plane are independent of the second principal stress. Denoting the normal and shear

Fig. 10. Shear stress versus the normal stress on the plane of localization. The solid dots represent the results from unit cell analyses, while the solid linerepresents the localization criterion defined by Eq. (61).

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 149

stress acting on the plane of maximum shear (α¼ 451) as

sms ¼ sIþsIII

2; ð57Þ

τms ¼sI�sIII

2; ð58Þ

we have

sn ¼ cos ð2αÞτmsþsms ð59Þ

τ¼ sin ð2αÞτms ð60ÞThe black dots in Fig. 10 show the shear stress τ as a function of the normal stress sn at the onset of localization. Note that

each data point fsn; τg represents the result from a unit cell analysis for a specific stress state. The striking outcome is that alldata points seem to lie on a smooth curve in that space. In particular, the plot is reminiscent of the criterion sketched byMohr (1900). Here, we propose an open convex envelope as criterion for the onset of shear localization,

jτjτ0

¼ 1� sns0

� �nð61Þ

with 0onr1; τ0 and s0 denote the respective limiting shear and normal stresses on the plane of localization. The blackcurve in Fig. 10 corresponds to a fit with τ0 ¼ 664 MPa, s0 ¼ 2048 MPa and n¼ 0:32.

Besides the good agreement with the simulation results in terms of stresses, it is worth noting that the above shearlocalization criterion also provides a relationship between the orientation of the plane of localization and the stress state.For a given stress state fη; θg, the mathematical problem determining the orientation α of the localization plane reads

αðη; θÞ ¼ argminα

fsðα; η; θÞg ð62Þ

where sðα; η; θÞ is given through the implicit equation

sτn�½1�ssn�n ¼ 0 ð63Þwith

snðα; η; θÞ ¼ cos ð2αÞ f IðθÞ� f IIIðθÞ2s0

þ 2ηþ f IðθÞþ f IIIðθÞ2s0

; ð64Þ

and

τnðα; η; θÞ ¼ sin ð2αÞ f IðθÞ� f IIIðθÞ2τ0

: ð65Þ

Fig. 11a shows the predicted angle α as a function of the stress triaxiality and the Lode angle parameter. Apart from theimmediate vicinity of the cut-off stress triaxiality, the model provides a reasonable approximation of the simulation results(Fig. 11b): the angle α decreases as a function of the stress triaxiality and the Lode angle parameter, with values aroundα¼ 451 (i.e. close to the plane of maximum shear).

Fig. 11. Orientation of the plane of localization (a) according to the localization criterion, and (b) according to the unit cell analyses.

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153150

5.2. Criterion in mixed stress–strain space

In the general framework provided by Mohr and Marcadet (submitted for publication), the solution

sf ðη; θÞ ¼ min fsðα; η; θÞg ð66Þof Eq. (62) corresponds to a conversion of the localization criterion from the two-dimensional space fsn; τg to the three-dimensional modified Haigh–Westergaard space fη; θ; sg. Using the isotropic hardening law, the localization criterion canthen also be converted into the mixed stress-strain space fη; θ; εp g,

εf ¼ k�1½sf ðη; θÞ� ð67Þwith k�1 denoting the inverse of the hardening function given by (14). The corresponding plot of the computed strain tofailure as a function of stress triaxiality and the Lode angle parameter is shown in Fig. 12a. It agrees well with that obtainedfrom localization analysis (Fig. 12b) which is another confirmation of the applicability of criterion (61) for predicting theonset of shear localization.

6. Discussion

One of the key features of the proposed localization criterion is its independence of the intermediate principal stress.It can thus be written in terms of two stress quantities only. However, the right pair of stress measures must be chosen, e.g.sI and sIII , or sn and τ. Attempts were made in the past to formulate phenomenological ductile fracture criteria in terms of εpand η (e.g. Johnson and Cook, 1985). For isotropic hardening Levy–von Mises materials, this corresponds to a criterion interms of s and sm in stress space. The poor agreement of such criteria with experimental data for non-axisymmetric stressstates led to the introduction of the Lode (angle) parameter. Based on the above findings (and the assumption that ductilefracture is imminent with the onset of localization), the introduction of a third stress measure into ductile fracture criteria isonly needed because the equivalent von Mises stress (and hence also the equivalent plastic strain) is not orthogonal to thesecond principal stress.

Fig. 12. Visualization in the mixed stress–strain space: (a) strain to failure according to the localization criterion as a function of the stress triaxiality andthe Lode angle parameter, (b) comparison of the analytical model predictions (solid curves) with the simulation results (solid dots).

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153 151

It is worth noting that unlike the von Mises stress, the Tresca stress τms is orthogonal to the second principal stress andcould be used in conjunction with the normal stress sms acting on the plane of maximum shear to formulate a fracturecriterion. As compared to using the normal and shear stresses on the plane of localization, the latter criterion would notrequire the identification of the critical plane. Preliminary computations actually showed that (61) also yields good resultswhen using the fsms; τmsg instead of fsn; τg.

7. Concluding remarks

Motivated by the assumption that ductile fracture is imminent with the onset of localization, a fully three-dimensionalunit cell model with a spherical void defect is built to investigate the effect of the stress state on the onset of localization in aLevy–von Mises solid. The particular feature of the present analysis is that it covers a rather dense grid of stress states, withstress triaxialities ranging from 0 to 1 and the full range of Lode parameters. The periodic boundary conditions areformulated such that the macroscopic stress triaxiality and Lode parameter are kept constant up to the onset of localization.The simulations are performed for co-rotational loading to compensate for the effect of unavoidable material rotation due tothe tangential loading along the unit cell boundaries.

The simulation results reveal that the macroscopic failure strain for monotonic proportional loading paths is amonotonically decreasing smooth function of the stress triaxiality, and a non-symmetric convex function of the Lodeparameter. This observation is in qualitative agreement with the simulations of Nahshon and Hutchinson (2008) whocarried out a localization analysis using an isotropic shear-modified Gurson model. A plot of the shear stress as a function ofthe normal stress acting on the plane of localization for all simulations suggests a simple open convex envelope as acriterion for predicting the onset of localization after monotonic proportional loading. The transformation of this criterioninto the mixed stress–strain space of stress triaxiality, Lode angle parameter and equivalent plastic strain leads to anaccurate analytical description of all simulation results. It is worth noting that the proposed criterion can be found as asketch in the early work of Mohr (1900), with the well-known Mohr–Coulomb model corresponding to a linearapproximation of the proposed localization criterion. The present micromechanical analysis therefore provides strongsupport for phenomenological ductile fracture models that postulate the existence of a Mohr–Coulomb type of failure

M. Dunand, D. Mohr / J. Mech. Phys. Solids 66 (2014) 133–153152

surface in stress space for proportional loading (e.g. Bai and Wierzbicki, 2010, Stoughton and Yoon, 2011, Mohr andMarcadet, 2014).

Acknowledgments

The partial financial support of the MIT Industrial Fracture Consortium and the French National Research Agency (GrantANR-11-BS09-0008, LOTERIE) is gratefully acknowledged.

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