Engineering Fracture Mechanics 147 (2015) 480–497

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Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

Anisotropic Hosford–Coulomb fracture initiation model:Theory and application

http://dx.doi.org/10.1016/j.engfracmech.2015.08.0040013-7944/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Solid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, École Polytechnique, Palaiseau, France.169335801.

E-mail address: mohr@mit.edu (D. Mohr).

Gongyao Gu a, Dirk Mohr a,b,c,⇑a Solid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, École Polytechnique, Palaiseau, Franceb Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USAcDepartment of Mechanical and Process Engineering (MAVT), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 February 2015Received in revised form 19 July 2015Accepted 7 August 2015Available online 4 September 2015

Keywords:AnisotropyDuctile fractureDamage indicator modelMohr–CoulombLinear transformation of stress tensor

An anisotropic extension of the Hosford–Coulomb localization criterion is obtainedthrough the linear transformation of the stress tensor argument. Unlike for isotropic mate-rials where the stress state is characterized through the stress triaxiality and Lode param-eter, the normalized Cauchy stress tensor is used to describe the stress state in ananisotropic solid. Based on experiments on extruded aluminum 6260-T6 covering stressstates from pure shear to equi-biaxial tension for different material orientations, it isshown that this phenomenological and uncoupled model is capable to provide reasonableengineering approximations of the strains and displacements to fracture for thirteen differ-ent loading conditions.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

It is important to accurately predict the ductile fracture of metallic materials for lightweight structure design. Damageevolution within the material is usually considered to be the reason for fracture initiation. Gurson [29] porous plasticitymodel describes the growth of voids and its effect on the plastic behavior. The model was later modified by taking the voidnucleation and coalescence into consideration [55,56]. Many recent efforts have been focused on improving the predictioncapability of porous plasticity models for shear-dominated softening and failure at low stress triaxiality [21,32,47,60].Another approach for modeling ductile fracture is the Continuum Damage Mechanics (CDM) framework where a phe-nomenological scalar damage variable is derived from a thermodynamic dissipation potential [10,11]; Chaboche, 1988;Lemaitre, 1985).

Besides the Gurson-like and CDM models, there are numerous so-called uncoupled phenomenological and empiricalmodels proposed for predicting ductile fracture initiation, in which the elastic and plastic behavior is assumed to be unaf-fected by the evolution of damage. As summarized by Wierzbicki et al. [59], phenomenological models can be simply strain-based [31,37], stress-based [4,9], or embedded into a damage indicator framework (e.g. [33,19]. A comparative study oneight damage indicator models were carried out by Bao and Wierzbicki [3] with the weighting functions of the modelsderived respectively from the work of McClintock [43], Rice and Tracey [51], Leroy et al. [38], Clift et al. [18] and Oh et al.[49]. It is common to utilize the stress triaxiality as the only stress state parameter in damage indicator models [3,48].

Tel.: +33

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 481

Recently, some more general phenomenological models, which take the effect of both the stress triaxiality and the Lodeangle parameter into consideration, were proposed by Bai and Wierzbicki [1,2], Coppola et al. [20], Gruben et al. [28], Louet al. [41], Lou and Huh [40], and Voyiadjis et al. [58], respectively. Ebnoether and Mohr [25] compared the originalstress-based Mohr–Coulomb model and the damage indicator model derived from the Mohr–Coulomb model with their pre-dicted results of the ductile fracture of low carbon steel sheets; it was found that the damage indicator model showed betteraccuracy in simulating the stamping experiment where the loading history was highly non-linear. Comparison of the predic-tive capabilities of uncoupled phenomenological, Gurson-like and CDM models were performed by Dunand and Mohr [24]and Li et al. [39].

Enhanced Gurson-type of models have been developed accounting for matrix anisotropy [6,7], spheroidal voids[26,27,12], a combination of both [46], and kinematic hardening (e.g. [44], Leblond et al., 1995). The effects of the void shapeand void distribution were also covered in the models proposed by Pardoen [50], Pardoen and Hutchinson (2003) and Ste-glich et al. [53]. Steglich et al. [54] developed another Gurson-like model incorporating the direction-dependent void growth.A summary of the anisotropic void growth and coalescence models can be found in the review by Benzerga and Leblond [8].By expressing the scalar phenomenological damage as a tensorial parameter, a series of studies on modeling anisotropicdamage and ductile fracture were performed within the framework of nonlinear continuum damage mechanics [13–17].Similar approaches with the variable of a damage tensor introduced into anisotropic CDM models can be also found inthe work by Hammi and Horstemeyer [30], Solanki et al. [52] and Voyiadjis and Dorgan [57].

Most of the previous studies on modeling anisotropic ductile fracture follow the approach of either the Gurson-like or theCDM model. For modeling the deformation-induced anisotropic fracture response of a Ti–6Al–4V alloy, Khan and Liu [35]proposed an uncoupled stress-based criterion, which was extended from an isotropic one based on the magnitude of stressvector [34] with a modified Hill anisotropic function [36]. Luo et al. [42] developed an anisotropic damage indicatormodel for describing the fracture behavior of the extruded sheets of 6260-T6 aluminum alloy, using the isotropic ModifiedMohr–Coulomb stress state weighting function [2] and the von Mises equivalent plastic strain definition after lineartransformation of the plastic strain tensor.

The present work proposes an anisotropic extension of the recently proposed micro-mechanically motivatedHosford–Coulomb fracture initiation model [45]. Anisotropy is introduced into the originally isotropic formulation throughthe linear transformation of the stress tensor. After discussing its direction dependent features, it is demonstrated that theanisotropic Hosford–Coulomb model is able to provide reasonable predictions of the strain and displacement to fracture inexperiments on highly anisotropic aluminum 6260-T6 extrusions subject to thirteen different nearly proportional loadingpaths in stress space.

2. Anisotropic plasticity model

2.1. Anisotropic yield function

The yield function is a 3D-extension of the anisotropic non-quadratic plane stress yield function Yld2000-2d of Barlatet al. [5]. It may be expressed in terms of an anisotropic equivalent stress measure, �ra, and a deformation resistance k,

f ½r� ¼ �ra � k ¼ 0: ð1Þ

The equivalent stress is defined as an anisotropic function of the Cauchy stress tensor r in the material coordinate system,

�ra½r� ¼ 1

21=d ð/0½s0� þ /00½s00�Þ1d ð2Þ

with

/0½s0� ¼ s011 � s022� �2 þ 4 s0212 þ s0213 þ s0223

� �h id2 ð3Þ

/00½s00� ¼ 32

s0011 � s0022� �2 þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis0011 � s0022� �2 þ 4 s0212 þ s0213 þ s0223

� �q� �d

þ 32

s0011 � s0022� �2 � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis0011 � s0022� �2 þ 4 s0212 þ s0213 þ s0223

� �q� �dð4Þ

and the linearly transformed deviatoric stress tensors s0 and s00. In vector notation,

r ¼ r11 r22 r33 r12 r23 r13f g ð5Þ

s0 ¼ s011 s022 s012 s023 s013f g ð6Þ

s00 ¼ s0011 s0022 s0012 s0023 s0013f g ð7Þ

482 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

the linear transformations read

s0 ¼ L0r ð8Þ

s00 ¼ L00r ð9Þ

with the transformation matrices

L0 ¼ 13

2a1 �a1 �a1 0 0 0�a2� 2a2 �a2 0 0 0

0 0 0 3a7 0 00 0 0 0 3a9 00 0 0 0 0 3a10

26666664

37777775

ð10Þ

L00 ¼ 19

�2a3 þ 2a4 þ 8a5 � 2a6 �4a4 � 4a6 þ a3 � 4a5 a3 þ 2a4 � 4a5 � 2a6 0 0 04a3 � 4a4 � 4a5 þ a6 �2a3 þ 8a4 þ 2a5 � 2a6 �2a3 � 4a4 þ 2a5 þ a6 0 0 0

0 0 0 9a8 0 00 0 0 0 9a11 00 0 0 0 0 9a12

26666664

37777775

ð11Þ

The anisotropic equivalent stress is therefore defined through the exponent d and the 12 transformation coefficients ai,i ¼ 1;2; . . . ;12. As discussed by Dunand et al. [22], the above yield function reduces to the Yld2000-2d model when planestress conditions prevail in the ðe1; e2Þ-plane (i.e. r13 ¼ r23 ¼ r33 ¼ 0).

2.2. Associated flow rule

The model assumes associated plastic flow,

dep ¼ dk@f@r

ð12Þ

with the anisotropic equivalent plastic strain

�ea ¼Z

dk: ð13Þ

defined as work-conjugate to the anisotropic equivalent stress,

r : ðdepÞ ¼ �raðd�eaÞ: ð14Þ

2.3. Isotropic hardening

For the aluminum 6260-T6 alloy, self-similar (isotropic) hardening behavior is described through the mixed Swift–Vocelaw (e.g. [45],

k½�ea� ¼ wASðe0 þ �eaÞnS þ ð1�wÞfk0 þ Qð1� e�b�ep Þg: ð15Þ

with the Swift parameters fAS; e0;nsg, the Voce parameters fk0;Q ; bg, and the weighting factor w.

3. Isotropic Hosford–Coulomb fracture initiation model

The formulation of the isotropic Hosford–Coulomb model as proposed by Mohr and Marcadet [45] is briefly recalled. Weadopt a slightly different notation in view of developing its anisotropic extension in Section 4. Note that the argument of afunction is always enclosed into square brackets, while parentheses are used to define the order of mathematical operations.

3.1. Starting point: isotropic Hosford–Coulomb criterion in stress space

Defining the Hosford equivalent stress as

�rHf ¼ 12

ðrI � rIIÞa þ ðrI � rIIIÞa þ ðrII � rIIIÞa� �� �1

a

ð16Þ

Mohr and Marcadet [45] proposed the Hosford–Coulomb criterion

gHC ½r� :¼ �rHf þ cðrI þ rIIIÞ ¼ b ð17Þ

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 483

to describe the onset of shear and normal localization in metals under proportional loading. The model features three param-eters, with a denoting the Hosford exponent, the cohesion b and the friction coefficient c. For a ¼ 1, the model reduces to aMohr–Coulomb model which is independent of the intermediate principal stress. For a–1, the Hosford–Coulomb modeldepends on all three principal stresses.

3.2. Isotropic fracture initiation model

For isotropic materials, the stress state is typically characterized through the stress triaxiality,

g ¼ rm

�rð18Þ

and the Lode angle parameter �h,

�h ¼ 1� 2p

arccos3

ffiffiffi3

p

2J3

ðJ2Þ3=2" #

: ð19Þ

For anisotropic materials, the orientation of the stress tensor with respect to the material directions must also be taken intoaccount (e.g. three Euler angles) in addition to the stress triaxiality and the Lode parameter. In other words, five parametersare needed to describe the stress state for anisotropic materials. As an alternative to working with Euler angles, we make useof the stress tensor in the material coordinate system as normalized by the von Mises equivalent stress, r=�r, to characterizethe stress state. Note that r=�r features five independent components only. In terms of the normalized stress tensor, criterion(17) reads,

�rprf ¼ b

gHC ½r=�r�: ð20Þ

with �rprf denoting the von Mises equivalent stress at the onset of fracture. Introducing a power law to define a bijective map-

ping from equivalent stress to equivalent strain space,

�rprf ¼ h �eprf

h i¼ A �eprf

nð21Þ

leads to an expression of the equivalent plastic strain to fracture as a function of the stress state:

�eprf ½r=�r� ¼ h�1 r*pr

f

h i¼ b

AgHC r=�r½ �� �1

n

¼ bA

� �1n 1

gHC ½r=�r�� �1

n

ð22Þ

In the above form the coefficients b and A both control the magnitude of the strain to fracture. We therefore substitute the

multiplier ðb=AÞ1=n through a single coefficient. Moreover, defining the new coefficient b such that it corresponds to the strainto fracture for uniaxial tension results in the final form of the Hosford–Coulomb fracture initiation model for proportionalloading:

�eprf ½r=�r� ¼ b1þ c

gHC ½r=�r�� �1

n

: ð23Þ

Note that we introduced the superscript ‘pr’ to indicate that the above expressions are applied to predict the onset of fractureunder proportional loading only. For non-proportional loading, the integral form

Z �ef

0

d�ep�eprf ½r=�r�

¼ 1 ð24Þ

is used, where the expression for the strain to fracture for proportional loading serves as weighting function. Note that theresult �ef ¼ �eprf ½r=�r� is obtained from (22) in the case of proportional loading.

3.3. Illustration

The above fracture initiation model features four parameters: a, b, c and n. In view of developing an anisotropic model, theabove equations have been expressed in terms of the normalized stress tensor r=�r. However, if the tensor r=�r is used as theargument of an isotropic function f iso½r=�r�, then there exists a function f �iso½g; �h� such that f iso½r=�r� ¼ f �iso½g; �h�. In the case of theisotropic Hosford–Coulomb criterion (21), we have

gHC ½r=�r� ¼ g�HC ½g; �h� ¼

12

ðf 1 � f 2Þa þ ðf 2 � f 3Þa þ ðf 1 � f 3Þa� �� �1

a

þ cð2gþ f 1 þ f 3Þ ð25Þ

(a)

(b)

prfε

θη

a=1.b=1.

c=0.1n=0.2

plan

e st

rain

u nia

xial

tens

ion

p ure

shea

r

unia

xial

com

pres

sion

equi

-bia

xial

a=1.b=1.

c=0.1n=0.2

biaxial tension valley

Fig. 1. Isotropic Hosford–Coulomb model for proportional loading: (a) strain to fracture as a function of the stress triaxiality and the Lode angle parameter;(b) fracture envelope for plane stress conditions.

484 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

with the Lode angle parameter dependent trigonometric functions

f 1½�h� ¼23cos

p6ð1� �hÞ

h i; f 2½�h� ¼

23cos

p6ð3þ �hÞ

h iand f 3½�h� ¼ �2

3cos

p6ð1þ �hÞ

h i:

Fig. 1a shows a plot of the isotropic Hosford–Coulomb fracture initiation model for proportional loading. The strain to frac-ture is a monotonically decreasing function of the stress triaxiality and a convex non-symmetric function of the Lode angleparameter. The corresponding strain to fracture for plane stress conditions (see black curve in Fig. 1a) is shown as a functionof the stress triaxiality only in Fig. 1b. Note that the apparent non-monotonic dependence on the stress triaxiality is deceiv-ing because of the underlying Lode angle effect that is due to the functional relationship �h ¼ �h½g� for plane stress,

�h ¼ 1� 2p

arccos �272g g2 � 1

3

� �� �for � 2=3 6 g 6 2=3 ð26Þ

The effect of the model parameters a and c is illustrated in Fig. 2. Note that b has always been chosen such that the strain tofracture for uniaxial tension equals 0.3. Observe from Fig. 2a that the lower the Hosford exponent, the deeper the ‘‘biaxialtension valley” between uniaxial and equi-biaxial tension. The friction parameter c controls the stress triaxialitydependency; the higher c, the greater the difference between the strain to fracture for pure shear ðg ¼ 0; �h ¼ 0Þ and plane

(a) (b)

(c) (d)

(e) (f)

a=1.0

a=1.3

a=1.6

a=1.9

c=0.05n=0.1

a=1.9

a=1.0

c=0.05n=0.1

c=0

c=0.1

c=0.2 a=1.3n=0.1

c=0.05

a=1.3n=0.1

c=0

c=0.2

n=0.4

n=0.1

n=0.05 c=0.05a=1.3

n=0.2

n=0.4

n=0.05

c=0.05a=1.3

Fig. 2. Illustration of the effect of the parameters a, c and n on the fracture envelope for plane stress conditions (left column) and the correspondingunderlying criteria in principal stress space (right column).

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 485

strain tension ðg ¼ 0:58; �h ¼ 0Þ, Fig. 3c. The effect of the parameter n is similar to that of the exponent a. It does not affect themonotonicity of the fracture envelope, but can amplify ðn ! 0Þ and de-amplify ðn ! 1Þ the differences in the strain to frac-ture for different stress states (Fig. 2e). Irrespective of the choice of the model parameters, the strain to fracture for uniaxial

Fig. 3. Isotropic fracture envelopes for different parameters a and n as polar plot with the strain to fracture as radial coordinate in the ðr11;r22Þ plane (leftcolumn) and the ðr12;rdÞ-plane (right column).

486 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

and equi-biaxial tension are always identical. This is due to the fact that the Hosford equivalent stress is equal to the max-imum principal stress for these two stress states. Furthermore, with the minimum principal stress being zero ðrIII ¼ 0Þ inboth cases, the Hosford–Coulomb model (Eq. (17)) reduces to maximum principal stress criterion for uniaxial and equi-biaxial tension.

3.4. Underlying model in stress space (for proportional loading)

The above formulation deviates from that proposed by Mohr and Marcadet [45] because of the power law transformation(19) from the f�r;g; �hg stress space to the f�ef ;g; �hg mixed strain–stress space, instead of using the isotropic hardening law(15). This deviation is necessary due to the fact that the bijective nature of the �r to �ep mapping defined by the isotropic hard-ening law is barely visible (i.e. almost the same stress is defined for different strains) in case of materials that feature pro-nounced saturation type of hardening, thereby making the model predictions sensitive to the numerical noise, in particularwhen using a single precision code.

However, it is emphasized that for proportional loading, using (21) is mathematically equivalent to using a localizationcriterion in stress space. The corresponding criterion in stress space is obtained by transforming the criterion from mixedstress–strain space fg; �h; �eprf g to the modified Haigh–Westergaard space fg; �h; �rpr

f g using the material’s isotropic hardening

law (15). For a Levy–von Mises material, we have �ep ¼ �ea and thus �rprf ¼ k �eprf

h i; the resulting criterion in stress space would

therefore read

�rprf ¼ k �eprf ½r=�r�

h i¼ k h�1 b

gHC ½r=�r�� �� �

with b ¼ Að1þ cÞbn ð27Þ

The corresponding criterion in stress space is the Hosford–Coulomb (HC) criterion given by Eq. (1) only if k½�ep� ¼ h½�ep�, i.e.only if the material’s hardening response follows a power law of exponent n. Otherwise a modified Hosford–Coulomb(mHC) criterion is obtained in stress space.

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 487

Fig. 2f serves as an illustration of the differences between the original and modified Hosford–Coulomb model. For exam-ple, assuming that the material’s hardening law is given by k½�ep� ¼ A�e0:1p , the red1 envelope with n ¼ 0:1 would represent theoriginal HC model in stress space (for a ¼ 1:3 and c ¼ 0:05). The other three envelopes (n ¼ 0:05, n ¼ 0:2 and n ¼ 0:4) wouldthen represent a modified HC criterion in stress space. Clearly, the differences are small in stress space (Fig. 2f), but thesecan be very significant in the mixed strain–stress space (Fig. 2e).

4. Anisotropic Hosford–Coulomb fracture initiation model

In the context of phenomenological modeling, a first step towards the formulation of an anisotropic fracture initiationmodel is substituting the von Mises equivalent plastic strain �ep by its anisotropic counterpart �ea. The ductile fracture modelthen reads

1 For

Z �ea;f

0

d�ea�epra;f ½r=�r�

¼ 1 with �epra;f ½r=�r� ¼ b1þ c

gHC ½r=�r�� �1

n

: ð28Þ

For ease of notation, we use the Hosford–Coulomb function gHC ½r�with the stress vector r as argument, gHC ½r� ¼ gHC ½r�. In Eq.(28), the entire anisotropy of the fracture model is inherited from the plasticity model. As shown by Luo (2012), thisapproach is not sufficient to account for the direction dependency of ductile fracture. The concept of linear stress tensortransformation has been successfully used in the past to derive anisotropic yield criteria from isotropic functions (Karafillisand Boyce, 1993, [5], Besson and Bron, 2004). Here, the same idea is exploited to introduce anisotropic stress state depen-dency into the fracture initiation model. WithM denoting a 6� 6 matrix, the weighting function of the anisotropic Hosford–Coulomb fracture initiation model is written as

Z �ea;f

0

d�ea�epra;f ½r=�r�

¼ with �epra;f ½r=�r� ¼ b1þ c

gHC ½Mr=�r�� �1

n

: ð29Þ

In view of modeling thin-walled structures, we chose a linear transformation matrix of the form

M ¼

1 M12 0 0 0 00 M22 0 0 0 00 0 1 0 0 00 0 0 M44 0 00 0 0 0 1 00 0 0 0 0 1

2666666664

3777777775

ð30Þ

with the transformation coefficients M12, M22 and M44. In close analogy with the discussion of the isotropic model(Section 3.4), it can be demonstrated that the proposed anisotropic model reduces to a criterion in stress space forproportional loading which is given through the implicit form

�rpra;f ¼ k b

1þ cgHC Mr=�r½ �

� �1n

" #ð31Þ

with �rpra:f denoting the anisotropic equivalent stress (defined by Eq. (2)) at the instant of onset of fracture.

4.1. Illustration

According to the proposed model, the equivalent plastic strain to fracture varies as a function of the loading path due to acombination of two effects:

1. Stress state dependency of the strain to fracture, i.e. the strain to fracture varies as a function of the stress triaxiality andthe Lode angle parameter which are both isotropic functions of the normalized stress tensor r=�r;

2. Anisotropy of the fracture model which is introduced through the linear transformation M of the stress tensor;

For illustration purposes, we consider the model response for proportional loading only. Firstly, we consider the modelresponse for biaxial tension in the ðr11;r22Þ-plane. Given that the stress state depends on the stress ratio r22=r11 only,we present the results in polar plots ðu; rÞ with the polar coordinates u ¼ arctanðr22=r11Þ and r ¼ �epra;f . The default param-

eters for the plots shown in Figs. 3–5 are: a ¼ 1:3, b ¼ 0:9, c ¼ 0, n ¼ 0:2, M12 ¼ 0, M22 ¼ 1 and M44 ¼ 1.

interpretation of color in Figs. 2 and 12, the reader is referred to the web version of this article.

Fig. 4. Isotropic fracture envelopes for different parameters b and c as polar plot with the strain to fracture as radial coordinate in the ðr11;r22Þ plane (leftcolumn) and the ðr12;rdÞ-plane (right column).

488 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

� As shown previously, the parameter a influences the model’s stress state sensitivity. The lower a, the greater the variationin the strain to fracture as a function of the loading path. All curves shown in Fig. 3a correspond to isotropic fractureenvelopes. Note that for a ¼ 2 the strain to fracture is both stress state and loading direction independent.

� The influence of the parameter n is similar to that of a, i.e. for a – 2, it amplifies the stress state dependency for n ! 0(Fig. 3c).

� For the anisotropic model, b is the strain to fracture for uniaxial tension along the 1-direction. For all loading paths, thestrain to fracture is proportional b (Fig. 4a).

� The parameters M12 and M22 control the anisotropic distortion of the fracture envelope. The strain to fracture for uniaxialtension along the 2-direction decreases as a function of M22 (Fig. 5a); it increases as a function of M12 (Fig. 5c).

The effect of the remaining parameters can be best seen in the plane ðr12;rdÞ with r11 ¼ rd=ffiffiffi2

pand r22 ¼ rd=

ffiffiffi2

p. This

particular plane is chosen as the condition r11 ffi r22 is often met in experiments along the 45�-direction.

� The parameter M44 results in an expansion/contraction along the in-plane shear direction (Fig. 5f).� The parameter c has a similar effect on the failure envelope (Fig. 4d). The key difference as compared to M44 is that c hasan isotropic effect and also changes the envelope in the ðr11;r22Þ-plane (compare Figs. 4c and 5e).

5. Application

The experiments performed by Dunand et al. (2013) and Luo et al. (2013) will serve as basis for calibrating and validatingthe anisotropic Hosford–Coulomb model.

In a first step, we revisit the calibration of the plasticity model before analyzing the fracture experiments.

5.1. Material and experiments

The experimental data used in this study had been obtained from experiments on aluminum 6260-T6 specimens. Allspecimens had been extracted from the 2 mm thick walls of an extruded automotive part. As emphasized by Luo et al.

Fig. 5. Anisotropic fracture envelopes for different parametersM22,M12 andM44 as polar plot polar plot with the strain to fracture as radial coordinate in theðr11;r22Þ-plane (left column) and the ðr12;rdÞ-plane (right column).

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 489

(2013), all specimens came from the same location within the extruded profiles. Following the discussion of Rousselier et al.(2013), only the results for full thickness specimens are considered in this study. The yield stress and Lankford ratios (slopeof the logarithmic plastic width strain versus plastic thickness strain plot) as determined from uniaxial tensile experimentsfor seven different in-plane orientations (15�-increments) are shown in Fig. 6a. The yield stress variations do not exceed�5%, while the Lankford ratios vary significantly as a function of the specimen orientation, from r30 ¼ 0:25 (minimum) tor90 ¼ 1:0 (maximum). Note that an r-value below 1 indicates that the thickness reduction is more pronounced that thereduction in width under uniaxial tension.

The fracture specimen geometries are shown in Fig. 7. The experimental program includes notched tensile specimenswith a minimum gage section width of 10 mm and three different notch radii: R ¼ 5 mm (NT5-specimen), R ¼ 10 mm

Fig. 6. Anisotropic plasticity of Al 6260-T6: (a) Lankford and yield stress ratios as a function of the material orientation, (b) stress–strain curve defining theself-similar (isotropic) hardening.

(a) NT20 (b) NT10 (c) NT5 (d) CH

Fig. 7. Fracture specimen geometries. The displacement measured by a 30 mm (20 mm) long virtual axial strain gage is reported for NT (CH) specimens.Note that all specimens feature 35 � 10 mm large gripping areas at the top and bottom.

490 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 491

(NT10-specimen) and R ¼ 20 mm (NT20-specimen). In addition, 20 mmwide tensile specimens with a R ¼ 5 mm radius cen-tral hole are tested (CH-specimens). Each type of specimen has been extracted for three different material orientations withrespect to the tensile axis. In the 0�-specimens (NT5-0, NT10-0, NT20-0 and CH-0), the tensile direction coincides with thetensile axis. Analogously, 90�-specimens (NT5-90, NT10-90, NT20-90 and CH-90) with the tensile axis being parallel to thetransverse direction, and 45�-specimens (NT5-45, NT10-45, NT20-45 and CH-45) along the diagonal direction have beenextracted from the extruded profiles through wire EDM cutting. In addition to these 12 tensile fracture experiments, we alsoinclude the result from a punch test on a disc specimen with a 45 mm diameter punch. All experiments had been performedunder static loading conditions at a strain rate of less than 10�2=s .

Figs. 8 and 9 provide a summary of all measured force–displacement curves (solid dots). Each curve represents the aver-age of two experiments. Note that the scatter in the experimental results for repeated experiments was small, i.e. the forcedisplacement curves lied on top of each other with variations in the displacement to fracture of less than 3% [42]. Thereported displacements correspond to the relative displacement of two points positioned on the respective upper and lowerspecimen shoulders at an initial distance of 30 mm and 20 mm for the NT and CH specimens, respectively; it has been mea-sured using planar digital image correlation.

5.2. Finite element models

A numerical simulation is performed of each fracture experiment to determine the so-called loading path to fracture, i.e.the evolution of the anisotropic equivalent plastic strain as a function of the stress state up to the instant of fracture initi-ation. We follow the recommendations of Luo et al. [42] and use eight first-order solid elements with reduced integration(element C3D8R from the Abaqus library) along half the specimen thickness. Details of the meshes become visible whenzooming into the contour plots shown in Fig. 8 and 9 (of the electronic version of this paper).

The finite element meshes for the tensile specimens discretize the specimen geometry between the points of displace-ment measurement. The displacement loading is then applied over at least 100,000 explicit time steps up to the experimen-tally measured displacement to fracture (assuming that fracture initiates with the first drop in force in the experiments). Afriction coefficient of 0.05 has been used in the simulations of the punch experiment. Due to the symmetry of the mechanicalproblem, only one eighth of the tensile specimens is modeled and one quarter of the punch specimen. A user materialsubroutine is used to simulate the anisotropic material response as outlined by the constitutive equations above.

5.3. Plasticity model calibration and validation

The same parameters ai as those identified by Dunand et al. [22] are chosen to describe the shape of the anisotropic yieldsurface (Table 1). This set of parameters provides an accurate description of the Lankford ratio variations (compare solid linewith solid dots in Fig. 6a). The basis for their identification were the yield stresses and Lankford ratios as determined fromtension experiments along the 0�, 45� and 90� directions. In addition, the yield stresses from shear experiments with twodistinct specimen orientations had been used. In an attempt to improve the model response in the post-necking range (ascompared to Dunand et al. [22]), the isotropic strain hardening function has been recalibrated using Eq. (15) instead of aSwift law only. The parameters fAS; e0;nsg are found from a fit of the Swift law to the uniaxial stress–strain curves (whichare valid up to a strain of about 0.05). Analogously, the parameters fk0;Q ; bg are found from a fit of the Voce law. As differ-ences between the Swift and Voce laws greater than 5% in stress only become apparent for strains greater than 0.1, we chosethe central hole tension experiment along the extrusion direction (where equivalent plastic strains above 0.5 can be reached)to identify the weighting factor w through inverse analysis. For this, we coupled the Abaqus solver with the optimizationtoolbox of Matlab and made use of a Nelder-Mead algorithm (function fminsearch) to minimize the difference betweenthe simulated and measured force–displacement curves. The final parameters are summarized in Table 2 and the corre-sponding stress–strain curve is shown in Fig. 6b. The predictions of the force–displacement curves for all other experimentsare shown as solid black lines in Figs. 8 and 9. Reasonable agreement is observed for all specimens, with the largest discrep-ancies for the NT5 experiments.

5.4. Loading paths to fracture

The computed contour plots of the equivalent plastic strain within the specimen gage section at the instant of onset offracture initiation are shown for each specimen in Figs. 8 and 9. The loading paths to fracture are extracted at the locationwhere the highest equivalent plastic strain is observed. Recall that the stress state for an anisotropic material is definedthrough the normalized tensor r=�r. The evolution of the entire Cauchy stress tensor is therefore extracted in addition tothe anisotropic equivalent plastic strain. The normalized stress tensor has five independent components. The complete load-ing path to fracture therefore describes an evolution in a six-dimensional space. However, for visualization purposes, we onlyshow two-dimensional projections of these loading paths:

� Fig. 10a shows the projections of the loading paths onto the ðr11;r22Þ-plane. Neglecting the contributions of the otherstress tensor components is a reasonable assumption for the punch (PU) and tensile experiments (NT and CH) alongthe 0�- and 90�-directions.

(a) NT20-0 (b) NT20-45 (c) NT20-90

(d) NT10-0 (e) NT10-45 (f) NT10-90

(g) N5-0 (h) NT5-45 (i) NT5-90

Fig. 8. Force–displacement curves for notched tension as measured experimentally (solid dots) and extracted from FE analysis. The scale bar for the contourplots of the equivalent plastic strain distribution near the specimen centers is included in Fig. 9.

492 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

(a) CH-0 (b ) CH-45

(c) CH-90 (d) PU

0.0 0.6aε

0.3

Fig. 9. Force–displacement curves for (a–c) tension with a central hole and (d) punch testing as measured experimentally (solid dots) and extracted from FEanalysis.

Table 1Calibrated coefficients for linear transformation of stress tensor for the plasticity model.

a1 a2 a3 a4 a5 a6 a7

0.901 1.042 0.658 0.933 1.044 1.021 0.754a8 a9 a10 a11 a12 a13 a14

1.230 1 1 1 1 0.901 1.042

Table 2Calibrated parameters for the isotropic hardening law.

A (MPa) e0 (–) n (–) k0 (MPa) Q (MPa) b (–) w (–)

325 0.007 0.1 205.2 48.4 23.7 0.054

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 493

(a)

(b)

CH-0

NT-0NT-45

PU

CH-90

NT-90

PUNT-45

CH-45

Fig. 10. Loading paths to fracture. The ends of the solid lines correspond to the strains to fracture as determined through hybrid numerical–experimentalanalysis. The groups of three curves for NT correspond to different notch radii. The strains to fracture as predicted by the calibrated fracture initiation modelare shown as solid dots.

494 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

� Fig. 10b shows the projection of selected loading paths on the diagonal ðrd;r12Þ-plane. Note that the shear stress com-ponent plays an important role in the tensile experiments on 45�-specimens.

Due to the particular choice of coordinates, the results for an isotropic material would be symmetric with respect to thebisector line ðr11 ¼ r22Þ in Fig. 10a. However, as demonstrated by the comparison of the loading paths for NT-90 and NT-0,and the comparison of the loadings paths for CH-90 and CH-0 demonstrate, a very strong anisotropy is observed in the mate-rial’s fracture response. For example, the anisotropic equivalent plastic strain to fracture for uniaxial tension along the 0�-

direction �e fa ¼ 0:86

is about 90% higher than that for the 90�-direction �e f

a ¼ 0:45

.

5.5. Fracture initiation model calibration and validation

The parameters fa; b; c;ng and fM22;M44;M12g describing the respective isotropic and anisotropic parts of the Hosford–Coulomb fracture initiation model are identified through inverse analysis:

� Step1: Identification of seed parameters for fa; b; c;ng assuming an isotropic fracture response. We imposeM22 ¼ M44 ¼ 1and M12 ¼ 0 and run a first round of identification using the results for 0�-specimens only (NT5-0, NT10-0, NT20-0, CH-0,PU).

G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497 495

� Step 2: Full identification of all parameters fa; b; c;n;M22;M44;M12g using the results from eight experiments: CH-0, CH-45, CH-90, NT10-0, NT10-45, NT10-90 and PU. The optimization is performed using a derivative-free simplex algorithm(Matlab) which minimizes the difference between the strains to fracture predicted by Eq. (29) with those measuredexperimentally.

The final set of parameters obtained after more than 150 iterations is given in Table 3. The model predictions of the strainto fracture are highlighted as solid dots in Fig. 10a and b. The comparison of the end points of the solid loading paths (exper-imental strain to fracture) with the corresponding solid dots shows the accuracy of the model predictions for all experiments.The maximum relative difference of 17% is observed for the NT5-90 experiment.

Another way of validating the model predictions is to compare the computed displacements to fracture with those mea-sured experimentally. Fig. 11 shows this comparison demonstrating good agreement with a maximum relative difference of7.3% for the CH-90 experiment. A ðr11;r22Þ polar plot of the identified fracture envelope for proportional loading is shown inFig. 12 (black solid curve) next to the isotropic envelopes for c = 0 (blue dots) and c = 0.1 (red dots). The calibrated envelopeexhibits a pronounced compression–tension asymmetry (strain to fracture for uniaxial compression about 2.5 times higherthan that for uniaxial tension) which is due to the normal stress effect that is introduced through the non-zero frictioncoefficient c. At the same time, the linear transformation of the stress tensor argument ensures that the strain to fracturefor uniaxial tension along the extrusion direction is more than twice as large as that for the transverse direction.

5.6. Comment on anisotropic fracture initiation model by Luo et al. [42]

The proposed model features the same number of parameters as the anisotropic MMC model proposed by Luo et al. [42].The comparison of the model predictions (red and gray columns in Fig. 11) demonstrates that both models can fit the presentexperimental data equally well. However, the modeling approach is fundamentally different: the anisotropic MMC model isbased on the linear transformation of the plastic strain tensor, while the anisotropic HC model is based on the linear trans-formation of the stress tensor. The latter approach is pursued here for conceptual reasons. The underlying fracture initiationmechanism is shear localization at the microscale. It has been demonstrated by Dunand and Mohr [23] through localizationanalysis that the onset of shear and normal localization in a Levy–von Mises solid under proportional loading can bedescribed through a Hosford–Coulomb type of criterion in stress space. It is thus expected that the onset of shear localization

Table 3Calibrated parameters for the anisotropic Hosford–Coulomb fracture initiation model.

M12 (–) M22 (–) M44 (–) a (–) b (–) c (–) n (–)

�0.086 1.182 1.114 1.035 1.015 0.1017 0.223

Disp

lace

men

t to

frac

ture

[mm

]

CH NT20 NT10 NT5

0

0.2

0.4

0.6

0.8

1

1.2EXP

HC

MMC

Fig. 11. Comparison of the displacement to fracture obtained from experiments (blue bars) and predicted by the proposed anisotropic Hosford–Coulombmodel (red columns) and the anisotropic Modified Mohr–Coulomb model (gray columns). For the HC model, the maximum relative difference (7.3%) isobserved for tension of a specimen with a central hole along the transverse direction (CH-90). For the punch loading (not shown), the HC simulationoverestimates the displacement to fracture by 5.2%. (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)

isotropic (c=0)

isotropic (c=0.1)

anisotropic

Fig. 12. Polar plot of the identified fracture envelope showing the strong loading direction dependency and stress state dependency of the anisotropicequivalent plastic strain to fracture (radial coordinate) for extruded aluminum 6260-T6.

496 G. Gu, D. Mohr / Engineering Fracture Mechanics 147 (2015) 480–497

(and hence the onset of fracture) in an anisotropic metal can be described through an anisotropic criterion in stress space.Consequently, a linear transformation of the stress tensor to account for anisotropy appeared to be conceptually more soundthan transforming the strain tensor. A clear assessment of the predictive capabilities of these modeling approaches willrequire either more experimental data or results from localization analysis.

6. Conclusions

An anisotropic extension of the Hosford–Coulomb localization criterion is proposed to provide a phenomenological modelfor predicting the initiation of fracture in large scale simulations. Instead of using isotropic stress state measures (such as thestress triaxiality and the Lode angle parameter) along with the three principal stress orientations with respect to the materialdirections, the Cauchy stress tensor as normalized by the von Mises equivalent stress is directly used to describe the stressstate in an anisotropic material. Starting with the formulation of the isotropic Hosford–Coulomb model in terms of the nor-malized Cauchy stress tensor, the anisotropic extension is developed through the linear transformation of the stress tensorargument. The influence of the seven model parameters is discussed in detail before their identification for extruded alu-minum 6260-T6. The identification procedure involves the detailed finite element analysis of fracture experiments for thir-teen different stress states (Luo et al., 2013) to extract the loading paths to fracture. The final set of model parameters isobtained though computational optimization. It is shown that the model is able provide a satisfactory engineering approx-imation of the observed fracture strains which ranged from 0.22 for notched tension along the transverse direction to valuesas high as 0.88 for uniaxial tension along the extrusion direction.

Acknowledgements

The support through the French National Research Agency (Grant ANR-11-BS09-0008, LOTERIE) and the Industrial Frac-ture Consortium is gratefully acknowledged. Special thanks are due to Dr. Meng Luo and Dr. Matthieu Dunand for providingthe experimental data and the user subroutine for the plasticity model.

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