Stress state dependence of ductile damage and fracture behavior: Experiments and numerical simulations

Stress state dependence of ductile damage and fracture behavior: Experiments and numerical simulations

Engineering Fracture Mechanics 141 (2015) 152–169 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.els...

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Engineering Fracture Mechanics 141 (2015) 152–169

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Stress state dependence of ductile damage and fracture behavior: Experiments and numerical simulations Michael Brünig ⇑, Daniel Brenner, Steffen Gerke Institut für Mechanik und Statik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany

a r t i c l e

i n f o

Article history: Received 16 December 2014 Received in revised form 8 May 2015 Accepted 10 May 2015 Available online 21 May 2015 Keywords: Ductile materials Damage and fracture Stress state dependence Biaxial experiments Scanning electron microscope images Numerical simulations

a b s t r a c t The paper deals with new experiments and corresponding numerical simulations to study the effect of stress state on damage and fracture behavior of ductile metals. Different branches of ductile damage criteria are considered corresponding to various mechanisms depending on stress intensity, stress triaxiality and the Lode parameter. New experiments with two-dimensionally loaded specimens have been developed covering a wide range of stress triaxialities and Lode parameters in the tension, shear and compression domains. Scanning electron microscope (SEM) analyses of the fracture surfaces show various failure modes corresponding to different stress states detected by numerical simulations of the experiments. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Modeling and numerical simulation of inelastic behavior, damage and fracture of materials are important topics in engineering mechanics, for example, in analyses of complex structural components, in prediction of structural reliability, or in development and optimization of structural design. Based on many experimental and numerical studies it is nowadays evident that during loading of ductile metals inelastic deformations occur which are accompanied by damage and local failure mechanisms on micro- and meso-scales. The accumulation of these processes may lead to fracture of structural elements. The damage and failure mechanisms on the micro-level depend on stress state of the material sample. For example, under tension dominated stress conditions (high positive stress triaxialities) damage in ductile metals is mainly caused by nucleation, growth and coalescence of voids whereas under shear and compression dominated stress states (small positive or negative stress triaxialities) evolution of micro-shear-cracks is the predominant damage mechanism. Furthermore, combination of both basic mechanisms occurs for moderate positive stress triaxialities whereas no damage in ductile metals has been observed for finite negative stress states. Therefore, to be able to develop phenomenological ductile material models it is important to analyze in detail and to understand these stress-state-dependent processes and mechanisms of damage and fracture acting on different scales. Different damage models have been proposed in the literature based on experimental observations as well as on multi-scale approaches [1–5]. In addition, important aspects concerning the choice of mechanical variables characterizing the damage process as well as their experimental identification have been discussed [6]. For example, the advantage of isotropic damage models considering scalar variables is the simple theoretical framework and simple identification of one ⇑ Corresponding author. Tel.: +49 89 6004 3415; fax: +49 89 6004 4549. E-mail address: [email protected] (M. Brünig). http://dx.doi.org/10.1016/j.engfracmech.2015.05.022 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

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Nomenclature a=c hydrostatic stress coefficient c; c0 yield stresses E; G; K elastic material parameters g pl ; g da potential functions da f damage condition pl f yield condition F1; F2 applied forces H hardening modulus I1 ; I1 ; J2 ; J2 ; J 3 invariants of (deviatoric) stress tensors n hardening exponent Te 1 ; Te 2 ; Te 3 principal stresses u displacement Ael ; Ada strain tensors _ pl ; H _ H _ el ; H _ da strain rate tensors H; M; N; N el

Q; Q ; Q o



normalized stress tensors pl

metric transformation tensors

R; R; R e S

damage tensors

e T; T; T a; b   d a ; b; _ c_ ; l_ k;

stress tensors damage mode parameters damage rule parameters rate of internal variables Poisson’s ratio stress triaxiality elastic-damage moduli equivalent damage stress von Mises equivalent stress mean stress Lode parameter

m g g1 . . .g4 r req rm x

deviatoric stress-related tensor

single parameter [7–10]. However, practical applicability of these isotropic models is very limited because especially in ductile metals anisotropic damage effects occur with large inelastic deformations which cannot be simulated by an isotropic approach. Thus, anisotropic damage models based on tensorial variables have been proposed, for example, by [5,11–15]. However, their practical applicability may be limited by large number of material parameters and difficulties in their identification. Furthermore, on the numerical side there may be remarkable problems to implement these approaches in computer codes and, thus, it seems to be difficult to realistically predict deformation and failure behavior of materials and structures in engineering applications. Therefore, a generalized and thermodynamically consistent, phenomenological continuum damage model has been proposed [16–18] which has been implemented as user-defined material subroutines in commercial finite element programs allowing analyses of static and dynamic problems in differently loaded metal specimens. The continuum model is based on kinematic definition of tensorial damage variables and considers various damaged and corresponding undamaged configurations where respective yield and stress-state-dependent damage criteria as well as constitutive rate equations are formulated. Furthermore, information on stress-state-dependent damage and failure mechanisms can be obtained by numerical simulations on the micro-level taking into account a large range of different loading conditions [19–26]. A distinct advantage of this microscopic approach is the detailed consideration of individual behavior of voids and micro-shear-cracks as well as their coalescence and accumulation to macro-cracks. With these numerical analyses taking into account a large range of stress states it was possible to detect different damage and fracture mechanisms which have not been exposed by experiments. In addition, the numerical results allowed proposal of equations for damage and fracture criteria as well as damage evolution laws showing remarkable dependence on stress triaxiality and – especially in regions with small or negative triaxialities – additional dependence on the Lode parameter or third deviatoric stress invariant [25,26]. However, the proposed stress-state-dependent criteria and evolution equations for damage and failure as well as the associated identification of material parameters are only based on numerical analyses on the micro-level whereas experimental validation is still required. In general, constitutive parameters of continuum models have to be identified by experiments with carefully designed specimens. For example, elastic material properties, yield stress and coefficients characterizing plastic hardening are

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determined from one-dimensional tension tests with smooth cylindrical or flat rectangular specimens. In addition, to be able to detect stress triaxiality dependence of the constitutive equations tension tests with differently pre-notched specimens and corresponding numerical simulations have been used [16,17,27–32]. These experiments with unnotched and differently notched flat specimens showed stress triaxialities (ratio of mean stress and von Mises equivalent stress: g ¼ rm =req ) between 0.33 and 0.6 which is only a small region in the positive triaxiality range. Larger values appear in tension tests with cylindrical (axi-symmetric) specimens but these can not be manufactured when the behavior of thin sheets is investigated. Therefore, it is necessary to develop new series of experiments with flat specimens where stress triaxialities larger than 0.6 will occur. Moreover, specimens with new geometries have been designed to be able to analyze stress states with nearly zero stress triaxiality. These specimens have been uniaxially loaded in tension tests leading to shear mechanisms in their centers [16,28,31,33]. However, in these experiments with specimens taken from thick sheets (more than 3 mm thickness) rotation of the center has been observed leading to increase in additional tension stresses which causes increase in stress triaxiality. To minimize this effect, additional notches in thickness direction have been proposed leading to earlier damage and failure of the specimens with dominant shear stress states in the central part [16,33]. Alternatively, shear-off-tests (punch tests) have been performed to investigate shear stress states [34]. Furthermore, to be able to take into account other ranges of stress states butterfly specimens have been developed [30,32,35] which can be uniaxially tested in different directions using special experimental equipment. Corresponding numerical simulations showed stress triaxialities between 0.33 and 0.60 in the critical regions. Moreover, large values of stress triaxialities can be taken into account performing combined tension–torsion experiments on notched hollow cylinders [31,36–40]. However, these tests are only possible for structures with moderate thickness but not for thin sheets. Therefore, in the case of thin structures, it is necessary to develop series of alternative experiments with new flat specimens taken from thin sheets which will reveal information on inelastic behavior, damage and fracture of ductile metals for a wide range of stress states not obtained by the experiments discussed above. For example, two-dimensional (2D) experiments and different geometries of cruciform specimens have been proposed [41–44] to analyze the anisotropic plastic behavior or martensite formation of metal sheets under various loading paths. The specimens have been tested in biaxial machines under biaxial planar loading conditions. However, further aspects for specimen’s design have to be taken into account to investigate stress-state-dependent ductile damage and fracture mechanisms. Thus, geometry of specimens undergoing shear-tension or shear-compression loading conditions is discussed in the present paper. These new specimens are tested in a biaxial machine and their deformation behavior as well as their ductile damage and fracture mechanisms are analyzed in detail. The present paper is organized as follows: Main ideas and equations of the phenomenological continuum damage model proposed by Brünig [5] are briefly discussed. Experiments on inelastic, damage and fracture behavior of an aluminum alloy are conducted. Results of uniaxial tests with smooth and pre-notched specimens are used to identify basic elastic–plastic material parameters as well as onset of damage. Furthermore, newly developed biaxial experiments with 2D-specimens up to final fracture will be discussed in detail. These combined shear-tension and shear-compression tests cover a wide range of stress states and will show different damage and fracture modes. Scanning electron microscope (SEM) analyses of fracture surfaces are also performed to detect the underlying stress-state-dependent mechanisms of ductile failure. Corresponding numerical simulations of these biaxial experiments will reveal different stress states in critical regions. Experimental and numerical data are used to propose and to validate criteria and evolution equations characterizing damage and fracture as well as to identify corresponding constitutive parameters.

2. Continuum damage model Large deformations as well as anisotropic damage and failure of ductile metals are predicted by the continuum model [5] which is based on experimental results and observations as well as on numerical simulations on the micro-level detecting information of microscopic mechanisms due to individual micro-defects and their interactions. The phenomenological approach is based on the introduction of damaged and corresponding fictitious undamaged configurations and has been implemented into finite element programs. The thermodynamically consistent framework is based on kinematic description of damage leading to definition of damage strain tensors. These tensors take into account isotropic as well as anisotropic effects providing realistic representation of ductile material degradation. In addition, free energy functions with respect to the undamaged and damaged configurations are introduced leading to elastic laws which are affected by increasing damage. Plastic behavior is governed by a yield condition and a flow rule and, in a similar way, damage behavior is characterized by a damage condition and a damage rule. An extended version of the continuum damage model [16,17] proposes a stress-state-dependent damage criterion based on experimental results of different tension and shear tests with smooth and pre-notched specimens and data from corresponding numerical simulations. Furthermore, the continuum approach takes into account numerical results of various analyses using unit cell models [24–26]. Based on these numerical results covering a wide range of stress states it was possible to propose damage equations as functions of the stress triaxiality and the Lode parameter and to estimate micro-mechanically based material parameters. However, these stress-state-dependent constitutive equations are mainly based on numerical calculations. Therefore, further experiments covering the considered stress states have to be performed which is the main purpose of the present investigation to be able to validate the continuum model.

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2.1. Kinematics The kinematic approach is based on the introduction of initial, current and elastically unloaded configurations each defined as damaged and fictitious undamaged configurations, respectively [5]. This leads to the multiplicative decomposition of the metric transformation tensor o



 pl R Q el Q ¼ R1 Q

ð1Þ o

 pl denotes the where Q describes the total deformation of the material body after loading, R represents the initial damage, Q 

plastic deformation of the fictitious undamaged body, R characterizes the deformation induced by evolution of damage and Q el represents the elastic deformation of the material body. In addition, different logarithmic strain tensors are defined: the elastic strain tensor

Ael ¼

1 ln Q el 2

ð2Þ

and the damage strain tensor

Ada ¼

 1 ln R 2

ð3Þ

describing the macroscopic deformation behavior caused by nucleation, growth and coalescence of micro-defects. Furthermore, the strain rate tensor

_ ¼ 1 Q 1 Q_ H 2

ð4Þ

is introduced, which using Eq. (1) can be additively decomposed

_ ¼H _ el þ R1 H _ da Q el _ pl R þ Q el1 H H

ð5Þ

into the elastic

_ el ¼ 1 Q el1 Q_ el ; H 2

ð6Þ

the plastic

 pl1 Q _ pl Q el ; _ pl ¼ 1 Q el1 Q H 2

ð7Þ

and the damage strain rate tensor  _ _ da ¼ 1 R1 R: H 2

ð8Þ

With the identity (see [5] for further details) 

R Q el ¼ Q el R

ð9Þ

the damage metric transformation tensor with respect to the current loaded configurations, R (see Eq. (5)), has been introduced. 2.2. Constitutive equations The effective undamaged configurations are considered to characterize the behavior of the undamaged matrix material. In particular, the elastic behavior of the undamaged matrix material is described by an isotropic hyperelastic law leading to the effective Kirchhoff stress tensor

   ¼ 2GAel þ K  2 G trAel 1 T 3

ð10Þ

where G and K represent the constant shear and bulk modulus of the undamaged matrix material, respectively. In addition, onset and continuation of plastic flow of ductile metals is determined by the yield condition

f

 pl   I1 ; J 2 ; c

¼

qffiffiffiffi   J 2  c 1  a I1 ¼ 0; c

ð11Þ

 and J 2 ¼ 1 devT   devT  denote invariants of the effective stress tensor T;  c is the strength coefficient of the where I1 ¼ trT 2 matrix material and a represents the hydrostatic stress coefficient.

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It has been observed in many experiments with ductile metals that plastic deformations are nearly isochoric. Thus, the plastic potential function

 ¼ g pl ðTÞ

qffiffiffiffi J 2

ð12Þ

is assumed to depend only on the second invariant of the effective stress deviator leading to the non-associated isochoric effective plastic strain rate pl

1 _ pl ¼ k_ @g ¼ k_ p  ¼ c_ N:  ffiffiffiffi devT H  @T 2 J 2

ð13Þ

 ¼ p1ffiffiffiffiffi devT  denotes the normalized deviatoric stress tensor and In Eq. (13) k_ is a non-negative scalar-valued factor, N  2J 2

 H _ pl ¼ p1ffiffi k_ represents the equivalent plastic strain rate measure used in the present continuum model. c_ ¼ N 2 Furthermore, the damaged configurations are considered to characterize the behavior of anisotropically damaged material samples. In particular, since damage remarkably affects the elastic behavior and leads to material softening, the elastic law of the damaged material sample is expressed in terms of both the elastic and the damage strain tensor, (2) and (3). The Kirchhoff stress tensor of the ductile damaged metal is given by

     

  2 T ¼ 2 G þ g2 trAda Ael þ K  G þ 2g1 trAda trAel þ g3 Ada  Ael 1 þ g3 trAel Ada þ g4 Ael Ada þ Ada Ael 3

ð14Þ

where the parameters g1 . . . g4 describe the deteriorating effect of damage on the elastic material properties. In addition, onset and continuation of damage is characterized using the concept of damage surface formulated in stress space at the macroscopic damaged continuum level. Following [25], the damage condition

f

da

pffiffiffiffi ¼ aI1 þ b J 2  r ¼ 0

ð15Þ

is formulated in terms of the stress invariants I1 ¼ trT and J 2 ¼  devT and the damage threshold r. In Eq. (15) the variables a and b represent damage mode parameters depending on the stress intensity (von Mises equivalent stress) pffiffiffiffiffiffiffi req ¼ 3J2 , the stress triaxiality 1 devT 2



rm I1 ¼ pffiffiffiffiffiffiffi req 3 3J2

with the mean stress



ð16Þ

rm ¼ 1=3I1 as well as on the Lode parameter

2 Te 2  Te 1  Te 3 with Te 1 P Te 2 P Te 3 Te 1  Te 3

ð17Þ

e1; T e 2 and T e3. expressed in terms of the principal stress components T Furthermore, evolution of macroscopic irreversible strains caused by the simultaneous nucleation, growth and coalescence of different micro-defects is modeled by a stress-state-dependent damage rule. Thus, the damage potential function

e ¼ g da ðI1 ; J ; J Þ g da ð TÞ 2 3

ð18Þ

e represents the stress tensor formulated in the damaged configuration which is work-conjugate to the is introduced where T _ da (see [5] for further details) and I1 ; J and J are corresponding invariants. This leads to the damdamage strain rate tensor H 2 3 age strain rate tensor

 da  da da da _ da ¼ l_ @g ¼ l_ @g 1 þ @g dev T e þ @g dev e S H e @I1 @J 2 @J 3 @T

ð19Þ

where l_ is a non-negative scalar-valued factor and

e e  2J 1 T dev e S ¼ dev Tdev 3 2

ð20Þ

represents the second order deviatoric stress tensor. Alternatively, the damage strain rate tensor (19) can be written in the form

  1 _ da ¼ l_ a  þ dM  pffiffiffi 1 þ bN H 3

ð21Þ

1 ffiffiffi e and M ¼ where the normalized deviatoric tensors N ¼ p dev T 2

J2

1

kdeve Sk

dev e S have been used. In Eq. (21) the

 and  ; b d are kinematic variables describing the portion of volumetric and isochoric stress-state-dependent parameters a damage-based deformations. The damage rule (21) takes into account isotropic and anisotropic parts corresponding to

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isotropic growth of voids and anisotropic evolution of micro-shear-cracks, respectively. Therefore, both basic damage mechanisms (growth of isotropic voids and evolution of micro-shear-cracks) acting on the micro-level are involved in the damage rule (21). In the isotropic case, where only isotropic growth of voids is considered, the parameter l_ corresponds to the rate of void volume fraction f_ used in damage approaches based on the Gurson model whereas in the anisotropic case, when void growth and formation of micro-shear-cracks are simultaneously active, the parameter l_ can only be seen as the rate of the scalar-valued internal damage variable denoting the rate of an equivalent damage strain. Similar ideas have been published in [45] discussing modification of the void volume fraction f incorporating a parameter which characterizes damage growth under low stress triaxiality conditions. In this phenomenologically motivated enhancement the parameter f no longer denotes the current void volume fraction and can only be seen as a damage parameter without further physical interpretation. In many engineering fracture approaches, critical values of scalar damage variables are used to characterize onset of ductile fracture. For example, the critical void volume fraction f cr is used in the extended Gurson model [46] and works well for tension loading as long as f denotes the void volume fraction (see discussion above). Alternatively, onset of macro-cracking is identified by the critical micro-defect area fraction based on Lemaitre’s isotropic continuum damage model [3,29]. However, characterization of onset of ductile fracture as a result of accumulation of growth and coalescence of different stress-state-dependent micro-defects requires a more sophisticated fracture criterion. It must take into account the current amount and orientation of anisotropic damage, for example expressed in terms of the damage strains Ada (3) introduced in the present continuum model. It is supposed that fracture conditions of the form fr

f ðAda ; afr Þ ¼ 0

ð22Þ

can adequately model onset of macro-cracking where afr represents the fracture threshold identified, for example, by numerical calculations on the micro-level. However, proposal of a fracture criterion of type (22) needs further investigation and is not subject of the present paper, it will be reported in a forthcoming one. 3. Identification of material parameters Identification of material parameters appearing in elastic–plastic-damage models is not an easy task. Only few parameters can be identified directly, other ones have to be determined by inverse procedures whereas those parameters of the phenomenological approach which are related to microscopic mechanisms can be obtained by interpretation of results taken from numerical calculations on the micro-level. Basic elastic–plastic material parameters are identified using experimental results from uniaxial tension tests with unnotched flat, bone-shape specimens manufactured from sheets with 4 mm thickness. For the aluminum alloy of the series 2017 investigated in the present paper, fitting of numerical curves and experimental data leads to Young’s modulus E = 65,000 MPa and Poisson’s ratio is taken to be m ¼ 0:3. In addition, from these uniaxial tension tests with smooth specimens equivalent stress–equivalent plastic strain curves are easily obtained from load–displacement curves as long as the uniaxial stress field remains homogeneous between the clip gauges fixed on the specimens during the experiments. In these tests no necking was observed up to the equivalent plastic strain c ¼ 0:15 and a smooth extrapolation has been used after this point. It is worthy to note that equivalent stress – equivalent plastic strain curves especially in the large inelastic deformation range have to be modeled accurately because numerical results for the inelastic material response for multi-axial loading conditions as well as numerical prediction of localization phenomena are very sensitive to the current plastic hardening modulus taken into account in the numerical simulations. For the aluminum alloy under investigation, the power law function for the equivalent stress–equivalent plastic strain function appearing in the yield criterion (11)

 c ¼ c0

Hc þ1 nc0

n ð23Þ

is used to model the work-hardening behavior. Good agreement of experimental data and numerical results is achieved for the initial yield strength c0 ¼ 174 MPa, the hardening modulus H = 1800 MPa and the hardening exponent n = 0.23. Furthermore, various parameters corresponding to the damage part of the constitutive model have to be identified. For example, onset of damage is determined by comparison of experimental load–displacement curves of the uniaxial tension tests with those predicted by numerical analyses based on the elastic–plastic model only [17,47]. In this procedure, the elastic–plastic finite element analysis takes into account geometry of the unnotched flat bone-shape specimen, clamped boundary conditions corresponding to stretching in the tension machine as well as the elastic–plastic constitutive law discussed above. Thus, elastic–plastic deformation and necking behavior during elongation of the specimen can be seen to be simulated in an accurate manner. First deviation of the experimental and numerically predicted curves is taken to be an indicator of onset of deformation-induced damage in the proposed phenomenological approach. Using this procedure, the damage threshold appearing in the damage condition (15) is r ¼ 295 MPa for the aluminum alloy under investigation. However, identification of further damage parameters is more complicated because most criteria and evolution equations are motivated by observations on the micro-level. Therefore, stress state dependence of the parameters in the damage condition (15) and in the damage rule (21) has been studied in detail performing numerical simulations with

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micro-void-containing representative volume elements under various loading conditions [25,26]. Based on these unit-cell calculations taking into account a wide range of stress triaxiality coefficients g and Lode parameters x different damage mode parameters have been identified for the aluminum alloy. In particular, the damage mode parameter a (see Eq. (15)) is given by

(

aðgÞ ¼

0 for 1 3

1 3

6g60

ð24Þ

for g > 0

whereas also based on further numerical calculations on the macro-level b is taken to be the non-negative function

bðg; xÞ ¼ b0 ðg; x ¼ 0Þ þ bx ðxÞ P 0;

ð25Þ

b0 ðgÞ ¼ 1:28g þ 0:85

ð26Þ

bx ðxÞ ¼ 0:017x3  0:065x2  0:078x:

ð27Þ

with

and

 and  ; b d in the damage rule Moreover, based on the results of the unit cell analyses the stress-state-dependent parameters a  (21) have been identified. The non-negative parameter a P 0 characterizing the amount of volumetric damage strain rates caused by isotropic growth of micro-defects is given by the relation

8 0 > > > > > > < 0:07903 þ 0:80117g a ðgÞ ¼ 0:49428 þ 0:22786g > > > 0:87500 þ 0:03750g > > > : 1

for g < 0:09864 for 0:09864 6 g 6 1 : for 1 < g 6 2

ð28Þ

for 2 < g 6 3:33333 for g > 3:33333

 is high for high stress triaxialities, smaller for moderate ones and zero for negative triaxialities. Dependence The parameter a on the Lode parameter x has not been revealed by the numerical simulations on the micro-scale. In addition, the parameter  characterizing the amount of anisotropic isochoric damage strain rates caused by evolution of micro-shear-cracks is given b by the relation

 g ; xÞ ¼ b 0 ðgÞ þ f ðgÞb  x ð xÞ bð b

ð29Þ

with

 g; xÞ ¼ bð

8 0:94840 þ 0:11965g þ ð0:0252 þ 0:0378gÞð1  x2 Þ for > > > > 2 > > < 1:14432  0:46810g þ ð0:0252 þ 0:0378gÞð1  x Þ for

g 6 13 g 6 23 for g62 for 2 < g 6 10 3 for g > 10 3

1:14432  0:46810g > > > 0:52030  0:15609g > > > : 0

1 6 3 1 < 3 2 < 3

ð30Þ

The parameter  d also corresponding to the anisotropic damage strain rates caused by the formation of micro-shear-cracks is given by the relation

dðg; xÞ ¼ f ðgÞdx ðxÞ d

ð31Þ

with

( dðg; xÞ ¼

ð0:12936 þ 0:19404gÞð1  x2 Þ for 0

1 3

6 g 6 23

for g > 23

ð32Þ

d only exists for small stress triaxialities and mainly depends on the Lode parameter It is worthy to note that this parameter 

x. Although its magnitude is small in comparison to the parameters a and b its effect on the evolution of macroscopic dam_ da is not marginal. More details on these functions (24)–(32), their visualization and interpretation are age strain rates H given in [25]. In summary, the material dependent parameters used in the numerical simulations are listed in Table 1. It is worthy to note that the damage related functions (24)–(32) are only based on numerical calculations on the micro-level and their validation has to be realized by experiments also covering a wide range of stress states. This is not possible by the uniaxial tests mentioned above and, therefore, new experiments have to be developed.

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4. Experiments with biaxially loaded specimens A new experimental program has been developed to propose new tests revealing the effect of stress state on inelastic behavior, damage and fracture in ductile metals. The experiments are performed using the biaxial test machine type LFM-BIAX 20 kN (produced by Walter & Bai, Switzerland) shown in Fig. 1. It contains four electro-mechanically, individually driven cylinders with load maxima and minima of 20 kN (tension and compression loading is possible). The respective specimens are fixed in the four heads of the cylinders where clamped or hinged boundary conditions are possible. In addition, Fig. 2 shows the geometry of the newly designed flat specimens as well as the loading conditions. In the center of the specimen a notch in thickness direction has been milled leading here to high stress parameters and localization of inelastic deformations during the tests. The geometry is similar to that one proposed in [33] where the specimens have been uniaxially loaded firstly in tension and after unloading subsequently in shear tests or vice versa, whereas in the experimental program discussed in the present paper the specimens are simultaneously loaded in both directions with F 1 and F 2 . In particular, the load F 1 leads to shear mechanisms in the center of the specimen and simultaneous loading with F 2 leads to superimposed tension or compression modes. These loads cause combined shear-tension or shear-compression deformation and failure mechanisms in the center of the specimen. This extension of former experimental work covers the full range of stress states corresponding to the damage and failure mechanisms discussed above with focus on low positive, nearly zero and negative stress triaxialities where the Lode parameter also plays an important role. First experimental results and corresponding numerical simulations have been presented in [47,48] to be able to validate the proposed damage criterion (15). In the present paper, the focus is on further evolution of damage for different loading conditions and the final fracture process. In this context, scanning electron microscope (SEM) analysis of fracture surfaces of these tests are also conducted to reveal the stress-state-dependent microscopic mechanisms of ductile damage and fracture and to validate the proposed continuum model. Since the stress states in the notched central part of the specimen are not expected to be homogeneous for the investigated loading conditions, three parts of the failed surface (see Fig. 3) are extracted. They are carefully analyzed microscopically with emphasis on the relation of the fracture modes and the stress triaxiality proposed by corresponding numerical simulation of the respective experiments. For the SEM images the amplification factor of 200 is used. 5. Experimental and numerical results Experimental and numerical results for the load ratios F 1 : F 2 ¼ 1 : 1; F 1 : F 2 ¼ 1 : 0 and F 1 : F 2 ¼ 1 : 1 will be shown and discussed. In particular, F 1 : F 2 ¼ 1 : 1 leads to shear with superimposed tension, F 1 : F 2 ¼ 1 : 0 characterizes nearly pure shear in the critical zone whereas F 1 : F 2 ¼ 1 : 1 leads to shear with superimposed compression. These tests will lead to different stress states corresponding to different damage and fracture mechanisms which will be analyzed in detail. In the experiments with the biaxially loaded specimen (Fig. 2) the stress and strain fields are not homogeneous due to the specimen’s geometry and only averaged quantities can be evaluated from these tests. Thus, corresponding numerical calculations have been performed to be able to get detailed information on distributions and amounts of different stress and strain Table 1 Material dependent parameters. E [MPa]

m

co [MPa]

a/c [MPa1]

H [MPa]

n

65,000

0.3

174

0.

1800

0.23

g1 ½MPa

g2 ½MPa

g3 ½MPa

g4 ½MPa

r ½MPa

100,000

50,000

12,500

2500

295

Fig. 1. Biaxial test machine.

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Fig. 2. Specimen and loading conditions.

Fig. 3. Parts A, B and C of the fracture surface.

measures especially in critical regions. The finite element analyses have been carried out using the program ANSYS enhanced by a user-defined material subroutine based on the continuum model discussed above. The specimen is discretized by 42,248 eight-node-elements of type Solid185. Remarkable refinement of the finite element mesh is shown in Fig. 4 in the central part of the specimen where high gradients of the stress and strain quantities are expected to occur.

5.1. Load ratio F 1 : F 2 ¼ 1 : 1 Deformation and failure behavior of the new specimen under shear-tension loading is investigated using experiments including SEM analyses and corresponding numerical simulations based on the continuum model discussed above. For example, for the load ratio F 1 : F 2 ¼ 1 : 1 the load–displacement curves F 1  u1 based on experiments and the corresponding numerical simulation are shown in Fig. 5 where u1 is the displacement of the grips. After elastic loading a large region with inelastic behavior including hardening occurs. Final fracture of the specimens occurred at u1 ¼ 0:69 mm but no remarkable softening behavior can be observed in the load–displacement curves before fracture. In addition, Fig. 5 shows numerically predicted curves taking into account the elastic–plastic model only and based on the continuum damage model proposed in the present paper, respectively. The diagram shows excellent agreement of experimental and numerical curves. Only small differences in the numerically predicted curves can be observed caused by localized occurrence of damage only in the central part of the specimen leading to slightly smaller loads. Therefore, the damage model does not seem to be important for the prediction of the load–displacement behavior of the biaxially loaded specimen (Fig. 2) but it is important to simulate localized failure behavior in the critical part of the specimen. Different parameters on the top surface in the notched part of the specimen are visualized in Fig. 6 at the end of the numerical calculation shortly before fracture occurs. In particular, the first stress invariant I1 shows its maxima with pffiffiffiffi 823 MPa at the boundaries of the notch of the specimen whereas the second deviatoric stress invariant J 2 has its maximum with 348 MPa in the center of the notch. This leads to different stress triaxialities g which are positive but nearly 0 only in the center and 0:165 6 g 6 0:33 in the remaining notched region with only few points with higher and lower values at the boundaries corresponding to the maxima and minima of the first stress invariant I1 . This distribution of stress triaxialities g will lead to mixed damage modes (2) with simultaneous growth of voids and formation of micro-shear-cracks. Fig. 6 clearly shows that occurrence of numerically predicted damage is restricted to the yellow zone in the notched part of the specimen where the equivalent damage strain l shows its maxima (about l ¼ 9%) at the boundaries of the notch but also about l ¼ 6% in the band between these points. This band of moderate equivalent damage strains nicely corresponds to the

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Fig. 4. Finite element mesh of the 2D specimen.

F1 [kN] 1.5

Experiment Simulation (elastic-plastic) Simulation (with damage)

1

0.5

0

u1 [mm] 0.25

0.5

0.75

Fig. 5. Load–displacement curves (F 1 : F 2 ¼ 1 : 1).

Fig. 6. First stress invariant I1 [MPa], second deviatoric stress invariant experimental fracture line (F 1 : F 2 ¼ 1 : 1).

pffiffiffiffi J2 [MPa], stress triaxiality g, damage mode, equivalent damage strain

l,

fracture line of the tested specimen also shown in Fig. 6. In addition, the numerically predicted geometry of the deformed specimen also corresponds to the geometry of the tested specimen shown in the photo. The fracture surfaces of the specimens loaded with F 1 : F 2 ¼ 1 : 1 are shown in the SEM images in Fig. 7. In particular, images of the upper (A) and lower sections (C) mainly show isotropic growth of voids as well as small effect of shear mechanisms whereas in the middle section (B) of the fracture surface, smaller pores can be seen with more predominant influence of shear. This corresponds to the numerically predicted stress triaxialities g shown in Fig. 6. For moderate stress triaxialities 0:25 6 g 6 0:5 damage is due to simultaneous pronounced growth of voids and small tendency of formation of micro-shear-cracks (regions (A) and (C) in Fig. 7) whereas for small stress triaxialities 0:0 6 g 6 0:25 damage is caused by

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Fig. 7. SEM images of the fracture surface (F 1 : F 2 ¼ 1 : 1).

small growth of voids and more predominant formation of micro-shear-cracks. It should be noted that the additional shear mechanisms which can be seen in Fig. 7(C) are caused by shearing of protruded parts of the surface during unloading after fracture occurred. This behavior was also observed in movies taken during the experiments. In conclusion, for the load ratio F 1 : F 2 ¼ 1 : 1 typical shear-tension mechanisms have been observed in the experiments in the central region of the biaxially loaded specimen. The experimental results correspond well with the numerical ones and many details of the ductile damage and fracture process have been revealed by the numerical simulations based on the proposed continuum model.

5.2. Load ratio F 1 : F 2 ¼ 1 : 0 Deformation and failure behavior of the new specimen under shear loading is examined next. Fig. 8 shows for the load ratio F 1 : F 2 ¼ 1 : 0 the load–displacement curves F 1  u1 based on experiments and the corresponding numerical simulations. In particular, after elastic loading a large region with inelastic behavior including hardening is again observed. Final fracture of the specimens occurred at u1 ¼ 1:10 mm but no remarkable softening behavior can be observed in the load–displacement curves of the experiments before fracture. The load–displacement curves of the corresponding numerical simulations based on the elastic–plastic as well as on the continuum damage model are also shown in Fig. 8 and the diagram shows good agreement of experimental and numerical results. The numerically predicted curves again only show small differences because localized formation of damage does not remarkably affect the global load–displacement behavior. However, consideration of damage is important to predict localized failure behavior in an accurate manner.

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F1 [kN] 2

1.5

Experiment Simulation (elastic-plastic)

1

Simulation (with damage) 0.5

0

u1 [mm] 0.25

0.5

0.75

1

Fig. 8. Load–displacement curves (F 1 : F 2 ¼ 1 : 0).

Fig. 9. First stress invariant I1 [MPa], second deviatoric stress invariant experimental fracture line (F 1 : F 2 ¼ 1 : 0).

pffiffiffiffi J2 [MPa], stress triaxiality g, damage mode, equivalent damage strain

l,

Fig. 9 shows distribution of different characteristic stress, strain and failure parameters on the top surface in the notched part of the specimen at the end of the numerical calculation shortly before fracture occurs. In particular, the first stress invariant I1 shows regions with large values up to I1 ¼ 741 MPa at the boundaries of the notch of the specimen whereas pffiffiffiffi the second deviatoric stress invariant has its maximum J 2 ¼ 390 MPa in the center of the notch but only pffiffiffiffi J 2 ¼ 220 MPa at the boundaries of the notch. This leads to different stress triaxialities g which are negative but nearly 0 in the center, 0:0 6 g 6 0:165 in the horizontal band and 0:165 6 g 6 0:33 in the remaining notched region near the boundaries of the notch, only few points show higher and lower values at the boundaries corresponding to the maxima and minima of the first stress invariant I1 caused by the deformation of the central part of the specimen. This distribution of stress triaxialities g will lead to a band with mixed damage modes (2) with simultaneous growth of voids and formation of micro-shear-cracks. However, in the center of the specimen a region with anisotropic damage (3) only caused by formation of micro-shear-cracks is numerically predicted. Thus, only in these zones damage is predicted to occur and the corresponding equivalent damage strain l is also shown in Fig. 9. This equivalent damage strain shows its maxima (about l ¼ 8%) at the boundaries of the notch but also small positive values in the band between these points. The fracture line of the tested specimens also shown in Fig. 9 runs through regions with damage mode (3) with elongation to the boundaries. In addition, the numerically predicted geometry of the deformed specimen again corresponds to the geometry of the failed specimen shown in the photo. The fracture surfaces of the failed specimens loaded with F 1 : F 2 ¼ 1 : 0 are shown in the SEM images in Fig. 10. In particular, images of the upper section (A) show small isotropic growth of voids as well as effects of shear mechanisms whereas in the middle section (B) of the fracture surface, nearly no voids and mainly shear mechanisms are observed. In the lower section (C), again some small voids can be seen in combination with formation of shear. This behavior corresponds to the numerically predicted stress triaxialities g shown in Fig. 9. For small positive stress triaxialities 0:1 6 g 6 0:3 damage is due to simultaneous growth of voids and formation of micro-shear-cracks (regions (A) and (C) in Fig. 10) whereas for small positive or negative stress triaxialities 0:15 6 g 6 0:1 damage is caused by small or marginal growth of voids and more pronounced formation of micro-shear-cracks. Compared with the SEM images based on the specimens loaded by F 1 : F 2 ¼ 1 : 1 shown in Fig. 7 where formation of larger pores has been observed the voids are much smaller after the experiments with F 1 : F 2 ¼ 1 : 0 (Fig. 10). This corresponds well to the assumed correlation of the stress triaxiality g and the

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Fig. 10. SEM images of the fracture surface (F 1 : F 2 ¼ 1 : 0).

damage modes taken into account in the mechanical model discussed above that hydrostatic tension will lead to growth of voids whereas isochoric shear will lead to formation of micro-shear-cracks. In conclusion, for the load ratio F 1 : F 2 ¼ 1 : 0 typical shear mechanisms have been observed in the experiments in the central region of the biaxially loaded specimen. The experimental results correspond well with the numerical ones and, again, many details of the ductile damage and fracture process have been revealed by the numerical simulations based on the proposed continuum model. 5.3. Load ratio F 1 : F 2 ¼ 1 : 1 Deformation and failure behavior of the new specimen under shear-compression loading is discussed in this subsection. Fig. 11 shows for the load ratio F 1 : F 2 ¼ 1 : 1 the load–displacement curves F 1  u1 based on experiments and the corresponding numerical simulation. In particular, after elastic loading a large region with inelastic behavior including hardening and some scattering effects during the fracture process are observed in the experiments. Final fracture of the specimens occurred at approximately u1 ¼ 0:75 mm. The load–displacement curves of the corresponding numerical simulations based on the elastic–plastic as well as on the continuum damage model are also shown in Fig. 11 and the diagram shows good agreement of experimental and numerical results. The numerically predicted curves again only show small differences because localized formation of damage does not remarkably affect the global load–displacement behavior. However, consideration of damage is important to predict localized failure behavior in an accurate manner. Distribution of different characteristic stress, strain and failure parameters on the top surface in the notched part of the specimen at the end of the numerical calculation shortly before fracture occurs is shown in Fig. 12. In particular, the first

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F1 [kN] 1.5

Experiment

1

Simulation (elastic-plastic) Simulation (with damage)

0.5

0

u1 [mm] 0.25

0.5

0.75

Fig. 11. Load–displacement curves (F 1 : F 2 ¼ 1 : 1).

Fig. 12. First stress invariant I1 [MPa], second deviatoric stress invariant experimental fracture line (F 1 : F 2 ¼ 1 : 1).

pffiffiffiffi J2 [MPa], stress triaxiality g, damage mode, equivalent damage strain

l,

stress invariant I1 shows regions with large positive values up to I1 ¼ 470 MPa as well as points with very large negative values up to I1 ¼ 920 MPa at the boundaries of the notch of the specimen. In addition, the second deviatoric stress invariant pffiffiffiffi has its maximum J 2 ¼ 350 MPa again in the center of the notch. This leads to different stress triaxialities g with a small band of negative values 0:33 6 g 6 0:165 surrounded by a region with smaller negative triaxialities up to nearly 0. Positive or higher negative values can be seen in small regions at the boundaries corresponding to the points with extreme values of the first stress invariant I1 . This distribution of stress triaxialities g will lead to a band with anisotropic damage mode (3) only caused by formation of micro-shear-cracks. Only in this band damage is predicted to occur and corresponding equivalent damage strains l are also shown in Fig. 12. Values up to l ¼ 3% can be seen in points at the boundaries which are connected by a band of small equivalent damage strains. The fracture line of the tested specimen also shown in Fig. 12 runs through the points of extreme values of l and the connecting band. In addition, the numerically predicted geometry of the deformed specimen also corresponds to the geometry of the tested specimen which can be seen in the photo. The fracture surfaces of the failed specimens loaded with F 1 : F 2 ¼ 1 : 1 are shown in the SEM images in Fig. 13. In particular, images of all sections (A–C) of the fracture surface mainly show shear mechanisms and only few very small voids. It is supposed that these voids were initial micro-defects which have been compressed during loading of the specimen. These images again correspond to the numerically predicted stress triaxialities g shown in Fig. 12 and the assumption that in regions with negative stress triaxialities damage is caused by formation of micro-shear-cracks. Compared with the SEM images based on the specimens loaded by F 1 : F 2 ¼ 1 : 1 and F 1 : F 2 ¼ 1 : 0 shown in Figs. 7 and 10, respectively, formation of voids is observed for shear-tension or shear loading conditions whereas for F 1 : F 2 ¼ 1 : 1 (Fig. 13) the voids are marginal and seem to be pre-existing but not during the loading process nucleated ones. This corresponds again well to the assumption taken into account in the mechanical model that isochoric shear will only lead to formation of micro-shear-cracks. In conclusion, for the load ratio F 1 : F 2 ¼ 1 : 1 typical shear-compression mechanisms have been observed in the experiments in the central region of the biaxially loaded specimen. The experimental results correspond well with the numerical ones and, again, many details of the ductile damage and fracture process have been revealed by the corresponding numerical simulation based on the proposed continuum model.

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Fig. 13. SEM images of the fracture surface (F 1 : F 2 ¼ 1 : 1).

5.4. Summary and discussion of experimental and numerical results The new specimen has been tested under different biaxial loading conditions covering a wide range of stress states in the tension, shear and compression range. Load–displacement curves are presented in Figs. 5, 8 and 11, respectively. The diagrams show good agreement of experimental and numerically predicted curves based on both the elastic–plastic and the continuum damage model. Surprisingly, formation of damage during loading of the specimens does not remarkably affect the global load–displacement behavior. Only slightly smaller loads are predicted compared to the simulation based on the simpler elastic–plastic model. As a consequence, no remarkable softening behavior usually observed in load–displacement curves of other specimens before fracture can be seen here caused by the special geometry of the investigated specimen. In Figs. 8 and 11 over-prediction of the load can be observed which at the end of the simulation is of about 10%. This discrepancy may be caused by imperfections in the geometry of the specimen (Fig. 2). Especially, small imperfections in shape, radius or depth of the notch in thickness direction remarkably affect the stiffness as well as the localized inelastic deformation, damage and failure behavior of the specimen. Another reason may be imperfections in material characteristics appearing during the production of the sheets or caused by milling of the notches in thickness direction leading to initial damage in this sensitive critical part of the specimen and associated softening behavior. Therefore, the authors plan to develop alternative specimens for biaxial experiments which will be less sensitive to these imperfections. Furthermore, in Figs. 5 and 8 the numerical simulations based on the continuum damage model stop at displacements less than the experimental ones or those numerically predicted using the elastic–plastic constitutive law. During these calculations few finite elements in the central part of the specimen fail due to formation of damage and their remarkably reduced stiffness leads to numerical problems resulting in sudden stop of the analysis. These difficulties may be avoided

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Fig. 14. Stress triaxialities covered by different specimens.

when a local fracture criterion will be taken into account and failed elements are eliminated by a special numerical procedure available in ANSYS. Moreover, different stress, strain and damage parameters in the notched part of the specimen predicted by numerical simulations based on the continuum damage model are shown in Figs. 6, 9 and 12. These pictures clearly demonstrate that a wide range of stress triaxialities can be covered by the biaxial tests with the new specimen corresponding to different damage modes. In all specimens, variation of the stress triaxialities is numerically predicted corresponding to different damage and fracture mechanisms which are validated by SEM images taken from the fracture surfaces showing different fracture modes. For example, in SEM images for the specimens loaded by F 1 : F 2 ¼ 1 : 1 (Fig. 7) formation and growth of voids and small effects of micro-shear-crack mechanisms are observed for this shear-tension loading condition whereas for F 1 : F 2 ¼ 1 : 0 (Fig. 10) the voids are smaller and more pronounced micro-shear-cracks occur. On the other hand, for F 1 : F 2 ¼ 1 : 1 (Fig. 13) the voids are marginal and seem to be pre-existing but not during the loading process nucleated ones whereas pronounced micro-shear-crack modes are present. These results correspond again well to the assumption taken into account in the mechanical model that hydrostatic tension will lead to formation and growth of voids whereas isochoric shear will only lead to formation and growth of micro-shear-cracks. In addition, for all tested specimens bands of equivalent damage strains correspond to photos with the experimental fracture lines. However, with these numerically predicted damage data it will not be possible to propose and to validate a stress state dependent fracture criterion taking into account different damage tensor components. This requires further investigation which will be subject of a forthcoming paper. Nevertheless, for all considered load ratios characteristic fracture mechanisms have been observed in the experiments in the central region of the biaxially loaded specimen. The experimental results correspond well with the numerical ones and many details of the ductile damage and fracture process have been revealed by the corresponding numerical simulations based on the proposed continuum model. 5.5. Stress triaxialities covered by the biaxially loaded specimen Fig. 14 shows stress triaxialities covered by different geometries of flat specimens which are uniaxially or biaxially loaded. In particular, as mentioned above for uniaxial experiments with unnotched flat specimens only the stress triaxiality g ¼ 1=3 (green point in Fig. 14) can be obtained whereas for uniaxial tests with differently notched flat specimens stress triaxialities pffiffiffi up to g ¼ 1= 3(red band in Fig. 14) can be reached. In addition, for uniaxially loaded shear specimens, the stress triaxialities g ¼ 0:3 and g ¼ 0:1 (blue points in Fig. 13) – depending on the notch in thickness direction [17,33] – are possible. However, further stress triaxialities especially in the negative or higher positive regime are not possible with uniaxially loaded flat specimens. This was the motivation to develop a new experimental procedure and to propose new geometries of specimens which are able to cover a much larger range of stress states in the compression, shear and tension regime as well as respective combinations. For the new biaxially loaded specimen tested in the new 2D-machine corresponding numerical simulations have shown stress triaxialities between 0:35 6 g 6 0:83 (gray points in Fig. 14). Thus, the new experimental program with further loading conditions will be able to cover a large regime of stress triaxialities (gray band in Fig. 14) at onset of fracture which can be used to detect further information on stress-state-dependent ductile damage and fracture mechanisms. 6. Conclusions The paper has discussed a series of new experiments with biaxially loaded specimens and corresponding numerical simulations. These studies have revealed the effect of different stress states on damage and fracture behavior of ductile metals. In this context, a phenomenological continuum model based on different branches of ductile damage and fracture criteria corresponding to various stress-state-dependent mechanisms on the micro-level has been discussed. Motivated by the fact that uniaxial tension tests with flat unnotched and differently pre-notched specimens only cover a small range of positive stress triaxialities, new 2D-experiments with biaxially loaded specimens have been developed and

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experimental results for shear-tension, shear and shear-compression conditions covering a large range of positive and negative stress triaxialities have been presented. Fracture surfaces of the failed specimens have been investigated by a scanning electron microscope (SEM) for the analysis of ductile fracture on the micro-level. The SEM images have demonstrated a clear tendency that tension loading leads to growth of voids whereas shear loading leads to formation of micro-shear-cracks. Corresponding to the experiments numerical simulations have been performed and detailed information on stress states and further parameters especially in the critical zones of the specimens have been obtained. The results of this experimental–numerical procedure allow validation of the stress-state-dependent constitutive equations evaluated from unit-cell calculations. Furthermore, it can be concluded that the fracture line occurs in the band of maximum equivalent damage strains. In addition, the correlation of stress triaxiality and damage modes in ductile metals has been confirmed: high positive stress triaxialities cause nucleation, growth and coalescence of micro-voids, nearly zero or negative stress triaxialities correspond to isochoric formation of micro-shear-cracks whereas for moderate positive stress triaxialities combination of these basic mechanisms occurs. Acknowledgement The SEM images of the fracture surfaces presented in this paper were performed by Wolfgang Saur and are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

Krajcinovic D. Damage mechanics. Mech Mater 1989;8:117–97. Tvergaard V. Material failure by void growth to coalescence. Adv Appl Mech 1990;27:83–151. Lemaitre J. A course on damage mechanics. Berlin, Heidelberg: Springer-Verlag; 1996. Voyiadjis G, Kattan P. Advances in damage mechanics: metals and metal matrix composites. Amsterdam: Elsevier; 1999. Brünig M. An anisotropic ductile damage model based on irreversible thermodynamics. Int J Plast 2003;19:1679–713. Lemaitre J, Dufailly J. Damage measurements. Engng Fract Mech 1987;28:643–61. Kachanov L. On rupture time under condition of creep. Otd Techn Nauk 1958;8:26–31. Rabotnov I. On the equations of state of creep. In: Progress in applied mechanics – the Prager anniversary volume. New York: MacMillan; 1963. p. 307–15. Lemaitre J. A continuous damage mechanics model for ductile fracture. J Engng Mater Technol 1985;107:83–9. Tai W, Yang B. A new microvoid-damage model for ductile fracture. Engng Fract Mech 1986;25:377–84. Betten J. Damage tensors in continuum mechanics. Journal de Mécanique Théorique et Appliquée 1983;2:13–32. Chaboche J. Continuum damage mechanics. Part I: general concepts. J Appl Mech 1988;55:59–64. Chaboche J. Continuum damage mechanics. Part II: damage growth, crack initiation, and crack growth. J Appl Mech 1988;55:65–72. Bruhns O, Schiesse P. A continuum model of elastic-plastic materials with anisotropic damage by oriented microvoids. Eur J Mech A/Solids 1996;15:367–96. Ju J. Isotropic and anisotropic damage variables in continuum damage mechanics. J Engng Mech 1990;116:2764–70. Brünig M, Chyra O, Albrecht D, Driemeier L, Alves M. A ductile damage criterion at various stress triaxialities. Int J Plast 2008;24:1731–55. Brünig M, Albrecht D, Gerke S. Numerical analyses of stress-triaxiality-dependent inelastic deformation behavior of aluminum alloys. Int J Damage Mech 2011;20:299–317. Brünig M, Gerke S. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. Int J Plast 2011;27:1598–617. Needleman A, Kushner A. An analysis of void distribution effects on plastic flow in porous solids. Eur J Mech A/Solids 1990;9:193–206. Brocks W, Sun D-Z, Hönig A. Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic material. Int J Plast 1995;11:971–89. Kuna M, Sun D. Three-dimensional cell model analyses of void growth in ductile materials. Int J Fract 1996;81:235–58. Zhang K, Bai J, Francois D. Numerical analysis of the influence of the Lode parameter on void growth. Int J Solids Struct 2001;38:5847–56. Chew H, Guo T, Cheng L. Effects of pressure-sensitivity and plastic dilatancy on void growth and interaction. Int J Solids Struct 2006;43:6380–97. Brünig M, Albrecht D, Gerke S. Modeling of ductile damage and fracture behavior based on different micro-mechanisms. Int J Damage Mech 2011;20:558–77. Brünig M, Gerke S, Hagenbrock V. Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. Int J Plast 2013;50:49–65. Brünig M, Gerke S, Hagenbrock V. Stress-state-dependence of damage strain rate tensors caused by growth and coalescence of micro-defects. Int J Plast 2014;63:49–63. Becker R, Needleman A, Richmond O, Tvergaard V. Void growth and failure in notched bars. J Mech Phys Solids 1988;36:317–51. Bao Y, Wierzbicki T. On the fracture locus in the equivalent strain and stress triaxiality space. Int J Mech Sci 2004;46:81–98. Bonora N, Gentile D, Pirondi A, Newaz G. Ductile damage evolution under triaxial state of stress: theory and experiments. Int J Plast 2005;21:981–1007. Bai Y, Wierzbicki T. A new model of metal plasticity and fracture with pressure and Lode dependence. Int J Plast 2008;24:1071–96. Gao X, Zhang G, Roe C. A study on the effect of the stress state on ductile fracture. Int J Damage Mech 2010;19:75–94. Dunand M, Mohr D. On the predictive capabilities of the shear modified Gurson and the modified Mohr–Coulomb fracture models over a wide range of stress triaxialities and Lode angles. J Mech Phys Solids 2011;59:1374–94. Driemeier L, Brünig M, Micheli G, Alves M. Experiments on stress-triaxiality dependence of material behavior of aluminum alloys. Mech Mater 2010;42:207–17. Xue Z, Pontin M, Zok F, Hutchinson J. Calibration procedures for a computational model of ductile fracture. Engng Fract Mech 2010;77:492–509. Mohr D, Henn S. Calibration of stress-triaxiality dependent crack formation criteria: A new hybrid experimental-numerical method. Exp Mech 2007;47:805–20. Zheng M, Hu C, Luo Z, Zheng X. Further study of the new damage model by negative stress triaxiality. Int J Fract 1993;63:R15–9. Barsoum I, Faleskog J. Rupture mechanisms in combined tension and shear – Experiments. Int J Solids Struct 2007;44:1768–86. Graham S, Zhang T, Gao X, Hayden M. Development of a combined tension-torsion experiment for calibration of ductile fracture models under conditions of low triaxiality. Int J Mech Sci 2012;54:172–81. Faleskog J, Barsoum I. Tension-torsion fracture experiments – Part I: Experiments and a procedure to evaluate the equivalent plastic strain. Int J Solids Struct 2013;50:4241–57. Haltom S, Kyriakides S, Ravi-Chandar K. Ductile failure under combined shear and tension. Int J Solids Struct 2013;50:1507–22.

M. Brünig et al. / Engineering Fracture Mechanics 141 (2015) 152–169

169

[41] Demmerle S, Boehler J. Optimal design of biaxial tensile cruciform specimens. J Mech Phys Solids 1993;41:143–81. [42] Müller W, Pöhland K. New experiments for determining yield loci of sheet metal. Mater Process Technol 1996;60:643–8. [43] Kuwabara T. Advances in experiments on metal sheet and tubes in support of constitutive modeling and forming simulations. Int J Plast 2007;23:385–419. [44] Kulawinski D, Nagel K, Henkel S, Hübner P, Fischer H, Kuna M, et al. Characterization of stress–strain behavior of a cast trip steel under different biaxial planar load ratios. Engng Fract Mech 2011;78:1684–95. [45] Nahshon K, Hutchinson J. Modification of the Gurson model for shear failure. Eur J Mech A/Solids 2008;27:1–17. [46] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [47] Brünig M, Gerke S, Brenner D. Experiments and numerical simulations on stress-state-dependence of ductile damage criteria. In: Altenbach H, Brünig M, editors. Inelastic behavior of materials and structures under monotonic and cyclic loading. Advanced structured materials. Berlin Heidelberg: Springer-Verlag; 2015. p. 17–34. [48] Brünig M, Gerke S, Brenner D. New 2D-experiments and numerical simulations on stress-state-dependence of ductile damage and failure. Procedia Mater Sci 2014;3:177–82.