A variational void coalescence model for ductile metals

DOI 1.17/s466-11-639-9 ORIGINAL PAPER A variational void coalescence model for ductile metals Amir Siddiq Roman Arciniega Tamer El Sayed Received: 21 February 211 / Accepted: 22 July 211 Springer-Verlag 211 Abstract We present a variational void coalescence model that includes all the essential ingredients of failure in ductile porous metals. The model is an extension of the variational void growth model by Weinberg et al. (Comput Mech 37:142 152, 26). The extended model contains all the deformation phases in ductile porous materials, i.e. elastic deformation, plastic deformation including deviatoric and volumetric (void growth) plasticity followed by damage initiation and evolution due to void coalescence. Parametric studies have been performed to assess the model s dependence on the different input parameters. The model is then validated against uniaxial loading experiments for different materials. We finally show the model s ability to predict the damage mechanisms and fracture surface profile of a notched round bar under tension as observed in experiments. Keywords Variational constitutive updates Void coalescence Constitutive model Ductile fracture 1 Introduction Ductile metals generally have inherent microvoids that play an important role during deformation. There are different available models which take into account void growth and coalescence in predicting failure [2 13]. McClintock [2] and Rice and Tracey [3] proposed very early models by including the evolution equations for void growth. The models were valid for cylindrical [2] and spherical [3] voids. Gurson [4] A. Siddiq R. Arciniega T. El Sayed (B) Computational Solid Mechanics Laboratory (CSML), Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia e-mail: tamer.elsayed@kaust.edu.sa proposed a model by assuming a concentric spherical void in a hollow sphere, making porosity the only microstructural variable. Plastic deformation in the matrix surrounding the voids was assumed to obey J2 plasticity theory. Tvergaard [14], Koplika and Needleman [6], Tvergaard [8], Pardoen et al. [7] modified the Gurson [4] constitutive model to describe void growth as well as damage initiation (onset of void coalescence) based on either the maximum effective stress or the critical porosity attained during deformation. These models have been examined for different materials and loading cases in [6 8]. It has been reported that the maximum effective stress criterion is valid for deformation states with low stress triaxiality while critical porosity requires detailed information of the microstructure and stress state. There are also other criteria based on the micromechanics of void coalescence, see for e.g. [9,1,12,13]; such criteria are based on void expansion and spacing. In ductile metals, once damage is initiated (onset of void coalescence), its evolution (void coalescence process) in the material results in damage induced softening [13]. There are different models that simulate void coalescence [5,11,13]. Tvergaard and Needleman [5] and Benzerga [11] proposed that the void coalescence process leads to the acceleration of the rate of increase of porosity. To simulate this trait, they defined an effective porosity parameter to control the speed of porosity evolution, i.e. void coalescence. Gologanu et al. [15] proposed a model that takes into account the growth and coalescence of voids by comparing the differences in void shapes during deformation. Horstemeyer et al. [16] and Jones et al. [17] proposed a model in which void coalescence occurs throughout the deformation, therefore, no explicit fracture criteria are needed. In this work, a fully variational porous plasticity model [1] is extended to account for void coalescence during failure. The void coalescence process is incorporated in the

model by the addition of a new internal variable that defines the percentage of voids undergoing coalescence at a given material point. The extended model contains all the deformation phases of a ductile porous metal, i.e. elastic deformation, plastic deformation including deviatoric and volumetric (void growth) plasticity followed by damage initiation and evolution due to void coalescence. The paper is organized as follows. In Sect. 2, we present the void coalescence model followed by a detailed description of the incremental solution scheme in Sect. 3. In Sect. 4, we present the parametric assessment of the proposed model. We validate the the model in Sect. 5 and then present a finite element based application in Sect. 6. We finally present the conclusions and future directions in Sect. 7. 2 Model formulation In this section, we describe the proposed void coalescence model for rate-independent porous plasticity. This is based on the original ideas of Ortiz and Molinari [18] and can be considered as an extension of the variational formulation for void growth given by Weinberg et al. [1]. 2.1 Constitutive model We start by postulating the existence of a free-energy density for the mechanical response of dissipative solids. That is A = A(E) (2.1) where E ={F, F p, Q} is a general set of external and internal variables. The tensor F denotes the total deformation gradient. In what follows, we adopt the classical multiplicative decomposition of the deformation gradient F = F e F p into the elastic part F e and the plastic part F p [19]. The quantity Q is the set of internal variables used to described the process. In the present framework for void coalescence we combine a macroscopic plasticity model of the Von Mises type with volumetric plastic expansion induced by void growth and coalescence. The set Q is given by Q ={ɛ p,θ p,η p } (2.2) where ɛ p is the effective deviatoric plastic strain and θ p is the volumetric plastic strain due to void growth. The variable η p accounts for the coalescence effect and it is defined as the percentage of voids undergoing coalescence at a given material point. It will be seen later that η p also characterizes the extra amount of plastic volumetric strain caused by void coalescence. According to this definition, η p is subject to the following constraints η p 1 (2.3) which means that when η p =, void growth is active without coalescence. Void coalescence exists when η p =. We assume that the plastic rate of deformation obeys the following flow rule Ḟ p F p 1 = ɛ p M + θ p N + β η p N (2.4) where the scalar internal variables ɛ p and θ p are subject to irreversibility constraints, i.e., ɛ p, θ p. Note that η p has the same flow direction as θ p due to volumetric deformation. β is a material constant that transforms η p into a strain like variable that measures the volumetric plastic strain due to void coalescence. It also scales the evolution of void coalescence fraction η p as will be shown later during the parametric assessment of the model. The tensors M and N are the directions of the deviatoric and volumetric plastic deformation rates, respectively. They are assumed to satisfy tr(m) =, M M = 3 2, N =±1 3 I (2.5) but are otherwise unspecified. The plus sign in N corresponds to void expansion, and the minus sign corresponds to void collapse [1,2]. Note that the choice of these non-holonomic constraints leads to a model of Von Mises type with volumetric expansion due to void growth and coalescence. Taking the trace of Eq. (2.4) wehave tr(d p ) = J p J p = d dt (ln J p ) =±( θ p + βη p ) (2.6) where D p is the symmetric part of the plastic velocity gradient L p and J p is the plastic Jacobian corresponding to the void growth and coalescence effects. For the particular case of monotonic volumetric expansion, Eq. 2.6 can be simplified to J p = c e (θ p +βη p ) (2.7) where c is a constant. 2.2 Void coalescence model In a typical void growth model, we assume the material to be a conglomerate of initially very small spherical voids inside spherical shells of the material matrix. Each void has a mean radius a and volume (4π/3)a 3. During the process of deformation, we assume that the plastic matrix is incompressible where the shells can grow but still retain their spherical shapes. In the deformed configuration, we define the void volume fraction (or porosity) f R as total volume of voids per unit of current volume, that is J p 4πa 3 f = N (2.8) 3

where N is the initial void density and J p is the plastic Jacobian. The present coalescence model is an extension of typical models for void growth. We still assume a conglomerate of very small spherical voids but now we assume that a portion of the total number of voids may grow and another may undergo coalescence. We denote η p as the percentage of voids with coalescence per unit of undeformed volume. The coalescence model is based on the following assumptions: For the onset of coalescence, we adopt the most widely accepted criterion [1,13] that states that void coalescence starts at a critical porosity f c which is regarded as a material property [5]. Many experimental and numerical works have assessed the validity of this criterion [6 8], which makes it reasonable enough for most ductile metals. The void volume fraction can be expressed as the sum of void growth and coalescence effects. Similarly, the volumetric part of the plastic stored-energy can be split into two contributions due to void growth and coalescence. Therefore f = f growth + f coal (2.9) where f growth and f coal are the void growth and coalescence void volume fractions, respectively. The void coalescence process is simulated by assuming an equivalent void growth model but using an artificially increased void radius ã. This idea is similar to the one [5] proposed for the coalescence process by accelerating the rate of increase of true porosity f using an effective porosity f in the following way f = f c + δ( f f c ) for f f c (2.1) where δ is a parameter that describes the degradation of the mechanical properties. In the present model we, instead, express the ratio (ã/a) as (ã ) 3 = 1 + α 1 ( f f c ) α 2 (2.11) a where α 1 and α 2 are material parameters that characterize the rate of increase in void radius once coalescence starts (see Eq. 2.11). It is clear from Eq. (2.11) that as the value of α 1 increases, the rate of increase in void radius will also increase causing the damage due to void coalescence to evolve faster. Similarly, an increase in α 2 will reduce the rate of increase in void radius resulting in slower damage evolution due to void coalescence. To ensure that the coalescence effect will cause a reduction in the load carrying capacity of the structure, the parameter α 1 must be positive. For the kinematics of the void coalescence model we neglect the elastic volume change of the voids. This assumption, together with the ones given above, lead to porosity contributions such that J p f growth = (1 η p 4πa 3 )N (2.12) 3 J p f coal = (η p 4πã 3 )N (2.13) 3 Notice that (1 η p )N can be considered as the current void density. After the onset of coalescence, the void density will decrease from the initial value N to zero where a state of full coalescence occurs. The current void volume fraction is J p 4πa 3 f = N 3 γ (2.14) where (ã ) 3 γ = (1 η p ) + η p (2.15) a Finally, the plastic Jacobian and the current void volume fraction are related via the following expression J p = 1 f (2.16) 1 f which is a one-to-one function when f f < 1. f is the initial local volume fraction of voids. 2.3 Energy functions The free energy density function is assumed to have an additive structure A(F, F p, Q) = W e (F e ) + W p (ɛ p,θ p,η p ) (2.17) where W e (F e ) and W p (ɛ p,θ p,η p ) are the elastic strain energy density and the plastic stored energy, respectively. Due to material frame indifference, W e must depend on F e through the elastic right Cauchy-Green tensor C e. Furthermore, the elastic strain-energy density can also be written in terms of Hencky s logarithmic strain measure E e = 1 2 log(ce ) (2.18) Consequently, W e = W e (E e ) (2.19) The use of the Hencky strain allows the development of finite strain constitutive models in small strain framework by operating purely at the level of kinematics (see [21]). The details of implementation of the logarithmic elastic strain formulation is given in Sect. 3.2.

A simple form of the elastic strain energy density is W e (E e ) = 1 2 κtr(ee ) 2 + μdev(e e ) dev(e e ) (2.2) where κ is the bulk modulus and μ is the shear modulus. The operator dev( ) takes the deviatoric part of a tensor. Note that the first term of Eq. (2.2) is the volumetric deformation contribution of the elastic energy while the second term is the deviatoric deformation. The plastic stored energy can be modeled by assuming an additive decomposition of deviatoric and volumetric components. In addition, the volumetric component can be split into void growth and coalescence parts. These considerations lead to W p (Q) = W p,dev (ɛ p ) + W p,vol growth (θ p,η p ) +W p,vol coal (θ p,η p ) (2.21) where the deviatoric part W p,dev is given by the conventional power-law hardening, i.e. W p,dev (ɛ p ) = nσ ɛ p n + 1 ( 1 + ɛ p ɛ p ) n (2.22) where n is the hardening exponent, σ is the yield stress and ɛ p is a reference deviatoric plastic strain. The volumetric strain-energy function is attributed to void growth and coalescence. Ortiz and Molinari [18] determined the plastic energy of a spherical void growth model in a power-law hardening material. Based on their ideas it can be shown that W p,vol growth (θ p,η p )= nσ ɛ p n + 1 (1 η p 4πa ) (N 3 ) ḡ(θ p,η p ) 3 (2.23) W p,vol coal (θ p,η p ) = nσ ɛ p n + 1 (η p ) where ḡ(θ p,η p ) = g(θ p,η p ) = 1/f 1 + 2 3ɛ p log 1 x 1 + 1/fã 1 + 2 3ɛ p log 1 x 1 + 4πã (N 3 ) g(θ p,η p ) 3 (2.24) x γ f f +e (θ p +βη p) 1 x γ f /(ã/a) 3 f +e (θ p +βη p) 1 n n dx (2.25) dx (2.26) where the variables (ã/a) and γ are defined in Eqs. (2.11) and (2.15), respectively. Notice that when η p = the present formulation reduces to the simple void growth model in [1]. During the void coalescence phase, the stored energy contribution toward void growth decreases while the plastic energy due to coalescence increases until a state of full coalescence is achieved (η p = 1). The quantity fã is determined by the incompressibility constraint of the material matrix, i.e. 1 = 1 ( ) 1 fã (ã/a) 3 f 1 + 1 (2.27) As discussed before, η p characterizes the fraction of voids undergoing coalescence at a given material point. The real sense of this internal variable can be understood through Eqs. (2.23, 2.24). The energy functions in Eqs. (2.23, 2.24) are motivated from the original idea of [18] for spherical void in a power-law hardening material. It must be noted that η p once multiplied by initial void density (N ) yields the number of voids undergoing coalescence and this way Eq. (2.24) gives the plastic energy contribution due to the number of voids undergoing coalescence. While its other counterpart (1 η p ) gives plastic energy contribution due to the fraction of voids undergoing void growth (Eq. 2.23). 3 Incremental solution In this Section, we present an incremental solution procedure for the time integration of the constitutive equations of the void coalescence model. We consider for that matter a generic time interval [t n, t ] and assume that the state of the material E n = {F n, Fn p, Q n } is known at the time t n. The deformation gradient F is also given at the time t. We want to determine the state of the material E ={F, F p, Q } at time t as well as the value of the first Piola-Kirchhoff stress tensor P. It has been shown [2,22] that a large set of dissipative materials can be modeled by assuming a pseudo-potential ψ that behaves like hyperelastic within the interval of a load increment [23], i.e. P = ψ(f ; E n ) (3.1) F The incremental update for a rate-independent porous plasticity material can be thus obtained from the incremental pseudo-potential by minimizing the internal variables. That is ψ(f ; E n ) = min F p,q {A(E ) A(E n )} (3.2) where A is the free-energy density given in Eq. (2.17). 3.1 Constitutive updates The plastic deformation is incrementally updated by the exponential mapping which has the property to preserve the isochoric character of the plastic deformation. The use of the exponential mapping technique with Hencky strain leads

to a simplified update procedure where small-strain update algorithms are easily extended to finite strain by recourse to a purely kinematical material-independent manipulation [21,24]. Void coalescence is a complex phenomenon where growing voids link together by two mechanisms: void impingement and void sheet mechanisms [25]. Void impingement is the growth of two neighboring voids until they coalesce while void sheet mechanism is related to shear band instabilities where the localized plastic shearing occurs among the growing voids inside a narrow shear band such that the process is affected by significant shear strains. In the present constitutive model, void impingement is related to the volumetric plastic strain θ p and fraction of void under going coalescence η p while plastic shear localization weakening mechanism is related to volumetric plastic strain θ p and deviatoric plastic strain ε p. In the model, complete failure (unstable crack growth) is characterized by comparing ε p and η p with their critical values εc p and ηc p, respectively. If, either ε p or η p reaches the critical value εc p or ηc p, respectively, the material is assumed to fail. It has been argued that after the onset of coalescence, the homogeneous void growth will be interrupted and the beginning of the final stage of damage will initiate[13,26 28]. Therefore, instead of having two mechanisms (void growth and coalescence) competing with each other, we assume that void coalescence will take over after its onset. With this in mind we consider that the void growth process, represented by the variable θ p, is interrupted through the coalescence phase. In mathematical terms, this means that θ p is kept constant once coalescence starts. The exponential mapping update before and after the onset of coalescence is thus given by F p = { exp ( ɛ p M + θ p N) F p n exp ( ɛ p M + β η p N) F p n if f f c if f > f c (3.3) where ɛ p = ɛ p ɛ p n, θ p = θ p θ p n and η p = η p η p n. Notice that before the onset of coalescence the formulation is the same as the one proposed by Weinberg et al. [1]. The update of the internal variables ɛ p, θ p and η p, and the directions of plastic flow M and N is performed by a variational minimization problem. The minimization of M and N is solved explicitly from the constraint conditions given in Eq. (2.5), and therefore, is not included in the numerical minimization. The incremental update of the pseudo-potential (see Eq. 3.2) during the coalescence phase can be written in the following variational form ψ(f ) = min F(F,ɛ p ɛ p,η p,m,n, θ p,ηp, M, N) (3.4) where the incremental objective function F, defined in terms of the elastic and plastic energies given in Eqs. (2.2) and (2.21), is F(F,ɛ p, θ p,ηp, M, N) = W e (E e ) + W p (ɛ p, θ p,ηp ) (3.5) The variable θ p denotes the volumetric plastic strain at the onset of coalescence and it is kept constant throughout the void coalescence process. It will be shown later that the logarithmic elastic strain E e depends on F, M, N, ɛ p and η p through the flow rule Eq. (2.4). The minimization problem Eq. (3.5) is subjected to the plastic irreversibility constraints ɛ p, η (3.6) 3.2 Numerical algorithm The implementation of the constitutive updates is based on the logarithmic elastic strain formulation [21,2]. First, we assume an elastic-predictor right Cauchy-Green tensor as C e,pre = Fp T n C F p 1 n (3.7) where C = F T F. IfM commutes with C e,pre and if we take the logarithms in Eq. (3.7) after using relation Eq. (3.3), we arrive to E e = Ee,pre ɛ p M β η p N (3.8) where E e,pre = 1 2 log ( C e,pre ) (3.9) is the trial logarithmic elastic strain. Notice that the constitutive updates exhibit an additive structure of elastic and plastic deformations identical to that of small-strain plasticity. Next, we carry out the minimization of the objective function F with respect to M and N subject to the constraints Eq. (2.5). This gives M = 3 2 S pre σ pre, N = 1 3 sgn ( ϱ pre ) I (3.1) where S pre = 2μ dev ( E e,pre ) pre, σ = (3/2) S pre Spre, ϱ pre = κ tr( E e,pre ) (3.11) Finally, minimization with respect to ɛ p and η p leads to a nonlinear system of differential equations F ɛ p F η p = ( σ pre 3μ ɛ p) + σ c = (3.12) = β ( ϱ pre κβ ηp) + ϱ c = (3.13)

for the case when σ pre σ c pre or ϱ ϱc, otherwise, we are in the elastic regime. Variables σ c and ϱc are given by σ c = W p ɛ p, c c = W p η p (3.14) respectively. The numerical solution of Eqs. 3.12 and 3.13 is thus carried out by a Newton Raphson iteration at each time step. 4 Parametric assessment of the model In this Section, we present the effect of the model parameters on the stress strain response of U-6%Nb alloy. It should be noted that the complete deformation process comprises elastic deformation and plastic deformation due to deviatoric and volumetric (void growth) contributions followed by damage initiation and evolution due to void coalescence. Parametric studies for the case of uniaxial and triaxial (volumetric) loading conditions are presented in the following. All the tests have been performed by varying five different parameters which include, material constants β, α 1, α 2 and the material inherent properties N initial void density, a initial void radius, f c critical porosity. The rest of the material parameters are kept constant and are given in Table 1. The parameters which are kept constant throughout the parametric assessment process have been taken from Weinberg et al. [1]. The values of the critical deviatoric plastic strain (ε p c =.55) and critical fraction of void undergoing coalescence (η p c = 1.) were kept constant throughout this section. Figure 1 shows the effect of different material parameters (discussed above) on the stress strain response for the test case. Effect of β on the stress strain response is shown in Fig. 1a. β is the parameter first introduced in Eq. (2.4) for the purpose of transforming the void fraction undergoing coalescence to a strain like variable and to scale the void coalescence rate. The model has been tested for a range of β values from.1 to 1.. It is found that an increasing β value reduces the slope of the damage evolution part of the stress strain response. That is, the void coalescence process is less sensitive to the increasing strain. Figure 1b shows the effect of the initial void density (number of voids per unit undeformed volume) on the stress strain response. Table 1 Material parameters used for parametric assessment E (GPa) ν σ (T ) (MPa) ε p n ρ (kg m 3 ) 165.15 6.1 2 17,3 a (μm) N (m 3 ) f c α 1 α 2 β 1 1 1.3 1. 1. 1. Initial void density (N ) is the inherent material property first introduced in Eq. (2.8) and describes the initial porosity of the material once multiplied by the volume of the spherical void. It is found that the variation in the initial void density changes the peak of the stress strain response. It must be noted that increasing the initial void density means increasing the initial porosity of the material. The higher the initial porosity of the material, the lower the ultimate strength (peak of the stress strain curve) of the material. Therefore, increasing the initial void density shows reduction in the peak strength of the material. Similar experimental results have been reported in [29] where it was shown that increasing the void fraction reduces the effective cross-sectional area under stress causing the tensile and yield strengths to decrease. It can also be inferred from Fig. 1b that as the initial void density increases the sharpness of the stress strain peak blunts and damage is initiated in a more ductile manner. Similar effects can be seen for the case when the initial void radius (see Eq. 2.8) is varied (see Fig. 1c). It can also be inferred from the above results that during the parameters identification process the peak stress can be increased or decreased by varying either the initial void radius or initial void density, if the initial porosity of the material is not known apriori. It was also reported in the literature [6,13,3 32] that the critical porosity (porosity at the onset of coalescence) is generally not a constant for a specific material, it rather depends on the shape, size, spacing of the voids as well as on the type of loading and stress state of the material during deformation. In the present constitutive model the critical porosity ( f c ) is chosen to be a material parameter that has to be identified from experimental data. The critical porosity ( f c ) is first introduced in Eqs. (2.1 and 2.11) as a criterion for the onset of void coalescence. Therefore, the value of the critical porosity parameter sets the criterion used for the onset of void coalescence. Figure 1d shows the effect of the critical porosity ( f c ) value on the stress strain response. Since the critical porosity ( f c ) is the criteria for the onset of void coalescence, changing the critical porosity only shifts the damage initiation point. It can be inferred from Fig. 1d that increasing the critical porosity ( f c ) simply increases the strain at which void coalescence starts. Void radius evolution parameters α 1 and α 2 are first introduced in Eq. (2.11). The effect of varying the void radius evolution parameters α 1 and α 2 are plotted in Fig. 1e, f. It can be inferred from Fig. 1e that as the value of α 2 increases, the slope of the damage evolution part of the stress strain curve decreases. This means that increasing the magnitude of α 2 makes the void coalescence process less sensitive to strain. Figure 1f shows the effect of α 1 on the stress strain response. It can be inferred that as the value of α 1 increases, the slope of the damage evolution part of the stress strain response increases. This shows that increasing the value α 1 causes the material

4e+9 2e+9 (a) Beta =.1 Beta =.5 Beta = 1. Beta = 1. 6e+9 5e+9 4e+9 3e+9 2e+9 (b) No = 2.39E9 No = 7.16E9 No = 1.19E1 No = 1.67E1 No = 2.15E1 No = 4.77E1-2e+9 1e+9 6e+9 5e+9 4e+9 3e+9 2e+9 1e+9 (c) 2 4 6 a = 7.94E-5 a = 1.E-4 a = 1.26E-4 a = 1.44E-4 a = 1.59E-4 a = 1.71E-4 a = 2.15E-4 5e+9 4e+9 3e+9 2e+9 1e+9.5 1 1.5 2 (d) fc =.1 fc =.2 fc =.3 fc =.4 fc =.5 5e+9 4e+9 3e+9 2e+9 1e+9.5 1 1.5 2 (e) α2 =.5 α2 = 1. α2 = 1.5 α2 = 2. α2 = 2.5 α2 = 3. α2 = 3.5 α2 = 4. α2 = 1. α2 = 2. α2 = 4..5 1 1.5 2 5e+9 4e+9 3e+9 2e+9 1e+9.5 1 1.5 2 2.5 (f) α1 =.1 α1 =.2 α1 =.3 α1 =.4 α1 =.8 α1 = 2. α1 = 3. α1 = 5. α1 = 6. α1 = 7..5 1 1.5 2 Fig. 1 Parametric study for the case of uniaxial loading: Effect of a β, b N, c a, d f c, e α 2, f α 1 to fail in a relatively brittle manner, i.e. with increasing the value α 1, void coalescence becomes more sensitive to the strain. Similar behavior is obtained for the case of volumetric loading and is not repeated here for brevity. 5 Numerical examples and validation Since the presented model is phenomenological and depends on a number of parameters (discussed in the previous section), it is necessary to identify these parameters through

Table 2 Material parameters identified using the inverse modeling approach Material A775-T6 [37] A224-T351 [37] A224-T851 [37] A661-T6 [38] E (GPa) 4 65 1 65 ν.33.33.33.33 σ (T ) (MPa) 16 15 21 19 ε p.1.1.1.1 n 3.8 3.5 5.2 6. a (μm) 1 1 1 1 N (m 3 ) 4.77 1 9 4.77 1 9 4.77 1 9 4.77 1 9 f c.28.27.18.2 α 1 2.1 12.1 2.5 5. α 2 1.1 1.7 1..9 β.8.1.8 1. ηc p.4.5.59.14 εc p.55.55.55.55 experiments. In this section, numerical examples are presented for different materials that undergo failure during deformation. The model predictions are validated against experiments by using the inverse modeling approach. In the presented constitutive model, material parameters can be divided into three categories, elastic parameters (Young s modulus E and Poisson s ratio ν), plasticity parameters (yield strength σ, reference plastic strain ε p, hardening exponent n, initial void radius a, initial void density N ) and damage parameters ( f c, α 1, α 2, β, ηc p, εc p ). The initial elastic and plastic hardening parts of the stress strain curves are fitted by varying the elastic and plastic parameters only. Then, based on the peak stress and damage evolution in the stress strain curves, the damage parameters are fitted. In the present work, these material parameters were identified by manually fitting the stress strain data, however the model can be interfaced with an optimization tool to perform automatic parameters identification [33 36] if manual fitting fails. Using the above discussed methodology, three tests are performed to calibrate the material parameters using experimental data reported in [37] for A775-T6, A224-T351, and A224-T851. The identified materials parameters are given in Table 2. The results are plotted in Fig. 2 showing a good agreement between the experimental and simulated responses especially during the void coalescence stage, i.e. the softening part of the stress strain response during the last stage of deformation. Experimental results of uniaxial test on a smooth round bar of Aluminum alloy (A661-T6) reported by Li et al. [38] are also used to calibrate the model parameters. The results after the calibration process are plotted in Fig. 3 showing a good agreement between the experimental and simulated responses. It must be noted that there are small differences in the experimental and simulated behaviors during the hardening part of the deformation curve due 7e+8 6e+8 5e+8 4e+8 3e+8 Experiment A775-T6 2e+8 Model Prediction A775-T6 Experiment A224-T351 Model Prediction A224-T351 1e+8 Experiment A224-T851 Model Prediction A224-T851.5.1.15.2 Fig. 2 Model predictions for three different aluminum alloy after parameters identification 5e+8 4e+8 3e+8 2e+8 1e+8 Experiment A661-T6 (Li et al., 21) Model Prediction A661-T6.5.1.15.2 Fig. 3 Model predictions for aluminum alloy (AA661-T6) after parameters identification

to the power hardening law used in the formulation, which gives only one parameter to adjust. However, the softening parts of the stress strain response show a good agreement which is due to the void growth followed by the coalescence process. 6 Finite element based application In order to demonstrate the model s ability to capture actual failure patterns during deformation, a finite element model of a notched round bar is constructed and tested under uniaxial loading condition. The notched round bar chosen for this study is based on the recent experimental work performed by Li et al. [38]. The loading velocity is 1. mm/min with a specimen length of 5 mm, diameter of 25 mm and a notch radius of 5 mm. Li et al. [38] reported that enlarged voids were present on the fracture surface. It was also reported that enlarged microvoids were located in nearly 9% of the fracture surface area and that slanted (shear) fracture occurs at the outer surface of the notched round bar. Based on the above experimental data, an axisymmetric finite element model of a notched round bar was constructed. The material model discussed in the previous section is interfaced with ABAQUS using a user defined material subroutine (VUMAT). The material used in the study was Aluminum alloy 661 with parameters given in Table 2. As discussed in the previous section, throughout the simulation, two criteria have been used simultaneously for element deletion. The first criterion is based on the critical void coalescence fraction (η p c ) given in Table 2. This fraction determines the percentage of coalesced voids after which the material fails completely without any further coalescence. In addition to this, a criterion based on deviatoric plastic strain is also included in the model. This criteria compares the deviatoric plastic strain (ε p ) with the critical deviatoric plastic strain (ε p c ) throughout the deformation. Both criteria are simultaneously checked during the deformation and element deletion is performed if wither one condition is met. Similar to values reported in the literature, a critical deviatoric plastic strain value of.55 was used in this simulation. Contour plots of three internal state variables: deviatoric plastic strain (ε p ), volumetric plastic strain (θ p ) and percentage of void undergoing coalescence (η p ) are shown in Fig. 4. The results are plotted at different stages of the deformation. It can be inferred from Fig. 4a that, in the early stages of deformation, deviatoric plastic strains are induced in the specimen with maximum values at the outer edge of the specimen. The process of plastic deformation governed by the deviatoric plastic strain progresses until a point is reached where the volumetric plastic strain due to void growth starts to evolve. Similar to the experimental findings, the model predicts that the volumetric plastic strain evolves from the center of the specimen as seen in Fig. 4b. As the deformation progresses, both the deviatoric and volumetric plastic strain continue to evolve from the outer surface and center of the specimen, respectively. As seen in Fig. 4c, the volumetric failure criterion is met first at the center of the specimen upon which damage initiates and evolves [28,38 4]. These results are in agreement with the experimental findings of [38], where it was reported that the central part of the notched specimen showed enlarged microvoids, i.e. failure due to void growth followed by coalescence. It can also be inferred from Fig. 4 that the volumetric damage evolution dominates the failure in the specimen until it reaches the point where the deviatoric damage criterion is met at the outer surface of the specimen causing the corresponding element to be deleted (Fig. 4). From this point onwards, both failure criteria (volumetric and deviatoric) are met at inner and outer elements, respectively, resulting in the deletion of those elements until the specimen is completely failed. As reported in the experimental study [38], the model predicts that the central part of the specimen fails due to the microvoid growth and coalescence and at the outer part fails due to deviatoric plastic deformation. Also, as reported during the experiments [38], around 9% of the fractured surface showed enlarged microvoids; the model prediction also shows that most of the fracture is dominated by the microvoid growth and coalescence. It is the last part of the deformation phase where deviatoric damage starts to evolve from the outer surface causing the crack path to deviate at an angle of 3. The fracture surface after complete failure is very similar to the one reported by Li et al. [38]. 7 Conclusions and future directions A fully variational void coalescence model has been developed by introducing a new internal state variable which accounts for the percentage of voids undergoing coalescence during deformation. Parametric studies have been performed to understand the effects of the input parameters on the stress strain response. Numerical validations have been presented based on parameters identification using uniaxial experimental data. The model shows a good agreement with experiments. A finite element based application of the model has been presented by simulating the uniaxial tension test of a notched round bar. The model captures the failure mechanisms as reported during experiments. The fracture surfaces also shows a very similar profile to the one reported in the experimental studies. Future directions of this work may include the introduction of rate effects by obtaining dual kinetic potentials for the existing ones via Legendre transformations.

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