2013 Mechanicsmats Vol 65.Pdf

Mechanics of Materials 65 (2013) 12–34

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Mechanics of Materials

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Adaptive concurrent multi-level model for multi-scale analysis of ductile fracture in heterogeneous aluminum alloys ⇑ Somnath Ghosh a, , Daniel Paquet b a Department of Civil Engineering, Johns Hopkins University, 203 Latrobe, 3400 N. Charles Street, Baltimore, MD 21218, United States b Department of Mechanical Engineering, The Ohio State University, Scott Laboratory, Columbus, OH 43210, United States article info abstract

Article history: This paper creates a novel adaptive multi-level modeling framework for rate-dependent Received 1 November 2012 ductile fracture of heterogeneous aluminum alloys with non-uniform microstructures. Received in revised form 22 April 2013 The microstructure of aluminum alloys is characterized by a dispersion of brittle heteroge- Available online 1 June 2013 neities such as silicon and intermetallics in a ductile aluminum matrix. These microstructural heterogeneities affect their failure properties like ductility in an adverse manner. The Keywords: multi-level model invokes two-way coupling, viz. homogenization for upscaled constitu- Ductile fracture tive modeling, and top-down scale-transition in regions of localization and damage. Adap- Heterogeneous aluminum alloys tivity is necessary for incorporating continuous changes in the computational model as a Adaptive multi-scale model LE-VCFEM consequence of evolving microstructural deformation and damage. The macroscopic ﬁnite Homogenization based plasticity-damage element analysis in regions of homogeneity incorporates homogenization-based contin- (HCPD) model uum rate-dependent plasticity-damage (HCPD) models. Transcending scales is required in regions of high macroscopic gradients caused by underlying localized plasticity and damage. Complete microscopic analysis using the LE-VCFEM is conducted in these regions, which follow the growth of microscopic voids and cracking to cause local ductile fracture. The macroscopic and microscopic simulations are done concurrently in a coupled manner. Physics-based level change criteria are developed to improve the accuracy and efﬁciency of the model. Numerical simulations are conducted for validations and ductile fracture in a real microstructure is demonstrated. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction strains in the vicinity of damaged inclusions lead to void nucleation in the matrix. Damage subsequently propa- Heterogeneous cast aluminum alloys, containing sili- gates with void growth and localizes in bands of intense con and intermetallic inclusions in the dendritic structure, plastic deformation between inclusions, until void coales- are widely used in automotive and aerospace structures. cence in the matrix leads to catastrophic failure. Experi- While higher strength may result from the presence of mental studies e.g. in Caceres et al. (1996), Caceres these heterogeneities in the microstructure, failure prop- (1999), Wang et al. (2003) have shown strong connec- erties like ductility are generally adversely affected. Major tions between morphological variations and microstruc- microstructural mechanisms responsible for deteriorating tural damage nucleation and growth. The studies have these properties include particulate fragmentation and shown that damage initiation is mainly controlled by matrix cracking. Ductile failure generally initiates with the shape of inclusions in the microstructure, while the second phase inclusion fragmentation. Large plastic rate of damage evolution at higher strains is controlled by the level of clustering. This makes predicting ductility quite challenging. ⇑ Corresponding author. Tel.: +1 410 516 7833; fax: +1 410 516 7473. E-mail address: [email protected] (S. Ghosh).

Computational studies have been conducted to study These homogenization methods capture relatively elastic–plastic deformation and ductile failure of heteroge- uniform macroscopic deformation fields without large neous materials in Steglich and Brocks (1997), Lim and gradients in the deformation fields or localized soften- Dunne (1996), Negre et al. (2003), Huber et al. (2005), ing behavior. Llorca and Gonzalez (1998), Llorca and Segurado (2004). Higher-order homogenization methods have been The predictive capability of some of these unit cell models developed e.g. in Kouznetsova et al. (2004), Luscher for failure properties is limited due to over-simplification et al. (2010), Vernerey et al. (2008), Fish and Kuznetsov of the microstructure. Quite often, critical local features (2010), Loehnert and Belytschko (2007) for transferring necessary to model strain to failure are lost in these mod- higher-order kinematics from the micro-scale to macro- els. Ductile fracture depends strongly on the extreme val- scale. This results in the introduction of a length-scale ues of microstructural characteristics, e.g. nearest parameter in the model, e.g. the size of the RVE. The neighbor distances, highest local volume fraction etc. and limited number of higher-order kinematic variables computational models must feature some of these charac- however restricts their applicability in the event of high teristics for accuracy. Additionally, many studies have fo- localization of deformation and damage. cused only on the initial stages of ductile damage, e.g. Continuous–discontinuous multi-scale models have been crack nucleation and have not considered their evolution developed in Massart et al. (2007), Belytschko et al. to failure. There is a paucity of image-based models that (2008), Song (2009) to capture the local softening aris- consider aspects of the real microstructural morphology ing in the microscopic scale due to damage. These and non-uniformities. Computational models developed methods are based on the partitioning of the RVE into by Ghosh et al. (Moorthy and Ghosh, 1998, Ghosh and sub-domains delineating regions in which the material Moorthy, 1998, Hu and Ghosh, 2008, Paquet et al., response is stable and unstable. Classical homogeniza- 2011a,b) have focused on realistic representation of micro- tion method is used for the overall behavior of the sta- structures with non-uniform dispersion of heterogeneities. ble sub-domain. The homogenization method for the The microstructural Voronoi cell finite element model unstable sub-domain is specific to the type of damage (VCFEM) developed in Moorthy and Ghosh (2000), Ghosh considered. The behavior and geometry of material (2011), Ghosh et al. (2000), Moorthy and Ghosh (1998), instabilities in the RVE need to be well defined. The Ghosh and Moorthy (1998) is very accurate and efficient accuracy of the method strongly depends on the for micromechanical analysis. Morphological non-unifor- assumptions made for the description of the equivalent mities in dispersions, shapes and sizes of micrographs are discontinuities at the macro-scale. readily modeled by this method. The method has been Multi-level, multi-scale models e.g. in Vemaganti and extended in the locally enhanced VCFEM (LE-VCFEM) to Oden (2001), Markovic and Ibrahimbegovic (2004), model stages of ductile fracture in Hu and Ghosh (2008), Temizer and Wriggers (2011), Kim et al. (2010), Miehe from inclusion fragmentation to matrix cracking in the and Bayreuther (2007), Fish and Shek (2000), Zohdi form of void nucleation, growth and coalescence. In and Wriggers (1999) require that complete microme- LE-VCFEM (Hu and Ghosh, 2008), the stress-based hybrid chanical analyses be performed in portions of the com- VCFEM formulation is adaptively altered. Regions of local- putational domain that are undergoing intense ized plastic flow are overlaid with finite deformation, dis- deformation and damage. The effects of microstructural placement-based elements to accommodate strain instabilities are fully captured with the micromechani- softening. LE-VCFEM has been demonstrated to be very cal model. Classical homogenization method is used effective for simulating ductile fracture for rate-indepen- for upscaling the mechanical behavior in the macro- dent and rate-dependent plasticity in Hu and Ghosh scopic sub-domains, for which there is no localization (2008), Paquet et al. (2011a,b), respectively. of deformation and damage. The effect of microstruc- Pure micromechanical analysis of an entire structure is tural damage in the microscopic sub-domains is trans- however computationally prohibitive due to the large ferred to the macroscopic sub-domains by coupling number of heterogeneities in the underlying microstruc- the two scales of analysis. ture. The need for multi-scale modeling, with micromechanisms of damage explicitly modeled only in regions of Ghosh and co-workers have developed adaptive multi- localization and cracking, is realized for these problems. level, multi-scale models for: (i) linear elastic composites Different classes of multi-scale models have been devel- (Raghavan et al., 2001, Raghavan and Ghosh, 2004, oped in the literature. Raghavan and Ghosh, 2004), (ii) elastic–plastic composite materials undergoing microstructural damage by inclusion Classical, first-order homogenization methods have cracking only (Lee et al., 1999; Ghosh, 2011), and (iii) com- been developed e.g. in Guedes and Kikuchi (1990), posites with interfacial debonding in Raghavan et al. Ghosh et al. (1995), Feyel and Chaboche (2000), Terada (2004), Ghosh et al. (2007). Developments of the multi- and Kikuchi (2001), Miehe and Koch (2002), Ghosh et al. scale analysis using VCFEM micromechanical analyses (1996), Lee and Ghosh (1996), Moulinec and Suquet have been discussed in Ghosh (2011). These multi-scale (2001), Zohdi et al. (1999), Oden and Zohdi (1997) models involve two-way coupling, viz. homogenization based on the asymptotic expansion theory of homoge- for upscaled constitutive modeling, and top-down scale- nization (Benssousan et al., 1978; Sanchez-Palencia et transition in regions of localization and damage. Multi- al., 1983). This theory assumes a complete separation scale characterization and domain partitioning have also of scales and does not incorporate length-scale effects. been developed as a pre-processor to multi-scale modeling 14 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 in Valiveti and Ghosh (2007), Ghosh et al. (2006) for deter- the microstructure and its evolution. The constitu- mining microstructural representative volume elements, tive material models are obtained by homogenizing as well as for identifying regions where homogenization the material response in the microstructural RVE breaks down. as discussed in Section 2.1.3.

The present paper, assimilates a variety of complemen- (2) Level-1: This computational subdomain Xl1 is used tary ingredients from the previous developments to create to establish criteria that necessitate switching over a novel multi-level modeling framework for ductile frac- from macroscopic to microscopic computations. ture. The adaptive multi-level computational model is Details of this ’’swing’’ level are discussed in Section developed for rate-dependent ductile fracture in heteroge- 2.2. neous aluminum alloys with non-uniform microstructures. (3) Level-2: This embedded computational subdomain

This extension incorporates complex mechanisms of dam- Xl2 requires pure microscopic analysis with delinea- age nucleation by particle cracking, subsequent void tion of the microstructure and its evolution. growth in the matrix and coalescence. Adaptivity is neces- (4) Transition interface layer: This computational subdo- sary for incorporating continuous changes in the computa- main Xtr is sandwiched between the macroscopic tional model as a consequence of evolving microstructural (level-0/level-1) and microscopic (level-2) domains deformation and damage. The macroscopic ﬁnite element and acts as an interface region to regularize kinetic analysis in regions of low gradients and homogeneity and kinematic incompatibilities. incorporates homogenization-based continuum rate- dependent plasticity-damage (HCPD) models. Transcend- Brief descriptions of each levels for ductile fracture are ing scales is required in regions of high macroscopic gradi- given next. ents caused by underlying localized plasticity and damage.

Complete microscopic analysis using the LE-VCFEM is con- 2.1. Computational subdomain level-0 (Xl0) ducted in these regions, which follow the growth of microscopic voids and cracking to cause local ductile fracture. The computational subdomain level-0 (Xl0) invokes The macroscopic and microscopic simulations are done macroscopic finite element analysis using homogenized concurrently in a coupled manner. The theory in this mul- constitutive relations. It assumes uniform macroscopic ti-scale formulation is a combination of small-strain and deformation and periodic microstructural variables to ad- large-strain formulation in different regions of the domain. mit homogenized response. Hierarchical models such as Small strain analysis formulation is applicable to the mac- the FE2 multi-scale methods (Feyel and Chaboche, 2000) roscopic plastic deformation regions that do not show solve the micromechanical RVE problem at every element much localization. A finite deformation formulation in a integration point in the computational domain. This can rotated Lagrangian system is developed for microstructural lead to prohibitively large computational costs. To over- regions of localized deformation and ductile fracture that come this limitation, Ghosh et al. (2009) have developed are modeled by the locally enhanced Voronoi cell elements macroscopic constitutive laws of elastic–plastic damage or LE-VCFEM. from homogenization of microscopic RVE response. Param- Appropriate level change criteria are developed to im- eters in these constitutive models represent the effect of prove the accuracy and efficiency of the model. The overall morphology, as well as evolving microstructural mecha- multi-level model for multi-scale analysis of ductile frac- nisms. These reduced-order constitutive models are highly ture is presented in Section 2. Adaptivity criteria for evolu- efficient as they do not have to account for the details of tion of the multi-level model are defined in Section 3, microstructural morphology or solve the micromechanical while coupling of the different levels are detailed in Sec- RVE problem in every step of an incremental process. This tion 4. Numerical simulations for validations are con- section summarizes a rate dependent homogenization- ducted in Section 5 and a real microstructure is based continuum plasticity-damage (HCPD) model for demonstrated in Section 6. macroscopic representation of ductile failure in porous viscoplastic materials containing brittle inclusions, that 2. Levels in the concurrent multi-scale modeling has been developed in Paquet et al. (2011), Dondeti et al. framework (2012).

The multi-level framework for multi-scale analysis It is developed from homogenization of a microscopic adaptively decomposes the heterogeneous computational RVE of a heterogeneous aluminum alloy with a non-uni- domain Xhet into a set of non-intersecting sub-domains, form dispersion of inclusions, as shown in Fig. 1, follow- denoted by level-0, level-1, level-2, and level-tr, i.e. ing the Hill-Mandel postulate. Micromechanical

Xhet ¼ Xl0 [ Xl1 [ Xl2 [ Xtr. Concurrent multi-scale anal- analyses of the elastic–plastic RVE with inclusion and ysis requires that all levels be coupled for simultaneous matrix cracking are conducted by LE-VCFEM (Hu and solving of variables in the different sub-domains. The lev- Ghosh, 2008; Paquet et al., 2011a). els are deﬁned as follows: The anisotropic rate-dependent HCPD model in Paquet et al. (2011), Dondeti et al. (2012) has the framework

(1) Level-0: In this subdomain Xl0, purely macroscopic of the Gurson–Tvergaard–Needleman model (Gurson, computations are executed using continuum consti- 1977; Chu and Needleman, 1980; Tvergaard, 1982; tutive models that implicitly represent the effect of Tvergaard and Needleman, 1984). An anisotropic plastic S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 15

Fig. 1. (a) Micrograph of a cast aluminum alloy W319 (192 lm 192 lm) showing dendritic regions, (b) typical SERVE of the microstructure in (a) obtained for a characteristic size of 48 lm (Micrograph: Courtesy Ford Research Laboratory).

flow potential is introduced in an evolving material- Dondeti et al. (2012). This region corresponds to an intra- damage principal coordinate system, in which parame- granular region containing secondary dendrite arms and ters evolve as functions of the plastic work. inter-dendritic regions containing particulates. The nomi- A macroscopic void nucleation law is developed by nal grain size of aluminum alloys is an order of magnitude homogenizing inclusion fragmentation results in the larger than this SERVE (typically 600—800 lm). Mechan- RVE. Rate effects and anisotropy are included in the ical properties in aluminum alloys have been shown to cor- nucleation law. relate better with the secondary dendritic arm spacing or SDAS rather than the grain size. Important tasks in the development of the HCPD model includes: (i) identification of a statistically equivalent 2.1.2. Micromechanical analyses by LE-VCFEM with RVE or SERVE; (ii) detailed micromechanical analyses by mechanisms of plasticity and damage LE-VCFEM including explicit mechanisms of plasticity The locally enhanced Voronoi cell finite element meth- and damage; (iii) homogenization with periodic boundary od or LE-VCFEM has been successfully developed in Hu and conditions for reduced-order modeling; (iv) framework Ghosh (2008), Paquet et al. (2011a), Ghosh (2011) to model development for rate-dependent anisotropic continuum the evolution of microstructural failure from particle frag- plasticity and damage; and (v) calibration of the evolving mentation to complete ductile failure by matrix cracking model parameter functions. due to void growth and coalescence. In LE-VCFEM, the stress-based hybrid VCFEM formulation is enhanced adap- 2.1.1. Identification of the SERVE size for homogenization tively in narrow bands of localized plastic flow and void The statistically equivalent representative volume ele- growth. These regions are locally embedded with finite ment or SERVE is the smallest volume element of the deformation, displacement-based elements to accommo- microstructure exhibiting the characteristics: (i) effective date strain softening in the constitutive behavior. A sum- constitutive material properties for the SERVE should equal mary of the constitutive and damage models for each those for the entire microstructure, and (ii) it should not phase in the microstructure of LE-VCFEM is given here. depend on the location in the microstructure. Following The inclusion phase in each Voronoi cell element is as- methods of SERVE identification from actual micrographs sumed to be isotropic, linear elastic. Instantaneous crack- of heterogeneous materials in Ghosh et al. (2009), Paquet ing and fragmentation of the inclusion follows a Weibull et al. (2011), Dondeti et al. (2012), the present study uses distribution based initiation criterion, where a crack is the multivariate marked correlation function MðrÞ to introduced when the probability function Pfrag at any point establish the SERVE. Marked correlation functions relate in the inclusion exceeds a critical value. The size-depen- any geometric or response field variable with the micro- dent probability function is expressed as: structural morphology. In Ghosh et al. (2009), the marked c m c v rI correlation function MðrÞ has been calculated with the Pfrag ðv; rI Þ¼1 exp ð1Þ v0 rw micromechanical plastic work Wp as marks. A high value of MðrÞ indicates a strong correlation between entities in m and rw are the Weibull modulus and characteristic the microstructure. MðrÞ stabilizes to near-unity values at strength respectively, v0 is a reference volume, v is the c a characteristic radius of convergence r0, which signifies inclusion size and rI is the maximum principal stress in the limit of correlated variables. For r P r0; MðrÞ1 and the inclusion. The matrix phase is modeled as a rate- the local morphology ceases to have any significant influ- dependent elastic–viscoplastic porous material, extending ence on the field variables beyond this characteristic radial the GTN model framework for rate dependent behavior distance. The RVE size is estimated as LRVE 2 r0, where in Paquet et al. (2011a). The total strain-rate in this model r0 corresponds to the local correlation length. For the cast is assumed to admit an additive decomposition into an aluminum alloy W319 microstructure in Fig. 1(b) the elastic and viscoplastic part, i.e. _ ¼ _ e þ _ p. For small

SERVE size has been established as LRVE ¼ 48 lmin strains, the rate of Cauchy stress is expressed as 16 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 r_ ¼ Ce : _ e, where Ce is the fourth order isotropic elasticity where tensor. The viscoplastic behavior of the porous ductile ma- Z 1 q trix is governed by the GTN yield function as: WðxÞ¼ wðjx xjÞdX and wðjxjÞ ¼ p Xm 1 þðjxj=LÞ 2 vp q 3q2p 2 ð8Þ / ¼ þ 2f q1 cosh ð1 þ q3f Þ¼0 r M 2r M Here p ¼ 8; q ¼ 2 and L > 0 is a material characteristic ð2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length. The weighting function wðjxjÞ ¼ 1atjxj¼0; 3 0 0 1 : > q ¼ 2 r : r and p ¼3 r : I are the Von-Mises equivalent wðjxjÞ ¼ 0 25 at jxj¼L and wðjxjÞ ! 0 8 jxj L. stress and the hydrostatic pressure respectively, r0 is the Finally, an acceleration function f is introduced in Eq. deviatoric stress and q1; q2; q3 are void growth related (2) to model the complete loss of material stress carrying parameters. f is a function of the void volume fraction f capacity due to void coalescence (Tvergaard and Needle- that is expressed in Eq. (9). The subscript M is used to des- man, 1984) as: ( ignate association with the pure matrix material without 6 fff c voids and r M is the equivalent matrix stress. For the visco- f ¼ fu fc ð9Þ fc þ ðf fcÞ f > fc plastic behavior (see Perzyna (1966)), an over-stress func- ff fc tion FM is defined as a measure of the excess stress over the fc is the critical void volume fraction at which void coales- rate-independent local yield strength r0, and is expressed cence first occurs and f is the value at final failure. As the as: f void volume fraction f ! ff , the acceleration function F ¼ r r ðW Þð3Þ M M 0 p f ! fu ¼ 1=q1. To avoid numerical difficulties, f ! 0:95f f is used in Eq. (9), at which f is frozen implying local ductile The plastic strain-rate for the porous matrix is governed by material failure. Details of implementation in LE-VCFEM the associated flow rule and is a function of the over-stress are given in Paquet et al. (2011a), Hu and Ghosh (2008). U ðF Þ, expressed as Paquet et al. (2011a): M M rffiffiffi @/vp 2 r @Uvp _ p ¼ k_ ¼ð1 f Þ M cU ðF Þ ð4Þ 2.1.3. Homogenization-based continuum plasticity-damage @r 3 @Uvp M M @r r : @r (HCPD) constitutive relations for level-0 elements The rate dependent homogenization-based continuum k_ is a viscoplastic multiplier that is derived in terms of the plasticity-damage (HCPD) model for porous viscoplastic matrix plastic strain-rate _ p using the Hill-Mandel energy M materials containing brittle inclusions is developed in equivalence and c is a temperature dependent viscosity Paquet et al. (2011), Dondeti et al. (2012) based on an coefficient. A power law expression is chosen for anisotropic GTN model framework. The homogenized Cau- U ðFÞ¼hF ip (Perzyna, 1966), where hiis the MacCauley M M chy stress rate is related to the elastic strain-rate tensor as: operator. A linear hardening law governs the evolution of e e e p the yield strength r0, expressed as: R_ ¼ C : e_ ¼ C : ðe_ e_ Þð10Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 2 where Ce is a homogenized fourth order anisotropic elas- r_ ¼ hðW Þ_ p ; where _ p ¼ _ p : _ p ¼ cU ðF Þ 0 p M M 3 M M 3 M M ticity tensor. The elastic anisotropy is due to the distribu- ð5Þ tion of the inclusions. The total homogenized strain-rate is assumed to be additively decomposed into homogenized where hðWpÞ is the instantaneous plastic modulus. The elastic and viscoplastic parts as e_ ¼ e_ e þ e_ p. / is the loading rate of evolution of the local void volume fraction f is di- surface in the stress space for the homogenized three- vided into growth and nucleation parts (Chu and phase material (voids and inclusions in the matrix). Fol- Needleman, 1980; Tvergaard and Needleman, 1984) as: lowing the structure of GTN models in Gurson (1977), _ _ _ f ¼ f growth þ f nucleation where Tvergaard and Needleman (1984), Ghosh et al. (2009), the effective flow potential / in the HCPD model is ex- f_ ¼ð1 f Þ_ p and growth kk pressed in terms of the hydrostatic (Rhyd) and deviatoric

(Req) parts of the homogenized Cauchy stress tensor as: ! _ p _ p p 2 hyd f nucleation ¼ AðMÞM; AðMÞ Req 3Q 2 R 2 "# / ¼ þ 2Q 1f cosh 1 ðQ 1f Þ ¼ 0 p 2 r 2 r f 1 c c ¼ pNffiffiffiffiffiffiffi exp M N ð6Þ ð11Þ sN 2p 2 sN where f is the homogenized void volume fraction. The Here N is the mean nucleation strain, sN is its standard parameters Q 1 and Q 2 are introduced to capture the effect deviation, and fN is the intensity of void nucleation. To avoid mesh sensitivity in LE-VCFEM (Paquet et al., 2011a; of void interaction (Ghosh et al., 2009). The flow potential Hu and Ghosh, 2008) a material length scale has been in Eq. (11) exhibits anisotropy emanating from two incorporated through a non-local model where the non-lo- sources, viz. (i) dispersion of brittle inclusions in the ma- cal growth rate of void volume fraction at a material point trix, and (ii) evolution of damage (voids) in the microstruc- x is given as: ture. Anisotropy is accounted for, through a homogenized Z equivalent stress R following Hill’s anisotropic yield 1 eq f_ non-local ¼ f_ðxÞwðjx xjÞdX ð7Þ function (Hill, 1948). Under plane strain condition, this is WðxÞ Xm expressed as: S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 17

2 2 2 2 2 be functions of the strain-rate ^e_, where Req ¼ FðRyy RzzÞ þ GðRzz RxxÞ þ HðRxx RyyÞ þ CRxy ^e ¼hAðh Þe þ Bðh Þe þ Cðh Þe i is an effective strain mea- ð12Þ p 1 p 2 p 3 sure in terms of the macroscopic principal strains

All variables in Eqs. (11) and (12) are expressed in an ei; i ¼ 1; 2; 3. The coefficients A; B and C are functions of evolving, material-damage principal (MDP) coordinate sys- direction of the maximum principal strain, represented tem. In the MDP coordinate system, the material is as- by an angle hp in 2D. sumed to retain its initial anisotropy (orthotropy in this The area fraction of cracked inclusions for a given case) throughout the deformation process. The orientation strain-rate is expressed in terms of the probability density of the material-damage coordinate system is computed in function of the inclusion size pðvÞ and the probability of _ each increment of the deformation process by enforcing inclusion fragmentation Pfragðv; ê; êÞ. For a discrete size dis- the orthotropy condition as shown in Ghosh et al. (2009). tribution in a finite sized SERVE, the area fraction is ex- The anisotropy parameters F; G; H and C have been found pressed as: 0 2 !31 to be functions of the evolving plastic work W in the m ê_ p XN ð Þ vi vi ê SERVE in Ghosh et al. (2009), Paquet et al. (2011). q ðê; ê_Þ¼ pð Þ@1 exp 4 5A c vi _ In Eq. (11), r c corresponds to the averaged stress in the i¼1 v0 v0 e0ðêÞ heterogeneous material consisting of matrix and inclu- ð17Þ sions, but without voids. The corresponding over-stress F in the viscoplasticity flow rule (Perzyna, 1966) is expressed N is the number of discrete inclusionP sizes vi in the N as: probability density function pðvÞ¼ i¼1dðv viÞpðviÞ, where dðv viÞ is the Dirac delta function. Eq. (17) yields F ¼ Y ðW Þð13Þ rc f p the area fraction of cracked inclusions for a constant where Y f is the rate-independent homogenized yield strain-rate. To account for variations in strain-rates, the strength of the heterogeneous material without voids, rate of evolution of the area fraction of cracked inclusions which depends on the plastic work W . The homogenized q is assumed to be governed by the relation: p ( viscoplastic strain-rate tensor, normal to the loading sur- dq ðê; ê_Þ k~ if k~ P 1 face /ðFÞ in the stress space, is derived as (Dondeti et al., q_ ¼ k~H c ê_; for k~H ¼ ð18Þ dê ~ 2012): 0ifk < 1 with k~ ¼ 1q . The factor k~H accounts for the instanta- @/ ð1 f Þrc @/ 1q ðê;ê_Þ _ p _ c e ¼ K ¼ C0UðFÞ ð14Þ neous change in strain-rate. The homogenized void nucle- @R R : @/ @R @R ation law in Eq. (15) is then expressed as: K_ is a homogenized viscoplastic multiplier obtained by _ _ enforcing the Hill-Mandel micro–macro energy equiva- f nucleation ¼ V pq ð19Þ lence condition in Hill (1972). C0 is a temperature depen- V p is a material parameter that relates the homogenized dent viscosity coefficient and the function UðFÞ is chosen nucleated void volume fraction to the area fraction of P to be of the power law form hFi , where hiis the cracked inclusions. MacCauley operator. Finally, the evolution equations for the homogenized plastic work Wp, yield stress Yf , and void 2.2. Computational sub-domain level-1 Xl1 volume fraction f are expressed as: The level-1 subdomain is an intermediate computa- _ p _ @Yf _ _ _ _ Wp ¼ R : e_ ; Yf ¼ Wp; f ¼ f growth þ f nucleation ð15Þ tional level that is used to facilitate a switch-over from @Wp macroscopic analysis in level-0 subdomains Xl0 using the _ _ p where f growth ¼ð1 f Þekk. The homogenized void nucle- HCPD model, complete micromechanical analysis in level- _ ation rate f nucleation follows directly from the inclusion 2 subdomains Xl2 by LE-VCFEM. This subdomain is seeded cracking statistics in the underlying microstructural in regions where macroscopic variables in level-0 simula- SERVE, that occurs in LE-VCFEM simulations. A strain- tions have locally high gradients. Level-1 subdomains serve based homogenized void nucleation model is proposed in as ‘‘transition or swing’’ regions, where macroscopic gradi- Dondeti et al. (2012), accounting for the effects of the ents as well as microscopic variables in the statistically underlying microstructural morphology and rate- equivalent RVE, are processed to assess whether homoge- dependency on damage nucleation. It invokes the Weibull nization holds. Two-level analysis similar to the FE2 meth- statistics probability function that is used to initiate ods, involving the regularized macroscopic analysis and inclusion cracking in Eq. (1). The macroscopic nucleation SERVE-based micromechanical analysis, is conducted in probability function Pfrag is written in terms of the homog- these subdomains. Major steps in level-1 element compu- enized strain tensor and its rate, as well as the inclusion tations are as follows. size v as: 2 !3 (1) Macroscopic finite element analysis using the HCPD mðê_Þ v ê constitutive model in Section 2.1, is performed and P ð ; ê; ê_Þ¼1 exp 4 5 ð16Þ frag v _ the macroscopic fields are updated. v0 e0ðêÞ (2) LE-VCFEM based micromechanical analysis of the where e0 and m are the Weibull parameters and v0 is a ref- microstructural SERVE as shown in Fig. 1(b) is con- erence volume. The Weibull parameters are determined to ducted as a post-processing operation with periodic 18 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34

boundary conditions and applied strain tensor e, tively. The microscopic equilibrated stress and strain M M tan tan obtained from the HCPD-based macroscopic increments are r and respectively. Sijkl and Eijkl are analysis. the instantaneous elastic-viscoplastic compliance and stiff- (3) Appropriate criteria (e.g. violation of boundary peri- ness tensors respectively. The last two terms in Eq. (20)

odicity) are developed and applied using the micro- incorporate the effect of the applied macroscopic strain eij. structural solution to signal transition from level-1 The homogenized Cauchy stress R, total strain e, rate of to level-2 elements. change of void volume fraction f and plastic work Wp are obtained by volume-averaging respective variables over Micromechanical LE-VCFEM analysis has been combined the SERVE as: with the asymptotic expansion homogenization (AEH) the- Z 1 ory in Ghosh et al. (2009); Paquet et al. (2011); Dondeti R ¼ r ðyÞdY ðaÞ ij Y ij et al. (2012) to calculate the microstructural response of Z Y Z 1 1 ÀÁÂÃ the SERVE Y when subjected to a homogenized strain eij ¼ ijðyÞdY þ ½ui nj þ uj ni d@Y ðbÞ Y Y 2Y @Y history e and periodic conditions on the boundary @Y. Z int Y-periodicity of any function in the SERVE is expressed as _ 1 f ¼ fdY_ ðcÞ ^f x; y ^f x; y kY , where k represents a 2 2 array of ð Þ¼ ð þ Þ Y Y Z integers. In an incremental formulation, the Y periodic _ 1 p Wp ¼ rij_ ðdÞð21Þ displacement conditions Duiðx; yÞ¼Duiðx; y þ kYÞ are ij Y Y applied on @Y and the macroscopic strain eij þ Deij is p imposed on Y. To implement the AEH theory in conjunction where is the microscopic plastic strain. The second term with LE-VCFEM based micromechanical analysis, a modifi- in Eq. (21)(b) corresponds to a discrete crack opening. cation of the formulation in Hu and Ghosh (2008), Paquet Since the fracture in this model is represented parametri- et al. (2011a) is needed for the energy functional in Y. cally, this term is omitted. The SERVE is tessellated into N Voronoi cell elements, each encompassing a region Ye and comprised of a boundary 2.3. Computational sub-domain level-2 (Xl2) e @Y e with outward normal n . Furthermore the matrix- inclusion interface is delineated as @Yc with outward nor- The sub-domain Xl1 is replaced by level-2 sub-domains c mal n , while the inclusion crack is @Ycr with outward nor- Xl2, where micromechanical analysis of detailed micro- mal ncr. Increments of microscopic displacements on the scopic regions is needed. The explicit morphology of the element boundary @Y e, inclusion-matrix interface @Yc and underlying microstructure is incorporated to be solved c cr crack boundary @Ycr are represented by Mu; Mu ; Mu concurrently by the microstructural LE-VCFEM (Hu and respectively. The corresponding incremental energy func- Ghosh, 2008; Paquet et al., 2011a; Ghosh, 2011) in these tional for each Voronoi cell element is written as Ghosh regions. High resolution microstructure morphology is re- et al. (2009): quired to solve the micromechanical boundary value prob- Z Z lem in Xl2. A method to reconstruct high resolution 1 P ¼ StanMr Mr dY Mr dY heterogeneous microstructures from limited micrograph e 2 ijkl ij kl ij ij ZYenYs YenYs information has been developed in Ghosh et al. (2006). M M e þ ðrij þ rijÞðui þ uiÞnj d@Y Z@Ye 2.4. Computational sub-domain level-tr (Xtr) ðrm þ Mrm rc Mrc Þðuc þ MucÞncd@Y ij ij ij ij i i j The interface between macroscopic level-0/level-1 ele- Z@Yc ments in Xl0=l1 and level-2 elements in Xl2 with explicit c M c cr M cr cr ðrij þ rijÞðui þ ui Þnj d@Y morphology requires satisfaction traction and displace- @Y Z cr Z ment continuity conditions. To facilitate smooth transition 1 tan s s s s Eijkl DijkldY rijDijdY of scales across the disparate element boundaries, a layer Ys 2 Ys Z of level-tr transition elements in Xtr is sandwiched be- s s s s s tween elements in level-0/level-1 and level-2 subdomains. þ ðrij þ DrijÞðui þ Dui Þnj d@Y Z@Ys Z Elements in Xtr are essentially level-2 elements that have M M M tan s compatibility and traction continuity constraints imposed þ ðeij þ eijÞ rijdY þ ðeij þ eijÞEijkl DkldY YenYs Ys on their interface with level-0/level-1 elements. Transition ð20Þ elements are located beyond the level-2 regions, away from critical hot-spots. It has been shown in Ghosh (2011); All variables in Eq. (20) represent association with both Ghosh et al. (2007) that if displacement constraints are di- x scales, differentiated by a factor ¼ y 1, where x and y rectly imposed on nodes of Voronoi cell finite elements in correspond to macroscopic and microscopic length scales the transition elements for compatibility with the adjacent respectively. In LE-VCFEM Ys is the region of displace- level-1 elements, spurious stress concentrations arise at the ment-based FE enrichment in regions of strain softening interface. This has been averted by incorporating a relaxed, within each Voronoi cell element Y e. Superscripts m; c; cr displacement-constraint method in Ghosh (2011); Ghosh and s correspond to variables associated with the matrix, et al. (2007). In this method, a weak form of the interface inclusion, crack and the enriched displacement-based fi- displacement continuity is implemented by using Lagrange nite element region in each Voronoi cell element respec- multipliers. It relaxes the level-0-transition element inter- S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 19 face displacement constraint to satisfy compatibility in a tion criterion is developed in terms of void volume fraction weak sense as described in Section 4. f and its gradients in Xl0. The transition of an element e in

Xl0 to Xl1 is conditioned upon the criterion: 3. Mesh adaptivity and level change criteria gdf H gdf H Ee fe P C2Emaxfmax ð23Þ H f f 0 There are two classes of error leading to adaptivity in where f ¼ is the normalized void volume fraction f 0 the proposed model. The first one is to reduce discretiza- with respect to the initial void volume fraction f 0. In Eq. gdf H tion error in the macroscopic scale due to inadequate (23), Ee is the norm of the local gradient of f expressed refinement of the level-0 mesh. This adaptation only refines as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the mesh to reduce error due to inappropriate FE discreti- H2 H2 zation. The error could however be an implicit indicator of gdf @fe @fe Ee ¼ þ ð24Þ where high gradients in the solution exist and have the @x @y secondary benefit of identifying regions, which lack ade- H gdf The quantities fmax and Emax are the maximum values of all quate resolution in the physics of the problem. The second H gdf fe and Ee . C2 < 1 is a prescribed factor determined from adaptivity is to explicitly reduce modeling error due to numerical experiments. The gradient of f H is computed inadequate representation of the physics of the problem by first interpolating over a patch of elements according through transcending scales. This leads to multi-scaling to the Zienkiewicz-Zhu super-convergent patch recovery and level transitions. The mathematical development of method (Zienkiewicz et al., 1992) and then differentiating. rigorous bounds for discretization and modeling error for highly nonlinear problems with damage is very difficult 3.3. Criteria for switching from level-1 to level-2 elements and hence rare in the literature. Adaptation criteria used in this paper are developed from physical considerations. Transition from level-1 to level-2 is activated for elements that fail the macroscopic uniformity and RVE peri- 3.1. Mesh refinement for level-0 elements odicity tests. Level-1 elements already correspond to those for which macroscopic nonuniformity has been Adaptive mesh refinement by h-adaptation of level-0 established according to the criterion in Eq. (23). Subse- elements is conducted to reduce the discretization error quently, departure from RVE periodicity is used as an indi- associated with critical variables in the solution. The h- cator for a switch from level-1 to level-2 elements. The adaptation procedure subdivides macroscopic elements switching criterion is developed in terms of a violation of into smaller elements to reduce a suitably chosen error, the traction anti-periodicity condition in the microstruc- as well as to identify regions of modeling error by zooming tural SERVE, quantified as: in on regions with evolving gradients. The adaptation crite- P R NSGPR iþ i iþ i rion is formulated in terms of the jump in traction across i¼1 C tx þ tx i þ ty þ ty j dC TRapt ¼ i ð25Þ adjacent element boundaries (Ghosh et al., 2007), which e NSEGX R i i is directly related to the local stress gradients (Bass and maxe t i þ t j dC Ci x y Oden, 1987). It states: Refine element e in Xl0 if the traction i¼1 jump error across the element boundary satisfies the NSGPR is the number of boundary segment-pairs on the condition: boundary of each SERVE (shown in Fig. 1(b)) over which tj tj the stresses should be anti-periodic and NSEG is the total E P C1E where e max R number of boundary segments on the SERVE. Tractions t 2 2 tj tj tj 2 @X ð½½Tx þ½½Ty Þd@X with superscripts + and in the numerator correspond E ¼ maxðE Þ and ðE Þ ¼ e R ð22Þ max e e e d@X to those on the segment pairs with anti-periodicity condi- @Xe tions. The numerator is a measure of the residual traction tj tj Emax is the highest of all Ee values for all level-0 elements violating the anti-periodicity condition. The denominator and the factor C1 < 1 is chosen from numerical experi- on the other hand corresponds to the maximum value of ments. Tx and Ty are the boundary traction components the absolute sum of all traction measures in all the SERVE’s in x and y directions and ½½ is the jump operator across apt of level-1 elements. TRe provides a measure of the lack of apt the element boundary @Xe. This criterion also has the sec- anti-periodicity of boundary tractions, since TRe ¼ 0if ondary benefit of identifying regions where damage is ex- and only if the boundary tractions are anti-periodic. A le- pected to localize expected due to local stress vel-1 element e in Xl1 is switched to a level-2 microscopic concentration and gradients. element if: TRapt P C ð26Þ 3.2. Criteria for switching from level-0 to level-1 elements e 3

where C3 < 1 is a constant determined from numerical The transition from level-0 to level-1 elements is aimed experiments. at identifying regions of departure from homogenizability, In addition, the level-0/level-1 to level-2 switching crite- due to the intensity of local deformation and deformation rion is also activated for elements undergoing significant gradients. For problems involving ductile deformation damage according to the criterion: and damage, localization of macroscopic void volume frac- q P q 8e 2 X ð27Þ tion is an indicator of this departure, and hence, the transi- e crit l1 20 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 where q is the area fraction of cracked inclusions in the where e Z Z SERVE of level-1 element e. A value of qcrit ¼ 0:1 has shown l0 nþ1 l0 l0 @du l0 l0 l0 to significantly reduce the sensitivity of the solution to the dP ¼ ðR þ DR Þ i dX ðt þ Dt Þdu dC Xl0 ij ij @x i i i Xl0 j Cl0 parameter C3 in Eq. (26). Z Z @dul1 dPnþ1 ¼ ðRl1 þ DRl1Þ i dX ðtl1 þ Dtl1Þdul1dC Xl1 ij ij i i i 3.3.1. Updating history-dependent state variables in X @xj C Z l1 Z l1 transformed level-2 elements @dul2 dPnþ1 ¼ ðrl2 þ Drl2Þ i dX ðtl2 þDtl2Þdul2dC Once the regions of level-2 and level-tr elements have Xl2 ij ij i i i X @xj C been identified, it is important to update the microstruc- Z l2 Z l2 @dutr tural information within each newly created element. This dPnþ1 ¼ ðrtr þ DrtrÞ i dX ðttr þ DttrÞdutrdC Xtr ij ij @x i i i includes the history of displacements, stress, elastic and Xtr j Ctr plastic strains, and damage. It should be noted that a simple mapping of level-0/level-1 variables onto level-2 Z nþ1 l0=l1 l0=l1 l0=l1 l0=l1 elements is not feasible. Their morphologies are very dif- dP ¼ d ðk þ Dk Þð i þ D i u Du ÞdC Cint i i v v i i ferent, one being homogeneous and the other heteroge- CZint neous. Alternatively, the history of all state and internal tr tr tr tr þ d ðk þ Dk Þðvi þ Dvi u Du ÞdC variables in the microstructure are obtained by incremen- i i i i Cint tally solving the local micromechanical LE-VCFEM bound- nþ1 ary value problem for each level-2 element from the start In the functional Phet (28), superscripts l0; l1; l2, and tr as- (increment 1) to the increment just before level transfor- sociate the variables to their respective sub-domain. Vari- nþ1 mation. The history of the macroscopic displacement solu- ables in Phet are evaluated at the end of the increment tion on the level-0/level-1 element boundary is utilized for n þ 1, i.e. ðÞnþ1 ¼ ðÞn þðD Þnþ1 where the prefix ‘D’ de- this purpose. This macroscopic displacement history is ap- notes increments of the respective variables in the incre- plied as boundary conditions to the micromechanical LE- mental solution process. The multi-level functional in Eq. l0=l1 VCFEM problem for each level-2 element. (28) couples macro-scale fields such as stresses Rij , trac- l0=l1 l0=l1 The local micromechanical analysis precedes the cou- tions ti and displacements ui in Xl0 [ Xl1 with their l2=tr l2=tr l2=tr pled concurrent analysis of the current step. Once the his- micro-scale counterparts rij ; ti and ui in Xl2 [ Xtr tory has been generated, the newly created level-2 element through the term Pnþ1 associated with Lagrange multi- Cint l0=l1 tr is connected to the multi-scale mesh and a relaxation step pliers ki and ki corresponding to macro-scale and mi- is performed to recover local equilibrium. cro-scale variables at the interface Cint respectively. The l0=l1 stress Rij is the homogenized macroscopic stress in Xl0 l2=tr 4. Coupling multiple levels in the concurrent setting and Xl1, while the stress rij is obtained from LE-VCFEM solution of the microstructural regions Xl2 and Xtr. l0 l1 l2 tr Concurrent multi-scale analysis requires that all levels ui ; ui ; ui , and ui are displacement degrees of freedom of consisting of different sub-domains Xl0; Xl1; Xl2, and Xtr elements in the Xl0; Xl1; Xl2, and Xtr sub-domains respec- l0 l1 l2 tr be coupled and solved simultaneously. The global stiffness tively. Tractions ti ; ti ; ti , and ti are prescribed on the matrix and load vectors are derived for the entire compu- boundaries Cl0; Cl1; Cl2, and Ctr respectively. On each seg- tational domain Xhet ¼ Xl0 [ Xl1 [ Xl2 [ Xtr. The corre- ment of Cint, the displacement field vi is interpolated using sponding domain boundary is partitioned as a suitable polynomial function, independent of the inter-

Chet ¼ Cl0 [ Cl1 [ Cl2 [ Ctr, where polations on @Xl0;@Xl1;@Xl2, and @Xtr. The functional Pnþ1 is used to enforce displacement continuity and trac- Cl0 ¼ Chet \ @Xl0; Cl1 ¼ Chet \ @Xl1; Cl2 ¼ Chet \ @Xl2, and Cint l0=l1 tr Ctr ¼ Chet \ @Xtr. The continuity of displacements and traction reciprocity on Cint in a weak sense using ki and ki , tions across the macro-scale sub-domain Xl0 [ Xl1 and mi- which are vector columns of macro-scale and micro-scale cro-scale sub-domain Xl2 [ Xtr is enforced using the Lagrange multipliers respectively. The Euler equations ob- l0=l1 tr relaxed, displacement-constraint method in Ghosh tained by setting the coefﬁcients of dki and dki to zero (2011); Ghosh et al. (2007). The method uses Lagrange respectively in Eq. (28) are: multipliers to enforce (in a weak sense) continuity of dis- Coefficient of dkl0=l1 : ul0=l1 þ Dul0=l1 ¼ þ D on C placements and tractions at the interface between i i i vi vi int tr tr tr macro-scale and micro-scale elements. The interface be- Coefficient of dki : ui þ Dui ¼ vi þ Dvi on Cint ð29Þ tween these two modeling scales is denoted as These correspond to displacement continuity across Cint. Cint ¼ð@Xl0 [ @Xl1Þ\ð@Xl2 [ @XtrÞ. Continuity of tractions across Cint is obtained by setting l0=l1 tr the coefﬁcients of dvi; du , and du to zero: 4.1. Weak form for the concurrent multi-scale model i i l0=l1 l0=l1 Coefficient of dvi : ki þ Dki The principle of virtual work for Xhet at the end of the tr tr ¼ðk þ Dk Þ on Cint ð30Þ increment n þ 1, associated with a virtual displacement i i

ﬁeld dui, is expressed as a sum of the individual contribu- l0=l1 : l0=l1 l0=l1 tions from each sub-domain as: Coefficient of dui ki þ Dki l0=l1 l0=l1 l0=l1 nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 ¼ðR þ DR Þn on C dP ¼ dP þ dP þ dP þ dP þ dP ¼ 0 ð28Þ ij ij j int het Xl0 Xl1 Xl2 Xtr Cint S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 21

tr tr tr tr tr tr Coefficient of dui : ki þ Dki ¼ðrij þ Drij Þnj on Cint In the present work, a Quasi-Newton iterative solver, spe- ciﬁcally the BFGS solver, is used to solve Eq. (36). For a gi- l0=l1 tr where nj is the unit normal vector and ki and ki corre- ven increment, the stiffness matrix @R is evaluated only @qi spond to the interfacial traction components on once, at the beginning of the iterative process. Introducing @Xl0=l1 \ Cint and @Xtr \ Cint respectively. The Euler equa- Eqs. (31)–(35)))) into Eq. (36) and taking derivatives, yields tions resulting from setting the coefﬁcients of dul0=l1 and i the system of algebraic equations to be solved for the gen- l2=tr dui to zero lead to the equilibrium equation within each eralized degrees of freedom qi (Ghosh et al., 2007): sub-domains and traction continuity conditions between the sub-domains.

2 3 I;I I;O i8 9 8 9 K K 000P 0 i I i l0=l1 l0=l1 l0=l1 > I > > DF > 6 7 > Dql0=l1 > > l0=l1 > 6 O;I O;O 7 > > > > 6 K K 000 007 > > > O > l0=l1 l0=l1 > DqO > > DF > 6 7 > l0=l1 > > l0=l1 > 6 I;I I;O 7 > > > > 6 00K K 00Ptr 7 > I > > I > 6 tr l2=tr 7 < Dqtr = < DF = 6 7 tr O;I O;O O O ð37Þ 6 00K K 0007 > Dq > ¼ > DF > 6 l2=tr l2=tr 7 > l2=tr > > l2=tr > 6 7 > > > > 6 00000Q Q 7 > Dqint > > DF > 6 l0=l1 tr 7 > > > int > T > > > > 6 T 7 > DKl0=l1 > > DF > 4 Pl0=l1 000Q l0=l1 005 :> ;> :> kl0=l1 ;> T T DKtr DFktr 00Ptr 0 Q tr 00

4.2. Finite element discretization for multi-scale analysis The superscript I in Eq. (37) is associated with nodes lying

on Cint, while O is associated with all other nodes. The use l0=l1 Displacements vi and the Lagrange multipliers ki and of two superscripts separated by a comma indicates the tr ki are interpolated from nodal values using suitable shape coupling effect. The stiffness sub-matrix ½Kl0=l1 and sub- functions as: vector fFl0=l1g for level-0 and level-1 elements are derived as: fvg¼½Lintfqint gð31Þ Z @Na @Rmn @Nb l0=l1 ðKl0=l1Þmanb ¼ dXðDFl0=l1Þma fk g¼½L l0=l1 fK gð32Þ @xk @ekl @xl k l0=l1 ZXl0[Xl1 Z

tr ¼ ðtm þ DtmÞNadC þ ðkm þ DkmÞNadC fk g¼½Lktr fKtrgð33Þ CZt Cint The displacements ul0 and ul1 in each level-0 and level-1 @Na i i ðR þ DR Þ dX ð38Þ mn mn @x elements are interpolated with the standard Legendre Xl0[Xl1 n polynomials based shape functions as: () Subscripts ðm; nÞ are associated with degrees of freedom, I while ða; bÞ correspond to node numbers in the element. l0 I O ql0 fu g¼½Nl0fq g¼½N N ð34Þ l0 l0 l0 qO These matrices and vectors are partitioned based on the l0 corresponding I and O nodes. Coupling between the level- () 0 and level-1 elements with level-2 and level-tr elements qI ful1g¼½N fq g¼½NI NO l1 ð35Þ is assured by the matrices ½P and ½Q defined as: l1 l1 l1 l1 O Z ql1 T ðPl0=l1Þmanb ¼ NmaðL l0=l1 ÞnbdC ð39Þ The nodal displacements in level-0 and level-1 elements are k Z Cint partitioned into two sets. Those for nodes in Xl0=l1 lying on T I ðPtrÞmanb ¼ NmaðLktr ÞnbdC the interface Cint are denoted as ql0=l1, while the other de- ZCint grees of freedom are qO . l0=l1 T ðQ l0=l1Þmanb ¼ ðLintÞmaðLkl0=l1 ÞnbdC 4.3. Iterative solution of the coupled multi-scale system Z Cint T ðQ trÞmanb ¼ ðLintÞmaðLktr ÞnbdC ð40Þ An iterative solver is used to solve the nonlinear alge- Cint braic equations obtained from the weak forms in Eq. (28) with their corresponding residual vectors fFg, defined as: by setting the residual R ¼ 0. In a Newton–Raphson itera- Z tive solver, the i-th update of q at increment n þ 1 can be T l0=l1 l0=l1 i ðDFintÞma ¼ ðLintÞaðk þ Dk ÞmdC obtained from the linearized form: ZCint @R T tr tr nþ1 nþ1 i ðLintÞaðk þ Dk ÞmdC ð41Þ R ¼ðR Þ þ Dqi ¼ 0 ð36Þ Cint @qi 22 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 Z T The microscopic stress ryy along the middle section, ob- ðDF l0=l1 Þ ¼ ðL l0=l1 Þ ½vm þ Dvm dC k ma k a tained for DU ¼ 0:192lm with the micromechanical and ZCint ÂÃmulti-level models with C modeled by (p ¼ 1; 4) polyno- T int þ ðLkl0=l1 Þa ðul0=l1Þm þ Dðul0=l1Þm dC mial shape functions, are plotted in Fig. 2(a) and 2(b) for Cint the two microstructures. Contour plots of ryy for the Z micromechanical and multi-scale analyses are also shown T ðDFktr Þma ¼ ðLktr Þa½vm þ Dvm dC in Figs. 3 and 4. These numerical results show that while ZCint higher order polynomials lead to a better continuity of ÂÃ T stresses across the interface, the resulting stress field is þ ðLktr Þa ðutrÞm þ DðutrÞm dC Cint not necessarily more accurate in comparison with lower order interfaces. The linear interface (p ¼ 1) representation Finally, the stiffness sub-matrix and load sub-vector asso- is deemed to adequately represent the interface in the re- ciated with level-2 and level-tr elements are obtained from laxed displacement constraint method and is henceforth LE-VCFEM calculations followed by a static condensation adopted. to retain the boundary terms only. 5.2. Calibration and validation of the level-1 to level-2 criteria 5. Numerical studies and validations This set of examples calibrates and validates the criteria Numerical studies are undertaken to: (a) analyze the ef- in Section 3.3 for identifying level-1 elements that should fect of the interface Cint on the solution, (b) to demonstrate be switched to level-2 elements. The geometry of the prob- the capabilities of the level-change criteria, and (c) to val- lem considered is a square plate with a square hole in its idate the multi-scale model for a problem for which a com- center with dimensions shown in Fig. 5(a). Only one quar- plete micromechanical analysis is tractable. ter of the plate is modeled and appropriate symmetry boundary conditions are prescribed. In addition to the

5.1. Effect of interface Cint between macro and micro sub- symmetry boundary conditions, a total prescribed dis- domains placement DUA is applied in the x direction on the left edge (x ¼ 0) of the plate. The bottom of the plate y ¼ 0 is free of This study investigates the effect of the order p of poly- any prescribed displacement and no traction boundary nomial shape functions used to interpolate displacement v conditions are applied. The silicon inclusion and aluminum on Cint. Appropriate shape functions must be used in the matrix parameters used for rate-independent microme- interpolation Eq. (31) to ensure that the interface retains chanical LE-VCFEM analysis are listed in Table 1 and Table accuracy with respect to microscopic displacements, stres- 2 respectively. The plastic hardening behavior of the alu- ses, and strains in Xl2 [ Xtr. Two different microstructures minum matrix without voids and without inclusions is with dimensions 48 lm 96 lm in the x and y directions plotted in Fig. 6. are considered for this demonstration. Uniaxial tension in First, the reference solution is obtained by solving a the y direction is applied by prescribing the following complete micromechanical analysis for the entire plate boundary conditions: with level-2 elements corresponding to the SERVE in

Uy ¼ DU at y ¼ 96 lm; Uy ¼ 0aty ¼ 0; Ux ¼ 0atx ¼ 0. Fig. 5(c). The next simulation models all macroscopic ele- The computational domain is discretized into 2 ments as level-1 elements in a fully coupled FE2 ﬁrst-order 48lm 48 lm macro-scale. The two underlying micro- homogenization scheme. At each increment, the value of apt structures of aluminum matrix containing silicon inclu- TRe in Eq. (25) is computed for each element and stored. apt sions analyzed are: Evolution of TRe is then compared with the evolution of microstructural damage in the microscopic computational apt (1) A single elliptical inclusion of volume fraction domain of LE-VCFEM analysis. Contour plots of TRe ob- V f ¼ 10%, aspect ratio a ¼ 2:0, and orientation tained with the homogenization scheme are shown in o h ¼ 108:8 with respect to the horizontal axis, in Fig. 7 for a total applied displacement of UA ¼2:6 lm. the square element. The corresponding contour plots of the microscopic stress

(2) Distributed microstructure from the micrograph of component rxx, obtained by micromechanical analysis are apt cast aluminum alloy. also shown. These contour plots demonstrate that TRe successfully identiﬁes the regions where macroscopic

The reference solutions are obtained by solving the micro- non-uniformities arise. To ascertain a value for C3 in Eq. mechanical problem with LE-VCFEM, for which the mate- (26), contour plots of rxx for the mutli-level model are rial parameters are given in Section 5.3. Inclusion compared with those for the fully micromechanical cracking is not considered in this study. The multi-level analysis. Two different values for C3 are considered, viz. computational domain is comprised of one level-0 macro- C3 ¼ 0:08 and 0:10. The contour plots of rxx at scopic element (top element) that uses a homogenized UA ¼2:6 lm obtained by the multi-level simulations in constitutive model and one microscopic level-tr element Fig. 8(a) and (b) are compared with those for the microme-

(bottom element). The polynomial order p is varied from chanical analysis in Fig. 7. C3 ¼ 0:08 gives a good agree- linear (p ¼ 1) to quartic (p ¼ 4). The microscopic stress ment with the micromechanical results. The Figs. 8(a) ﬁeld obtained from the multi-level analysis is compared and (b) indicate a strong sensitivity of the solution to the with the micromechanical LE-VCFEM solution. selected value C3 in Eq. (26). To mitigate this sensitivity S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 23

0.2

0.18

0.16 (GPa) yy

σ 0.14

0.12 micromechanics p=1 p=4 0.1 0 5 10 15 20 25 30 35 40 45 50 x (μm)

0.2

0.18

0.16 (GPa) yy

σ 0.14

0.12 micromechanics p=1 p=4 0.1 0 5 10 15 20 25 30 35 40 45 50 x (μm)

Fig. 2. Microscopic stress ryy (GPa) distribution along the interface Cint for the micromechanical and multi-scale analyses: (a) single inclusion microstructure, and (b) distributed aluminum microstructure.

Fig. 3. Contour plot of microscopic stress component ryy (GPa) for the single inclusion microstructure by: (a) micromechanical analysis, (b) multi-level analysis with p ¼ 1, and (c) multi-level analysis with p ¼ 4. in the multi-level analysis, Eq. (27) is also activated for this for the micromechanical analysis in Fig. 7. The combina- level change. A value qcrit ¼ 0:1 increases the robustness of tion of criteria (26) and (27) makes the multi-scale algo- the multi-scale model by reducing the sensitivity of the rithm very reliable and accurate. solution to the parameter C3. This is shown in Fig. 8(c) Finally to validate the effectiveness of Eqs. (26) and for multi-level analysis with C3 ¼ 0:10 and qcrit ¼ 0:10. (27), the multi-level model is solved for the boundary va- Good agreement is found between these results and those lue problem of cast aluminum alloy in Fig. 5. The 24 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34

Fig. 4. Contour plot of microscopic stress component ryy (GPa) for the distributed aluminum alloy microstructure by: (a) micromechanical analysis, (b) multi-level analysis with p ¼ 1, and (c) multi-level analysis with p ¼ 4.

Fig. 5. (a) Geometry of the square plate with a center square hole, with applied symmetry boundary conditions, (b) underlying microstructure taken from a micrograph of a W319 cast aluminum alloy, and (c) statistically equivalent representative volume element (SERVE) (48 lm 48 lm).

microstructure is taken from a micrograph in Fig. 5(b). Table 1 Fig. 9 shows the contour plot of the microscopic stress Inclusion elastic and cracking properties used in LE-VCFEM simulations. rxx for an applied displacement UA ¼4:32 lm by the E (GPa) m r (MPa) m 2 cr w v0 (lm ) Pfrag ð%Þ adaptive multi-level model and a micromechanical simula- 165 0.27 680 2.4 8.29 55 tion. The two results agree very well, which conﬁrms the effectiveness of the multi-level model in predicting ductile fracturing.

Table 2 Aluminum matrix elastic, plastic and void evolution properties used in LE-VCFEM simulations.

1 1 E (GPa) m c0 (GPa s )pf0 fc ff N sN fN 70 0.35 8.086 1 0.01 0.15 0.25 0.2 0.075 0.08 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 25

0.8 reported in Tables 3 and 4 and the plastic hardening curve Hardening curve for pure aluminum matrix of the aluminum matrix is shown in Fig. 6. Rate effects are 0.7 not considered in this example. The prescribed boundary

0.6 conditions are: Ux ¼DU at y ¼ 576 lm; Ux ¼ 0at x ¼ 576 lm, and Uy ¼ 0atðx; yÞ¼ð576 lm; 0Þ, where DU 0.5 is incremented till fracture. Results of multi-scale analysis by the multi-level and (GPa) 0 0.4 σ micromechanical models are depicted in the Figs. 10 and 11. Fig. 10(b) shows the evolved multi-level mesh at the 0.3 end of the loading sequence. When the homogenized stress 0.2 in a level-2 element decreases sharply, it indicates rapid propagation of microstructural damage in the matrix. At 0.1 total failure, the level-2 element is replaced by a sealed 0 0.5 1 1.5 2 2.5 3 3.5 4 macroscopic element (black elements in Fig. 10(a)) with εp M near-zero constant stresses. This sealing feature in the wake of a crack signiﬁcantly speeds up the multi-level sim- Fig. 6. Stress–strain behavior of the aluminum matrix used in LE-VCFEM simulations. ulations. Fig. 10(a) shows a comparison of the total reaction force at the ﬁxed edge x ¼ 576 lm as a function of

the applied displacement Ux by the multi-level and micro- 5.3. Validation of the multi-level model against mechanical simulations. A very good match is obtained be- micromechanical analysis tween the two models. Each drop in the multi-level model results from failure of the underlying level-2 microstruc- This study tests the accuracy of the multi-level model in ture and sealing with macroscopic elements. Contour plots p simulating ductile fracture by comparing its predictions of void volume fraction f and equivalent plastic strain M with those obtained by a LE-VCFEM-based micromechani- by the two methods at ﬁnal fracture are shown in the Figs. cal analysis. The macroscopic computational domain, a 11. The two contour plots in Figs. 11(b) and (c) concur in L-shaped connector link subjected to bending forces, is dis- predicting the ductile crack path in the underlying micro- cretized into 108 QUAD4 elements as shown in Fig. 5(a). structure. The contour plots for the multi-level model The underlying microstructure is kept simple for ease of shows some discontinuities due to the element sealing micromechanical analysis by LE-VCFEM. Each macro-scale process. This example demonstrates the accuracy of the element is assumed to comprise a single silicon inclusion multi-level model for solving the multi-scale ductile frac- of volume fraction V f ¼ 10%, aspect ratio a ¼ 2:0, and ori- ture problems. entation h ¼ 0o with respect to the horizontal axis, embed- As a concluding example, the effect of loading rates on ded in a square aluminum matrix (this constitutes the RVE ductile fracture is investigated. For this, the boundary too). Material parameters for individual phases are value problem is solved for two different loading rates,

apt Fig. 7. Contour plot of TRe indicating the intensity of departure from traction anti-periodicity within each element of the fully macroscopic level-1 apt simulation and the corresponding contour plot of rxx (GPa) for the fully microscopic level-2 simulation: (a) TRe at UA ¼2:6 lm, (b) rxx at UA ¼2:6 lm. 26 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34

apt Fig. 8. Microscopic stress component rxx (GPa) with UA ¼2:6 lm, for which transition from level-1 to level-2 elements occurs at: (a) TRe ¼ 0:10, (b) apt apt TRe ¼ 0:08, and (c) TRe ¼ 0:10 and q > 0:10.

_ corresponding to prescribed velocities Ux ¼1:5 lm=s and This indicates a strong sensitivity of the ductile behavior _ Ux ¼15:0 lm=s respectively. Fig. 12(c) shows the corre- to the applied loading rate. sponding force–displacement response. For the higher rate, the total applied displacement at ﬁnal fracture is signiﬁ- 6. Ductile failure of a cast aluminum tension bar cantly higher even though the high strain-rate in the microstructure result in early inclusion cracking. This in- The multi-level model is now applied for multi-scale crease of ductility is attributed to stress redistribution that ductile fracture analysis of an Al-Si-Mg cast aluminum al- delays the evolution of plasticity in the neighborhood of loy W319 rectangular bar, for which the micrograph is the cracked inclusions, thus reducing void nucleation and shown in Fig. 1(a). It comprises age-hardened ductile alu- growth rates in those regions. Contour plots showing the minum matrix, strengthened by Mg/Si and a dispersion distributions of the equivalent plastic strain in the micro- of brittle Si particulates. Dimensions of the rectangular structure for the two rates, are shown in Fig. 12(a) and computational domain are: 384 lm 1536 lm in the hor- (b). The viscoplastic response of the ductile matrix induces izontal (x) and vertical (y) directions respectively. High res- higher stresses in the silicon inclusions, resulting in a dis- olution microstructure of the alloy is mapped on the tribution of damage that is much more diffuse in compar- rectangular specimen following procedures detailed in ison with the rate-independent behavior of the structure. Ghosh et al. (2006). Since it is impossible to analyze the S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 27

apt Fig. 9. Microscopic stress rxx (GPa) for an applied displacement UA ¼4:32 lm by: (a) adaptive multi-level simulation with TRe ¼ 0:10 and q > 0:10, and (b) fully micromechanical simulation.

Table 3 Inclusion elastic and cracking properties used in LE-VCFEM simulations. entire domain using a micromechanics model, the multi- 2 cr E (GPa) m rw (MPa) m v0 (lm ) Pfrag ð%Þ level model is initiated with a level-0 computational domain X for the entire bar. The specimen is discretized into 165 0.27 300 2.4 230.4 95 l0

Table 4 Aluminum matrix elastic, plastic and void evolution properties used in LE-VCFEM simulations.

E (GPa) m 1 1 p f f f s f c0ðGPa s Þ 0 c f N N N 70 0.35 8.086 1 0.001 0.15 0.25 0.10 0.075 0.08

0.02 Multiscale 0.018 Micromechanics 0.016

0.014

0.012 m) μ 0.01

F (N/ 0.008

0.006

0.004

0.002

0 0 10 20 30 40 50 |U | (μm) x

Fig. 10. (a) Comparing the total reaction force F at x ¼ 576 lm as a function of the applied displacement Ux by multi-level and micromechanical analyses; (b) evolved adaptive multi-level mesh at the end of simulation. (Legend: level-0 (turquoise), level-1 (blue), level-2 & level-tr (red), sealed elements (black)). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) 28 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34

p Fig. 11. Contour plots showing the ﬁnal crack path at complete failure of the structure: (a) void volume fraction f and (b) equivalent plastic strain M by the p multi-level model, and (c) equivalent plastic strain M by the micromechanical model.

8 32 ¼ 256 macro-scale elements, each of which has the 6.1. Parameters in the HCPD model for level-0 elements dimensions 48 lm 48 lm as shown in Fig. 15. The constitutive model for this domain is the rate-dependent Parameters for the rate-dependent HCPD model are cal- homogenization-based continuum plasticity-damage or ibrated for the SERVE in Fig. 1(b) in Dondeti et al. (2012). HCPD model described in Section 2.1.3. The statistically The anisotropy parameters F; G; H and C in Eq. (12) and equivalent representative volume element for this micro- the yield strength Yf ðWpÞ in Eq. (13) are derived in Dondeti structure has been determined in Ghosh et al. (2009) to et al. (2012) as functions of the evolving plastic work Wp as be of the size 48 lm as shown in Fig. 1(b). The HCPD model shown in Fig. 13. The homogenized viscoplastic parame- 1 1 has been developed in Dondeti et al. (2012) by homogeniz- ters in Eq. (14) are C0 ¼ 6:13GPa s and P ¼ 1. The ing results of LE-VCFEM-based micromechanical analysis, parameters in Eq. (11) are estimated as Q 1 ¼ 1:89 and for which the inclusion and matrix material parameters Q 2 ¼ 1:01. The parameters e0 and m in Eqs. (16) are plotted are listed in Tables 1 and 2 respectively and plotted in in Figs. 14 as functions of the local strain-rate. The anisot- Fig. 6. ropy parameters A; B and C have functional forms of an el-

lipse in terms of the principal strain angle hp. The values S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 29

0.05

0.045 Rate-independent 0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0 0 102030405060

p _ _ Fig. 12. Contour plots of the equivalent plastic strain M for different loading rates: (a) Ux ¼1:5 lm=s, (b) Ux ¼15:0 lm=s, and (c) reaction force F at x ¼ 576 lm as function of the applied displacement for two loading rates.

0.25

0.2

0.15 (GPa) f Y 0.1

0.05

0 0 0.002 0.004 0.006 0.008 0.01 W (GPa) p

b b b Fig. 13. (a) Evolution of anisotropy parameters F; G and H, (b) yield stress in shear Yf with plastic work for SERVE in Fig. 1(b). 30 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34

14 Values calibrated from Values calibrated from 0.01 Avg. VCFEM response Avg. VCFEM response 12 Functional fit Functional fit

0.008 10

. . 8 . . 0.006 ^ ^ ^ ^ 0 −0.3108e −21.4e −30.22e −0.03124e m e e =0.009852e −0.003372e m=12.37e +2.927e 0 6 0.004 4

0.002 2

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 . 0.5 0.6 0.7 0.8 . e^ e^

_ Fig. 14. Evolution of parameter (a) e0, and (b) m with local strain-rate ^e.

Fig. 15. Evolution of the adaptive multi-level mesh for the multi-scale analysis of a rectangular specimen loaded in tension at: (a) U ¼ 0, (b) U ¼ 7:8 lm, (c) U ¼ 10:3 lm, (d) U ¼ 13:2 lm, (e) U ¼ 13:5 lm, (f) U ¼ 13:7 lm. (Legend: level-0 (turquoise), level-1 (blue), level-2 & level-tr (red), sealed elements (black)). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

corresponding to the major and minor axes are respec- following boundary conditions: Uy ¼ DU at tively Aðhp ¼ 0 Þ¼0:0; Aðhp ¼ 90 Þ¼1:01; Bðhp ¼ 0 Þ¼ y ¼ 1536 lm; Uy ¼ 0aty ¼ 0, (Ux ¼ 0atðx; yÞ¼ð0; 0Þ. 0:67; Bðhp ¼ 90 Þ¼0:66, and Cðhp ¼ 0 Þ¼0:66; Cðhp ¼ The necessity for combining multi-scale characteriza- 90Þ¼0:69. tion with multi-scale modeling has been advocated in ear- lier papers by Ghosh in Ghosh et al. (2006); Ghosh et al. (2009); Ghosh et al. (2008). This is realized in this problem 6.2. Initial model and level changes in a tension test as well. Simulation of the tensile specimen with an initial computational domain containing only macroscopic level- In the tensile test, the specimen is loaded in uniaxial 0 elements leads to uniform states of stress and strain for tension along the vertical (y) direction by prescribing the the entire domain. Consequently, the level change criteria S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 31

the initial computational model using results of microstructural characterization. In Paquet et al. (2011a,b),it has been established that regions characterized by high values of local inclusion volume fraction and clustering have critically low ductility. A parameter ~f , deﬁned in

terms of the inclusion volume fraction V f and cluster contour index i as: i ~f ¼ ð42Þ 0:929 1:83V f

has been shown in Paquet et al. (2011a) to be a very good indicator of local ductility in a microstructure. This parameter is used to seek out the critical regions in the computa-

tional domain Xl0 where ductile fracture is likely to initiate in the microstructure. Prior to the multi-scale analysis, ~f is computed for the local microstructure of each of the 256 level-0 elements. Six critical elements (regions) are identi- ﬁed as ’’hot-spots’’ for premature nucleation of ductile cracks. Correspondingly, in a concurrent setting, these are modeled as level-2 elements from the start. The remainder of all elements in the initial multi-level mesh are level-0 as shown in Fig. 15(a). Adaptive level change is now activated based on the criteria in Eqs. (23), (26), and (27). The con-

stants used in the criteria are C2 ¼ 0:2; C3 ¼ 0:1, and

C4 ¼ 0:1. Mesh adaptivity with h-adaptation algorithm is not considered in this example.

6.3. Simulation through complete micro-cracking

With increasing deformation, non-uniformities arise in

the microscopic sub-domains Xl2 due to cracking of silicon inclusions and plastic deformation of the aluminum ma- Fig. 16. (a) Underlying microstructure of the level-2 elements in the trix, thus altering the initial homogeneity of the macro- multi-level mesh, and (b) contour plot of microscopic stress ryy (GPa) for the computational domain of the tensile specimen at an applied scale stress and strain ﬁelds. This leads to high gradients displacement Uy ¼ 13:0 lm. in the void volume fraction f that switch the surrounding level-0 elements to level-1 elements, and subsequently pre- cipitates level-2 elements in these regions. This adaptive in Eqs. (23), (26), and (27) are not able to identify the re- transition of macroscopic to microscopic elements, shown gions at which microscopic damage ﬁrst initiates, grows in Fig. 15, continues with the evolution of damage primar- and coalesces. This limitation is overcome by modifying ily in a direction perpendicular to the applied load. These

0.3 0.12

0.25 0.1

0.2 0.08 m) μ

(GPa) 0.15 0.06 yy Σ F (N/

0.1 0.04

0.05 0.02

0 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 e U (μm) yy y

Fig. 17. (a) Local averaged stress–strain response of the ﬁrst level-2 element sealed during multi-scale simulation, and (b) overall response of the tensile specimen obtained with the multi-scale model. The total reaction force F at y ¼ 1536 lm is plotted as function of the applied displacement Uy. 32 S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34

Fig. 18. Contour plots showing the ﬁnal crack path at complete failure of the ﬁrst level-2 element during the multi-scale simulation: (a) void volume p fraction f, and (b) equivalent plastic strain M .

microscopic elements in Xl2 coalesce with the evolving stress has been reached. The multi-scale simulation leads ductile deformation and damage as shown in Fig. 15(f) to tensile plots having the same characteristics as those ob- for an applied displacement of Uy ¼ 10:3 lm. Subse- tained from experiments for the aluminum alloy W319 in quently, the evolution of microscopic sub-domains stabi- Hu and Ghosh (2008) and the aluminum alloy AS7GU in lizes until localization of damage within a level-2 Chisaka (2009). microstructure leads to its complete failure. The first com- This numerical example clearly demonstrates the effec- plete level-2 element failure and consequent sealing occurs tiveness of the multi-level model in capturing multi-scale at an applied displacement Uy ¼ 13:2 lm. Sealing of a ductile fracture mechanisms in cast aluminum alloys, con- level-2 element due to complete failure of its microstruc- sisting of damage nucleation by inclusion cracking, fol- ture is activated when a sharp drop in the homogenized lowed by void nucleation, growth, and coalescence in the stress–strain response of the element is detected. The con- matrix. It averts the need for introducing artificial non-uni- tour plot of the microscopic stress component ryy at the formities such as local variations of void volume fraction, onset of local failure of the level-2 element is given in or geometric discontinuities such as notch, for capturing Fig. 16(b). The figure shows localization of stresses due to damage localization in the microstructural domain. Locali- local ductile failure. An image of the underlying micro- zation of damage naturally arises due to heterogeneities in structure for the Xl2 sub-domain is given in Fig. 16(a). the microstructure morphology. The abrupt drop in the averaged stress response of a level-2 element is associated with the unstable growth of damage in the microstructure. The homogenized stress– 7. Summary and conclusions strain response of the first sealed element is plotted in Fig. 17(a). The local softening of the element causes a drop This paper assimilates a variety of complementary in the force–displacement response of the tensile specimen ingredients to create a novel two-way multi-level model- as seen in the Fig. 17(b). Contour plots of microscopic void ing framework that is necessary for modeling ductile frac- volume fraction and equivalent plastic strain for the newly ture in heterogeneous aluminum alloys. a unique feature is sealed element are also shown in Fig. 18. These plots the incorporation of detailed microstructural information clearly demonstrate that a dominant ductile crack has acquired from micrographs. The two-way multi-level propagated through the level-2 microstructure. After the model involves both bottom-up and top-down structure– first element sealing has occurred, an increase in local material coupling. Bottom-up coupling invokes hierarchi- stresses and strains in the remaining level-2 elements re- cal multi-scaling and implements homogenization to yield sults in an acceleration of damage growth. This leads to reduced-order constitutive relations for efficient computa- the propagation of sealed elements as seen in the Figs. tions at the higher scales. Top-down coupling is facilitated 15(d)–(f). It is also observed in Fig. 15(j) that a second in a concurrent way, such that detailed lower scale (micro- macro-crack emerges from the microstructure. This failure mechanical in this case) models can be directly embedded process continues until the macro-cracks become extre- in the higher-scale models at regions of evolving intense mely unstable and the final failure of the structure occurs. localization and damage. This facility successfully captures The force–displacement response in tension of the speci- micromechanisms of ductile fracture in a deforming mate- men shows a very low ductility after the maximum tensile rial. The adaptive capability enables this top-down S. Ghosh, D. Paquet / Mechanics of Materials 65 (2013) 12–34 33 coupling for the evolutionary problems in an automatic References fashion, without user intervention. Micromechanical analysis in microscopic levels of this Caceres, C., Griffiths, J., 1996. Damage by the cracking of silicon particles in an Al–7Si–0.4Mg casting alloy. Acta Mater. 44, 25–33. framework is performed by the locally enhanced Voronoi Caceres, C., 1999. Particle cracking and the tensile ductility of a model Al– cell FEM or LE-VCFEM for ductile fracture in heterogeneous Si–Mg composite system. Aluminum Trans. 1, 1–13. materials, previously developed by the authors in Paquet Wang, Q., Caceres, C., Griffiths, J., 2003. Damage by eutectic particle et al. 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Foundation NSF Div Civil and Mechanical Systems Division Ghosh, S., Lee, K., Moorthy, S., 1996. Two scale analysis of heterogeneous through the GOALI Grant No. CMS-0308666 (Program elastic–plastic materials with asymptotic homogenization and Director: Dr. M. Dunn), by the Department of Energy Alu- Voronoi cell finite element model. Comput. Methods Appl. Mech. minum Visions program through Grant No. A5997. This Eng. 132, 63–116. Lee, K., Ghosh, S., 1996. Small deformation multi-scale analysis of sponsorship is gratefully acknowledged. The authors are heterogeneous materials with the Voronoi cell finite element model grateful to Drs. Xuming Su, and James Boileau of Ford Re- and homogenization theory. Comput. Mater. Sci. 7, 131–146. search Laboratory for invaluable discussions and for micro- Moulinec, H., Suquet, P., 2001. A computational scheme for linear and nonlinear composites with arbitrary phase contrasts. Int. J. Numer. graphs and samples used. 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