On Finite Strain Micromorphic Elastoplasticity

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International Journal of Solids and Structures 47 (2010) 786–800

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International Journal of Solids and Structures

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On ﬁnite strain micromorphic elastoplasticity

Richard A. Regueiro *

Department of Civil, Environmental, and Architectural Engineering, University of Colorado at Boulder, USA

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Article history: In the micromorphic continuum theory of Eringen, it was proposed that microstructure of materials could Received 14 December 2008 be represented in a continuum framework using a micro-deformation tensor governing micro-element Received in revised form 8 November 2009 deformation, in addition to the deformation gradient governing macro-element deformation. The paper Available online 22 November 2009 formulates finite strain micromorphic elastoplasticity based on micromorphic continuum mechanics in the sense of Eringen. Multiplicative decomposition into elastic and plastic parts of the deformation gra- Keywords: dient and micro-deformation are assumed, and the Clausius–Duhem inequality is formulated in the inter- Micromorphic elastoplasticity mediate configuration B to analyze what stresses, elastic deformation measures, and plastic deformation Multiplicative decomposition rates are used/defined in the constitutive equations. The resulting forms of plastic and internal state var- Finite strain iable evolution equations can be viewed as phenomenological at their various scales (i.e., micro-continuum and macro-continuum). The phenomenology of inelastic mechanical material response at the

various scales can be different, but for demonstration purposes, J2 flow plasticity is assumed for each of three levels of plastic evolution equations identified, with different stress, internal state variables, and material parameters. All evolution equations and a semi-implicit time integration scheme are formulated in the intermediate configuration for future coupled Lagrangian finite element implementation. A simpler two-dimensional model for anti-plane shear kinematics is formulated to demonstrate more clearly how such model equations simplify for future finite element implementation. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction grain/particle/fiber scale (sometimes called the ‘meso’-scale) to the macro-continuum scale of the engineering application, without los- There currently is great interest in accounting for underlying ing salient kinematic structure and micro-stresses. The finite strain microstructural response at the grain/particle/fiber scale on the micromorphic elastoplasticity model framework presented in the overall continuum mechanical behavior of heterogeneous materi- paper is meant to bridge the mechanics between the grain/parti- als—such as polycrystalline metals, ceramics, concrete, masonry, cle/fiber and macro-scales: to do so not only in a hierarchical infor- geomaterials (soils and rocks), asphalt, bone—in terms of predicting mation-passing (homogenization) multi-scale fashion, but also for their damage, fracture nucleation, and localized deformation. Much concurrent multiscale modeling (Fish, 2006). Concurrent multiscale research has been done on traditional macro-continuum inelastic modeling, in our case, would involve retaining the grain/particle/fi- constitutive modeling such that a wide range of books are available ber scale resolution in spatial regions of interest—for instance where to reference (Hill, 1950; Desai and Siriwardane, 1984; Lubliner, damage/micro-cracking nucleates or at the interface between 1990; Maugin, 1992; Simo and Hughes, 1998; Simo, 1998; Nemat- contacting materials—while transitioning to a ‘far’-field macro-scale Nasser, 2004). Likewise, research has been done and is ongoing on continuum representation via a micromorphic continuum region simulating directly the inelastic microstructural mechanical (Fig. 1). The additional degrees of freedom (dofs) and constitutive response—at the grain/particle/fiber scale—and reported in the liter- richness of the micromorphic continuum mechanics and plasticity ature (e.g., for polycrystalline metals (Vogler and Clayton, 2008), equations provide a more plausible transition than standard ceramics (Maiti et al., 2005; Molinari and Warner, 2006; Sadowski macro-continuum mechanics. We note that these additional dofs et al., 2007), concrete (Caballero et al., 2006), masonry (Formica are still fewer than if a micromechanical finite element simulation et al., 2002), geomaterials (soils (Nezami et al., 2007) and rocks (Mor- is attempted for the whole spatial domain, which is the argument ris et al., 2006)), asphalt (Birgisson et al., 2004; Dai et al., 2005), bone in favor of concurrent multiscale models: that is, using high-fidelity (Chevalier et al., 2007; Lee et al., 2007)). One of the current research materials modeling where needed (e.g., at a crack tip or large shear challenges, however, is how to bridge these length scales, from deformation interface, Fig. 1), and less fidelity where not needed. The paper proposes a phenomenological bridging-scale consti- * Fax: +1 303 492 7317. tutive modeling framework in the context of finite strain micro- E-mail address: [email protected] morphic elastoplasticity based on a multiplicative decomposition

Fig. 1. Illustration of concurrent computational multi-scale modeling approach in the contact interface region between a bound particulate material and deformable solid body. The grains (binder matrix not shown) of the micro-structure are ‘meshed’ directly (direct numerical simulation (DNS)) using discrete elements (DEs) and/or ﬁnite elements (FEs) with cohesive surface elements (CSEs) at their interfaces. The open circles denote micromorphic continuum FE nodes that have prescribed degrees of freedom b (dofs) D based on the underlying grain-scale response, while the solid circles denote micromorphic continuum FE nodes that have free dofs D governed by the micromorphic continuum elasto-plastic model.

of the deformation gradient F and micro-deformation tensor v into uum approach by Eringen and Suhubi (1964) is that the integral- elastic and plastic parts. In addition to the three translational dis- averaging of certain stresses, body forces, and micro-inertia terms placement vector u degrees of freedom (dofs), there are nine dofs is already part of the formulation. This will become especially use- associated with the unsymmetric micro-deformation tensor v (mi- ful when computationally homogenizing underlying microstruc- cro-rotation, micro-stretch, and micro-shear). We leave the formu- tural mechanical response (e.g., provided by a microstructural lation general in terms of v, which can be further simplified finite element or discrete element simulation) in regions of inter- depending on the material and associated constitutive assump- est, such as overlapping between micromorphic continuum and tions (see Forest and Sievert (2003, 2006)). The Clausius–Duhem grain/particle/fiber representations for a concurrent multiscale inequality formulated in the intermediate configuration yields modeling approach (Fig. 1). the mathematical form of three levels of plastic evolutions equa- Forest and Sievert (2003, 2006) established a hierarchy of elasto- tions in either (1) Mandel stress form (Mandel et al., 1974), or plastic models for generalized continua, including Cosserat, higher (2) an alternate ‘metric’ form. For demonstration of the micromor- grade, and micromorphic at small and finite strain. Specifically with phic elastoplasticity modeling framework, J2 flow plasticity and regard to micromorphic finite strain theory, Forest and Sievert linear isotropic elasticity are assumed. A semi-implicit time inte- (2003) follows the approach of Germain (1973), which leads to dif- gration scheme is also presented for implementation in a coupled ferent stress power terms in the balance of energy and, in turn, Clau- Lagrangian finite element code in the future. A two-dimensional sius–Duhem inequality than presented by Eringen (1999). Also, the strict anti-plane shear model is presented to discuss a simpler invariant elastic deformation measures do not match the sets (39) model with future finite element implementation. and (119) proposed by Eringen (1999). Upon analyzing the change The formulation presented in the paper differs from other 2 in square of micro-element arc-lengths ds0 2 dS0 between cur- works on finite strain micromorphic elastoplasticity that consider ð Þ a multiplicative decomposition into elastic and plastic parts (San- rent B and intermediate configurations B (cf. Appendix A), then sour, 1998; Forest and Sievert, 2003, 2006) and those that do not either set (39) or (119) is unique. Forest and Sievert (2003, 2006) pro-

(Lee and Chen, 2003; Vernerey et al., 2007). posed to use a mix of the two sets, i.e. (39)1, (119)2, and (119)3,in Sansour (1998) considered a finite strain Cosserat and micro- their Helmholtz free energy function. When analyzing morphic plastic continuum, redefining the micromorphic strain 2 ðds0Þ2 dS0 , they would also need (119) as a fourth elastic defor- measures (see (119) in Appendix B) to be invariant with respect 1 to rigid rotations only, not also translations. Sansour did not extend mation measure. As Eringen proposed, however, it is more straight- his formulation to include details on a finite strain micromorphic forward to use either set (39) or (119) when representing elastic elastoplasticity constitutive model formulation, as this paper does. deformation. The paper analyzes the use of both sets. Mandel stress Sansour proposed to arrive at the higher-order macro-continuum tensors are identified in Forest and Sievert (2003, 2006) to use in the by integrally averaging micro-continuum plasticity behavior using plastic evolution equations. This paper presents additional Mandel computation. Such an approach is similar to computational stresses and considers also an alternate ‘metric’-form oftentimes homogenization, as proposed by Forest and Sievert (2006) to esti- used in finite deformation elastoplasticity modeling. mate material parameters for generalized continuum plasticity Vernerey et al. (2007) treated micromorphic plasticity modeling models. On a side note, one advantage to the micromorphic contin- similar to Germain (1973) and Mindlin (1964), which leads to 788 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 different stress power terms and balance equations than in Eringen 2. Micromorphic kinematics (1999). The resulting plasticity model form is thus similar to Forest and Sievert (2003), although does not use a multiplicative decom- Fig. 2 illustrates the mapping of the macro-element and micro- position and thus does not assume the existence of an intermediate element in the reference configuration to the current configuration configuration. An extension presented by Vernerey et al. is to through the deformation gradient F and micro-deformation tensor consider multiple scale micromorphic kinematics, stresses, and v. The macro-element continuum point is denoted by balance equations, where the number of scales is a choice made PðX; NÞ and pðx; n; tÞ in the reference and current configurations, by the constitutive modeler. A multiple scale averaging procedure respectively, with centroid C and c. The micro-element continuum is introduced to determine material parameters at the higher point centroid is denoted by C0 and c0 in the reference and current scales based on lower scale response. configurations, respectively. The micro-element is denoted by an In general, in terms of a multiplicative decomposition of the assembly of particles, but in general represents a grain/particle/fi- deformation gradient and micro-deformation, as compared to ber microstructural sub-volume of the heterogeneous material. recent formulations of finite strain micromorphic elastoplasticity The relative position vector of the micro-element centroid with re- reported in the literature (just reviewed in preceding paragraphs), spect to the macro-element centroid is denoted by N and nðX; N; tÞ we view our approach to be more in line with the original concept in the reference and current configurations, respectively, such that and formulation presented by Eringen and Suhubi (1964, 1999), the micro-element centroid position vectors are written as (Fig. 2) which provide a clear link between micro-element and macro-ele- (Eringen, 1999) (in terms of contravariant components) ment deformation, balance equations, and stresses. Thus, we be- 0K K K k k k lieve our formulation and resulting elastoplasticity model X ¼ X þ N ; x0 ¼ x ðX; tÞþn ðX; N; tÞð1Þ framework is more general than what has been presented previ- Eringen and Suhubi (1964) assumed that for sufficiently small ously. The paper by Lee and Chen (2003) also follows closely Erin- lengths kNk1(kk is the L norm), n is linearly related to N gen’s micromorphic kinematics and balance laws, but does not 2 through the micro-deformation tensor v, such that treat multiplicative decomposition kinematics and subsequent constitutive model form in the intermediate configuration, as this k k K n ðX; N; tÞ¼vK ðX; tÞN ð2Þ paper does. We demonstrate the formulation for three levels of J2 plasticity and linear isotropic elasticity, and numerical time inte- where then the spatial position vector of the micro-element cen- gration by a semi-implicit scheme, as well as restriction to strict troid is written as anti-plane shear kinematics. 0k k k K Index notation will be used throughout so as to be as clear as x ¼ x ðX; tÞþvK ðX; tÞN ð3Þ possible with regard to details of the formulation. Some sym- This is equivalent to assuming an affine, or homogeneous, deforma- bolic/direct notation is also given, such that ðabÞ ¼ a bj , ik ij k tion of the macro-element differential volume dV (but not the body ða bÞ ¼ a b ; ða cÞij ¼ aimcj . Boldface denotes a tensor or ijkl ij kl k mk B; i.e., the continuum body B is expected to experience heteroge- vector, where its index notation has been given uniformly through- neous deformation because of v, even if boundary conditions (BCs) out the paper. Generally, variables in uppercase letters and no are uniform). It also simplifies considerably the formulation of the overbar live in the reference configuration (such as the refer- B0 micromorphic continuum balance equations as presented in Erin- ence differential volume dV), variables in lowercase live in the cur- gen and Suhubi (1964) and Eringen (1999). This micro-deformation rent configuration (such as the current differential volume dv), B v is analogous to the small strain micro-deformation tensor w in and variables in uppercase with overbar live in the intermediate Mindlin (1964), physically described in his Fig. 1. Eringen (1968) configuration (such as the intermediate differential volume B also provides a physical interpretation of v generally, but then dV). The same applies to their indices, such that a differential line simplies for the micropolar case. For example, v can be interpreted segment in the current configuration dxi (contravariant component i of dx ¼ dx gi) is related to a differential line segment in the reference configuration dXI through the deformation gradient: i i I dx ¼ FIdX (Einstein’s summation convention assumed (see, Erin- gen, 1962; Holzapfel, 2000)). In addition, the multiplicative decomposition of the deformation gradient is written as i ei pI e p FI ¼ FI FI ðF ¼ F F Þ, where superscripts e and p denote elastic and plastic parts, respectively. Subscripts ðÞ;i; ðÞ;I and ðÞ;I imply covariant differentiation in the reference, intermediate, and cur- i rent configurations, respectively, whereas oiðÞ ¼ oðÞ=ox denotes a partial derivative, in this case in B. With an eye toward eventual continuum finite element implementation of a resulting micromorphic elastoplasticity model, the reference, intermediate, and current configurations will be assumed Cartesian in the future. A superscript prime symbol ðÞ0 denotes a variable associated with the micro-element (see Section 2 on kinematics). Superposed dot ð_Þ¼DðÞ=Dt denotes material time derivative. An outline of the remainder of the paper is as follows: Section 2 describes micromorphic kinematics based on multiplicative decompositions of F and v into elastic e and plastic p parts; Section 3 the balance equations; Section 4 the Clausius–Duhem inequality; Section 5 constitutive form and reduced dissipation inequality; Section 6 form of plastic evolution equations; Section 7 constitu- Fig. 2. Map from reference B0 to current configuration B accounting for relative position N, n of micro-element centroid C0, c0 with respect to centroid of macro- tive equations for linear isotropic elasticity and J plasticity; Sec- 2 element C, c. F and v can load and unload independently (although coupled through tion 8 semi-implicit time integration; Section 9 anti-plane shear constitutive equations and balance equations), and thus the additional current model; and Section 10 conclusions. configuration is shown. R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 789 as calculated from a micro-displacement gradient tensor U as ‘ ¼ F_ eFe1 þ FeLpFe1 ¼ ‘e þ ‘p ð7Þ v ¼ 1 þ U, where U is not actually calculated from a micro-dis- l _ el e1 A el pB e1 C el pl 0 0 ¼ F ðF Þ þ F L ðF Þ ¼ ‘ þ ‘ placement vector u , but a u can be calculated once v is known. v;k A k B C k k k For finite element implementation, U will be interpolated at the pB _ pB p1 B L ¼ F B ðF Þ nodes, providing an additional nine dofs because it is unsymmetric. C C e e1 e v;p e1 e p The micro-element spatial velocity vector (holding X and N fixed) is m ¼ v_ v þ v L v ¼ m þ m ð8Þ then written as ml ¼ v_ el ðve1ÞA þ vel Lv;pBðve1ÞC ¼ mel þ mpl k A k B C k k k v;pB pB B 0k ¼ k þ n_ k ¼ k þ mknl ð4Þ L ¼ v_ ðvp1Þ v v v l C B C In the next section, the Clausius–Duhem inequality requires the where k is the macro-element spatial velocity vector, v covariant derivative of the micro-gyration tensor, which will be mk ¼ v_ k ðv1ÞK ðm ¼ vv_ 1Þ the micro-gyration tensor, similar in form l K l split into elastic and plastic parts based on (8). Thus, it is written as k _ k 1 K _ 1 to the velocity gradient v ¼ F K ðF Þ l ð‘ ¼ FF Þ. ;l e p Now we take the partial spatial derivative of (3) with respect to $m ¼ $m þ $m ð9Þ the reference micro-element position vector X0K , to arrive at an l el pl mm;k ¼ mm;k þ mm;k 0k expression for the micro-element deformation gradient FK as mel ¼ v_ el ðve1ÞA mel ven ðve1ÞD ð10Þ (see Appendix C) m;k A;k m n D;k m o k pl el pC el pE el v;pF pG 1 A pl ea e1 A 0k k vLðX; tÞ L m ¼ v v_ þ v v_ v L v ðv Þ m v ðv Þ ð11Þ F ¼ F ðX; tÞþ N m;k C;k A E A;k F G A;k m a A;k m K K oXK The covariant derivative (i.e., the gradient) of the elastic microde- ovk ðX; tÞ oNA þ vk ðX; tÞFk ðX; tÞ M NM ð5Þ formation tensor $ve is analogous to the small strain micro-defor- A A o A o K X X mation gradient N in Mindlin (1964), and its physical e 1 where the deformation gradient of the macro-element is interpretation in Fig. 2 of Mindlin (1964). For example, ðv Þ1;2 is k o k o K 0k FK ¼ x ðX; tÞ= X . The micro-element deformation gradient FK an elastic micro-shear gradient in the x2-direction based on a mi- 0k 0k 0K maps micro-element differential line segments dx ¼ FK dX and cro-stretch in the x1-direction. Furthermore, just as differential volumes dv0 ¼ J0dV 0, where J0 ¼ det F0 is the micro-element Jacobian macro-element volumes map as of deformation. This is presented for generality of mapping stresses dv ¼ JdV ¼ JeJpdV ¼ JedV ð12Þ between B0 and B; B0 and B; B and B, as shown starting in e e p p (22), but will not be used explicitly in the constitutive equations where J ¼ det F and J ¼ det F , then micro-element differential in Section 7. volumes map as We now assume a multiplicative decomposition of the defor- 0 0 0 e0 p0 0 e0 dv ¼ J dV ¼ J J dV ¼ J dV 0 ð13Þ mation gradient (Lee, 1969) and micro-deformation (Sansour, 1998; Forest and Sievert, 2003, 2006)(Fig. 3), such that where Je0 ¼ det Fe0 and Jp0 ¼ det Fp0. Fe0 and Fp0 have not been defined from (5), and are not required for formulating the final e p e p F ¼ F F ; v ¼ v v ð6Þ constitutive equations. Likewise, according to micro- and macro- Fk ¼ FekFpK ; vk ¼ vekvpK element mass conservation, mass densities map as K K K K K K q ¼ qJ ¼ qJeJp ¼ qJp ð14Þ Given the multiplicative decompositions of F and v, the velocity 0 0 0 0 0 e0 p0 0 p0 gradient and micro-gyration tensors can be expressed as q0 ¼ q J ¼ q J J ¼ q J ð15Þ

Fig. 3. Multiplicative decomposition of deformation gradient F and microdeformation tensor v into elastic and plastic parts, and the existence of an intermediate configuration B. Since Fe; Fp; ve; and vp can load and unload independently (although coupled through constitutive equations and balance equations), additional configurations are shown. The constitutive equations and balance equations presented in the paper will govern these deformation processes, and so generality is preserved. 790 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 R This last result was achieved by using a volume-average definition klm 112 def 011 2 0 011 m , we have m ¼ð1=dvÞ dv r n dv , where r is the normal 2 relating macro-element mass density to micro-element mass den- micro-element stress in the x1-direction, and n is the shear couple sity as in the x2-direction. Z Z Z The remainder of the paper focusses on the Clausius–Duhem def 0 0 def 0 0 def 0 0 qdv ¼ q dv ; q0dV ¼ q0 dV ; qdV ¼ q dV ð16Þ inequality mapped to the intermediate configuration to identify dv dV dV evolution equations for various plastic deformation rates that must This volume averaging approach by Eringen and Suhubi (1964) is be defined constitutively, and their appropriate conjugate stress used extensively in formulating the balance equations and Clau- arguments in B, and then an example for J2 plasticity and linear sius–Duhem inequality. isotropic elasticity, as well as an anti-plane shear form.

3. Micromorphic balance equations and Clausius–Duhem 4. Clausius–Duhem inequality in B inequality From a materials modeling perspective, it is oftentimes pre- Details regarding the formulation of micromorphic balance ferred to write the Clausius–Duhem inequality in the intermediate equations and the Clausius–Duhem inequality are given in Eringen configuration B, which is considered elastically unloaded, and for- and Suhubi (1964) and Eringen (1999) and thus are not repeated mulate constitutive equations there. The physical motivation lies here. The equations are summarized over the current configuration with earlier work by Kondo (1952), Bilby et al. (1955), Kröner B for balance of linear and first moment of momentum, balance of (1960), and others, who viewed dislocations in crystals as defects energy, and the Clausius–Duhem inequality, respectively, as with associated local elastic deformation, where macroscopic elastic deformation could be applied and removed without disrupting rlk þ qðf k akÞ¼0 ð17Þ ;l the dislocation structure of a crystal. More recent models extend rml sml þ mklm þ qðklm xlmÞ¼0 ð18Þ Z;k Z this concept, such as papers by Clayton et al. (2005, 2006) and references therein. The intermediate configuration can be consid- ml def 0ml 0 klm def 0kl m 0 0 B s dv ¼ r dv ; m nkda ¼ r n nk da ered a ‘‘reference” material configuration in which fabric/texture dZv da Z anisotropy and other inelastic material properties can be defined. lm def 0 0l m 0 lm def 0€l m 0 qk dv ¼ q f n dv ; qx dv ¼ q n n dv Thus, details on the mapping to B are given in this section. Recall dv dv that the Clausius–Duhem inequality in (20) was written using qe_ ¼ðskl rklÞm þ rkl þ mklmm þ qk þ qr ð19Þ lk vlk lm;k ;k localization of an integral over the current configuration B, such as 1 Z _ _ kl kl klm k 1 qðw þ ghÞþr ðvl;k mlkÞþs mlk þ m mlm;k þ q h;k P 0 ð20Þ _ _ kl kl klm k h qðw þ ghÞþr ðvl;k mlkÞþs mlk þ m mlm;k þ q h;k dv P 0 B h where rlk are contravariant components of the unsymmetric Cauchy ð21Þ stress, q is the mass density, f k is the body force vector per unit mass, 0l k 0kl e0k 0K L e0l e0 f is the body force vector per unit mass over the micro-element, a is Using the Piola transform r ¼ FK S FL =J , the following map- the acceleration, sml are the contravariant components of the sym- pings of the volume-averaged micro-stress and higher order couple klm stress terms are obtained as metric micro-stress, m are the contravariant components of the Z Z higher order couple stress, klm the body force couple per unit mass, kl def 0kl 0 1 e0k 0K L e0l e0 0 ek el K L lm s d ¼ r d ¼ F S F J dV ¼ F F R dV x the micro-spin inertia per unit mass, e is the internal energy v v e0 K L K L dv dV J i Z per unit mass, mlk ¼ glim the covariant components of the micro- k K L def e1 K e1 L e0k e0l I J i R dV ¼ðF Þ ðF Þ F F S0 dV 0 ð22Þ gyration tensor, vl;k ¼ gliv;k the covariant components of the velocity k l I J i Z Z dV gradient, mlm;k ¼ glimm;k the covariant components of the covariant m mklmd ¼ m FekFel emMK L MdV derivative of the micro-gyration tensor, gil are covariant coefficients lm;k v lm;k K LvM of the metric tensor in the current configuration B, qk is the heat flux B B Z vector, r is the heat supply per unit mass, w is the Helmholtz free K L M def e1 K e1 L e0k e0l 0I J M 0 M dV ¼ðF Þi ðF Þj FI FJ S N dV ð23Þ energy per unit mass, g is the entropy per unit mass, and h is the abso- dV K L lute temperature. Note that the balance of first moment of momen- where S0 is the symmetric second Piola–Kirchhoff stress in the mi- tum is more general than the balance of angular momentum (or cro-element intermediate configuration (over dV), RK L is the sym- ‘‘moment of momentum” Eringen, 1962), such that its skew-symmet- metric second Piola–Kirchhoff micro-stress in the intermediate ric part is the angular momentum balance of a micropolar continuum. configuration B, MK L M is the higher order couple stress written in 0ml Recall that the Cauchy stress r over the micro-element is symmet- the intermediate configuration, and NK are the covariant compo- ric because the balance of angular momentum is satisfied over the mi- nents of the unit normal on dA. In general, Fe0–Fe, but the constitu- e cro-element (Eringen and Suhubi, 1964). tive equations in Section 7 do not require that F 0 be defined or Physically, the micro-stress s defined in (18) as the volume solved. We have used a volume-average definition for the higher or- average of the Cauchy stress r0 over the micro-element, can be der couple stress (rather than an area-average), such as Z Z interpreted in the context of its difference with the unsymmetric mklmd def¼ r0klnm d 0 ¼ Fe0kFe0lvem S0K LNM dV 0 ð24Þ Cauchy stress as s r (Mindlin (1964) called this the ‘‘relative v v K L M dv dV stress”). This is the energy conjugate driving stress for the micro- ¼ FekFelvemMK L MdV ð25Þ deformation v through its micro-gyration tensor m ¼ vv_ 1 in (19), K L M and also the reduced dissipation inequality in the intermediate The result for expressing the Clausius–Duhem inequality in the configuration (45) and (48) as R S (the analogous stress differ- intermediate configuration is the same whether we use an ence in B). In fact, we do not solve for s or R directly, but consti- area- or volume-average definition for mklm. The volume-average tutively we solve for the difference s r or R S (see (68)). The definition becomes useful when homogenizing directly micro-ele- higher order stress m is analogous to the double stress l in Mindlin ment stress r0kl and relative position vector nm over a representative (1964) with physical components of micro-stretch, micro-shear, volume to calculate mklm, say in a multiscale modeling method. and micro-rotation shown in his Fig. 2. For example, m112 is the Using the mappings for q and dv, and the Piola transform on qk, higher order shear stress in the x2-direction based on a stretch in the Clausius–Duhem inequality can be rewritten in the intermedi- the x1-direction. Using the volume average definition in (24) for ate configuration as R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 791 Z h ÀÁ o _ _ e kl e kl K L qw kl e1 L q w þ gh þ J r ð l;k mlkÞþJ s mlk v S ¼ g ðF Þl ð34Þ B o ek FK ek el em K L M 1 K þmlm;k FK FLv M þ Q h;K dV P 0 ð26Þ ÀÁ ÀÁ M h o qw o qw RK L ¼ gklðFe1ÞL þðFe1ÞK vec gabðFe1ÞL oFek l c A ovea b Individual stress power terms in (26) can be additively decomposed K ÀÁ A into elastic and plastic parts based on (7)–(9). Using (9), the higher o qw þðFe1ÞK ved gflðFe1ÞL ð35Þ order couple stress power can be written as o d M;E ovef l ek el em K L M K L M el _ ea e en M;E mlm;k F F v M ¼ M F glav mlnv elastic K L M L M;K M;K 9 ÀÁ K L M el p en > o w þM F m v = K L M q kl e1 L L ln M;K M ¼ g ðF Þ ð36Þ hi plastic ovek l > M;K þg vea v_ pC þ veav_ pD vea Lv;pB vpE ðvp1ÞA ; la C;K A D A;K B E A;K M ÀÁ ð27Þ o qw qg ¼ ð37Þ where the covariant derivative with respect to the intermediate con- oh figuration can be defined as def Fek; e g ea,and B ðÞ;K ¼ðÞ;k K mln ¼ lamn kl p pa where g are contravariant metric coefficients on B. For compari- m ¼ g m n . The other stress power terms using (7) and (8) are written ln la son to the result reported in Eq. (6.3) of Eringen and Suhubi as (1964), we map these stresses to the current configuration, using e kl el _ ek K L e pB K L ÀÁ J r vl;k ¼ F g F S þ C L S ð28Þ |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}L lk K |fflfflfflfflfflffl{zfflfflfflfflfflffl}L B K 1 1 o qw rkl ¼ FekSK LFel ¼ Fek gal ð38Þ elastic plastic e K L e K o ea J J FK 0 1 e kl el e ek K L v;pE e1 F ek K L ÀÁ ÀÁ ÀÁ J r mlk ¼ F m F S þ W e L ðv Þ F S ð29Þ L lk K L E F k K o o o |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} kl 1 ek K L el 1 @ ek qw al ek qw al ek qw flA s ¼ e F R F ¼ e F ea g þ v ea g þ v g elastic plastic J K L J K oF A ov M;E ovef K A M;E e kl el e ek K L e v;pE e1 F ek K L ÀÁ J s mlk ¼ F mlkF R þ W L ðv ÞkF R ð30Þ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}L K |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}LE F K 1 1 o qw mklm ¼ FekFelvemMK L M ¼ galFekvem elastic plastic Je K L M Je ovea K M M;K where Ce ¼ Felg Fea are the covariant components of the right elas- L B L la B e eT e e el ea The equations match those in (6.3) of Eringen and Suhubi (1964) if tic Cauchy–Green tensor C ¼ F F in B, and W ¼ FLglav are the LE E e covariant components of an elastic deformation measure in B as elastic, i.e. F ¼ F; ve ¼ v. We prefer, however, to express the Helm- We ¼ FeT ve (cf. Appendix A). holtz free energy function in terms of invariant—with respect to rigid body motion on the current configuration B—elastic deformation 5. Constitutive model form and reduced dissipation inequality measures, such as the set proposed by Eringen and Suhubi (1964) as

Ce ¼ Fekg Fel; We ¼ Fekg vel; Ce ¼ Fekg v ð39Þ Similar to Eringen and Suhubi (1964) for a micromorphic elastic K L K kl L K L K kl L K L M K kl L;M material, the Helmholtz free energy function in B is assumed to e p take the following functional form for micromorphic elastoplastic- We have good physical interpretation of F (and F ) from crystal lat- ity as tice mechanics (Bilby et al., 1955; Kröner, 1960; Lee and Liu, 1967, 1969), while the elastic micro-deformation ve has its interpretation qwðFe; ve; $ve; Z; Zv; $Zv; hÞð31Þ in Fig. 3 of this paper (elastic deformation of micro-element) and qw Fek; vek; vek ; ZK ; ZvK ; ZvK ; h also Fig. 1 of Mindlin (1964) for small strain theory. The covariant K K M;K ;L derivative (i.e., gradient) of elastic micro-deformation $ve has it K vK where Z is a vector of macro-strain-like ISVs in B, Z is a vector of physical interpretation in Fig. 2 of Mindlin (1964), and was earlier vK micro-strain-like ISVs, and Z is a covariant derivative of a vector of e 1 ;L in this paper described, for example, as ðv Þ1;2 is the micro-shear micro-strain-like ISVs. Then, by the chain rule gradient in the x -direction based on a stretch in the x -direction ÀÁ ÀÁ ÀÁ ÀÁ 2 1 ek D w o w o w o w Dðv Þ (although directions are not exact here because of the covariant q q _ ek q q M;K ¼ F þ v_ ek þ derivative with respect to the intermediate configuration ). The ek K o ek K o ek B Dt oF v v Dt K K M;K Helmholtz free energy function w per unit mass is then written as ÀÁ ÀÁ ÀÁ vK ÀÁ o qw o qw o qw D Z o qw _ K _ vK ;L _ qw Ce; We; Ce; Z; Zv; $Zv; h ð40Þ þ Z þ Z þ þ o h ð32Þ oZK oZvK oZvK Dt h ;L qw Ce ; We ; Ce ; ZK ; ZvK ; ZvK ; h where an artifact of the ‘‘free energy per unit mass” assumption is K L K L K L M ;L that and the constitutive equations for stress result from (34)–(36) as ÀÁ ÀÁ ÀÁ ÀÁ ÀÁ D qw ÀÁJ_p ÀÁJ_p D qw o qw o qw L A o qw L A ¼ q_ w þ qw_ ¼ qw þ qw_ ) qw_ ¼ qw þ SK L ¼ 2 þ Ce1 We þ Ce1 Ce ð41Þ Dt Jp Jp Dt oCe oWe A B oCe A B C K L K B K B C ð33Þ "# ÀÁ ÀÁ o w o w L A _ p _p p K L q q e1 e where we used the result q ¼ Dðq0=J Þ=Dt ¼qJ =J . Substituting R ¼ 2 þ 2sym C W e e A B (27)–(30) and (32) and (33) into (26), and using the Coleman and oC oW K L "#ÀÁ K B Noll (1963) argument for independent rate processes (independent o L A qw e1 e F_ ek; v_ ek; Dðvek Þ=Dt, and h_), the Clausius–Duhem inequality is satis- þ 2sym C C ð42Þ K K M;K oCe A B C fied if the following constitutive equations hold: K B C 792 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 ÀÁ o def K L M qw far in the constitutive equations. For example, if G C (Clayton M ¼ ð43Þ K L ¼ K L e oC p L M K et al., 2004)(Appendix A), then the covariant components of L are where sym[] denotes the symmetric part, and notice that the met- Lp ¼ G LpB ¼ Ce LpB in (48). However, this choice also leads to zero L K L B K L B K ric coefficients cancel, i.e. galg da. These stress equations (41)– kl ¼ k elastic strain Ee ¼ 0, and thus we will assume the intermediate (43) when mapped to the current configuration are the same as K L Eqs. (6.9)–(11) in Eringen and Suhubi (1964) if there is no plasticity, configuration B is Cartesian in the future. If the intermediate con- i.e. Fe F and ve v. To consider another set of elastic deformation figuration is assumed Cartesian, Lp ¼ G LpB ¼ d LpB, and there is ¼ ¼ LK L B K L B K measures and resulting stresses, refer to Appendix B. no difference between superscripts and subscripts. The thermodynamically conjugate stress-like ISVs are defined as ÀÁ ÀÁ ÀÁ o o L o def qw v def qw rv def qw 6. Form of plastic evolution equations Q K ¼ ; Q ¼ ; Q ¼ ð44Þ oZK K oZvK K oZvK ;L Based on (45), in order to satisfy the reduced dissipation which will be used in the evolution equations for plastic deforma- inequality, we can write plastic evolution equations to solve for tion rates, as well as multiple scale yield functions in Section 7.2, pK pK pK FK ; vK ; and v in Mandel stress form as where we will assume scalar Z; Zv; $Zv, and Q; Qv; Q rv. The K;L stress-like ISVs in Section 7 will be physically interpreted as yield LpL ¼ HL SCe; Q solve for FpK and Fek ¼ Fk ðFp1ÞK ð49Þ stress Q and Qv for macro-plasticity (stress S calculated from elas- K K K K K K tic deformation) and micro-plasticity (stress difference R S calcu- rv 1 lated from elastic deformation), respectively, while Q is a higher Lv;pL ¼ HvL Cv;e WeT R S We; Q v order yield stress for micro-gradient plasticity (higher order stress K K M calculated from gradient elastic deformation). pK ek k p1 K solve for v K and v ¼ vK ðv Þ ð50Þ The remaining terms in the Clausius–Duhem inequality lead to K K the reduced dissipation inequality expressed in localized form in ÀÁ ÀÁ v;pD v;pD C F D two ways: (1) Mandel form with Mandel-like stresses (Mandel L 2skw L We1 Ce ¼ Hrv MWe; Q rv M;K C F M K M K et al., 1974), and (2) an alternate ‘metric’ form. Each will lead to different ways of writing the plastic evolution equations, and stres- solve for vpK and vek ¼ vk vekvpA ðvp1ÞK ð51Þ K;L K;L K;L A K;L K ses that are used in these evolution equations. The reduced dissipation inequality in Mandel form is written as where the arguments in parentheses ( ) denote the Mandel stress vK and stress-like ISV to use in the respective plastic evolution equa- ÀÁJ_p 1 L D Z v rv K _ K v _ vK rv ;L tion, where H; H , and H denote tensor functions for the evolution qw p þ Q h;K Q K Z Q Z Q J h K K Dt equations, chosen to ensure that convexity is satisfied, and the dis- K N sipation is positive. This can be seen for the evolution equations in þ SK BCe LpL þ Cv;e1 We RA B SA B We Lv;pL B L K A N B L K (52)–(54) by the constitutive definitions in (70) and (74) and (78) in ÀÁ terms of stress gradients of potential functions (i.e., the yield func- v;pD v;pD e1C F þ MK L MWe L 2skw L W Ce P 0 ð45Þ tions for associative plasticity). In an alternate ‘metric’ form, from L D M;K C F M K (48), we can solve for the plastic deformation variables as K N ÀÁ v;e1 e1 K kn e1 N e1 C F e1 C ia e1 F where C ¼ðv Þkg ðv Þn; W ¼ðv Þi g ðF Þa, skw[] denotes the skew-symmetric part defined as Ce LpB ¼ H S; Q solve for FpK and Fek ¼ Fk ðFp1ÞK ð52Þ L B K L K K K K K ÀÁ ÀÁ def C F D G 2skw½ ¼ Lv;pD We1 Ce Lv;pB We1 Ce ð46Þ C F M K M G B K FN We Lv;pE Cv;e1 We ¼ Hv R S; Q v L E F K N L K and the covariant derivative of the micro-scale plastic velocity gra- solve for vpK and vek ¼ vk ðvp1ÞK ð53Þ dient is K K K K hi v;pD pD B pD v;pD pB B ÀÁ _ p1 _ p1 C F L ¼ vB ðv Þ M ¼ v L v ðv Þ M ð47Þ e v;pD e v;pD e1 e rv rv M;K ;K B;K B B;K W L 2W skw L W C ¼ H M; Q L D M;K L D C F M K LM K K N pK ek k ek pA p1 K K B e v;e1 e A B A B e The Mandel stresses are S C , C W R S W , and solve for v and v ¼ v v v ðv ÞK ð54Þ B L A N B L K;L K;L K;L A K;L MK L MWe , where the first one is well-known as the ‘‘Mandel stress”, L D We use this ‘metric’ form in defining evolution equations in Section whereas the second and third are the relative micro-Mandel stress 7.2. and the higher order Mandel couple stress, respectively. We rewrite the reduced dissipation inequality (45) in an alternate ‘metric’ form Remark 1. The reason that we propose the third plastic evolution equation (51) or (54) to solve for vpK directly (not calculating a as K;L pK covariant derivative of the tensor vK from a finite element vK p ÀÁJ_p 1 L D Z interpolation of v ) is to potentially avoid requiring an additional K _ K v _ vK rv ;L qw p þ Q h;K Q K Z Q Z Q balance equation to solve in weak form by a nonlinear finite J h K K Dt element method (refer to Regueiro et al. (2007) and references FN þ SK L Ce LpB þ RK L SK L We Lv;pE Cv;e1 We cited therein). With future finite element implementation and L B K L E F K N numerical examples, we will attempt to determine whether (51) or ÀÁ pK K L M e v;pD e v;pD e1 C F e (54) leads to an accurate calculation of v . In Section 9, a simpler þ M W L 2W skw L W C P 0 ð48Þ K;L L D M;K L D C F M K anti-plane shear version of the model demonstrates the two ways for calculating $vp, either by an evolution equation like in (51) or Appendix A considers various choices for covariant metric coeffi- (54), or a finite element interpolation for vp and corresponding K L p cients GK L, (and contravariant metric G ), that have appeared thus gradient calculation $v . Note that in Forest and Sievert (2003), for R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 793

their equation (1553), they also propose a direct evolution of a same. Note that A is the typical linear isotropic elastic tangent mod- gradient of plastic microdeformation. ulus tensor, and k and l are the Lamé parameters. After some alge- bra using (41)–(44) and (55), it can be shown that the stress constitutive relations are 7. Constitutive equations SK L ¼ AK L M NEe þ DK B M NEe M N M N hi The constitutive equations for linear isotropic elasticity and J L A 2 þ DK B M NEe þ BK B M NEe ðCe1Þ Ee þ G plasticity with scalar ISV hardening/softening are formulated. We M N M N A B A B define a specific form of the Helmholtz free energy function, yield L Q þ CK B CN P Q Ce Ce1 Ce ð60Þ functions, and evolution equations for ISVs, and then conduct a N P Q Q B C numerical time integration presented in Section 8. Future work en- tails implementing the model in a coupled Lagrangian finite ele- RK L ¼ AK L M NEe þ DK B M NEe ment method. MnN M N hi 2sym DK L M NEe BK B M N e Ce1 L A e G þ M N þ EM N ð Þ EA B þ A B 7.1. Helmholtz free energy and stresses L Q CK B CN P Q Ce Ce1 Ce 61 þ N P Q Q B C ð Þ Assuming linear isotropic elasticity and linear relation between stress-like and strain-like ISVs, a quadratic form for the Helmholtz free energy function results as MK L M CK L M N P Q Ce 62 ¼ N P Q ð Þ

def 1 1 qw ¼ Ee AK L M NEe þ Ee BK L M NEe 2 K L M N 2 K L M N Q ¼ H Z ð63Þ 1 1 1 þ Ce CK L M N P Q Ce þ Ee DK L M NEe þ H Z2 2 K L M N P Q 2 K L M N 2 Qv ¼ HvZv ð64Þ 1 ÀÁ2 1 ÀÁK L þ Hv Zv þ Zv Hrv Zv ð55Þ ;K ;L 2 2 L Qv ¼ HrvZv GA L ð65Þ Note that the ISVs are scalar variables in this model, which will be ;A related to scalar yield strength of the material at two scales, macro Note that the units for Hrv are stress length2 (Pa m2), thus there is and micro, and H and Hv are scalar hardening/softening parame- ÀÁK L a built in length scale to this hardening/softening parameter for the ters, and Hrv is a symmetric second order hardening/softening higher order stress-like ISV. Assuming elastic deformations are modulus tensor, which we will assume is isotropic as ÀÁK L small, we ignore quadratic terms in (60) and (61), leading to the Hrv ¼ HrvGK L. Elastic strains are defined as (Suhubi and Erin- simplified stress constitutive equations for SK L and RK L as gen, 1964)2Ee ¼ Ce G and Ee ¼ We G . The elastic mod- K L K L K L K L K L K L uli are defined for isotropic linear elasticity, after manipulation of SK L ¼ AK L M N þ DK L M N Ee þ BK L M N þ DK L M N Ee equations in Suhubi and Eringen (1964) as M N M N M N e K L K N L M e K L M N K L M N K M L N K N L M ¼ðk þ sÞ G E G þ 2ðl þ rÞG G E A ¼ kG G þ l G G þ G G ð56Þ M N M N GM N e GK L GK MGL N e GK NGL M e 66 þ g EM N þ j EM N þ m EM N ð Þ BK L M N ¼ðg sÞGK LGM N þ jGK MGL N þ mGK NGL M r GK MGL N þ GK NGL M ð57Þ RK L ¼ðk þ 2sÞ GM NEe GK L þ 2ðl þ 2rÞGK NGL MEe M N MN þð2g sÞ GM NEe GK L þ 2ðk þ m rÞsym GK MGL NEe K L M N P Q K L M N P Q K Q L M N P M N M N C ¼ s1 G G G þ G G G ð67Þ K L M P N Q K M L Q N P þ s2 G G G þ G G G Note that the stress difference used in (53) then becomes K L M Q N P K N L M P Q þ s G G G þ s G G G K L K L M N e K L K N L M e 34 R S ¼ s G E G þ 2rG G E M N M N þ s GK MGL NGP Q þ GK PGL MGN Q 5 GM N e GK L GK MGL N e þðg sÞ EM N þðm rÞ EM N K M L P N Q K N L P M Q þ s6G G G þ s7G G G GL MGK N e 68 þðj rÞ EM N ð Þ K P L Q M N K Q L N M P þ s8 G G G þ G G G

K N L Q M P K P L N M Q þ s9G G G þ s10G G G 7.2. Yield functions and evolution equations

K Q L P M N þ s11G G G ð58Þ In this section, three levels of plastic yield functions are defined based on the three conjugate stress-plastic-power terms appearing DK L M N ¼ sGK LGM N þ r GK MGL N þ GK NGL M ð59Þ in the reduced dissipation inequality (48), with the intent to define the plastic deformation evolution equations such that (48) is where AK L M N and DK L M N have major and minor symmetry, while satisfied. Recall the plastic power terms in (48) come naturally BK L M N and CK L M N P Q have only major symmetry, and the elastic from the kinematic assumptions F ¼ FeFp and v ¼ vevp, and from parameters are k, l, g, s, j, m, r, s1,...,s11. Note that the units for the Helmholtz free energy function dependence on the invariant 2 2 e e e s1; ...; s11 are stress length (Pa m ), thus there is a built in length elastic deformation measures C ; W ; C , and the plastic strain-like scale to these elastic parameters for the higher order stress. The ISVs Z; Zv; and $Zv. elastic modulus tensors AK L M N; BK L M N ; and DK L M N are not the same as in Eringen (1999) because different elastic strain measures were 7.2.1. Macro-scale plasticity used, but the higher order elastic modulus tensor CK L M N P Q is the For macro-scale plasticity, we write the yield function F as 794 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 ÀÁ def def F S; a ¼ devS a 6 0 ð69Þ Frv M; arv ¼ devM arv 6 0 ð77Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi devS ¼ devS : devS I J 1 devMI J K ¼ MI J K Ce1 Ce MA B K 3 A B I J devS : devS ¼ devS devSI J where arv is the micro-gradient yield strength (stress-like ISV I J M N rv def rv ¼ devS GM IGNJ devS Q ¼ a ). Note that at the gradient micro-scale, the yield strength can be determined separately from the micro- and macro-scale 1 I J devSI J SI J Ce SA B Ce1 parameters, which is a constitutive assumption. The definitions of ¼ A B 3 the covariant derivative of micro-scale plastic velocity gradient def where a is the macro-yield strength (i.e., stress-like ISV Q ¼ a). $Lv;p and strain-like ISV then follow as The definitions of the plastic velocity gradient Lp and strain-like ÀÁ rv ISV then follow as C F def oF We Lv;pD 2We skw Lv;pD We1 Ce ¼ c_ rv ð78Þ L D M;K L D C F M K K L M def oF oM Ce LpB ¼ c_ ð70Þ L B K oSK L o oFrv devMP I J F b A B ¼ GP K GI LGJ M ¼ GK A N GB L o K L M oSK L M devM b devSA B NA B ¼ v D Z rv ÀÁrv;B devS def oF a ;A ¼ c_ rv ¼ c_ rv G ð79Þ Dt orv;A kkarv B A _ def oF a Z ¼ c_ ¼ c_ ð71Þ oa arv;L ¼ HrvZv GA L ð80Þ ;A a ¼ H Z ð72Þ where c_ rv is the micro-plastic gradient multiplier. where c_ is the macro-plastic multiplier. Remark 3. The main advantage to defining constitutively the 7.2.2. Micro-scale plasticity evolution of the covariant derivative of micro-scale plastic velocity v v;p For micro-scale plasticity, we write the yield function F as gradient $L in (78) separate from the micro-scale plastic velocity v;p pK v v def v gradient L in (74) (i.e., no PDE in v_ ) is to avoid finite element F R S; a ¼ dev R S a 6 0 ð73Þ K solution of an additional balance equation in weak form. One could v;p _ v allow $L and $Z to be defined as the covariant derivatives of 1 I J v;p _ v dev RI J SI J ¼ RI J SI J Ce RA B SA B Ce1 L and Z , respectively, but then the plastic evolution equations are 3 A B PDEs and require coupled finite element implementation (such as in where av is the micro-yield strength (stress-like ISV Qv def¼ av). Note Regueiro et al. (2007)). We plan to implement the model, after time that at the micro-scale, the yield strength can be determined sepa- integration in Section 8, within a coupled finite element formulation rately from the macro-scale parameter a. for the coupled balance of linear and first moment of momentum, and thus avoiding another coupled equation to include in the finite Remark 2. We use the same functional forms for macro- and element equations is desired. We will assess the accuracy of calcu- micro-plasticity (Fv with similar functional form as F, but different lating $vp by this direct time integration. Further discussion is ISVs and parameters), but this is only for the example model provided in Section 9 for an anti-plane shear version of the model. presented here. It is possible for the functional forms to be different when representing different phenomenology at the Remark 4. With these evolution equations in , (70) can be inte- micro- and macro-scales. More micromechanical analysis and B grated numerically to solve for Fp and in turn Fe, (74) can be inte- experimental data are necessary to determine the micro-plasticity grated numerically to solve for vp and in turn ve, and (78) can be functional forms in the future. integrated numerically to solve for $vp and in turn $ve. Then, the The definitions of the micro-scale plastic velocity gradient Lv;p stresses S; R S; and M can be calculated and mapped to the cur- and strain-like ISV then follow as rent configuration to update the balance equations for finite element nonlinear solution. Such numerical time integration will be carried FN oFv out in Section 8, and finite element implementation is ongoing work. We Lv;pE Cv;e1 We def _ v 74 L E K N ¼ c ð Þ F o RK L SK L o v 8. Numerical time integration F b vA B ¼ GK A N GB L o RK L SK L The constitutive equations in Section 7 are integrated numerically in time following a semi-implicit scheme (Moran et al., dev RA B SA B b vA B 1990). The advantage of such a scheme is the simplicity for inte- N ¼ dev R S grating complex constitutive models while maintaining frame indifference of the integration. The disadvantage is that it is condi- v _ def oF Zv ¼ c_ v ¼ c_ v ð75Þ tionally stable and thus care must be taken in choosing a stable oav time step. We will solve for plastic multiplier increments v v v a ¼ H Z ð76Þ Dc and Dcv in a coupled fashion (if yielding is detected at both scales; see Box 3), and multiplier Dcrv afterward because it is where c_ v is the micro-plastic multiplier. uncoupled. The plastic multipliers Dc and Dcv are uncoupled from Dcrv because of the assumption of small elastic deformations and 7.2.3. Micro-scale gradient plasticity dropping the quadratic terms in (60) and (61). For micro-scale gradient plasticity, we write the yield function We assume a deformation-driven time integration scheme with- Frv as in a coupled finite element program solving the isothermal coupled R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 795 balance of linear momentum and first moment of momentum equations (17) and (18), respectively, such that deformation gradient Fnþ1 4. Update stresses: and micro-deformation tensor v are given at time t , as well as nþ1 nþ1 ÀÁ e e their increments DF ¼ F F and Dv ¼ v v . We as- Snþ1 ¼ A þ D : Enþ1 þ B þ D : Enþ1 nþ1 nþ1 n nþ1 nþ1 n sume a time step Dt ¼ t t . Boxes 1-2 provide summaries of e e e nþ1 n R S ¼ sðtrEnþ1Þ1 þðg sÞðtrEnþ1 Þ1 þ 2rðEnþ1 Þ1 nþ1 the semi-implicit time integration of the stress and plastic evolution e eT þðm rÞEnþ1 þðj rÞEnþ1 equations, respectively, in symbolic form. For $X vnþ1 in Box 2, because U is a nodal degree of freedom in a finite element solution 5. Integrate strain-like ISVs, and update stress-like ISVs: and thus interpolated in a standard fashion, its spatial gradient can be calculated. Znþ1 ¼ Zn þ Dcnþ1; anþ1 ¼ H Znþ1 Box 3 summarizes the algorithm for solving the plastic multipli- v v v v v v Znþ1 ¼ Zn þ Dcnþ1; anþ1 ¼ H Znþ1 ers from evaluating the yield functions at time tnþ1. It involves multiple plastic yield checks, such that macro- and/or micro-plasticity could be enabled, and/or micro-gradient plasticity. Because the v 6. Solve for Dcnþ1 and Dcnþ1 using Newton–Raphson in Box macro- and micro-plasticity yield functions F and Fv, respectively, 3. are decoupled from the micro-gradient plastic multiplier c_ rv,we will solve first for the micro- and macro-plastic multipliers, as indicated by (I) in Box 3, and then for the micro-gradient plastic multiplier in (II) afterward. Once the plastic multipliers are calculated, the stresses and ISVs can be updated as indicated in Boxes 1-2.

Box 2 Semi-implicit numerical integration of micro-gradient plastic evolution equations. Box 1 e e e e p p p Given:ÀÁÀÁ$X vnþ1; Fnþ1; Cnþ1; vnþ1; Wnþ1; Fnþ1; Fn; vnþ1; Semi-implicit numerical integration of macro- and micro- p p v rv vn; $v n; $Z n; an plastic evolution equations. e e p p v v 1. Calculate trial values and yield function: Using Given: Fnþ1; vnþ1; Cn; Wn; Fn; vn; Zn; Zn ; an; an def ðÞ ¼ ðÞ Fp1K 1. Calculate trial values and yield functions: ;K ;K K hiÀÁ e p1 e p p1 etr p1 $v ¼ ðÞ$X v F v $v v F ¼ Fnþ1Fn Cetr ¼ FetrT Fetr then . Eetr ¼ Cetr 1 2 ÀÁ hiÀÁ etr p1 e p p1 etr p1 $v $ v F v $v v v ¼ v v ¼ðX nþ1Þ nþ1 nþ1 n nþ1 nþ1 n ÀÁ Wetr ¼ FetrT vetr etr eT etr C ¼ Fnþ1 $v etr etr E ¼ W 1 . ÀÁ tr . etr tr etr etr M ¼ C.C S ¼ A þ D : E þ B þ D : E ÀÁ rv;tr rv tr rv tr tr F ¼ F M ; a R S ¼ sðtrEetr Þ1 þðg sÞðtrEetr Þ1 þ 2rðEetr Þþðm sÞEetr þðj rÞEeT n p tr tr etr 2. Integrate micro-plastic gradient tensor $v : First, from F ¼ F S ; C ; an nþ1 Eq. (47), tr v;tr v etr v F ¼ F R S ; C ; an ÂÃÀÁ p $Lv;p ¼ $v_ p Lv;p $vp vp1 2. Integrate plastic part of deformation gradient Fnþ1 and micro-deformation tensor vp : ÀÁ ÀÁ nþ1 where it can be shown that $v_ p ¼ D $vp Dt þ $vp Lp. oF e _ p p1 _ Second, recall from Eq. (78), CnFnþ1Fn ¼ cnþ1 "#oStrT ÀÁ o rv ÂÃ o v;p v e1 F v;p e1 e p e1 F p _ r $L ¼ c Wn þ 2skw L W C Fnþ1 ¼ 1 þ Dcnþ1Cn Fn o T oStrT M Third, set the previous two equations equal to each other o v p v F p We v_ vp1Cv;e1WeT ¼ c_ toÀÁ come up with an evolution equation for $v as n nþ1 n n n nþ1 trT o D $vp ÀÁ oFrv ÂÃ R S ¼ c_ rv We1 vp þ 2skw Lv;pWe1Ce vp 2 3 Dt o T ÀÁ MÀÁ p p v;p p 6 oFv 7 $v L þ L $v p 6 v e1 eT v;e7 p v ¼ 41 þ Dc W T W C 5v nþ1 nþ1 n tr n n n Then, the semi-implicit time integration is written as o R S ÀÁ ÀÁ o rv rv F p 3. Update elastic deformation: $vp ¼ $vp þ Dc We1 v nþ1 n nþ1 nþ1 o trT nþ1 . ÂÃM ÀÁÀÁ v;p e 1 e p p p Fe ¼ F Fp1; Ce ¼ FeT Fe ; Ee ¼ Ce 1 2 þ 2skw DL W C v $v DL nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 ÀÁÀÁnþ1 nþ1 n nþ1 n nþ1 v;p p e p1 e eT e e e þ DLnþ1 $v n vnþ1 ¼ vnþ1vnþ1; Wnþ1 ¼ Fnþ1vnþ1; Enþ1 ¼ Wnþ1 1 p p p1 v;p p p1 where DLnþ1 ¼ðDFnþ1ÞFnþ1 and DLnþ1 ¼ðDvnþ1Þvnþ1. 796 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800

This micromorphic elastoplasticity model numerical integration 3. Update elastic deformation and stress: scheme will fit into a coupled Lagrangian finite element formula- ÀÁ hiÀÁ tion and implementation of the balance of linear momentum and $ve $ v Fp1 ve $vp vp1 nþ1 ¼ðX nþ1Þ nþ1 nþ1 nþ1 nþ1 first moment of momentum. Such work is ongoing. ÀÁ Ce FeT ve nþ1 ¼ nþ1 $ nþ1 . 9. Anti-plane shear model . e Mnþ1 ¼ C.Cnþ1 To present a simpler version of the general three-dimensional 4. Integrate strain-like ISVs, and update stress-like ISVs: 0 1 model discussed up to this point, we consider a strict anti-plane shear elasto-plasticity model in the context of micromorphic con- ÀÁ ÀÁ B arv C $Zv ¼ $Zv þ Dcrv @ n ACe stitutive modeling presented in this paper. Such model is useful to nþ1 n nþ1 rv nþ1 an better understand the finite element implementation, since it is ÀÁ easier to implement, but is limited physically to an unrealistic arv Hrv Zv Ce1 nþ1 ¼ $ nþ1 nþ1 material. For example, for a single shear plane, the material will look like a deck of cards being sheared as in Fig. 4. Details of such rv 5. Solve for Dcnþ1 using Newton–Raphson in Box 3. model for a different elasto-plasticity model, and coupled nonlinear finite element formulation, are presented in Regueiro et al. (2007). The micromorphic model presented here can be implemented in a similar manner as described in Regueiro et al. (2007), although with an additional balance of first moment of momentum equation, and different constitutive equations. We assume a Cartesian coordinate system for all configurations (reference, intermediate, and current). The deformation for anti-plane shear is given through the current coordinates Box 3 x1 ¼ X1 Check for plastic yielding and solve for plastic multipliers. x2 ¼ X2 ð81Þ (I) Solve for macro- and micro-plastic multipliers x3 ¼ X3 þ uðx1; x2Þ Dc and Dcv:

where xi are the current coordinates, XI are the reference coordi- Consider three cases: nates, and uðx1; x2Þ is the out-of-plane shear displacement. Thus, the problem reduces to a two-dimensional domain. The deformation gradient and micro-deformation tensor are then written as tr v;tr v (i) If F > 0 and F > 0, solve for Dcnþ1 and Dcnþ1 using Newton–Raphson on coupled equations: ox F ¼ ¼ 1 þ s c; s c ¼ 0 ð82Þ ÀÁ oX e v 2 3 2 3 F Snþ1; Cnþ1; anþ1 ¼ F Dcnþ1; Dcnþ1 ¼ 0 ou=oX1 0 o 6 7 6 7 ÀÁ u o o v e v v v c ¼ ¼ 4 u= X2 5; s ¼ 4 0 5 ð83Þ F R S ; Cnþ1; anþ1 ¼ F Dcnþ1; Dcnþ1 ¼ 0 oX nþ1 0 1 T v ¼ 1 þ s /; s / ¼ 0; / ¼ ½/1 /2 0 ð84Þ

tr v;tr v (ii) If F > 0 and F < 0, solve for Dcnþ1 with Dcnþ1 ¼ 0 where c is the displacement gradient vector, s the out-of-plane nor- using Newton–Raphson: mal and direction of shear displacement, and / the micro-displacement gradient vector. Note that ou=oX ¼ ou=ox. The anti-plane shear kinematics are strict (Cermelli and Gurtin, 2001) since we assume Fe; Fp; ve; and vp take the same form as F F S ; Ce ; a ¼ FðDc ; Dcv ¼ 0Þ¼0 nþ1 nþ1 nþ1 nþ1 nþ1 and v, such that

tr v;tr v (iii) If F < 0 and F > 0, solve for Dcnþ1 with Dcnþ1 ¼ 0 using Newton–Raphson: ÀÁ v e v v v F R S ; Cnþ1; anþ1 ¼ F Dcnþ1 ¼ 0; Dcnþ1 ¼ 0 nþ1

(II) Solve for micro-gradient plastic multiplier Dcrv, given Dc and Dcv:

rv;tr rv If F > 0, solve for Dcnþ1 using Newton–Raphson: rv rv rv rv F Mnþ1; anþ1 ¼ F Dcnþ1 ¼ 0

Fig. 4. Strict anti-plane shear for single slip with unit normal m and displacement

u ¼ uðx1; x2Þs, where s ¼ e3, similar to shearing a deck of playing cards. R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 797

Fe :¼ 1 þ s ce; s ce ¼ 0 the reduced dissipation inequality (45), and can show because of Fp :¼ 1 þ s cp; s cp ¼ 0 ð85Þ small elastic deformations that these Mandel stresses are e p c :¼ c þ c 1 T ÂÃ SCe S; Cv;e We R S We R S; MWe M ð101Þ e e e T p p p T c ¼ ½c1 c2 0 ; c ¼ c1 c2 0 The plastic evolution equations assume a number of slip systems a ve :¼ 1 þ s /e; s /e ¼ 0 with unit normal vector m a in the plane. For macro-scale plasticity, p p p v :¼ 1 þ s / ; s / ¼ 0 ð86Þ the evolution equations are / :¼ /e þ /p X ÂÃ ÂÃ p a a e e e T p p p T c_ ¼ c_ m ð102Þ / ¼ /1 /2 0 ; / ¼ / / 0 1 2 a "# m a where ce and cp are macro-elastic and plastic vectors, respectively, a a hS i c_ ¼ c_ 0 and /e and /p the micro-elastic and plastic vectors. Note that the Qa inverses are e Sa ¼ S : ðs m aÞ¼½ðl þ r þ mÞce þ j/ m a

e 1 p 1 _ ðF Þ ¼ 1 s ce; ðF Þ ¼ 1 s cp ð87Þ Q a ¼ Hc_ a ðveÞ1 ¼ 1 s /e; ðvpÞ1 ¼ 1 s /p ð88Þ where hi is the Macauley bracket. For micro-scale plasticity, the The elastic deformation tensors, and plastic velocity gradients in the evolution equations are intermediate conﬁguration can then be derived from these previous X equations. For elastic deformation, we can show that /_ p ¼ c_ v;am a ð103Þ a 2DE3 e eT e e e a m C ¼ F F ¼ 1 þ c s þ s c ð89Þ R S v;a v;a6 7 v;e eT e e e c_ ¼ c_ 4 5 C ¼ v v ¼ 1 þ / s þ s / ð90Þ 0 Qv;a e eT e e e W ¼ F v ¼ 1 þ c s þ s / ð91Þ a a e e a e e o e R S ¼ R S : ðs m Þ¼½jc þðm rÞ/ m T ov o/ / Ce ¼ Fe ¼ s ; Ce ¼ s L ð92Þ o oX K L M K oX _ .X M Qv;a ¼ Hvc_ v;a Ee ¼ Ce 1 2 ¼ ðÞce s þ s ce =2 ð93Þ

e o _ p o o _ e o o _ o o _ p o Ee ¼ We 1 ¼ ce s þ s / ð94Þ We need / = X in order to calculate / = X ¼ /= X / = X, and in turn M. For micro-scale gradient plasticity, we have two _ p where since we assume elastic deformations are small choices, as mentioned in Remark 1, to solve for o/ =oX: (1) calculate e _ p _ p p (kceke and k/ ke; e a small positive number), we ignore qua- o/ =oX directly as a gradient of / , where / is treated as an addi- dratic terms in ce and /e, and also can show that o/e=oX o/e=oX. tional nodal dof in the finite element implementation, or (2) define The index notation for Ce is useful when writing the higher order an evolution equation. These two choices are summarized as fol- stress M in (97). We loosely use subscripts K in the intermediate lows and will be considered in the finite element implementation configuration B on variables in the current configuration, such as for future work: s, because coordinates are nearly the same given strict anti-plane _ p shear kinematics and small strain elastic deformation assumption. 1. Direct calculation of gradient o/ =oX. This would require The stresses can then be derived as expressing the micro-scale plasticity equation in weak form for finite element solution (Regueiro et al., 2007): S ¼ðl þ rÞðce s þ s ceÞþjðce s þ s /eÞ X p v;a a e /_ c_ m ¼ 0 ð104Þ þ mðs ce þ / sÞð95Þ a In this case, there is no yield function Frv from which to solve for a R S ¼ rðce s þ s ceÞþðj rÞð/e s þ s ceÞ micro-scale gradient plastic multiplier c_ rv;a for slip system a, nor a e e _ þðm rÞðc s þ s / Þð96Þ gradient stress-like ISV Q rv;a for which to evolve. 2. Separate evolution equation: e e e M ¼ s1d C þ s4C d þ s7C K L M K L M QQ K QQ L M K L M o _ p X e e e e / rv;a a a þ s8 C þ C þ s9C þ s10C _ m n 105 M K L L M K K M L L K M o ¼ c ð Þ X a þ s Ce ð97Þ 11 M L K o/=oX na ¼ m a kko/=oX "# For plastic velocity gradients, we can show that m Ma rv;a rv;a p 1 c_ ¼ c_ Lp ¼ F_ pðF Þ ¼ðs c_pÞð1 s cpÞ¼s c_p ð98Þ 0 Q rv;a v;p p p 1 p p p L ¼ v_ ðv Þ ¼ðs /_ Þð1 s / Þ¼s /_ ð99Þ Ma M s m a I ¼ J K I J K p 1 o/_ _ v;a v v;a a $Lv;p 2skw Lv;pWe Ce $Lv;p ¼ s ð100Þ Q r ¼ Hr c_ r n oX In this case, a separate micro-scale gradient plastic multiplier c_ rv;a _ where for the third equation, gradient of micro-scale plastic velocity for each slip system a is solved, and a gradient stress-like ISV Q rv;a, gradient, we again used the assumption of small elastic deformation to solve for o/_ p=oX from an evolution equation (105). to ignore quadratic terms in ce and /e, or their multiples. In the evo- The numerical time integration, and finite element implemen- lution equations for plastic deformation, we use the Mandel form of tation, of these constitutive equations will follow the procedure 798 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 described in Regueiro et al. (2007). An additional balance equation, i.e. K is second and k is first. The multiplicative decomposition is then written as the balance of first moment of momentum (18), will likewise be hihi formulated for anti-plane shear and implemented by a coupled fi- F ¼ FeFp ¼ Fekg GK FpJ G GK ¼ FekFpK g GK ð107Þ nite element method in future work. K k K J K K k K where G are the contravariant basis vectors in B, and GJ the covar- 10. Conclusions iant basis vectors in B. Recall the transpose operation as (Marsden and Hughes, 1994) The paper formulated finite strain micromorphic elastoplasticity K based on a multiplicative decomposition of the deformation gradi- eT K L el F ¼ G F Lgkl ð108Þ ent F and micro-deformation tensor v in the sense of micromorphic k continuum mechanics by Eringen (1999). The Clausius–Duhem Knowing gkl ¼ gkgl, it can be shown that the right elastic Cauchy– inequality written in the intermediate configuration provided con- Green tensor is hihi stitutive forms for the plastic evolution equations to solve uniquely e eT e ek el K L ek el K L p p p C F F F g F G G F g F g G G 109 for plastic deformations F ; v ; and $v , and in turn their elastic ¼ ¼ K kl L ¼ K k L l ð Þ counterparts and the stresses and internal state variables. The Clau- Clayton et al. (2004) proposed the following definition for the sius–Duhem inequality was written in two forms (Mandel stress and covariant basis vectors in B as an alternate ‘metric’ form), and thus it becomes a constitutive modeler’s choice which form to use. Either approach will solve for G def¼ Fekg ð110Þ Fp; vp; and $vp, but their solutions could be different depending K K k on which form is used and the particular constitutive equations cho- such that sen for the heterogeneous material of interest. The additional plastic evolution equation for $vp in (51) or (54) removes the need for an e K L K L C ¼ GK GL G G ¼ GK LG G ð111Þ additional weak form equation for finite element solution of vp, but such assumption will be compared to a direct computation of $vp which defines the covariant metric coefficients on B as from a finite element interpolation of vp. In the metric form, evolu- G def¼ Ce ¼ Fekg Fel. In summary, we have K L K L K kl L tion equations for J2 plasticity were assumed, and also linear isotropic elasticity. A semi-implicit time integration scheme in the covariant metric in : G def Ce Fekg Fel 112 B K L ¼ K L ¼ K kl L ð Þ intermediate configuration was presented for future coupled Lagrangian finite element implementation. Another paper (Regue- K L K L def e1 e1 K kl e1 L iro, 2009) presents non-associative Drucker-Prager micromorphic contravariant metric in B : G ¼ C ¼ðF Þkg ðF Þl elastoplasticity mapped to the current configuration with semi-im- ð113Þ plicit time integration. An anti-plane shear model is presented to demonstrate a simpler two-dimensional form to implement in the LM M where we used the orthogonality condition GK LG ¼ dK to derive future by the finite element method. GK L (Eringen, 1962). Normally, we assume the reference and current configurations are Cartesian for continuum finite element imple- Acknowledgements mentation (unless a general curvilinear finite element shell formulation is required). Another choice of covariant metric on B would The support of National Science Foundation Grant CMMI- be to use the right elastic micro-deformation tensor as 0700648, Army Research Office Grant W911NF-09-1-0111, and hihi the Army Research Laboratory is gratefully acknowledged. The Cv;e ¼ veT ve ¼ ekg elGK GL ¼ ekg elg GK GL ð114Þ vK klvL vK k vL l author also acknowledges Dr. A.C. Eringen, who even in his later where then G def¼ Cv;e ¼ vekg vel. The corresponding contravariant years continues to inspire young researchers in materials mechan- K L KL K kl L ics and continuum physics. The author also acknowledges the con- metric could likewise be reached through the orthogonality structive comments of the anonymous reviewers. condition. Remark 5. If elastic deformation measures in (39) or (119) are Appendix A. Deformation measures and metric coefficients used, then metric coefficients on B would need to be defined. The two choices of forms for plastic evolution equations in the paper The paper by Clayton et al. (2004) summarizes the derivation of outline all the required terms, regardless of choice of metric deformation measures associated with the multiplicative decom- coefficients. But we note that for the elastic strain Ee to not equal K L position of F and the existence of a non-Euclidean intermediate zero, then G –Ce , and thus the choice by Clayton et al. (2004) in configuration . It was shown that the covariant components of K L K L B (110) and (111) would lead to Ee ¼ 0. Thus, for future finite certain tensors in B contain the covariant metric coefficients G . K L K L element implementation, we will assume B is Cartesian and A short discussion proceeds here to put into context the formula- G ¼ d . tion presented in this paper. It is reasonable to assume Cartesian K L K L coordinates for reference B0 and current B configurations for a Remark 6. It was mentioned in the introduction that the change in continuum finite element implementation, but for now the metric 2 0 2 kl square of micro-element arc-lengths ðds Þ dS0 should include coefficients gkl and g on B that appear in the formulation are left undefined, to be defined by a constitutive modeler later (and useful only three unique elastic deformation measures (the two sets pro- if implementing in a curved shell finite element method). posed by Eringen (1999) and considered in this paper for finite The deformation gradient, a second-order, two-point tensor F is strain elastoplasticity). Here, we write directly written as (Marsden and Hughes, 1994) 0 2 0 0 0k 0l ðds Þ ¼ dx dx ¼ dx gkldx ð115Þ F Fk g GK 106 ¼ K k ð Þ where k where F are the mixed components associated with the covariant k K dx0 ¼ FekdXK þ vek NK dXL þ vekvpK NK dXL þ vekdNK ð116Þ K K K;L K K;L K basis vectors gk in B and contravariant basis vectors G in B0, and the dot in front of K in the subscript denotes the order of indices, Then R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800 799 D M and ðds0Þ2 ¼ Ce þ 2sym Ce NB þ Ce Ce1 Ce NANB K L K B L D A K M B L o o X N B C 0 ¼ 1 0 ð126Þ þ2sym We Ce1 Ce vpE NANE oX oX B E C A K E;L o o 0 o o A B It is possible to show that N= X N= X, starting with þWe Ce1 We vpD vpE NDNE þ 2sym We vpE NE dXK dXL AD B E D;K E;L K E E;L oN oN oX 0 ¼ 0 ð127Þ B C oX oX oX 2 We We Ce1 Ce NA þ K L þ B L C A K which using (126) leads to A B pD þWe Ce1 We v ND dXK dNL 1 A L B D oN oN oN D;K 1 128 0 ¼ þ o o ð Þ A B oX X X þ We Ce1 We dNK dNL ð117Þ A K B L If we assume the gradient of microstructure is small across a mate- o o and rial kkN= X 1 (for the region of interest where the micromorphic continuum model is used), then 2 dS0 ¼ G dXK dXL þ 2dXK dNL þ dNK dNL ð118Þ K L oN 1 oN 1 þ 1 ð129Þ It can be seen that the first set in (39) appears exclusively as elastic oX oX deformation in (117); there are also some plastic terms, which do where then not appear in (1.5.8) in Eringen (1999). Eq. (117) could likewise be expressed in terms of the elastic set in (119). But one or the other set is oN oN oN oN ¼ 1 ð130Þ unique, as outlined by Eringen (1999) for micromorphic elasticity, oX0 oX oX oX here put into context for finite strain micromorphic elastoplasticity. The expression for F0 then results as in (5).

Appendix B. Another set of elastic deformation measures References

Here, another set of elastic deformation measures, (1.5.11) in Bilby, B., Bullough, R., Smith, E., 1955. Continuous distributions of dislocations: a Eringen (1999), are considered in their covariant components as new application of the methods of non-Riemannian geometry. Proc. Roy. Soc. Lond. Ser. A 231, 263–273. Cv;e ¼ vekg vel; e ¼ G ðve1ÞA Fea; Birgisson, B., Soranakom, C., Napier, J., Roque, R., 2004. Microstructure and fracture K L K kl L K L A L a K in asphalt mixtures using a boundary element approach. J. Mater. Civil Eng. 16 (2), 116–121. Pe ¼ G ðve1ÞA vea ð119Þ K L M A K a L;M Caballero, A., Lopez, C., Carol, I., 2006. 3D Meso-structural analysis of concrete specimens under uniaxial tension. Comput. Methods Appl. Mech. Eng. 195 (52), Thus, the Helmholtz free energy function is written as 7182–7195. Cermelli, P., Gurtin, M.E., 2001. On the characterization of geometrically necessary qw Cv;e; e ; Pe ; ZK ; ZvK ; ZvK ; h ð120Þ dislocations in finite plasticity. J. Mech. Phys. Solids 49, 1539–1568. K L K L K L M ;L Chevalier, Y., Pahr, D., Allmer, H., Charlebois, M., Zysset, P., 2007. Validation of a voxel-based fe method for prediction of the uniaxial apparent modulus of and the constitutive equations for stress result from (34)–(36) as ÀÁ human trabecular bone using macroscopic mechanical tests and o qw nanoindentation. J. Biomech. 40 (15), 3333–3340. SK L ¼ G ðve1ÞC gklðFe1ÞL ð121Þ Clayton, J., Bammann, D., McDowell, D., 2004. Anholonomic configuration spaces and o e B C k l metric tensors in finite elastoplasticity. Int. J. Non. Linear Mech. 39 (6), 1039– K B ÀÁ 1049. ÀÁo ÀÁ Clayton, J., Bammann, D., McDowell, D., 2005. A geometric framework for the K L e1 K qw e1 L R ¼ 2 ð122Þ kinematics of crystals with defects. Philos. Mag. 85 (33–35), 3983–4010. A oCv;e B ÀÁ A B Clayton, J., McDowell, D., Bammann, D., 2006. Modeling dislocations and disclinations o qw with finite micropolar elastoplasticity. Int. J. Plasticity 22 (2), 210–256. MK L M ¼ G ðve1ÞB gklðFe1ÞL ð123Þ Coleman, B.D., Noll, W., 1963. The thermodynamics of elastic materials with heat oPe IB k l conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178. IM K Dai, Q., Sadd, M., Parameswaran, V., Shukla, A., 2005. Prediction of damage ÀÁ behaviors in asphalt materials using a micromechanical finite-element model K e1 K where e1 ¼ðF Þ vek. These stress equations take a somewhat and image analysis. J. Eng. Mech. 131 (7), 668–677. A k A simpler form than in (41)–(43), but now the metric coefficients Desai, C., Siriwardane, H., 1984. Constitutive Laws for Engineering Materials: With Emphasis on Geologic Materials. Prentice-Hall, Englewood Cliffs, NJ. appear directly. Thus, it becomes a choice of the modeler how the Eringen, A., 1962. Nonlinear Theory of Continuous Media, first ed. McGraw-Hill, specific constitutive form of the elastic part of the Helmholtz free en- New York. ergy function is written, i.e. in terms of (40) or (120). Eringen (1999) Eringen, A., 1968. Theory of micropolar elasticity. In: Liebowitz, H. (Ed.), Fracture: An Advanced Treatise, vol. 2. Academic Press, New York, pp. 622–729. advocated the use of (120) for micromorphic elasticity, whereas Suh- Eringen, A., 1999. Microcontinuum Field Theories I: Foundations and Solids. ubi and Eringen (1964) used (40). We use (40) in Section 7. Springer, Berlin. Eringen, A., Suhubi, E., 1964. Nonlinear theory of simple micro-elastic solids – I. Int. J. Eng. Sci. 2 (2), 189–203. Appendix C. Derivation of F0 Fish, J., 2006. Bridging the scales in nano engineering and science. J. Nanoparticle Res. 8 (5), 577–594. Forest, S., Sievert, R., 2003. Elastoviscoplastic constitutive frameworks for The formulation of (5) is presented in this appendix, and we will generalized continua. Acta Mech. 160 (1–2), 71–111. use direct notation. To start, we recognize that Forest, S., Sievert, R., 2006. Nonlinear microstrain theories. Int. J. Solids Struct. 43 (24), 7224–7245. ox0 ox0 oX F0 ¼ ¼ ð124Þ Formica, G., Sansalone, V., Casciaro, R., 2002. A mixed solution strategy for the oX0 oX oX0 nonlinear analysis of brick masonry walls. Comput. Methods Appl. Mech. Eng. 191 (51–52), 5847–5876. where Germain, P., 1973. The method of virtual power in the mechanics of continuous media. II. Microstructure. SIAM J. Appl. Math. 25 (3), 556–575. ox0 ov oN Hill, R., 1950. The Mathematical Theory of Plasticity. Clarendon Press, Oxford. ¼ F þ N þ v ð125Þ Holzapfel, G.A., 2000. Nonlinear Solid Mechanics: A Continuum Approach for oX oX oX Engineering. Wiley, New York. 800 R.A. Regueiro / International Journal of Solids and Structures 47 (2010) 786–800

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