A nonlocal strain gradient plasticity theory for finite deformations

A nonlocal strain gradient plasticity theory for finite deformations

International Journal of Plasticity 25 (2009) 1280–1300 Contents lists available at ScienceDirect International Journal of Plasticity journal homepa...

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International Journal of Plasticity 25 (2009) 1280–1300

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

A nonlocal strain gradient plasticity theory for finite deformations Castrenze Polizzotto * Università di Palermo, Dipartimento di Ingegneria Strutturale e Geotecnica, Viale delle Scienze, 90128 Palermo, Italy

a r t i c l e

i n f o

Article history: Received 18 March 2008 Received in final revised form 24 September 2008 Available online 10 October 2008

Keywords: Elastic–plastic continua Finite deformations Gradient plasticity Nonlocal continuum thermodynamics

a b s t r a c t Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of PDEs and related higher order boundary conditions (including those associated with the moving elastic/ plastic boundary). Two fourth-order projection tensor operators, functions of the elastic and plastic strain states, are shown to relate the skew-symmetric parts of the Mandel stress and back stress to the related symmetric parts. Consistent with the thermodynamic restrictions therein derived, the flow laws for rate-independent associative plasticity are formulated in a six-dimensional tensor space in terms of symmetric parts of Mandel stresses and related work-conjugate generalized plastic strain rates. A simple shear problem application is presented for illustrative purposes. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction After the pioneering works of Aifantis (1984, 1987), plenty of papers have been written on the subject of gradient plasticity. Some of them are oriented towards computational aspects aiming at solving * Tel.: +39 091 6568411; fax: +39 091 6568407. E-mail address: [email protected] 0749-6419/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2008.09.009

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strain localization problems, see e.g. Lasry and Belytscko (1988), de Borst and Muhlhaus (1992), de Borst and Pamin (1996) and Abu Al-Rub and Voyiadjis (2005). Some others report on laboratory experiments related to size effect and scale length mesurements, see e.g. Stelmanshenko et al. (1993), Fleck et al. (1994), Ma and Clarke (1995), Poole et al. (1996), Stolken and Evans (1998), Abu Al-Rub and Voyiadjis (2004) and Kysar et al. (2007). But in the most part these papers are devoted to various aspects of gradient plasticity with the primary intent to provide a sound theoretical formulation basing either on micromechanics considerations, or on thermodynamics and other energy concepts, see e.g. Acharya and Bassani (2000), Bassani (2001), Aifantis (2003), Abu Al-Rub and Voyiadjis (2006), Brinckmann et al. (2006), Muhlhaus and Aifantis (1991), Fleck and Hutchinson (1993, 1997, 2001), Shizawa and Zbib (1999), Shu et al. (2001), Gao et al. (1999), Huang et al. (2000, 2004), Hwang et al. (2002), Liu et al. (2004), Gurtin (2000, 2002, 2003, 2004), Gurtin and Anand (2005a,b), Gurtin and Needleman (2005), Cermelli and Gurtin (2001, 2002), Gudmundson (2004), Polizzotto and Borino (1998), Polizzotto (2003a, 2007), Borino and Polizzotto (2007), Abu Al-Rub et al. (2007), Abu Al-Rub (2008) and Peerlings (2007). Only a relatively small number of papers has been devoted to gradient plasticity with finite deformations, perhaps because some fundamental questions of finite deformation plasticity are still under debate. Shizawa and Zbib (1999) provided a thermodynamically rigorous strain gradient plasticity theory for large deformations, which requires more than one intermediate configuration and introduced strain gradient effects through the concept of dislocation density tensor. Gurtin (2000, 2002) advanced original theories of strain gradient plasticity for large deformations which are strictly related to single crystal models, for which the use of an intermediate microstructural configuration (equivalent to Mandel’s isoclinic one) constitutes a natural choice. Gurtin and Anand (2005b) proposed a strain gradient plasticity theory for continua in large deformations, which employes the concept of Burgers tensor (burrowed from Cermelli and Gurtin (2001)) and adopts spinless, or irrotational, intermediate configurations, what renders it particularly appealing computationally. Hwang et al. (2004) advanced a generalization to finite deformation of the mechanism-based gradient plasticity theory by Gao et al. (1999) and Huang et al. (2000). Chambon et al. (2004) proposed a finite deformation phenomenological strain gradient plasticity which is similar to Fleck and Hutchinson (1997), but includes the first and second gradients of the displacements, for which multiplicative and additive decomposition rules are adopted, respectively, while the plasticity flow laws are formulated in the strain space. None of the above papers seems to address a few questions arising with strain gradient plasticity in finite deformations. Namely, in local plasticity, one does not care about the locations of the individual plastically deformed particles, each being in an intermediate (elastically relaxed) configuration. Indeed, for a simple material, the constitutive equations—written in the particle intermediate configuration—do depend on the state variables value, not on their spatial gradients, which thus do not need being computed. On the contrary, in gradient plasticity, the constitutive equations also depend on the plastic strain gradient, which thus has there to be mathematically meaningful. As a consequence, the totality of such locations must belong to a space domain (say, domain of intermediate configurations), such that the plastic strain variable may there be a regular field. Additionally, this plastic strain field can vary considerably from one such domain to another as a consequence of the rigid-body rotations of the individual material particles. So the problem arises about which domain of intermediate configurations is to be considered for the computation of a plastic strain gradient being physically meaningful, that is, able to realistically capture phenomena as size effects, strain localization and the like. Also, given for granted that such a domain exists, if in place of it another one is used, there correspondingly exists the necessity for some kind of objective plastic strain gradient, analogous to the well-known corotational (time) rates. It is in the purpose of the present paper to give an answer to the above questions within the framework of nonlocal strain gradient plasticity. For this aim, a continuum model will be used, in which the elasticity symmetries are insensitive to plastic deformations and the nonlocality sources are macroscopically represented by the first-order (spatial) gradient of plastic strain with the role of internal variable. It will emerge that, in gradient plasticity with finite deformations, the substructure of the material particles system possesses a geometry, and that thus the substructure and the continuum

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must there be distinguished from each other not only for their different kinematics (as in local plasticity), but also for their different geometries. The outline of the paper is as follows. Section 2 is devoted to the essentials of the finite deformation kinematics, which are basic for the subsequent developments. The concept of domain of (particle) intermediate configurations is devised as the (translated) undeformed body domain with a plastically deformed particle appended at every point of it. In Section 3, the classical concept of objectivity of the time derivatives of vectors and tensors is reminded and extended to the spatial derivatives with the introduction of the notion of corotational gradient. Section 4 is devoted to thermodynamics considerations, whereby the notion of energy residual (previously devised by the author and co-workers) is rediscussed. Also, the concept of globally simple material is introduced, featured by the insulation condition, the bilinear dissipation condition and the locality recovery condition. In Section 5, with reference to a generic domain of intermediate configurations, the relevant restrictions on the constitutive equations are derived in the simplifying hypothesis of isothermal conditions. In Section 6, the evolution laws for rate independent associative plasticity are established in a form respectful of the derived thermodynamic restrictions. Two fourth-order projection tensor operators are devised, which make it possible to formulate the plasticity flow laws in a six-dimensional tensor space, in which the symmetric parts of the Mandel stress and back stress play the role of dissipative driving stresses and are work-conjugate of some generalized plastic strain rates. In Section 7 a simple illustrative example is worked out. Conclusions are drawn in Section 8. Notation. A compact notation is used as a rule, except at some specific points (where the index notation is used for more clarity). An orthogonal Cartesian co-ordinate system is used as reference system. The index contraction operation between vectors and tensors is denoted with as many dots as the number of contracted index pairs. For example, denoting by u ¼ fui g, v ¼ fvi g, E ¼ fEij g, S ¼ fSij g, T ¼ fT ijk g, A ¼ fAijkh g some vectors and tensors, one can write: u  v ¼ ui vi , S : E ¼ Sij Eji , . A..T ¼ fAijkh T hkj g, A : E ¼ fSijkh Ehk g. The mark  is sometimes used to denote the tensor product, i.e. u  v ¼ fui vj g. The symbol S  E indicates a second kind of tensor product between two second-order tensors, with components ðS  EÞijmn ¼ Sim Ejn . The symbol ðÞT indicates the transpose of (). An upper dot over a quantity denotes its material time derivative, A_ ¼ DA=Dt. The symbol r denotes the spatial gradient operator, ru ¼ fui;j g ¼ foj ui g. Other symbols will be defined in the text at their first appearance.

2. Kinematics Let a continuous (finite) body undergo a transformation x ¼ xðx0 ; tÞ8x0 2 V 0 , which leads the body from its initial (undeformed) configuration at time t ¼ 0, say B0 , to its actual (deformed) configuration at the generic time t, say B. x0 and x are position vectors of a material particle in B0 and B, respectively, these configurations being referred to the same Cartesian orthogonal co-ordinate system. The symbols C0 and C will be used to denote the corresponding configurations of an individual particle, with the indication of the particle location, say CðxÞ, whenever necessary. Let F ¼ ox=ox0 denote the deformation gradient relative to the mapping x ¼ xðx0 ; tÞ. Following Lee and Liu (1967) and Lee (1969), a multiplicative decomposition rule into elastic and plastic parts is adopted for F, that is:

F ¼ Fe  Fp;

F ij ¼ F eik F pkj

ð1Þ 0

where the second index of F ij refers to B , the first one to B. The decomposition (1) implies that every material particle is elastically unloaded from C to an intermediate (relaxed) configuration, say C, 1 through the (local) transformation F e , or plastically deformed from C0 to C through the (local) transp formation F . This maps the undeformed particle at x0 2 V 0 into the plastically deformed particle at  ¼ F p  dx0 of C. , such that the generic fiber dx0 of C0 is transformed into the fiber dx some point x

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For a given particle and for assigned C0 and F, the particle’s intermediate configuration resulting from the decomposition (1), say C, is not uniquely determined, since in fact it can be arbitrarily rotated and the new intermediate configuration is still an elastically relaxed configuration obtainable from C by means of an elastic unloading procedure, see e.g. Mandel (1971). Therefore, for a fixed C0 and for every given transformation F, there exists a continuous set of intermediate (relaxed) particle configurations, which differ from one another by a (local) rigid-body rotation only. As a consequence of (1), the total Green–Lagrange elastic strain tensor, E, is decomposed as in the following:

E :¼

1 T T ðF  F  IÞ ¼ Ep þ F p  Ee  F p 2

ð2Þ

where

Ep :¼

 1  pT p F F I ; 2

Ee :¼

 1  eT e F F I : 2

ð3Þ

Analogously, the velocity gradient of the material particle in C, i.e. L ¼ ov=ox, proves to be decomposed as 1 L ¼ F_  F 1 ¼ Le þ F e  Lp  F e ;

ð4Þ

1 Lp :¼ F_ p  F p ;

ð5Þ

where 1 Le :¼ F_ e  F e :

p

Since in general L is not symmetric, it can be decomposed as

Lp ¼ Dp þ W p

ð6Þ

p

p

p

where D (plastic deformation rate) is the symmetric part of L and W (total plastic spin) its antisymmetric part, that is:

  1 Dp ¼ LpS ¼ F_ p  F p ; S

  1 W p ¼ LpA ¼ F_ p  F p : A

ð7Þ

W p is known in the literature with different names, e.g. material plastic spin in Dafalias (1987, 1998), or continuum plastic spin in Aravas (1994), whereas the name total plastic spin is used in this paper. Another plastic strain tensor, alternative to Ep , is the Eulerian plastic strain given by

ep :¼ where F p relations

T

 1 T 1 I  Fp  Fp ; 2 

:¼ F p T

1

T

ð8Þ

. The tensor ep , here introduced for subsequent use, is related to Ep by the

ep ¼ F p  Ep  F p :

ð9Þ

Many of the tensor quantities previously introduced depend on the particular choice of the particle intermediate configuration, C. They nevertheless are objective, meaning that on changing C by a rigidbody rotation, they vary according to the tensorial rules. After Mandel (1971, 1973), the director vectors constitute a special reference axes triad, say ðaÞ ðaÞ d ; ða ¼ 1; 2; 3Þ, attached to the material particle substructure. In practice, the d triad may specify the elasticity symmetries of the material, the directions of reinforcement fibers in composite materials, or even the principal directions of deformation, see e.g. Dafalias (1987, 1998), Aravas (1992, 1994) and Pereda et al. (1993). In the present context, in which the substructure irreversible deformation processes consist in the inherent plastic deformation, the director vector triad is taken coincident with ðaÞ ðaÞ the plastic stretch principal direction triad, say d ¼ N ðaÞ in C0 and d ¼ m ðaÞ in C. b in which the The Mandel isoclinic configuration is a particular intermediate configuration, say C, ^ ðaÞ , but here, in the context of gradient director vector triad has a constant orientation at any time, say d ðaÞ ^ ðaÞ triad coincides plasticity, d is assumed constant also in space. For simplicity, we assume that the d b is denoted with the general co-ordinate basis triad. The (plastic) deformation gradient associated to C

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b p , whereas the location of the generic plastically deformed particle is x ^. The transformation which F b into C obviously consists in a (local) rigid-body rotation, say R, (plus an inessential glotransforms C ðaÞ ^ ðaÞ . The rotation R is assumed to be a regular function of the locabal translation), such that d ¼ R  d b , or, equivalently, of the location x ^2V  2 V. tion x 3. Objectivity of time and space derivatives The constitutive equations must be invariant under change of observer, hence all vector and tensor quantities there involved must be objective (Mandel, 1971, 1973; Loret, 1983; Dafalias, 1987, 1988, 1998; Aravas, 1992). The latter requisite is satisfied by the quantities F p , Ep , Ee , ep , but likely not by their time and space derivatives, nor by the tensors L, Lp , Le previously introduced. This issue is investigated hereafter. 3.1. Corotational rates b to a generic interDenoting by R the particle rigid-body rotation from the isoclinic configuration C mediate one, C, and taking the time derivative of the latter, we can write:  _ b p ¼ RT  F p ; F

ð10Þ

where we have introduced the tensor quantities 

F p :¼ F_ p  x  F p ;

x :¼ R_  RT :

ð11Þ

The (skew-symmetric) tensor x denotes the rigid-body spin of the continuum in C with respect  to the b it thus coincides with the spin of the dðaÞ triad in C. The new tensor quantity F p is the continuum in C, well-known corotational (time) rate of F p , that is, the material time derivative of F p in the relative motion of the continuum with respect to the substructure, see e.g.Dafalias (1987, 1988, 1998) and Aravas (1992). b e ¼ F e  R, we can write Analogously, since F  _ b e ¼ F e R; F



F e :¼ F_ e þ F e  x

 e

ð12Þ  p

e

where F is the corotational rate of F and has a meaning similar to F . Also, proceeding in a similar way, we can easily obtain:  _ b e ¼ RT  Ee R; E

 _ e^p ¼ RT  ep  R;

 e

ð13Þ

 p

where the tensors E and e are defined as 

Ee :¼ E_ e  x  Ee þ Ee  x;



ep :¼ e_ p  x  ep þ ep  x e

ð14Þ

p

and denote the corotational rates of E and e , respectively. Following Dafalias (1987, 1988, 1998), let the objectivity features of Lp , Dp and W p of Eqs. (5)1 and (7) be also considered. By (11)1 we can replace Lp with the tensor 

1

Lpr :¼ Lp  x ¼ F p F p ;

ð15Þ

which represents the relative plastic velocity gradient of the material particle in the generic intermediate configuration, C, in the relative motion of the continuum being in the latter configuration with respect to the material substructure. For the material particle being in the isoclinic configuration, the L p , and is related to Lpr relative plastic velocity gradient coincides with the standard one, that is, b L pr ¼ b by

b L p ¼ RT  Lpr  R: Taking the symmetric and antisymmetric parts of (15) gives, by (7),

ð16Þ

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   1 1 Dp ¼ F_ p  F p ¼ F p F p S   S pr p p p 1 W :¼ W  x ¼ F F :

ð17Þ ð18Þ

A

Eqs. (17) and (18) show that Dp and W pr are objective. W pr (called plastic spin by Dafalias (1985, 1987), but here called relative plastic spin to distinguish it from the total plastic spin, W p ) expresses the spin of the principal direction triad of Dp in C in the relative motion of the latter triad with respect ðaÞ to the substructure (spinning as the director vector triad d embedded in it, i.e. x). On concluding this subsection, let us remark that, if the actual deformed configuration C is subjected to an arbitrary (local) rigid-body rotation, say Q, the transformation F e changes into  F e ¼ Q  F e , whereas the Green–Lagrange elastic strain Ee of Eq. (3)2 remains unchanged. Since no other deformation tensor among those considered previously is affected by Q, we can conclude that the latter tensors are objective also with respect to the actual deformed configuration. 3.2. Corotational gradients Gradient plasticity theories ground on the assumption that plastic strain gradients provide an effective macroscopic representation of a class of physical phenomena that are sensitive to them, as size effects, plastic strain localization, and the like. For small deformations, this assumption can be practically applied without difficulties, but, as anticipated in Section 1, it is not so for finite deformations. In the latter case, the necessity there arises of suitably choosing the locations of the plastically deformed material particles, each being in an intermediate configuration, in such a way that, correspondingly, the notion of plastic strain gradient be meaningful, both mathematically and physically. For this purpose one may use, as in Gurtin (2000, 2002) and Gurtin and Anand (2005b), the domain V0 of the undeformed body, but anyway there remains the problem of how to obtain an objective measure of the plastic strain gradient in a generic domain of particle intermediate configurations.  of the plastically deformed From the mathematical point of view, it is sufficient that the location x :x  ¼ x0 þ  c8x0 2 V 0 g, obtained from V0 by a particle belong to a continuous space domain, say V ¼ fx simple (inessential) translation  c . V is the support domain of the plastically deformed particle system  is, by B :¼ fCg, every particle being appended to a point of V. The plastic strain of the particle at x  2 V, hence also of x0 2 V 0 , and thus its gradient there exists. assumption, a C1-continuous function of x On the other hand, the assumption is made that the domain of intermediate configurations more suitable to the plastic strain gradient computation identifies with the one of the isoclinic configurab , featured by a substructure director vector triad being constant both in space and time. tions, say V b the plastic strain gradient proves This choice stems from the consideration that, correspondingly, in V to be coincident with the standard gradient evaluated with respect to the plastic stretching principal directions. This conjecture seems to agree with the way in which measurements are often accomplished in laboratory experiments. ^ ðaÞ envelope straight lines which b , the substructure director vector triads d Let us note that, in V (by assumption) coincide with the general co-ordinate lines, whereas in V, as a consequence the ðaÞ (assumed) regularity of R, the substructure director vector triads, d , varies regularly in space, such as to envelope three families of mutually orthogonal lines. These lines happen to coincide with the plastic stretching principal direction lines. Indeed, as already noted in Section 1, in gradient plasticity the particle system substructure proves to be endowed not only with a kinematics (as in local plasticity), but also with a geometry; its way of interacting with the continuum is likely more intricate. It is paramount, at this point, to establish how the (first order) plastic strain gradient computed in b of the the domain V of the intermediate configurations is related to the one computed in the domain V isoclinic configurations. This question is here addressed by considering, in place of the relevant plastic b¼A b ðx ^; tÞ be defined strain tensor, a generic second-order tensor variable, say A. For this purpose, let A b , A ¼ Aðx b to C, we can write ; tÞ in V. Thus, denoting with R the particle rigid-body rotation from C in V

b ¼ RT  A  R: A

ð19Þ

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b and A are assumed to be C1-continuous fields in V b and V, respectively. The rotation R as a Both A b and V differ each from V0 for an arbi^, or of x , is also C1-continuous. (Note: since V function of either x b and V with V0; however, we prefer maintain them distinct for trary translation, we may identify V more clarity.) b be computed. Making b and let the gradient r bA b ¼ f^ ^2V op g ¼ fo=o^ xp g denote the gradient at x Let r use of the indicial notation for more clarity, by (19) we can write:

b ij ¼ ^op ðRki Akh Rhj Þ ¼ Rki ^op Akh Rhj þ ^op Rki Akh Rhj þ Rki Akh ^op Rhj ^op A ¼ Rki ^op Akh Rhj þ dir ^op Rkr Akh Rhj þ Rki Akh ^op Rhs dsj :

ð20Þ

b equals an analogous displacement dx ^ in V  ¼ R  dx ^ in V, Let us note that a particle displacement dx xq . Thus, in virtue of the identities dir ¼ Rmi Rmr , dsj ¼ Rns Rnj , and such that ^ op ¼ Rqp oq , where oq ¼ o=o modifying the indices of the last member of (20) by the substitutions k ? m, h ? n in the first term, h ? n in the second one and k ? m, h ? k, s ? r in the third one, Eq. (20) can be rewritten as:

b ij ¼ Rmi ðop Amn þ op Rkr Rmr Akn þ Amk op Rkr Rnr ÞRnj Rqp : ^op A

ð21Þ

Next, let us pose:

Gpkn :¼ op Rkr Rnr ;

ð22Þ

and let us observe that, since Rkr Rnr ¼ dkn , hence op ðRkr Rnr Þ ¼ 0, it is Gpkn ¼ Gpnk ; that is, the just introduced coefficients Gpkn prove to be skew-symmetric with respect to the k, n indices. With the above notation, Eq. (21) can thus be rewritten in the notable form:

b ij ¼ Rmi ðoq Amn  Gqmk Akn þ Amk Gqkn ÞRqp Rnj ^op A

ð23Þ

where the expression within parentheses, i.e.





rA

:¼ oq Amn  Gqmk Akn þ Amk Gqkn

ð24Þ

mnq

is symmetric with respect to the m; n indices and is in this paper called the corotational gradient of A in V. With the notation in (24), Eq. (23) can thus be written as follows:



b bA r

 ijp

  b ij ¼ Rmi Rnj r A :¼ ^op A

Rqp ;

ð25Þ

mnq

or, in compact notation,





b¼r b ðRT  A  RÞ ¼ ðRT  RT Þ : r A  R: bA r

ð26Þ

b is the pulled-back counterpart of the corobA From the foregoing analysis, it can be concluded that r constitutes an objective entity. tational gradient of A in V and that therefore the corotational gradient  b ¼ rA. In the domain V bA b of b to C), obviously it is G ¼ 0, hence r A ¼ r For R ¼ I (no rotation from C the isoclinic configurations, the corotational gradient coincides with the standard gradient. For subsequent use, the following is shown to hold. Let B ¼ fBpij g denote a third-order tensor of the C configuration and let us note the following:

  Bpij r A

¼ Bpij ðop Aij  Gpik Akj þ Aik Gpkj Þ ¼ op ðBpij Aij Þ  op Bpij Aij  Bpij Gpik Akj þ Bpij Aik Gpkj

ijp

¼ op ðBpij Aij Þ  op Bpij Aij  Bprj Gpri Aij þ Bpir Gpjr Aij ¼ op ðBpij Aij Þ  ðop Bpij  Gpir Bprj þ Bpir Gprj ÞAij :

ð27Þ

This, written in compact notation, reads:

    .  B.. r A ¼ r  ðB : AÞ  r B : A

ð28Þ

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where r B is a divergence-like tensor defined as





r B

:¼ op Bpij  Gpir Bprj þ Bpir Gprj :

ð29Þ

ij

We close this subsection noting that, with the notion of corotational gradient herein devised, the computation of the gradient of a tensor field is accomplished in the domain of the particle intermediate configurations, V, taking into account the substructure director vector triad orientation, which varies from point to point in V. The notion of corotational gradient is thus similar to the notion of covariant gradient of tensor analysis, and the three-index coefficients Gpkn are similar to the Christoffel symbols (Sokolnikoff, 1962). 4. Thermodynamics considerations Considering isothermal conditions for simplicity, let the material be endowed with an internal  b^ b j; g , all arguments being specified over be; ^ ep ; j; r ep ; r energy potential per unit mass, say u ¼ u E b , g is the enb of the isoclinic configurations; r b denotes the gradient operator in V the domain V tropy density per unit mass and j is some scalar measure of plastic strain. Both variables ^ ep and j are related to a same plastic deformation state of the material, but they are nevertheless treated as independent of each other as far as the relevant plasticity flow laws remain unspecib . The function uðÞ is continuous, at least fied. Both ^ ep and j are assumed C1-continuous in V twice differentiable with respect to its arguments; also, it is an isotropic function (Dafalias, 1987, 1988, 1998). Within the context of nonlocal (gradient) continua, the classical thermodynamics principle of the local action does not hold (Edelen and Laws, 1971; Edelen et al., 1971; Eringen, 2002) due to the long distance energy interchanges promoted by the irreversible deformation processes and the consequent microstructure rearrangements. These long distance interchanges can macroscopically be accounted for by means of a scalar variable, say P, called (nonlocality) energy residual, or simply residual, which has the meaning of the power density (per unit mass) transmitted to the generic material particle by all other particles in the body. The concept of energy residual, introduced in the context of generalized continua (Kunin, 1968; Edelen and Laws, 1971; Edelen et al., 1971; Eringen and Edelen, 1972; Eringen, 2002), and exploited by Dunn and Serrin (1985) and Maugin (1990), was subsequently reproposed in the form shown in the present paper by Polizzotto and Borino (1998), Polizzotto et al. (1998), Borino et al. (1999), Polizzotto (2001, 2003a,b, 2007), Liebe and Steinmann (2001) and others. The first thermodynamics principle can be set in point-wise form as follows:

1

q

r : D þ P ¼ u_

ð30Þ

where q is the mass density in the B configuration; r, the Cauchy stress; and D ¼ LS ¼ ðF_  F 1 ÞS . For P = 0, the first principle (30) takes on its classical format for simple materials; hence, a simple material can be qualified as one for which the residual P is identically vanishing under any deformation mechanism. In a body made of simple material no long distance particle interactions occur and Eq. (30) holds with P  0 correspondingly. On the other hand, a body made of nonsimple material as a whole forms a constitutively insulated (or closed) system, for which Eq. (30) holds with PX0, but P has to disappear on integration of (30) over the domain, V, of the body, that is,

Z V

r : D dv ¼

Z

qu_ dv:

ð31Þ

V

This means that the first principle saves its classical format if written globally for the entire body. A nonsimple material particle collection (or body) obeying (30) and (31) is here referred to as a globally simple material (or body); a featuring property of it is the (global) insulation condition (Polizzotto and Borino, 1998), that is,

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Z

qP dv ¼ 0

ð32Þ

V

for all deformation mechanisms. In addition to (32), the residual P has to satisfy the so-called locality recovery condition, namely, P  0 for any deformation mechanism with plastic strain carrying no nonlocality sources, such that the gradient material model behaves as a simple one correspondingly (Polizzotto et al., 2006). In the present context, this condition requires that the plastic strain field be featured by a vanishing corotational gradient in a domain of intermediate configurations, or, equivalently, it has to be uniform in the domain of isoclinic configurations. It is to be noted that condition (32) continues to hold also in the presence of a boundary layer with surface energy of the type considered by Gudmundson (2004), Abu Al-Rub et al. (2007), Abu Al-Rub (2008), provided that this layer—like an internal interface—be a part of the considered closed system (Polizzotto, 2008). By assumption, the presence of such boundary layers is excluded. The free energy potential, say w, is related to u by the Legendre transformation u ¼ w þ hg, where h P 0 is the absolute temperature. Since h_ ¼ 0 by assumption, the first principle (30) can be rewritten as

hg_ ¼

1

q

r : D  w_ þ P:

ð33Þ

The second thermodynamics principle states that the entropy production is nonnegative, that is (in the present context of isothermal conditions), g_ P 0 everywhere in V. This property was assumed to hold both for simple and nonsimple materials for reasons explained e.g. in Polizzotto (2007) (hence not repeated here). This is in contrast to Edelen and Laws (1971), Edelen et al. (1971), Eringen and Edelen (1972), who instead introduced an energy residual also for the entropy inequality. As a consequence, we obtain from (33)

C :¼

1

q

r : D  w_ þ P P 0;

ð34Þ

where the scalar C ¼ hg_ provides the plastic dissipation power density per unit mass. Eq. (34) is the Clausius–Duhem inequality written in its nonlocal form featured by the presence of the residual P; it holds for any deformation mechanism, with the equality sign if, and only if, the underlined deformation process is reversible. For nonreversible deformation mechanisms, as a consequence of the Onsager reciprocity principle, which is assumed to hold also in the present context, C proves to be an essentially positive ‘‘bilinear form” in terms of independent plastic strain rates (or fluxes) and of associated ‘‘plastic” driving forces (or affinities). This property, referred to as bilinear dissipation condition in the following, was not considered by Edelen and Laws (1971), Edelen et al. (1971), Eringen and Edelen (1972), Dunn and Serrin (1985), Maugin (1990). 5. Restrictions upon the constitutive equations A thermodynamic procedure, previously elaborated by Polizzotto and Borino (1998), Polizzotto et al. (1998), Polizzotto(2003a, 2007) in the framework of gradient plasticity with small deformations, is in this section rediscussed within the present context and used for the assessment of the restrictions upon the constitutive equations. In this procedure a central role is played by inequality (34) together with the aforementioned insulation condition, locality recovery condition and bilinear dissipation condition. Consistent with the classical constitutive equation theory (Truesdell and Noll, 1965; Colemann and Gurtin, 1967; Germain et al., 1983), the procedure at hand develops entirely within (nonlocal) thermodynamics and provides the state equations, the right expressions of the dissipation power density and of the residual, as well as the pertinent higher order boundary conditions associated with the differential-type state equations. For the moment, the plasticity flow laws are left unspecified. In the following, the procedure is expounded with reference to the domain V of the intermediate configurations. This procedure is achieved in a number of steps.

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Step 1. The material time derivative of the Green–Lagrange strain, E, decomposed as in Eqs. (2) and (3) with reference to C, can be set in the form (details are skipped for brevity):

  1 T T E_ ¼ F p  Ee þ Lpr  C e þ C e  Lpr  F p : 2

ð35Þ

Here, the (symmetric) tensor C e (right Cauchy–Green elastic strain tensor) is defined as T

C e :¼ F e  F e ¼ I þ 2Ee ;

ð36Þ  e

pr

whereas L is given by Eq. (15) and E by Eq. (14). The second Piola–Kirchhoff stress tensor in C, say S, is expressed as



q e 1 T F  r  Fe q

ð37Þ

 is the mass density in C. Hence, since E_ ¼ F T  D  F, by Eqs. (35) and (37), we easily obtain where q

1

q

r:D¼

  1 1 T S : Ee þ Lpr  C e þ C e  Lpr : 2 q

ð38Þ

Step 2. Let the free energy w be assumed in the form

    b e þ w ^ep ; r b ^ep ; j; r bj ; w ¼ we E p

ð39Þ

b . Temperature h does not where all arguments are referred to the domain of isoclinic configurations, V appear as argument of w due to the assumed isothermal conditions. Since

b e ¼ RT  Ee  R; E

^ep ¼ RT  ep  R;

ð40Þ

the potential w can be rewritten as

b ðRT  ep  RÞ; j; r b jÞ: w ¼ we ðRT  Ee  RÞ þ wp ðRT  ep  R; r

ð41Þ

On expanding the time derivative of w and with the aid of Eqs. (12) and (25), we can write the following:



 owe : RT be o E   owp : RT o^ep

  owe T  R : Ee be  oE  ow  _ p  ep  R ¼ R  p  RT : ep ^ oe _  Ee  R ¼



R

1 0 1   . .  T p  owp ow _ p T . b R  ep  R ¼ @  b R e R @  A. r A.. r b ^ep b ^ep o r o r 0 1 2 3T   .   ow p A  RT 5 .. r ep : ¼ 4ðRT  RT Þ : @  b ^ep o r

ð42Þ ð43Þ

0

ð44Þ

Next, with the positions

Y Y

ð0Þ

ð45Þ 1

3

ow  4ðR  RÞ : @  p A  RT 5 :¼ q b ^ep o r 0 1 owp T @ owp A ð1Þ  R    ; :¼ q ; v :¼ q oj bj o r

ð1Þ

vð0Þ

ow  R  p  RT :¼ q ^p 0 2 oe

ð46Þ

ð47Þ

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we also can write

     ow 1h w_ ¼ R  e  RT : Ee þ Y : ep þ vj_ þ r  Y be q oE

ð1Þ

i  : ep þ vð1Þ j_

ð48Þ

where, using the notation of (29),

Y :¼ Y

ð0Þ



 r Y

ð1Þ

;

v :¼ vð0Þ  r  vð1Þ :

ð49Þ p

Step 3. The time derivative of the Eulerian plastic strain e of (8) can be set in the form

e_ p ¼

 1  pT p p L  b þ b  Lp 2

ð50Þ

p

where b is a (symmetric) Cauchy–Green type plastic strain tensor as p

b :¼ F p

T

 Fp

1

¼ I  2ep :

ð51Þ

By (14)2 and (15) we then easily obtain the relation 

ep ¼

 1  pr T p p L  b þ b  Lpr 2

ð52Þ

 p

where e is a field as regular as ep . Step 4. Substituting (52) into (48), and then the latter together with (38) into (34), we can rewrite Eq. (34) as follows:



    1 ow 1 1 T S  R  e  RT : E e þ S : Lpr  C e þ C e  Lpr  Y  b e 2 2 q q q oE   1    1 T p p : Lpr  b þ b  Lpr  vj_  r  Y ð1Þ : ep þ vð1Þ j_ þ P P 0: q q



ð53Þ

Since this inequality holds for arbitrary elastic–plastic deformation mechanisms, even when these are purely elastic and thus P is zero together with all plastic strain variables, inequality (53) implies

R  S¼q

owe T ow  ee : R ¼q b e oE oE

ð54Þ

This is the pertinent elasticity law. Assuming that (54) remains valid also in the presence of plastic deformations, inequality (53) reduces to the expression



q C ¼ ðR  PÞ : Lpr  vj_  r  Y

ð1Þ

  P P 0 : ep þ vð1Þ j_ þ q

ð55Þ

where R and P are the (in general nonsymmetric) Mandel stresses defined as

R :¼ S  C e ;

p

P :¼ Y  b :

ð56Þ

The latter stresses are called ‘‘Mandel stresses” because R coincides with a stress tensor devised by Mandel (1971, 1973). Eq. (56) poses restrictions whereby R and P must be each the simple product of two symmetric second-order tensors. We ignore these restrictions for the rest of this section, hence consider R and P as general second-order tensors (but we shall return to this point in the next section). Step 5. Next, let us note that, in the generic deformation mechanism, the flux set ðLpr ; j_ Þ comprises 10 (independent, by assumption) plastic strain rate components. Hence, by the bilinear dissipation condition previously stated, we can set

q C ¼ R : Lpr þ v j_ 

ð57Þ pr

where R is a second-order stress tensor, v a scalar, both being (unknown) functions of the set ðL ; j_ Þ. On subtracting (57) from (55), we then obtain 



q P ¼ ðR  R þ PÞ : Lpr  ðv  vÞj_ þ r  Y

ð1Þ

  : ep þ vð1Þ j_ :

ð58Þ

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Thus the global residual in the domain V, after application of the divergence theorem, proves to be

Z

q P dv ¼

Z

V

½ðR  R þ PÞ : Lpr  ðv  vÞj_  dv þ

V

Z

 n Y

ð1Þ

  : ep þ vð1Þ j_ da

ð59Þ

B

where B :¼ oV and n is the outward normal to B. Note that, in writing (59), the vector 

w :¼ Y ð1Þ : ep þ vð1Þ j_ has been tacitly considered continuous throughout V. The volume integral on the right-hand side of (59) describes the mutual long distance interactions between all the material particles within V, hence it has to vanish identically (interactions between every two particles cancel each other), for any deformation mechanism and for any possible choice of the evolution laws, hence for arbitrary Lpr and j_ fields. This implies the following equalities:

R ¼ R  P;

v ¼ v

in V;

ð60Þ

hence Eq. (57) becomes

q C ¼ ðR  PÞ : Lpr  vj_ P 0;

ð61Þ

which substantially coincides with a result by Mandel (1971, 1973), whereas Eqs. (58) and (59) simplify as follows:



  : ep þ vð1Þ j_ Z Z    q P dv ¼ n  Y ð1Þ : ep þ vð1Þ j_ da: ð1Þ

q P ¼ r  Y V

ð62Þ ð63Þ

B

Step 6. In general the plastic deformation mechanism is active only in a subdomain V p V, such  that ep ¼ 0, j_ ¼ 0 and P ¼ 0 in the instantaneously elastic subdomain V e ¼ V n V p . Hence, the boundary surface Bp :¼ oV p splits into two parts, i.e. Bp ¼ BpðintÞ [ BpðextÞ , where BpðintÞ is the moving elastic/ plastic boundary that separates V p from V e , whereas BpðextÞ 2 B. As a consequence of the C1-continuity  of ep , ep and j_ in V, it obviously is 

ep ¼ 0; j_ ¼ 0 

on BpðintÞ

p

re ¼ 0; rj_ ¼ 0

ð64Þ

on BpðintÞ : ð1Þ

By (64) and with the positions qðnÞ :¼ n  Y

Z

q P dv ¼

Z

V

q P dv ¼

Vp

Z

BpðextÞ

ð65Þ ð1Þ

ð1Þ

, pðnÞ :¼ n  vð1Þ , Eq. (63) can be rewritten as

  ð1Þ  ð1Þ qðnÞ : ep þ pðnÞ j_ da;

ð66Þ

which means that the long distance power flux across BpðintÞ is null identically. By the insulation condition (32), the integral on the left-hand side of (66) has to vanish for any deformation mechanism and for any choice of the evolution laws, therefore the following higher-order boundary conditions must be satisfied: 

ep ¼ ep ¼ 0;  ð1Þ

qðnÞ ¼

ð1Þ qðnÞ

¼ 0;

ð1Þ j_ ¼ j ¼ 0 on Bð1Þ pc Bc ; ð1Þ p_ ðnÞ

¼

ð1Þ pðnÞ

¼ 0 on

ð1Þ Bpf



ð67Þ ð1Þ Bf :

ð68Þ

Conditions (67) and (68), being related to points lying on the external boundary B, hold both in (coroð1Þ ð1Þ tational) rate form and in time finite form. In (67) and (68) the decomposition B ¼ Bc [ Bf has been ð1Þ adopted, where Bc is the portion of B where some idealized (higher-order) ‘‘hard” constraints (capað1Þ ble to impede the onset of plastic strains during deformation) are located, whereas Bf is the remaining part free from such constraints. The boundary conditions (67) and (68) correspond to the ‘‘microclamped” (or ‘‘microhard”) and ‘‘microfree” boundary conditions advanced by Gurtin (2000, 2002) and Gurtin and Needleman (2005).

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ð1Þ

The time derivatives of qðnÞ and pðnÞ can be set in the following forms:

  .    ð1Þ  qðnÞ ¼ Q 1 : ep þ Q 2 .. r ep þ Q 3 j_ þ Q 4  rj_ ;   .    ð1Þ p_ ðnÞ ¼ P 1 : ep þ P 2 .. r ep þ P 3 j_ þ P 4  rj_ :

ð69Þ ð70Þ

Here, the quantities Q r , P r , (r = 1, 2, 3, 4), are expressed in terms of the partial derivatives of   ð1Þ ð1Þ qðnÞ and pðnÞ , respectively, and are in general not identically vanishing. The tensor r ep , by Eq. (24), has the components





r ep





mnq





¼ oq epmn  Gqmk epkn þ epmk Gqkn ;

ð71Þ



ð1Þ ð1Þ hence, by (64) and (65), we can state that qðnÞ ¼ 0 and p_ ðnÞ ¼ 0 on the moving elastic/plastic boundary, BpðintÞ . Conversely, under the conditions



 ð1Þ

ð1Þ

ep ¼ qðnÞ ¼ 0; j_ ¼ p_ ðnÞ ¼ 0 on BpðintÞ ;

ð72Þ

we have from (69) and (70):

  .   Q 2 .. r ep þ Q 4  rj_ ¼ 0;

  .   P 2 .. r ep þ P 4  rj_ ¼ 0;

ð73Þ

 

which imply r ep ¼ rep ¼ 0 and rj_ ¼ 0 on BpðintÞ correspondingly. It thus follows that the higher-order boundary conditions (67) and (68) can equivalently be replaced by (72). Step 7. By the locality recovery condition, the residual P of (62) has to vanish everywhere in V for any plastic strain mechanism with a plastic strain field being uniform in the isoclinic configuration, hence with a vanishing corotational gradient in any intermediate configuration. By (62), the following conditions hold:

rY

ð1Þ

¼ 0; r  vð1Þ ¼ 0 in V:

ð74Þ

These, by (46) and (47)2, impose restrictions upon the function wp . Namely, wp has to depend on b j homogeneously with a degree greater than one (Polizzotto, 2007). b^ ep and r r At this stage, we have found the state equations (45)–(47), (49) and (54), the correct expressions of the dissipation C, (61), and of the residual P, (62), as well as the HO boundary conditions (67), (68) and (72). As shown by (54) and (61), the relevant phenomenological constitutive model exhibits an elastic–plastic behavior with a mixed kinematic and isotropic hardening. Namely, S is elastically related to Ee by (54), whereas P and v play the role of back-stress and drag-stress, respectively, with hardening laws (49). The latter laws consist in a second-order PDE system which, together with the boundary conditions (67), (68) and (72), can in principle be solved with respect to ep and j in terms of any specified Y and v fields in given domain V p . As to the constitutive parameters, those incorporated in the Hessian matrix of we , say K, can be considered known if K is identified with the standard elasticity moduli fourth-order tensor, whereas those incorporated in the Hessian matrix of wp , i.e. the hardening moduli tensor H, need being specifically determined by laboratory experiments and/or suitable identification procedures. In contrast to the external boundary surface, BpðextÞ , in which there are only seven higher-order boundary conditions at every point, either the kinematic ones, (67), or the static ones, (68), in the internal moving elastic/plastic boundary surface, BpðintÞ , there are fourteen higher-order boundary conditions at every point, (72), seven of kinematic type and seven of static type, all in time rate form. The extra boundary conditions can be thought of to be aimed at determining the current location of BpðintÞ within V.

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6. Evolution laws The dissipation power density (61) interprets the material plastic constitutive behavior in the tendimensional tensor space in which the relative stress variables R :¼ R  P (nine components) and v, as well as the related work-conjugate kinematic variables Lpr and j_ , are imbedded. Such full dimension description of the material plastic behavior has been rendered possible by ignoring—in the thermodynamic procedure of the previous section—the existence of conditions (56). These in fact pose restrictions upon the Mandel stresses R and P, which consequently cannot be considered as general second-order tensors, but instead belong each to a six-dimensional manyfold of the nine-dimensional stress space, Lubliner (1986) and Aravas (1992). This restrictive property of the Mandel stresses makes them unable to effectively work against some plastic strain rate modes of Lpr , which thus turn out to be redundant, in the sense that no plastic work is done correspondingly. This also means that any plasticity flow laws based upon the Mandel stresses as driving dissipative stresses cannot lead to a complete evaluation of Lpr . The purpose of this section is to construct the evolution laws for the material at hand taking into account all thermodynamic restrictions previously derived, including (56). Since, as noted by Lubliner p (1986), the tensors C e  R and b  P are symmetric by (56), we can write the conditions p

C e  R  RT  C e ¼ 0;

p

b  P  PT  b ¼ 0

ð75Þ

which must be satisfied by any R and any P. As shown in Appendix A, the above conditions provide specific relations between the symmetric and skew-symmetric parts of R and P, respectively. Namely, with the positions:

T :¼ RS ;

e :¼ RA ; T

X :¼ PS ;

e :¼ PA ; X

ð76Þ

the following relations hold true, i.e.

e ¼ V e : T; T

e ¼ V p : X; X

ð77Þ

n o where V e ¼ V eijab and V p ¼ fV pijab g are fourth-order tensors (referred to as projection tensors in the following), skew-symmetric in the first index pair ði; jÞ, but symmetric in the second one ða; bÞ. The relations in (77) can be interpreted stating that the skew-symmetric parts of the Mandel stresses are uniquely determined as (specific) skew-symmetric projections of the respective symmetric parts. (We use the adjective ‘‘specific” to mean that the projection operation is centered upon the use of a p p specific (strain) tensor, either C e , or b .) Notice that V e and V p are constructed with C e and b , respecp e e e e tively. For C ¼ I (i.e. null elastic strain, E ¼ 0), it is V ¼ 0, T ¼ 0, R ¼ RS ¼ S; analogously, for b ¼ I e ¼ 0, P ¼ PS ¼ Y. (i.e. null plastic strain, ep ¼ 0), it is V p ¼ 0, X With the aid of Eqs. (76) and (77), the dissipation density (61) becomes: 0

00

q C ¼ T : Dp  X : Dp  vj_ P 0

ð78Þ

where we have posed 0

T

Dp :¼ Dp þ V e : W pr ;

00

T

Dp :¼ Dp þ V p : W pr :

ð79Þ

The above means that the symmetric parts of the Mandel stresses, T and X, play the role of independent driving dissipative stresses, which induce independent generalized plastic strain rates, 0 00 Dp and Dp , each of which is the sum of the standard plastic strain rate, Dp , with a specific symmetric T T projection of the relative plastic spin, that is, the symmetric strain rates V e : W pr and V p : W pr , respectively. On the basis of the expression (78) of the dissipation power density, we see that only the symmet0 00 ric strain rates Dp and Dp need being modelled. Hence, the evolution laws (with associative flow rule) can be set, using a standard notation, as:

f ¼ /ðT; XÞ  v  ry 6 0; p0

D ¼ j_ /;T ;

p00

D

¼ j_ /;X

j_ P 0; j_ f ¼ 0

ð80Þ ð81Þ

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Here, f is the (convex, smooth) yield function, ry > 0 the (constant) yield stress and /;T :¼ o/=oT and 0 00 analogously for /;X . Any variable set ðT; X; Dp ; Dp ; j_ Þ complying with (80) and (81) satisfies the sign constraint of Eq. (78). The flow laws (80) and (81) admit a maximum dissipation principle, but this point is skipped for brevity. It is worth noting that the circumstance whereby Eqs. (80) and (81) do not comprise a specific flow law for the relative plastic spin, W pr , does not mean that there is no need for such a law. Indeed, the restrictive nature itself of the Mandel stresses makes the relative plastic spin be evaluated only through its specific symmetric projections, as shown by (79). For a complete evaluation of the strain rate tensors Dp and W pr there would be the necessity to treat the Mandel stresses as general secondorder tensors (nine independent components), hence to ignore the constraints (56), but this would be not consistent with thermodynamic requirements.

7. A simple application As a simple illustrative example, a well-known finite deformation shear model is used in this section (see e.g. Dafalias, 1983; Obata et al., 1990; Aravas, 1992; Pereda et al., 1993). For simplicity, here we assume that the material exhibits isotropic hardening, but no kinematic hardening. As usual, the starting point is the mapping:

x ¼ x0 þ uðyÞ; 0

0

y ¼ y0 ;

z ¼ z0 ;

ð0 6 y 6 HÞ

ð82Þ

0

where ðx ; y ; z Þ is the initial position of a particle, ðx; y; zÞ its actual position, whereas uðyÞ denotes the displacement function. On posing c :¼ du=dy ¼ ce þ cp , we can write:

0

1 1 c 0 B C F ¼ @ 0 1 0 A; 0 0 1

0

1 ce B F ¼ @0 1 0 0

0

0

1

1 cp B F ¼ @0 1 0 0

C 0 A; 1

e

0

1

C 0 A:

p

ð83Þ

1

The plastic strain rate and total plastic spin components are:

Dp12 ¼ Dp21 ¼ c_ p =2;

W p12 ¼ W p21 ¼ c_ p =2

ð84Þ

all other components being identically vanishing. The nonzero components of the rigid-body spin of the plastic stretch principal directions are



x12 ¼ x21 ¼ h_ ¼ c_ p = 4 þ c2p



ð85Þ

where the relation tan 2h ¼ 2=cp has been accounted for, and thus pr W pr 12 ¼ W 21 ¼

h   i 2 þ c2p =2 4 þ c2p c_ p

ð86Þ

The elastic tensors C e and Ee prove to be

0

1

B C e ¼ @ ce 0

ce 1 þ c2e 0

1 0 C 0 A; 1

0 Ee ¼

0

1B @ ce 2 0

1

ce 0 c2e 0 C A 0

ð87Þ

0

and thus the second Piola–Kirchhoff stress tensor is of the form

0

S11 B S ¼ @ S21 0

S12 S22 0

0

1

C 0 A: S33

ð88Þ

S and Ee are mutually related by Hooke’s law for isotropic materials; both are constant in space.

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Next, with C e we can compute the elastic projection tensor V e , whose not trivially vanishing components are:

9 V e1211 ¼ V e1222 ¼ ce =ð2 þ c2e Þ > > > = V e1212 ¼ c2e =2ð2 þ c2e Þ ; e 2 2 > V 2323 ¼ ce =ð4 þ ce Þ > > ; V e2331 ¼ V e3123 ¼ ce =ð4 þ c2e Þ

ð89Þ

besides the components obtained from the latter ones by interchanging the indices in the first and in the second index pairs. With the latter coefficients, together with S and C e , we can compute the symmetric and antisymmetric parts of the Mandel stresses as

T 11 ¼ S11 þ ce S12 ;

 9 T 12 ¼ T 21 ¼ 12 ce ðS11 þ S22 Þ þ ð2 þ c2e ÞS12 > =

T 22 ¼ ce S12 þ ð1 þ c2e ÞS22 ; T 33 ¼ S33  T~ 12 ¼ T~ 21 ¼ 12 ce ðS11  S22 Þ þ c2e S12

> ;

ð90Þ

all other components being trivially vanishing. The generalized plastic strain rate proves to have the nonzero components:

)  0 0 Dp11 ¼ Dp22 ¼ 12 ce =ð4 þ c2e Þ c_ p  0 0 Dp12 ¼ Dp21 ¼ 12 1 þ c2e =ð4 þ c2e Þ c_ p

ð91Þ

such that

j_ ¼ Gðce Þc_ p ;

h i1=2 pffiffiffi

2 4 þ c2e : Gðce Þ :¼ 5c2e þ ð4 þ c2e Þ2

ð92Þ

In the assumed hypothesis of isotropic hardening, by (47) we can write:

v ¼ hðj  ‘2 d2 j=dy2 Þ

ð93Þ

where h > 0 is the (constant, by hypothesis) hardening modulus and ‘ is an internal length parameter. Hence, by the yield condition, we can write the differential equation:

j  ‘2 d2 j=dy2 ¼ A :¼ ½/ðTÞ  ry =h:

ð94Þ

Since the right hand member is independent of y, and since jð0Þ ¼ jðHÞ ¼ 0, Eq. (94) can be integrated to obtain, with the position f :¼ H=‘ (size ratio):



jðyÞ ¼ A

  y  y cosh f  1  cosh f þ1 : sinh f sinh f H H

ð95Þ

Substituting the latter (written in rate form) into (92), we then write

  _  y  y du_ A_ cosh f  1 S12 ¼ sinh f  cosh f þ1 þ dy G sinh f H H l

ð96Þ

where l is the elastic shear modulus. Since uð0Þ ¼ 0, the latter equation can be integrated to give:

  y  sinh f y y  S_ y _ A_ cosh f  1  uðyÞ 12 H cosh f 1  þ : ¼ þ G f sinh f H H l H H f

ð97Þ

For a complete rate solution of the problem we need to evaluate T_ and S_ 12 . This in turn would require to determine the tangent stiffness tensor, what indeed is not a straightforward task in the gradient plasticity framework (this in general requires solving a PDE system). This issue actually requires further investigation, what we shall do in a future work. Here instead we drastically simplify the shear problem at hand by considering the material to be rigid-plastic, i.e. we assume ce  0, hence C e ¼ I, pffiffiffi e V ¼ 0, T ¼ S, G ¼ 1= 2. Then, with the position g :¼ uðHÞ=H, we finally obtain:

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  y  sinh f y y  uðyÞ cosh f  1  H ¼ KðfÞ cosh f 1  þ g; H f sinh f H H f   1 pffiffiffi cosh f  1 2A ¼ KðfÞg; KðfÞ :¼ 1  2 ; f sinh f /ðTÞ h ¼ 1 þ pffiffiffi KðfÞg: ry 2ry

ð98Þ ð99Þ ð100Þ

The latter relation shows the size effects on the ‘‘effective stress” /ðTÞ, which—at g constant—increases with increasing ‘. In Figs. 1 and 2 some response functions are plotted for different values of the size ratio f. The obtained results qualitatively agree with analogous results given by Shu et al. (2001) for small deformations. 8. Conclusions The strain gradient plasticity theory herein presented is based upon known concepts of finite deformation plasticity, as the multiplicative strain decomposition rule (Lee and Liu, 1967; Lee, 1969), the isoclinic configuration and substructure director vector triad (Mandel, 1971, 1973), the (relative) plastic spin (Dafalias, 1985, 1987, 1988, 1998; Zbib and Aifantis, 1988). In addition, the assumption has been made that a class of physical phenomena sensitive to plastic strain gradients (as size effects, plastic strain localization and the like) can be macroscopically represented by means of the (first order) plastic strain gradient computed in the domain of isoclinic configurations, in which the substructure director vector triad has constant orientation both in time and space. The objective notion of corotational gradient has been devised to make it possible to compute the plastic strain gradient in a generic domain of intermediate configurations. In this way, the result by Dafalias (1987, 1988, 1998), whereby the constitutive equations can be invariantly written in any intermediate configuration, has been extended to gradient plasticity. It emerged that, in finite deformation nonlocal gradient plasticity, the particle system substructure interacts with the continuum not only through its kinematics (characterized by the director vector triad spin), but also through its geometry (characterized by the director vector envelope lines). The corotational gradient constitutes a means by which the substructure geometry is accounted for in the computation of the plastic strain gradient. A thermodynamic procedure employing the so-called (nonlocality) energy residual (a concept already used by the author and co-workers elsewhere in the framework of small deformation theories, see e.g. Polizzotto, 2007), has been rediscussed and applied to derive the thermodynamic restrictions upon the constitutive equations. These restrictions include, besides the elasticity law, the correct constitutive forms of the plastic dissipation power density and of the energy residual, as well as the (mixed kinematic/isotropic) hardening law. These prove to be in the form of PDEs with related higher

Fig. 1. Rigid-plastic shear model. Shear strain profiles for different values of the size ratio H=‘ and for g :¼ uðHÞ=H ¼ 0:5.

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Fig. 2. Rigid-plastic shear model. Adimensionalized effective stress, /ðTÞ= pffiffiffi ry , as a function of the adimensionalized displacement, g :¼ uðHÞ=H, for different values of the size ratio H=‘, with h= 2ry ¼ 2.

order boundary conditions, which include the ones associated to the moving elastic/plastic boundary. In this formulation, a crucial role has been played by the concept of globally simple material, i.e. a particle collection being constitutively insulated with respect to the long distance energy interactions from the exterior world. By the above thermodynamic procedure, it has been possible to arrive at a known expression (Mandel, 1971, 1973) of the dissipation power density in which the (nonsymmetric) Mandel stress tensor is multiplied by the (relative) plastic velocity gradient. As a consequence of the nature itself of the Mandel stress and back stress tensors, two fourth-order projection tensor operators have been envisioned, functions of the current elastic and plastic deformation state, respectively. These have permitted one to express the skew-symmetric parts of the Mandel stresses and back stresses in terms of the corresponding symmetric parts, and thus to formulate the plasticity flow laws in a six dimensional stress space in terms of the symmetric stresses. These flow laws admit a maximum dissipation principle, but this point has been skipped for brevity. The author believes that this research study provides several original contributions to finite deformation plasticity, both for local and nonlocal gradient materials. A simple example has been presented as an illustration of the method, from where the problem of the evaluation of the tangent stiffness tensor has emerged as a typical issue of nonlocal gradient plasticity. A numerical implementation of the proposed theory will be addressed in a separate paper to follow. Appendix A. In this appendix the relations (77) are proved. For this purpose, a procedure similar to analogous procedures by Dienes (1979), Obata et al. (1990), Nemat-Nasser (1990) and Mehrabadi and NematNasser (1987) is devised here. The symmetry conditions in (75) are the starting point. Since

e; R ¼ RS þ RA ¼ T þ T

~ P ¼ PS þ PA ¼ X þ X;

ðA:1Þ

the conditions in Eq. (75) can be written as follows:

eþT e  C e ¼ ðC e  T  T  C e Þ Ce  T p ~ ~  bp ¼ ðbp  X  X  bp Þ b X þX

ðA:2Þ ðA:3Þ

e and X ~ each of which is a set of three algebraic equations. These equations can be solved in terms of T as functions of T and X, respectively. Considering Eq. (A.2) first, let us note that both members of this equation are skew-symmetric. Using the index notation for more clarity, the mentioned equation reads:

C eik T~ kj þ T~ ik C ekj ¼ Xeij ;

Xeij :¼ C eik T kj  T ik C ekj :

ðA:4Þ

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C. Polizzotto / International Journal of Plasticity 25 (2009) 1280–1300

Let the skew-symmetric tensors T~ ij and Xeij be expressed in terms of their respective axial vectors, i.e.

T~ ij ¼ ijm /m ;

Xeij ¼ ijn wn ;

ðA:5Þ

where  ¼ fijm g is the Ricci alternating tensor, whereas /m and wn are the mentioned axial vectors. Then, Eq. (A.4)1 transforms as



 C eik kjm þ C ekj ikm /m ¼ ijn wn :

ðA:6Þ

Next, let the latter equality be multiplied by ijr and let us take into account the following identities, i.e.

ijr kjm ¼ dik drm  dik drk ; ijr ikm ¼ djk drm  djm drk ijr ijn ¼ 2drn :

ðA:7Þ ðA:8Þ

Then we obtain from (A.6):



C ekk drm  C erm /m ¼ C erm /m ¼ wr ;

ðA:9Þ

where we have set

C e :¼ trðC e ÞI  C e :

C eij :¼ C ekk dij  C eij ;

ðA:10Þ

Since, by (A.5)2, it is wr ¼ 12 rab Xab , Eq. (A.9) becomes

/k ¼



 1  1   e1  rba C eam T mb  C ebm T ma ¼ C e rab C ebm T ma C kr kr 2

ðA:11Þ

and this, with the index substitutions m ! b, b ! s, transforms as

 1  /k ¼ C e

kr

ras C esb T ab ¼

1   e1  ras C esb þ rbs C esa T ab : C kr 2

ðA:12Þ

Next, substituting (A.12) into (A.5)1 gives

T~ ij ¼ V eijab T ab ;

e ¼ V e : T; T

ðA:13Þ

where

 1  1

V eijab :¼ ijk C e ram C emb þ rbm C ema ; kr 2

ðA:14Þ

or, in compact notation,



1 V e :¼   C e    C e S :

ðA:15Þ

This fourth-order tensor is symmetric in the second index pair, but skew-symmetric in the first one. In a similar way, starting from (A.3), we obtain

~ ij ¼ V p X ab ; X ijab

~ ¼ V p : X; X

ðA:16Þ

where

 1  1

p V pijab :¼ ijk b ram bpmb þ rbm bpma ; kr 2

ðA:17Þ

or, in compact notation,

 p1    bp : V p :¼   b S p

ðA:18Þ e

V has the same symmetry features as V . References Abu Al-Rub, R.K., 2008. Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals. Int. J. Plasticity 24, 1277–1306.

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