Variational formulations for single-crystal strain-gradient plasticity at large deformations

Transcription

1 GAMM-Mitt. 36, No. 2, (2013) / DOI /gamm Variational formulations for single-crystal strain-gradient plasticity at large deformations. D. Reddy 1, 1 University of Cape Town, Department of Mathematics and Applied Mathematics and Centre for Research in Computational and Applied Mechanics, 7701 Rondebosch, South Africa Received 08 October 2012, revised 21 December 2012, accepted 27 January 2013 Published online 07 October 2013 Key words strain-gradient plasticity, dissipation function, single-crystal plasticity, large deformations, variational problem MSC (2000) 74C05,74C15,74S05 Variational formulations are constructed for rate-independent problems in single-crystal straingradient plasticity in the framework of large deformations. Provision is made for energetic and dissipative microstresses, and a hardening law based on accumulated generalized slips is introduced. The variational formulations use the flow rule either in terms of the dissipation function or a generalized normality law. 1 Introduction The literature on strain-gradient theories of plasticity has expanded considerably over the last two decades, with a number of important theoretical and experimental contributions. The interest in gradient theories is largely accounted for by their ability to model size effects at the mesoscale, and the underlying behaviour of geometrically necessary dislocations. Some representative and important works include [1 6]. In recent work, the author [7,8] has undertaken an investigation of the theoretical properties of problems of strain-gradient plasticity. The models studied are rate-independent versions of those developed by Gurtin and co-authors (see for example, [4, 9, 10]), and are confined to the small-strain regime. Results have been presented on variational formulations and wellposedness for both polycrystalline and single-crystal plasticity. In a subsequent work [11], the earlier variational framework has been extended to the large-deformation regime for problems of single-crystal strain-gradient plasticity. In that work the kinematic framework adopted includes as primary variables the displacements and slip rates and their gradients on the individual slip systems. The conjugate quantities are a scalar microforce and vector microstress respectively. Flow relations for each slip system are constructed in terms of a yield function and equivalently in terms of a dissipation function that depends on slip rate and slip-rate gradient. The latter option serves as the point of depature for construction of the variational problem. Corresponding author daya.reddy@uct.ac.za, Phone: Fax:

2 150. D. Reddy: Single-crystal strain-gradient plasticity at large deformations Related work of a computational nature includes the investigation in [12] for the special case of microstresses of energetic type. Furthermore, the viscoplastic problem is studied numerically in [13] with both energetic and dissipative hardening, and in the context of large deformations. The objectives of this contribution are, first, to provide an overview of the rate-independent problem for strain-gradient plasticity of single crystals. A new hardening law, based on one developed in [14] in the small-strain regime and which makes use of the accumulated generalized slips, is introduced. The variational problem is presented in two forms: first, using the dissipation function as the basis for the flow law, and referred to as the primal form; and second, a dual formulation that uses a generalized form of the yield function and normality law. These two formulations are equivalent. Details of the first have been presented in [11], while the dual formulation is new in this context. For the primal formulation, the associated incremental problem may be formulated as a minimization problem. The relationship of the dual formulation to classical versions is highlighted. The structure of the rest of this work is as follows. In Section 2 the governing equations for the large-deformation problem are set out. The flow law is formulated as a normality relation and, equivalently, in terms of the dissipation function. Section 3 is concerned with the primal variational formulation of the problem. The dual problem is discussed in Section 4. Notation. Direct notation is largely used. Vectors and second-order tensors are denoted respectively by lower- and upper-case boldface letters. The scalar product of two vectors a and b and of two second-order tensors A and are defined with respect to their components in an orthonormal basis by a b = a i b i, A : = A ij ij. The summation convention for repeated indices is used. The magnitudes of a and A are therefore obtained from a =(a a) 1/2 and A =(A : A) 1/2. Other non-standard notation will be defined when first introduced. 2 Governing equations In [15], Gurtin has presented a thermodynamically consistent large-deformation theory of strain-gradient single-crystal plasticity. A cornerstone of the theory is the accounting for power expenditures associated with slip rates and their gradients, the conjugate quantities in the power being respectively a scalar microforce and vector microstress. The theory in [15] is one of viscoplasticity; in [11], Reddy has developed a rate-independent version of the theory. In the rest of this section we follow [15] and, for aspects of the rate-independent theory, [11]. 2.1 Kinematics Let the domain 0 in R 3 denote the undeformed, unstressed reference configuration of a material body. Its boundary is denoted by 0. The current configuration is denoted by, with boundary. The current position x of a material point X 0 is defined through the motion motion ϕ : 0 (0,T] for some final time T,sothatx = ϕ(x,t):=ϕ t (X). The map ϕ is assumed to be invertible and continuously differentiable with a continuous

3 GAMM-Mitt. 36, No. 2 (2013) 151 inverse. The deformation gradient F is defined by F = X ϕ t (X). The right Cauchy-Green tensor C and Green-St. Venant strain E are then given by C = F T F and E = 1 2 (C I). For an elastic-plastic body the standard multiplicative decomposition [16,17] is assumed: that is, F = F e F p, (2.1) in which F e is the elastic part of the deformation, accounting for stretch and rotation of the lattice, while F p represents the irreversible plastic distortion. Corresponding to the Cauchy-Green and strain tensors, elastic analogues may be defined according to C e = F et F e, E e = 1 2 (Ce I). (2.2) The velocity gradient is given by L = v, in which denotes the spatial gradient operator. The velocity gradient may be decomposed additively into elastic and plastic parts L e and L p as a result of the decomposition of F : that is, using the relation L = ḞF 1 together with (2.1), we define so that L e = Ḟ e [F e ] 1, L p = Ḟ p [F p ] 1 (2.3) L = L e + F e L p [F e ] 1. (2.4) Plastic deformation is assumed to isochoric in nature, so that the assumption det F p =1is made. Motion of dislocations is assumed to take place on a prescribed set of A slip systems. The αth slip system is defined by the slip-plane normal m α and slip direction s α. These two vectors constitute an orthonormal pair. Then the cumulative effect of slip on the A systems is determined through the slip rates ν α via the relation L p = α ν α s α m α. (2.5) Here and henceforth the notation α will denote the sum A α=1. Furthermore, the summation convention does not apply to lower-case Greek indices relating to slip systems. It is useful to define also the unit vector l α which together with the slip-plane normal and slip direction forms an orthonormal triad in the sense that l α = m α s α. (2.6) The slip-plane normal and slip direction are defined with respect to the intermediate or lattice configuration. Their spatial counterparts m α and s α are defined by m α =[F e ] T m α, s α = F e s α. (2.7) Likewise, the vector l α pushed forward to the current configuration is given by lα = F e l α. (2.8)

4 152. D. Reddy: Single-crystal strain-gradient plasticity at large deformations We consider edge dislocations, with line direction l α and slip direction s α ;andscrewdislocations, which have both line and slip directions s α. These are geometrically necessary dislocations. Their rates of change are related to the spatial slip rate gradient ν α : in particular, if the densities of edge and screw dislocations are denoted respectively by ρ e and ρ s,then their evolution is governed by the equations 1 ρ α e = s α ν α, ρ α s = l α ν α. (2.9) We also define the array of accumulated dislocation densities ρ acc = {ρ 1 acc,...,ρ A acc} by ρ α acc = [ ρ α e 2 + ρ α s 2] 1/2. (2.10) Here and henceforth, for any variable μ α associated with the slip systems, μ denotes the set {μ α } A α=1. It is shown in [15] (eqn (2.32)) that the accumulated dislocation density is related to the slip rate by ρ α acc = P α F et α tanν α, (2.11) in which P α := I m α m α is the projection onto the α th slip plane, and the tangential gradient α tan is defined by α tanμ =( μ s α ) s α +( μ l α ) l α. 2.2 Forces and balance equations We will require the resolved shear stress acting on the αth slip plane [16, 18]. This is defined in the current configuration by τ α = T m α s α (2.12) in which T is the symmetric Cauchy stress. The conventional equation of balance of linear momentum or equilibrium takes the form div T + b = 0, (2.13) in which b is a prescribed body force. Here div denotes the spatial divergence operator. The gradient theory makes provision in addition for a scalar microforce π α and vectorial microstress ξ α, which are power-conjugate respectively to the slip rate ν α and its gradient ν α. Then from a principle of virtual power it follows that these quantities are related to each other and to the resolved shear stress through the equation div ξ + τ α π α =0. (2.14) 1 Note that we make use of the definition of dislocation densities as quantities with dimensions of length, measured per unit area, so that they carry the dimension [length] 1. See [10], page 591 for further details.

5 GAMM-Mitt. 36, No. 2 (2013) Free energy and dissipation inequality The dissipation inequality takes the form J 1 ψ J 1 S e : Ė e α [π α ν α + ξ α ν α ] 0, (2.15) with J =detf.hereψ is the free energy. The quantity S e is analogous to the conventional second Piola-Kirchhoff stress in that it is related to the Cauchy stress through T = J 1 F e S e F et. (2.16) The first Piola-Kirchhoff stress P is related to the Cauchy stress by P = JTF T. (2.17) The free energy ψ is assumed to be decomposed additively in the form of an elastic energy ψ e, dependent on E e, a defect energy ψ d which depends, possibly via other variables, on the dislocation densities, and a term ψ h that depends on a set of internal-hardening variables η α (α =1,...,A) associated with each slip system. Thus ψ = ψ e (E e )+ψ d ( ρ)+ψ h ( η). (2.18) In (2.18), the set ρ includes both edge and screw dislocation densities. With the assumption that the stress S e is a function of the elastic strain E e, the use of the dissipation inequality in the usual way leads to the form S e = ψe E e =2 ψe C e (2.19) of the elastic relation. The energetic microstress ξ α en is defined by ψ d = α ξ α en ν α (2.20) so that, from (2.9), ( ) ξ α en = J 1 ψd s α ρ α + ψd e ρ α lα. (2.21) s A number of examples of defect energies ψ d and microstresses ξ α en may be found in [11]. With the decomposition ξ α = ξ α dis + ξ α en (2.22) into dissipative and energetic components, we obtain the reduced dissipation inequality [π α ν α +(ξ α ξ α en) ν α + g α η α ] 0. (2.23) α

6 154. D. Reddy: Single-crystal strain-gradient plasticity at large deformations The quantity g α is a conjugate force to the hardening variable η α,definedby g α 1 ψh = J η α. (2.24) It is convenient to define the generalized dissipative stress S α dis and slip rate Γ α by [ ] [ ] S α π dis = α l 1, Γα ν = α d ξα dis l d ν α, (2.25) where l d is a dissipative length scale. Then the reduced dissipation inequality (2.23) can be written in the compact form [S α dis : Γ α + g α η α ] 0. (2.26) α The inner product S α dis : Γ α is defined componentwise; that is, S α dis : Γ α = π α ν α + ξ α dis ν α. Denote the unit normal vector to the α th slip plane in the current configuration by ˆm α,and the orthogonal projection operator onto this plane by P α ;thatis, P α = I ˆm α ˆm α. (2.27) We make the assumption (see [15]) that ξ α normal := (I P α )ξ α = 0. (2.28) It follows that the reduced dissipation inequality (2.26) now holds with Γ α redefined as [ ] Γ α ν = α l d α tanν α. (2.29) We define the accumulated generalized slip Γ α acc by Γ α acc = Γ α = ν α 2 + l 2 d α tanν α 2, (2.30) and set Γacc := (Γ 1 acc, Γ 2 acc,...,γ A acc). (2.31) The quantity γacc α defined by γ acc α = ν α, γ α acc t=0 =0 may be viewed as a measure of accumulated glide dislocations, so that ν α measures the accumulation rate of glide dislocations on α. In this spirit, and given the definition (2.11) of the accumulation rate of dislocation densities ρ α acc and their relationship to the slip-rate gradients and hence the rate of change of geometrically necessary dislocations (GNDs), the quantity Γ α acc is a measure of the net accumulation rate of GNDs and glide dislocations for the αth slip system.

7 GAMM-Mitt. 36, No. 2 (2013) Yield function, flow and hardening relations The work [11] sets out in detail the flow relations in terms either of a yield function or a dissipation function. We briefly summarize the key concepts and relations. In [14] a theory of plastic flow has been developed in the context of infinitesimal strains, and in which the hardening relation depends on the array (2.31) of accumulated generalized slips. Furthermore, full account is taken in the theory of the contributions of latent and selfhardening. These latter details are omitted for convenience in what follows, in which the formulation is extended to the large-strain situation. Introduce the hardening function Y α (Γ α acc) with the properties Y α (Γ α acc) 0 and Y α (0) = 0,andset g α =ĝ α (Γ α acc) = Y α (Γ α acc). (2.32) The yield function is given by f α (S α dis,gα )= S α dis + gα Y 0 0. (2.33) The quantity Y 0 is an initial yield stress, assumed here to be constant on all slip systems, so that Y 0 g α represents the current yield stress on the αth slip system. Assuming rate-independent behaviour and an associative flow law the generalized slip rate and hardening rate, which are conjugate to S α dis and gα respectively, are given by the normality relation Γ α = λ α fα S α = λ α Sα dis dis S α dis, (2.34) η α = λ α fα g α = λα, together with the complementarity conditions f α 0, λ α 0, λ α f α =0. (2.35) At flow, from (2.30) and (2.34) we have λ α = Γ α acc = Γ α = η α. (2.36) Thus we see in particular that η α may be equated with Γ α acc. We will assume this equivalence in what follows, retaining in general the notation η α to emphasise that the hardening variable is treated as an independent variable, notwithstanding (2.36). Following a well-established path (see for example [19] (Section 4.2)), it can be shown [8] that the flow relations (2.34) and (2.35) may be written in the equivalent form D α ( Γ α ) D α ( Γ α )+τ α ( ν α ν α )+ξ α dis ( ν α ν α )+g α ( η α η α ), (2.37) in which the dissipation function D α is given by { D α ( Γ α, η α Y0 Γ )= α for all ( Γ α, η α ) such that Γ α η α, + otherwise. (2.38) Here, the sets of admissible generalized plastic strains Γ α and hardening variables η α comprise those which satisfy Γ α η α.

8 156. D. Reddy: Single-crystal strain-gradient plasticity at large deformations 3 The variational problem oundary conditions. The macroscopic boundary conditions are assumed to be u = 0 on D and Tn= t on N, (3.1) where D and N are complementary parts of the boundary of the domain. The boundary conditions for the slip rate and microstress are assumed to take the form ν α =0 on H, (3.2a) ξ α n =0 on F, (3.2b) in which H and F are complementary parts of the boundary. These boundary conditions are referred to respectively as micro-hard and micro-free conditions (see [20]). The weak or variational problem is shown in [11] to be one of finding the displacement u and slip rates ν α that satisfy the macroscopic equilibrium equation T : ũ dx = b ũ dx + t ũ ds, N (3.3) and the flow relation D α ( Γ α ) dx D α ( Γ α ) dx + T :( s α m α )( ν α ν α ) dx ξ α en ( να ν α ) dx + g α ( η α η α ) dx, (3.4) with the Cauchy stress T given by (2.16) with (2.19), and ξ α en by (2.21). The incremental problem. The variational problem (3.3) (3.4) does not have an equivalent formulation as a minimization problem. It is however possible to formulate the corresponding incremental problem as an unconstrained minimization problem. The time interval of interest 0 t T is partitioned according to 0=t 0 <t 1 < <t N = T with Δt = t n t n 1 being assumed uniform. The value of a quantity w at time t n is denoted by w n,andan increment in w over the time interval Δt is denoted by Δw = w n w n 1. Rate quantities are approximated by an Euler backward difference approximation, with ẇ Δw/Δt. The dislocation densities (2.9) at time t n can be expressed in terms of the slip rate νn α at time t n, using an Euler backward approximation Δρ α =Δt p α n ν α n. (3.5) Here p α = s α or m α for edge and screw dislocations, respectively. Then ρ α n = ρ α n 1 +Δt p α n νn α (3.6) { = ρ α F e n 1 +Δt n s α νn α for edge dislocations, +F e nl α νn α (3.7) for screw dislocations. In the same way, from (2.11) we can write Δρ α acc =Δt P α F et n α tan (n) να n (3.8)

9 GAMM-Mitt. 36, No. 2 (2013) 157 in which the time-discrete tangent α tan (n) is defined by α tan (n) να =( νn α s α n) s α n +( νn α l α n) l α n. (3.9) The time-discrete version of the defect energy is ψ d ( ρ) =ψ d ( ρ n 1 +Δ ρ n ):= ˆψ d (Δ ρ n ). (3.10) We make use of the exponential map [21] F p n matrix-valued exponential, and = (expλ n )F p n 1 in which exp is the Λ n =Δt α ν α n sα m α. (3.11) The incremental problem is then one of finding (u n,νn α,ηn)(α α =1,...,A) that satisfy, for α =1,...,A, T n : ũ dx = b n ũ dx + t n ũ ds, (3.12) N D α ( Γ α ) dx D α (Γ α n ) dx + T n :( s α m α )( ν α νn α ) dx ξ en(n) ( ν α νn α ) dx + gn α ( ηα Δη α ) dx. (3.13) Here T n is found from (2.19) and (2.16), and gn α and ξ en(n) are obtained by evaluating (2.24) and (2.21) respectively at time t n.furthermore,γ α n = (νn α)2 + l 2 d α tan να n 2. It has been shown [11] that if (u, ν, η) solves the problem of finding u n,νn α,ηα n that minimize the functional [ J(ũ, ν, η) = ψ e (Ẽe )+ ˆψ ] d ( ρ)+ψ h ( η) dx +Δt D α ( Γ α n ) dx b n ũ dx t ũ ds (3.14) N α over all admissible ũ, ν α and η α (α =1,...,A),then(u, ν, η) solves the variational problem (3.12) (3.13). 4 The dual problem The formulation in the previous section is based on the flow relations expressed in terms of the dissipation function. We examine here the equivalent and alternative variational formulation, which is based on the normality law (2.34) and complementarity conditions (2.35). For convenience we confine attention to problems without hardening, so that η α is absent as a variable; the extension to include hardening is straightforward. First, making use of the notation (S, T) := S : T dx, with a similar componentwise usage, we note that the relations (2.34) and (2.35) may be written concisely as ( Γ α, S α S α dis) 0 for all S E. (4.1)

10 158. D. Reddy: Single-crystal strain-gradient plasticity at large deformations Here E denotes the elastic region and yield surface: that is, E = { S : S α Y 0 0}. This inequality may be expanded in terms of the components of the terms appearing in it, to give (ν α, π α π α )+( α tanν α, ξ α ξ α dis) 0. (4.2) The weak form of the microforce balance equation (2.14), which is incorporated in the variational inequality (3.13), is given by (τ α π α, ν α )=(ξ α, α tan ν α ) (4.3) for all ν α satisfying the homogeneous micro-hard boundary condition (3.2a) and with ξ α satisfying the micro-free condition (3.2b). Now restrict the arbitrary microforces and resolved shear stresses to the subset of those satisfying the weak microforce balance equation and micro-free boundary condition then we have ( τ α π α, ν α )=( ξ α, α tan να ). (4.4) The system (4.2) (4.4) is required to be solved in conjunction with the equilibrium equation (3.3). This system has some interesting special cases. Consider first the case in which there is no defect energy, so that ξ α en 0, ξα = ξ α dis and we have a purely dissipative problem: then it is readily shown that (4.2) (4.4) may be combined to give (ν α, τ τ α ) 0. (4.5) This is precisely the form of the flow law for the case of conventional plasticity. It should be noted however that the interpretations are different for the two cases: for the classical problem (4.5) is a local inequality, posed over all τ α that satisfy the local yield condition τ α Y 0 0. (4.6) In the gradient case, on the other hand, the stresses τ α are required to satisfy the microforce balance equation (4.4) and, indirectly, the yield condition (2.33) with g α =0. A further special case is that in which there are no dissipative microstresses so that ξ α dis = 0. ThenS dis reduces to π α, the yield condition is now simply π α Y 0 0 and the normality law becomes (ν α, π α π α ) 0. (4.7) Given the relationship (2.14) between the resolved shear stress and the micro forces, and assuming the microstresses to be sufficiently smooth, the yield condition may be rewritten in the form τ α div ξ α Y 0 0, (4.8) so that the role of the microstress ξ α as a back-stress is clear. This notion is given a mathematically rigorous interpretation in [12], in the context of the small-strain problem.

11 GAMM-Mitt. 36, No. 2 (2013) Solution of the incremental problem The incremental problem could be solved as the unconstrained (but non-smooth) minimization problem (3.14). An alternative approach, more in line with the conventional predictorcorrector strategy, is that based on the dual problem discussed in this section. The key step is that of generalizing the standard return map for determining the slip rates. This has been carried out in [12] for the small-strain problem with energetic microstress only: it is shown there that the approach for the classical theory can be generalized, and the return map applied locally, albeit at element level rather than at integration points. The key to the corrector step is the definition of a scalar variable ζ h, defined in the small-strain case for a quadratic defect energy ψ d ( γ) = 1 2 S 0l 2 α γα 2 by (ζ h, γ α )=S 0 l 2 ( γ α, γ α ), (4.9) where γ α is the total slip and γ α an arbitrary but kinematically admissible slip. Here S 0 and l are respectively a hardening modulus and energetic length scale. Thus ζ α can be interpreted as the back-stress div ξ α. In the context of finite element approximations, for piecewise-constant approximations of ζ α, equation (4.9) allows for computation of ζ α at element level. The slips, and in the large-strain context the slip rates ν α, would typically be approximated in a conforming framework by continuous functions. The remainder of the algorithm amounts to solving the flow rule for Δγ α = ν α n from Π h Δγ α = λ α n sgn πα n, (4.10) where Π h indicates the projection onto piecewise constants at element level, and complementarity conditions λ α n 0, πα Y 0 0, λ α n ( πα Y 0 )=0, (4.11) which are the local forms of (4.7) and (4.8). Here λ α n is a plastic multiplier. It is shown in [12] that the corrector step may be interpreted as a closest-point projection of the trial stress, as in the conventional case. The extension of that work to the general large-strain case is the subject of current work. 5 Concluding remarks It has been shown that the problem of strain-gradient single-crystal plasticity may be placed in alternative variational settings that are amenable to further analysis and computation. The problem considered here is the rate-independent version of model proposed in [15, 20], the small-strain case of which has been analysed in [8]. For the present large-deformationproblem the relationship between the primal variational problem and the minimization problem for the time-discrete or incremental case is made explicit. Details of the variational formulation for the dual problem, involving both the slip rates and resolved stresses and microstresses, have been presented. This form of the problem is important in the development of predictor-corrector algorithms: as has been shown in [12], for example, the corrector step, which involves solving an incremental form of the dual problem, is equivalent to finding the closest-point projection of a trial stress.

12 160. D. Reddy: Single-crystal strain-gradient plasticity at large deformations A full analysis of the large-strain problem awaits attention. Likewise, while algorithms for the problem with both dissipative and energetic hardening have been developed for the viscoplastic problem in [13], the rate-independent problem awaits a similar treatment. Acknowledgements The research reported here has been supported by the Department of Science and Technology and the National Research Foundation through the South African Research Chair in Computational Mechanics. This support is gratefully acknowledged. References [1] N. A. Fleck and J. W. Hutchinson, Advances in Applied Mechanics 33, (1997). [2] N. A. Fleck and J. W. Hutchinson, Journal of the Mechanics and Physics of Solids 49, (2001). [3] P. Gudmundson, Journal of the Mechanics and Physics of Solids 52, (2004). [4] M. E. Gurtin and L. Anand, Journal of the Mechanics and Physics of Solids 53, (2005). [5] A. Menzel and P. Steinmann, Journal of the Mechanics and Physics of Solids 48, (2000). [6] W. D. Nix and H. Gao, Journal of the Mechanics and Physics of Solids 46, (1998). [7]. D. Reddy, Continuum Mechanics and Thermodynamics 23, (2011). [8]. D. Reddy, Continuum Mechanics and Thermodynamics 23, (2011). [9] M. E. Gurtin, International Journal of Plasticity 24, (2008). [10] M. E. Gurtin, E. Fried, and L. Anand, The Mechanics and Thermodynamics of Continua (Cambridge University Press, 2010). [11]. D. Reddy, Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) (2013), (in press). [12]. D. Reddy, C. Wieners, and. Wohlmuth, International Journal for Numerical Methods in Engineering 90, [13] S. argmann and. D. Reddy, European Journal of Mechanics - A/Solids 30, (2011). [14] M. E. Gurtin and. D. Reddy, In Review. Reddy2012.pdf (2013). [15] M. E. Gurtin, International Journal of Plasticity 26, (2010). [16] E. Kröner, Archive for Rational Mechanics and Analysis 4, (1960). [17] E. H. Lee, Journal of Applied Mechanics 36, 1 6 (1969). [18] J. Mandel, CISM Courses and Lectures, Springer 97 (1971). [19] W. Han and. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Second Edition (Springer, 2013). [20] M. E. Gurtin, Journal of the Mechanics and Physics of Solids 50, 5 32 (2002). [21] J. C. Simo, Numerical Analysis and Simulation of Plasticity, in: Handbook of Numerical Analysis, Volume 6, edited by P. G. Ciarlet and J. L. Lions (North-Holland, 1998), pp