Some Aspects of a Discontinuous Galerkin Formulation for Gradient Plasticity at Finite Strains

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1 Some Aspects of a Discontinuous Galerkin Formulation for Gradient Plasticity at Finite Strains Andrew McBride 1 and B. Daya Reddy 1,2 1 Centre for Research in Computational and Applied Mechanics, University of Cape Town, 7701 Rondebosch, South Africa Andrew.McBride@uct.ac.za 2 Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa Daya.Reddy@uct.ac.za 1 Introduction Motivated in large part by the inability of classical theories to model material behaviour at the meso-scale level, various plasticity theories that incorporate size-dependence via the inclusion of strain gradients have been developed. These theories include in a natural way a length scale, and permit phenomena such as shear banding to be captured. For example, in the early work of Aifantis [1], the yield function is augmented by a term involving the Laplacian of the equivalent plastic strain and possibly other higher-order terms. An overview and critical comparison of the features, both mathematical and physical, of various gradient plasticity formulations is given in [2, 3]. The nonstandard higher-order contributions arising in gradient plasticity formulations render the conventional framework of classical finite elements inappropriate. The discontinuous Galerkin (DG) finite element method [4] allows the higherorder contributions to be treated in an elegant and effective manner as was demonstrated by the authors in [5, 6] for the aforementioned model of gradient plasticity [1] restricted to the infinitesimal strain regime. In this work we consider the extension of the gradient plasticity model to the finite deformation regime. The extension uses a logarithmic hyperelasticplastic model [7] that preserves the essential ingredients of the return mapping algorithms of the infinitesimal theory. The simplicity of this model of plasticity has been exploited by others as a basis for classical and non-local plasticity formulations (see e.g. [8] and references therein). This work treats the formulation of the problem in the finite-strain context. Details of the thermodynamic framework, not previously discussed in detail, are presented. In addition, algorithmic and computational aspects are summarised, and two example problems discussed. Further results and details are presented in [9].

2 2 Andrew McBride and B. Daya Reddy 2 The governing equations for the problem Let Ω be a bounded domain in R 2 with boundary Ω, and which is occupied by an elastoplastic body in its undeformed configuration. The current placement resulting from a motion ϕ(x,t) is denoted S. A material point in Ω is denoted X. Let Th 0 = {K0 } be a shape-regular subdivision of the reference domain Ω where K 0 are quadrilateral elements. For the space of displacements we construct conforming basis functions via functions Nϕ A defined on the reference element = [ 1, 1] [1, 1] with coordinates ξ. Then the interpolation of the reference domain and the displacement field over a typical element are given by X X h = n e node A=1 N A ϕ (ξ)x e A and u u h = n e node A=1 N A ϕ (ξ)u e A, (1) where n e node denotes the number of nodes per element, Xe A the reference coordinates and u e A the displacement field associated with node A. For the sake of convenience, the displacement field is approximated using conforming, bilinear elements. The classical deformation gradient is interpolated across an element from the nodal displacement d e A := X e A + u e A by GRAD X [ϕ e h] = n e node A=1 d e A(t) GRAD X [N A ϕ], (2) where GRAD X [( )] := ( )/ X is the gradient operator with respect to the reference configuration. In order to circumvent volumetric locking associated with low-order elements we use an enhanced assumed strain formulation [10], in which the deformation gradient is additively composed of the Galerkin approximation (2) and an enhanced part as F h = GRAD X [ϕ h ] + }{{} }{{} F h Galerkin enhanced. (3) The structure of F h chosen here is based on recommendations made in [11]. With P denoting the first Piola Kirchhoff stress, we denote by τ := PF T h the Kirchhoff stress which is related to the Cauchy stress σ by τ = det[f h ]σ. Following [10], the weak form of the governing equations are P h : GRAD[v h ] dx = B v h dx + T v h ds, (4a) Ω Ω Γ T P h : g h dx = 0, (4b) Ω where v h is an arbitrary test function, g h an arbitrary enhanced assumed strain, and B is the body force per unit reference volume.

3 A DG Formulation for Gradient Plasticity at Finite Strains 3 The right and left Cauchy Green tensors are respectively defined by C h = F h T F h and b h = F h F h T. (5) The deformation gradient is assumed to be multiplicatively decomposed into an elastic F e h and plastic part F p h as F h = F e hf p h. The multiplicative decomposition of F h motivates the definition of the elastic left, the plastic right, and the elastic right Cauchy Green tensors as b e h = F e hf e T h, C p h = F pt h F p h and C e h = F et hf e h. (6) The super- or subscript h will be dropped subsequently where convenient. 2.1 Constitutive relations and the flow law The system is characterised by a free energy function W additively composed of an elastic part W e and plastic part W p. The classical plastic free energy function, see e.g. [10], is extended to the gradient regime by setting W = W (F e,ξ, ξ) = W e (C e ) + (1/2)k 1 ξ 2 + (1/2)k 2 ξ 2, (7) }{{} W p (ξ, ξ) where ξ is the isotropic hardening parameter, k 1 is the isotropic hardening constant and k 2 > 0 is the gradient hardening constant that effectively introduces a length scale l = abs[k 2 /k 1 ] into the formulation. The classical plasticity formulation is recovered by setting k 2 = 0. The problem of classical plasticity with hardening k 1 > 0 is well-posed. For the case of softening classical plasticity, i.e. when k 1 < 0, the problem is illposed. It is well documented that finite element approximations of softening materials are pathologically dependent upon the resolution of the discretisation as this provides the only length scale. Well-posedness of the gradient plasticity model analysed here for small strain softening problems was proven in [5]. A primary motivation for gradient plasticity formulations of this form is the ability to obtain mesh-independent solutions in the softening regime, see [6] and references therein. Following classical thermodynamic arguments [12], the dissipation inequality τ : d Ẇ 0 yields 0 τ : d Ẇ τ : (de + d p ) W e (C e ) C e : Ċ e + ḡ ξ + m ξ ( [ W τ 2F e e (C e ) ) ]F C e et : d e + τ : d p + ḡ ξ + m ξ, (8) where ḡ := W/ ξ and m := W/ ξ, and d is the symmetric rate of deformation tensor, which is additively decomposed into elastic and plastic parts d e and d p. The notation ( ) denotes the material time derivative of an

4 4 Andrew McBride and B. Daya Reddy arbitrary function ( ). From the dissipation inequality (8) we obtain the elastic relation τ = 2F e W e (C e ) C e F et, (9) and the reduced dissipation inequality τ : d p + ḡ ξ + m ξ 0. (10) We assume an isotropic elastic response, in which case the dependence of W e on C e or equivalently b e can be expressed via the elastic principal stretches λ e A where (λe A )2 (A = 1,2,3) are the principal stretches of b e. We assume a Hencky model for the elastic stored energy function, that is ( W e (b e ) = w e (ε e A) := (Λ/2)(ε e 1 + ε e 2 + ε e 3) 2 +µ (ε e 1) 2 + (ε e 2) 2 + (ε e 3) 2), (11) where Λ > 0 and µ > 0 are the Lamé constants and ε e A := ln[λe A ] are the logarithmic elastic stretches. The constitutive relation for the components of τ in the principal directions of b e, denoted τ A, are obtained from (9) as τ A = Λ (ε e 1 + ε e 2 + ε e 3) + 2µε e A, A = 1,2,3. (12) We note that although w is not a polyconvex function of F h, the Hencky model has been shown to be acceptable for all but extreme elastic strains [7]. The gradient plasticity model considered here assumes, pointwise, an elastic domain E with boundary E, the yield surface, and a generalised normality law. For definiteness E is assumed here to be defined by the von Mises condition restricted to isotropic hardening and extended to the gradient regime. Plastic flow is considered incompressible, i.e. det [F p ] = 1. The region of admissible generalised stresses is defined as the set (τ,g) that satisfies f(τ,g) := dev[τ] 2/3(κ (ḡ div[m]) ) 0, (13) }{{} g where κ is a constant related to the initial tensile yield stress. The generalised plastic strains are thus defined via the flow law and Kuhn Tucker conditions d p = λ f τ, (14) ξ = λ f g = 2/3 λ, (15) λ 0, f(τ,g) 0, and λf(τ,g) = 0, (16) where λ is the plastic consistency parameter. The equivalent plastic strain γ(t) is related to the plastic consistency parameter by γ = λ. The generalised normality law (14) can be restated in an equivalent dual form [13] by using the dissipation function D, which is given here by

5 A DG Formulation for Gradient Plasticity at Finite Strains 5 D(d p, ξ) { 2/3κ d p if d p ξ, + otherwise. (17) Then, for arbitrary plastic deformations q and isotropic hardening parameter rates η, the flow rule becomes D(q,η) D(d p, ξ) + τ : (q d p ) + g(η ξ). (18) The weak form of (18) leads, after substitution for g, integration by parts and application of the boundary conditions ξ = 0 on S 1, and m n = k 2 ξ n = 0 on S 2 (19) where S 1 and S 2 are complimentary subsets of S, to the expression ( D (q,η) dx D d p, ξ ) dx + τ : (q d p ) dx S S S ( + ḡ η ξ ) (20) dx + m [η ξ] dx, S which forms the basis for the analysis of the small strain problem in [5]. S 2.2 Time-discrete approximation of the flow law We discretise the plastic flow law in time using a backward-euler scheme. We denote an arbitrary function ψ evaluated at time t n as ψ n. Consider a partition of the time interval [0,T] into N subintervals with node points t n = nk, 0 n N, where t = t n+1 t n = T/N is the step-size. Following [10], we define A p := [C p ] 1 = F 1 h. (21) ( Then for any τ [t n,t n+1 ], Ȧ p = 2 h be F T 1 f ξf h ) τ F h A p, which motivates the } {{ } Q approximation A p τ (exp[q]) A p n, and this in turn implies that b e exp [ 2 ξ f τ ] ( ) F h A p nf T h, (22) }{{} b e where b e is the trial elastic left Cauchy Green tensor obtained by assuming that plastic flow is frozen at t n. The principal trial stress τ is determined from the constitutive relationship (12) using the elastic principal stretches. Furthermore, we define the trial logarithmic strain components ε e A := log [λe A ] where (λ e A )2 are the eigenvalues of τ. One then obtains the time-discrete approximation to the flow law as [7]

6 6 Andrew McBride and B. Daya Reddy ε e n+1 = ε e γ f( τ,g) dev[ τ] = εe γν, (23a) ξ n+1 = ξ n + γ f( τ,g) = ξ n + 2/3 γ, (23b) g γ 0, f( τ n+1,g n+1 ) 0, γ f( τ n+1,g n+1 ) = 0, (23c) where the normal to the yield surface is denoted ν and the von Mises yield function in the principal directions is denoted f. 3 A discontinuous Galerkin formulation We denote by P k (K) the space of polynomials of degree at most k 0 on K. Let T h = {K} be a shape-regular subdivision of the current domain S where, here, K are quadrilaterals. We consider here the subdivision of the current configuration as this is the placement in which the non-local expression of the flow rule is defined, see (20). Let E h = {e} denote the unique set of the edges of T h, and Eh int = E h \ S the unique set of interior edges. We associate with each edge e of an element the outward unit normal vector n (e). For an edge that lies on the boundary S of the domain, n (e) is defined to be the outward normal to S. The jumps and averages, denoted and { } respectively, of a scalar η and a vector v across an interior edge e 12 common to elements K 1 and K 2 are defined as η = η 1 n 1 + η 2 n 2, {η } = 1/2(η 1 + η 2 ), v = v 1 n 1 + v 2 n 2, {v } = 1/2(v 1 + v 2 ). (24) Following [6] for the small strain problem, we obtain the non-local expression of the discrete consistency condition f n+1 = 0 for γ from the linearised, fully-discrete symmetric interior penalty DG formulation [4] of (20) as T ((( 2µ k 1 ) γ ) k 2 [ γ] E h 2 3 k 2 ( { [ γ] } + { } γ ) ds = ) k 2 β dx + γ ds E h h e f( τ,g n ) dx, (25) where the parameter β is a positive penalty term and h e the distance between the centroids of two elements that share a common edge. Homogeneous Neumann type boundary conditions are assumed here on the external boundary of the current domain, i.e. m n = 0 on S 2. While the choice of appropriate boundary conditions for the internal hardening parameter field is motivated by physical considerations, the emphasis in this work is to demonstrate the salient features of the gradient plasticity formulation. We remark however that any choice of boundary condition could be implemented. Also, motivated by the structure of the variational problem, we solve the non-local statement of the consistency condition over the complete current domain rather than over some plastic subset thereof. T

7 A DG Formulation for Gradient Plasticity at Finite Strains 7 4 Solution procedure and numerical examples A predictor-corrector type solution algorithm is used to solve the gradient plasticity problem for the increment in displacement and plastic deformation during a time-step of duration t = t n+1 t n. The complete system state is assumed known at t n. In the predictor step, the increment in the nodal displacements is obtained by making an assumption as to the evolution of plastic flow and solving the governing equations (4a) (4b). The purpose of the corrector step is to then determine the stress state based on the strain field arising from the predictor step taking into account the possible evolution of plastic deformation. For classical plasticity the corrector step involves checking the trial yield condition f := f(τ,g n ) at each quadrature point. If the quadrature point is active, i.e. f > 0, we determine the increment in plastic flow by solving the algebraic consistency condition locally. The non-local expression for the increment in plastic deformation arising in the gradient formulation can not be solved at the level of the quadrature point. Instead, a search is performed to determine the active quadrature points and (25) solved to determine the increment in the internal hardening parameter. The performance of the predictor corrector solution scheme is dependent on the form of approximation as to the evolution of plastic flow in the predictor step. Following the approach pioneered in [7] for the classical problem at finite strains and drawing on the approach adopted in [6] for the gradient problem under the assumption of infinitesimal strain, we utilise a consistent tangent formulation rederived for the gradient plasticity problem under consideration. For details see [9]. The gradient plasticity formulation is applied to two example problems. The first, a rectangular plate with a small initial imperfection subjected to compressive loading wherein the material undergoes softening [14], is chosen to assess the ability of the gradient plasticity formulation to overcome the pathological mesh-dependence associated with classical formulations involving softening, i.e. k 1 < 0. The second example problem demonstrates features of the gradient formulation for a hardening problem involving more significant deformation. Plane strain conditions are assumed applicable. Example 1: Softening response of a rectangular plate. The formation of the shear band is induced via the introduction of a 10 mm square region in the lower left hand corner of the plate, see Fig. 1(a), wherein the yield strength is reduced by 10 % relative to the rest of the domain. The well-documented pathological localisation of the shear band to the scale of the discretisation (i.e. the mesh size) for the classical problem is evident in Fig. 1(d). The relationship between the resulting force and the applied displacement on the upper edge of the plate, shown in Fig. 1(e), illustrates, once again, the mesh dependent response. The post-peak response of the force displacement curve is governed by the discretisation: the finer the mesh, the greater the rate at which the material looses residual strength.

8 8 Andrew McBride and B. Daya Reddy The ability of the gradient plasticity formulation to produce solutions independent of the mesh for softening problems is demonstrated in Figs 1(b) 1(c). As the mesh resolution is increased so the width of the shear band converges to a constant value prescribed by the internal length scale. It is evident that the pathological localisation demonstrated by the classical formulation has been overcome. 120 Imperfection (a) Domain (b) gradient (c) gradient Reaction force on top edge Classical 6 12 classical classical classical 6 12 gradient gradient gradient skewed gradient Displacement of top edge (d) classical (e) Applied force versus the resulting displacement Fig. 1. Schematic of the domain, the deformed domain obtained using various discretisations, and the resulting force versus displacement relationship Example 2: Indentation test. The second example problem entails a rectangular domain subjected to loading via a frictionless rigid indenter. The domain is composed of a hardening elastoplastic material. The indenter moves downwards at a constant rate into the specimen resulting in significant deformation. The final indentation depth is 8% of the initial specimen height.

9 A DG Formulation for Gradient Plasticity at Finite Strains 9 The results predicted using the classical and gradient plasticity formulations at the final stage of the deformation process are compared in Fig. 2(a). The results obtained using the gradient plasticity formulation with an internal length scale of 2.83 mm are mirrored around the symmetry axis for the purpose of direct comparison. The positions of the quadrature points are shown and the subset of those quadrature points that are active indicated. The relationship between the resulting force on the indenter and the imposed displacement is shown in Fig. 2(b). The results obtained using the gradient formulation are clearly influenced by the relative scale of the problem, that is the ratio of the internal length scale to a characteristic dimension of the domain. The material offers increased resistance to deformation with increasing internal length scale Gradient Classical 7000 Gradient l = 2.83 mm 6000 Reaction force on indenter (N) Classical Gradient l = 2 mm Quadrature points Active quadrature points Difference active gradient and classical quadrature points Displacement of indenter (mm) (a) Deformed domain using gradient (left) and classical (right) plasticity formulations (b) Resulting force on the indenter and the imposed displacement Fig. 2. Deformed domain and the force versus displacement relationship for the indentation test 5 Conclusion There remain a number of avenues which merit further exploration and study. First among these would be the extension of the present work to three space dimensions, the framework for which exists in much of the work reported here. The major advantage of adopting a DG approach is that it offers the flexibility to accommodate more complex gradient plasticity models. The gradient model considered here, for example, could be readily extended to include a biharmonic term in the yield condition by using a continuous/discontinuous Galerkin formulation rather than a cumbersome classical approach based on Hermite interpolation. Again, the groundwork for such a model has been presented here and in previous works.

10 10 Andrew McBride and B. Daya Reddy Further questions worth addressing include the extension of the work reported here to other models of strain gradient plasticity. For example, the class of problems that involve the Burgers vector, or curl of plastic strain, represents a group that is worth studying computationally. (see, e.g., [15]). References 1. E. C. Aifantis. On the microstructural origin of certain inelastic models. J. Eng. Mater. Tech., 106: , P. Gudmundson. A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids, 52: , R. A. B. Engelen, N. A. Fleck, R. H. J. Peerlings, and M. G. D. Geers. An evaluation of higher-order plasticity theories for predicting size effects and localisation. Int. J. Solids Struct., 43: , D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal, 19: , J. K. Djoko, F. Ebobisse, A. T. McBride, and B. D. Reddy. A discontinuous Galerkin formulation for classical and gradient plasticity - Part 1: Formulation and analysis. Comput. Methods Appl. Mech. Engrg., 196: , J. K. Djoko, F. Ebobisse, A. T. McBride, and B. D. Reddy. A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: Algorithms and numerical analysis. Comput. Methods Appl. Mech. Engrg., 197:1 21, J. C. Simo. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput. Methods Appl. Mech. Engrg., 99:61 112, M. G. D. Geers. Finite strain logarithmic hyperelasto-plasticity with softening: a strongly non-local implicit gradient framework. Comput. Methods Appl. Mech. Engrg., 193: , A. T. McBride and B. D. Reddy. A discontinuous Galerkin formulation for gradient plasticity at finite strains. In review. 10. J. C. Simo and F. Armero. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Methods Eng., 33: , S. Glaser and F. Armero. On the formulation of enhanced strain finite elements in finite deformations. Eng. Comput., 14(7): , B. D. Coleman and M. Gurtin. Thermodynamics with internal variables. J. Chem. Phys, 47: , W. Han and B. D. Reddy. Plasticity: Mathematical Theory And Numerical Analysis, volume 9. Springer, J. Pamin. Gradient-Dependent Plasticity in Numerical Simulation of Localization Phenomena. PhD thesis, Delft University of Technology, M. E. Gurtin and L. Anand. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. Int. J. Plasticity, 53: , 2005.