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Finite deformation analysis of mechanism-based strain gradient ...

Finite deformation analysis of mechanism-based strain gradient ...

236 K.C. Hwang et al. /

236 K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251either as an independent measure of deformation besides the strain (e.g. Fleck andHutchinson, 1993, 1997; Shu, 1998; Gao et al., 1999; Shu and Fleck, 1999; Huang etal., 2000a,b; Shuand Barlow, 2000; Gurtin, 2001), or as an internal variable toincrease the plastic work hardening modulus (e.g. Nilsson, 1998; Acharya and Bassani,2000; Acharya and Beaudoin, 2000; Chen and Wang, 2000; Dai and Parks,2001; Meissonnier et al., 2001). There are also early works on strain gradient plasticitythat were proposed in order to avoid a spurious solution for the localized zoneand an excessive mesh dependence in classical plasticity (e.g. Aifantis, 1984, 1992;Lasry and Belytschko, 1988; Zbib and Aifantis, 1988; Muhlhaus and Aifantis, 1991;de Borst and Muhlhaus, 1991, 1992; Sluys et al., 1993).Since the aforementioned micron and submicron scale experiments involve largedeformation, Hwang et al. (2001) generalized Fleck and Hutchinson’s (1997) andGao et al.’s (1999) strain gradient plasticity theories from infinitesimal to finitedeformation. The equilibrium equations and boundary conditions are established inthe current configuration, while the constitutive relations are obtained by nominalgeneralization from the corresponding infinitesimal deformation theories. The infinitesimalstrain " ij and its gradient " ij,k are replaced by the Green–Lagrange strainE IJ and the corresponding gradient in the reference configuration, E IJ,K , respectively.Such a nominal generalization, however, does not always ensure the appropriatedecomposition of deformation into the volumetric and deviatoric parts. Forexample, " kk is the volumetric strain in infinitesimal deformation, but its nominalgeneralization E KK does not represent the volumetric strain in finite deformation.Instead, only the determinant J=det(F) of the deformation gradient F= @x@X representsthe volume change in finite deformation, where x and X denote the coordinatesof a material point in the current and reference configurations, respectively.The purpose of this paper is to develop a finite deformation strain gradient plasticitytheory that ensures the proper decomposition of volumetric and deviatoricdeformation. It is a generalization of a classical finite deformation plasticity theoryin Section 2 that excludes the volumetric deformation. We focus on the finitedeformation theory of mechanism-based strain gradient plasticity (Gao et al., 1999;Huang et al., 2000a,b), though the approach can also be applied to Fleck andHutchinson’s (1997) strain gradient plasticity theory. We then present the analyticalsolution for a thin cylinder under torsion, and numerical study of the crack tip field.2. A finite deformation theory of classical plasticityWe first present a finite deformation theory of classical plasticity that excludes thevolumetric deformation. Let X and x denote the coordinates of a material point inthe reference (undeformed) and current (deformed) configurations, respectively. Thedeformation gradient F= @x@X can be expressed in terms of the principal stretches l 1,l 2 and l 3 asF ¼ l 1 n 1 N 1 þ l 2 n 2 N II þ l 3 n 3 N III ;ð1Þ

K.C. Hwang et al. / International Journal of Plasticity 19 (2003) 235–251 237where n 1 ; n 2 ; n 3 and N I ; N II ; N III are the principal stretch directions in the currentand reference configurations, respectively. The volumetric deformation is characterizedby the determinant of F;J ¼ detðFÞ ¼ l 1 l 2 l 3 :ð2ÞThe right and left Cauchy–Green strain tensors C and B and the Green straintensor E are defined byC ¼ F T F; B ¼ FF T ; E ¼ 1 ð2 C 1 Þ; ð3Þwhere 1 is the second-order identity tensor.In order to exclude the volumetric deformation from F, we define a modifieddeformation gradientF ¼ J 1 3 F;ð4Þ which has no volumetric deformation since F =1. The isochoric right and leftCauchy-Green strain tensors, C and B , and the isochoric Green strain tensor E ,which all exclude the volumetric deformation, can be defined similar to (3) via F byC ¼ F T F; B ¼ F F T ; E ¼ 1 2 C 1 : ð5ÞIt can be shown that E degenerates to deviatoric strain tensor for infinitesimaldeformation. The invariants of C and B areI 1 ¼ I C ¼ I B ¼ ðl 1 l 2 l 3 Þ 2 3 l21 þ l2 2 þ 3 l2 ; ð6ÞI 2 ¼ II C ¼ II B ¼ ðl 1 l 2 l 3 Þ 2 3 l21 þ l 22 þ l 2 3 ;and III C ¼ III B =1. The effective strain is defined by" 2 ¼ 2 3 E : E ¼ 1 6 I 2 1 2I 1 2I 2 þ 1 2 ;ð7Þð8Þwhich degenerates to the von Mises effective strain for infinitesimal deformation.For a deformation theory of plasticity, the strain energy density function U (perunit volume in the reference configuration) of an isotropic solid generally takes theform U ¼ U J; I 1 ; I 2¼ U V ðÞþU JD ðÞ "ð9Þ

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