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

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 large**deformation**, Hwang et al. (2001) generalized Fleck and Hutchinson’s (1997) andGao et al.’s (1999) **strain** **gradient** plasticity theories from infinitesimal to finite**deformation**. 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 infinitesimal**strain** " ij and its **gradient** " ij,k are replaced by the Green–Lagrange **strain**E 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 coordinates**of** 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 deviatoric**deformation**. It is a generalization **of** a classical finite **deformation** plasticity theoryin Section 2 that excludes the volumetric **deformation**. We focus on the finite**deformation** 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. The**deformation** **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 **strain**tensor 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 modified**deformation** **gradient**F 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 infinitesimal**deformation**. 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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