Mechanism-based strain gradient plasticityÐII. Analysis

A mechanism-based theory of strain gradient (MSG) plasticity has been proposed in Part I of this paper. The theory is based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient. This theory is motivated by our recent analysis of indentation experiments which strongly suggest a linear dependence of the square of plastic ̄ow stress on strain gradient. Such a linear dependence is consistent with the Taylor plastic work hardening model relating the ̄ow stress to dislocation density. This part of this paper provides a detailed analysis of the new theory, including equilibrium equations and boundary conditions, constitutive equations for the mechanism-based strain gradient plasticity, and kinematic relations among strains, strain gradients and displacements. The theory is used to investigate several phenomena that are in ̄uenced by plastic strain gradients. In bending of thin beams and torsion of thin wires, mechanism-based strain gradient plasticity gives a signi®cant increase in scaled bending moment and scaled torque due to strain gradient e€ects. For the growth of microvoids and cavitation instabilities, however, it is found that strain gradients have little e€ect on micron-sized voids, but submicron-sized voids can have a larger resistance against void growth. Finally, it is shown from the study of bimaterials in shear that the mesoscale cell size has little e€ect on global physical quantities (e.g. applied stresses), but may a€ect the local deformation ®eld signi®cantly. # 1999 Elsevier Science Ltd. All rights reserved. 0022-5096/00/$ see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0022 -5096 (99)00022 -8 Journal of the Mechanics and Physics of Solids 48 (2000) 99±128 * Corresponding author. Tel.: +1-217-265-5072; fax: +1-217-244-6534. E-mail address: [email protected] (Y. Huang) Keywords: Strengthening mechanisms; Strain gradient plasticity; Bending; Torsion; Void growth; Cavitation instabilities; Bimaterials 1. Summary of the mechanism-based strain gradient plasticity Based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient, we have derived the constitutive equations of the mechanism-based strain gradient (MSG) plasticity in Part I of this paper (Gao et al., 1999). The MSG plasticity constitutive equations, as well as equilibrium equations and kinematic relations among strains, strain gradients and displacements are summarized in this section to constitute a complete set of governing equations for mechanism-based strain gradient plasticity. We use MSG plasticity to investigate several phenomena that are in ̄uenced by plastic strain gradients, including bending of thin beams, torsion of thin wires, growth of microvoids, cavitation instabilities, and bimaterials in shear. The deformation theory of the mechanism-based strain gradient plasticity is summarized in this section, while the MSG ̄ow theory will be published elsewhere. For simplicity, elastic deformation and compressibility of materials are ignored in this paper. In addition, the theory and analysis are restricted to small deformation only. 1.1. Generalized stresses and strains in a strain gradient theory Both strains and strain gradients are introduced in higher-order continuum theories of elasticity (Toupin, 1962; Koiter, 1964; Mindlin, 1964, 1965) and plasticity (Fleck and Hutchinson, 1993, 1997; Fleck et al., 1994; Gao et al., 1999). In a Cartesian reference frame xi, the strain tensor Eij and strain gradient tensor Zijk are related to the displacement ui by (Fleck and Hutchinson, 1997) Eij ˆ 1 2 …ui,j ‡ uj,i † …1†