Method of finite spheres solution of micron-scale plasticity based on a strain gradient formulation

Continuum “strain gradient” theories of plasticity have been developed to account for the size-dependence of micron-scale metallic materials undergoing inhomogeneous plastic deformation. A particularly promising theory has been recently proposed by Gurtin and co-workers [Anand L, Gurtin ME, Lele SP, Gething C. A one-dimensional theory of strain gradient plasticity: formulation, analysis, numerical results. J Mech Phys Solids 2005;53(7):1789–826] which has several attractive features including the ability to predict isotropic internal variable hardening, energetic hardening associated with plastic-strain gradients, and dissipative strengthening associated with plastic-strain-rate gradients which results in size-dependence of the yield stress. However, using the traditional finite element method to solve the resulting boundary value problem leads to a rapid deterioration of the solution results with increase in strain gradient. In this paper, we propose a solution to this problem by developing a computational scheme based on the meshfree method of finite spheres [De S, Bathe KJ. The method of finite spheres. Comput Mech 2000;25(4):329–45]. In this method, the shape functions are generated using the partition of unity paradigm [Yosida K. Functional analysis, vol. 5. Berlin, Heidelberg: Springer-Verlag; 1978] and are compactly supported on n-dimensional spheres. Excellent convergence rates are observed for problems in one- and two-dimensional analysis which are attributed to the higher order continuity of the approximation spaces used in this method.