ANALYSIS OF STRAIN LOCALIZATION BY ENERGY EQUIVALENCE:Ⅱ.FINITE ELEMENT SOLUTION

Founded on the nonlocal plasticity and the state space theories, a new approach is proposed to find the meshindependent solution of the strain localization problems by equating the rates of plastic energy dissipation in the local and nonlocal state spaces. Following the previous paper by the authors, general formulas are developed for the solution of the nonlocal internal variables in the two-and more than two-dimensional problems. A stress updating algorithm is proposed to integrate the rate form constitutive equations in the finite element context. To verify the proposed approach, a one-dimensional model problem and three two-dimensional plane strain problems are solved numerically by the finite element method. Numerical results show that the plastic strain distributions and the load-displacement curves stably converge with refinement of the finite element mesh. The size of the localization zone depends only on the internal length scale and is insensitive to the mesh size. For the one-dimensional problem, numerical solutions converge to the analytical ones. For the two-dimensional problems, although no analytical solutions are available, the numerical solutions converge toward the unique ones. The width and the inclination are almost not changed as the mesh size is reduced. Also, the distribution of the plastic strains and the deformation patterns are smooth in the entire domain. A slope stability problem and a plane strain test of a coal specimen are also solved numerically to demonstrate the robustness of the proposed approach. It is well shown that the proposed approach can overcome the drawbacks of the classical continuum theory and lead to physically meaningful, mesh-independent solution of strain softening problems. Because only C⁰ continuity is needed between element boundaries, the proposed approach is easy to be incorporated into the existing finite element code without substantial modification.