Numerical Simulation Of Strain Localization By Energy Equivalence Between Local And Nonlocal State Spaces

It is frequently observed that a lot of engineering materials exhibit the phenomenon of strain localization when loaded close to failure.When we solve the strain localization problems with the classical continuum theory or the general integral-type nonlocal plasticity theory,the results will be mesh-dependent.It means that the shear band becomes vanishingly small with refinement of the finite element mesh.In order to get mesh-independent results,a new method for solving the strain localization problems is proposed by equating the plastic energy dissipation rates in the local and nonlocal state spaces,which is beased on the first law of the thermodynamics and the nonlocal plasticity theory.Founded on state space theory,two state spaces,local and nonlocal state space,are defined for each material point in this method.The internal variable of the yield material point can be mapped from the local state space to the nonlocal state space by weighted averaging of the local internal variables.Based on the first law of thermodynamics,it is found that the plastic part and the elastic part of the internal energy in the nonlocal state space are equal to the corresponding parts in the local state space.The general formula of this new method is derived and the stress updating algorithm is developed in this paper.Numerical examples for a one-dimensional and three two-dimensional strain softening problems are carried out to verify the proposed method.Analytical solution of one dimensional strain localization problem is presented and the quantitative relationship between strain localization size LcP and the internal length scale l is derived.The numerical solutions for the one-dimensional and two-dimensional examples are given by the finite element method,and the plastic strain distributions and the load-displacement curves are obtained.The numerical results converge to the analytical ones stably with the refinement of finite element mesh for all the examples.It is shown that the numerical value of the localization zone size is consistent with the theoretical one,and it does not change with the refinement of mesh.This affirms that the proposed method can lead to mesh-independent solution of the strain localization problems.The influence of the internal length scale on the strain localization zone and the load-displacement curves is also analyzed through the slope stability problem and a coal specimen under plane strain test.Because only C0 continuity is needed between element boundaries and there is no need to introduce new parameters in the model,the proposed method can be easily incorporated into the existing finite element code without substantial modification.