FE-formulation of a nonlocal plasticity theory

A nonlocal continuum plasticity theory is presented. The nonlocal field introduced here is defined as a certain weighted average of the corresponding local field, taken over all the material points in the body. Hereby, a quantity with the dimension of length occurs as a material parameter. When this so-called internal length is equal to zero, the local classical plasticity theory is regained. In the present model, the yield function will depend on a nonlocal field. The consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers. The integral equations are classified as Fredholm equations of the second kind and the existence of a solution will be commented upon. After discretization, a matrix equation is obtained, and an algorithm for finding the solution is proposed. For a generalized von Mises material, a plane boundary value problem is solved with a FE-method. Since the nonlocal quantities are integrals, C-0-continuous elements are sufficient. The solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations. In the numerical simulations shear band formation is analysed and the results display mesh insensitivity.