A variational principle for gradient plasticity

Abstract We elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by Aifantis and co-workers. The generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher-order spatial gradients of the equivalent plastic strain in the yield condition. First it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands. Due to the presence of higher-order gradients, additional boundary conditions for the equivalent plastic strain are required. This question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work. We demonstrate here that upon assuming a certain type of additional boundary conditions, the structural symmetries of the gradient-dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain. The variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method. Explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given.

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