A quasi-extremal energy principle for non-potential problems in rate-independent plasticity

Abstract

The rate-problem of continuing equilibrium is examined for a general class of rate-independent elastoplastic solids, without assuming the normality flow rule or symmetry of the tangent stiffness matrix. Accordingly, the problem addressed is of non-potential type, for which the usual stationarity or minimum principles for a governing potential do not apply. It is shown that the rate-problem can nevertheless be formulated as a quasi-extremal energy principle. It is characterized by explicit dependence of the minimized energy function or functional not only on variables undergoing variations but also, although only in a particular way, on an unknown solution as a parameter. To enable transparent and mathematically simple presentation of the main concept, the energy function is defined in a finite-dimensional setting for a spatially discretized material body with generalized velocities and a number of plastic multipliers as unknowns. If a solution is non-unique then incrementally stable solutions can be selected using the quasi-extremal principle in which the minimized energy function includes the second-order terms. Examples and extensions concern an elastic-plastic continuum obeying a non-associative plastic flow rule, without or with a higher-order gradient term in the loading function. The issue of selection of active slip-systems in a single crystal of a non-symmetric slip-system interaction matrix is also addressed.