Theoretical Basis and a Finite Element Formula for the Direct Calculation of Steady Plastic States

Abstract

Direct methods discussed in the literature address mainly the determination of an elastic shakedown boundary, and insufficient attention has been paid to the practically important case of inelastic steady cyclic structural response. To fill this gap, this paper extends a systematic direct approach to safety factor and cyclic state assessment beyond elastic shakedown, which is applicable for any kind of steady cyclic elastic-plastic deformation. In order to construct the limit state boundaries, the basic theorems of elastic and inelastic shakedown accounting for material hardening are formulated uniformly for all the possible combinations of ratcheting and alternating plasticity. A special emphasis is put on practical material description for each kind of limit state. After the cyclic state under prescribed loading is classified, a direct approach created on the same theoretical framework is employed to determine the evolution of the strain and stress fields over the cycle. Mathematically, this results in a convex constrained optimization problem, which is formulated making use of finite element discretization. The proposed optimization approach is validated through the Bree problem, with the results agreeing well with the analytical solution to the problem.