The linear matching method for the shakedown analysis of geotechnical problems

We investigate the performance of a sequential programming method, based on the linear matching method, for the direct evaluation of limit loads and shakedown limits for elastic-perfectly plastic bodies subjected to complex histories of loading. This end is achieved by solving a sequence of linear problems defined with spatially varying moduli, which relates properties of the yield condition to those of the linear problems. The method provides a sequence of upper bounds that monotonically reduces and converges to the least upper bound associated with the chosen class of displacement fields, provided a sufficient condition is satisfied. We applied this method to a class of isotropic yield condition that depends not only on the von Mises effective stress but also on the hydrostatic pressure. This is followed by a set of examples of finite element solutions including the problem of a circular Hertzian contact region that repeatedly traverses the surface of an elastic-perfectly plastic half-space. Numerical examples demonstrate that the convergence is still obtained when the sufficient condition as given by Ponter et al. (Eur. J. Mech. 2000; 19:401–421) is violated, i.e. the condition is sufficient but not a necessary condition for convergence.