Limit theorems for gradient-dependent elastoplastic geomaterials

In this paper, the limit theorems proposed by Drucker and co-workers are reformulated to address a class of gradientdependent elastoplastic geomaterials. The gradient effects are accounted for by incorporating strain gradients and their conjugate higher-order stresses into the constitutive descriptions. Gradient-dependent equilibrium equations and higher-order boundary conditions are defined for both statically admissible and kinematically admissible states, and the associated lower bound and upper bound theories are recast for the gradient-enhanced limit theorems. A generalised Drucker–Prager yield criterion, that includes the gradient influence on the deviatoric stress, is proposed and then employed to investigate the collapse loads for soil layers under conditions of generalised plane-strain, simple shear and uniaxial compression. The corresponding lower and upper bounds are found for these problems. It is demonstrated that the predicted collapse bounds are generally dependent on both the conventional and gradient properties, with normalised length scale(s) being present in the results. This feature enables us to give physically reasonable interpretations for size effects and shear banding during material collapse. Comparisons are also made between the gradient-dependent bounds and those obtained through conventional plasticity theories. Influences of model parameters and sample dimensions on the predictions are also discussed. It is shown that the proposed gradient-dependent limit theorems can be used to provide physically meaningful predictions for general geotechnical applications.