Reproducing kernel formulation for multisurface plasticity and continuum damage mechanics in geotechnical materials

A reproducing kernel (RK) formulation for geotechnical materials considering pressure sensitive multisurface plasticity coupled with strain-based damage mechanics based on an effective stress concept is first developed. In this approach, the problem domain is discretized by a set of particles, and the particle shape functions are constructed based on an RK approximation that does not require connectivity of an explicit mesh. The particle shape functions have compact supports and can be customized for desired regularity and consistency in the numerical approximation. The RK discretization is formulated under a Galerkin framework, and both displacement-based formulation and mixed formulation of an assumed strain method are developed and studied. A stable return mapping algorithm and the consistent tangent operators of the model problem are introduced.;To remedy the loss of ellipticity in strain localization due to material instability, a moving least square (MLS) and reproducing kernel (RK) strain smoothing are introduced as a means of regularization for strain localization problems. Compared to the conventional nonlocal method, MLS/RK strain smoothing further introduces enrichment of monomial basis functions to ensure higher-order solution accuracy for the solution prior to bifurcation and for the solution in the non-localized regions. Further, by imposing appropriate reproducing conditions in the RK nonlocal strain smoothing, the method can recover the gradient operator of any type. The analytical derivation and numerical examples show that MLS/RK strain smoothing can be tailored to provide a gradient typed regularization without dealing with the additional boundary conditions that cannot be justified physically in the gradient method.;Application of the proposed methods to geotechnical materials with strain softening is studied using a set of numerical examples, which show that the displacement-based RK formulation (1-level regularization) is sufficient to remedy mesh-sensitivity in elastic damage induced strain localization. For strain localization associated with plasticity, a 2-level RK regularization in displacement and strain is necessary for a mesh-insensitive solution. Adaptivity procedures using the wavelet solution as a resolution indicator are also introduced in meshless computation to demonstrate the advantages of a meshless approach compared to conventional finite element-based methods.