Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation.

*(English)*Zbl 1231.74023Summary: Gradient theories have found wide applications in modeling of strain softening phenomena. This paper presents a higher order stress-strain theory to describe the damage behavior of strain softening materials. In contrast to most conventional gradient approaches for damage modeling, the present higher order theory considers strain gradients and their conjugate higher-order stress such that stable numerical solutions may be achieved. We have described the derivation of the required constitutive relationships, the governing equations and its weak form for this higher-order theory. The constitutive coefficients were obtained from a granular media approach such that the internal length scale parameter reflects the natural granularity of the underlying microstructure. The weak form was discretized using an element-free Galerkin (EFG) formulation that readily admits approximation functions of higher-order continuity. We have also discussed the implementation of essential boundary conditions and linearization of the derived discrete equations. Finally, the applicability of the derived model is demonstrated through two examples with different imperfections designed to initiate dislocation bands and shear bands, respectively.

##### MSC:

74A45 | Theories of fracture and damage |

74S05 | Finite element methods applied to problems in solid mechanics |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |