An anisotropic gradient damage model for quasi-brittle materials Original Research Article

An anisotropic continuum damage model based on the microplane concept is elaborated. Scalar damage laws are formulated on several individual microplanes representing the planes of potential failure. These uniaxial constitutive laws can be cast into a fourth-order damage formulation such that anisotropy of the overall constitutive law is introduced in a natural fashion. Strain gradients are incorporated in the constitutive equations in order to account for microstructural interaction. Consequently, the underlying boundary value problem remains well-posed even in the softening regime. The gradient continuum enhancement results in a set of additional partial differential equations which are satisfied in a weak form. Additional nodal degrees of freedom are introduced which leads to a modified element formulation. The governing equations can be linearized consistently and solved within an incremental-iterative Newton–Raphson solution procedure. The capability of the present model to properly simulate the localized failure of quasi-brittle materials will be demonstrated by means of several examples.