2D finite elements for the computational analysis of crack propagation in brittle materials and the handling of double discontinuities

Crack growth simulations by way of the traditional Finite Element Method claim progressive remeshing to fit the geometry of the fracture, severely increasing the computational effort. Methods such as the eXtended Finite Element Method (XFEM) allow to overcome this limitation by means of nodal shape functions multiplied by Heaviside step function to enrich finite element nodes. Through the medium of a discontinuous field, the entire geometry of the discontinuity can be modelled regardless of the mesh, avoiding remeshing. In this paper two shell-type XFEM elements (a three-node triangular element and a four-node quadrangular element) to evaluate crack propagation in brittle materials are presented. These elements have been implemented into the widespread opensource framework OpenSees to evaluate crack propagation into a plane shell subjected to monotonically increasing loads. Moreover, in the perspective of fracture propagation simulations, the problem of managing multiple cracks without remeshing or operating subdivisions on the integration domain has been investigated and a four-node quadrangular finite element for the computational analysis of double crossed discontinuities by the means of equivalent polynomials is presented in this paper. Equivalent polynomials allow to overcome inaccuracies on the results when performing standard numerical integration (e.g. Gauss-Legendre quadrature rule) over the entire domain of XFEM elements, without the need of defining integration subdomains. The presented work and the computational strategy behind it may be extremely useful not only in the field of fracture mechanics, but also to solve complex geometry problems or material discontinuities.