Analysis of 3D Crack Boundary Problems by Means of the Enriched Scaled Boundary Finite Element Method

Abstract

In fracture mechanics, a demanding challenge is the analysis of truly three-dimensional crack boundary value problems. For instance laminate structures composed of fiber-reinforced plies are typically prone to the formation of inter-fiber cracks because of the given strongly anisotropic stiffness and strength properties. These inter-fiber cracks commonly run through complete plies but are stopped at the ply interfaces. This leads to non-standard three-dimensional crack configurations with locally singular stress fields which should be investigated in regard of their criticality. For that purpose, the Scaled Boundary Finite Element Method turns out to be an appropriate and effective analysis method that permits solving linear elastic mechanical problems including stress singularities with comparably little effort. Only the boundary is discretized by two-dimensional finite elements while the problem is considered analytically in the direction of the dimensionless radial coordinate ξ. A corresponding separation of variables representation for the displacement field employed in the virtual work equation leads to a system of differential equations of Cauchy-Euler type. This differential equation system can be converted into an eigenvalue problem and solved by standard eigenvalue solvers for non-symmetric matrices. Depending on the given load the respective 3D stress singularities may lead to subsequent crack initiation and propagation, i.e. secondary moving cracks (with correspondingly moving boundaries). For the analysis of these secondary 3D crack configurations the Scaled Boundary Finite Element Method has been extended by an appropriate enrichment of the displacement representations. This leads to a clearly better numerical performance and computational efficiency compared to the standard Scaled Boundary Finite Element Method.