Sascha Hell

The scaled boundary finite element method for the analysis of 3D crack interaction

Abstract The scaled boundary finite element method (SBFEM) can be applied to solve linear elliptic boundary value problems when a so-called scaling center can be defined such that every point on the boundary is visible from this scaling center. From a more practical point of view, this means that in linear elasticity, a separation of variables ansatz can be used for the displacements in a scaled boundary coordinate system. This approach allows an analytical treatment of the problem in the scaling direction. Only the boundary needs to be discretized with finite elements. Employment of the separation of variables ansatz in the virtual work balance yields a Cauchy–Euler differential equation system of second order which can be transformed into an eigenvalue problem and solved by standard eigenvalue solvers for nonsymmetric matrices. A further obtained linear equation system serves for enforcing the boundary conditions. If the scaling center is located directly at a singular point, elliptic boundary value problems containing singularities can be solved with high accuracy and computational efficiency. The application of the SBFEM to the linear elasticity problem of two meeting inter-fiber cracks in a composite laminate exposed to a simple homogeneous temperature decrease reveals the presence of hypersingular stresses.

THE USE OF ENRICHED BASE FUNCTIONS IN THE THREE-DIMENSIONAL SCALED BOUNDARY FINITE ELEMENT METHOD

The Scaled Boundary Finite Element Method (SBFEM) is a semi-analytical method which combines the advantages of the Boundary Element Method (BEM) and the Finite Element Method (FEM). Like in the BEM, only the boundary needs to be discretized. On the other hand, the SBFEM is based on the virtual work principle and does not need any fundamental solutions. If the scalability condition is fulfilled, a separation of variables representation can be employed leading to a quadratic eigenvalue problem and a linear equation system which can be solved by standard methods. The SBFEM has proven its high efficiency and accuracy in the presence of stress singularities, especially in 2D fracture mechanics when the singularity is entirely located within the considered domain. However, in 3D elasticity problems, there can also be singularities on the discretized boundary itself. Then, the SBFEM suffers from drawbacks also well known from the standard FEM, i.e. moderate accuracy and bad convergence. To overcome these deficiencies in such 3D cases, we propose the enrichment of the standard separation of variables representation with analytical fields which are known to exactly fulfill the local boundary conditions: u = N(η1, η2)u(ξ) } {{ } standard + F(r, φ)a(ξ) } {{ } enrichment . (1) The examples of a single plane crack and two perpendicularly meeting cracks in an isotropic continuum are considered. It is demonstrated that the method’s original excellent accuracy and convergence are regained, at a minimum cost of additional degrees of freedom (DOF). The normalized errors in the solution of the quadratic eigenvalue problem already become negligibly small for very coarse boundary meshes. The obtained convergence orders are often optimal and sometimes even superconvergence is observed.

Hypersingularities in Three-Dimensional Crack Configurations in Composite Laminates

Composite laminates meanwhile are of common use, especially in aerospace engineering. Inter-fiber cracks within a laminate ply are often accepted in practice to be still within failure tolerance, although the structural mechanics of this situation is not fully understood. The situation gets even more complex when the interaction of inter-fiber cracks in neighboring plies is considered. In this work, such three-dimensional crack configurations in composite laminates involving inter-fiber cracks and the influence of the laminate free-edge effect are studied by means of the Scaled Boundary Finite Element Method (SBFEM). The SBFEM is an efficient semi-analytical method that permits the analysis of linear elasticity problems including stress singularities or infinite domains. It is shown that in crack configurations in composite laminates so-called hypersingularities (or supersingularities) can occur, i.e. stress singularities which are of higher order than the classical crack singularity. This indicates that the laminate failure risk induced by certain considered crack configurations is not to be underestimated.