Cracking elements method with a dissipation-based arc-length approach

Abstract

The dissipated strain energy, representing a monotonically increasing state variable in nonlinear fracture mechanics, can be used to develop an arc-length constraint equation for tracking the energy dissipation path instead of the elastic unloading path of the response of a structure. This was the motivation for the development of a dissipation-based arc-length method, followed by its implementation in the framework of the recently proposed Global Cracking Elements Method (GCEM). The dissipated energy is extracted with the help of the crack openings and tractions, i.e. by means of the displacement jumps and the cohesive forces between the two surfaces of a crack. The stiffness factor of the arc-length constraint equation is obtained in the solution process by means of the Sherman-Morrison formula. Several numerical tests are performed. The results demonstrate the robustness of the proposed method. It captures both global and local peak loads and all snap-back parts of the force-displacement responses of loaded structures with multiple cracks. • The dissipated energy is directly determined by the crack openings and tractions as opposed to indirect determination by forces and displacements. • Path following and crack propagating are treated simultaneously without a special strategy. • The Sherman-Morrison formula is used for implementation of the solution process and the coding efforts are reduced. • The stiffness factor of the arc-length constraint is obtained during, rather than before, the solution process.