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Spatial convergence of crack nucleation using a cohesive finite-element model on a pinwheel-based mesh. (English) Zbl 1110.74854
Summary: We consider the use of initially rigid cohesive interface models in a two-dimensional dynamic finite-element solution of a fracture process. Our focus is on convergence of finite-element solutions to a solution of the undiscretized medium as the mesh spacing $$\Delta x$$ (and therefore time-step $$\Delta t$$) tends to zero. We propose the use of pinwheel meshes, which possess the ’isoperimetric property’ that for any curve $$C$$ in the computational domain, there is an approximation to $$C$$ using mesh edges that tends to $$C$$ including a correct representation of its length, as the grid size tends to zero. We suggest that the isoperimetric property is a necessary condition for any possible spatial convergence proof in the general case that the crack path is not known in advance. Conversely, we establish that if the pinwheel mesh is used, the discrete interface first activated in the finite-element model will converge to the initial crack in the undiscretized medium. Finally, we carry out a mesh refinement experiment to check convergence of both nucleation and propagation. Our results indicate that the crack path computed in the pinwheel mesh is more stable as the mesh is refined compared to other types of meshes.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74R10 Brittle fracture
Triangle
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