A contact algorithm for frictional crack propagation with the extended finite element method

Abstract

SUMMARYWe present an incremental quasi-static contact algorithm for path-dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from nonlinear contact mechanics and embedded within a localized element by penalty method. Newton-Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called 'LATIN' for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution.

Finite deformation formulation for embedded frictional crack with the extended finite element method

“…The above expression is the same as that developed by Liu and Borja [20] and Borja [19] in the previous work.…”

“…Following the developments in [19,20,30,42], we consider a body B bounded externally by surface *B and internally by a pair of crack faces S − and S + in the reference configuration (the choice of S − and S + is irrelevant in the present case, as shown later with a numerical example). Without loss of generality, we assume that the crack is initially closed so that S − = S + ≡ S. Furthermore, we denote the subdomains on the negative and positive sides of the crack by B − and B + , respectively.…”

“…and f h (X), called blending function in [19,20], is any arbitrary smooth function that satisfies the requirements…”

“…We refer to [19,20] for similar expressions developed for the infinitesimal case. The FE matrix equation consistent with the Galerkin approximation (65) is (with a slight abuse in the notation by using s for both the vector and the tensor of Kirchhoff stresses)…”

“…Unfortunately, their solution is hampered by the very slow convergence rate of the iteration. Very recently, Borja [19] and Liu and Borja [20,21] reformulated a variational equation in terms of the relative displacement of the crack faces and showed that improved efficiency can be achieved with Newton iteration. However, their work has been limited to infinitesimal deformation.…”