A multi-dimensional space method for dynamic cracks problems using implicit time scheme in the framework of the extended finite element method

Abstract

The extended finite element method (XFEM) has been widely used to investigate the moving cracks without any remeshing. The degrees of freedom (DOFs) of nodes around crack surfaces and crack tips are added to represent the discontinuous fields and asymptotic fields. However, the XFEM encounters some challenges in dynamic problems because the total number of DOFs increases with the crack growth, which leads to difficulty in iteration algorithm of time integration. In this paper, based on the XFEM, a multi-dimensional space method is developed to simulate the crack growth under dynamic loads. Every node, not limited to the nodes around crack tips and crack surfaces, has 12 DOFs in the domain containing one crack (12n DOFs in the domain containing n cracks). The generalized boundary condition is proposed to modify the matrices of stiffness, mass and damping. The modified matrices of stiffness, mass and damping are highly sparse, the additional central processing unit time is little and the use of internal memory does not increase compared with the traditional XFEM. Moreover, the implicit Newmark time integration scheme is introduced to solve the dynamic problems of XFEM. Compared with the other time integration schemes, the time step of the present method is much longer. The convergence property of the present method is demonstrated in theoretical and numerical studies. In order to verify the robustness and the precision of the present method, three numerical cases are presented.

Fatigue Deformation and Energy Change of Single-Joint Sandstone After Freeze-Thaw Cycles and Cyclic Loadings

“…In the past, numerical simulations were typically used to research fractured rock because of sampling difficulties (Wong and Zhang, 2014;Cheng and Zhou, 2015). Park and Bobet (2010), Lee and Jeon (2011); Tang et al (2001), and Zhou et al (2014) used gypsum materials or rock-like mix materials to prepare fractured rocks and obtained good test results.…”

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

“…Actually, because of the multiple enrichment scheme, the enriched elements cover up the real crack tip. Although the multiple enrichment scheme has the ability to improve the accuracy of the XFEM in some degree . It is considered that the multiple enrichment scheme has a disadvantage in solving the problem of mixed mode I and II crack.…”

“…In this section, the familiar multiple enrichment schemes for the XFEM were discussed. 15,[40][41][42][43] In many previous works, it was applied to improve the accuracy of the…”

“…Although the multiple enrichment scheme has the ability to improve the accuracy of the XFEM in some degree. 15,[40][41][42][43] It is considered that the multiple enrichment scheme has a disadvantage in solving the problem of mixed mode I and II crack.…”

Simulation of cracking behaviours in interlayered rocks with flaws subjected to tension using a phase‐field method

“…Numerical methods can be classified into two categories: continuum-based methods and discontinuum-based methods. In terms of the continuum-based methods, finite element method (FEM), 16,17 finite difference method (FDM), 18 boundary element method (BEM), 19 extended finite element method (XFEM), [20][21][22] phantom node method, 23 strain softening elements, 24 and specific meshfree methods, [25][26][27] such as bond particle method (BPM), 28,29 peridynamics (PD), 30-33 smoothed particle hydrodynamics (SPH), 34 and general particle dynamics (GPD), 35,36 were developed to investigate interlayers and crack problems. In terms of the discontinuum-based methods, distinct lattice spring model (DLSM), 37 discrete element method (DEM), [38][39][40][41][42] discontinuous deformation analysis (DDA), [43][44][45] and numerical manifold method (NMM) [46][47][48] were developed.…”