A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

Abstract

Abstract The numerical manifold method (NMM) can be viewed as an inherent continuous-discontinuous numerical method, which is based on two cover systems including mathematical and physical covers. Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants. Therefore, higher-order NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy. In addition, it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics. Thus, some other enriched local approximations are introduced to model the stress singularity at the crack tip. Generally, higher-order NMM, especially first-order NMM wherein local approximations are first-order polynomials, has the linear dependence problems as other partition of unit (PUM) based numerical methods does. To overcome this problem, an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method (FEM), which has no linear dependence issue. Meanwhile, the stresses at the nodes of mathematical mesh (the nodal stresses in FEM) are continuous and the degrees of freedom defined on the physical patches are physically meaningful. Next, the extended NMM is employed to solve multiple crack propagation problems. It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation. Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM. The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions. Thus the effectiveness and correctness of the developed NMM have been validated.