Incompatible numerical manifold method for fracture problems

Abstract

The incompatible numerical manifold method (INMM) is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack.improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given.

Stress Intensity Factor for Interfacial Cracks in Bi-Materials Using Incompatible Numerical Manifold Method

“…In this paper, we develop an INMM for bi-material interfacial cracks. This work extends the capabilities of the INMM [8,9] to the analysis of cracks that lie at the interface of two elastically homogeneous isotropic materials. A detailed description of the numerical implementation of the INMM for 2-D crack modeling in isotropic media and for a crack normal to a bi-material interfacial is given in Reference [8].…”

“…In INMM [9], the additional displacement e λ N is used to the displacement functions TD . The displacement functions can be expressed as…”

Numerical Manifold Method for the Forced Vibration of Thin Plates during Bending

“…The application research on NMM mainly focuses on certain fields, such as geotechnical engineering [ 3 , 4 ] and crack propagation [ 5 – 8 ]. However, NMM has recently been successfully applied to fatigue failure [ 9 ], fluid calculations [ 10 ], and seepage [ 11 ]. In theoretical research, most NMMs adopt shape functions in the finite element method as the weight functions for the manifold method and use the local cover functions of high-order polynomials to improve the continuity and interpolation accuracy of elements [ 12 – 14 ].…”