Meshless manifold method for dynamic fracture mechanics

Abstract

In the paper, the meshless manifold method (MMM) is utilized to analyze transient deformations in dynamic fracture. The MMM is based on the partition of unity method and the finite coverage approximation which provides a unified framework for solving problems involving both continuums and dis-continuums. The method can treat crack problem easily because the shape function is not affected by the discontinuity in the domain. For localization problems at the tip of the discontinuity, these shape functions are more effective than those used in other numerical methods. The method avoids the disadvantages of other meshless methods in which the tip of a discontinuous crack is not considered. In meshless manifold method, the finite coverage approximation is used to construct the shape functions that overcome influences of the interior discontinuities in the displacement. Consequently, the meshless manifold method has some advantages in solving the discontinuity problems when the discontinuities are complex. When the dynamic fracture mechanics is analyzed by the MMM, the weak formulation of the partial differential equation for elastic dynamics is derived from the method of weighted residuals (MWR). The discrete space of the domain is used for the MMM. The Newmark family of methods is used for the time integration scheme. At last, the validity and accuracy of the MMM are illustrated by two numerical examples of which the numerical results agree with the analytical solution.