ANALYSIS OF ELASTODYNAMIC DEFORMATIONS NEAR A CRACK-NOTCH TIP BY THE MESHLESS LOCAL PETROV–GALERKIN (MLPG) METHOD

Abstract

The Meshless Local Petrov-Galerkin (MLPG) method is used to analyze transient deforma- tions near either a crack or a notch tip in a linear elastic plate. The local weak formulation of equations govern- ing elastodynamic deformations is derived. It results in a system of coupled ordinary differential equations which are integrated with respect to time by a Newmark family of methods. Essential boundary conditions are imposed by the penalty method. The accuracy of the MLPG so- lution is established by comparing computed results for one-dimensional wave propagation in a rod with the an- alytical solution of the problem. Results are then com- puted for the following two problems: a rectangular plate with a central crack with plate edges parallel to the crack axis loaded in tension, and a double edge-notched plate with the edge between the notches loaded by compres- sive tractions. Stresses at points near the crack/notch tip computed from the MLPG solution are found to agree well with those obtained from either the analytical or the finite element solution of the same problem. The index of stress singularity is ascertained from a plot of log (stress) vs. log ( r) where r is the distance from the crack tip. It is found that, for the double-edge notched plate, the mode-mixity of deformations near a notch-tip in an or- thotropic plate can be adjusted by suitably varying the in-plane moduli of the material of the plate. The varia- tion of shear stress with r exhibits a boundary layer effect near r= O. tegrals appearing in the local weak formulation of the problem. Atluri et al. (1999) have pointed out that the Galerkin approximation can also be adopted that leads to a symmetric stiffness matrix. Atluri and Zhu (2000) solved elastostatic problems by the MLPG method, and Lin and Atluri (2000) introduced the upwinding scheme to analyze steady convection-diffusion problems. Ching and Batra (200 1) enriched the polynomial basis functions with those appropriate to describe singular deformation fields near a crack tip and used the diffraction criterion to find stress intensity factors, the J-integrals and sin- gular stress fields near a crack tip. Gu and Liu (2001) used the Newmark family of methods to study forced vi- brations of a beam. The problem of bending of a thin plate has been studied by Long and Atluri (2002). War- lock et al. (2002) have analyzed elastostatic deforma- tions of a material compressed in a rough rectangular cavity analytically by the Laplace transformation tech- nique and numerically by the MLPG method. Atluri and Shen (2002a,b) have demonstrated the use of different weight functions and have compared their performance with that of the Galerkin finite element method. By choosing a Heaviside step function as the test function, they eliminated the domain integration in the local weak form. Thus only boundary integrals over local subdo- mains remained in the local weak form. For elastostatic problems, this was shown to be more efficient than both the finite element and the boundary element methods. The paper is organized as follows. Section 2 gives the MLPG formulation including the local weak form, the moving least squares approximation, the discrete gov- erning equations and the time integration scheme. Cal- culations of the dynamic stress intensity factors from the near-tip stress fields are also described. Numerical ex- amples are presented in Section 3. The MLPG results are compared with either analytical or finite element so- lutions. Section 4 summarizes the conclusions.