Analytical and numerical evaluation of crack-tip plasticity of an axisymmetrically loaded penny-shaped crack

Abstract

Analytical and numerical approaches are used to solve an axisymmetric crack problem with a refined Barenblatt-Dugdale approach. The analytical method utilizes potential theory in classical linear elasticity, where a suitable potential is selected for the treatment of the mixed boundary problem. The closed-form solution for the problem with constant pressure applied near the tip of a penny-shaped crack is studied to illustrate the methodology of the analysis and also to provide a fundamental solution for the numerical approach. Taking advantage of the superposition principle, an exact solution is derived to predict the extent of the plastic zone where a Tresca yield condition is imposed, which also provides a useful benchmark for the numerical study presented in the second part. For an axisymmetric crack, the numerical discretization is required only in the radial direction, which renders the programming work efficient. Through an iterative scheme, the numerical method is able to determine the size of the crack tip plasticity, which is governed by the nonlinear von Mises criterion. The relationships between the applied load and the length of the plastic zone are compared for three different yielding conditions.

Penny-Shaped Dugdale Crack in a Transverse Isotropic Medium

“…which is consistent with that predicted by Chaiyat et al (2008). The CSD due to the external load (10) can be generally expressed in terms of special functions as well.…”

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

“…An illuminating discussion on using the harmonic function method to solve axisymmetric mixed boundary value problems and many useful expressions have been provided by Barber (1983). Also, some important contributions to fracture mechanics have recently been made by Chaiyat et al (2008) and Jin et al (2008) utilizing this method. The current formulation of non-adhesive and adhesive contact problems is greatly facilitated by the findings of these studies.…”

“…(3) and an expression similar to Eq. (4) (containing sin(xt) rather than cos(xt)) were employed earlier to solve penny-shaped and other internal crack problems (Kassir and Sih, 1975;Chen and Keer, 1993;Chaiyat et al, 2008). This harmonic function is adopted in the current study to solve non-adhesive and adhesive contact problems involving axisymmetric punches of arbitrary shapes and BCs of both types.…”

“…On the other hand, the harmonic potential function method for axisymmetric elasticity problems developed by Green (1949) and Collins (1959Collins ( , 1963 and elaborated in Green and Zerna (1968) has been found to exhibit significant advantages in solving axisymmetric half-space problems with mixed boundary conditions (e.g., Barber, 1983;Chaiyat et al, 2008;Jin et al, 2008). However, this method has not been systematically explored for its use in studying adhesive contact problems of axisymmetric elastic bodies, for which the JKR model, the DMT model, and the M-D model and its variants have been developed using the Hertz theory (1882) or Sneddon's (1965) solution.…”

Extended displacement discontinuity method for nonlinear analysis of penny-shaped cracks in three-dimensional piezoelectric media

“…This model assumes that the elastic-plastic material is ideal, and that the stress in the yielding zone is constant and equal to the yield value. Fracture mechanics based on the Dugdale model have been studied extensively and have been applied to twodimensional (2D) [5][6][7][8][9][10][11], as well as three-dimensional (3D) situation [12][13][14]. Extending the Dugdale model to piezoelectric materials, Gao and Barnett [15] and Gao et al [16] proposed a strip polarization saturation (PS) model in which the piezoelectric material is treated as mechanically brittle and electrically ductile and the electric displacement is equal to the polarization saturation value in the electric yielding zone.…”