The inapplicability of variational methods in the cohesive crack problem for initially rigid traction‐separation relation and its solution using integral equations

Abstract

A cohesive crack problem with initially rigid traction-separation relation has been considered here. It has been shown that the total potential energy is not differentiable at the crack-tip. This makes the application of a variational operator over this total potential energy theoretically incorrect in the general sense. It further implies that variational methods are not applicable in this situation. One way to overcome this problem is to introduce a penalty of initial stiffness in the traction-separation relationship. However, this means modifying the material property of a solid. To avoid variational operator, an integral equation form of this problem has been numerically solved. The results thus obtained conform to the present understanding of cohesive stresses. This method is theoretically more accurate to solve a cohesive crack problem whenever the total potential energy is non-differentiable.