A physical approach to asymptoticallysolving some integral equations occurring in solid mechanics

Abstract

We use the Comninou model of the interface crack to illustrate a method of solving aclass of integral equations containing a small parameter. We need to know the inverse of the integral equation when this small parameter is set equal to zero. This inverse, when applied to the original integral equation, leads to an expression for the unknown which is an asymptotic series in the small parameter, provided the field point x is outside the boundary layer. Next we need to know the inverse of an integral equation which models the given integral equation for points x in the boundary layer. When this inverse is applied to the original integral equation, one obtains an expression for the unknown which is an asymptotic series in the small parameter, provided the field point x is inside the boundary layer. This solution and the former solution are often referred to as the inner and outer solutions, respectively. They each contain unknown constants. These constants are determined by matching the inner and outer solutions on an overlap region where both solutions are valid. A uniform expression for the solution is obtained by adding the inner and outer solutions and subtracting off the matching terms.