A three-dimensional solution is presented for the transient response of an infinite plate which contains a rectangular crack. The Laplace and Fourier transforms are used to reduce the problem to a pair of dual integral equations. These equations are solved with the series expansion method. The stress intensity factors are defined in the Laplace transform domain, and they are inverted numerically in the physical space.
1. Introduction
IN RECENT years there has been a mounting interest in solving the dynamic crack problems. It occasionally occurs that the structural members, which are safe under static loads, break down when dynamic loads act on them. This is due to the facts that fracture toughness values of the materials decrease as the rate of loading increases [ l] and that dynamic stress intensity factors exceed the static ones [2]. Therefore, to prevent brittle fracture of mechanical structures, which leads to serious damage, we must take care whether the dynamic stress intensity factor is smaller than the dynamic fracture toughness value of the material or not.
Previous works on dynamic crack problems have been mainly concerned with a single crack in an infinite medium [2]. It is mathematically difficult to obtain a solution for two or more cracks, and when we try to ascertain the boundary effects on the dynamic stress intensity factors, it is also attended with similar difficulties. However, dynamic stress intensity factors around two coplanar Griffith cracks in an infinite elastic medium have been investigated by Jain and Kanwal[31 and by Itou[4-61. Recently, analytical treatments on the transient crack problem including the effect of boundaries have been worked by Chen[7-91 and by Itou[lO] and by Sih and Chen[ll, 121.
When catastrophic structural failure initiates from an embedded crack, three-dimensional dynamic problems must be studied for a finite sized crack such as a penny-shaped crack or a rectangular crack. Mixed boundary value problems for the penny-shaped crack geometry can be easily solved by using the Hankel transforms. If the crack has not such symmetry, it is very difficult to obtain analytical solutions for the dynamic stress intensity factors. However, the author has resolved some dynamic crack problems which concern rectangular-shaped crack(s) in an infinite elastic body[13-181. In those papers, method of solution is quite simple and accuracy of solution is equivalent to that given by Sih and Loeber [ 191.
An analytical approach to the dynamic crack problem including the effect of boundaries involves great difficulties even in the two-dimensional treatment. However, the author has studied the threedimensional impact response of a rectangular crack in a semi_space [20]. In this investigation, the elastodynamic stress intensity factors of a rectangular crack in an infinite plate are obtained. The surfaces of the centrally located crack is perpendicular to the plane boundaries of the plate and they are opened by internal pressure with the Heaviside-function time dependence. Laplace and Fourier transforms are used to reduce the mixed boundary-value problem to a set of dual integral equations. To solve these equations, the crack surface displacement is expanded in a series of functions which are automatically zero outside of the crack. To obtain the coefficients accompanied in that series, the Schmidt method is used. The Laplace inversion is accomplished using a numerical procedure[21] and the results of stress intensity factors are compared with those in an infinite elastic medium[l6].