Let the domain be occupied by an elastic material. The crack S is a smooth non-self-intersecting curve. The geometry of S is known. The elastic displacement field produced by the incident wave is obtained as the solution to the following initial boundary value problem:
where and are the displacement field (unknown) and the incident field (known), , are Lamé's constants, and is the density. The incident wave is a solution of
As in the Laplace case, the discontinuity of the solution across S is denoted by and is called the crack opening displacement.
The solution to this direct problem is written as follows:
where is the double layer kernel for elastodynamics defined in terms of the fundamental solution by
where , and and are velocities of P and S waves, respectively.
The function in (24) is obtained as the solution to the following integral equation:
which is derived from (24) and the traction - free boundary condition in (20). As in the Laplace case, the spatial shape functions for used in the conventional collocation method for solving (25) must have continuity at least at the collocation points, while the use of elements is permissible with the variational method. We therefore see that the variational approach is more accurate and versatile than collocation, although the conventional approaches with collocation may be faster in small problems. We shall therefore test both Galerkin (this section) and collocation (next section) approaches in time domain crack determination problems.
To formulate Galerkin, we start with the observation that the kernel function in (25) can be written in terms of the stress function as follows[28,4]:
This result shows that a variational equation equivalent to (25) can be written as
where is the permutation symbol, is a test function on S such that is satisfied at , , are kernel functions defined by
As the crack opening displacement is obtained, the scattered field is computed as
The P wave component of the scattered field (the part of which propagates with the P wave velocity ) allows a far field approximation given by , where
T is a time variable defined by and is the unit vector directed towards the point of the observation viewed from the origin. In the sequel, we shall call the quantity as the far field produced by and observed in the direction given by .