Solution of flat crack problem by using variational principle and differential–integral equation

Variational principle is used to solve some flat crack problems in three-dimensional elasticity. In the formulation, the strain energy is evaluated by multiplying the crack opening displacement (COD) by the boundary traction. The boundary traction is related to the COD function by a differential–integral representation. By using an integration by part, the portion of the strain energy of the potential functional can be expressed by a repeated integral. In the integral all the integrated functions are non-singular. Letting the functional be minimum, the solution is obtained. In the actual solution, the COD function is represented by a shape function family in which several undetermined coefficients are involved. Using the variational principle, the coefficients are obtained. Several numerical examples are given with the stress intensity factors calculated along the crack border