A boundary integral equation(BIE) formulation for design sensitivity with changing boundary conditions is developed using the material derivative concept and the direct differentiation method, for elliptic boundary value problems. The change of boundary conditions is described using the tangential component of the velocity field in the study of shape optimal design. The normal velocity component is not included in the sensitivity formula since the shape change is not of interest. An arbitrary rigid body motion is considered to remove the singularities occuring from differentiation of the fundamental solutions, thus avoiding the difficulties with the numerical integration. The derived sensitivity formula has the same form as the original BIE with the exception of the difference in state variables and some additional terms. So the same numerical procedures employed in the solution of the original BIE are used to solve the sensitivity equation.
For the numerical verification of the derived sensitivity formula, a heat flow problem and a semi-infinite plate problem are considered. To obtain the change of boundary conditions, the boundary segments of insulation and of uniform load are forced to change for the former and the latter problem, respectively. The numerical soutions are compared with those by finite difference method and the analytic ones.
The formulation is then applied to calculate energy release rate and stress intensity factors as a new method of computational fracture mechanics. The basic idea stems from the fact that a crack extension can be interpreted as a change of boundary condition and this can be described using the tangential component of the design velocity field. Thus, the energy release rate which is the total derivative of the strain energy with respect to crack length, is recognized as corresponding to the design sensitivity analysis with changing boundary conditions. Three types of cracked plate problems are treated as numerical examples, mode-I, mixed mode of mode-I and mode-II and interface cracked plate problem. For all problems, traction singular quarter-point boundary elements are used on either side of crack tip to represent the singular behavior at the crack tip accurately. A multi-region boundary element method is adopted from the geometric asymmetry in mixed mode cracked plate problem, and material asymmetry in interface cracked plate problem. For the self-similar extension of mode-I and interface cracked plate problem, nonzero tangential velocity values are assigned to the node at the crack tip and two mid-nodes next to the crack tip with a linearly varying function. For the mixed mode cracked plate problem, crack kinking is considered. Due to the difficulties in defining a suitable velocity field for the infinitesimal kinked crack, a finite length crack emanating from the main crack is considered and analyzed, and limit is taken to make it of infinitesimal length.
The energy release rates for all problems are calculated using the derived sensitivity formula. The stress intensity factors can also be calculated form Irwin formula in the case of mode-I cracked plates. For the mixed mode cracked plates including interface cracked plate, an unknown load parameter method is proposed and numerically applied to a single edge slanted cracked plate problem. The results agree very well with the analytic solutions in the literature. The method is based on analytic formulas and the amount of calculation when implemented is minimal requring two solutions of system equations. This suggests that the approach of using the proposed sensitivity analysis formula with changing boundary conditions can be a new powerful tool in the compuatational fracture mechanics.
