13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- where Tj are Chebyshev polynomials of the first kind and cj are unknown constants. By satisfying the boundary condition at selected collocation points along the crack surfaces, the parameters cj can be determined in terms of the boundary stress as, (8) where [S] is a known matrix, {A}={c1, c2, … } T and {f} is a matrix containing the boundary stresses at the collocation points along the crack surfaces. From this solution, the stress and displacement field caused by this crack can be calculated in terms of {A}. 2.2 Interaction problem The solution of other single defect problems can also be determined and the solution can be expressed in the similar format as shown in Eq. (8). When multiple defects are involved, for defect Aj, as shown in Fig.2(a), all scattered waves from other defects will become an incident wave, i.e. the pseudo-incidenr wave (uj p). Therefore, defect Aj is subjected to both the original incident wave and the pseudo-incident wave and results in a scattered wave, as shown by Fig.2(b). (a) (b) Figure 2 Illustration of pseudo-incident waves (a) scattering from other defects, (b) total incident wave for a defect Based on the relation between defects discussed above, for defect Aj the solution can be expressed as (9) where [Sj] is the matrix given by (8) for Aj, and the two terms on the right hand side represent the original incident wave and the pseudo-incident wave. If Eq. (9) is applied to all the defects and the pseudo-incident waves are represented in terms of the scattered waves, the governing equation for the interaction problem can be determined, (10) where [Qj] are determined by the scattered waves of the defects, {fj} are the original incident wave at different defects. It should be mentioned that both [S] and [Q] matrices are obtained from the analytical solution of single defects. By solving this linear equation, the solution of the interaction problem can be determined.
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