Eigenpairs for Two-Dimensional Elasticity

Abstract

The two-dimensional elastic solution in the vicinity of a singular point has the same characteristics as presented for the heat conduction solution, namely, it can be expanded as a linear combination of eigenpairs and their coefficients:

$$u = \sum\limits_{i=0}^{I} \sum\limits_{j=0}^{J} \sum\limits_{l=0}^{L} A_{{i}{j}{r}}r^{\alpha_{{i}+j}} 1n^{l}(r)s_{{i}{j}{l}}(\theta) + u^{reg}$$

r and _ being the polar coordinates of a system located in the singular point, and ?i and s._/ are the eigenpairs; M is zero except for cases in which the boundary near the singular point is curved (see Appendix C), and logarithmic terms may be present J ¤ 0 only for special cases for which m multiple eigenvalues exist with fewer than m corresponding eigenvectors (the algebraic multiplicity is grater than the geometric multiplicity), or when inhomogeneous BCs are prescribed on the V- notch faces. This case is not rigorously discussed in this chapter, but several remarks are provided at the end of it and it is further addressed in the chapters that compute the eigenpairs numerically.