Singular solutions of a transmission problem in plane linear elasticity for wedge-shaped regions

Abstract

In plane elasticity, when two different wedge-shaped elastic materials (isotropic, homogeneous) are bonded together along a common edge and subject to tractions on the boundary, the stress field ? will become infinite at the apex. In fact, asymptotically, the displacementu satisfies $$u \approx \sum\limits_{j = 1}^n {r^{ - i\lambda _j } } \sum\limits_{k = 0}^{\eta _j - 1} {\ln ^k } (r)\Phi _{jk} (\theta )asr \to 0,$$ where (r, ?) are the polar coordinates with origin at the wedge apex and the complex numbers? j are the roots with positive imaginary part and multiplicity? j of an explicitly given transcendental equation:G(?)=0. For applications it is desirable to understand how the number of terms that give rise to unbounded stresses in the above expansion depends on the elasticity constants and on the wedge angles. Setting $$N(q) = \sum\limits_{0< \operatorname{Im} (\lambda _j )< q} {\eta _j } $$ , the problem is then to studyN(1) as a function of the material constants and wedge angles. A rigorous analysis is presented here for the case in which the wedge angles are equal (symmetric domains). Using the fact thatG is an entire function of ? which depends continuously on the parameters of the problem, we develop a continuity method which enables us to evaluate the functions $$N\left( {\frac{\pi }{a}} \right)$$ and $$N\left( {\frac{\pi }{{2a}}} \right)$$ wherea is the measure of each wedge angle. This yields bounds onN(1) as well as some information on the zeros ofG that correspond to higher order singularities.