5.10 THE WILLIAMS EIGENFUNCTION ANALYSIS

The family of solutions associated with the wedge is of special interest in the theory of elasticity. By considering properties of these solutions, it is possible to make valuable deductions about the influence of plane geometry (e.g., the wedge angle) on the stress state in the vicinity of the apex.

Following Williams (1952), we carry out an analysis of the eigenfunctions and eigenvalues for the wedge (plane elastic problem). Wedge geometry ??< ? of Figure 5.1 is once again considered. Solutions must satisfy traction-free boundary conditions ? _?? = ? _{r ?}=0 for ?= ?. In this section we search for such solutions that have the Airy stress function in the variable-separable form

The requirement of biharmonicity of A ₀ (r, ?) leads to

The solution of this equation for the unknown function A ₀ (r, ?) has the form

Now the Beltrami tensor potential can be built and the stress tensor calculated using the familiar procedure of equation (5.3). The stress components ? _?? and ? _{r ?} assume the forms

These must satisfy traction-free boundary conditions on the edges ?= ?, which lead to four linear algebraic equations for the four unknown coefficients a ₁ , a ₂ , a ₃ , a ₄. The system matrix has the form

An eigenfunction of the problem can be found if this system has a nontrivial solution, which happens only...