Torsion problem for elastic cylinder with inserts and holes

An integral equation method to solve the classical torsion problem for an elastic cylinder with inserts and holes is treated. The bounded region outside the inserts and the holes will be termed a matrix. As is well-known the solution depends on finding plane harmonic functions in the matrix and inserts such that (a) on the outer boundary of the matrix and the boundaries of the holes the harmonic function in the matrix takes the values $\text{1}\text{2}\text{(x}{}^2\text{+y}{}^2\text{)+c}{}_{\mathrm{j}}$, and (b) on the interfaces of the matrix and the inserts relations exist between the harmonic functions and between their normal derivatives. Here (x, y) are the coordinates of the point on the boundary and c_j, are unknown constants. The usual methods are cumbersome and lengthy. In this paper a straightforward method is presented which is easily programmable. The numerical solution is obtained by evaluating a few integrals either analytically or numerically and solving a system of linear simultaneous equations. An example of a cylinder with an eccentric insert is given which substantiates the theory developed in this paper and is found to agree with known results. However, the method is general and may be applied to a variety of problems.