Question

What are the methods or approaches for the analytical solutions of boundary value problems in the theory of elasticity for orthotropic materials?

Answer 1

Analytical methods of solution are typically restricted to two-dimensional geometries, see for example Applications of symmetry methods in basic problems of orthotropic elasticity (1999)

We discuss basic problems of orthotropic elasticity in a plane domain whose boundary is a piecewise-algebraic curve. First, by means of bi-analytic functions, a basic problem is reduced to a boundary value problem for analytic functions. Then, by use of the generalized symmetry principle for algebraic curves, a boundary value problem for analytic functions is converted to a problem on a Riemann surface; then the solution to the original problem is obtained in closed form for a domain with algebraic boundaries having genus $\rho\geq 0$.

Numerical methods in three dimensions are discussed in A matrix decomposition method for orthotropic elasticity problems (1989).