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

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The same methods as for other partial differential equations. –  Michael Renardy Dec 11 '12 at 15:09

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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).

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