Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.