Derivation of an elastic shell model of order h(n) as n tends to infinity

Starting from three-dimensional linear elasticity and performing the dimensional reduction by integration over the thickness, we derive a general form of the areal strain energy density for elastic shells. To obtain the new constitutive model, we do not approximate the deformation fields as polynomials in the thickness coordinate, but rather we keep all terms in the thickness-wise series expansions. As a result, we deduce the explicit form of the shell strain energy density in which the constitutive coefficients are expressed as integrals depending on the thickness h and on the initial curvature. Then, to obtain the shell model of order O(h(n)), we expand the integral coefficients in the strain energy function as power series of h and truncate them to the power hn. In the case n = 3, we recover the classical strain energy density for combined bending and stretching of linear shells, which leads to the Koiter model. Finally, we prove that the proposed shell strain energy function is coercive for any n >= 3, as well as for the general case n -> infinity.