Extension, bending and torsion of cylindrical Cosserat shells made from a porous elastic material

We consider the relaxed Saint-Venant’s problem in the linear theory of elastic shells made from a porous material. For our purpose, we use the model of Cosserat surfaces and the Nunziato-Cowin theory of elastic materials with voids [

1

].We extend the method employed in [

2

], [

3

] for the case of Cosserat shells with two porosity fields: one field characterizes the volume fraction variations along the middle surface of the shell, while the other accounts for the changes in volume fraction along the shell’s thickness.

We obtain the solution of the extension, bending and torsion problem in closed form, for both open and closed cylindrical shells of arbitrary cross-section. The solutions determined are shown to be minimizers of the strain energy functional on certain classes of solutions to the relaxed Saint-Venant’s problem for shells, by analogy with the well-known characterizations of the classical Saint-Venant’s solutions from the three-dimensional theory of elasticity. We observe that the torsion of cylindrical shells does not affect the porosity fields, so that the torsion problem reduces to the previously known results for the purely elastic case. On the other hand, for the extension and bending problem, we remark the influence of the material’s porosity on the deformation of the shell. In the particular case of porous Cosserat plates, we compute the solution of the extension and bending problem and show that this solution is in agreement with the corresponding results obtained in [

1

] and [

4

] using three-dimensional approaches.

The solutions determined in this paper are exact and they prove useful in solving many practical problems and for the comparison with related results obtained by various computational methods.