Restrictions on nonlinear constitutive equations for elastic shells
It is well known that the problem of the writing the constitutive equations of a shell-like structure is very difficult if the shell theory is based on some Cosserat model, even if the nonlinear three-dimensional constitutive laws are known. The reason comes from the coupling of the three-dimensional laws with the geometrical properties of the shell model. The approach used by the authors, the first one being as a recall one of the pioneers of the model, lies in the search of some restrictions given by the model with respect to the three-dimensional theory. After having described the two-dimensional Cosserat theory (a surface endowed with one director), the restrictions are found in the assumed and simplified case of homogeneous deformations considered as a necessary condition. Besides, the main classical assumption is the existence of a nonlinear strain energy as a function of the deformation unknowns. The definitions are recalled in a rather intrinsic method, some useful change of variables is made, and the associate surface stresses are established through the dynamic equilibrium equations. Some formulae give their values as functions of the Piola-Kirchhoff stresses, the three-dimensional corresponding strains, and the three-dimensional strain energy in the homogeneous deformation case. It is assumed an additional part of the strain energy, so that the restrictions are written for that part owing to the homogeneous strain assumption. A case of simple strain energy function is written for a shell made of a homogeneous material with a general nonlinear anisotropic elastic resonse. First, applications are developed in the case of an infinite plate with a circular hole and compared to the Reissner or the Mindlin plate theory. The second application concerns the torsion of a rectangular plate and is compared with the exact three-dimensional solution. Both comparisons, which are either in statics or in dynamics by matching the first frequencies of free vibrations, show a very good accuracy. The case of a spherical shell in the linear range is also carefully studied, but the results are not so satisfactory. All the theory is developed thanks to formulae of differential geometry recalled in 3 appendices. In conclusion, in spite of this interesting step in the improving of the model, it must be estimated that the research seems to be still open.
scientific article; zbMATH DE number 807330