On simple shear for incompressible isotropic linear elastic materials

In this paper we discuss some problems involving simple shear in incompressible isotropic linear elastic materials within the framework of the linearized finite theory of elasticity. First we obtain for a simple shear a universal relation in terms of components of the first Piola-Kirchhoff stress tensor. Afterwards for a rectangular block deformed by a simple shear we evaluate the absolute error and the relative error both for the Piola-Kirchhoff tractions and the Cauchy tractions calculated by classical linear elasticity. Finally we discuss two dead load problems corresponding to different Piola-Kirchhoff tractions by using both the linearized finite theory of elasticity and the classical linear elasticity. The first problem can be solved only in linearized finite theory of elasticity and the solution is a simple shear. The second problem admits a simple shear as a solution in both theories, so that we can compare the solutions.