On the Derivation of Hooke’s Law for Plane State Conditions

Abstract

We discuss the derivation of Hooke’s law for plane stress and plane strain states from its general three-dimensional representation. This means that we consider the anisotropic case to ensure a certain generality of our representation. Thereby, two approaches are examined, namely the tensorial representation involving fourth-order tensors over a three-dimensional vector space, and the Voigt-Mandel-Notation involving second-order tensors over a six-dimensional vector space. The latter reduces to a vector-matrix notation common in engineering applications. It turns out that both approaches have their merits: The tensorial approach is easier to handle symbolically, the matrix approach is easier to handle numerically. Both procedures are applicable for arbitrary material symmetries. Finally, we answer the question why a material under the assumptions of a plane stress state behaves softer and why it behaves stiffer under a plane strain state compared to the three-dimensional state.