Governing Differential Equations for the Mechanics of Undamageable Materials

In this work the mathematical foundations of the mechanics of elastic undamageable materials are presented. In particular the governing differential equations are derived for both the scalar and tensorial cases. In the isotropic case it is found that the resulting scalar differential equations are simple and easy to solve. However, in the anisotropic case the tensorial differential equations are complicated and unsolvable at this time. The current work presents the solution in the form of explicit nonlinear stress-strain relations for the simple one-dimensional case. However, the general solution of the three-dimensional case remains unattainable at the present time. Only the governing tensorial differential equations are derived for this latter case. It is to be noted that the term “undamageable” is reflected in the context of the material stiffness and not the property of indestructibility due to various loading conditions. Thus, the undamageable material reflects that no microcracks or microvoids occur as well as no plastic yielding in the material. To illustrate this concept, a last section is added on applications.