Nonlinear Material Properties

Abstract

The formulation of Hooke's law rests on the assumption of infinitesimally small deformations. Its application to the simple model of a mass connected with a spring results in a linear force law and to the well known harmonic oscillation. Investigating even with very modest means the behavior of a real system of this sort shows that the limits of accuracy of this simple description are quite narrow indeed. A more general and accurate description will have to be a nonlinear one. This, in fact turns out to be true for all material properties, e.g. dielectric properties and the simple relation (4.2) is valid only for small fields and is an approximation in the same way as Hooke's law (3.51). If we are looking close enough we find that all phenomena actually are nonlinear, which means that the response of even simple systems to an external influence cannot be precisely described by a direct proportionality.The nonlinearity of the response, in general, is due to two different causes: The first one lies in the fact that the body under consideration changes its shape and size by the application of an external stimulus. Therefore, in the course of this process all quantities that are referred to unit volume (in fact obtained by dividing it by the volume of the body), or to unit area of a surface will be influenced by the continuous change of volume or of surface area, leading to a deviation from truly linear behavior. Effects of this sort are called geometrical nonlinearities.On the other hand we know that interatomic forces strongly depend on the separation of the particles and can be described by a linear force law only in the lowest approximation but are essentially nonlinear and this affects the response of the material itself such that we speak of material nonlinearity. A proper treatment of nonlinear effects has to take into account both of these nonlinearities.Let us consider Hooke's law as stated in (3.51'). The stiffness coefficients c λμ are clearly defined as constants. As a first step of approaching nonlinear behavior we could consider leaving the form of the law as it is, but interpreting the stiffness coefficients as functions of the strain S λμ . Such a procedure is frequently adopted in engineering practice. It has its place there and it has its merits for the sake of qualitative discussions and as a visualizing tool and we will use it in this sense later. The natural measure of nonlinearity then becomes the first derivative of this function. This approach clearly has its limits and a more general procedure (involving this 101