Material nonlinearity

The next higher level of complexity involves the use of nonlinear relations between stress and strain components. Even when computing displacement gradients to first order, only the higher order material relationship introduces higher order effects in the differential equation. Material nonlinearity might easily be coupled with geometric nonlinearity and drawing a clear boundary between them might be hard in many cases. This kind of nonlinearity introduces new parameters into the differential equation, namely new material constants (higher order elastic parameters). In many practical cases we might not have knowledge of these parameters or they might be hard to estimate.

Negligence of the second and higher order cross terms ensures that a wave equation is linear. If these terms are neglected in the Taylor expansion of $d{\bf u}$, we are still confined to examine only a small neighborhood around a reference point. Assuming that products of the displacement gradients are small, assures that the principle of superposition is valid. The validity of this principle can be proved by applying two deformations consecutively. The order of application of the deformations has no effect on the final observed deformation. The principle of superposition is a fundamental property of the linear theory of elasticity.

We have to use equations (1) and (3) as soon as we consider finite displacements of the medium. It is practical to start with this nonlinear description since it introduces only small modifications to existing modeling programs, while admitting some degree of nonlinearity. Algorithms which are based on linear assumptions fail in this case and other methods have to be used to solve the problem. The wave equation therefore will be nonlinear in the spatial domain, in the time domain the principle of superposition is still valid. In order to investigate the impact of nonlinearity on radiation patterns and signal wave forms, numerical examples can be calculated in which nonlinear wave propagation is modeled using finite differences in time and space. Such a model should serve as a lower limit as to what we can expect from nonlinearities.