2.1 Origin of Nonlinearities

First, we recall the origins of nonlinearities in the frame of mechanics. It is well known that one can the origin of nonlinearities a follows. First, there is the geometrical origin of nonlinearities. The simplest example of such a nonlinear behaviour is provided by large oscillations of a pendulum. Geometrical nonlinearities arise in the field of continuum mechanics due to the nonlinear quadratic terms that occur in the relation between the displacement and the strain tensors [Eringen et. al. (1975)], [Sedov (1971)].

Second, nonlinearities may occur from constitutive laws: nonlinear elasticity for example [Antman (1995)] or any other nonlinear constitutive laws [Palmov (1998)] such as plasticity, viscoelastoplasticity, etc. It is well known that mathematical expressions of nonlinear constitutive laws do not always use smooth mathematical expressions but on the contrary must often involve nonsmooth terms [Alber (1998)].

The third origin of nonlinearities is due to links or joints that are needed to describe physical systems: for example, two shells can be linked together using screws; locally if screws are locked not enough, friction occurs. Clearly, even if the behaviour of each shell is linear, this joint may be highly nonlinear; or a beam clamped at one extremity can impact on an obstacle at the other one. In some sense the beam is linked to an external system consisting of the obstacle.

2.2 Smooth and Nonsmooth Nonlinearities

We distinguish between smooth and nonsmooth nonlinearities.