Nonlinear Continuum Models

Modeling of continuous structures is addressed. First, a metamodel is introduced, in which linear and nonlinear operators appear, of generic nature (algebraic, differential and/or integral) describing the form of the equations governing the motion of a large class of systems, which are of interest to the analysis to be developed ahead. The metamodel accounts for inertial, viscous damping (external and internal), elastic and external forces. It is successively specialized to a number of basic structural elements, as strings, beams, membranes and plates, for each of which a minimal model is built-up, amenable to an analytical treatment, and able to capture the main qualitative mechanical behavior. One-dimensional models of strings, cables and beams, are expressed by one field equation in the transverse displacement only, after the longitudinal displacement has been condensed under simplifying hypotheses. In contrast, two-dimensional models of membranes and plates, since condensation is no longer viable, are governed by coupled field equations in the out-of-plane and in-plane displacements.