Stability of discretized nonlinear elastic systems

These notes give a short introduction to the methods for the study of stability of elastic structures. We consider only the finite-dimensional case, where the state of the system is represented by a discrete set of variables. The core of the exposition focuses on the illustration of energetic methods where equilibrium and stability are found by studying the point of stationarity and minima of a scalar function of the state variables. After three introductory sections presenting the links between stability and energy minimization (Section 2), potential energy (Section 3) and discretization methods (Section 4), we detail the mathematical methods required to minimize a function of

n

variables (Section 5-8). We include the theory and recipes to deal with equality and inequality constraints, providing several examples of applications to simple structures. We then show how to classify regular and singular points (bifurcations) in force-displacement diagrams (Section 9) and give a fully worked example with several degrees of freedom (Section 10). Section 11 presents, through an example, the dynamical theory of stability including Floquet theory for systems with periodic solutions. Finally, Section 12 shows how energetic methods can be applied to the study of material instabilities, by considering the case of springs with irreversible damage.