Stability of discretized nonlinear elastic systems

Abstract : The goal of these notes is to 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 parameters. The core of the exposition focuses on the illustration of energetic methods where the equilibrium and stability are found by studying the point of station-arity and minima of a scalar function of the state parameters. After three introductory sections presenting the links between stability and energy minimisations (Section 2), potential energy (Section 3) and discretisation methods (Section 4), we detail the mathematical methods required to minimise 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. Hence, we 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 coefficients. 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.
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Arnaud Lazarus, Corrado Maurini, Sébastien Neukirch. Stability of discretized nonlinear elastic systems. Extremely Deformable Structures, 562, Springer-Verlag Wien, 2015, CISM International Centre for Mechanical Sciences, 978-3-7091-1876-4. ⟨10.1007/978-3-7091-1877-1⟩. ⟨hal-01447329⟩

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