This paper presents a class of arc-length methods for the quasi-static analysis of problems involving material and geometric nonlinearities. A constraint equation accounting for geometric and dissipative requirements is adopted: the geometric part refers to the Riks and Crisfield equations, while the dissipative one refers to the dissipated energy. The approach allows for a continuous variation of the nature of the constraint, and a switch criterion is not needed to trace the elastic and the dissipative parts of the equilibrium paths. To illustrate the robustness and the efficiency of the methods, three examples involving geometric and material nonlinearities are discussed
Discussed is the implementation of a continuation technique for the analysis of nonlinear structural problems, which is capable of accounting for geometric and dissipative requirements. The strategy can be applied for solving quasi-static problems, where nonlinearities can be due to geometric or material response. The main advantage of the proposed approach relies in its robustness, which can be exploited for tracing the equilibrium paths for problems characterized by complex responses involving the onset and propagation of cracks. A set of examples is presented and discussed. For problems involving combined material and geometric nonlinearties, the results illustrate the advantages of the proposed hybrid continuation technique in terms of efficiency and robustness. Specifically, less iterations are usually required with respect to similar procedures based on purely geometric constraints. Furthermore, bifurcation plots can be easily traced, furnishing the analyst a powerful tool for investigating the nonlinear response of the structure at hand.
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