Nonlinear behavior and instabilities of a hyperelastic von Mises truss

Abstract

Recent decades have witnessed a renewed interest in the field of structural stability due to new applications involving smart and deployable structures, micro- and nanocomponents and mechanical metamaterials, among others. In many of these structures multistable behavior is desirable, which can be accomplished by traditional and new materials capable of undergoing large elastic deformations. In this paper the nonlinear behavior, bifurcations and instabilities of a hyperelastic von Mises truss exhibiting multistable behavior is investigated. Most papers dealing with the von Mises truss are restricted to linear elastic materials. Here, the nonlinear equilibrium equations are derived considering elasticity in the fully non-linear range and the incompressible Mooney–Rivlin constitutive law is adopted to model the hyperelastic material. The nonlinear equations are solved by using the Newton–Raphson method and continuation techniques. Then, all equilibrium paths and bifurcation points are obtained and their stability is investigated using the energy criterion. A detailed parametric analysis of shallow and nonshallow trusses under horizontal and vertical loads is conducted. Load and geometric imperfections are considered and their influence on the bifurcation scenario and the truss load carrying capacity is clarified. The influence of the material parameters on the nonlinear response is also examined. The results show that the simultaneous presence of geometric and material nonlinearities leads to several equilibrium paths, some of which are not expected for linear elastic materials or found in the existing literature on nonlinear materials, resulting in several coexisting stable and unstable solutions and a complex potential energy landscape, thus clarifying the influence of the constitutive hyperelastic model on the results. Analytical expressions for the normalized snap-through and pitchfork bifurcation loads are derived as a function of the material parameters, truss geometry and imperfections for practical applications. The influence of Eulerian buckling on the truss load carrying capacity is also investigated and formulas to evaluate the buckling load under both vertical and horizontal loads are derived. The present results may help in the development of new engineering applications where multistability and large deformations are desired.