Treating inextensibility constraints in hyperelastic materials with inequality level sets

Abstract

Elastomers are viscoelastic polymers with low Young's modulus and high failure strain that are used in many c ivil e ngineering applications, including bridge bearings, seismic isolators for buil dings and resilient rail wheels. Their constitutive behaviour i s characterized by a nonlinear stress - strain relation with an extensibility limit . This contrasts with materials that have instead a limit on the tensile stresses, such as mild steel. This MSc thesis is concerned with the numerical modeling of elastomers. T his involves dealing with a medium with two phases: a constrained region, wher e the particles have reached their maxi mum allowable deformation, and a free region, where the inextensibility constraint is still inactive. Moreover, one can think of an interf ace splitting the two phases of the medium. If the focus is put in obtaining methods to locate and evolve such interface, then a two phase medium with a moving interface is considered. From the mathematical point of view, this is a constrained minimization problem. One of the strategies to solve it is to turn the minimization problem into a shape equilibrium one. This approach has b een successfully employed for an interface location problem in small strains and serves as the starting point of this work. T hu s, t he main purpose of this thesis is to extend this formulation to a large strains interface locating and evolving scenario. A first analysis of the problem reveals two sources of nonlinearity : the inextensibility constraint and the kinematics in large st rains. A simple but thorough one - dimensional study of the problem is then developed to find methods to sort out both nonlinearities. Following this, explicit iterative schemes to locate and evolve one or multiple interfaces are straightforwardly obtained i n 1D linear elasticity. However, the same ideas applied to a simple St.Venant - Kirchhoff hyperelasticity model , evidences that even very simple 1D problems become rather complex and cannot be solved as directly and explicit as before. Numerical examples are provided throughout this analysis and they are also useful to conclude that both locating and evolving the interface can be essentially seen as the same problem, but with different driving effects. After that, an extension of the one - dimensional schemes t o two or more dimensions is explored . Although the same ideas can be applied, more sophisticated modeling tools are required, namely, the X - FEM and Level set methods, the shape sensitivity analysis and the Arbitrary Lagrangian - Eulerian methods . A complemen tary numerical implementation of the pr oposed strategy is to show its computational benefits. In particular, a combination of the three previous techniques shall make unnecessary a stepwise update of the Level set. T he work presented here may not be limite d to this particular case and be r elevant to other engineering problems involving moving interfaces and boundaries , such as plasticity analysis or the saturation of a porous medium.