Numerical Evaluation of Finite Element Methods for Rubber Parts

Abstract

<div class="htmlview paragraph">A finite element method of the perturbed mixed type for rubber elasticity applications is presented. The method is novel in the sense that the perturbation parameter appearing in this method is not artificial but can be identified with the measurable bulk modulus of a nearly incompressible material. The usage of elements with local pressure variables makes the method equivalent to a penalty method of the projection penalty type.</div> <div class="htmlview paragraph">An Updated Lagrangean formulation of the method is presented.</div> <div class="htmlview paragraph">A numerical investigation including convergence and stability tests on the computed pressure for a choice of two-dimensional isoparametric quadrilateral elements indicates good agreement with previously published data and theoretical predictions.</div> <div class="htmlview paragraph">The application of the method to some simple engineering problems is demonstrated.</div> <div class="htmlview paragraph">A perturbed Lagrangean multiplier method of the Key [<span class="xref">7</span>] type is derived for isotropic nearly incompressible hyperelastic solids. The presented generalization of the Key principle to rubber elasticity is novel in the sense that the perturbation parameter can be identified with the measurable bulk modulus of the nearly incompressible material. The frequently used assumption that the (mean) pressure is only a function of the volumetric strain is adopted here.</div> <div class="htmlview paragraph">With this method as a basis and taking the mean pressure to be local to each finite element a penalty method of the projection type [<span class="xref">3</span>] is derived.</div> <div class="htmlview paragraph">These results are obtained using the Updated Lagrangean formalism. The use of the Updated Lagrangean formulation in rubber elasticity applications is considered as novel.</div> <div class="htmlview paragraph">The numerical evaluation of the methods put forward indicates good agreement with theoretical predictions regarding the convergence of the computed pressure and regarding the stability (in the Brezzi sense [<span class="xref">13</span>]) of the different two-dimensional element types investigated. Relevant numerical results obtained by others are confirmed.</div> <div class="htmlview paragraph">The experience with engineering problems is still limited. The results obtained so far are encouraging. These indicate that it is important that the element used is insensitive to distortions due to large deformations of the mesh. The simple 4-node constant pressure element seems to meet the requirement of being insensitive to severe distorslons, despite its bad stability properties in some cases (in the Brezzi sense).</div>