A numerical study of finite element calculations for incompressible materials under applied boundary displacements

Abstract

In this thesis, numerical experiments are performed to test the numerical stability of the finite element method for analyzing incompressible materials from boundary displacements. The significance of the study relies on the fact that incompressibility, or density preservation during deformation, is an important property of materials such as rubber and soft tissue.It is well known that the finite element analysis (FEA) of incompressible materials is less straightforward than for materials which are compressible. The FEA of incompressible materials using the usual displacement based finite element method results in an unstable solution for the stress field. Hence, a different formulation called the mixed u-p formulation (u– displacement, p – pressure) is used for the analysis. The u-p formulation results in a stable solution but only when the forces and/or stress tractions acting on the structure are known. There are, however, certain situations in the real world where the forces or stress tractions acting on the structure are unknown, but the deformation (i.e. displacements) due to the forces can be measured. One example is the stress analysis of soft tissues. High resolution images of initial and deformed states of a tissue can be used to obtain the displacements along the boundary. In such cases, the only inputs to the finite element method are the structural geometry, material properties, and boundary displacements. When finite element analysis of incompressible materials with displacement boundary conditions is performed, even the mixed u-p formulation results in highly unstable calculations of the stress field. Here, a hypothesis for solving this problem is developed and tested. Theories of linear and nonlinear stress analysis are reviewed to demonstrate that it may be possible to determine the von Mises stress uniquely in spite of the numerical instability inherent in the calculations.To validate this concept, four different numerical examples representing different deformation processes are considered using ANSYS®: a plate in simple shear; expansion of a thick-walled cylinder; a plate in uniform strain; and Cook’s membrane. Numerical results show that, unlike the normal stress components Sx, Sy, and Sz, the calculated values of the von Mises stress are reasonably accurate if measurement errors in the displacement data are small. As the measurement error increases, the error in the von Mises stress increases approximately linearly for linear problems, but can become unacceptably large in nonlinear cases, to the point where solution process encounter fatal errors. A quasi-Dirichlet patch test in association with this problem is also introduced.