The development of a three-dimensional least-squares ÿnite element technique suitable for deformation analysis was presented. By adopting a spatial viewpoint, a consistent rate formulation that treats deformation as a process was established. The technique utilized the least-squares variational principle that minimizes the squares of errors encountered in any attempt to meet the ÿeld equations exactly. Both velocity and Cauchy stress rate ÿelds were discretized by the same linear interpolation function. The discretization always yields a sparse, symmetric, and positive-deÿnite coe cient matrix. A conjugate gradient iterative solver with incomplete-Choleski preconditioner was used to solve the resulting linear system of equations. Issues such as ÿnite element formulation, mesh design, code e ciency, and time integration were addressed. A set of linear elastic problems was used for patch-test; both homogeneous and non-homogeneous deformations were considered. Additionally, two ÿnite elastic deformation problems were analysed to gauge the overall performance of the technique. The results demonstrated the computational feasibility of a three-dimensional least-squares ÿnite element technique for deformation analysis.
Shear banding is an outstanding problem in lubricant rheology where a thin film of lubricant is under high pressure and high shear stress. To study such a phenomenon, a viscoelastic-plastic model is proposed. It is postulated that shear bands appear when the character of the field equations changes from elliptic to hyperbolic. The model is derived based on a rate formulation which combines a Maxwell fluid model and a compressible rate-independent plasticity model. The model gives the constitutive relation of a compressible viscoelastic-plastic fluid for which the assumption of Stokes condition becomes unnecessary. It also accounts for the elastic coupling effect of both the viscous and the rate-independent behavior of a lubricant. It provides a novel feature that the development of shear bands can be tracked rather than being determined after they are fully developed. Hence, the process of shear banding may be articulated. The analysis focuses on identifying necessary and sufficient conditions which, if achieved in a deformation, would produce a change in character of the governing field equations. Through the use of Drucker-Prager criterion as the component that governs the rate-independent behavior of a lubricant, the analysis gives results that are in accord with experimental observations. Parametric results are given graphically.
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