(Dr. David McDowell, advisor)

"Development of Three-Dimensional Least-Squares Finite Element Technique for Inelastic Deformation Analysis"

Abstract

In this research, a three-dimensional mixed finite element technique to analyze inelastic deformation of porous metallic solids is presented. The finite element technique is based on a variational principle of the least--squares type. There are several unique features to the least-squares technique which sets it apart from the conventional finite element method based on the Galerkin-displacement and other mixed formulations. One feature is that the technique always yields a symmetric and positive-definite system of linear equations, regardless of level of work-hardening. Second, the technique is not sensitive to the character of the governing equations, thus the same problem-solving procedure can be used for problems governed by mixed elliptic-hyperbolic differential equations such as problems involving shear banding. Another feature is that the mixed least-squares technique circumvents the LBB condition imposed on saddle-point formulations such as the Galerkin mixed method. This will simplify the finite element implementation by allowing equal-order finite element spaces to be used for all dependent variables.

In the present formulation, deformation is viewed as a continuous process and an incremental approach is adopted for the analysis. The three velocity and six Cauchy stress rate components constitute the set of nodal dependent variables. Linear interpolation functions are used for their finite element approximation. Due to the nature of three-dimensional modeling and the nine degrees of freedom per node, the resulting system coefficient matrix is extremely sparse. A sparse data storage scheme is implemented to minimize computer memory. Since the least-squares technique always generates a symmetric and positive-definite system of linear equations, an efficient and robust preconditioned conjugate gradient iterative solver is chosen to solve the linear system. The deformation and stress histories are determined by simple explicit time integration schemes.

To explore the basic performance of the least-squares technique and to verify the computer program, a comprehensive set of benchmark problems were treated. They included patch-test and simple linear elastic test cases. The Saint Venant's torsion of a prismatic bar was used to study the error distribution and convergence rates. Results for two finite elastic problems involving uniaxial and rectilinear simple-shear were in good agreement with solutions from analytic or other numerical methods. Several elasto-plastic test problems were run to gauge the program's ability to satisfy the plastic consistency condition. To further demonstrate the utility of the computer program, two common metal-forming processes were investigated. The first study involved the necking behavior of a flat titanium alloy tensile specimen. Necking was initiated by a set of asymmetric boundary conditions. The results showed signs of plastic strain localizing along an oblique line inclined to the loading axis. It was observed that the relative density decreases while the stress triaxiality increases rapidly at the neck region. The second study involved an open-die upset forging of a porous aluminum rectangular block. The growth of the plastic zone was predicted accurately. The forging load required to produce a given height reduction and the spread of the block at the equatorial planes compared favorably with published experimental results.