#### Date of Graduation

1994

#### Document Type

Dissertation/Thesis

#### Abstract

A nonlinear 'velocity-pressure' mixed finite element formulation for three dimensional rigid-viscoplastic analysis of metal forming problems is presented. It adequately takes into account the nonlinear material behavior and the large deformations that occur during metal forming operations. A ten-noded tetrahedral finite element with a quadratic interpolation in velocity and linear interpolation in pressure is used to discretize the domain of the deforming body. The salient feature of the formulation is the enforcement of the incompressibility requirement necessitated by the use of the rigid-viscoplastic constitutive law by modifying the constitutive equation to include a pressure variable and adding an additional constraint equation to the equations of equilibrium. A Newton-Raphson procedure is adopted for iterative incremental solution of the nonlinear stiffness equations and the details of the consistent linearization procedure for obtaining a tangent stiffness matrix are provided. The Newton-Raphson algorithm requires the repeated solution of a set of linear equations with a symmetric, non-singular co-efficient matrix. A methodology is presented to render the coefficient matrix positive definite in order to use a Preconditioned Conjugate Gradients (PCG) algorithm to iteratively solve the resulting system. The algorithm is implemented utilizing an Element-By-Element (EBE) strategy so that the formation and factorization of a global stiffness matrix is avoided. The details of formulating a suitable preconditioner which approximates the global stiffness matrix are presented. Finally, the finite element formulation is applied to several benchmark problems and the solutions of the problems are presented to indicate the potential of using tetrahedral finite elements for metal forming analysis.

#### Recommended Citation

Rachakonda, V B., "Development of a stable tetrahedral finite element for metal forming analysis." (1994). *Graduate Theses, Dissertations, and Problem Reports*. 9614.

https://researchrepository.wvu.edu/etd/9614