| The development of a numerical simulation for steady-state, elastic, viscous flows in two dimensions is presented. A mixed finite element method is used to couple the rate-equilibrium and the rate-constitutive equations by using successive substitution to solve for the velocity field and the stress field simultaneously. The method is applied to the flow analysis of co-rotational Maxwell (CRM), upper convective Maxwell (UCM), and four-element UCM fluid models.; A flow through contraction problem is analyzed for the CRM, UCM and four-element UCM models. For both the CRM and UCM models, the purely elastic case is compared to a solution found by using a linear formulation and it is found to compare favorably. For the purely viscous case, comparison is made with results obtained using the mixed formulation for velocity and pressure. Again, the results compare quite favorably. For the four-element UCM model, it is compared with the conventional UCM model. For the purely elastic and the purely viscous cases both models compare very well, and two examples are given for simulating both the Oldroyd-B and the Kelvin-Voigt models. From this illustration, the four-element UCM model is shown to work well for a wide range of constitutive behaviors.; A second example presents an analysis of a metal forming rolling problem in the presence of free surfaces. For the purely elastic case, a decent solution of the velocity and stress distributions in the control volume for both CRM and UCM fluids is found. However, when the viscous effect is increased in the material, the free surface exhibits a continual swelling on the downstream side and the accuracy of the stress distribution deteriorates. In spite of the progress made on the free surface problem, the solutions for free surface problem presented in this dissertation are not yet of sufficient accuracy to be directly applicable to practical forming process design or analysis.; The final example is an application of the rate-equilibrium formulation for elastic, nonlinear-viscous flows. It is shown that the method converges to reasonable solutions and hence the procedure has possible applications to metal forming processes. |
