A control volume unstructured grid approach to the solution of the elastic stress-strain equations
The aim of the research is to develop and apply algorithms for the solution of elastic stress-strain equations based upon a control volume approach on unstructured meshes. The purpose is to integrate the solution of solid mechanics with that of fluid flow and heat transfer within the context of solidification problems in the casting of metals. The conventional way of solving stress-strain problems is to use a finite element approach which yields a set of linear equations relating loads to displacements. This approach works very well and can also deal well with temperature loading, but becomes problematic when flow and change of phase are included in a transient context. A major set of projects are under way to develop an integrated suite of algorithms to model the flow-cooling-solidification-residual stress development process. This project concerns the component associated with the development of stress-strain distributions under temperature and other loads. A control volume formulation solution procedure for the elastic stress-strain equations in two-dimensions has been developed that solves directly for displacements. The formulation works for mixtures of quadrilateral and triangular elements in an unstructured mesh. The control volume finite element code has been tested on a range of problems, such as a cantilever loaded at one end, a beam with a thermal gradient applied and a multi-material mixed element non-regular shape with a load applied. The results for these test cases have been compared to those obtained by standard finite element codes and analytical solutions where available. Besides the plane stress and plane strain options, the model has been extended to include axisymmetric problems. Two examples are used to test the validity of the algorithm for axisymmetric problems. The results compare very well against other numerical results and analytic solutions for the three special two-dimensional cases. Another problem considered at an inteiface is that of friction between two solids. This non-linear boundary condition has been included in the model. An example of this is Silica in a mould being pressed, results for the stress-strain code are compared to previous results. The control volume stress-strain code has been integrated with the solidification heat transfer code. Problems of simple castings have been considered. When a liquid solidifies it may deform away from the mould, so possible air gap formation at the mould/metal inteiface has been included in the model. Prediction of hole formation in solidification, in the form of volumetric shrinkage and porosity, has been included into the coupled heat transfer stress code via a simple model. Examples show encouraging results when compared to experimental porosity results.