Eulerian volume of solid (VOS) approach in solid mechanics and metal forming

The commonly known volume of fluids (VOF) method in fluid mechanics applications is extended to applications in solid mechanics. For this extension, we propose the name volume of solid (VOS) method in solid mechanics. The derivation of the Eulerian finite element equation is highlighted for quasi-static analysis. The VOF method is adapted to solid mechanics to track the free solid surface in large strain metal forming problems. The theoretical development of the VOS method includes the calculation of the fractional volume value of the solid in each element and defining a free surface directional vector to find the location and shape of the free surface. The method is presented for uniform and non-uniform Cartesian grids. All issues related to the use of Eulerian finite element (FE) formulation and VOF method such as the connectivity of the free surface, tracking material point properties and updating the values for newly added nodes during the analysis, are discussed in this paper. The implementation of the VOS method in metal forming applications is presented using two processes; compressions between wedge-shaped dies and backward extrusion.