Mixed finite element analysis with application to spot welding

Abstract

A mixed finite element method is introduced in this thesis by two or three first-order C0 stress functions for plane or axisymmetric problems respectively, which satisfy the force equilibrium equations, along with a constraint to impose the moment equilibrium equations. The stresses so expressed are equivalent to those in terms of the higher order Airy or Love stress function. With compatibility condition satisfied in the same way as in a displacement finite element (FE) method, the remaining constitutive relation in elasticity, i.e. Hooke's law, is satisfied by minimizing a mixed functional, with variables of the displacement vector and two/three first-order stress functions. Some elementary problems in plane and axisymmetric elasticity are solved by this method. It is found that for an incompressible solid and a solid with a crack, the mixed model yields better results than the conventional FE method. The effects of Gaussian integration and Poisson's ratio on the solution are discussed in detail. Special attention is paid in bending a beam and a disc, where the importance of the constraint to enforce moment equilibrium is studied. For rigid-perfect-plasticity, the Levy-Mises flow rule and the corresponding yield condition are satisfied by another extremum principle. By substituting the plastic part of the elasto-plastic strain into the extremum for rigid plasticity, and the elastic part of the elasto-plastic strain into the extremum for elasticity, an extremum principle for elasto-plasticity is established straightaway. Applications of this method to some wellknown examples are discussed. In comparison with the conventional displacement method and/or analytical solution, this method offers very satisfactory results and good convergence of the solution. An interesting feature of this method is that the value of each functional indicates in some degree the solution error at a giving point or region. This may provide useful information for accuracy control or a remeshing procedure. A more sophisticated problem is solved by a so-called mixed fluid-FE model, which is the simulation of the flow of an adhesive between two aluminium sheets squeezed by a pair of electrodes in spot-process. The effects of various factors on the formation of the entrapment of the adhesive in the central area of faying surface are studied in detail. Very close results between displacement method and the mixed method are obtained in this study.