FEM simulation of the size- and constraining effect in lead- free solder joints with the theory of strain gradient elasticity

Reliability studies of electronic parts prevalently involve FEM simulations of solder joints under service conditions. However, simulations performed with classical continuum mechanics lead to strain singularities at the surface of material transitions. In consequence, the desired independence of the results from mesh size can usually not be achieved. Therefore, we propose a novel version of strain gradient elasticity which consequently removes strain singularities. Our approach shows similarities to a strain gradient theory which was developed already in the 1960s. But in our version of the theory it is required that the stress tensor of equilibrium states is always symmetric. This approach is implemented in the commercial FEM code ABAQUS through user subroutine UEL. Thus, it is demonstrated that in the new approach mesh convergence is achieved. Furthermore, simulations for solder joints of different sizes predict a mechanical size effect in the sense "smaller is stronger".