Geometric Nonlinear Effect on Biaxial Bending Strength of Thin Silicon Die in the Ball-on-Ring Test

Abstract

Abstract The ball-on-ring (BoR) test, one of the most popular biaxial bending tests, is thoroughly investigated in this study for determining the bending strength of thin silicon dies. The application of this test method with a linear theory to the thin dies is also reevaluated using a nonlinear finite element method (NFEM) by taking into account the geometric nonlinearities, including large-deflection (global) and contact (local) nonlinearities. Mechanics of the BoR test is also discussed in terms of geometric linearity and nonlinearity. It is found that the bending strength calculated by the existing linear theory for the BoR test is still valid for the nonthin die specimens, but not for thin ones. The reason is that the thin-die specimens in the test suffer a contact-nonlinearity effect, due to a maximum applied stress moving away from the loading pin center during the loading process. The global geometric nonlinear (large-deflection) behavior occurring in the three-point bending test is not observed in the test. For applications, the fitting equations of the maximum stress in terms of applied load are proposed based on the NFEM results. Those fitting equations only depend on the specimen thickness, the head radius of the loading pin, and the elastic modulus of the specimen, but not on the specimen radius, a supporting ring radius and the head radius of the ring. The 110 μm and 160 μm-thick silicon dies in the BoR test are also demonstrated with the related fitting equations.