Mathematical Assessment of the Effects of Parabolic and Spherical Surface Topographies on the Interfacial State of Stress

Abstract

A mathematical procedure to utilize the complementary energy method was developed, by minimization, in order to find an approximate analytical solution to the 3-D stress distributions in bonded interfaces of dissimilar materials. In order to incorporate the effects of surface topography, the interface was expressed as a general surface in Cartesian coordinates, i.e., F (x, y, z) = 0. The 3-D stress functions were used to produce 3-D stress components in dissimilar materials. At the interface, the internal tractions in each of the coordinate directions were balanced by the mathematical procedure. By using a penalty function method of the optimization theory, integration of the complementary energy produced the necessary equations to solve the 3-D stress distribution problem at the interface. A noticeable advantage of our method is that the stress jumps at the interface predicted in elasticity theory are captured, while standard finite element analysis (FEA) methods usually have difficulty to show such stress jumps at interfaces. The 3-D mathematical procedure we developed for obtaining the stress components at bonded bi-material interfaces offers significant promise in solving interface problems with different surface topographies, which can be described mathematically. Thus, the procedure developed provides an efficient tool to optimize interface construction by various methods such as chemical (etching), mechanical (machining, roughening, etc.), and other novel methods, such as laser ablation, currently becoming available to achieve desired interface stress distributions for bonded materials. In this paper, the parabolic interface problem, i.e., y = x², and the spherical interface problem, i.e., y = x²/4 + z²/4, are considered for an aluminum/epoxy interface.

Keywords:

• Adhesive joints
• Bi-material interfaces
• Bonding surfaces
• Interface topographies
• Interfacial stresses
• Mechanical adhesion
• Parabolic interfaces
• Peel stresses
• Shear stresses
• Spherical interfaces
• Stress jump
• Surface roughening

Notes

Presented in part at the 2nd International Conference on Advanced Computational Engineering and Experimenting (ACE-X 2008), Barcelona, Spain, 14–15 July, 2008.