Thermal stress in a bi-material assembly with a 'piecewise-continuous' bonding layer: theorem of three axial forces

We consider a bi-material assembly with a 'piecewise-continuous' bonding layer. The layer is characterized by different elastic constants of its 'pieces' (segments) and is assumed to be thin. Young's moduli of all the 'pieces' of the bonding layer are significantly lower than the moduli of the adherend materials. In such a situation the coefficient of thermal expansion (CTE) of the bonding material need not be accounted for. Only the interfacial compliance of the bonding layer is important. This is indeed the case for the majority of electronic, opto-electronic or photonic assemblies. We consider the situation when the assembly is manufactured at an elevated temperature and is subsequently cooled down to a low (say, room) temperature. The objective of the analysis is to develop a simple, easy-to-use and physically meaningful analytical ('mathematical') predictive model for the evaluation of the interfacial shearing stresses that arise at the boundaries of the 'pieces' (segments) of the bonding layer and at the assembly edge. The basic equation is obtained for the thermally induced forces acting in the adherends' cross-sections that correspond to the boundaries between the dissimilar portions of the bonding layer. This equation has the form of the theorem of three (bending) moments in the theory of multi-span beams lying on separate simple supports and could therefore be called the 'theorem of three axial forces'. We show, as an illustration, how this equation could be employed to design a bi-material assembly with an inhomogeneous bonding layer and with low interfacial shearing stresses. Low shearing stresses will certainly result in lower peeling stresses as well. The numerical example is carried out for an assembly with a relatively high-modulus bonding material in its mid-portion (aimed primarily at providing good adhesion and, if necessary, good heat transfer as well) and a low-modulus material in its peripheral portions (aimed primarily at bringing down the interfacial stresses). The maximum interfacial shearing stress in the assembly with the inhomogeneous bonding layer turned out to be only about 30% of the maximum shearing stress in an assembly with a homogeneous bonding layer having throughout its length the same modulus as the bonding material in the mid-portion of an assembly with an inhomogeneous bond. We would like to emphasize that the inhomogeneous bonding material addressed in this analysis does not necessarily have to be an epoxy one. Many other bonding materials with different moduli in the mid-portion and at the peripheral portions of the assembly could be considered for various practical applications. For instance, a high-modulus solder material such as, say, a tin-silver-copper (SAC) solder can be employed in the mid-portion of an assembly and a low-modulus solder (e.g. an indium-based alloy) at its peripheral portions. In other applications a high-modulus solder material could be considered for the mid-portion of the assembly and a low-modulus epoxy adhesive at its peripheral portions. We would also like to point out that the developed concept can be easily generalized for the situations when the bonding material is not a continuous one, but is of the ball-grid-array (BGA) or a pad-grid-array (PGA) type. Our concept could also be easily generalized for a two-dimensional case, for the situation when bending deformations should be accounted for, for non-uniform distribution of temperature (i.e. for assemblies with temperature gradients in the through-thickness and/or in the longitudinal direction), for situations when time-dependent effects (visco-elasticity, creep, stress relaxation) in the bonding material are important and for numerous other more complicated and practically important situations that might be encountered in engineering practice.