Numerical Example
Input data
Consider a trimaterial body comprised of 0.125 mm dia glass fiber (£0 = 7400 kg/mm2, v0 = 0.2, a„ = 0.5 x 10^1/°С, r0 = 0.0625 mm) as the zero (“core”) component; 2 mil = 0.0508 mm thick ring of the silver-tin solder eutectic (melting point TmeU = 221°C, £| =650 kg/mm2, v, =0.35, a, = 29.0x 10'61/°C, r, =0.1133 mm) as the intermediate,” bonding,” component #1; and the silicon “enclosure”
(£, = 12300 kg/mm2, v2 = 0.27, a2 = 2.6x 10'61/°C) as the outer (#2) material. The “enclosure” is considered thick, so that its axial compliance X2 can be put equal to zero.
Calculated data
Because both the interfacial radial stresses are positive, the solder ring is in tension, and the radial stress at its interface with silicon is about 14% higher than the radial stress at its interface with the glass. One has to make sure that these stresses are still low enough in order not to cause excessive inelastic deformations in the solder and/or delaminations at its interface.
The radial stress in the solder at its inner interface is
the tangential stress in the solder at its interface with the fiber is
which is about 40% higher than the radial stress; the radial and tangential stresses in the solder at its interfaces with silicon are
respectively (the latter stress is by about 11% higher than the radial stress); the radial and the tangential stresses in the silicon at its interface with the solder ring are
respectively.
Thus, the highest lateral normal stresses are the tangential (circumferential) stresses at the solder-fiber interface.
The computed axial forces and he normal stresses are:
(their sum is certainly zero) and
Thus, while the axial normal stresses in the glass are low, and the axial stress in the silicon is only about 4% of the stress in the glass, the axial stress in the solder is significant. It is of the same order of magnitude as the lateral stresses are.
In order to evaluate the interfacial shearing stresses one has to compute first the longitudinal interfacial compliances. These are:
The partial parameters of the interfacial shearing stresses areh
and the computed parameter of the interfacial stress for the entire body is к = 24.6085 mm'1. Assuming that the body is long enough, so that the product kl exceeds considerably the kl = 3 value (indeed, with such a significant к value, a body
of the length 21 = — = 0.244 mm is long enough to be considered long), the com-
К
puted maximum shearing stresses are
These stresses are substantially lower than the lateral or the axial normal stresses, but, still on the same order of magnitude, and should be considered in the stress-strain and reliability analyses of the bodies of the type in question.
Conclusion
Thermal stresses in a bonded elongated elastic cylindrical trimaterial body of finite length are predicted based on the developed simple, easy-to-use, and physically meaningful analytical (mathematical) model. The numerical example is carried out in application to structures, in which a glass fiber is soldered into a silicon chip (silicon photonics technology) using a “soft” silver-tin solder. It is concluded that the appropriate solder material and its thickness should be selected based on the calculations using the developed model.
