A Conjectured Solder Fatigue Law Based on Nonlinear Dynamics

Abstract

A discipline of nonlinear dynamics (Chaos Theory), not traditionally used in the study of solder, is applied here in developing a conjectured general form for fatigue based on creep. Solder joint data are analyzed which demonstrates two strain-rate regimes. In the literature, this is recognized as related to the two modes of grain boundary and matrix creep. In this paper, the behavior is treated as a “bifurcation” and a quadratic normal form identified which addresses qualitative features of the observed data. The resulting mathematical model is transformed into a quadratic map (difference equation) which is a classic paradigm for chaotic motion. This model addresses not only the two distinct rate regimes but also an unstable intermediate range of erratic motion which is observed in the data. Correlation between solder fatigue behavior and onset of model instability regarding effects of solder dwell (model relaxation) and solder grain size (assumed to be associated with model time increment) has been discussed in previous work. This has motivated examination of the classic Coffin-Manson (C.M.) low cycle fatigue law for elastic, plastic deformation in terms of appropriate quadratic maps. The result is a “curve-fit” representation of the C.M. forms in terms of map parameters. In particular, a simple relationship appears to exist between the “universal slopes” of the C.M. law and the “universal” Feigenbaum constant δ = 4.66920..., so called because it is associated with a wide class of maps which includes the quadratic as a special case. With this approach, a conjectured form of solder fatigue law under creep deformation is generated. General properties of the resulting form are discussed and compared with some of the fatigue models currently in use.