Probabilistic Palmgren–Miner rule, with application to solder materials experiencing elastic deformations

Abstract

It has been recently shown that there are effective ways not only to reduce the interfacial stresses in electronic packaging assemblies with solder joint arrays as the second level of interconnections, but to do that to an extent that inelastic strains in the peripheral joints, where the induced thermal stresses and strains are the highest, are avoided. While various and numerous modifications of the empirical Coffin–Manson relationship are used to predict the fatigue life of solder materials experiencing inelastic strains and operated in low cycle fatigue conditions, the Palmgren–Miner rule of the linear accumulation of fatigue damages, although suggested many decades ago, is still viewed by many material scientists and reliability physicists as a suitable model that enables one to quantify the cumulative fatigue damage in metals experiencing elastic strains. In this analysis the Palmgren–Miner rule is extended for the case of random loading, and a simple formalism is suggested for the evaluation of the remaining useful lifetime for a solder material subjected to random loading and experiencing elastic thermally induced shearing deformations. Special highly focused and highly cost effective accelerated tests have to be conducted, of course, to establish the S–N curve for the given solder material. In the future work we intend to extend the suggested methodology to take into account various aspects of the physics-of-failure: the role of the growth kinetics of intermetallic compound layers; the random number, size and orientation of grains in the joints; position of the joint with respect to the mid-cross-section of the assembly (peripheral joints are more prone to elevated interfacial stresses); assembly size, etc. All this effort, important as it is, is, however, beyond the scope of this analysis, which is aimed at the extension of the classical Palmgren–Miner rule for the case of random loading.