C. Basaran Assistant Professor and Director, Electronic Packaging Lab, 212 Ketter Hall, SUNY at Buffalo, Buffalo, NY 14260 cjb@eng.buffalo.edu C.-Y. Yan Staff Engineer, Pico Design Inc., Detroit, MI A Thermodynamic Framework for Damage Mechanics of Solder Joints Damage mechanics describes the degradation process that takes place in materials and structures. Traditionally, Coffin-Manson type empirical curves are used to deter- mine the fatigue life. Damage mechanics allows us to determine the fatigue life without the need for empirical curves. The main problem in damage mechanics has always been a lack of universally agreed upon definition of a damage metric. In this paper a damage metric based on the second law of thermodynamics and statistical mechanics is presented. The proposed thermodynamic framework treats a solid body as a thermodynamic system and requires that the entropy production be nonnegative. Verification of the damage model has been performed by extensive comparisons with laboratory test data of low cycle fatigue of Pb40/Sn60 solder alloy. Introduction In his pioneering paper, Kachanov (1958) introduced the field variable ~b, called continuity, which is considered by many as the starting point of continuum damage mechanics (CDM). Over the years D = ( 1 - ~,) has become accepted as a thermo- dynamical internal state variable. D = 0 identifies the virgin state, and D = 1 is used to identify the total failure. In reality, both extremes, D = 0 and D = 1, are asymptotic states that can never be attained physically. Most materials and structures have initial microcracks and flaws from the manufacturing pro- cess, so the initial value of D is not zero. Moreover, most structures collapse long before D reaches a value of 1. Finding a universal metric to quantify D has always been the biggest challenge in CDM. Lemaitre (1996) has presented an excellent physical explana- tion of damage mechanics at the most basic level. To briefly summarize his definition of damage, at the atomic level cohesive forces that hold the materials together are due to the interaction of electronic fields. The onset of damage process and plastic microstrains is the result of debonding at the atomic level. In metals regular arrays of atoms compose the structure except where there are dislocations on many lines where atoms are missing. Under external perturbations, the dislocations may move by the displacement of bonds thus creating a plastic strain by slip without any debonding. In the case where the disloca- tions can not propagate due to a microdefect, a constrained zone is created in which another dislocation may be stopped. This second process cannot occur without debonding. Several arrests of dislocations nucleate a microcrack. Other damage mechanisms in metals are intergranular de- bonding and decohesion between inclusions and the matrix. Essentially, damage occurs as a result of debonding process starting from the atomic level to the onset of microcracking at the mesoscale. The purpose of this ongoing project is to develop a thermody- namic framework to model the evolution of the damage mecha- nism and propose a metric to quantify the damage. It is assumed that the reorganization that takes place in the microstructural configuration of a material during the damage process is, in general, irreversible. There are different schools of thought about what happens to tension microcracks during Contributed by the Electrical and Electronic Packaging Division for publication in the JOURNALOF ELECTRONICPACKAGING.Manuscript received by the EEPD March 3, 1998; revision received July 6, 1998. Associate Technical Editor: S. M. Heinrich. a following compression cycle. Discussion of this current unre- solved debate is outside the scope of this paper. During the self-organization process, the entropy that is a measure of disorder in the system must increase according to the second law of thermodynamics. Direct measurement of ani- sotropic damage has not been easy to accomplish up to now. Usually, damage-related parameters have been used to quantify accumulative damage in the laboratory. Different metrics have been proposed in the literature to quantify damage in a material or a structure such as the plastic strain or the maximum deflec- tion. In the laboratory, damage can be measured by total crack area, degradation of elasticity module, variation in ultrasonic wave propagation speed, variation of the microhardness, change in electrical conductivity, thermal resistance ratio, variation of density, cyclic response ultimate stress amplitude drop (a.k.a., load drop), acoustic properties, tertiary creep response and oth- ers. Most of these measurement methods are discussed by Le- maitre (1996) in great detail. On the other hand, in numerical analysis procedures, such as the finite element method, irrevers- ible deformations or the energy dissipated in the system have been the most commonly used damage metrics in the literature. Kachanov ( 1958, 1986), Rabotnov, (1969), Valanis ( 1971, 1997), Chaboche (1988), Murakami (1988), Onat, and Leckie (1988), Krajcinovic (1989), Ju (1990), Bazant (1991), Chow and Chen(1992), Muhlhaus et al. (1994), Voyiadjis and Thiya- garajan (1996), Lemaitre (1996), and others have presented damage mechanics in a thermodynamic framework using the plastic strain, the plastic strain rate, or the energy as the primary damage metric. Using the plastic strain as a damage criterion may, under special loading conditions, lead to reasonable qualitative esti- mates, Ozmat (1990). But many researches, Dasgupta et al. (1994) and others, have shown that the maximum plastic strain and damage (microcracks) can localize at different locations in the material. Many researchers have also shown that plastic strain is not a reliable damage metric and different stress paths to the same final state of stress yield different plastic strain values. Solomon and Tolksdorf (1996) and others have shown that using dissipated energy alone does not lead to a unique damage value. Loading and strain rates significantly vary the energy dissipated in the system. The proposed model uses entropy as a damage metric. The first question comes to mind is that entropy is not a primitive quantity. Valanis (1971) used the concept of thermodynamic internal variables to render entropy a state function. Valanis Journal of Electronic Packaging Copyright © 1998 by ASME DECEMBER 1998, Vol. 120 / 379 Downloaded 25 Apr 2008 to 128.205.19.117. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm