Temperature-dependent variability in lifetime prediction of thermally activated systems
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2/4/04
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Temperature-Dependent Variability in Lifetime Prediction of Thermally Activated Systems R. RAJ, J.S. KONG, D.M. FRANGOPOL, and I.E. RAJ The lifetime of high-temperature systems is often controlled by thermally activated mechanisms such as diffusion. The variability in the lifetime of such systems is analyzed when the operating temperature varies according to a normal (Gaussian) distribution. Linear approximation analysis is employed to obtain simple closed form results for the probability density function (pdf) for the lifetime. The Gaussian variation in temperature is shown to transform to a log-normal distribution for the lifetime. The standard deviation of the log-normal distribution can be predicted from the activation energy, the peak temperature, and the standard deviation of the temperature distribution. Higher activation energy and lower operating temperature increase the variability of the lifetime. This approximate result is compared with the exact transformation. Lifetime experiments with incandescent tungsten lamps are compared to the theoretical prediction.
I. INTRODUCTION
THE influence of temperature on the lifetime of hightemperature systems can, nearly always, be traced to a thermally activated atomistic process. The reason is that most mechanisms that control the rate of “damage” accumulation at high temperatures are dependent upon thermally activated transport processes such as diffusion. An athermal process such as fast fracture or shear localization may dominate the last few moments of failure, but the vast majority of time is spent in slow accumulation of damage such as creep, void growth, and slow crack growth. The study of specific mechanisms of fracture and their regime of dominance (in the field of stress and temperature) has been the hallmark of fundamental materials science research for many years. Referring to Figure 1, these studies have concentrated on the “A” branch of the flow diagram. Typically, the study occurs in three steps: first, a distinct phenomenon is identified through experiments and observation; next, a mechanistic model is developed that links the measurement to the microstructure and the fundamental material parameters; and finally, controlled experiments are carried out to validate the model. For example, consider the phenomenon of grain-boundary sliding, which was described in articles as early as 1963.[1] Several models were developed to describe the phenomena,[2,3] but a clear confirmation of the model was obtained only in one case, where the sliding is controlled by diffusional accommodation,[4,5] which is described by the following equation: 8 ss # lDV pdDB # u p kT h2
[1]
# where u is the sliding rate, s is the applied shear stress across the grain boundary, is the atomic volume, is the R. RAJ, Professor, and I.E. RAJ, Undergraduate Intern, Department of Mechanical Engineering, and D.M. FRANGOPOL, Professor, Department of Civil Architectural and Environmental Engineering, are with the University of Colorado, Boulder, CO 80309
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