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# 4Applications of NE Thermodynamic Degradation Science to Mechanical Systems: Accelerated Test and CAST Equations, Miner’s Rule, and FDS

## 4.1 Thermodynamic Work Approach to Physics of Failure Problems

We are now in a position to assess a number of reliability degradation problems. In this chapter we will continue to use the thermodynamic work to assess the entropy damage to the system and the amount of free energy left for numerous key problems in reliability. As we found in Chapter 3, the thermodynamic work is perhaps the most directly measurable and practical quantity to use in assessing a system’s free energy and its degradation [1–3]. This chapter provides many examples for deriving physics of failure aging laws and their acceleration factors in mechanical systems using the thermodynamic work principle.

## 4.2 Example 4.1: Miner’s Rule

We derived the Miner’s rule using an entropy approach in Section 2.1.1. Here we derive it using the thermodynamic work approach [1–3]. Equation (3.25) for cyclic damage may be used to derive Miner’s empirical rule , commonly used for accumulated fatigue damage and a number of other useful expressions in damage assessment of devices and machines.

Consider a system undergoing fatigue as in the paper clip example, where we bend it back and forth a certain distance for three cycles. To find the actual work we need to sum the cyclic work area of each bend cycle since both the stress σ and strain e will change slightly as the paper clip fatigues. ...

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