What could and should be done differently: failure-oriented-accelerated-testing (FOAT) and its role in making an aerospace electronics device into a product

Abstract

High operational reliability of an electronic material or a device intended for aerospace applications is critical, and, in the author’s opinion, cannot be assured, if the underlying physics of failure is not well understood and the never-zero probability of failure is not predicted and made adequate for the particular material, device and application. The situation is the same in some other areas of electronics materials engineering, such as military, medical, or long-haul communications, where high level of reliability is required. The situation is different in today’s commercial electronics, where cost and time-to-market are typically more important than high reliability. Failure-oriented-accelerated-testing (FOAT) of aerospace electronics materials and products and its role in making a viable device into a reliable product is addressed and discussed vs. very popular today highly-accelerated-life-testing (HALT). The differences of the two accelerated test procedures and objectives is briefly discussed. FOAT is an essential part of the recently suggested probabilistic design for reliability (PDfR) approach in electronics engineering. It is argued that high (adequate) reliability level of aerospace electronics materials and devices cannot be achieved and assured, if their never-zero probability-of-failure is not quantified for the given (anticipated) combination of the loading conditions (stresses, stimuli) and time in operation. It is the application of the FOAT, the heart of the highly effective and highly flexible PDfR concept, that should be employed and mastered, when high reliability of a material or a device is imperative. The general concepts are illustrated by numerical examples. They are based on an analytical modeling approach, as the FOAT models are.

Appendix: Entropy of the Arrhenius model

If the lifetime in the Arrhenius equation is interpreted as the mean time to failure (MTTF) and if it is assumed that the (single-parametric) exponential law or reliability takes place, then the probability of non-failure at the moment t of time can be found as the double-exponential probability distribution. The rate of the probability of non-failure is therefore

$$\frac{{dP}}{{dt}}= - \frac{{H(P)}}{t},\,\frac{{dP}}{{dT}}=\frac{{H(P)}}{T}\ln \frac{\tau }{{{\tau _0}}}$$

(1)

where $H(P)= - P\ln P$ is the entropy of the distribution. The following conclusions can be drawn from the formula (1):

• (1) The exponentially distributed probability $P$of non-failure is equal to one at the initial moment of time and decreases exponentially with an increase in time; $P \to 0,$ when $t \to \infty ;$

• (2) The entropy $H(P)$ of the distribution of the probability of non-failure is zero at the initial moment of time, increases to its maximum value of $\frac{1}{e}=0.3679$, when this probability decreases from one to the same $\frac{1}{e}=0.3679$ value, decreases with the further decrease in the probability of non-failure, and becomes zero when this probability becomes zero;

• (3) The lifetime $\tau$in the formula (1) corresponds to the maximum entropy of ${H_{\hbox{max} }}=\frac{1}{e}=0.3679$ and to the probability of non-failure of $P=\frac{1}{e}=0.3679.$ Indeed, from (2) we obtain:

  $$t= - {\tau _0}\exp \left( {\frac{U}{{kT}}} \right)\ln P= - \tau \ln P.$$

  (2)

  Since $\ln P= - 1,$ when $P=\frac{1}{e}=0.3679,$ we have $t=\tau .$;

• (4) The rate $\frac{{dP}}{{dt}}$ in the decrease in the probability $P$ of non-failure with an increase in time $t$ is proportional to the entropy of the distribution of the probability of non-failure at this moment of time to the time that elapsed from the beginning of operation; this conclusion is also true for all the higher time derivatives of the probability $P$ of non-failure;

• (5) The rate $\frac{{dP}}{{dT}}$ in the decrease in the probability $P$ of non-failure with an increase in temperature $T$ is proportional to the entropy of the distribution of the probability of non-failure at the given moment of time to the temperature level; the relationship

  $$\frac{{dP}}{{dT}}=\frac{{H(P)}}{T},$$

  (3)

  i.e., the relationship similar to the first formula in (A-11), when the mean stress-free energy level $kT$ becomes equal to the stress barrier ${U_0}$; since the factor $\ln \frac{\tau }{{{\tau _0}}}$ is much larger than unity, the rate $\frac{{dP}}{{dT}}$ is always significant, especially at low temperature conditions.