Template:Temperature non-thermal relationship

=Temperature-NonThermal Relationship=

Introduction
When temperature and a second non-thermal stress (e.g. voltage) are the accelerated stresses of a test, then the Arrhenius and the inverse power law relationships can be combined to yield the Temperature-NonThermal (T-NT) relationship. This relationship is given by:


 * $$L(U,V)=\frac{C}$$

where: •	 $$U$$ is the non-thermal stress (i.e., voltage, vibration, etc.) •	 $$V$$ is the temperature (in absolute units). •$$B,$$  $$C,$$   $$n$$  are the parameters to be determined. The T-NT relationship can be linearized and plotted on a Life vs. Stress plot. The relationship is linearized by taking the natural logarithm of both sides in the T-NT relationship or:


 * $$\ln (L(V,U))=\ln (C)-n\ln (U)+\frac{B}{V}$$

Since life is now a function of two stresses, a Life vs. Stress plot can only be obtained by keeping one of the two stresses constant and varying the other one. Doing so will yield the straight line described by the above equation, where the term for the stress which is kept at a fixed value becomes another constant (in addition to the $$\ln (C)$$  constant). When the non-thermal stress is kept constant, then the linearized T-NT relationship becomes:


 * $$\ln (L(V))=const.+\frac{B}{V}$$

This is the Arrhenius equation and it is plotted on a log-reciprocal scale. When the thermal stress is kept constant, then the linearized T-NT relationship becomes:


 * $$\ln (L(U))=const.-n\ln (U)$$

This is the inverse power law equation and it is plotted on a log-log scale. In the next two figures, data obtained from a temperature and voltage test were analyzed and plotted on a log-reciprocal scale. In the first figure, life is plotted versus temperature, with voltage held at a fixed value. In the second figure, life is plotted versus voltage, with temperature held at a fixed value.



A look at the Parameters $$B$$ and  $$n$$
Depending on which stress type is kept constant, it can be seen from the linearized T-NT relationship that either the parameter $$B$$  or the parameter  $$n$$  is the slope of the resulting line. If, for example, the non-thermal stress is kept constant then $$B$$  is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of the product's life to the temperature. In other words, $$B$$  is a measure of the effect that temperature has on the life and  $$n$$  is a measure of the effect that the non-thermal stress has on the life. The larger the value of $$B,$$  the higher the dependency of the life on the temperature. Similarly, the larger the value of $$n,$$  the higher the dependency of the life on the non-thermal stress.

Acceleration Factor
The acceleration factor for the T-NT relationship is given by:


 * $${{A}_{F}}=\frac=\frac{\tfrac{C}{U_{u}^{n}}{{e}^{\tfrac{B}}}}{\tfrac{C}{U_{A}^{n}}{{e}^{\tfrac{B}}}}={{\left( \frac{{{U}_{A}}}{{{U}_{u}}} \right)}^{n}}{{e}^{B\left( \tfrac{1}-\tfrac{1} \right)}}$$

where: •	 $${{L}_{USE}}$$ is the life at use stress level. •	 $${{L}_{Accelerated}}$$ is the life at the accelerated stress level. •	 $${{V}_{u}}$$ is the use temperature level. •	 $${{V}_{A}}$$ is the accelerated temperature level. •	 $${{U}_{A}}$$ is the accelerated non-thermal level. •	 $${{U}_{u}}$$ is the use non-thermal level.

The acceleration factor is plotted versus stress in the same manner used to create the Life vs. Stress plots. That is, one stress type is kept constant and the other is varied.


 * $$\begin{align}

& \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt = & \int\limits_{0}^{\infty }t\cdot \frac{C}{{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dt = & \frac{C} \end{align}$$