The equation of thermal stress is:
Stress =$\frac{F}{A}$ = -$E$ $a$ $dT$, where $E$ is Young's Modulus, $a$ is the coefficient of linear thermal expansion, and $dT$ is the change in temperature.
I can't think of an intuitive reason for which $E$, $a$, and $dT$ would be multiplied together, and I haven't been able to find anything online. In this case, I would assume that a derivation would explain the properties of this formula. Does anyone have an idea as to what this derivation would look like?
Best Answer
From the definition of Young's Modulus, we have the following expression: $$ Y = \frac{Stress}{Strain}=\frac{\frac {F}{A}}{\frac {\Delta l}{l_0}} $$
Also from the definition of coefficient of thermal expansion, we have $$ \alpha =\frac{l-l_0}{l_0t} $$ or, $$ \Delta l = l - l_0 = l_0 \alpha t $$
Substituting the value of $\Delta l$ in the first expression, we have $$ Y = \frac{\frac {F}{A}}{\frac { l_0 \alpha t}{l_0}} =\frac {F}{\alpha A t}$$ or $$ \frac{F}{A} = Y \alpha t$$