# Relation between thermal expansion and stress

• raju_pro
In summary, the question is whether initial stress affects thermal expansion coefficient in a metallic component. The answer is no, as shown by the example of two identical springs under different stresses and the explanation of superposition. However, some manipulation of partial derivatives shows that the coefficient of thermal expansion is dependent on stress, as demonstrated in a Journal Paper from MIT-Cambridge. The authors also confirmed this through experiments. If the Young's Modulus is assumed to not vary with temperature, then the variation of \alpha with respect to stress can be disregarded.
raju_pro
I have been trying to find an answer for this for quite some time and found this forum which could help me.

Does an initial stress (elastic strain induced in a metallic component) change its thermal expansion coefficient? if yes then what is the relationship?

In simpler terms, does 2 identical parts one under load (elastically strained) and other free state would expand through the same length when the surrounding temprature is increased?

No, it does not.

Imagine this example. You have two identical springs. They are both extension springs with loops on each end, they are hanging from pegs. One of the springs is under tension because there is a weight hung from it. This weight is not enough to plastically deform the spring. The other spring has no additional load, it just hangs by its own weight. One spring, the one with the weight, is under greater stress.

So you take this set up and put it in an oven and increase its temperature by 100 degrees. Both springs will, if they are perfect ideal springs, expand equally due to the normal thermal expansion effects.

The explanation is just common supper-position. Both effects just are independent and ignore each other.

Thanks for your response. I had the same thought earlier, but then came across this Journal Paper from MIT-Cambridge (although it is quite an old reference)

http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5121477

It is often assumed that the coefficient of thermal expansion is stress-independent, but some manipulation of partial derivatives shows that that's not strictly true. Consider a long rod (with length L and spring constant $k=dF/dL$) that can be axially loaded with force F; we'd like to know whether the linear thermal expansion coefficient

$$\alpha_\mathrm{L}=\frac{1}{L_0}\left(\frac{\partial L}{\partial T}\right)_F$$

is dependent on the load. This dependence can be written as

$$\frac{\partial}{\partial F}(\alpha_\mathrm{L})=\frac{\partial}{\partial F}\left[\frac{1}{L_0}\left(\frac{\partial L}{\partial T}\right)_F\right]_T=\frac{1}{L_0}\frac{\partial}{\partial F}\left[\left(\frac{\partial L}{\partial T}\right)_F\right]_T=\frac{1}{L_0}\frac{\partial}{\partial T}\left[\left(\frac{\partial L}{\partial F}\right)_T\right]_F=-\frac{1}{L_0k^2}\left(\frac{\partial k}{\partial T}\right)_F$$

which shows that the temperature dependence of stiffness is somehow involved!

The authors did essentially the same calculations for the more general 3-D case of stress and strain, then confirmed their predictions by experiment. Does this answer your question?

Mapes, thanks for your reply. So in this case if I had assumed that the Young's Modulus does not vary with temperature i.e., $$\frac{\partial\bf{E}}{\partial\bf{T}}$$ = 0, then I do not have to worry about the variation of $$\alpha$$ wrt to temperature ?

If you mean $\alpha$ with respect to stress or force, correct; the two effects are linked. (But $\alpha$ will still vary with temperature in a way that's material-specific.)

## 1. What is thermal expansion?

Thermal expansion is the tendency of a material to expand or contract in response to changes in temperature.

## 2. How does thermal expansion relate to stress?

When a material undergoes thermal expansion, it exerts stress on its surroundings. This stress can lead to deformations or even failure of the material if it exceeds its strength.

## 3. What factors affect the degree of thermal expansion in a material?

The degree of thermal expansion in a material is affected by its composition, crystal structure, and temperature range. Different materials have different coefficients of thermal expansion, which measure the rate of expansion per degree change in temperature.

## 4. How is thermal expansion measured and quantified?

Thermal expansion is typically measured using techniques such as dilatometry, interferometry, or strain gauges. The coefficient of thermal expansion can then be calculated by dividing the change in length or volume by the original length or volume and the change in temperature.

## 5. Can thermal expansion be controlled or prevented?

Thermal expansion cannot be completely prevented, but it can be controlled by incorporating materials with different coefficients of thermal expansion in a design, or by using techniques such as pre-stressing or thermal compensation. Proper design and material selection can minimize the effects of thermal expansion on a structure or device.

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