Now we are going further to start our discussion to understand the concept of “Thermal stress and strain”, in subject of strength of material, with the help of this post.

### Let us see here the concept of thermal stress and strain

First we will understand here the concept of thermal stress and after that we will see here, in this post, thermal strain.

As we know that if we are going to heat any material, there will be increase in temperature of the material and hence there will be increase in dimensions of the material. Similarly if we are going to cool the material, there will be decrease in temperature of the material and hence there will be decrease in dimensions of the material.

Let us consider that we are going to heat or cool any material, as we have discussed, there will be changes in temperature of the material and therefore there will be changes in dimensions of the material.

We can simply say that there will be free expansion or free contraction in the material according to the rising or lowering of temperature of the material. If free expansion or free contraction of the material due to change in temperature is restricted partially or completely, there will be stress induced in the material and this stress will be termed as thermal stress.

We must note it here that if free expansion or free contraction of the material due to change in temperature is not restricted i.e. expansion or contraction of the material is allowed, there will no stress developed in the material.

### So what we have concluded for thermal stress and thermal strain?

Thermal stress will be basically defined as the stress developed in the material due to change in temperature and free expansion or free contraction of the material, due to change in temperature, will be restricted.

Respective strain developed in the material will be termed as the thermal strain.

### Derivation for thermal stress and thermal strain formula

Let us consider one metal bar AB as shown in following figure. Let us assume that initial temperature of the metal bar is T1 and we are heating the metal bar, as shown in figure, at one end to achieve the final temperature of the metal bar T2. Let us assume that initial length of the metal bar is L.
As we have discussed above that if we are going to heat the material, there will be increase in temperature of the material and therefore there will be increase in dimensions of the material too.

Increase in temperature of the metal bar, ΔT = T2-T1
Let us think that free expansion of the material, due to rise in temperature, is not restricted i.e. free expansion of the material is allowed and it is displayed in figure as BC.

Let us assume that free expansion, as a result of increase in temperature of the metal bar due to heating, is δL. Free expansion will be given by the following formula as displayed here.

δL = α. L. ΔT
Thermal strain will be determined, as written here, with the help of following result
Thermal strain, Ԑ = δL/L

#### Thermal strain, Ԑ = α. ΔT

Let us think that if one load P will be exerted at B as shown in figure and hence as we have discussed above, free expansion or free contraction of the material due to change in temperature will be restricted and hence there will be stress induced in the material i.e. thermal stress will be developed in the material.

Thermal stress, σ = thermal strain x Young’s modulus of elasticity

#### Thermal stress, σ = α. ΔT. E

We must have to note it here that thermal stress is also known as temperature stress and similarly thermal strain is also termed as temperature strain.
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### Reference:

Strength of material, By R. K. Bansal