We were discussing the “Elongation of uniformly tapering circular rod” and “Elongation of uniformly tapering rectangular rod” and also we have seen “Stress analysis of bars of composite sections” and “Stress analysis of bars of varying sections” with the help of
previous posts.

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 T

_{1}and we are heating the metal bar, as shown in figure, at one end to achieve the final temperature of the metal bar T_{2}. 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 = T

_{2}-T_{1}
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*

*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*

*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.

Do you
have any suggestions? Please write in comment box

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. Thermal stresses in composite bars, in the category of strength of material, in our
next post.

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