Thermal Stress

When an object is supplied with a finite amount of heat, the atom presents in the lattice structure starts agitating, and they tend to leave their stable position and starts to oscillate about their mean position. Due to the inter-atomic forces of attraction the neighbouring atoms also starts vibrating with the frequency equal (ideal case) or less (non-ideal case) than that of the previous atom. This phenomenon leads the transfer of heat.

There are many processes by which an abject can be heated, but the mode of transfer of heat inside a body remains same.

Due to this agitation, the body tries to undergo either a linear, surface or volumetric expansion depending upon the dimensions we choose. Under this expansion, whenever the body is subjected to and hindrance, stresses are developed and this stress is called thermal stress.

Consider the following situation:

Body absorbing heat

Suppose a block is placed upon a smooth frictional-less surface and heat is being transferred by the means of radiation. If we consider the surrounding atmosphere of the block to be replaced by vacuum then the block will go a free volumetric expansion. By the word “free” we mean that there is no hindrance created by surrounding, either by atmospheric pressure or by friction, in the expansion of block. So no stresses are produced.

However if the surface was rough, the block had to do work against the friction in order to expand and thermal stress would have been generated in this expansion process.

Expression

Consider the following setup:

A bar element of length L fixed at its one end.

Suppose the temperature changes by ∆T due to which the block undergoes a thermal expansion.

Let α be the coefficient of linear expansion and dell being the change in length.

Then, change in length (δ L) due to thermal expansion is given by:

                              ∆L=α (∆T) L

In the limits  δ changes to ∆