If we clamp the ends of a rod rigidly to prevent its expansion or contraction and then change the temperature, tensile or compressive stress developed is called thermal stress. The rod like to expand or contract but clamps won’t allow it. As shown in fig.
To calculate the thermal stress in a clamped rod , we have to find how much the rod would expand if not held rigidly and then find the stress needed to compress it back to its original length. Suppose that the rod with length L0 and cross-sectional area A is held at constant length while the temperature is reduced causing tensile stress. The fractional change in length if the rod were free to contract would be
(ΔL/L0 )thermal= αΔT………….(i)
Both ΔL and ΔT are negative . the tension must increase by an amount F that is just enough to produce an equal and opposite fractional change in length (ΔL/L0 )thermal from the definition of Young’s modulus (Y),
Therefore Y =( F/A)/(ΔL/L);
So (ΔL/L0 )thermal= F/AY …………(II)
If the length is to be constant , the total fractional change in length must be zero. This means that
(ΔL/L0 )thermal + (ΔL/L0 )thermal = 0
αΔT + F/AY = 0
F/A = – YαΔT…………………….(III)
For a decrease in temperature , ΔT is negative , so F and F/A are positive . This means that a tensile force and stress are needed to maintain the length. If ΔT is positive , F and F/A is negative , and the required stress and force are compressive.
If there are temperature difference within a body , non uniform expansion or contraction will result and thermal stress can be induced .
You can break a glass bowl by pouring hot water into it . The thermal stress between the hot and cold parts of the bowl exceeds the breaking stress of the glass, causing cracks . The same phenomenon make ice cube crack when dropped into warm water.