Question

The thermal stress in terms of modules of elasticity(E), co-efficient of thermal expansion(\(\alpha\)), Temperature(T) is given by

• A. \(E\alpha T\)
• B. \(ET/\alpha\)
• C. \(E\alpha/T\)
• D. \(1/E\alpha T\)

Hint:

Thermal stress only develops when thermal expansion or contraction is constrained.
The formula is a direct combination of Hooke's Law ($\sigma = E\epsilon$) and the thermal expansion strain equation ($\epsilon = \alpha T$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the formula that defines the thermal stress induced in a fully constrained body subjected to a change in temperature.

Step 2: Key Formula or Approach:
Thermal strain (\(\epsilon_{\text{th}}\)) when a body undergoes a temperature change \(T\) is:
\[ \epsilon_{\text{th}} = \alpha T \] By Hooke's Law, if the expansion is completely prevented, the resulting thermal stress (\(\sigma_{\text{th}}\)) is:
\[ \sigma_{\text{th}} = E \cdot \epsilon_{\text{th}} = E \alpha T \]

Step 3: Detailed Explanation:

• When a homogeneous material is heated, it expands. The linear expansion strain is directly proportional to the temperature change \(T\) and the material's coefficient of thermal expansion \(\alpha\).

• If the material is free to expand, no stress is generated in the body (\(\sigma = 0\)).

• However, if the material is constrained by rigid supports at both ends, mechanical stress is induced to suppress the thermal deformation.

• The magnitude of this induced stress is the product of Young's Modulus \(E\) and the thermal strain \(\alpha T\).

• This derivation yields the classic equation: \(\sigma = E \alpha T\).

Step 4: Final Answer:
The thermal stress is given by the expression \(E\alpha T\).