In solid mechanics and machine design, thermal (temperature) stress in a fully or partially restrained member depends on which factors? Select the most complete description.

Introduction / Context:
Thermal stress appears when a component is heated or cooled but is restrained from freely expanding or contracting. This topic is fundamental in mechanical design, civil structures, piping, and pressure vessels where temperature cycles can produce significant stresses even without external loads.

Given Data / Assumptions:

• Uniform temperature change ΔT is imposed.
• The member experiences some degree of restraint (full or partial).
• Material properties: elastic modulus E and linear thermal expansion coefficient α are relevant.

Concept / Approach:
For a fully restrained, homogeneous member, the classic relation for thermal stress is:
sigma_thermal = E * α * ΔTPartial restraint scales this result by the degree of fixity. The relation shows direct proportionality to all three: temperature change ΔT, stiffness E, and expansivity α.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional reasoning confirms: α (per °C) * ΔT (°C) gives strain; multiplying by E (stress per strain) yields stress, consistent with the formula.

Why Other Options Are Wrong:
ΔT only / E only / α only: Thermal stress is a product of all three; omitting any factor is incomplete.None of these: Contradicts the standard thermal stress relation.

Common Pitfalls:
Ignoring restraint (no restraint means no stress); using plastic modulus instead of elastic modulus for elastic-range estimates; forgetting that property values E and α vary with temperature.

Final Answer:

All of these (ΔT, E, and α)