A composite rod of total length L at temperature θ₀ consists of two perfectly bonded...

A composite rod of total length L at temperature θ₀ consists of two perfectly bonded sections A and B, each of uniform cross-sectional area A but different materials.

Section A has length L₁, Young’s modulus E₁, and coefficient of linear expansivity α₁.

Section B has length L₂, Young’s modulus E₂, and coefficient of linear expansivity α₂, where L₁ + L₂ = L.

The rod is rigidly fixed between two immovable walls such that no net change in total length is allowed. The temperature of the system is increased uniformly by Δθ.

Assume:

Both materials remain within their elastic limits.

There is no slipping at the interface.

Thermal stresses are fully developed.

(a)

Derive an expression for the common thermal stress σ developed in the composite rod in terms of
α₁, α₂, E₁, E₂, L₁, L₂, and Δθ.

(b)

Determine the internal axial force in each section and prove that the forces are equal in magnitude but opposite in effect to the free expansion tendencies.

(c)

Find the condition under which section A is in compression while section B is in tension, and explain the physical significance of this result.

(d)

If E₁α₁L₁ = E₂α₂L₂, show mathematically that the thermal stress in the system becomes independent of the Young’s moduli and interpret what this means from a materials engineering perspective.

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In Physics
1 Answers Available
Asked by ROCKMAN on 18th January, 2026