A solid cube of side 1 {m} is kept at a room temperature of 32^{\circ} {C}. The coefficient of linear thermal expansion of the cube material is 1 \times 10^{-5} \/{ }^{\circ} {C} and the bulk modulus is 200 {GPa}. If the cube is constrained all around and heated uniformly to 42^{\circ} {C}, then the magnitude of volumetric (mean) stress (in MPa) induced due to heating is ___________.

GATE Mechanical Engineering (ME) 2019 Shift 1 Solved Paper

Question : 11 of 65

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A solid cube of side

1 {m}

is kept at a room temperature of

32^{∘} {C}

. The coefficient of linear thermal expansion of the cube material is

1 × 10^{-5} \/{ }^{∘} {C}

and the bulk modulus is

200 {GPa}

. If the cube is constrained all around and heated uniformly to

42^{∘} {C}

, then the magnitude of volumetric (mean) stress (in MPa) induced due to heating is ___________.

Solution:

side of cube

= 1 m

Temperature rise

(∆ T)=10^{∘} {C}

Coefficient of linear thermal expansion

(α)=1 × 10^{-5} {/}{ }^{∘} {C}

Bulk modulus

( {K})=200 {GPa}

E_{V}=E_{x}+E_{y}+E_{z}=\;{1-2 µ}/{E}(σ_{x}+σ_{y}+σ_{z})

Since, the cube is symmetric, therefore,

σ_{x}=σ_{y}=σ_{z}=σ

E_{V}=\;{1-2 µ}/{E}(3 σ)

This volumetric strain will be equal to

3 α ∆ T

So,

\;{1-2 µ}/{E}(3 σ)=3 α ∆ T

E=3 K(1-2 µ)

So,

\;{1-2 µ}/{E}(3 σ)=\;{3 σ}/{3 K}=3 α ∆ T

Or,

σ=3 K α ∆ T=3 × 200 × 10^{3} × 10^{-5} × 10=60

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