Suppose three stresses P, Q and R act perpendicular to the three facesABCD, ADHE and ABFE of a cube of unit volume (Fig.). Each of these stresses will produce an extension in its own direction and a compression along the other two perpendicular directions.

*Relation between the three moduli of elasticity*

*Suppose three stresses P, Q and
R act perpendicular to the three facesABCD, ADHE and ABFE of a cube of unit
volume (Fig.). Each of these stresses will produce an extension in its own
direction and a compression along the other two perpendicular directions. If λ
is the extension per unit stress, then the elongation along the direction of P
will be λP. If µ is the contraction per unit stress, then the contraction along
the direction of P due to the other two stresses will be µQ and µR.*

*The net change in dimension
along the direction of P due to all the stresses is e = λP - µQ - µR.*

*Similarly the net change in
dimension along the direction of Q is f = λQ - µP - µR and the net change in
dimension along the direction of R is g = λR - µP - µQ.*

*Case (i)*

*If only P acts and Q = R = 0
then it is a case of longitudinal stress.*

*∴** Linear strain = e = λP*

*Young's modulus q =linear
stress /linear strain = P / λP*

*q= 1/λ*

* or*

* λ = 1/q
……………....(1)*

*Case (ii)*

*If R = O and P = - Q, then
the change in dimension along P is e = λP - µ (-P)*

*(i.e) e = (λ + µ) P*

*Angle of shear θ = 2e* = 2 (λ
+ µ) P*

*Rigidity modulus*

*n = P/θ*

* = P/ 2(λ+ µ)P
……………..(2)*

*Case (iii)*

*If P = Q = R, the increase in
volume is = e + f + g*

*= 3 e = 3 (λ − 2µ) P (since e
= f = g)*

*∴** Bulk strain = 3(λ−2µ) P*

*Bulk modulus k = P / 3(λ -
2µ)P*

*Or (λ − 2µ)= 1/3k ……………(3) *

*From (2), 2(λ + µ) = 1/n *

*2λ + 2µ = 1/n*

*From (3), (λ − 2µ) = 1/3k*

*Adding (4) and (5)*

*3λ = 1/n + 1/3k*

*λ = 1/3n + 1/9k*

*∴** From (1), *

*1/q = 1/3n + 1/9k*

*9/q = 3/n + 1/k*

*This is the relation between
the three moduli of elasticity.*

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