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# 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.

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 � is the contraction per unit stress, then the contraction along the direction of P due to the other two stresses will be �Q and �R. The net change in dimension along the direction of P due to all the stresses is e = λP - �Q - �R.

Similarly the net change in dimension along the direction of Q is f = λQ - �P - �R and the net change in dimension along the direction of R is g = λR - �P - �Q.

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 - � (-P)

(i.e) e = (λ + �) P

Angle of shear θ = 2e* = 2 (λ + �) P

Rigidity modulus

n = P/θ

= P/ 2(λ+ �)P   �����..(2)

Case (iii)

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

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

Bulk strain = 3(λ−2�) P

Bulk modulus k = P / 3(λ - 2�)P

Or   (λ − 2�)= 1/3k            �����(3)

From (2), 2(λ + �) = 1/n

2λ + 2� = 1/n

From (3), (λ − 2�) = 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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