Elastic Constants Relation

Any general elastic deformation of a homogeneous isotropic solid can be regarded as comprising of a change in size (volume) plus a change in shape (shear). Such an analysis gives an inter-relation between different elastic constants as discussed below.
    
(i) Relation between, Y, K and σ: Consider a cube of unit length and let a force f acts normally outwards on each of its six faces. Thus, we produce a uniform deformation in all directions in the cube.

Now, consider for example the force acting along X-axis. Let this force produce a longitudinal extension along X-axis. Hence, we have



The same force also produces a lateral strain in Y and Z directions. If lateral contraction is along normal direction, then we have



Hence, we get = σ = σ/V F

Similarly, force along Y-axis produces an extension in Y-direction and contractions in X and Z-axis produces an extension along Z-direction and contractions along X and Y directions. The net change in length e_x, e_y and e_z along X, Y and Z directions are,

e_x = e_y = e_z = – 2 = F/Y (1 – 2 σ) = e (say)

The change in volume of the cube is therefore

Δ V = (1 + e)³ – 1 = 3 e

Hence, the bulk modulus K of the solid is given by



Or, Y = 3 K (1 – 2 σ)

The above expression relates the ordinary volume compressibility of a solid to the values of Young’s modulus and Poisson’s ratio.

(ii) Relation between Y, η, and σ. Consider the cube of unit length as above, and let a compression stress F acts on it along X-direction and an extension stress F acts along Y-direction.

The force along X-axis produces a longitudinal compression along X-direction and a lateral extension along Y-direction. Similarly, force along Y-axis produces a longitudinal extension along Y-direction and a lateral compression along X-direction. Hence, the net change in length e_x and e_y along X and Y directions are,



Also, e_z = 0

Because while compressive stress produces lateral extension, the extensive stress produces equal lateral compression along Z-direction.

Now, a force F of compression along X-axis and an equal force of extension along Y-axis are equivalent to a shear stress of magnitude F at 45° to X and Y axes. Latter would produce a shearing strain of angle in θ XY-plane, such that

θ = 2 e_x    (or 2 e_y)

Hence, modulus or rigidity of the body is given by



Since K and η are positive constants, relation (iii) shows that – 1 < σ < ½.

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