Corresponding to three types of strain, there are three types of modulus of elasticity:
(i) YOUNG'S MODULUS OF ELASTICITY (Y): It is defined as the ratio of normal stress to the longitudinal strain within the elastic limit. Thus, Y =normal stress/longitudinal strain
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Consider a metal wire AB of lengthl, radius r and of uniform area of cross-section a and having negligible volume. Let it be free from a hard support at A, as given in figure.
Let a normal force F be operated at its last end B and let its length increase by Dl (= BB¢ )
Then, longitudinal strain = Δl/l
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Normal stress =F/a =F/πr2 ( a = πr2)
Y = (F/πr2)/ (Δl/l)
(ii) BULK MODULUS OF ELASTICITY (K): It is defined as the ratio of normal stress to the volumetric strain, within the elastic limit. Thus, K = normal stress/Volumetric strain
Consider a spherical solid body of volume V and field area a. In order to suppress the object, let a force F be operated normally on the entire surface of the body and suppose that its volume decreases by DV as shown in figure. Then, the volumetric strain = - ΔV/V
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Here negative sign shows that volume is decreasing when force is operated. Because modulus is a positive value.
Normal stress = F/a
K = (F/a)/ΔV/V
If p shows the increase in pressure operated on the spherical body then F/a = p
k = -pV/ΔV
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(iii) COMPRESSIBILITY: The reciprocal of the bulk modulus of a material is called its compressibility.
Compressibility b = 1/K; For water b = 45.8 ´ 10-1 Pa-1
(iv) MODULUS OF RIGIDITY (h): It is the ratio of tangential stress to the shearing strain, within the elastic limit. It is also known as shear modulus of rigidity.
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Thus, η = Tangential stress/Shearing strain
η = (F/A)/E
Here A is the surface area of the horizontal surface on which tangential force is applied.
σ=-(Δr/r)/(Δl/l)
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(v) Poisson's Ratio: It is the ratio of lateral strain to longitudinal strain
For most of the materials s finds between 0.18 to 0.25, though theoretically it can be lied between -1 and +0.5. So far no substance has been found having negative Poisson's ratio. Negative quantity of Poisson's ratio would mean that when the wire is related to stress, its radius will increase.
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