Name: Poisson's Ratio
Uploaded: 2019-07-19
Duration: 9 min 46 s
Description: In this video, let us discuss Poisson's ratio, the most interesting and important measure of a property of a material. Watch this to explore.

Question

Explain fully the terms stress, strain, and define young's modulus of elasticity.

Accepted Answer

The strain produced in a body is measured as the ratio of change in configuration to the original configurations of the body.

$Strain = \frac{change in dimension}{original dimension}$

The strain has no unit and is a dimensionless quantity.

Types of strains:

The stain is of Linear or Tensile strain

(i) Longitudinal or Linear or Tensile strain

(ii) Volume strain or Bulk stain

(iii) Shearing strain.

Longitudinal strain

Within elastic limit, it is the ratio of change in length to the original length.

Longitudinal strain $= \frac{change in length}{original length} = \frac{∆ L}{L}$

$\Rightarrow$ Longitudinal strain is possible only in solids. Volume strain

Within elastic limit. it is the ratio of change in volume the original volume.

Volume strain $= \frac{change in volume}{Original in volume} = \frac{∆ V}{V}$

The volume strain produced in solids and liquids is very small but it is very large in goses.

Shearing strain.

When a deforming force in applied on a body along its surface in such a way that the volume of the body remains constant but the shape is changed, then the body is said to be sheared and the strain produced in the body is called the shearing strain.

Within elastic limit, it is measured by the ratio of the relative displacement of one plane to its distance from the fixed plane.

Shearing strain $= \frac{AA' \leftarrow are}{AB \leftarrow radius} = ϕ$

$ϕ = \frac{AA'}{AB}$

$ϕ$ is very small so we can write $ϕ = \tan ϕ$

Shearing strain $= ϕ = \tan ϕ = \frac{AA'}{AB} = \frac{relative displacement of two surfaces}{Distance \frac{b}{w} two surface}$

Thus, the shearing strain is also defined as the ratio of displacement of a surface under a tangential force to the perpendicular distance of the displaced surface.

When an external or deforming force is applied to a body, it gets deformed.

The internal restoring force acting on a unit area of cross-section of a deformed body is called the stress.

Stress $= \frac{F}{A}$

were $A =$ Area of cross-section

$F =$ force

Unit of stress $= \frac{unit of force}{unit of area} = \frac{N}{m^{2}}$

Dimensional formula $\Rightarrow$ stress $= \frac{t}{A} = \frac{{MLT}^{- 2}}{{ML}^{- 1} T^{- 2}}$

Kinds of stress

The stress is of four kinds

(i) Tensile stress

(ii) Compressional stress

(iii) Tangential or shearing stress

(iv) Hydrostatic stress

(i) Tensile stress: It is the restoring force developed per unit cross-sectional area of a body when a length of the body increases in the direction of the deforming force It is also called longitudinal stress Tensile stress $= \frac{F}{A}$

(ii) Compressional stress: It is the restoring force developed per unit cross-sectional area of the body when the body is compressed i.e. when its length decreases under the action of deforming force.

(iii) Tangential stress: When a deforming acts tangentially to the surface of a body, it produces a change in the shape of the body. The tangential force applied per unit area is equal to the tangential stress.

(iv) Hydrostatic stress: It a body is subjected to a uniform force from all sides, then the corresponding static stress.

Young modulus of elasticity

Within the elastic limit, the ratio of longitudinal stress to the longitudinal strain is called young's modulus of the material of the wire.

As shown in fig. suppose a wire of length increase in length $∆ l$ under a force $F$ acting along its length $l$ . Then Young's modulus is given by

$Y = \frac{Longitudinal stress}{Longitudinal strain}$

$Y = \frac{\frac{F}{A}}{\frac{∆ l}{l}} = \frac{F}{A} \cdot \frac{l}{∆ l}$

If the wire has a circular cross-section of radius $r$ then

$Y = \frac{F}{{πr}^{2}} \cdot \frac{l}{∆ l}$

If $l = 1 m$ , $A = 1 m^{2}$ and $∆ m = 1 m$ then $Y = F$ Thus Young's modulus of elasticity is equal to the force required to extend a wire of unit length and unit area of cross-section by unit amount i.e. the force required to double the length of the wire. Unit and dimension of $Y$ S.I. unit $= N / m^{2}$

$Y = \frac{F}{A} \cdot \frac{l}{∆ l} = \frac{{MLT}^{- 2}}{L^{2}} \cdot \frac{L}{L} = {ML}^{- 1} T^{- 2}$

Explain fully the terms stress, strain, and define young's modulus of elasticity.

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