Elasticity

Introduction
$\to \textbf{Elasticity}$ is the tendency of restoring / recovering the original shape and size of deformed body after the deforming force is removed.$\\$$\to \textbf{Perfectly elastic body}$ regains its original shape and size completely after removal of deforming force. Eg: Quartz fiber, Phosphor bronze etc.$\\$$\to \textbf{Perfectly plastic body}$ does not regain its original shape and size at all. Eg: mud, wax, plastics etc.$\\$

$\textbf{Stress} \\$$\to$Restoring force applied in a unit area$\\$$Stress(S) = \frac{ Force(F)}{Area(A)}=FA$$\\$$\to$Stress is a tensor quantity with dimension $[M L^{-1} T^2]$and unit $N/m^2$.$\\$

$\textbf{ Types of stress are:} \\$

$\to$Normal stress: Normal force per unit cross-section area.$\\$$\to$Tangential or Shearing stress:Tangential force applied parallel to surface per unit area of that surface producing shearing (change in shape).$\\$$\to$Volume stress: Force per unit total surface area producing change in volume.$\\$

$\textbf {Strain} \\$The change in dimension of body per unit original dimension is strain.$\\$Strain = $\frac{change -in -configuration}{ original- configuration} \\$Strain is unit less and dimension less physical quantity.

$\textbf{Types of strain are:}\\$$\to \textbf{Longitudinal strain :}$ Change in length per unit initial length. Strain (long.) $=\frac{ \Delta L}{L} \\$$\to$ $\textbf{Volumetric Strain :}$ Change in volume per unit original volume and is given by $\frac{\Delta V}{V} \\$$\to \textbf{Shearing Strain :}$ Change in position per unit original position. It is generally represented by angle θ through which a body initially fixed is twisted under the action of tangential deforming force. Shearing force $= \theta = \frac{x}{L}$

Hooke's law

It states that, "Within elastic limit, extension produced is directly proportional to force applied."$\\$i.e. $e \propto F$ $\to$ F = -k.e ; where k is force constant.

It also states, within elastic limit, stress is directly proportional to the strain$\\$i.e. $Stress \propto Strain$ so, $\frac{stress}{strain}= E$ (modulus of elasticity)

$\textbf {Types of modulus of elasticity:} \\$$\to \textbf {Young's modulus of elasticity (Y):} \\$It is the ratio of normal stress to the longitudinal strain, within the elastic limit.$\\$$Y = \frac{normal -stress}{ longitudinal -strain} =\frac{\frac{F}{A}} {\frac{\Delta L}{L}}= \frac{FL}{A \Delta L}$

$\textbf{NOTE :} \\$$\to$It is property of solid body only.$\\$$\to$Young's modulus of elasticity increases on mixing the impurity in the solid and decreases on increasing temperature of solid body.$\\$

$\to \textbf {Bulk modulus of elasticity (B):} \\$Ratio of normal stress to volumetric strain within elastic limit.$\\$$B = \frac{normal -stress}{ volumetric -strain} =\frac{\frac{F}{A}} {\frac{-\Delta V}{V}}= \frac{\Delta P}{\frac{-\Delta V}{V}} \\$( -ve sign indicates decreasing of volume on increasing of applied force)$\\$

$\textbf{NOTE :} \\$$\to$Bulk modulus is property of solid, liquid and gases.$\\$$\to$Gases have isothermal and adiabatic bulk modulus of elasticity.$\\$$\to$K is minimum for gases (maximum volume change) and maximum for solid (minimum volume change)$\\$$\to$Reciprocal of bulk modulus of elasticity is Compressibility.$\\$

$\textbf{Modulus of rigidity (η)} \\$$\to$Ratio of tangential stress to shearing strain within elastic limit.$\\$$\to$ $η = \frac{tangential- stress}{shearing -strain}= \frac{\frac{F}{A}}{\theta} =\frac{F}{A \theta }$
$\to$It is property of solid materials only. And its value is zero for liquid.$\\$

$\textbf{Poisson's ratio (σ) :} \\$$\to$ The ratio of lateral strain to longitudinal strain$\\$$\to$ $σ = \frac{lateral- strain}{ longitudinal- strain} =\frac{\frac{-\Delta r}{r}} {\frac{\Delta L}{L}}= \frac{-\Delta r l}{\Delta l r}\\$
(-ve sign indicates decrease of radius in expense of increasing length for constant volume)$\\$$\to$ Theoretical value of σ lies in between -1 to $+ \frac{1}{2}$ but practical lies between 0 to $+ \frac{1}{2} . \\$$\to$ Fractional change in volume : $\frac{\Delta V}{V} =(1-2σ)x \frac{\Delta L} {L}$ ( so, change in volume is zero when the value of σ 0.5 )

Energy stored during elastic deformation

A step towards visualization

Think of a triangle with extension (x) as its base and the force applied on body to produce that extension be represented by height of triangle. Where, force $(F) = kx$ And $k$ is the force constant of that elastic body. Then, area of that triangle represents the work done in stretching the body by $x$.$\\$

So, elastic potential energy stored $( E_{pe} ) = \frac{1}{2} k x^2$

Elastic P.E in a stretched wire(U)$\\$$= \frac{1}{2}*stretching -force*extension \\$$= \frac{1}{2}*F*e \\$$= \frac{1}{2}*\frac{F}{A}*\frac{e}{l}*A.l \\$$=\frac{1}{2} * Stress * Strain * Volume \\$

Energy density / elastic energy per unit volume $U/volume = \frac{1}{2}* Stress * Strain = \frac{1}{2}*\frac{(stress)^2}{ Y} \\$$= \frac{1}{2}*Y*strain^2 [ Y = \frac{Stress}{Strain} ]$

Extension of rope by its own weight

For small element 
$dx$, the small extension produced on the rope above the element will be:

$de =\frac{dm . g . (L-x)}{A. Y} \\$[ where, $\textbf{dm}$ is mass of element and $\textbf{L}$ is length of rope ]$\\$$dm = \frac{M}{L} * dx \\$so,$de = \frac{M . g . (L-x)L}{.A. Y} dx$ now, for total extension on the rope due to its own weight, we have to integrate from bottom to top of rope $e = \int_0^L{de} = \int_0^L \frac{ M . g . (L-x)L}{A. Y} dx =\frac {M g L^2}{2LAY} =\frac{M g L^2}{ 2A Y}=\frac {ρ L^2g}{2 Y} \\$

FEEL THE POINTS

Y for elastic body is infinity.

Y for perfectly plastic body is zero.

Young's modulus (Y) measures extensibility, bulk modulus (B) measures compressibility while shear modulus (η) measures rigidity.

On increasing temperature Y, β, η all decrease on account of decrease in intermolecular force.

Thermal force set up due to thermal expansion = F = Y θαA

Breaking stress depends upon material only while breaking strength depends on material and cross-section area but independent of length.$\\$Breaking stress $\propto$ Y and Breaking Strength $\propto$ A

The area under the stress – strain graph represents the work done per unit volume.