Properties of Solids

SOME IMPORTANT DEFINITION

(i) Deforming Force

If a force applied to a body causes a change in the normal positions of the molecules of the body, resulting in a change in the configuration of the body either in length, volume or shape, then the force applied is deforming in nature.

(ii) Elasticity

The property of a body by virtue of which the body regains its original configuration (length, volume or shape) when external deforming forces are removed is called elasticity.

(iii) Cause of Elasticity

Molecules in a body are bound to each other by bonds. When elongated the bonds behave like springs due to inter molecular interaction. When subjected to compression beyond a certain distance or less than a certain inter molecular separation, the molecules, instead of attracting each other, repel.

(iv) Interatomic force constant (k)

Interatomic force constant is defined as ratio of interatomic force to change in interatomic distance.

$K=\frac{\mathrm{int}eratomic\text{ force }}{change\text{ in interatomic distance }}$

$K=Youn{g}^{\prime }s\text{ modulus }\times \text{ interatomic distance }$

(v) Perfectly Elastic Body

A body which regains its original configuration immediately and completely after the removal of deforming force from it, is called perfectly elastic body. Quartz and phosphor bronze are the examples of nearly perfectly elastic bodies.

(vi) Perfectly Plastic Body

A body which does not regain its original configuration at all on the removal of deforming force, however small the deforming force may be is a perfectly plastic body.

(vii) STRESS

Stress is measured by the deforming force per unit normal area.

Stresses set up in a body due to the action of forces can be classified into three broad categories

(a) Tensile stress, which produces elongation, is the result of forces acting on the body along a single direction.

(b) Shear stress, which produces "shear or bending - being the result of equal and opposite torques acting on a body.

(c) Volume stress (Bulk Stress) which produces a change in the volume of the body - the resulting strain is known as volume strain (Bulk Strain).

(d) Unit of stress = $\frac{F}{A}=$N/m²

(e) Dimension of stress = ML^-1T^-2

(viii) STRAIN

When deforming forces are applied to a body, there is a change in the shape of the body. The body is said to be strained or deformed. The ratio of change in dimension to the original dimension is called strain. i.e.

Strain = $\frac{Changeindimension}{Originaldimension}$

Strain being the ratio of two like quantities has no units and dimensions. There is three type of strain

(a) Longitudinal strain = $\frac{Changeinlength}{Originallength}$ = $\frac{\Delta L}{L}$

(b) Volume strain = $\frac{Changeinvolume}{Originalvolume}$=$\frac{\Delta v}{V}$

(c) Shear strain =q where q is the angle between the deformed surface and original surface.

(ix) ELASTIC LIMIT

Elastic limit is the upper limit of deforming force up to which, if deforming force is removed, the body regains its original form completely and beyond that limit, if deforming force is increased, the body loses its property of elasticity and gets permanently deformed.

Elastic limit is the property of a body whereas elasticity is the property of the material of a body.

HOOKE'S LAW

Hooke's law is valid for only small deformation. Hooke's law states that the extension produced in the wire is directly proportional to the load applied within elastic limit. i.e. within elastic limits, extension µ load applied.

Later on it was found that this law is applicable to all types of deformation such as compression, bending, twisting etc and thus a modification form of Hooke's law was given as stated below.

"Within elastic limit, the stress developed is directly proportional to the strain produced in a body"

i.e. stress µ strain or, stress = E $\times$ strain

where E is a constant and is known as the modulus of elasticity of the material.

STRESS, STRAIN RELATIONSHIP FOR A WIRE

The relationship between stress and strain in a wire is illustrated by the graph shown in the adjoining figure. The regions are referred to as follows:

OA = Elastic region, Hooke's law is valid.

AB = Elastic region, Hooke's law is not true

A = proportional limit

B = Elastic limit(strain £1 %) or yield point

BD = Plastic deformation

OE = Permanent set

CD = Plastic flow

D = Fracture point (Corresponding stress is called breaking stress/ tensile strength)

ELASTOMERS

The substances which can be stretched to large values of strain are called elastomers. e.g. elastic tissue of aorta, the largest artery carrying blood from the heart.

MODULUS OF ELASTICITY

According to Hooke's law, within elastic limit,

stress µ strain

or stress = E $\times$ strain

or $\frac{stress}{strain}$ = E = a constant,

where E is known as coefficient of elasticity or modulus of elasticity of a body which depends upon the nature of material of the body and the manner in which the body is deformed. Thus, modulus of elasticity or coefficient of elasticity of a body is defined as the ratio of the stress to the corresponding strain produced, within the elastic limit.

TYPES OF MODULE OF ELASTICITY

Corresponding to three types of strain, there are three types of module 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 = $\frac{normalstress}{longitudinalstrain}$

Consider a metal wire AB of length $\ell$, radius r and of uniform area of cross-section a and having negligible mass. Let it be suspended from a rigid support at A, as shown in figure.

Let a normal force F be applied at its free end B and let its length increase by $\Delta$$\ell$ (= BB¢ ) Then, longitudinal strain = $\frac{\Delta \ell }{\ell }$Normal stress = $\frac{F}{a}$ = $\frac{F}{\pi r^2}$;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;($\because$ a =;$\pi$r²) $\therefore$ Y = $\frac{F/\pi r^2}{\Delta \ell /\ell }=\frac{F\ell }{\pi r^2\Delta \ell }$

(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 = $\frac{normalstress}{volumetricstrain}$;; Consider a spherical solid body of volume V and surface area a.;In order to compress the body, let a force F be applied normally on the entire surface of the body and suppose that its volume decreases by $\Delta$v as shown in figure.;Then, the volumetric strain;= - $\Delta$v/VHere negative sign shows that volume is decreasing when force is applied. Normal stress;= F/a $\therefore$ k = $\frac{F/a}{\Delta v/V}=-\frac{FV}{a\Delta v}$If p represents the increase in pressure applied on the spherical body then F/a = p;;;;;;;;;;;k = -pV/$\Delta$vCOMPRESSIBILITYThe reciprocal of the bulk modulus of a material is called its compressibility.Compressibility;= $\frac{1}{k}$

(iii);;;;;;MODULUS OF RIGIDITY ($\eta$); It is the ratio of tangential stress to the shearing strain, within the elastic limit.;It is also called shear modulus of rigidity.;Thus;;;;;;;;;;;$\eta$ = $\frac{\tan gentialstress}{shearingstrain}$;;;;;;;;;;;;;;;;;;;;;$\therefore$;;;;;;;;;$\eta$ = $\frac{F/a}{\theta }$;;;;;;;;;;;$\frac{F}{a\theta }=\frac{FL}{a\Delta L}$
EFFECT ON ELASTICITY BY DIFFERENT CAUSES(i);;;;;;;;Effect of temperature:;(a) In general elasticity decreases as the temperature increases. ;;;;;;;;;;;(b) Invar is an exception. There is no effect of temperature on elasticity of invar. Invar is in facet a short form of invariable. (ii) ;;;;;;Effect of impurities: (a) If the impurity is more elastic, the elasticity of the material increases. (b) If the impurity is more plastic, the elasticity of material decreases. (iii);;;;;;On hammering or rolling elasticity increase. (iv);;;;;On annealing i.e. on alternate heating and cooling elasticity decreases.;
ELASTIC POTENTIAL ENERGY IN A STRETCHED WIRE;When a wire is stretched, some work is done against the internal restoring forces acting between the particles of the wire.;This work done appears as the elastic potential energy of the wire.
Consider a wire of length $\ell$ and area of cross section a.;Let F be the stretching force applied on the wire and let $\Delta$$\ell$ be the increase in length of the wire. Initially, the internal restoring force was zero but when length is increased by $\Delta$$\ell$, the average internal restoring force for an increase in length $\Delta$$\ell$ of the wire = $\frac{0+F}{2}$ = $\frac{F}{2}$

Hence, work done on the wire, w = average force $\times$ increase in length = $\frac{F}{2}$ $\times$ $\Delta$$\ell$ This is stored as elastic potential energy U in the wire.$\therefore$ U = $\frac{1}{2}$ F $\times$ $\Delta$$\ell$ = $\frac{1}{2}$ $\frac{F}{a}$ $\times$ $\frac{\Delta \ell }{\ell }$ $\times$ a$\ell$

= $\frac{1}{2}$ (stress) $\times$ (strain) $\times$ volume of the wire

$\therefore$ Elastic potential energy per unit volume of the wire (= energy density)

u = $\frac{U}{a\ell }=\frac{1}{2}$(stress) $\times$ (strain)

=$\frac{1}{2}$(Young's modulus $\times$ strain) $\times$ strain

($\therefore$ Young's modulus = stress / strain)

$\therefore$ u = $\frac{1}{2}$(Young's modulus ) $\times$ (strain)²

ELASTIC FATIGUE

It is the lack of elastic strength of a substance when subjected to repeated stresses and strain. If the substance is kept undisturbed for some time, the previous properties are restored. Thus elastic fatigue is a tempeorary phenomenon.

SEARLE'S APPARATUS

Y = $\frac{MgL}{\pi r^{2}l}$

(b) To calculate the maximum load to be applied :

Breaking weight for the experimental wire = breaking stress × $\pi r^2$

Maximum load to be applied = $\frac{1}{3}$× breaking stress