Elasticity

aElasticityElasticity is the property of the body by virtue of which a deformed body regains its original shape, size and position after the removal of deforming force• The elasticity of the object generally decreases with increase in temperature.• The elasticity of molten cadmium ( 80 ) and molten copper increases with increase in temperature.• The elasticity of invar is unaffected with the change in temperature.Deforming forceA force is said to be a deforming force if it can change shape, size and position of the object.Restoring forceA force is said to be restoring force if it brings the object back to its original shape, size and position after the removal of deforming force.Types of object on the basis of elasticity1. Elastic object:An object is said to be elastic if it regains its original shape, size and position after the removal of deforming force.For example; quartz, rubber, steel, iron etc.2. Plastic object:An object is said to be plastic if it doesn't regain its shape, size and collision after the removal of deforming force.For example: polythene, wax, etc.3. Rigid object:An object is said to be rigid if it doesn't get reformed by an external forceFor example: bricks, stones, etc.StressStress is defined as the force per unit area that acts on a material.Stress is given by the following formula:𝜎=F

ATypes of Stress1. Normal StressStress is said to be Normal stress when the direction of the deforming force is perpendicular to the cross-sectional area of the body. There are two types of normal stress. a. Tensile StressThe external force per unit area of the material resulting in the stretch of the material is known as tensile stress. b. Compressive StressCompressive stress is the force that is responsible for the deformation of the material, such that the volume of the material reduces.

2. Shearing Stress or Tangential StressWhen the direction of the deforming force or external force is parallel to the cross-sectional area, the stress experienced by the object is called shearing stress or tangential stress. This results in the change in the shape of the body.3. Bulk Stress or Volume StressWhen the deforming force or applied force acts from all dimensions resulting in the change of volume of the object then such stress in called volumetric stress or Bulk stress. In short, when the volume of body changes due to the deforming force it is termed as Volume stress. StrainStrain is defined as the ration of change in configuration (length, shape, volume) of a body to the oringinal configuration of the body. strain = Change in configuration

original configurationTypes of Strain1. Longitudional Strain : Longitudional strain is defined as the change in length per unit length per unit original length due the the deforming force.Longitudional strain = Change in Length (Δl)

original configuration(l)2. Volumetric strain (Bulk strain) : It is defined as change in volume per unit original volume of the body due to deforming force.Bulk strain = Change in volume (ΔV)

original volume(V)3. Shear strain: It is defined as the angle through which the face of the body originally perpendicular to the ffixed fface is turned when it is under the shearing stress.

tan 𝜃= x

lFor small 𝜃𝜃=x

l Elasticity and perfectly elastic bodyAbility of a deformed material body to return to its original shape and size when the forces causing the deformation are removed is known as elasticity. A perfectly elastic body is an object that returns to its original shape and size instanteneously after a deforming force is removed. In practice, no material is perfectly elastic, but some materials are nearly perfectly elastic, such as quartz and phosphor bronze. Plasticity and perfectly plastic bodyThe property of a deformed material body not to return to its original shape and size when the forces causing the deformation are removed is known as elasticity. A perfectly plastic body is an object that returns to its original shape and size at all after a deforming force is removed. In practice, no material is perfectly plastic, but some materials are nearly perfectly elastic, such as clay, wax, polethene etc Elastic LimitWhen a material is stressed below its elastic limit, it will return to its original shape and size when the stress is removed. When a material is stressed beyond its elastic limit, it will permanently deform and will not return to its original shape and size when the stress is removed. So Elastic limit is the upper limit of deforming force upto which if deforming force is removed, the body regains its original form completely. Elastic ModulusHooke's Law can be modified in terms of stress and strain. It states that within elastic limit, the stress developed is directly proportional to the strain produced in a body. i.e a stress ∝ strainor, stress = E × strainor, stress

Strain= EWhere, E is a constant of proportionality known as modulus of elasticity of the material. Higher the value of E, the material is more elastic. Variation of strain with stress (Stress-Strain curve) curves representing the relationship between stress and strain in any form of deformation can be regarded as stress–strain curves.

When a graph between stress and strain is ploted for a wire, it is observed thatI. Upto point A, Hooks law is followed, i.e stress ∝strain. a. So, point A is called Proportional limit.II. Up to B, From A to B, large strain is seen for smaller value of stress but the wire returns to its original length when stress is removed.a. So, B is called elastic limitIII. Up to C, If stress is increased beyond B up to C, the wire doesnot terrutn to its length on removable of stress and there is permanent deformation equal to OO'. In this region wite shows both elastic and plastic behaviour.a. So, point C is called yielding point.IV. Up to D, If stress is increased beyond C, there is a large strain in the wire up to point D.If increased beyond D, the wire will break down.a. So, Point D is called breaking point.Ductile SubstanceDuctility refers to a material's ability to plastically deform before it breaks down. In graph ductile substance have wider region. (from elastic limit, B to breaking point). e.g metals like wrot iron, copper, silver , gold etcBrittle substance: The substance which breakdown just after elastic limit is known as brittle substance. In graph, the substance has narrow region (from elastic limit, B to breaking point). Comparision of Stress-Strain graph

1. Young Modulus : We know E =Stress

Strain = slope of graph So Material A has greater slope and is more elastic.2. Ductility: The region between elastic limit and breakdown pint is more in A So, Material A is more ductile.3. Brittle: Breaking stress (S) of material B is less less than that of A(P). So B can easily breakdown hence is more brittle.4. Breaking point of A (s) is higher so A is more stronger. Types of Modulus of elasticity 1. Young Modulus of elasticity (Y)Young Modulus of elsticity is defined as the ratio of normal stress to the longitudional strain within the elastic limit. Young Modulus (Y) = Normal Stress

Longitudional Strain

Let us consider a wire of radius r and length l. Force F is applied on the wire along its length normal to its cros section area. 'e' be the change in length of the wire, thennormal stress = F

Alongitudional stress = e

lThen,aYoung modulus (Y) = F/A

e/lor Y = Fl

eA Y is the property of material and is indepandent of the dimentsion (length, Area) of the body.For a fixed amount force applied on a wire of constant area A and length l, Y ∝ 1

eHence, the material which elongates more, has less modulus of elasticy and is more elastic. (e.g, rubber is more elastic than iron because it is streatched (elongated) more than iron) 2. Bulk Modulus of elasticity (k)It is defined as the ratio of normal stress to the volumetric strain i.e.,K= Normal stress

Votumetric strain Let us consider a spherical object of volume V is compressed by the forces and its volume is decreased by 𝛥v.

Now, Bulk modulus of elasticity (K)= Normal stress

Volumetric strain Or, K=F/A

𝛥V/VOr, K=FV

𝛥VANote: Compressibility (C) is the reciprocal of bulk modulus of elasticity i.e, C=1

𝜅(iii) Bulk Modulus Or Modulus of rigidity (𝜂) : Modulus of rigidity is defined as the ratio of tangential stress to shear strain i.e. Modulus of rigidity (𝜂)= Tangenkial sfress

Shear sfrain

From figure tangential stress =F orce

Area=F

A And, shear strain (𝜃)=x

T Now, Modulus of rigidity (𝜂)= Tangential sfress

Shear strain a

	𝜂=F/A 𝜃
	∴𝜂=F 𝜃A

Measurement of Young's Modulus of Elasticity (Y)Young's Modulus of elasticity Y of a material (wire) can be measured in the laboratory by using vernier apparatus as shown in the figure.

A reference wire A is stretched by dead load wich contains main scale. Experimental wire B is attached nearby which contains vernier scale. When weight is suspended at the end, wire B is stretched and the stretched amount (Δl) is observed with the value of deforming force / weight (F = mg). The graph obtained is a straight line passing through origin as shown in figure.

The stress in the wire B =F

AStrain = ΔL

LThen, Young Modulus aY = stress

Strain =F/A

ΔL/La= F

ΔL×L

A= slope × L

A∴ Y = slope ×L

AThus, obtaining the slope from experimental data, we can determine the Young's Modulus of elasticity of a material. Poisson's RatioIt is observed that when lenth of a rod or wire is stretched there is reduction in the width. The change in reduction of width with original width is called latteral strain. It is perpendicular to the direction of stress applied.

In the fugure above,longitudional strain (𝛼) = dL

Llateral strain (𝛽) = d

DPoisson studied the change in lateral width with the change in length and found that the lateral strain is directly proportional to the longitudional strain.i.e, Lateral strain ∝ longitudional strain 𝛽∝ 𝛼 𝛽 = 𝜎 𝛼 𝜎 = 𝛽

𝛼This ratio of lateral strain to the longitudional strain is known as poisson's ratio. It is dimenssionless quantity. 𝜎= d/D

dl/L 𝜎 = d

D×L

dL Elastic potential energyWhen a wire is stretched, some work is done against the restoring force. This work is stored in the stretched wire in the form of potential energy.

Let's consider a wire of length L and cross section area A. On applying force F, it elongates by l Then, stress = Force

Area=F

A strain = l

Lwe know,Young modulus,Y=FL

lAor, F = YAl

LLet dl be the small amount of extension and dw be the small amount of force for extension dlthen, dW= force ×dl a =FAl

L ×dl Now, total work done for extension dl is obtained by integrating dw fro l= 0 to l=l W= ∫dW= l∫0YAl

L ×dl =YA

Ll∫ol dl = YA

L [l2

2]l0 = YAl2

2L ∴W =YAl2

2L Or, W=1

2×YAl

L×lor, W= 1

2×Force ×extension The area under the graph of force with the extension shows the amount of work done by the deforming force.

This work done is stored in the wire in the form of elastic potential energy. So elastic potential energy P.E =YAl2

2L Now, the potential energy per unit volume = YAl2

2L×1

Volume = YAl2

2L×1

AL =1

2×Y×l2

L2 = 1

2stress

strain ×strain2 = 1

2×stress ×strainElastic After EffectElastic after effect refers to the phenomenon where an elastic body does not immediately return to its original shape after the deforming force is removed. Instead, there is a temporary delay before it fully recovers its original configuration.Elastic after effect occurs when the strain in a material lags behind the applied stress. Some materials return to their original state immediately, while others take longer. For example, quartz fiber and phosphor bronze returns to its normal state immediately, but glass fiber can take hours.note: Spring used to suspend coil in galvanometer is made up of phosphor bronze because it has negligible elastic after effect, it returns quickly to its original position. Elastic Hysteresis A stress -strain (force-extension) curve plotted after the it has been stretched for several times is shown below. The graph shows that the extension is elastic because it has returned to its original configuration on removing the deforming force. But extension is not directly proportional to the deforming force. Rather the path followed while increasing the load (loading) and while removing the load (unloading) is also different. Such phenomena is known as hysteresis.

We know the area under the force vs extension curve gives the workdone. Here, Area under the loading line (blue) is greater than area under unloading line (red). Which shows the work done while loading is greater than work done while unloading. Some amount of work is lost during the process. Greater the area of hysteresis loop, means more amount work/energy is lost during the process. Note: Rubber with large elastic hysteresis is very useful for absorbing vibration such as in engine mouns and shock absorber busting for cars. Elastic fatigueDeforming forces are external forces that change a material's shape or size, such as stretching, squeezing, or sliding. When these forces are repeatedly applied and removed, the material goes through cycles of deformation. Over time, the material may reach a point where it can't return to its original shape and size after the forces are removed. This is elastic fatigue.The condition when a material loses its elasticity over time due to repeated deforming forces is known as elastic fatigue.Examples:• A thin wire that gradually loses its elastic properties and eventually breaks when repeatedly bent in an alternating direction • Bridges that are declared unsafe after a long time of use • Spring balances that show wrong readings after being used for a long time