MECHANICAL PROPERTIES OF SOLIDS

Mohamed Sherif K, HSST Physics, GHSS Athavanad

Deforming Force & Restoring Force

A force which changes the shape and size of a body is called deforming force

When a deforming force is applied, the body may get deformed. Then the force developed inside the body, which try to bring the body back to its original shape and size is called restoring force.

Elasticity

It is the property of a body by virtue of which it tends to regain its original size and shape after the applied force is removed.Examples of elastic materials − quartz fibre, phosphor bronze

Plasticity

It is the inability of a body in regaining its original status on the removal of the deforming forces.Examples of plastic materials − Wax, mud

9.2 ELASTIC BEHAVIOUR OF SOLIDS

Spring-ball model for the illustration of elastic behaviour of solids

9.3 STRESS AND STRAIN

StressThe restoring force or deforming force experienced by a unit area is called stress. S.I unit = Nm−2

Dimension is [ML-1T-2]

A] Normal Stress When the elastic restoring force or deforming force acts perpendicular to the area, the stress is called normal stress. Normal stress can be sub-divided into the following categories1) Tensile Stress 2) Compressive Stress 3) Volume Stress

B] Tangential or Shearing StressWhen the elastic restoring force or deforming force acts parallel to the surface area, the stress is called tangential stress

9.3 STRESS AND STRAIN

Normal Stress

9.3 STRESS AND STRAIN

Tangential or Shearing Stress

9.3 STRESS AND STRAIN

Strain•Ratio of change in configuration to the original configuration

•Strain =

•It is a dimensionless quantity

Types of Strain1. Longitudinal Strain 2. Volumetric Strain 3. Shearing Strain

9.3 STRESS AND STRAIN

Longitudinal Strain

Longitudinal Strain =

9.3 STRESS AND STRAIN

Volumetric Strain

Volumetric Strain =

9.3 STRESS AND STRAIN

Shearing Strain

Shearing Strain

9.4 HOOKE’S LAW

Hooke’s law state that within the elastic limit, stress is directly proportional to strain

Stress α Strainstress = k × strainwhere k is known as modulus of elasticity

9.5 STRESS-STRAIN CURVE

9.5 STRESS-STRAIN CURVE

Part OAThis part OA obeys Hooke’s law. The point ‘O’ is called elastic limit or yield point. Corresponding stress is called yield strength. In this region the material behaves as elastic material.

Part ABIn this region a small increase in stress produce a large change in strain. At any point between AB, if the deforming force is removed, the body will never return to the original length. But results a permanent change in length (Eg: OO’). This is known as permanent set.

9.5 STRESS-STRAIN CURVE

Part BC :The stress corresponding to the point ‘C’ is called Ultimate tensile strength (Su). This is the maximum stress that can be applied to a wire.

Part CD :Beyond the point C, additional strain is produced even by a reduced stress and the wire breaks at D

Ductile SolidsMaterials have a large CD region is called ductile solids. So these materials can be drawn into thin wires eg: copper , AluminumBrittle MaterialsIf C and D are very close, the material is said to be brittle. It suddenly breaks as soon as the ultimate strength (C) is crossed Eg : Glass.

Elastomer

Substances which can stretch to large values of strain are called elastomer.These materials does not obey Hooke’s lawEg: Rubber band, Aorta

Elastic Moduli

According to Hooke’s law, within elastic limit,Stress Strain∝Stress = k × Strain

k =

known as modulus of elasticity

Types of modulus of elasticity1. Young’s Modulus of Elasticity (Y)2. Bulk Modulus (B)3. Rigidity Modulus (G)

Young’s Modulus (Y)

Y =

Y =

∴ Y =

Where,F - Force appliedr - Radius of the wirel - Original lengthΔl - Change in lengthUnit → Nm−2 or Pascal (denoted by Pa)

Bulk Modulus (B)

If P is the increase in pressure applied on the spherical body, thenP = F/A

B =

∴ B =

V - Original volumeΔV - Change in volumeUnit → Nm−2 or Pascal

Compressibility (k) − Reciprocal of bulk modulus of elasticity (B) i.e., k = 1/B

Rigidity Modulus (G)

G =

G =

Where,F - Force applieda - AreaL- Original lengthΔL - Change in lengthUnits → Nm−2 or Pascal

Stress, Strain & Modulii of Elasticity

Applications of Elastic Behaviour of Materials

Construction of a BeamConsider a bar of length l, breadth b, and depth (height) ‘d ‘ . Let a load ‘W’ be applied at the mid point of the bar. The mid-point will sag by anamount ‘δ’ is given by

[Given; the beam has to support a maximum load ‘W’ and Beam length or span is ‘l’]

Applications of Elastic Behaviour of Materials

From the expression it is clear that, to reduce bending of a Steel Bar; Use material of large Young’s modulus (Y) . It is better to increase thedepth (δ α 1/d3) rather than the breadth (δ α 1/b).

Applications of Elastic Behaviour of Materials

As depth increases the chance of buckling also increases. So to avoid buckling and bending I section beams are used. This shape reduces the weight of the beam without sacrificing thestrength and hence reduces the cost.

Height of a Mountain

At the base of the mountain pressure (hρg) is exerted. Where h-height of mountain, ρ- density of rocks(3×103). But the elastic limit of typical rock at the bottom of mountain is 30 × 107 .Equating; hρg = 30 × 107

h = 10kmFrom the calculations it is clear that maximum height of mountain must be less than 10 km.

9.6.5 Poisson’s Ratio

The strain perpendicular to the applied force is called lateral strain

Simon Poisson pointed out that within the elastic limit; lateral stain is directly proportional to the longitudinal strain.

The ratio of the lateral strain to the longitudinal strain in a stretched wire is called Poisson’s ratioPoisson’s ratio = (Δd/d)/(ΔL/L) = (Δd/ΔL) (L/d)

it is a pure number and has no dimensions or unitsFor steels the value is between 0.28 and 0.30, and for aluminum alloys it is about 0.33

9.6.6 Elastic Potential Energy in a Stretched Wire

When a wire is put under a tensile stress, work is done against the inter-atomic forces.This work is stored in the wire in the form of elastic potential energyF = YA (l/L).work done dW = F dl or YAld l /L

W = ½ × Young’s modulus strain2 volume of the wire = ½ × stress strain volume of the wire

This work is stored in the wire in the form of elastic potential energy (U). Therefore the elastic potential energy per unit volume of the wire (u) is