Page 1 :
Mechanical Properties of Solids, , 1. Introduction, A rigid body generally means a hard solid, object having a definite shape and size. But, in reality, bodies can be stretched,, compressed and bent. Even the appreciably, rigid steel bar can be deformed when a, sufficiently large external force is applied on, it. This means that solid bodies are not, perfectly rigid. A solid has definite shape, and size. In order to change (or deform) the, shape or size of a body, a force is required., , 2. Deforming Force, A force which produces a change in, configuration (size or shape) of the object, on applying it, is called a deforming force., , 3. Elasticity, Elasticity is that property of the object by, virtue of which it regains its original, configuration after the removal of the, deforming force., For example, if we stretch a rubber band, and release it, it snaps back to its original, length., , 4. Perfectly Elastic Body, , If a body does not regains its original size, and shape completely and immediately, after the removal of deforming force, it is, said to be a plastic body and this property is, called plasticity., , 6. Perfectly plastic body, That body which does not regain its original, configuration at all on the removal of, deforming force are called perfectly plastic, bodies.Putty and paraffin wax are nearly, perfectly plastic bodies., , 7. Stress, If a body gets deformed under the action of, an external force, then at each section of, the body an internal force of reaction is set, up which tends to restore the body into its, original state., 7.1 Definition, The internal restoring force set up per unit, area of cross section of the deformed body, is called stress., 7.2 Mathematical Form, Stress =, , Applied Force, 𝐴𝑟𝑒𝑎, , Those bodies which regain its original, configuration immediately and completely, after the removal of deforming force are, called perfectly elastic bodies. The nearest, approach to a perfectly elastic body is, quartz fibre., , Its unit is N/m2 or Pascal., , 5. Plasticity, , 1. Longitudinal Stress, , Its dimensional formula is [ML-1T -2 ]., 7.3 Types of stress, There are three different types of stress

Page 2 :
Mechanical Properties of Solids, If deforming force is applied normal to the area, of cross section, then the stress is called, longitudinal stress. It is further categorized in, , 8.1 Mathematical Equation, Strain =, , two types, (a) Tensile stress If there is an increase, in length of the object under the effect, of applied force, then stress is called, tensile stress., (b) Compressional stress If there is a, decrease in length of the object under, the effect of applied force, then stress is, called compression stress., , change in dimension, original dimension, , It has no unit and it is a dimensionless, quantity., According, to, the, change, in, configuration, the strain is of three, types, (1) longitudinal strain=, (2) Volumetric strain =, , change in length, original length, change in volume, Original volume, , applied force, (3) 𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑠𝑡𝑟𝑎𝑖𝑛 tangential, Area of face, , 9. Hooke’s Law, , 2. Tangential or Shearing Stress, If deforming force 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, called tangential stress., 3. Normal Stress, If a body is subjected to a uniform force, from all sides, then the corresponding, stress is called hydrostatic stress., , 8.Strain, When a deforming force acts on a body,, the body undergoes a change in its, shape and size. The fractional change in, configuration is called strain., , Robert Hook found that within the, elastic limit, the stress is directly, proportional to strain. Thus we have, 𝒔𝒕𝒓𝒆𝒔𝒔 ∝ 𝒔𝒕𝒓𝒂𝒊𝒏, or, 𝒔𝒕𝒓𝒆𝒔𝒔 = 𝑲. 𝒔𝒕𝒓𝒂𝒊𝒏, where K is the constant of, proportionality called “Elastic Modulus”, of the material., There are some materials that do not, obey Hooke’s law like rubber, human’s, muscle., 9.1 Types of Modulus of rigidity, 9.1.1 Young’s Modulus of rigidity (Y)

Page 3 :
Mechanical Properties of Solids, , It is defined as the ratio of normal stress, to the longitudinal strain within the, elastic limit., Y=, , 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠, Longitudinal strain, , It has same units as stress because, strain does not have any unit. Y is, measured in N/m2 or Pa., Metals generally have large values of, Young’s modulus compare to other, materials. In scientific terms, the higher, the Young’s modulus of the material the, more elastic it is., 9.1.2 Bulk Modulus of Rigidity, κ=, or, , 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠, Volumetric strain, , κ=, , −𝐹/𝐴, ∆V/V, , Compressibility, Compressibility of a material is the, reciprocal of its bulk modulus of elasticity., Compressibility (C) = 1 / k, Its SI unit is N-1m 2 and CGS unit is dyne-1, cm2 ., , 9.1.3 Modulus of rigidity or shear, Modulus (𝜼), 𝜼=, , 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠, shear strain, 𝐹, 𝐹, 𝜼= 𝐴 =, ϒ 𝐴ϒ, 𝜼=, , 𝐹, Aϒ, , = - pV/∆𝑉, , The SI unit of bulk modulus is N/m2, , The SI unit of shear modulus is N/m2, The shear modulus of a material is, always considerably smaller than the, Young’s modulus for it., , 10. Limit of elasticity, The maximum value of deforming force for, which elasticity is present in the body is, called its limit of elasticity.

Page 4 :
Mechanical Properties of Solids, , 11.Stress- strain Curve, , Figure shows the stress-strain curve for a, metal wire which is gradually being loaded., (a) The initial part OA of the graph is a, straight line indicating that stress is, proportional to strain. Upto the point A,, Hooke’s law is obeyed. The point A is called, the proportional limit. In this region, the, wire is perfectly elastic., (b) After the point A, the stress is not, proportional to strain and a curved portion, AB is obtained. However, if the load is, removed at any point between O and B, the, curve is retraced along BAO and the wire, attains its original length. The portion OB of, the graph is called elastic region and the, point B is called elastic limit or yield point., The stress corresponding to B is called yield, strength., (c) Beyond the point B, the strain increases, more rapidly than stress. If the load is, removed at any point C, the wire does not, come back to its original length but traces, dashed line. Even on reducing the stress to, , zero, a residual strain equal to OE is left in, the wire. The material is said to have, acquired a permanent set. The fact that, stress-strain curve is not retraced on, reversing the strain is called elastic, hysteresis., (d) If the load is increased beyond the point, C, there is large increase in the strain or the, length of the wire. In this region, the, constrictions ( called necks and waists), develop at few points along the length of, the wire and the wire breaks ultimately at, the point D, called the fracture point., In the region between B and D, the length, of the wire goes on increasing even without, any addition of load. This region is called, plastic region and material is said to, undergo plastic flow or plastic deformation., The stress corresponding to the braking, point is called ultimate strength or tensile, strength of the material., , 12. Elastic after Effect, The bodies return to their original state on, the removal of the deforming force. Some, bodies return to their original state, immediately after the removal of the, deforming force while some bodies take, longer time to do so. The delay in regaining, the original state by a body on the removal, of the deforming force is called elastic after, effect., , 13. Elastic Fatigue, The property of an elastic body by virtue of, which its behavior becomes less elastic

Page 5 :
Mechanical Properties of Solids, , under the action of repeated alternating, deforming force is called elastic fatigue., , 14. Ductile Materials, The materials which have large plastic range, of extension are called ductile materials., Such materials undergo an irreversible, increase in length before snapping. So they, can be drawn into thin wires. For e.g., copper, silver, iron, aluminium etc., , When a deforming force is applied at the, free end of a suspended wire of length 1, and diameter D, then its length increases by, ∆l but its diameter decreases by ∆𝐷. Now, two types of strains are produced by a, single force., (i) Longitudinal strain =, (ii) Lateral strain = –, , ∆𝑙, 𝑙, , −∆𝐷, D, 𝐿𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛, , ∴ Poisson’s Ratio (σ) = 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛=, , 15 Brittle Materials, The materials which have very small range, of plastic extension are called brittle, materials. Such materials break as soon as, the stress is increased beyond the elastic, limit. For e.g. cast iron, glass, ceramics etc., , The materials for which strain produced is, much larger than the stress applied, with in, the limit of elasticity are called elastomers,, e.g., rubber, the elastic tissue of aorta, the, large vessel carrying blood from heart. etc., Elastomers have no plastic range., , Energy, , of, , When a wire is stretched, interatomic, forces come into play which opposes the, change. Work has to done against these, restoring forces. The work done in, stretching the wire is stored in it as its, elastic potential energy., , 18. Poisson’s Ratio, , −∆𝐷, D, ∆𝑙, 𝑙, , =−, , 𝑙∆𝐷, 𝐷∆𝑙, , The negative sign shows that longitudinal, and lateral strains are in opposite sense., As Poisson’s ratio is the ratio of two strains,, it has no units and dimensions., , 16. Elastomers, , 17. Elastic Potential, stretched wire, , =, , The theoretical value of Poisson’s ratio lies, between – 1 and 0.5. Its practical value lies, between 0 and 0.5, , 19. Applications of elasticity, The elastic behavior of materials plays an, important role in everyday life. All, engineering designs require precise, knowledge of the elastic behavior of, materials. For example while designing a, building, the structural design of the, columns, beams and supports require, knowledge of strength of materials used., A bridge has to be designed such that it can, withstand the load of the flowing traffic, the, force of winds and its own weight. Similarly,, in the design of buildings use of beams and

Page 6 :
Mechanical Properties of Solids, , columns is very common. In both the cases,, the overcoming of the problem of bending, of beam under a load is of prime, importance. The beam should not bend too, much or break. Let us consider the case of a, beam loaded at the centre and supported, near its ends as shown in Fig., , A bar of length l, breadth b, and depth d, when loaded at the centre by a load W sags, by an amount given by δ = W l 3/(4bd3Y), Bending can be reduced by using a material, with a large Young’s modulus Y. Depression, can be decreased more effectively by, increasing the depth d rather than the, breadth b. But a deep bar has a tendency to, bend under the weight of a moving traffic,, hence a better choice is to have a bar of Ishaped cross section. This section provides, a large load bearing surface and enough, depth to prevent bending. Also this shape, reduces the weight of the beam without, sacrificing its strength and hence reduces, the cost.