Elastic Properties of Solids MODULE - 2 Mechanics of Solids and Fluids 8 Notes ELASTIC PROPERTIES OF SOLIDS In the previous lessons you have studied the effect of force on a body to produce displacement. The force applied on an object may also change its shape or size. For example, when a suitable force is applied on a spring, you will find that its shape as well as size changes. But when you remove the force, it will regain original position. Now apply a force on some objects like wet modelling clay or molten wax. Do they regain their original position after the force has been removed? They do not regain their original shape and size. Thus some objects regain their original shape and size whereas others do not. Such a behaviour of objects depends on a property of matter called elasticity. The elastic property of materials is of vital importance in our daily life. It is used to help us determine the strength of cables to support the weight of bodies such as in cable cars, cranes, lifts etc. We use this property to find the strength of beams for construction of buildings and bridges. In this unit you will learn about nature of changes and the manner in which these can be described. OBJECTIVES After studying this lesson, you should be able to : z distinguish between three states of matter on the basis of molecular theory; z distinguish between elastic and plastic bodies; z distinguish between stress and pressure; z study stress-strain curve for an elastic solid; z define Young’s modulus, bulk modulus, modulus of rigidity and Poisson’s ratio; and z derive an expression for the elastic potential energy of a spring. PHYSICS 213 MODULE - 2 Elastic Properties of Solids Mechanics of Solids and Fluids 8.1 MOLECULAR THEORY OF MATTER : INTER- MOLECULAR FORCES We know that matter is made up of atoms and molecules. The forces which act between them are responsible for the structure of matter. The interaction forces between molecules are known as inter-molecular forces. Notes The variation of inter molecular Repulsion forces with inter molecular separation is shown in Fig. 8.1. When the separation is large, Force F the force between two distance R molecules is attractive and O weak. As the separation Attraction R0 decreases, the net force of attraction increases up to a Fig. 8.1 : Graph between inter-molecular force and particular value and beyond Inter molecular separation. this, the force becomes repulsive. At a distance R = R 0 the net force between the molecules is zero. This separation is called equilibrium separation. Thus, if inter-molecular separation R > R0 there will be an attractive force between molecules. When R < R0 , a repulsive force will act between them. In solids, molecules are very close to each other at their equilibrium separation –10 ( 10 m). Due to high intermolecular forces, they are almost fixed at their positions. You may now appreciate why a solid has a definite shape. In liquids, the average separation between the molecules is somewhat larger –8 ( 10 m). The attractive force is weak and the molecules are comparatively free to move inside the whole mass of the liquid. You can understand now why a liquid does not have fixed shape. It takes the shape of the vessel in which it is filled. In gases, the intermolecular separation is significantly larger and the molecular force is very weak (almost negligible). Molecules of a gas are almost free to move inside a container. That is why gases do not have fixed shape and size. Ancient Indian view about Atom Kanada was the first expounder of the atomic concept in the world. He lived around 6th century B.C. He resided at Prabhasa (near Allahabad). According to him, everything in the universe is made up of Parmanu or Atom. They are eternal and indestructible. Atoms combine to form different molecules. If two atoms combine to form a molecule, it is called duyanuka and a triatomic molecule is called triyanuka. He was the author of “Vaisesika Sutra”. 214 PHYSICS Elastic Properties of Solids MODULE - 2 Mechanics of Solids The size of atom was also estimated. In the biography of Buddha (Lalitavistara), and Fluids the estimate of atomic size is recorded to be of the order 10–10 m, which is very close to the modern estimate of atomic size. 8.2 ELASTICITY You would have noticed that when an external force is applied on an object, its Notes shape or size (or both) change, i.e. deformation takes place. The extent of deformation depends on the material and shape Bow of the body and the external force. When the deforming forces are withdrawn, the body tries to regain its original shape and size. string You may compare this with a spring loaded with a mass or a force applied on the string of a bow or pressing of a rubber ball. If you apply a force on the string of the bow to pull Fig 8.2 : Force applied on the it ( Fig 8.2), you will observe that its shape string of a bow changes it shape changes. But on releasing the string, the bow regains its original shape and size. The property of matter to regain its original shape and size after removal of the deforming forces is called elasticity. 8.2.1 Elastic and Plastic Bodies A body which regains its original state completely on removal of the deforming force is called perfectly elastic. On the other hand, if it completely retains its modified form even on removing the deforming force, i.e. shows no tendency to recover the deformation, it is said to be perfectly plastic. However, in practice the behaviour of all bodies is in between these two limits. There exists no perfectly elastic or perfectly plastic body in nature. The nearest approach to a perfectly elastic body is quartz fiber and to the perfectly plastic is ordinary putty. Here it can be added that the object which opposes the deformation more is more elastic. No doubt elastic deformations are very important in science and technology, but plastic deformations are also important in mechanical processes. You might have seen the processes such as stamping, bending and hammering of metal pieces. These are possible only due to plastic deformations. The phenomenon of elasticity can be explained in terms of inter-molecular forces. 8.2.2 Molecular Theory of Elasticity You are aware that a solid is composed of a large number of atoms arranged in a definite order. Each atom is acted upon by forces due to neighbouring atoms. PHYSICS 215 MODULE - 2 Elastic Properties of Solids Mechanics of Solids and Fluids Due to inter-atomic forces, solid takes such a shape that each atom remains in a stable equilibrium. When the body is deformed, the atoms are displaced from their original positions and the inter-atomic distances change. If in deformation, the separation increases beyond their equilibrium separation (i.e., R >R0), strong attractive forces are developed. However, if inter–atomic separation decreases (i.e. R < R0), strong repulsive forces develop. These forces, called restoring forces, drive atoms to their original positions. The behaviour of atoms in a solid Notes can be compared to a system in which balls are connected with springs. Now, let us learn how forces are applied to deform a body. 8.2.3 Stress When an external force or system of forces is applied on a body, it undergoes a change in the shape or size according to nature of the forces. We have explained that in the process of deformation, internal restoring force is developed due to molecular displacements from their positions of equilibrium. The internal restoring force opposes the deforming force. The internal restoring force acting per unit area of cross-section of a deformed body is called stress. In equilibrium, the restoring force is equal in magnitude and opposite in direction to the external deforming force. Hence, stress is measured by the external force per unit area of cross-section when equilibrium is attained. If the magnitude of deforming force is F and it acts on area A, we can write restoring force deforming force (F ) Stress = = area area (A ) F or Stress = (8.1) A The unit of stress is Nm–2 . The stress may be longitudinal, normal or shearing. Let us study them one by one. (i) Longitudinal Stress : If the deforming forces are along the length of the body, we call the stress produced as longitudinal stress, as shown in its two forms in Fig 8.3 (a) and Fig 8.3 (b). FF (a) FF (b) Fig. 8.3 (a) : Tensile stress; (b) Compressive stress 216 PHYSICS Elastic Properties of Solids MODULE - 2 Mechanics of Solids (ii) Normal Stress : If the deforming forces are applied uniformly and normally and Fluids all over the surface of the body so that the change in its volume occurs without change in shape (Fig. 8.4), we call the stress produced as normal stress. You may produce normal stress by applying force uniformly over the entire surface of the body. Deforming force per unit area normal to the surface is called pressure while restoring force developed inside the body per unit area normal to the surface is known as stress. Notes F F F F F F F F F F F F F F (a) (b) Fig. 8.4 : Normal stress (iii) Shearing Stress : If the deforming forces act tangentially or parallel to the surface (Fig 8.5a) so that shape of the body changes without change in volume, the stress is called shearing stress.