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Notes On Elasticity

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Published in: Physics
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Notes on Physics

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  1. Elasticity Introduction Elasticity is the branch of physics that studies the effect of forces on solid bodies, which cause a deformation or change of shape and the subsequent return to the original shape after the removal of the deforming force. You might have noticed that when an external force is applied on an object, its shape or size (or both) change, i.e., deformation takes place. The extent of deformation depends on the material and shape of the body and the external force. When the deforming forces are withdrawn, the body tries to regain its original shape and size. 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 it, you will observe that its 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. Fundamental to elasticity of solid bodies are the concepts of 'stress' and 'strain'. These have been discussed below in details. 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 fibre and to the perfectly plastic is ordinary putty. The phenomenon of elasticity can be best explained in terms of inter-molecular forces. 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. 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 interatomic distances change. If in deformation, the separation increases beyond their equilibrium separation (i.e., R >RO), strong attractive forces are developed. However, if inter—atomic separation decreases (i.e. R < IRO), strong repulsive forces develop. These forces, called restoring forces, drive atoms to their original positions. The behaviour of atoms in a solid 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.
  2. STRESS, STRAIN AND THE BASIC DEFORMATIONS The study of elasticity is concerned with how bodies deform under the action of pairs of applied forces. In this study there are two basic concepts: stress and strain. The pairs of forces act in opposite directions along the same line. Thus, there is no resulting acceleration (change of motion) but there is a resulting deformation or change in the size or shape of the body. This is described in terms of strain. The strain is the relative change in dimensions of a body resulting from the external forces. As a result of the deformation, internal forces are set up and these give rise to stresses. In many simple cases, these stresses are simply related to the external forces, because when these two are in balance the deformation will be maintained without further change. For these simple cases we make the following definition. The stress is the external force divided by the area over which this force is applied. 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 Its unit is N/m2 or Nm -2 Area (A) There are three particular cases we will consider — longitudinal or linear, normal and shearing. V z. 12-11 sutlB.•ttn an an-aunt. nrtewhat of Figure 1 (a), (b), (c)
  3. Figure 1 shows three ways in which a solid might change its dimensions when forces act on it. In Fig. la, a cylinder is stretched. In Fig. 1b, a cylinder is deformed by a force perpendicular to its long axis, much as we might deform a pack of cards or a book. In Fig. lc, a solid object placed in a fluid under high pressure is compressed uniformly on all sides. What the three deformation types have in common is that a stress, or deforming force per unit area, produces a strain, or unit deformation. In Fig. 1, tensile stress (associated with stretching) is illustrated in (a), shearing stress in (b) and hydraulic stress in (c). The stresses and the strains take different forms in the three situations of Fig. 12-11, but— over the range of engineering usefulness—stress and strain are proportional to each other. The constant of proportionality is called a modulus of elasticity, so that (ii) Stress = Modulus x Strain 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 2 (a) and (b). Figure 2 (a) Tensile Stress F Figure 2 (b) Compressive Stress Normal Stress: If the deforming forces are applied uniformly and normally all over the surface of the body so that the change in its volume occurs without change in shape (Fig. 3), 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. Figure 3 Normal Stress Shearing Stress: If the deforming forces act tangentially or parallel to the surface (Fig. 4a) so that shape of the body changes without change in volume, the stress is called shearing stress. An example of shearing stress is shown in Fig 4 (b) in which a book is pushed side ways. Its opposite face is held fixed by the force of friction. Figure 4. Shearing Stress.
  4. Tension and Compression For simple tension or compression, the stress on the object is defined as F/A, where F is the magnitude of the force applied perpendicularly to an area A on the object. The strain, or unit deformation, is then the dimensionless quantity AL/L, the fractional (or sometimes percentage) change in a length of the specimen. If the specimen is a long rod and the stress does not exceed the yield strength, then not only the entire rod but also every section of it experiences the same strain when a given stress is applied. Because the strain is dimensionless, the modulus in Eq. (ii) has the same dimensions as the stress—namely, force per unit area. The modulus for tensile and compressive stresses is called the Young's modulus and is represented in engineering practice by the symbol E. Equation (ii) becomes F AL (iii) The strain AL/L in a specimen can often be measured conveniently with a strain gage, which can be attached directly to operating machinery with an adhesive. Its electrical properties are dependent on the strain it undergoes. Although the Young's modulus for an object may be almost the same for tension and compression, the object's ultimate strength may well be different for the two types of stress. Concrete, for example, is very strong in compression but is so weak in tension that it is almost never used in that manner. Shearing In the case of shearing, the stress is also a force per unit area, but the force vector lies in the plane of the area rather than perpendicular to it. The strain is the dimensionless ratio Ax/L, with the quantities defined as shown in Fig. 1b. The corresponding modulus, which is given the symbol G in engineering practice, is called the shear modulus. For shearing, Eq. (iii) is written as Eq. (iv) F AX (iv) Shearing occurs in rotating shafts under load and in bone fractures due to bending. Hydraulic Stress In Fig. lc, the stress is the fluid pressure p on the object, and, as you will see in Chapter 14, pressure is a force per unit area. The strain is 'V/ V, where V is the original volume of the specimen and 'V is the absolute value of the change in volume.
  5. The corresponding modulus, with symbol B, is called the bulk modulus of the material. The object is said to be under hydraulic compression, and the pressure can be called the hydraulic stress. For this situation, we write Eq. (ii) as AV (v) The bulk modulus is 2.2 x 109 N/m2 for water and 1.6 x 1011 N/m2 for steel. The pressure at the bottom of the Pacific Ocean, at its average depth of about 4000 m, is 4.0 x 107 N/m2. The fractional compression AV/V of a volume of water due to this pressure is 1.8%; that for a steel object is only about 0.025%. In general, solids—with their rigid atomic lattices—are less compressible than liquids, in which the atoms or molecules are less tightly coupled to their neighbours. Strain Deforming forces produce changes in the dimensions of the body. In general, the strain is defined as the change in dimension (e.g. length, shape or volume) per unit dimension of the body. As the strain is ratio of two similar quantities, it is a dimensionless quantity. Depending on the kind of stress applied, strains are of three types: (i) linear strain, (ii) volume (bulk) strain and (iii) shearing strain. (i) Linear Strain: If on application of a longitudinal deforming force, the length I of a body Change in Length AL changes by Al (Fig. 5), then Linear Strain = (vi) Original Length — L Figure 5. Linear Strain (ii) Volume Strain: If on application of a uniform pressure AP, the volume V of the body changes by AV (Fig 6) without change of shape of the body, Change in Volume AV (vii) then Volume Strain = Original Volume V Figure 6. Volume Strain åp .åp (iii) Shearing strain: When the deforming forces are tangential (Fig 7), the shearing strain is given by the angle 0 through which a line perpendicular to the fixed plane is turned due to AX deformation. (The angle 0 is usually very small.) Then we can write 0 = — (viii) Figure 7. Shearing Strain Fi-*bå
  6. Some More Related Terms Here, we shall discuss some more relevant and related terms to Elasticity before moving on to the next section. Elastic body: An elastic body is one that returns to its original shape after a deformation. Elastic collision: An elastic collision loses no energy. The deformation on collision is fully restored. Inelastic body: An inelastic body is one that does not return to its original shape after a deformation. Inelastic collision: In an inelastic collision, energy is lost and the deformation may be permanent. Tensile stress: A tensile stress occurs when equal and opposite forces are directed away from each other. Compressive stress: A compressive stress occurs when equal and opposite forces are directed towards each other. Elastic Limit: The elastic limit is the maximum stress a body can experience without becoming permanently deformed. Ultimate Strength: The ultimate strength is the greatest stress a body can experience without breaking or rupturing. Hooke's Law Objects, like springs, that you can stretch but that return to their original shapes are called — it means you can use springs for all kinds of elastic. Elasticity is a valuable property applications: as shock absorbers in lunar landing modules, as timekeepers in clocks and watches, and even as hammers of justice in mousetraps. As long ago as the 1600s, Robert Hooke, a physicist from England, undertook the study of elastic materials. He created a new law, not surprisingly called Hooke's law, which states that stretching an elastic material gives you a force that's directly proportional to the amount of stretching you do. For example, if you stretch a spring a distance x, you'll get a force back that's directly proportional to x: (vii) where k is the spring constant. In fact, the force F resists your pull, so it pulls in the opposite direction, which means you should have a negative sign here:
  7. (viii) F = —kx In 1678, Robert Hooke obtained the stress-strain curve experimentally for a number of solid substances and established a law of elasticity known as Hooke's law. According to this law: Within elastic limit, stress is directly proportional to corresponding strain. i.e. stress a strain Stress . = Constant (E) or Stram This constant of proportionality E is a measure of elasticity of the substance and is called modulus of elasticity. As strain is a dimensionless quantity, the modulus of elasticity has the same dimensions (or units) as stress. Its value is independent of the stress and strain but depends on the nature of the material. Hooke's law is valid as long as the elastic material you're dealing with stays elastic — that is, it stays within its elastic limit. If you pull a spring too far, it loses its stretchy ability, for example. In other words, as long as a spring stays within its elastic limit, you can say that F = —kx, where the constant k is called the spring constant. The constant's units are Newtons per meter. When a spring stays within its elastic limit, it's called an ideal spring. The negative sign in Hooke's law for an elastic spring is important: F = —kx The negative sign means that the force will oppose your displacement, as you see in Figure 8, which shows a ball attached to a spring. L Figure 8. The direction of force form a spring. As you see in Figure 8, if the spring isn't stretched or compressed, it exerts no force on the ball. If you push the spring, however, it pushes back, and if you pull the spring, it pulls back. The force exerted by a spring is called a restoring force. It always acts to restore itself toward equilibrium. Poisson's Ratio You may have noticed that when a rubber tube is stretched along its length, there is a contraction in its diameter (Fig. 9). (This is also true for a wire but may not be easily visible.) While the length increases in the direction of forces, a contraction occurs in the perpendicular direction. The strain perpendicular to the applied force is called lateral strain. Poisson pointed
  8. out that within elastic limit, lateral strain is directly proportional to longitudinal strain, i.e., the ratio of lateral strain to longitudinal strain is constant for a material body and is known as Poisson's ratio. It is denoted by a Greek letter o (sigma). If a and ß are the longitudinal strain and lateral strain respectively, then Poisson's ratio o = ß / a. If a wire (rod or tube) of length I and diameter d is elongated by applying a stretching force AL by an amount Al and its diameter decreases by Ad, then longitudinal strain a = — AD lateral strain ß= AD/D AD/L and Possion's ratio o = (ix) AL/L - AL/D Since Poisson's ratio is a ratio of two strains, it is a pure number. i Figure 9. A stretched rubber tube. 1 The value of Poisson's ratio depends only on the nature of material and for most of the substances, it lies between 0.2 and 0.4. When a body under tension suffers no change in volume, i.e., the body is perfectly incompressible, the value of e Poisson's ratio is maximum, i.e., 0.5. Theoretically, the limiting values of Poisson's ratio are — 1 and 0.5. Take two identical wires. Make one wire to execute torsional vibrations for some time. After some time, set the other wire also in similar vibrations. Observe the rate of decay of vibrations of the two wires. You will note that the vibrations decay much faster in the wire, which was vibrating for longer time. The wire gets tired or fatigued and finds it difficult to continue vibrating. This phenomenon is known as elastic fatigue. Some other facts about elasticity: 1. If we add some suitable impurity to a metal, its elastic properties are modified. For example, if carbon is added to iron or potassium is added to gold, their elasticity increases. 2. The increase in temperature decreases elasticity of materials. For example, carbon, which is highly elastic at ordinary temperature, becomes plastic when heated by a current through it. Similarly, plastic becomes highly elastic when cooled in liquid air. 3. The value of modulus of elasticity is independent of the magnitude of stress and strain. It depends only on the nature of the material of the body. Applications of Elastic Behaviour of Materials Elastic behaviour of materials plays an important role in our day-to-day life. Pillars and beams are important parts of our structures. A uniform beam clamped at one end and loaded at the other is called a Cantilever [Fig. 9].
  9. Figure 9. A cantilever. The hanging bridge of Laxman Jhula in Rishkesh and Howrah Bridge in Kolkata are supported on cantilevers. A cantilever of length l, breadth b and thickness d undergoes a depression 5 at its free end when it is loaded by a weight of mass M: 4Mg13 (x) Ybd3 It is now easy to understand as to why the cross-section of girders and rails is kept I-shaped (Fig. 10). Other factors remaining same, 5 a $3. Therefore, by increasing thickness, we can decrease depression under the same load more effectively. This considerably saves the material without sacrificing strength for a beam clamped at both ends and loaded in the middle (Fig. 11), the sag in the middle is given by, Mg13 (xi) 4Ybd3 Thus for a given load, we will select a material with a large Young's modulus Y and again a large thickness to keep 5 small. However, a deep beam may have a tendency to buckle (Fig 12). To avoid this, a large load-bearing surface is provided. In the form I-shaped cross-section, both these requirements are fulfilled. Figure 10 Figure 11 Figure 12 In cranes, we use a thick metal rope to lift and move heavy loads from one place to another. To lift a load of 10 metric tons with a steel rope of yield strength 300 mega Pascal, it can be shown, that the minimum area of cross section required will be 10 cm or so. A single wire of this radius will practically be a rigid rod. That is why ropes are always made of a large number of turns of thin wires braided together. This provides ease in manufacturing, flexibility and strength. The maximum height of a mountain on earth can be 10 km or else the rocks under it will shear under its load.

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