ELASTIC PROPERTIES OF SOLIDS
In our study of mechanics thus far, we have assumed that objects remain undeformed when external forces act on them. In reality, all objects are deformable. That is, it is possible to change the shape or the size of an object (or both) by applying external forces. As these changes take place, however, internal forces in the object resist the deformation.
We shall discuss the deformation of solids in terms of the concepts of stress and strain. Stress is a quantity that is proportional to the force causing a deformation; more specifically, stress is the external force acting on an object per unit cross-sectional area. Strain is a measure of the degree of deformation. It is found that, for sufficiently small stresses, strain is proportional to stress; the constant of proportionality depends on the material being deformed and on the nature of the deformation. We call this proportionality constant the elastic modulus. The elastic modulus is therefore the ratio of the stress to the resulting strain:
In a very real sense it is a comparison of what is done to a solid object (a force is applied) and how that object responds (it deforms to some extent). 
A plastic model of an arch structure under load conditions. The wavy lines indicate regions where the stresses are greatest. Such models are useful in designing architectural components.
We consider three types of deformation and define an elastic modulus for each:
We shall discuss the deformation of solids in terms of the concepts of stress and strain. Stress is a quantity that is proportional to the force causing a deformation; more specifically, stress is the external force acting on an object per unit cross-sectional area. Strain is a measure of the degree of deformation. It is found that, for sufficiently small stresses, strain is proportional to stress; the constant of proportionality depends on the material being deformed and on the nature of the deformation. We call this proportionality constant the elastic modulus. The elastic modulus is therefore the ratio of the stress to the resulting strain:
In a very real sense it is a comparison of what is done to a solid object (a force is applied) and how that object responds (it deforms to some extent). A plastic model of an arch structure under load conditions. The wavy lines indicate regions where the stresses are greatest. Such models are useful in designing architectural components.We consider three types of deformation and define an elastic modulus for each:
Young’s Modulus: Elasticity in Length
Consider a long bar of cross-sectional area A and initial length Li that is clamped at one end, as in Figure 12.13. When an external force is applied perpendicular to the cross section, internal forces in the bar resist distortion (“stretching”), but the bar attains an equilibrium in which its length Lf is greater than Liand in which the external force is exactly balanced by internal forces. In such a situation, the bar is said to be stressed. We define the tensile stress as the ratio of the magnitude of the external force F to the cross-sectional area A. The tensile strain in this case is defined as the ratio of the change in length ΔL to the original length Li . We define Young’s modulus by a combination of these two ratios:
Figure 12.13 A long bar clamped at one end is stretched by an amount ΔL under the action of a force F.
Young’s modulus is typically used to characterize a rod or wire stressed under either tension or compression. Note that because strain is a dimensionless quantity, Y has units of force per unit area. Typical values are given in Table 12.1. Experiments show (a) that for a fixed applied force, the change in length is proportional to the original length and (b) that the force necessary to produce a given strain is proportional to the cross-sectional area. Both of these observations are in accord with Equation 12.6.
The elastic limit of a substance is defined as the maximum stress that can be applied to the substance before it becomes permanently deformed. It is possible to exceed the elastic limit of a substance by applying a sufficiently large stress, as seen in Figure 12.14. Initially, a stress–strain curve is a straight line. As the stress increases, however, the curve is no longer straight. When the stress exceeds the elastic limit, the object is permanently distorted and does not return to its original shape after the stress is removed. Hence, the shape of the object is permanently changed. As the stress is increased even further, the material ultimately breaks.
Figure 12.14 Stress-versus-strain curve for an elastic solid.
Consider a long bar of cross-sectional area A and initial length Li that is clamped at one end, as in Figure 12.13. When an external force is applied perpendicular to the cross section, internal forces in the bar resist distortion (“stretching”), but the bar attains an equilibrium in which its length Lf is greater than Liand in which the external force is exactly balanced by internal forces. In such a situation, the bar is said to be stressed. We define the tensile stress as the ratio of the magnitude of the external force F to the cross-sectional area A. The tensile strain in this case is defined as the ratio of the change in length ΔL to the original length Li . We define Young’s modulus by a combination of these two ratios:
Figure 12.13 A long bar clamped at one end is stretched by an amount ΔL under the action of a force F.Young’s modulus is typically used to characterize a rod or wire stressed under either tension or compression. Note that because strain is a dimensionless quantity, Y has units of force per unit area. Typical values are given in Table 12.1. Experiments show (a) that for a fixed applied force, the change in length is proportional to the original length and (b) that the force necessary to produce a given strain is proportional to the cross-sectional area. Both of these observations are in accord with Equation 12.6.
The elastic limit of a substance is defined as the maximum stress that can be applied to the substance before it becomes permanently deformed. It is possible to exceed the elastic limit of a substance by applying a sufficiently large stress, as seen in Figure 12.14. Initially, a stress–strain curve is a straight line. As the stress increases, however, the curve is no longer straight. When the stress exceeds the elastic limit, the object is permanently distorted and does not return to its original shape after the stress is removed. Hence, the shape of the object is permanently changed. As the stress is increased even further, the material ultimately breaks.
Figure 12.14 Stress-versus-strain curve for an elastic solid.
Shear Modulus: Elasticity of Shape
Another type of deformation occurs when an object is subjected to a force tangential to one of its faces while the opposite face is held fixed by another force (Fig. 12.15a). The stress in this case is called a shear stress. If the object is originally a rectangular block, a shear stress results in a shape whose cross-section is a parallelogram. A book pushed sideways, as shown in Figure 12.15b, is an example of an object subjected to a shear stress. To a first approximation (for small distortions), no change in volume occurs with this deformation.
Figure 12.15 (a) A shear deformation in which a rectangular block is distorted by two forces of equal magnitude but opposite directions applied to two parallel faces. (b) A book under shear stress.
We define the shear stress as F/A, the ratio of the tangential force to the area A of the face being sheared. The shear strain is defined as the ratio Δx/h, where Δx is the horizontal distance that the sheared face moves and h is the height of the object. In terms of these quantities, the shear modulus is
Values of the shear modulus for some representative materials are given in Table 12.1. The unit of shear modulus is force per unit area.
Another type of deformation occurs when an object is subjected to a force tangential to one of its faces while the opposite face is held fixed by another force (Fig. 12.15a). The stress in this case is called a shear stress. If the object is originally a rectangular block, a shear stress results in a shape whose cross-section is a parallelogram. A book pushed sideways, as shown in Figure 12.15b, is an example of an object subjected to a shear stress. To a first approximation (for small distortions), no change in volume occurs with this deformation.
Figure 12.15 (a) A shear deformation in which a rectangular block is distorted by two forces of equal magnitude but opposite directions applied to two parallel faces. (b) A book under shear stress.We define the shear stress as F/A, the ratio of the tangential force to the area A of the face being sheared. The shear strain is defined as the ratio Δx/h, where Δx is the horizontal distance that the sheared face moves and h is the height of the object. In terms of these quantities, the shear modulus is
Values of the shear modulus for some representative materials are given in Table 12.1. The unit of shear modulus is force per unit area.
Bulk Modulus: Volume Elasticity
Bulk modulus characterizes the response of a substance to uniform squeezing or to a reduction in pressure when the object is placed in a partial vacuum. Suppose that the external forces acting on an object are at right angles to all its faces, as shown in Figure 12.16, and that they are distributed uniformly over all the faces. As we shall see in Chapter 15, such a uniform distribution of forces occurs when an object is immersed in a fluid. An object subject to this type of deformation undergoes a change in volume but no change in shape. The volume stress is defined as the ratio of the magnitude of the normal force F to the area A. The quantity P = F/A is called the pressure. If the pressure on an object changes by an amount ΔP = ΔF/A, then the object will experience a volume change ΔV. The volume strain is equal to the change in volume ΔV divided by the initial volume Vi . Thus, from Equation 12.5, we can characterize a volume (“bulk”) compression in terms of the bulk modulus, which is defined as
A negative sign is inserted in this defining equation so that B is a positive number. This maneuver is necessary because an increase in pressure (positive ΔP) causes a decrease in volume (negative ΔV ) and vice versa.
Table 12.1 lists bulk moduli for some materials. If you look up such values in a different source, you often find that the reciprocal of the bulk modulus is listed. The reciprocal of the bulk modulus is called the compressibility of the material.
Note from Table 12.1 that both solids and liquids have a bulk modulus. However, no shear modulus and no Young’s modulus are given for liquids because a liquid does not sustain a shearing stress or a tensile stress (it flows instead).
Bulk modulus characterizes the response of a substance to uniform squeezing or to a reduction in pressure when the object is placed in a partial vacuum. Suppose that the external forces acting on an object are at right angles to all its faces, as shown in Figure 12.16, and that they are distributed uniformly over all the faces. As we shall see in Chapter 15, such a uniform distribution of forces occurs when an object is immersed in a fluid. An object subject to this type of deformation undergoes a change in volume but no change in shape. The volume stress is defined as the ratio of the magnitude of the normal force F to the area A. The quantity P = F/A is called the pressure. If the pressure on an object changes by an amount ΔP = ΔF/A, then the object will experience a volume change ΔV. The volume strain is equal to the change in volume ΔV divided by the initial volume Vi . Thus, from Equation 12.5, we can characterize a volume (“bulk”) compression in terms of the bulk modulus, which is defined as
A negative sign is inserted in this defining equation so that B is a positive number. This maneuver is necessary because an increase in pressure (positive ΔP) causes a decrease in volume (negative ΔV ) and vice versa.Table 12.1 lists bulk moduli for some materials. If you look up such values in a different source, you often find that the reciprocal of the bulk modulus is listed. The reciprocal of the bulk modulus is called the compressibility of the material.
Note from Table 12.1 that both solids and liquids have a bulk modulus. However, no shear modulus and no Young’s modulus are given for liquids because a liquid does not sustain a shearing stress or a tensile stress (it flows instead).
Prestressed Concrete
If the stress on a solid object exceeds a certain value, the object fractures. The maximum stress that can be applied before fracture occurs depends on the nature of the material and on the type of applied stress. For example, concrete has a tensile strength of about 2 x 106 N/m2, a compressive strength of 20 x 106 N/m2, and a shear strength of 2 x 106 N/m2. If the applied stress exceeds these values, the concrete fractures. It is common practice to use large safety factors to prevent failure in concrete structures.
Concrete is normally very brittle when it is cast in thin sections. Thus, concrete slabs tend to sag and crack at unsupported areas, as shown in Figure 12.17a. The slab can be strengthened by the use of steel rods to reinforce the concrete, as illustrated in Figure 12.17b. Because concrete is much stronger under compression (squeezing) than under tension (stretching) or shear, vertical columns of concrete can support very heavy loads, whereas horizontal beams of concrete tend to sag and crack. However, a significant increase in shear strength is achieved if the reinforced concrete is prestressed, as shown in Figure 12.17c. As the concrete is being poured, the steel rods are held under tension by external forces. The external forces are released after the concrete cures; this results in a permanent tension in the steel and hence a compressive stress on the concrete. This enables the concrete slab to support a much heavier load.
Figure 12.17 (a) A concrete slab with no reinforcement tends to crack under a heavy load. (b) The strength of the concrete is increased by using steel reinforcement rods. (c) The concrete is further strengthened by prestressing it with steel rods under tension.
If the stress on a solid object exceeds a certain value, the object fractures. The maximum stress that can be applied before fracture occurs depends on the nature of the material and on the type of applied stress. For example, concrete has a tensile strength of about 2 x 106 N/m2, a compressive strength of 20 x 106 N/m2, and a shear strength of 2 x 106 N/m2. If the applied stress exceeds these values, the concrete fractures. It is common practice to use large safety factors to prevent failure in concrete structures.
Concrete is normally very brittle when it is cast in thin sections. Thus, concrete slabs tend to sag and crack at unsupported areas, as shown in Figure 12.17a. The slab can be strengthened by the use of steel rods to reinforce the concrete, as illustrated in Figure 12.17b. Because concrete is much stronger under compression (squeezing) than under tension (stretching) or shear, vertical columns of concrete can support very heavy loads, whereas horizontal beams of concrete tend to sag and crack. However, a significant increase in shear strength is achieved if the reinforced concrete is prestressed, as shown in Figure 12.17c. As the concrete is being poured, the steel rods are held under tension by external forces. The external forces are released after the concrete cures; this results in a permanent tension in the steel and hence a compressive stress on the concrete. This enables the concrete slab to support a much heavier load.
Figure 12.17 (a) A concrete slab with no reinforcement tends to crack under a heavy load. (b) The strength of the concrete is increased by using steel reinforcement rods. (c) The concrete is further strengthened by prestressing it with steel rods under tension.Reference: http://www.kshitij-iitjee.com/elastic-modulus
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