12.4 Elastic Properties of Solids
In the previous discussion we have mengansumsikan that the object will remain rigid whenthere are external forces acting on it. But in fact, all objects can change shape.
Stress is the external force acting on the body per unit cross-sectional area / cross.
Strain is the result of Stress
The Stress required to produce a particular strain depends on the state of the material is pressed.
Comparison between stress and strain, strain or stress unity, called the elastic modulus.
The elastic modulus in general relates what is done to a solid object (a force is applied) to how that object responds (it deforms to some extent). The greater the elastic modulus, the greater the stress needed for a particular strain. We consider three types of deformation and deﬁne an elastic modulus for each:
Young’s modulus, which measures the resistance of a solid to a change in its length
Shear modulus, which measures the resistance to motion of the planes within a solid parallel to each other
Bulk modulus, which the resistance of solids or liquids to changes in thei volume.
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 12.14. When an external force is applied perpendicular to the cross section, internal forces in the bar resist distortion (“stretching”), but the bar reaches an equilibrium situation in which its ﬁnal length Lf is greater than Li and 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
to the original length Li. We deﬁne Young’s modulus by a combination of these two ratios:
Shear Modulus: Elasticity of Shape
Another type of deformation occurs when an object is subjected to a force parallel to one of its faces while the opposite face is held ﬁxed by another force (Fig. 12.16a). 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 12.16b, 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. 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 /h, where 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. Like Young’s modulus, the unit of shear modulus is the ratio of that for force to that for area.
Bulk Modulus: Volume Elasticity
Bulk modulus characterizes the response of an object to changes in a force of uniform magnitude applied perpendicularly over the entire surface of the object, as shown in Figure 12.17. (We assume here that the object is made of a single substance.) As we shall see in Chapter 14, 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 total force F exerted on a surface to the area A of the surface. The quantity P =F/A is called pressure, which we will study in more detail in Chapter 14. If the pressure on an object changes by an amount =/A,then the object will experience a volume change . The volume strain is equal to the change in volume 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
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 12.18a. The slab can be strengthened by the use of steel rods to reinforce the concrete, as illustrated in Figure 12.18b. 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 12.18c. 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.