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MECHANICAL PROPERTIES OF SOLIDS 169 in its volume without any change of its compression and shear stress may also be geometrical shape. obtained. The stress-strain curves vary from The body develops internal restoring forces material to material. These curves help us to that are equal and opposite to the forces applied understand how a given material deforms with by the fluid (the body restores its original shape increasing loads. From the graph, we can see and size when taken out from the fluid). The that in the region between O to A, the curve is internal restoring force per unit area in this case linear. In this region, Hooke’s law is obeyed. is known as hydraulic stress and in magnitude The body regains its original dimensions when is equal to the hydraulic pressure (applied force the applied force is removed. In this region, the per unit area). solid behaves as an elastic body. The strain produced by a hydraulic pressure is called volume strain and is defined as the ratio of change in volume (∆V) to the original volume (V). V Volume strain = (8.5) V Since the strain is a ratio of change in dimension to the original dimension, it has no units or dimensional formula. 8.3 HOOKE’S LAW Stress and strain take different forms in the situations depicted in the Fig. (8.1). For small deformations the stress and strain are proportional to each other. This is known as Fig. 8.2A typical stress-strain curve for a metal. Hooke’s law. Thus, In the region from A to B, stress and strain are not proportional. Nevertheless, the body still stress ∝ strain stress = k × strain (8.6) returns to its original dimension when the load where k is the proportionality constant and is is removed. The point B in the curve is known known as modulus of elasticity. as yield point (also known as elastic limit) and the corresponding stress is known as yield Hooke’s law is an empirical law and is found strength ( σ ) of the material. to be valid for most materials. However, there y are some materials which do not exhibit this If the load is increased further, the stress linear relationship. developed exceeds the yield strength and strain increases rapidly even for a small change in the stress. The portion of the curve between B and 8.4 STRESS-STRAIN CURVE The relation between the stress and the strain D shows this. When the load is removed, say at some point C between B and D, the body does for a given material under tensile stress can be not regain its original dimension. In this case, found experimentally. In a standar d test of even when the stress is zero, the strain is not tensile properties, a test cylinder or a wire is zero. The material is said to have a permanent stretched by an applied force. The fractional change in length (the strain) and the applied set. The deformation is said to be plastic deformation. The point D on the graph is the force needed to cause the strain are recorded. ultimate tensile strength ( σu) of the material. The applied force is gradually increased in steps Beyond this point, additional strain is produced and the change in length is noted. A graph is even by a reduced applied force and fracture plotted between the stress (which is equal in magnitude to the applied force per unit area) and occurs at point E. If the ultimate strength and fracture points D and E are close, the material the strain produced. A typical graph for a metal is said to be brittle. If they are far apart, the is shown in Fig. 8.2. Analogous graphs for material is said to be ductile. Reprint 2026-27