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168 PHYSICS elastic behaviour and mechanical properties of solids which would answer many such Longitudinal strain = L (8.2) questions. L However, if two equal and opposite deforming 8.2 STRESS AND STRAIN forces are applied parallel to the cross-sectional When forces are applied on a body in such a area of the cylinder, as shown in Fig. 8.1(b), mannerthatthebodyisstillinstaticequilibrium, there is relative displacement between the itisdeformedtoasmallorlargeextentdepending opposite faces of the cylinder. The restoring force upon the nature of the material of the body and per unit area developed due to the applied the magnitude of the deforming force. The tangential force is known as tangential or deformation may not be noticeable visually in shearing stress . many materials but it is there. When a body is As a result of applied tangential force, there subjected to a deforming force, a restoring force is a relative displacement ∆x between opposite is developed in the body. This restoring force is faces of the cylinder as shown in the Fig. 8.1(b). equal in magnitude but opposite in direction to The strain so produced is known as shearing theappliedforce.Therestoringforceperunitarea strain and it is defined as the ratio of relative isknownasstress. IfFistheforceappliednormal displacement of the faces ∆x to the length of the to the cross–section and A is the area of cross cylinder L. section of the body, x Magnitude of the stress = F/A (8.1) Shearing strain = = tan θ (8.3) The SI unit of stress is N m–2 or pascal (Pa) L and its dimensional formula is [ ML T1 –2]. where θ is the angular displacement of the There are three ways in which a solid may cylinder from the vertical (original position of the change its dimensions when an external force cylinder). Usually θ is very small, tan θ acts on it. These are shown in Fig. 8.1. In Fig.8.1(a), a cylinder is stretched by two equal is nearly equal to angle θ , (ifθ = 10°, for forces applied normal to its cross-sectional area. example, there is only 1% difference between θ and tan θ). The restoring force per unit area in this case is It can also be visualised, when a book is called tensile stress. If the cylinder is pressed with the hand and pushed horizontally, compressed under the action of applied forces, the restoring force per unit area is known as as shown in Fig. 8.2 (c). compressive stress. Tensile or compressive Thus, shearing strain = tan θ ≈θ (8.4) In Fig. 8.1 (d), a solid sphere placed in the fluid stress can also be termed as longitudinal stress. under high pressure is compressed uniformly on In both the cases, there is a change in the all sides. The force applied by the fluid acts in length of the cylinder. The change in the length perpendicular direction at each point of the ∆L to the original length L of the body (cylinder in this case) is known as longitudinal strain. surface and the body is said to be under hydraulic compression. This leads to decrease (a) (b) (c) (d) Fig. 8.1 (a)Acylindricalbodyundertensilestresselongates∆L(b)Shearingstressonacylinderdeformingitby anangle θ(c)Abodysubjectedtoshearingstress(d)Asolidbodyunderastressnormaltothesurfaceat everypoint(hydraulicstress).Thevolumetricstra∆nV/V,butthereisnochangeinshape. Reprint 2026-27