Stress, Strain and Hooke's Law

Transcript

0:00Introduction

Hello everyone! Quick question before we start. Take a rubber band and a steel wire. Which one is more elastic? Rubber, obviously, right? It stretches so easily! Hold on to that answer; by the end of this lesson, you may change it. Today: stress, strain and Hooke's law, how solids stretch and spring back. This comes up often in JEE, so let's begin.

0:27Stress: restoring force per unit area

So, first, stress. Pull a wire, and it pulls back from inside. Stress is this internal restoring force per unit area: sigma equals F over capital A, okay? Sigma, this Greek letter, means stress here, not surface tension; F is the force, and capital A is the area of cross-section. Now tell me: force in newtons, area in square metres. What is the unit of stress? Yes, exactly! Newton per square metre, written N m to the power minus two. Its special name is the pascal, written capital P, small a. Clear? And the dimensions, inside square brackets: M, L to the power minus one, T to the power minus two, for mass, length and time. With me?

1:20Strain: fractional deformation

Now, strain: the fractional deformation, the change as a fraction of the original size. Stretch or squeeze a rod along its length: that is longitudinal strain, tensile or compressive. It is delta L over L, okay? The little triangle, delta, means change: change in length over original length. Right? Squeeze a body from all sides, and its volume changes. Volume strain is delta V over capital V, the change in volume over the original volume. Got it? Third, shear. Push the top face of a block sideways, with the bottom fixed. The side face turns through a small angle, phi, this Greek letter. Phi is approximately delta x, how far the top moves, over L, the height; the wavy equals sign means approximately. Now quickly: what is the unit of strain? None at all! Length over length, volume over volume: the units cancel. So strain has no unit and no dimensions. Okay?

2:26Hooke's law and the stress–strain curve

Now the main law of this chapter: Hooke's law. Within the elastic limit, stress is proportional to strain. This sign, like a stretched alpha, means is proportional to. Never drop the condition: within the elastic limit. Clear? So stress equals a constant times strain, and that constant is called the modulus of elasticity. Now look at the graph, for a ductile metal pulled in tension. It starts as a straight line, from the origin O up to point A, okay? So, the slope of this straight part: what does it give? Slope is stress over strain: exactly the modulus! So the slope of the straight part O A, for stretching, is Young's modulus Y, okay? Point A is the proportional limit: up to here, stress is exactly proportional to strain. Just above it, point B is the elastic limit, also marked as the yield point. Up to B, remove the load and the wire returns to its original shape. So, strictly, the straight-line law holds up to A; B is so close to A that books usually state Hooke's law within the elastic limit. With me? Beyond B is the plastic region: remove the load there, and the wire keeps a permanent set. Point D is the ultimate strength, the highest stress, and at E the wire breaks: fracture. Okay?

3:49Three moduli of elasticity

So, three kinds of strain give three moduli. First, Young's modulus Y: the stress, F over capital A, divided by the longitudinal strain, delta L over L; the slash simply means over. Got it? Second, the bulk modulus, capital B, for a change of volume. Here the stress is the extra pressure, delta P, okay? So capital B equals minus delta P, over the volume strain. Why the minus sign? More pressure, less volume: delta V is negative, so the minus keeps the bulk modulus positive. Clear? Third, the shear modulus: eta, this Greek letter, equals the stress, F over capital A, divided by the shear angle phi. Careful: eta here is the shear modulus, not viscosity. Right? Notice: every modulus is stress over strain, and strain has no unit. So every modulus has the units and dimensions of stress. With me? Liquids and gases resist only a change of volume, so they have only a bulk modulus. And a perfectly rigid body shows no strain for any stress, so its moduli are infinite. Okay?

5:03Steel or rubber: which is more elastic?

Now, back to our puzzle. Let us compare Young's moduli. Steel: about two times ten to the power eleven pascal. Got it? Rubber: only ten to the power six to ten to the power seven pascal; this wavy line, the tilde, means of the order of. The modulus is the stress needed per unit strain. So, which is more elastic: steel or rubber? Steel! For the same strain, steel needs a far larger stress, and in physics, that is what more elastic means. Rubber stretches easily because its modulus is small. So if you said rubber at the start, that was the trap. Clear?

5:44Example 1: stress, strain and Young's modulus of a wire

Alright, Example one. Read it with me: a wire of length two metres and area one square millimetre stretches by one millimetre under a load of one hundred newtons. Find the stress, the strain, and Young's modulus. Pause the video, and try it first. Done? Okay, let's check. One millimetre is ten to the power minus three metre. So one square millimetre is how many square metres? Ten to the power minus six square metre! Square the conversion too. Oh no, careful: write ten to the power minus three here, and the stress and Young's modulus both come out wrong by a factor of a thousand. Okay? Stress: F over capital A, one hundred over ten to the power minus six. Ten to the power eight pascal. Right? The strain, delta L over L: ten to the power minus three metre, over two metres. Five times ten to the power minus four, with no unit. With me? And Young's modulus is stress over strain: ten to the power eight, over five times ten to the power minus four. Two times ten to the power eleven pascal. Exactly the value we saw for steel. Very good!

7:04Example 2: longest steel wire hanging under its own weight

Now a harder one. Steel has density seven thousand eight hundred kilogram per cubic metre, and breaking stress seven point eight times ten to the power eight pascal. What is the longest steel wire that can hang vertically without breaking under its own weight? Take g as ten metre per second squared. Pause the video and try it. Done? First, a question. The wire hangs from the ceiling. Where is the stress greatest: at the top, in the middle, or at the bottom? At the top, yes, exactly! The top holds up the weight of the whole wire below it, while the bottom end holds nothing. Clear? Now, weight is density times volume times g: rho, the Greek letter for density, times capital A, times L, times g. Divide by the area, capital A, and the stress at the top is just rho g L, okay? The wire breaks when this reaches the breaking stress, sigma with a small b below. So L max, with max written below, equals sigma b over rho g: seven point eight times ten to the power eight, over seven thousand eight hundred times ten. The denominator is seven point eight times ten to the power four: so ten to the power four metres, ten kilometres! Now, would a thicker wire hang longer? No, no, careful! Capital A cancelled. The thicker wire is stronger, but heavier in the same ratio. So the longest length does not depend on the thickness. Got it?

8:43Quick revision: stress, strain, Hooke's law, Y, B and eta

So, quick revision. Stress is force over area, in pascal; strain is a ratio, with no unit. Hooke's law: within the elastic limit, stress is proportional to strain. Young's modulus Y, bulk modulus B and shear modulus eta: each is stress over strain, and steel is more elastic than rubber. Now go and try the practice problems on this page. All the best!