doPhysics · Elasticity
Elastic Energy, Thermal Stress and Real Materials
Young's modulus measured in the laboratory. Stress that changes along a hanging, pulled or spinning rod. A wire as a stiff spring, the energy it stores, and the stress that appears when heating is stopped. The whole stress-strain curve of real materials, and how engineers design with it.
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Why this chapter
Why this chapter exists
Hooke's law gave a wire one number, Young's modulus. This chapter puts that number to work. It shows how a school laboratory measures a stretch of a hundredth of a millimetre, and what happens when the tension changes along a rod. It treats a wire as a spring that stores energy, and asks what a load let go on it does. Then it heats a rod that cannot grow, and follows real materials past the elastic limit to fracture. Rails, cables, bridges and cranes are designed with exactly these ideas.
You will learn
By the end of this chapter you will
- Explain how Searle's apparatus cancels warming and the sag of the support, and find \(Y\) from the slope of a load-extension line.
- Judge which measurement limits the accuracy of Young's modulus, and why the wire is long and thin.
- Find the tension along a rod hanging under its own weight, pulled on a smooth table or turning about one end, and add up its extension.
- Treat a wire as a spring of constant \(YA/L\). Cut it, join it, share a load between wires, and quote the period of a load bobbing on it.
- Find the strain energy of a wire and its energy per unit volume in three forms. Show that a load let go on a taut wire overshoots to twice its resting stretch.
- Find the thermal stress \(Y\alpha\,\Delta T\) in a clamped rod, with or without a gap, and give its sign for heating and for cooling.
- Read a stress-strain curve to fracture, tell ductile, brittle and rubbery materials apart, and use a safety factor in a design.
Before you begin
Before you begin
From the chapter on Stress, Strain and Hooke's Law, you should know stress \(F/A\), strain \(\Delta L/L\), Young's modulus \(Y\) and the elastic limit. From earlier chapters you should know the spring law \(F = kx\) with springs in series and in parallel, the work of a variable force as the area under its graph, the energy \(\tfrac12 kx^2\), and the centripetal force \(m\omega^2 r\). You should be able to integrate a power of \(x\).
Lessons
1
Measuring Young's Modulus: Searle's Experiment
A school laboratory measures a stretch of one millimetre to a hundredth of a millimetre. Two wires cancel every disturbance but the load, a line through the readings gives Young's modulus, and a long thin wire keeps the errors small.
Searles apparatusReading the load extension lineChoosing the wire and the loads
2
Stress That Changes Along a Rod
When a rod hangs under its own weight, is pulled from one end or spins about one end, the tension differs from section to section. Cut the rod, find the tension at the cut, and add up the stretch of every small piece.
Elongation under own weightRod pulled by a force on a smooth tableTension in a rotating rod
3
A Wire Is a Spring
Hooke's law for a wire has the form \(F = k\Delta L\) with \(k = YA/L\). So a wire is a very stiff spring. Cut it, double it or join it to another, and the rules for springs give the answer; a load on it bobs with the period of a spring.
Wire as a springCutting and joining wiresA load oscillating on a wire
4
Elastic Potential Energy
The work done in stretching a wire is the area under its force-extension line, \(\tfrac12 F\Delta L\), and the wire stores it as strain energy. Per unit volume it is \(\tfrac12\,\text{stress} \times \text{strain}\). A load let go on a just-taut wire overshoots to twice its resting stretch.
Work done in stretchingStrain energy densityLoad released suddenly
5
Thermal Stress
A rod heated between rigid walls cannot take its new length, so the walls squeeze it back and a stress \(Y\alpha\Delta T\) appears, whatever its length. Heating a clamped rod compresses it and cooling stretches it; rail gaps and bridge joints exist to stop both.
Thermal stress in a clamped rodHeating compresses cooling stretches
6
Real Materials: the Stress-Strain Curve to Fracture
Pull a specimen until it breaks, and its stress-strain curve tells its whole story. Ductile metals yield, flow, neck and break; brittle ones snap; rubber stretches far and returns on a lower curve. Engineers design with these curves.
Stress strain curve to fractureDuctile brittle and elastomersDesigning with elasticity
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- brittle
- Breaking at or near the elastic limit with almost no plastic region, and so without warning. Glass and cast iron are brittle.
- buckling
- The sudden sideways bowing of a long rod or rail under a large compressive stress, such as a rail heated with no room to lengthen.
- coefficient of linear expansion
- The fractional increase in length of a solid for each kelvin rise in temperature, \(\alpha = \Delta L/(L\,\Delta T)\). For steel it is about \(1.2 \times 10^{-5}\,\text{K}^{-1}\).
- dead load
- A fixed load hung on a wire before any reading is taken, to pull out its kinks. Every extension is measured from the dead-loaded length.
- ductile
- Having a long plastic region between the elastic limit and fracture, so that the material can be drawn into wire or hammered into sheet. Copper and mild steel are ductile.
- elastic fatigue
- The loss of strength of a material loaded and unloaded many times, even below its breaking stress, as small cracks spread with each cycle.
- elastic hysteresis
- The return of a material along a lower curve on unloading than on loading. The loop between the curves is energy turned to heat in each cycle.
- elastic limit
- The greatest stress from which a material returns to its original size when the load is removed. Beyond it the material keeps a permanent set.
- elastomer
- A material such as rubber that can be stretched to several times its length and still return, though not in proportion to the stress.
- expansion joint
- A gap in a long structure such as a bridge deck or a pipeline, often bridged by interlocking steel teeth. It lets the structure lengthen and shorten with the temperature without stress.
- fracture point
- The point of the stress-strain curve at which the specimen breaks.
- least count
- The smallest change a measuring instrument can read. For a screw of pitch 0.5 mm with 50 divisions on its head it is 0.01 mm.
- micrometer screw
- A screw of known pitch with a divided head, used to measure small distances. Turning it until the spirit level is flat measures the extension.
- necking
- The rapid local thinning of a ductile specimen after the ultimate stress, in which the stretching concentrates in one short region that finally breaks.
- permanent set
- The strain that remains when a body loaded beyond its elastic limit is unloaded.
- proportional limit
- The greatest stress up to which stress is proportional to strain, so that Hooke's law holds.
- reference wire
- The wire in Searle's apparatus that carries a fixed load throughout. It shares every disturbance with the test wire, such as warming and the sag of the support, so that these cancel.
- safety factor
- The ratio of the stress a part can safely bear, such as its elastic limit, to the working stress it is designed for. It is often about 10 for ropes and cables.
- Searle's apparatus
- An apparatus for Young's modulus. A test wire and a reference wire hang from one support, joined by a spirit level resting on a micrometer screw. Only the extra stretch of the test wire is measured.
- spirit level
- A sealed tube of liquid with an air bubble, which runs to the higher end when the tube tilts. In Searle's apparatus it shows when the two wire ends are level.
- stiffness
- The force needed per unit extension, \(k = F/\Delta L\), in N/m. For a wire within its elastic limit \(k = YA/L\), so a wire behaves as a spring.
- strain energy
- The elastic potential energy stored in a stretched or compressed body, equal to the work done in deforming it. For a wire within its elastic limit it is \(\tfrac12 F\,\Delta L\).
- strain energy density
- Strain energy per unit volume, \(u = \tfrac12\,\text{stress} \times \text{strain} = (\text{stress})^2/2Y = \tfrac12 Y(\text{strain})^2\), in J/m³.
- test wire
- The wire in Searle's apparatus that carries the added loads and whose extension is measured.
- thermal stress
- The stress in a body that is prevented from expanding or contracting with a change of temperature; \(Y\alpha\,\Delta T\) in a rod clamped between rigid walls.
- ultimate tensile stress
- The greatest stress on a material's stress-strain curve, the most it can carry, calculated with the original area. Also called the tensile strength.
- yield point
- The stress at which a metal begins to flow plastically, its strain growing with little or no increase in stress.
- Young's modulus
- The ratio of stress to strain for stretching or compression within the proportional limit, \(Y = FL/(A\,\Delta L)\). It is a property of the material; steel has about \(2 \times 10^{11}\,\text{Pa}\).