doPhysics · Elasticity

Stress, Strain and Hooke's Law

Why this chapter

Why this chapter exists

Until now every string was inextensible and every block was rigid. Real solids give a little under every load, and they push back. That small give holds up every lift, bridge and crane, and it decides how thick a cable must be. This chapter measures the give with two numbers, stress and strain, and finds that for small loads they are proportional. One constant per material, a modulus, then answers the practical questions. How far will a wire stretch? How much does the sea squeeze a litre of water? How far does a rubber pad slide?

You will learn

By the end of this chapter you will

Explain elastic, plastic and rigid bodies with the spring model of a solid.
Find tensile, compressive and shearing stress, and longitudinal, volume and shear strain.
State Hooke's law, and read the proportional limit, the elastic limit, the yield point, the ultimate strength and the fracture point on a stress-strain curve.
Explain the permanent set of a wire loaded beyond its elastic limit, and use a breaking stress with a safety factor.
Use Young's modulus to find the extension of a wire and to compare wires of different length and diameter.
Solve two wires joined end to end and hung side by side.
Use the bulk modulus and compressibility to find the change in volume and density of a body under pressure.
Use the shear modulus and Poisson's ratio, and decide which modulus governs a given deformation.

Before you begin

From Dynamics you should know Newton's laws, weight \(mg\), tension in a string and Hooke's law for a spring, \(F = kx\). From Rotational Mechanics you should know the condition for a body not to turn, that the torques about any point balance. You should be able to work with powers of ten and convert mm² and cm² into m².

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Lessons

Why Solids Push Back

Every solid gives a little under a load and pushes back. Elastic, plastic and rigid bodies, the spring model of a solid, stress as the restoring force per area, and strain as the fractional change in size or shape.

Deforming and restoring forcesStress as force per areaStrain as a fractional change

13 min0%

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Hooke's Law and the Elastic Limit

For small loads stress is proportional to strain. Past the proportional limit the line bends, past the elastic limit the wire no longer comes back, and a wire unloaded from there keeps a permanent set. The ultimate strength, fracture, and ductile and brittle materials in brief.

Hookes law stress and strainProportional and elastic limitsPermanent set beyond the limit

14 min0%

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Young's Modulus

Young's modulus is the ratio of tensile stress to longitudinal strain, a property of the material. The extension of a wire, \(\Delta L = FL/AY\), and how it grows with the length and falls with the square of the diameter. Two wires end to end and side by side.

Youngs modulus of a wireExtension with length and areaComposite wires in series and side by side

15 min0%

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Bulk Modulus and Compressibility

A pressure from every side squeezes a body's volume. The bulk modulus, \(B = -\Delta P/(\Delta V/V)\), and its inverse, the compressibility. Gases squeeze easily, liquids hardly, solids least. A body under pressure grows denser by \(\Delta P/B\).

Volume stress and bulk modulusCompressibilityDensity under pressure

12 min0%

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Shear Modulus and Poisson's Ratio

A force along a face slides one layer over another; the shear modulus \(G = F/(A\theta)\) measures the resistance. A stretched wire also grows thinner, by Poisson's ratio. Which modulus governs a cable, a sea-bed or a shaft, and why a liquid has only a bulk modulus.

Shear strain and shear modulusPoissons ratioWhich modulus governs

14 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

breaking stress: The stress at which a material breaks. The breaking load of a wire is its breaking stress times its area of cross-section.
brittle: Breaking soon after the elastic limit with almost no permanent stretch, like glass or cast iron.
bulk modulus: The ratio of the increase in pressure on every side of a body to its fractional decrease in volume, \(B = -\Delta P/(\Delta V/V)\); about \(2 \times 10^9\,\text{Pa}\) for water.
compressibility: The inverse of the bulk modulus, \(1/B\): the fractional loss of volume for each pascal of extra pressure.
compressive stress: Stress produced by a normal force that pushes on a face, as in a pillar. It shortens the body.
deforming force: A force that changes the size or the shape of a body.
ductile: Able to stretch a long way between the elastic limit and fracture, like copper or mild steel, so that it can be drawn into wires.
elastic limit: The largest stress from which a body returns to its original size and shape when the load is removed.
elasticity: The property of a body by which it returns to its original size and shape when the deforming force is removed.
fracture point: The point of a stress-strain curve at which the material breaks.
Hooke's law: Within the proportional limit, stress is proportional to strain. The constant ratio is the modulus of elasticity of the material.
lateral strain: The fractional change in the width or diameter of a body, measured across the direction of the stretch, \(\Delta d/d\).
longitudinal strain: The change in length divided by the original length, \(\Delta L/L\).
permanent set: The strain that remains in a body after it has been loaded beyond its elastic limit and then unloaded.
plasticity: The property of a body by which it keeps its deformed size or shape when the deforming force is removed, as clay or dough does.
Poisson's ratio: Minus the ratio of the lateral strain to the longitudinal strain, \(\sigma\); about 0.3 for steel and nearly 0.5 for rubber.
pressure: The normal force per unit area that a fluid exerts on a surface, measured in pascals. The chapter on Pressure in Fluids studies it.
proportional limit: The largest stress up to which stress is proportional to strain, the end of the straight part of a stress-strain curve.
restoring force: The internal force that a deformed body develops against the deformation. At rest it balances the deforming force.
rigid body: An idealised body that does not deform at all under any force. Real solids are only close to rigid.
safety factor: The number by which the breaking load is divided to give the largest load a structure is allowed to carry, often 4 to 10.
shear modulus: The ratio of shearing stress to shear strain, \(G = F/(A\theta)\), also written \(\eta\) and called the modulus of rigidity.
shear strain: The angle \(\theta\), in radians, through which the side faces of a body turn in shear; \(\theta \approx x/L\).
shearing stress: Stress produced by a force along a face, a tangential force. It slides one layer of the body over another.
strain: The fractional change in the size or shape of a body, such as \(\Delta L/L\). It is a pure number with no unit.
stress: The restoring force per unit area of a cross-section of a deformed body, \(F/A\) at rest, measured in pascals (Pa).
tensile stress: Stress produced by a normal force that pulls on a face, as in a loaded wire. It lengthens the body.
ultimate tensile strength: The largest stress a material can bear, the highest point of its stress-strain curve.
volume strain: The change in volume divided by the original volume, \(\Delta V/V\).
yield point: The stress at which a material begins to flow, the strain growing a lot for little extra stress. For many metals it is close to the elastic limit.
Young's modulus: The ratio of tensile or compressive stress to longitudinal strain, \(Y = FL/(A\,\Delta L)\); \(2 \times 10^{11}\,\text{Pa}\) for steel.