A sample’s stress–strain graph curves. Must some deformation remain after unloading?

No. Recovery determines elasticity; a straight graph is required only for Hooke’s law.

What must divide the diameter decrease before you compare the two strains?

The original diameter. Poisson’s ratio compares fractional changes, so raw changes alone cannot give it.

Two identical wires share a load equally. What fraction belongs in each wire’s extension calculation?

Half. Each wire’s extension depends on its own share of the total load.

On matching axes, A starts steeper. B reaches higher stress. Which has greater ultimate strength?

B. Ultimate strength comes from peak stress, while the initial slope measures stiffness.

A block’s bottom stays fixed while its top shifts sideways. Which modulus describes its resistance?

Shear modulus G. The displacement distorts the block sideways rather than stretching it along its length.

At fixed temperature, two materials receive equal small pressure increases. Which loses the smaller volume fraction?

Whichever has larger bulk modulus. Fractional compression equals the pressure increase divided by bulk modulus.

An initially unstretched wire obeys Hooke’s law. Gradual loading to 100 N produces 2 mm extension. How much energy is stored?

0.1 J. The average force is 50 N, acting through an extension of 0.002 m.

A centre-loaded rectangular beam doubles in depth. Load, span, breadth and material stay fixed. What sag remains?

One-eighth of the original. Doubling depth multiplies depth cubed by eight, reducing sag by that factor.

Teach Mechanical Properties of Solids in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper, a pencil and a flexible plastic ruler.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Rubber usually recovers after small stretches.
Coil springs extend mainly through twisting of their wire.
Deeper beams sag less when the other conditions stay unchanged.

What to teach

How to teach it

Describe and compare deformation

Stress–strain graph showing straight and curved elastic sections, with loading and unloading arrows.

1. Elasticity and Hooke’s law

Elasticity means recovery. Hooke’s law requires stress proportional to strain.

Show: Look at the diagram’s straight section, then its curved section. Follow the unloading arrows back to zero strain.
Ask: On this curved section, will doubling stress necessarily double strain?
Say: Doubling stress will not necessarily double strain. Recovery shows elasticity, even along this curved section. Only the straight section through the origin follows Hooke’s law.
Used in: Rubber seals recover after small squeezes, helping keep containers sealed.

If your child says…“A curved graph means permanent stretching.”

Ask: After release, what change remains?
What’s true: Permanent deformation remains after unloading.

Unloaded and stretched rods with end pulls, area, original dimensions and dimensional changes.

2. Stress, strain and Poisson’s ratio

Stress measures force per area. Strains compare changes with original dimensions.

Show: Look at the diagram’s unloaded rod, then its stretched shape. Find the end pulls, cross-sectional area and changes from the original dimensions.
Ask: For this long, thin rod, will raw narrowing divided by extension give Poisson’s ratio?
Say: Stress is the pull at one end divided by cross-sectional area. Each strain divides a size change by that direction’s original size. Poisson’s ratio divides fractional narrowing by fractional lengthening.
Used in: Rubber washers spread sideways when compressed, helping seal gaps around taps.

If your child says…“I just divide narrowing by lengthening.”

Ask: Are its original width and length equal?
What’s true: The two changes need their own original dimensions.

$\Delta L=\dfrac{FL_0}{AY}$

$\Delta L=\dfrac{100\times2.0}{(1.0\times10^{-6})(2.0\times10^{11})}\,\mathrm{m}$

3. Young’s modulus and elastic extension

Young’s modulus measures material stiffness. Dimensions and load determine extension.

Show: Write these lines for steel obeying Hooke’s law. Name F: load, L₀: original length, A: area, Y: Young’s modulus.
Ask: How far will this wire stretch?
Say: The extension is 1.0 mm.
Used in: Shorter or thicker supporting wires stretch less under equal loads, reducing movement.

If your child says…“Each support carries the whole load.”

Ask: What must the loads carried by all the supports add up to?
What’s true: Identical supports under a symmetric load carry equal shares of that load.

Stress–strain curve showing stiffness, yielding, peak strength, fracture and strain remaining after unloading.

4. Stress–strain graphs

Initial slope measures stiffness. Peak stress measures ultimate strength.

Show: Look at the diagram’s initial slope, yield point, peak and endpoint. Follow the unloading arrow to the remaining strain. Stress here uses the original cross-sectional area.
Ask: Would this curve’s steep start alone tell us how much stress the material withstands?
Say: The initial slope measures stiffness, while the peak gives ultimate strength. Yielding starts permanent deformation, shown by strain remaining after unloading. The endpoint marks fracture.
Used in: Cable design considers stiffness and strength to limit stretching and avoid failure.

If your child says…“The stiffest material is strongest.”

Ask: Does a curve’s starting slope fix its peak?
What’s true: High stiffness does not guarantee high strength.

Change shape or volume

Block with a fixed base, sideways top force, face area, height and displacement.

5. Shear modulus

Shear modulus measures resistance to sideways distortion.

Show: Look at the diagram’s fixed base and sideways force on the top. Find the loaded face’s area, block height and sideways shift.
Ask: Within the linear elastic range, will doubling this block’s sideways force double its shift?
Say: Shear stress divides sideways force by the loaded face’s area. Shear strain divides the top’s sideways shift by the block’s height. Within the linear elastic range, their ratio gives shear modulus, which also governs coil-spring twisting.
Used in: Coil springs cushion jolts because their wire twists as the springs shorten or extend.

If your child says…“Young’s modulus handles every deformation.”

Ask: Are we pulling lengthwise or pushing sideways?
What’s true: Axial stretching and sideways distortion use different moduli.

Uniform inward pressure compresses a sphere, reducing its volume.

6. Bulk modulus and compressibility

Bulk modulus measures resistance to volume change. Compressibility is its reciprocal.

Show: Compare the diagram’s inward arrows and sphere volumes at fixed temperature.
Ask: Would greater bulk modulus mean more fractional shrinkage under this pressure increase?
Say: Greater bulk modulus means less fractional shrinkage.
Used in: Hydraulic brakes use fluid that compresses very little, reducing pedal travel needed to apply braking force.

If your child says…“Bigger bulk modulus means easier squeezing.”

Ask: Does greater resistance allow more volume loss or less?
What’s true: Under equal small pressure increases, larger bulk modulus means less fractional volume loss.

Store energy and carry loads

Force–extension graph with a shaded triangle representing stored elastic energy.

7. Elastic potential energy

Elastic deformation stores energy that can be released during recovery.

Show: Look at the diagram for a wire stretched slowly within Hooke’s law. Follow the rising force and find the shaded triangle beneath it.
Ask: For this graph, would final force times extension overestimate or underestimate the stored energy?
Say: It would overestimate the stored energy. The force grows from zero, so its average is half its final value. Stored energy equals this average force times extension.
Used in: A wind-up toy’s spring stores energy that later drives its wheels.

If your child says…“The final force acts throughout.”

Ask: Was the force already largest at the start?
What’s true: For gradual Hooke’s-law stretching from zero, the average force is half the final force.

Two identical rulers across a gap, one lying flat and one standing on its edge.

8. Load limits and beam shape

Safe loads depend on material limits, cross-sectional area and structural shape.

Show: Hold the ruler near its ends and bend it slightly through its thin direction. Turn it onto its edge. After the prediction, try bending in the same direction with similar gentle effort.
Ask: Will the ruler bend as easily on its edge?
Say: The ruler resists bending more on its edge.
Used in: I-shaped beams resist bending economically by placing material where it helps.

If your child says…“Double depth means half the sag.”

Ask: Turning the ruler changes breadth too, so what must stay fixed to test depth alone?
What’s true: With everything else fixed, doubling a rectangular beam’s depth reduces its sag to one-eighth.

A real problem

Howrah Bridge combined steel strength with structural design.

The earlier pontoon bridge could no longer carry the growing load.

Howrah Bridge entered service in 1943.

Its steel truss uses a balanced cantilever structure.

Its construction used high tensile, low alloy steel.

Wikipedia The Asiatic Society

Everyday traffic beneath Howrah Bridge’s steel trusses.

Concept Lab · Common misconception

The tempting mistake in Mechanical Properties of Solids

Open standalone misconception page →

Does elastic mean easy to stretch?

Wrong

Rubber has a larger Young’s modulus than steel because it stretches easily.

Why it feels right. Everyday “elastic” means stretchy.

Right

A larger Young’s modulus means greater stiffness in the linear elastic range.

Worked example

The dashed ends show equal starting lengths. Under the same small tensile stress, the rubber strip extends farther.

Under the same small tensile stress, compare the strains of steel and rubber.
Steel changes length by a smaller fraction.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

A sample’s stress–strain graph curves. Must some deformation remain after unloading?

No. Recovery determines elasticity; a straight graph is required only for Hooke’s law.
What must divide the diameter decrease before you compare the two strains?

The original diameter. Poisson’s ratio compares fractional changes, so raw changes alone cannot give it.
Two identical wires share a load equally. What fraction belongs in each wire’s extension calculation?

Half. Each wire’s extension depends on its own share of the total load.
On matching axes, A starts steeper. B reaches higher stress. Which has greater ultimate strength?

B. Ultimate strength comes from peak stress, while the initial slope measures stiffness.
A block’s bottom stays fixed while its top shifts sideways. Which modulus describes its resistance?

Shear modulus G. The displacement distorts the block sideways rather than stretching it along its length.
At fixed temperature, two materials receive equal small pressure increases. Which loses the smaller volume fraction?

Whichever has larger bulk modulus. Fractional compression equals the pressure increase divided by bulk modulus.
An initially unstretched wire obeys Hooke’s law. Gradual loading to 100 N produces 2 mm extension. How much energy is stored?

0.1 J. The average force is 50 N, acting through an extension of 0.002 m.
A centre-loaded rectangular beam doubles in depth. Load, span, breadth and material stay fixed. What sag remains?

One-eighth of the original. Doubling depth multiplies depth cubed by eight, reducing sag by that factor.

15 more questions with answers, 2 with pictures

Teach Mechanical Properties of Solids in eight steps.

Why it matters