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Chapter 11 · Part I · Mechanics
Revised on ____________________

Chapter 11 · Properties of Matter

How much a solid gives, and when it stops coming back

sheets
diagrams
results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

Stress and strain

STRESS HAS UNITS, STRAIN HAS NONE

I.Stress σ = FA Pa; the same units as pressure

II.Longitudinal strain ε = ΔLL

III.Shear strain φ = ΔxL an angle, in radians

IV.Volume strain ΔVV

V.Hooke's law σ = E ε only up to the proportional limit

VI.Poisson's ratio ν = − lateral strainlongitudinal strain 0 < ν ≤ 0.5

The three moduli

EACH ONE HAS ITS OWN GEOMETRY

I.Young's modulus Y = F LA ΔL stretch, one direction

II.Shear (rigidity) G = F_∥A φ solids only — fluids have none

III.Bulk modulus B = − ΔPΔV/V uniform squeeze

IV.Compressibility K = 1B

V.Elongation of a wire ΔL = F LA Y a spring of stiffness k = AY/L

VI.Under its own weight ΔL = ρ g L²2 Y half the load acts at the centre

III

The stress–strain curve

WHERE THE MATERIAL STOPS OBEYING

I.Proportional limit the end of the straight line Hooke's law fails beyond

II.Elastic limit the last point that returns to zero strain

III.Yield point permanent deformation begins

IV.Ultimate tensile strength the highest stress the sample carries

V.Fracture point where it parts brittle: close to the UTS

VI.Ductile vs brittle a long plastic region vs almost none

Elastic energy

THE AREA UNDER THE CURVE

I.Energy stored U = 12 F² LA Y = 12 F ΔL

II.Energy density u = 12 σ ε = σ²2 Y J m⁻³

III.Toughness the whole area, to fracture

IV.Thermal stress σ = Y α ΔT clamped ends, no strain allowed

V.Composite rod, parallel the same strain, forces share as A Y

Steel, Y

2.0×10¹¹ Pa

Copper, Y

1.1×10¹¹ Pa

Rubber, Y

≈ 10⁶ Pa

Water, B

2.2×10⁹ Pa

Steel, ν

0.29

Breaking stress, steel

≈ 4×10⁸ Pa

What the numbers mean

READING A MATERIAL OFF ITS CONSTANTS

A large Y means stiff, not strong. Steel stretches less than copper under the same load; whether it breaks first is a separate number — the breaking stress.

Rubber has a tiny Y and an enormous elastic range: it stores far more energy per kilogram than steel before it fails, which is why catapults are not made of steel.

Where marks are lost in this chapter

1Comparing wires by breaking force. Compare the stress σ = F/A before deciding which fails first.

2Comparing extensions instead of strain. A longer wire of the same material extends more.

3Series and parallel backwards. End-to-end: same force. Side by side: same extension.

4Dropping the ½ in U = ½ F ΔL. The force builds from zero, so the average is F/2.