STRENGTH OF MATERIALS John Parkinson ©

THE GOLDEN ORB SPIDER’S SILK IS TWICE AS STRONG AS MILD STEEL AND CAN STRETCH BY UP TO 30% 0F ITS ORIGINAL LENGTH

He suspended various masses from springs, and measured the extension. STIFFNESS first investigated by ROBERT HOOKE, 1648 m m He suspended various masses from springs, and measured the extension.

Hooke's Law m e.g. for a spring or wire e F = mg “Up to a certain level of force, the extension (e) produced is proportional to the applied force (F).”

Hooke's Law k is called the spring constant EXTENSION Force e/m F/N 0.00 0.12 1 0.24 2 0.36 3 0.48 4 0.60 5 0.72 6 0.84 7 0.96 8 1.08 9 1.20 10 k is called the spring constant k is equal to the gradient of the graph = 8.3 Nm-1 k is a measure of the spring’s stiffness

WORK DONE IN STRETCHING / ENERGY STORED IN A STRETCHED SPRING OR WIRE 1. Special case: when F is proportional to e force extension e F Work Done = Average Force x Extension W = ½ Fe which is the area under the graph W = ½ ke2 or, as F = k e

dw = f de 2. The general case de WORK DONE IN STRETCHING / ENERGY STORED IN A STRETCHED SPRING OR WIRE 2. The general case force extension f dw = f de e de which again is the area under the graph

STRETCHING OF A POLYMER e.g. RUBBER force extension F Initially it is easy to merely straighten the polymer chains. F It then becomes much harder to separate the atoms of the carbon chain F C F

work done in stretching STRETCHING OF RUBBER force extension loading work done in stretching HYSTERESIS LOOP unloading energy recovered energy LOST as internal energy The word HYSTERESIS is used to refer to the energy lost in a cyclic process.

Extension depends upon: (i) Material Properties Steel is approximately 1½ STIFFER than copper. Which is graph below shows the stretching of a copper bar and which a steel bar under loading? A force extension B ANSWER WE CANNOT TELL! Extension depends upon: (i) Material Properties (ii) Geometry of Bar (i.e. L and A)

s ε l STRESS and STRAIN = A = F e TENSILE STRESS and TENSILE STRAIN are used to compare the stiffness of different materials. TENSILE STRESS, σ = force applied per unit cross sectional area A F = s A F TENSILE STRAIN, ε = fractional change in length under load = extension ÷ original length l e = ε Original length l Extension,e

Strain can also be expressed as a percentage. STRESS and STRAIN UNITS A F = s N m-2 = Pa STRESS l e = ε NO UNITS STRAIN Strain can also be expressed as a percentage.

WHICH IS COPPER, A or B? ANSWER B must be COPPER Steel is approximately 1½ STIFFER than copper. stress strain A B WHICH IS COPPER, A or B? ANSWER B must be COPPER

YOUNG’s MODULUS OF ELASTICITY - E E measures stiffness stress strain E = gradient of the stress – strain graph e A F l = E UNITS of E are Pa (GPa)

YOUNG’s MODULUS OF ELASTICITY TYPICAL VALUES MATERIAL E ( MPa ) Carbon Fibre 270 Steel 210 Copper 130 Kevlar 124 Bone 28 Rubber 0.02

Copper – a Ductile Material STRESS – STRAIN GRAPHS Copper – a Ductile Material stress L Y F U L = Limit of Proportionality Y = Yield Point U = Ultimate Tensile Strength (UTS) F = Fracture strain

(permanent deformation) STRESS – STRAIN GRAPHS Copper – a Ductile Material stress strain L Y F U Strain hardening Necking UTS Yield Strength PLASTIC REGION (permanent deformation) ELASTIC REGION

If the load is removed a Permanent Set remains in the sample. STRESS – STRAIN GRAPHS stress strain Some of a specimen’s elasticity may be lost before the yield point is reached. Y Permanent Set If the load is removed a Permanent Set remains in the sample.

AREA UNDER STRESS – STRAIN GRAPHS l A Energy stored per unit volume Energy stored per unit volume = ½ STRESS X STRAIN

STEEL STRESS – STRAIN GRAPHS stress Y = (UPPER) YIELD POINT LOWER YIELD POINT is the point at which the main plastic deformation begins. Copper for comparison strain

FRACTURE AT THEIR ELASTIC LIMIT STRESS – STRAIN GRAPHS BRITTLE SUBSTANCES stress e.g. Cast Iron or Glass X BRITTLE SUBSTANCES FRACTURE AT THEIR ELASTIC LIMIT strain

YOUNG’S MODULUS APPARATUS Two identical wires are suspended from the same support. This counteracts any sagging of the beam and any expansion due to change in temperature. Extensions, e, are measured with the vernier scale. Diameter of the wire, D, is measured with a micrometer. BEAM l support wire test wire main scale venier scale fixed load keeps system taut F = mg Plot F vs. e variable load F e