MaterialsPHYA2

MATERIALSDENSITY, SPRINGS, STRESS AND STRAIN

Topics 11, pp.162–173

Studying materials• in terms of their

– bulk properties– and tensile strength

• Materials scientists then go on to link these bulk properties to the atomic/ molecular level, but we don’t get that far in this course.

Density

• Density = mass per unit volume

• ρ = m/V

• Units kg m–3 (in Chemistry g cm–3. Factor different?)

• How do you find the density of (i) regular solids, (ii) irregular solids, (iii) liquids, (iv) gases?

• Now do Summary Q1, p.163

Measuring density• Displacement methods to find irregular

volumes

• Density of alloys: – mass of A, mA = ρAVA, mass of B, mB = ρBVB

• Total volume = V

• Total density = ρAVA + ρBVB

V

• Use this method for layers, eg:

Have a go at summary Q4

Hooke’s Law• The force needed to stretch a spring is

directly proportional to the extension of the spring from its natural length.

• F = kΔL

• k is Hooke’s constant for the spring= spring constant = stiffness

• How can you measure k?

                  

Hooke’s Law

• A graph of force vs extension is…?

• a straight line

• through the origin

• i.e., directly proportional

• Gradient = kextension

forc

e

Note: We have exchanged usual arrangement of axes for dependent/independent variables. Why?

Hooke’s Law

• What mass would you need to hang from a spring of stiffness (k) 49 Nm–1 to achieve an extension of 0.6 m?

• What is the extension of a spring with spring constant of 120 Nm–1 if a mass of 40 g is added?

Spring combinations• Two springs in parallel, equal extension

• What is the new effective stiffness k?

• Force on P:

• Force on Q:

• Total Force:

• So

                  

LkF PP LkF QQ

LkkW

LkLkFF

QP

QPQP

)(

QP kkk

Spring combinations• Two spring in series• What is the new effective stiffness k?• Both springs loaded with force W• Extension of P:• Extension of Q:• Total extension:

• So

                   PP k

WL

QQ k

WL

QP

QPQP

kkWL

k

W

k

WLL

11

QP kkk

111

Energy stored in a spring

• Elastic PE is stored in a stretched (or compressed) spring

• Energy stored = work done

• Work to extend length by

L is area under graph

2

2

1

2

1LkLFW

extension

forc

e

L

Some jargon to learn

• Tensile deformation: stretching

• Compressive deformation: squashing

• Stiff: hard to deform

• Flexible: easy to deform

• Elastic: returns to original shape when force is removed

• Plastic: permanently deformed when force is removed

Some more jargon to learn…

• Plastic materials are also:– Ductile: can be drawn into wires– Malleable: can be hammered into sheets

• Malleable materials are tough:– They give way gradually and need a lot of

energy to break them

• The opposite of tough is brittle:– These materials do not deform plastically but

crack or shatter suddenly

Deformation of solids

• How much a sample deforms depends on:– The material– The force applied– The dimensions of the sample

Stress and strain

• To remove the dependency on dimensions it is useful to define:

• Tensile stress ():– Tension (force) per cross-sectional area– Units Nm–2 or Pa (like pressure)

• Tensile Strain ():– Fractional extension– No units (a ratio)

• Hooke’s Law:

A

T

L

L

Young’s Modulus

• We can now define a material property similar to stiffness but independent of geometry:The Young Modulus

unit: Pa

• E is given by the gradient of the elastic part of a stress-strain graph

ELA

TL

L

L

A

TE

Measuring Young’s Modulus

• Measure extension of wire for different loads• What else do you need to measure?

– How are you going to measure them?

• Obtain a value for ECu and estimate its uncertainty

Stress-strain curve for steel

Elastic deformation (gradient = E)

Limit of proportionality

Elastic limit

Yield point

Ultimate tensile strength

fracture

Onset of work hardening

ExamplesMaterial UTS E

Aluminium 80 Mpa 71 GPa

Gold 120 MPa 71 GPa

Mild steel 460 MPa 210 GPa

Steel piano wire 3 000 MPa 210 GPa

Lead 15 MPa 18 GPa

Bone 140 MPa 345 GPa

Concrete ~4 MPa 14 GPa

Glass ~100 MPa 71 GPa

LD polythene 13 MPa 0.18 GPa

Exam questions

[8], June ‘07

Different ways of deforming

• Describe how these materials behave under stress as shown in the graph.

• What might they be?• Have a go at

Summary Q4, p.169 (careful with axes!)

• The diagram shows tensile stress-strain curves for three different materials X, Y and Z.

• For each material named below, state which curve is typical of the material, giving the reasoning behind your choice.

(a) copper(b) glass(c) hard steel

Loading and Unloading

a. On unloading, the elastic deformation is reversed, but plastic deformation remains

b. Unloading curve below loading, remains elastic (“hysteresis”).

c. Mainly plastic deformation.

Strain energy• We do work to stretch a metal wire

stretched within its elastic limit– This is stored as elastic energy– The amount is equal to the area under the

load-extension graph– When the stress is released all the stored

energy can be recovered

• Beyond the elastic limit, some energy cannot be recovered

– Where has it gone?

Stretching a rubber band

• Is the area greater under the loading or unloading curve?

• What does this tell you about the work done and energy released?

• How can we work out and account for the difference?

• Practice of stress and strain: SQs p.169

• Practice of curves: SQs p.171

• Now make sure you can do the exam-style questions on pages 172-173

• After that you can try Materials Revision 1