11.3 - deformation of solids

what is needed to stretch, twist, or compress a material?

a force pair (the deformative force and tension)

what is the force pair involved in deforming a material?

the deformative force and tension

what is elasticity?

a solid material’s ability to regain its shape after it has been deformed or distorted after the forces deforming it have been released

what is a tensile force?

deformative stretching force

what is a compressive force?

deformative compressing force

which is a deformative stretching force - tensile or compressive force?

tensile

which is a compressive stretching force - tensile or compressive force?

compressive

how do you test the ease at which material stretches?

1. material held at its upper end and loaded by hanging weights on its lower end

2. weight of load increased in steps, then decreased to zero

3. set square / pointer attached to the bottom of the weights used to measure material extension

4. extension-weight graph can be drawn from the results

here

what is an extension-weight graph for the test for material extension?

• y = extension of strip of material at each step = increase in length from its unloaded length (L)

• x = weight (= tension in the material)

• area under graph = elastic potential energy in material

• gradient = hooke’s law (only if extension ∝ weight)

what is the area under an extension-weight graph?

elastic potential energy

when testing material extension, what is the tension?

equal to the weight of the load

what is the extension-weight graph for a steel spring?

here

steel spring

what is the extension-weight graph for a rubber band?

here

• at first it extends easily when its stretched

• however, it becomes fully stretched and very difficult to stretch further when it has been lengthened considerably

here

rubber band

why does the extension-weight graph for a steel spring look like this?

here

its in accordance with hooke’s law as its extension ∝ weight

what is the extension-weight graph for a polythene strip?

here

• ‘gives’ and stretches easily after its initial stiffness is overcome

• however, after ‘giving’ easily, it extends a little and becomes difficult to stretch

polythene strip here

polythene strip

why does the extension-weight graph for a polythene strip look like this?

here

• it ‘gives’ and stretches easily after its initial stiffness is overcome

• however, after ‘giving’ easily, it extends a little and becomes difficult to stretch

where is the extension measured from?

the original (unstretched) object length

which material ‘gives’ the easiest?

polythene strip

which material doesn’t ‘give’ easily?

steel spring

how do we measure the extension of a wire under tension?

• searle’s apparatus

• similar apparatus using a vernier scale

how does a searle’s apparatus work?

here

1. micrometer attached to control wire is adjusted so spirit level between control and test wire is horizontal

2. when test wire is loaded, it extends slightly, causing the spirit level to drop on one side

3. the micrometer is then readjusted to make the spirit level horizontal again

4. the change of micrometer reading is therefore equal to the extension

5. the extension may be measured for different values of tension by increasing the test weight in steps

in searle’s apparatus, how is the spirit level adjusted?

so the spirit level between the control and test wire is horizontal

what is the extension in in searle’s apparatus?

the micrometer reading due to displacement of spirit level due to test weight

in searle’s apparatus, what is the change in micrometer reading equal to?

extension

in searle’s apparatus, what is the micrometer attached to?

the control wire

what are the features of a searle’s apparatus?

• test wire

• supports

• control wire

• spirit level

• hinge

• control weight

• test weight

• micrometre

what is tensile stress?

• tension (= force applied) per unit cross-sectional area

• σ = T / A

what is the unit for tensile stress?

Pascal (Pa) or newtons per metre squared (N m^-2)

what is the symbol for tensile stress?

σ

σ = T / A

tensile stress

what is the equation for tensile stress?

σ = T / A

• σ = tensile stress

• T = tension in material

• A = cross-sectional are of material

what is tensile strain?

• extension per unit length

• measure of the deformation of a material when subjected to tensile stress

• ratio of lengths

• ε = Δ L / L

ε = Δ L / L

tensile strain

what is the symbol for tensile strain?

ε

what is the unit for tensile strain?

it doesn’t have one cuz its a ratio

what is the equation for tensile strain?

ε = Δ L / L

• ε = tensile strain

• Δ L = extension of wire

• L = original length

what is the cross-sectional area of a wire?

area of a circle, (λ d²) / 4

what is toughness?

a measure of the energy needed to break a material

what is a measure of the energy needed to break a material?

toughness

what is the tensile stress-strain graph for a wire?

here

label the points on this tensile stress-strain graph for a wire

here

here

what are the points you’ll find on a tensile stress-strain graph for a wire?

• limit of proportionality (P)

• limit of elasticity (E)

• yield point (Y₁)

• Y₂

• ultimate tensile stress (UTS)

• breaking point (B)

what happens at the limit of elasticity?

the point where the material loses its elasticity, and can no longer regain its shape after being deformed. it is plastically deformed

here

tensile stress-strain graph for a wire

what is young’s modulus?

• a constant of stress / strain for a material

• gradient of the proportional part of a stress-strain graph

• E = T L / A Δ L

E = T L / A Δ L

young’s modulus

what is the equation for young’s modulus?

E = T L / A Δ L

• E = young modulus

• T L = tensile stress

• A Δ L = tensile strain

derive E = T L / A Δ L

• young’s modulus = tensile stress / tensile strain

• = σ / ε

• = (T / A) ÷ (Δ L / A Δ L)

• = T L / A Δ L

how do we notate young’s modulus?

E

what does E notate?

young’s modulus

what happens at the limit of proportionality?

the point at which stress is no longer proportional to strain, and the gradient of the graph stops showing young’s modulus

how do we notate limit of proportionality?

P

what does P notate?

limit of proportionality

what happens at the yield point, Y₁?

the wire weakens temporarily

what happens at Y₂?

a small increase in tensile stress causes a large increase in tensile strain as the material of the wire undergoes plastic flow

why is there a sudden large increase in tensile strain at Y₂?

because the material of the wire undergoes plastic flow

what happens at the ultimate tensile stress (UTS)?

• the wire loses its strength, extends, and becomes narrower at its weakest point. there is an increase of tensile stress

• maximum tensile strength, i.e., the strength of a material

why is there an increase of tensile stress at the UTS?

because the cross-sectional area of the material has reduced

what happens at the breaking point?

the wire breaks (oh, mr nobel … i’m ready to collect my prize…)

how else is the UTS referred to?

the breaking stress

what is the breaking stress?

the UTS

at which point on a stress-strain graph does the stress stop being proportional to the strain?

limit of proportionality

at which point on a stress-strain graph does young’s modulus stop being followed?

limit of proportionality

at which point on a stress-strain graph is the material plastically deformed?

elastic limit

at which point on a stress-strain graph does the wire weaken temporarily?

yield point, Y₁

at which point on a stress-strain graph does a small increase in tensile stress cause a large increase in tensile strain?

Y₂

at which point on a stress-strain graph does the material of the wire undergo plastic flow?

Y₂

at which point on a stress-strain graph does the wire lose all strength, extends, and becomes narrower at its weakest point?

UTS

at which point on a stress-strain graph does the cross-sectional area of the wire reduce?

UTS

at which point on a stress-strain graph does the wire break?

breaking point (/ stress)

how can we compare the stiffness of different materials?

using the gradient (when proportional) of their respective stress-strain graphs

the larger the young’s modulus, the stiffer / elastic the material?

the larger the young’s modulus, the stiffer the material

when comparing stiffness between materials, why do we compare the gradient when proportional?

because at that point the gradient is young’s modulus (stress / strain). the young’s modulus is a constant measure of stiffness for that material, so it can be used for comparison

what is the strength of a material?

its UTS, which is its maximum tensile stress

what is the maximum tensile strength?

• ultimate tensile strength (UTS)

• strength of a material

why is steel stronger than copper?

because it has a higher maximum tensile strength

what determines the strength of a material?

its maximum tensile strength

when talking about strength, do we use the term ‘ultimate’ or ‘maximum’ tensile strength?

maximum tensile strength

what is a brittle material?

material that snaps without any noticeable yield, e.g., glass

what is a ductile material?

material that can be drawn into a wire

which is more ductile - copper or steel?

copper

what is the stress-strain graph for steel?

here

• large stress range, therefore stiff

• high UTC, therefore strong

here

stress-strain graph for steel

what is the stress-strain graph for glass?

here

• small strain range, therefore brittle

• breaking point soon after limit of elasticity, therefore brittle

• low UTC, therefore weak (for this particular glass)

here

stress-strain graph for glass

what is the stress-strain graph for copper?

here

• large strain range, therefore ductile

• no obvious yield points

• low UTC, therefore weak

here

stress-strain graph for copper

what does it mean for a material to have a low strain range on a stress-strain graph?

it is brittle

what does it mean for a material to have a high strain range on a stress-strain graph?

it is ductile

what does it mean for a material to have a high stress range on a stress-strain graph?

it has a greater stiffness

what does it mean for a material to have a low stress range on a stress-strain graph?

it’s not that stiff

what does it mean for a material to have a high UTC on a stress-strain graph?

it is strong

what does it mean for a material to have a low UTC on a stress-strain graph?

it is weak