1. Solid Materials

2. Index Properties of Solid Materials Uses of Solid Materials Maths Help
Hardness
Hooke’s Law
Finding the Gradient of a Graph
Stiffness
Force – extension curves
Toughness
Elastic Strain Energy
What does proportional mean?
Brittle
Stress
Strong
Strain
Area Under A Graph
Malleable
The Young Modulus
Ductile
Energy Density
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3. Properties of Solid Materials
Hardness is a surface phenomenon. The harder the material, the more difficult it is to indent or scratch the surface
The Mohs scale of hardness grades minerals from talc (1) to diamond (10), i.e. From softest to hardest.
BHN is the British Hardness Number used in engineering.

4. Properties of Solid Materials
Stiffness A stiff material deforms very little even when subject to large forces.
Stiff materials have a high Young Modulus, i.e. a high gradient on a stress – strain curve

5. Properties of Solid Materials
Toughness A tough material is able to absorb energy from impacts and shocks without breaking
Tough metals often undergo considerable plastic deformation in order to absorb this energy
Car tyres use a mixture of rubber and steel so the can absorb the energy from the impacts with road surfaces.
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6. Properties of Solid Materials
Brittle objects will shatter or crack easily when they are subjected to impacts and shocks
Brittle materials undergo very little plastic deformation before they break
Glass is often brittle – it cracks or breaks with relatively small amounts of energy
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7. Properties of Solid Materials
Strong objects can withstand large forces before they break
The strength of a material depends on its size and so therefore is defined in terms of its breaking stress
(where stress = force / area)

8. Properties of Solid Materials
Malleable materials can be hammered out into thin sheets or beaten into shape
Gold leaf is very malleable and so can be easily made into gold leaf

9. Properties of Solid Materials
Ductile materials can be drawn into wires.
Copper is very ductile and so copper wires are often used as electrical cables.
The copper is drawn out from cylinders until it reaches the desired diameter.
Ductile materials are often malleable however, malleable materials will often break when extended.

10. What does proportional mean?
Hooke’s Law
A material obeys Hooke’s Law if the extension produced by a force is directly proportional to that force.
F α Δx
F = k Δx Δx F
K = spring constant or stiffness, in N m-1
Maths Help
What does proportional mean?

11. Force – Extension Curves
When copper wire is extended, you may get a curve like this
From O to A Hooke’s Law is Obeyed
This means that the wire is behaving elastically and so loading and unloading are reversible
The bonds between atoms are stretched like springs but return to their original lengths when the deforming force is removed Δx F A O
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12. Force – Extension Curves
When copper wire is extended, you may get a curve like this
Beyond point B, the wire is no longer elastic
Although the wire may shorten when the load is removed it will not return to its original length
It has gone past the point of reversibility has therefore undergone permanent deformation Δx F A B O
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13. Force – Extension Curves
When copper wire is extended, you may get a curve like this
As the load is increased the wire yields and will not contract at all if the load is reduced
This is the yield point (C on the graph)
The wire is now plastic (it can be pulled like plasticine until it breaks)
In the plastic region the bonds are no longer being stretched
Layers of atoms stretch across each other with no restoring forces C Δx F A B O
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14. Force – Extension Curves
If the load is now removed from the copper wire during the plastic phase and reloaded, the following graph could be produced F Δx O
reloading
unloading
The wire regains its ‘springiness’ and has the same stiffness as before
The ability of some metals to be deformed plastically and then regain elasticity is very useful
Sheets of mild steel can be pressed into the shape of a car door and then pressed again when the stiffness and elasticity of the steel are regained

15. Elastic Strain Energy
Elastic strain energy is sometimes referred to as elastic potential energy
It is analogous to gravitational potential energy
It is therefore the ability of a deformed material to do work as it regains its original dimensions
For example the energy stored in a catapult is transferred to kinetic energy as it is released
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16. Elastic Strain Energy Remember work done = energy transferred
The work done, and therefore the elastic strain energy can be worked out from a force extension graph
For an object obeying Hooke’s Law the following graph would be obtained Δx F
Work done = force x displacement
i.e ΔW= FaveΔx
This is equal to the area under the curve and so
i.e ΔW= ½ FmaxΔx
Fmax
Maths Help
Area of a triangle

17. Stress Stress is the force (or tension) per cross- sectional area
i.e Stress = tension
cross-sectional area
Units of stress are N m-2 = Pa (pascal)
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18. More Stress
This is sometimes referred to as tensile stress when a material (e.g. A wire) as a force pulling it or compressive stress when a material has a force squashing it
The stress needed to break a material is called the breaking stress or ultimate tensile stress

19. Strain
Strain is the ratio of the extension of a wire to its original length when a stress is applied
Strain = extension
length
Strain has no units, although it may sometimes be referred to as a %age
Careful! The extension is sometimes x, Δx or Δl

20. The Young Modulus
The Young modulus of a material is a property of materials that undergo tensile or compressive stress
Young modulus = stress
strain
Since stress is in Pa and strain has no units, the Young Modulus also has the units of Pa
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21. The Young Modulus
The Young Modulus is a measure of the stiffness of a material, i.e. a stiff material has a high Young Modulus
Since ε σ
If the sample breaks or behaves elastically, the line won’t be straight. However the Young Modulus can still be found from the gradient of the straight line section of the graph
Maths Help
Calculating Gradient

22. Energy Density - 1 Remember work done = energy transferred
The energy density is the work done in stretching a specimen (or the strain energy stored) per unit volume of the sample.
For a wire that obeys Hooke’s Law
energy density = work done
volume
Therefore energy density = stress/strain
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23. Energy Density - 2 From the last slide, energy density = stress/strain
It is therefore the area under a stress-strain graph
If it is under the Hooke’s Law area of a graph, you need to find the area of the triangle
So energy density = ½ x stress x strain
= ½ x σ x ε σ ε
Maths Help
Area of a triangle
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24. Energy density - 3
If you have a more complicated curve, you could find the area under the graph by counting squares
If you do this, be careful to convert your square count correctly, i.e. Calculate how much one square is ‘worth’ σ ε
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25. Energy Density - 4
The ability of a material to absorb a large amount of energy per unit volume (i.e. it has a large energy density) before fracture is a measure of the toughness of the material, e.g. mild steel
Materials which fracture with little plastic deformation and so the area under the stress-strain graph is small (i.e. It has a low energy density) is brittle, e.g. glass

26. Back To The Young Modulus
Gradient of a graph
The gradient of a graph can be found be ‘dividing the up by the across’ or in mathematical terms by
Gradient = change in y – axis = Δy
change in x – axis Δx y Δy Δx
Back To The Young Modulus x

27. Maths Help – What Does Proportional Mean?
Direct proportionality is shown by a straight line graph through the origin
As x is doubled, y is doubled, etc
The proportionality sign is α, so
we can write y α x
To turn this into an equation without the
proportionality sign we need a constant,
so it becomes
y=kx
k is the constant of proportionality and
is represented by the gradient of the graph x y
Back To Hooke’s Law

28. Maths Help – Area of a triangle
The area of a triangle = ½ x base x height
This is useful for finding the area under straight line graphs x y
Area under the graph
= area of the triangle
= ½xy
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