Hardness Ductile Force extension curves Malleable Brittl

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Solid Materials
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Index
Properties of Solid Materials
Uses of Solid Materials
Maths Help
Hardness
Hookes Law
Finding the Gradient of a Graph
Force extension curves
Stiffness
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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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.

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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

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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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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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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)

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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

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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.

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Hookes Law

• A material obeys Hookes Law if the extension
  produced by a force is directly proportional to
  that force.

F a ?x F k ?x
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 Hookes 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

O
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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

O
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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
O
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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

• 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

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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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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 Hookes Law the following
  graph would be obtained

• Work done force x displacement
• i.e. ?W Fave?x
• This is equal to the area under the curve and so
• i.e. ?W ½ Fmax?x

Fmax
Maths Help Area of a triangle
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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)

More
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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

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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, ?x or ?l
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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

More
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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

If the sample breaks or behaves elastically, the
line wont 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
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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 Hookes Law
• energy density work done
• volume
• Therefore energy density stress/strain

More
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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 Hookes Law area of a graph,
you need to find the area of the triangle So
energy density ½ x stress x strain ½ x s
x e
Maths Help Area of a triangle
More
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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
More
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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

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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 ?y
• change in x axis ?x

Back To The Young Modulus
x
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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 a, so
• we can write y a x
• To turn this into an equation without the
• proportionality sign we need a constant,
• so it becomes
• ykx
• k is the constant of proportionality and
• is represented by the gradient of the graph

Back To Hookes Law
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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

• Area under the graph
• area of the triangle
• ½xy

Back To Elastic Strain Energy
Back To Energy Density