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Solid Materials

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**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 Clicking this button returns to this page

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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. Back To Energy Density

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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 Back To Energy Density

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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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**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?

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**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 More

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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 Δx F A B O More

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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 Δx F A B O More

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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 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

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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 More

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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 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

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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, Δx or Δl

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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 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

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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 Hooke’s 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 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 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’ σ ε 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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**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

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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 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

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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 x y Area under the graph = area of the triangle = ½xy Back To Energy Density Back To Elastic Strain Energy

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ENGR 225 Section 3.1 - 3.8.

ENGR 225 Section 3.1 - 3.8.

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