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Hardness Ductile Force – extension curves Malleable Brittle Toughness Stiffness Strong Hooke’s Law Elastic Strain Energy Stress Strain The Young Modulus Energy Density Finding the Gradient of a Graph What does proportional mean? Area Under A Graph Properties of Solid Materials Uses of Solid Materials Maths Help Clicking this button returns to this page

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

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When copper wire is extended, you may get a curve like this ΔxΔx F A O 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 More

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When copper wire is extended, you may get a curve like this ΔxΔx F A B O 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 More

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When copper wire is extended, you may get a curve like this ΔxΔx F A B O 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 More

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If the load is now removed from the copper wire during the plastic phase and reloaded, the following graph could be produced F ΔxΔx 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 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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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 ΔxΔx F Work done = force x displacement i.e. ΔW= F ave Δx This is equal to the area under the curve and so i.e. ΔW= ½ F max Δx F max Maths Help Area of a triangle

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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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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 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 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 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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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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From the last slide, energy density = stress/strain It is therefore the area under a stress-strain graph σ ε 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 Maths Help Area of a triangle More

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If you have a more complicated curve, you could find the area under the graph by counting squares σ ε 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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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 steeltoughness 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. glassbrittle

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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 x y ΔyΔy ΔxΔx Back To The Young Modulus

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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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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 Elastic Strain Energy Back To Energy Density

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