Section 5 - Materials Flashcards
What is density?
The mass of a material per unit volume.
What is the equation for density?
Density (kg/m³) = Mass (kg) / Volume (m³)
p = m / v
What is the symbol for density?
ρ - rho (looks like a ‘p’)
What are the units for density?
g/cm³ or kg/m³
Convert 1 g/cm³ to kg/m³.
1 g/cm³ = 1000 kg/m³
Is density affected by size or shape?
No, just the material.
What determines whether a material floats?
- The relative average densities.
* If a solid has a lower density than a fluid, it will float in the fluid
What is the density of water?
1 g/cm³ (which is 1000 kg/m³)
What is Hooke’s law?
- The extension of a stretched object (Δl) is proportional to the load (F)
- F = k x Δl
What is the equation for Hooke’s law?
Force (N) = Stiffness constant (N/m) x Extension (m)
F = k x Δl
What are the units for the spring constant, k?
N/m
What is k?
- The stiffness constant for a material being stretched
* With springs, it is usually called the spring constant
Describe the forces acting on a metal wire being stretched by a load.
- Load pulls down on the end of the wire with force F
- Support pulls up on the top of the wire with an equal reaction force R
- F = R
(See diagram pg 66 of revision guide)
Does Hooke’s law only work for tensile forces?
No, it also works for compressive forces.
See diagram pg 66 of revision guide
What things obey Hooke’s law?
• Springs
• Metal wires
• Most other materials
(Up to a point!)
What types of forces does Hooke’s law work for?
- Tensile (stretching)
* Compressive
Does Hooke’s law involve just one force?
- No, there must be two equal and opposite forces at the ends of the object.
- They can be tensile of compressive.
(See diagram pg 66 of revision guide)
Is the value of k the same with tensile and compressive forces?
- In springs - the same.
* In other materials - not always.
A material with only deform (stretch, bend, twist, etc.) if…
…there’s a pair of opposite forces acting on it.
Describe the forces acting on a fixed spring that has a compressive force acting on the base.
- The compressive force, F, pushes up onto the spring
- The support exerts an equal and opposite reaction force, R, down onto the spring
- F = R
How is Hooke’s law illustrated on a graph?
- Graph of force (y) against extension (x)
* Gradient of straight part is the value of k
Does Hooke’s law always work?
No, it stops working when the force is great enough (the limit of proportionality).
Why is a force-extension graph plotted with extension on the x axis?
So that the gradient gives k.
Describe the force-extension graph for a typical metal wire.
- Straight-line from origin up to the limit of proportionality (P)
- Line curves slightly towards x-axis up to elastic limit (E)
- Line curves more towards the x-axis
What is the limit of proportionality on a force-extension graph?
- The point at which the line starts to curve
* Hooke’s law works up to this point
How is Hooke’s law illustrated on a graph?
- Graph of force (y) against extension (x)
* Gradient of straight part is the value of k
Does Hooke’s law always work?
No, it stops working when the force is great enough (the limit of proportionality).
Why is a force-extension graph plotted with extension on the x axis?
So that the gradient gives k.
Describe the force-extension graph for a typical metal wire.
- Straight-line from origin up to the limit of proportionality (P)
- Line curves slightly towards x-axis up to elastic limit (E)
- Line curves more towards the x-axis
What is the limit of proportionality on a force-extension graph?
- The point at which the line starts to curve
* Hooke’s law works up to this point
What is the elastic limit on a force-extension graph?
The force beyond which the material will be permanently stretched and will no longer return to its original shape.
Does rubber obey Hooke’s law?
Yes, but only for really small extensions.
On a force-extension graph for a metal wire, where are P (limit of proportionality) and E (elastic limit)?
- P -> Where the line starts curving
* E -> After P
What are the two types of stretching?
- Elastic
* Plastic
What is elastic deformation?
When a material returns to its original shape and size ice the forces are removed.
What is plastic deformation?
When a material is stretched so that it cannot return to its original shape or size and is permanently deformed.
Describe elastic deformation in terms of atoms.
1) Under tension, the atoms in the material are pulled apart.
2) They move short distances from their equilibrium positions without changing positions in the material.
3) Once load is removed, atoms can return to their equilibrium distances apart.
Describe plastic deformation in terms of atoms.
1) Certain atoms move position relative to one another.
2) When the load is removed, the atoms don’t return to their equilibrium position.
When do elastic and plastic deformation happen?
- Elastic deformation -> Below the elastic limit
* Plastic deformation -> Beyond the elastic limit
Describe the energy transfers when an object is deformed elastically.
- All the work done to stretch is stored as elastic strain energy
- When the force is removed, the stored energy is transferred to other forms (e.g. kinetic energy)
Describe the energy transfers when an object is stretched plastically.
- The work done to separate atoms is not stored
* It is mostly dissipated (e.g. as heat)
What type of deformation occurs in crumple zones in cars and why?
- Plastic
* Energy goes into changing the shape of the vehicle’s body -> Less is transferred to the people inside
What is a tensile force?
A stretching force.
What is tensile stress?
The force applied per unit cross-sectional area of a material.
What is the equation for stress?
Stress (N/m²) = Force (N) / Cross-sectional area (m²)
Stress = F / A
What are the units for stress?
N/m² or Pa
What is tensile strain?
The change in length of a material divided by the original length when stretching.
What is the equation for strain?
Strain = Change in length (m) / Original length (m)
Strain = ΔL / L
What are the units for strain?
- There are no units.
* It’s just a decimal or percentage.
Which is correct?
• A stress causes a strain
• A strain causes a stress
A stress causes a strain.
Do stress and strain equations apply with both tensile and compressive forces?
Yes, except tensile forces are considered positive, while compressive are negative.
In stress and strain calculations, what are the signs of tensile and compressive forces?
- Tensile - Positive
* Compressive - Negative
What is the difference between stress and strain?
- Stress is the force applied per unit area.
* Strain is the change in length over original length.
What is breaking stress?
The point at which the material being stretched will break.
Describe the effect of stress on atoms in a material.
- Stress pulls the atoms apart slowly.
* Eventually the atoms separate completely and the material breaks -> This is the breaking point.
What is the ultimate tensile stress?
The maximum stress that a material can withstand.
How is the stretching of a material illustrated on a graph?
- Graph of stress (y) against strain (x)
* Gradient is straight part in the Young modulus
Are ultimate tensile stress and breaking stress fixed values?
No, they depend on conditions, such as temperature.
On a stress-strain graph, where is the ultimate tensile stress?
The highest point reached by the line.
On a stress-strain graph, where is the breaking stress?
At the end of the line.
How can you find the (elastic strain) energy from a force-extension graph?
- It is the area under the straight part of the line.
* Because “W = F x d”.
Why does the area under a force-extension graph represent the elastic strain energy stored in a material?
- Work has to be done to stretch the material
* Before the elastic limit, all the work done is stored as elastic strain energy in the material.
What equation gives the energy required to stretch a material?
• E = 1/2 x F x ΔL
OR
• E = 1/2 x k x (ΔL)²
Where F = Final force
(NOTE: This is only if Hooke’s law is obeyed!)
Why is the work done is stretching a material equal to 1/2 x F x d, even though W = F x d?
- Because the force required to stretch a material increases from zero to the final force (F).
- So the average force is used.
Derive the two equations for the energy required to stretch a material.
• W = F x d So • E = Average force x ΔL • E = 1/2 x F x ΔL (Equation 1) • F = k x ΔL So • E = 1/2 x k x ΔL
The energy stored in a material by stretching it is equal to…
…the work done in stretching it.
Are strain and stress on a material proportional?
Only up to the limit of proportionality.
What is the Young Modulus of a material?
- The stress divided by the strain below the limit of proportionality.
- Measure of stiffness.
What is the symbol for the Young modulus?
E
Give the equation for the Young modulus.
E = Stress / Strain
E = (F x L) / (A x ΔL)
Where: F = Force (N) A = Cross-sectional area (m²) L = Original length (m) ΔL = Extension (m)
Derive the equation for Young modulus.
- E = Stress / Strain
- E = (F / A) / (ΔL / L)
- E = (F x L) / (A x ΔL)
What are the units for the Young modulus?
N/m² or Pa
Same as stress, since strain has no units.
What is Young modulus s measure of?
Stiffness of a material.
What is the Young modulus used for?
Engineers use it to ensure that materials they are using can withstand sufficient forces.
Describe an experiment to find the Young modulus of a wire.
1) Measure the diameter of a thin wire using a micrometer in several places and take an average.
2) Find the cross-sectional area of the wire using “A = πr²”.
3) Clamp the wire with a clamp at one end and over pulley at the other end, so that weights can be hung on the wire.
4) Align a ruler with the wire and attach a marker.
5) Start with the smallest weight to straighten the wire (but ignore this weight in calculations).
6) Measure the unstretched length of the wire from clamped end of the string to the marker.
7) Add 100g weights to the string and measure the extension.
8) Plot a stress (y) against strain (x) graph of your results. The gradient of the straight part is the Young modulus.
(See diagram pg 70 of revision guide)
Name some ways in which the experiment to find the Young modulus of a wire is made more accurate.
- Using a long, thin wire -> Reduces uncertainty
- Taking several diameter readings and finding an average
- Using a thin marker on the wire
- Looking directly at the marker and ruler when measuring extension
Why is a stress-strain graph plotted, even though the stress is the independent variable?
On a stress-strain graph, the gradient gives the Young modulus.
How can you find the Young modulus from a stress-strain graph?
- It is the gradient of the straight part of the line.
* This is because E = Stress / Strain
On a stress-strain graph, what does the area under the straight part of the line represent?
- The strain energy stored per unit volume.
* i.e. The energy stored per 1m³ of wire
What are the units for elastic strain energy stored per unit volume?
J/m³
Why does the area under the straight part of the line on a stress-strain graph give the elastic energy stored in the wire?
- Area = 1/2 x Stress x Strain
- Area = 1/2 x N/m² x No units
- Area = 1/2 x N/m²
- Area = 1/2 x N x m / m³
- Area = 1/2 x F x d / V
- Area = 1/2 x Work done / Volume
What equation gives the elastic energy per unit volume of a stretched wire?
Energy per unit volume = 1/2 x Stress x Strain
As long as Hooke’s law is obeyed!
On a force-extension graph, what do the gradient and area under the line give?
- Gradient = Spring constant (k)
* Area under line = Work done (or elastic energy stored)
On a stress-strain graph, what fit the gradient and area under the line give?
- Gradient = Young modulus
* Area under line = Elastic energy stored per unit volume
On a force-extension and stress-strain graph, what do the gradient and area under the line give?
FORCE-EXTENSION:
• Gradient = Spring constant (k)
• Area under line = Work done (or elastic energy stored)
STRESS-STRAIN:
• Gradient = Young modulus
• Area under line = Elastic energy stored per unit volume
Describe a typical stress-strain graph for a DUCTILE material being stretched, with all the important points.
- Straight line up until the limit of proportionality.
- Curves towards the x-axis slightly until the elastic limit
- Curves more towards the x-axis until the yield point
- After yield point, the line goes down slightly
- There may be a second peak before the breaking stress
- The UTS is the highest stress reached, usually on the second peak
Do force-extension and stress-strain graphs show Hooke’s law?
Yes - straight lines through the origin on both show Hooke’s law.
If a material was stretched to the limit of proportionality, would it return to its original size and shape?
Yes, as long as the elastic limit is not exceeded.
What are the important points along a stress-strain graph?
- Limit of proportionality (P)
- Elastic limit (E)
- Yield point (Y)
- Ultimate tensile stress (UTS)
- Breaking stress (B)
What is the yield point on a stress-strain graph?
- The point beyond which the material starts to stretch without any extra load.
- i.e. A large amount of plastic deformation occurs with constant or reduced load
Remember to practise labelling a stress-strain graph.
Find a diagram on the internet.
Describe the shape of a typical stress-strain graph for a ductile material.
- Two peaks
- Second peak is higher than the second
- Goes through origin
Where is the limit of proportionality on a stress-strain graph?
Where the line starts curving.
Where is the elastic limit on a stress-strain graph?
Soon after the limit of proportionality.
Where is the yield point on a stress-strain graph?
When the line suddenly goes down (usually the peak of the first bump).
On a stress-strain graph, which area represents the energy stored in the material per unit volume?
The area under the curve up to point P (the limit of proportionality).
Do brittle materials obey Hooke’s law?
Yes
Describe the stress-strain graph for a brittle material.
- Straight line through origin.
- Reaches breaking point without curving.
(See diagram pg 72 of revision guide)
Give an example of a brittle material.
Ceramics (e.g. glass and pottery)
What is the difference between a force-extension and stress-strain graph?
- Force-extension is specific to the tested object and depends on the dimensions
- Stress-strain describes the general behaviour of a material, because stress and strain are independent of dimensions
What is the opposite of brittle?
Ductile
Describe the loading-unloading force-extension graph for a metal wire stretched to below its elastic limit.
The loading and unloading lines are the same and both go through the origin.
Describe the loading-unloading force-extension graph for a metal wire stretched beyond its elastic limit.
- The loading line curves towards the x-axis until unloading starts.
- The unloading line is parallel to the loading line and crosses the x-axis at a positive extension value.
(See diagram pg 73 of revision guide)
On a force-extension graph, why is the unloading line parallel to the loading line?
The stiffness constant (k) is still the same since the forces between the atoms are the same as they were during loading.
On a loading-unloading force-extension graph, how can you find the work done to deform the wire (i.e. the energy lost)?
It is the area between the two lines.
What type of material will have two peaks on a stress-strain graph?
Ductile
Add flashcards about determining the difference between a strong/weak and brittle/ductile material from a stress-strain graph.
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