S-Cool Revision Summary

S-Cool Revision Summary

Stress and Strain

The problem with force - extension graphs is that they only give information about the exact object and material that you are examining.

Stress and strain are measurements that allow us to compare behaviour of materials and objects no matter what size or shape they are because the force and extension are multiplied up or down to find out what the force would be if it was spread over 1 m2 or what the extension would be per metre of the original material.

Stress and Strain, Definitions

Stress is defined as the force per unit area of a material.

Stress = force / area

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Units: Nm-2 or Pa.

Strain is defined as extension per unit length.

Strain = extension / original length.

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Strain has no units.

A useful tip: In calculations stress is usually a very large number and strain is usually a very small number. If it comes out much different than that, you've done it wrong!

Young Modulus

Instead of drawing a force - extension graph, if you plot stress against strain for an object showing (linear) elastic behaviour, you get a straight line.

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This is because stress is proportional to strain. The gradient of the straight-line graph is the Young's modulus, E

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Units of the Young modulus E: Nm-2 or Pa.

Note: The value of E in Pa can turn out to be a very large number. It is for this reason that, some times the value of E may be given MNm-2.

Note: Because 'stress' and 'strain' are (uniquely) related to force and extension, it is not surprising that the two graphs, stress v/s strain and force v extension, have similar shapes and characteristics. (Of course, there is a quantitative difference because the units of plots on the two graphs are different.).

Stress - strain graph beyond elastic behaviour

In this 'quick learn' so far, we have drawn stress-strain graphs for the elastic behaviour of a material. In the elastic region the stress-strain graph is a straight line. We can, however draw a stress-strain graph beyond the elastic region. The graph, then becomes non-linear because Hooke's law is not obeyed and stress is not proportional to strain. Here are schematic stress-strain graphs of copper and glass.

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Note: the gradient of a stress - strain graph = stress / strain.

Density

We know that some materials are light while some are very heavy. For instance, a cube of polythene will weigh a lot less than the same size cube of steel.

We can systematically compare the amount of matter in of different materials by defining a property called density (ρ).

Density = mass per unit volume

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Where:

M is the mass of an object, and V is its volume.

Units of ρ : kgm-3.

Deformation and fracture

When looking at different materials for mechanical purposes we use 'stress-strain' curves. We saw that for materials obeying Hooke's law the stress strain graph is a straight line. However, this straight line forms just a part of the stress strain curve. The whole of the stress-strain curve of a material is an invaluable aid to describing its mechanical behaviour.

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  1. Linear elastic region (region I)
  2. Non-linear elastic region (region II)
  3. Yield region (region III)
  4. Beyond the lower yield point (region IV)
  5. If the material is stretched further beyond the UTS (region V)

Equations

Stress/ Strain

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Symbols

Stress/ Strain

E - Young Modulus, Nm-2 or Pa

F - Force, N

A - Area, m2

σ - stress, Nm-2 or Pa

ε - strain, no units

e - extension, m

l - original length of material, m

Molecular volume (Vmolecule) Average volume occupied by each molecule in a substance by using the following formula
Molar mass mass of one mole of the material.
Mole amount of material containing 6.023x1023 molecules
Plastic behaviour Behaviour of a material where deformation remains after the forces are removed (irreversible deformation)
Resilience Ability of a material to be repeatedly stressed without plastic deformation and without losing strength
Spring constant Force per unit extension (a constant of proportionality in Hooke's law)
Strain Extension per unit length.
Strain energy density (ρε) Energy stored per unit volume in a material when elastically deformed. (units: Jm-3)
Stress Force per unit area of a material (units: Nm-2 or Pa)
Strain energy density (ρε) Strain energy stored per unit volume (units:Jm-3)
Ultimate tensile strength (UTS) The maximum tensile stress a material can stand
Yield Suddenly increased deformation
Young's Modulus (E) A constant indicating stiffness of a material, given by the ratio: stress/strain (units: Nm-2 or P)