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MATTER

 

Elasticity

 

Young's Modulus

Hooke's law

strain energy

elastic hysteresis

 

 

 

Young's Modulus

 

This is a typical stress-strain curve of a ductile material.

 

A brittle material would have a much smaller strain value before breaking(making EB shorter).

 

 

stress-strain curve

 

 

L - limit of proportionality


E - elastic limit


Y - yield point


X - stress removed here, body has permanent strain 0X'


B - breaking stress

 

 

 

For a given material,

 

Young's Modulus (E) is the ratio of stress* to strain*, provided the limit of

 

proportionality is not exceeded.


* sometimes referred to as stress σ (sigma), strain ε (epsilon)

 

 

Young's Modulus in terms of stress and strain

 

 

The gradient at any point on a stress - strain graph is the Young's Modulus (E).

 

 

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Hooke's Law and Molecular Separation

 

Hooke's Law states that the extension(x) of a spiral spring(or other elongated body) is proportional to the applied force(F), provided the limit of proportionality is not exceeded.

 

Hooke's Law equation

 

k is the constant of proportionality(a measure of 'stiffness')

 

 

So force-extension curves of different materials are straight lines through the origin, with gradient k.

 

 

Hooke's Law & stiffness

 

 

On a stress-strain graph the Young's Modulus is constant for the portion of the graph where Hooke's Law applies.

 

This can be easily shown by substituting for k = F/x into the equation for E.

 

 

proof that E is constant for Hooke's Law

 

E is a constant because l, A and k are all constant.

 

 

The force F between molecules is directly proportional to small displacements either side of the equilibrium position ro .

 

This translates in the bigger picture to applied force being proportional to extension.

 

 

Hooke's law & molecular separation

 

 

 

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Strain Energy

 

To obtain an expression for the strain energy(work done) in stretching a wire, consider a wire of original length l0 where a force F produces an extension x .

 

It is assumed that the wire obeys Hooke's Law.

 

 

Hooke's Law energy stored in a wire - diagram

 

 

Now, let the force F cause a further extension δx, , where δx is so small that F may be considered constant.

 

Since,

 

work done = force x distance force moves

 

strain energy derivation equation #1

 

The total work done when the wire is stretched from 0 to x is the area under the F - x curve between these two limits.

 

strain energy equation #2

 

Since the wire obeys Hooke's Law,

 

strain energy equation #3

 

Substituting for F in the integral expression(above):

 

strain energy equation #4

 

strain energy equation #5

strain energy equation #3,         strain energy equation #6

 

 

energy stored in a stretched wire

 

 

As can be seen from the graph, the area under the curve is half the product of F and x .

 

 

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Elastic Hysteresis

 

 

elastic hysteresis

 

 

The graph illustrates how, for a given force, the extension is greater for unloading than loading.


Looking at this another way, for a given extension the loading force is greater than the unloading force.


The stretching produces an increase in temperature(loading).  

 

When the strain is reduced(unloading) the temperature drops.

 

However, some heat is retained to keep the material above its initial temperature.

 


For a complete cycle, the increase in heat energy is the area of the hysteresis loop.

 


The graph is for rubber, but metals also exhibit this property, though the effect is considerably smaller.

 

 

 

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