Elastic and plastic behaviour

AS Level Physics · Deformation of solids. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

In elastic deformation the object returns to its original length when the load is removed. In plastic deformation it does not: it stays permanently longer. The elastic limit is the largest load (or stress) for which the object still returns to its original length.

The elastic limit is usually a little beyond the limit of proportionality. Between the two, the graph is no longer straight but the wire still goes back to its original length.

The area under a force–extension graph is the work done in stretching. Within the limit of proportionality the graph is a straight line through the origin, so the area is a triangle and all of this work is stored as elastic potential energy.

Rules to remember

Common mistake

Students use energy = Fx and forget the ½. The force rises from zero to F as the spring stretches, so the average force is ½F and the energy is ½Fx.

Worked example

A spring obeys Hooke's law and extends by 6.0 cm under a load of 9.0 N. How much elastic potential energy does it store when its extension is 10 cm?

  1. Spring constant k = F ÷ x = 9.0 ÷ 0.060 = 150 N m−1
  2. Change the new extension to metres: 10 cm = 0.10 m
  3. Ep = ½kx2 = ½ × 150 × 0.102 = 0.75 J

Answer: 0.75 J

Practice questions

Try each one, then open the answer.

1. On a force–extension graph for a wire loaded then unloaded within its elastic region, the loading and unloading lines coincide. What does this show?

  1. A
    the wire has deformed plastically
  2. B
    all the energy stored on loading is returned on unloading
  3. C
    the wire has broken
  4. D
    the material obeys Hooke's law only on unloading
Show answer

Answer: B. Coinciding lines mean no energy is lost: elastic behaviour. A loop between the lines (hysteresis, as in rubber) shows energy dissipated.

2. A metal wire is stretched beyond its elastic limit and the load is removed. What happens?

  1. A
    it returns to its original length
  2. B
    it keeps a permanent extension (plastic deformation)
  3. C
    it breaks immediately
  4. D
    it becomes shorter than before
Show answer

Answer: B. Beyond the elastic limit the deformation is plastic: the wire does not fully recover when unloaded.

3. A spring of spring constant 50 N m−1 is stretched from an extension of 0.10 m to an extension of 0.30 m. How much extra work is done? The spring obeys Hooke's law throughout.

  1. A
    0.25 J
  2. B
    1.0 J
  3. C
    2.0 J
  4. D
    2.25 J
Show answer

Answer: C. Work = ½k(x22 − x12) = ½ × 50 × (0.090 − 0.010) = 2.0 J. Using ½k(x2 − x1)2 gives 1.0 J, which is wrong.

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