Elastic and plastic deformation

​​In a nutshell

When forces are applied to a material it will undergo elastic deformation up to its elastic limit. Elastic deformation means the object will return to its original shape after the forces are removed. After the elastic limit, the object will deform plasticly which means the atomic structure is rearranged and the object will become permanently deformed even after the forces are removed. The work done on deforming the object will be stored as elastic potential energy.

Equations

Elastic and plastic deformation

When a material is deformed, it can either undergo elastic or plastic deformation. In most cases, the material will always undergo an elastic deformation before a plastic deformation.

Elastic deformation

When a force is applied to a material, the atoms in the material move away from their equilibrium position. The amount that the atoms can move away from their equilibrium position depends on the size of the forces and the composition of the material. Up to a certain point, the forces can be removed and the atoms will return to their equilibrium, meaning the material will return to its original shape.

This point is called the elastic limit. 

Beyond the elastic limit, the material will undergo plastic deformation.

Plastic deformation

Plastic deformation occurs beyond the elastic limit. If the forces are too large for the atoms, the atoms can shear and permanently move away from their equilibrium position.  When the forces are removed after a plastic deformation the object will not return to its original shape and will be permanently deformed. 

Plastic deformation is the last deformation before the object undergoes failure and breaks.   

Elastic potential energy

When a work is done to deform an object, the energy is transferred to elastic potential energy. 

Recall the equation for work done:

$$W = Fx$$​​​​

The force is the force being applied and the distance is the extension of the object. However, as the applied force is not a constant force and increases from $$0\,N$$​, an average force needs to be calculated. 

​$$average \ force = \dfrac {final \ force - initial\ force}{2}$$​​

​$$average \ force = \dfrac {F- 0}{2}$$​​

​$$average \ force = \dfrac {F}{2}$$​​

Substituting into the work done equation, where work done is the elastic potential energy:

​$$elastic \ potential \ energy=\frac 12 Fx$$

This can also be expressed another way by substituting in Hooke's law, $$F=kx$$:

$$elastic \ potential \ energy = \frac 12 kx^2$$​​

Graphically, consider the graph for Hooke's law.

A Force B Extension 1 Limit of proportionality

The area under the graph is equal to the work done, $$W=Fx$$. Up to the limit of proportionality the shape is a triangle and the area can be calculated as:

​$$elastic \ potential \ energy = \frac 12 Fx$$

Loading and unloading graphs

Materials can display different characteristics when the force is being loaded as opposed to removing the force. Different materials will also display different behaviours. There are three you need to know: A metal wire, polyethene and rubber.

Note: Loading means when force is being applied to an object and unloading means when forces are being removed from an object.

The work done is the area in between the loading and unloading curves. The work done is transferred to changing the molecular structure and/or thermal energy. 

Curiosity: This is why if an elastic band is repeatedly stretched in quick succession, the rubber will become warm. 

A metal wire

A Force B Extension 1 Limit of proportionality 2 Elastic limit 3 Permanent deformation length

Polyethene

A Force B Extension 1 Permanent deformation length

Rubber

A Force B Extension

Note: Rubber returns to its original shape as it returns to the origin of the graph. This particular shape of graph is called a hysteresis loop.​