6.2 Elastic and plastic behaviour
- Syllabus
- 9702–2028–2029
- Topic
- 6.2
- Level
- AS
Learning objectives
6.2.1The terms elastic deformation, plastic deformation and elastic limit• understand and use the terms elastic deformation, plastic deformation and elastic limit6.2.2The area under the force–extension graph represents the work done• understand that the area under the force–extension graph represents the work done6.2.3The elastic potential energy of a material deformed within its limit of• determine the elastic potential energy of a material deformed within its limit of proportionality from the area under the force–extension graph6.2.4EP = 1/2 Fx = 1/2 kx2 for a material deformed within its limit of• recall and use EP = 1/2 Fx = 1/2 kx2 for a material deformed within its limit of proportionality
Elastic deformation is reversible, while plastic deformation leaves a permanent change
Elastic deformation disappears when the load is removed; plastic deformation remains. The elastic limit is the greatest load or stress before permanent deformation begins.
Separate the elastic limit from the limit of proportionality: a material may stop being linear before it becomes permanently deformed.
A metal wire unloaded within its elastic region returns to its original length; loaded beyond its elastic limit it remains longer.
Elastic does not mean perfectly linear, and plastic does not mean the object has already broken.
Force–extension area represents work done
During a small extension Δx at approximately constant force F, work done is FΔx. Adding these narrow strips over the loading path gives the total area under the force–extension graph.
Force is in newtons and extension in metres, so graph area has unit N m = J. For straight segments, calculate and add rectangle, triangle or trapezium areas; for a curve, estimate the area under it.
If force rises linearly from 10 N to 30 N while extension increases by 0.20 m, the trapezium area is ½(10 + 30) × 0.20 = 4.0 J of work.
Final force × total extension is the whole rectangle and is generally not the work for a changing force. Follow the actual loading graph and its axis order.
Elastic potential energy is the proportional loading area
When a material is loaded within its limit of proportionality, deformation is reversible and the work done is stored as elastic potential energy. On a force–extension graph, this is the area under the loading line.
The proportional line runs from the origin to final point (x, F), so the area is a triangle: elastic potential energy = ½ × extension × final force.
A spring reaches 40 N at extension 0.20 m on a straight line from the origin. Stored energy = ½ × 0.20 × 40 = 4.0 J.
This triangular-area result requires loading within the limit of proportionality. For a non-linear graph, determine the actual area rather than assuming a triangle.
For a linear spring, elastic potential energy is ½Fx or ½kx²
Within the proportional region, E_P=½Fx=½kx² because the average force during loading is F/2.
Use final force and extension only for a straight-line force–extension graph from zero. For a non-linear graph, find the area under the curve.
A spring with k=200 N m⁻¹ stretched 0.10 m stores 1.0 J.
Do not use Fx for a spring whose force changes during loading; that would overestimate the work by a factor of two in the linear case.