understand that the area under the force–extension graph represents the work done

Elastic and Plastic Behaviour

1. Elastic Behaviour

When a material is deformed by an applied force, it may return to its original shape once the force is removed. This reversible deformation is called elastic behaviour. In the elastic region the relationship between the applied force \(F\) and the resulting extension \(x\) is linear and is described by Hooke’s law:

• \(F = kx\)
• \(k\) is the spring constant (stiffness) of the material.
• The proportionality holds only up to the elastic limit.

2. Plastic Behaviour

Beyond the elastic limit the material undergoes permanent deformation. This irreversible change is known as plastic behaviour. Key points:

• Yield point: the stress at which plastic deformation begins.
• Ultimate tensile strength: the maximum stress the material can withstand.
• After unloading, the material does not return to its original length.

3. Work Done and the Force–Extension Graph

The work done \(W\) in stretching a material from \(x=0\) to \(x=x_f\) is the integral of the force over the displacement:

\[

W = \int{0}^{xf} F(x)\,\mathrm{d}x

\]

On a force–extension graph, the area under the curve between \(x=0\) and \(x=x_f\) represents this work. For a linear elastic region the area is a triangle; for a plastic region it is a trapezoid or more complex shape.

4. Calculating Work from the Graph

1. Identify the region of interest on the \(F\)–\(x\) plot.
2. Determine the shape of the area (triangle, trapezoid, etc.).
3. Calculate the area using geometric formulas or integration.
4. Use the area as the numerical value of \(W\) (units: joules).

5. Example Problem

Given a steel wire with spring constant \(k = 2000\;\text{N/m}\). The wire is stretched to \(x = 0.02\;\text{m}\). The force–extension graph shows a linear elastic region up to \(x = 0.01\;\text{m}\) and a plastic region from \(0.01\;\text{m}\) to \(0.02\;\text{m}\) where the force remains constant at \(F = 20\;\text{N}\).

1. Elastic work: area of triangle with base \(0.01\;\text{m}\) and height \(F_{\text{elastic}} = kx = 2000 \times 0.01 = 20\;\text{N}\).

   \(W_{\text{elastic}} = \frac{1}{2} \times 20 \times 0.01 = 0.1\;\text{J}\).

2. Plastic work: area of rectangle with width \(0.01\;\text{m}\) and height \(20\;\text{N}\).

   \(W_{\text{plastic}} = 20 \times 0.01 = 0.2\;\text{J}\).

3. Total work: \(W_{\text{total}} = 0.1 + 0.2 = 0.3\;\text{J}\).

6. Summary

• Elastic deformation is reversible and follows Hooke’s law.
• Plastic deformation is permanent and occurs beyond the elastic limit.
• The work done in stretching a material equals the area under the force–extension curve.
• In the elastic region the area is triangular; in the plastic region it is typically rectangular or trapezoidal.
• Integration of \(F(x)\) provides a general method for any force–extension relationship.

7. Practice Questions

1. For a rubber band with \(k = 500\;\text{N/m}\), stretched to \(x = 0.05\;\text{m}\), calculate the work done assuming purely elastic behaviour.
2. In a tensile test, the force–extension graph shows a yield point at \(F = 15\;\text{N}\) and an ultimate tensile strength at \(F = 30\;\text{N}\). Sketch the expected shape of the graph and describe the work done up to the yield point.
3. Explain why the area under the force–extension curve is a measure of energy input, and discuss what happens to this energy when the material returns to its original shape in the elastic regime.
4. Given a material that exhibits a linear elastic region up to \(x = 0.02\;\text{m}\) and a plastic region where the force remains constant at \(25\;\text{N}\) until \(x = 0.05\;\text{m}\), calculate the total work done.
5. Describe how the concept of the area under the curve can be applied to determine the toughness of a material.

8. Key Terms