PHYSICS 9702: Study Notes - Deformation of Solids (6.2)

Welcome to Elastic and Plastic Behaviour!

Hello future engineers and physicists! This chapter connects the forces you learned about with the real-world performance of materials. Ever wonder why a bridge doesn't just snap, or why a paperclip stays bent after you twist it? The concepts of elasticity and plasticity explain exactly how materials react to stress and strain.

Don't worry if this seems tricky at first—we will break down the crucial graphs and definitions step-by-step. Let's make sure you can differentiate between temporary stretching and permanent damage!

Section 1: Defining Deformation Types

When a force (load) is applied to a solid object, it changes shape. This change is called deformation. We categorize this change into two main types: elastic and plastic.

1. Elastic Deformation

Definition: A material undergoes elastic deformation if, when the deforming force is removed, the material returns exactly to its original size and shape.

  • Analogy: Think of a perfect rubber band. When you stretch it and let go, it immediately springs back.

  • In this phase, the work done (energy input) to stretch the material is completely stored as elastic potential energy (\(E_p\)) and is fully recoverable.

  • Key Concept: For many materials (like metal wires) during the initial elastic phase, the extension (\(x\)) is directly proportional to the force (\(F\)). This is where Hooke's Law (\(F=kx\)) applies (as covered in syllabus section 6.1).

2. The Elastic Limit

The key distinction between elastic and plastic behaviour is the elastic limit.

  • Definition: The elastic limit is the maximum stress or force a material can withstand without undergoing permanent deformation.

  • If you apply a force below the elastic limit, the material is elastic (it returns to normal).

  • If you apply a force beyond the elastic limit, the material enters the plastic region.

Memory Aid: Think of a stretched slinky toy. Stretch it a little (elastic) and it goes back. Stretch it too far (beyond the elastic limit), and those coils are permanently misshapen (plastic).

3. Plastic Deformation

Definition: A material undergoes plastic deformation if, when the deforming force is removed, the material does not return to its original shape and size. It retains a permanent extension.

  • Analogy: Bending a paperclip. Once you bend it past a certain point, it stays bent. The internal bonds have slipped past each other permanently.

  • In this phase, energy is expended not just to stretch the material, but also to cause the internal changes (like atoms rearranging or crystal planes sliding). This energy is converted into heat and is not recoverable.

Quick Review:
Elastic: Temporary, recoverable, Hooke’s Law might apply.
Plastic: Permanent, non-recoverable, requires force beyond the elastic limit.

Section 2: Work Done and Energy Stored

When you apply a force to stretch a wire or spring, you are doing work. This work done is transferred into the material as energy.

1. Work Done and the Force-Extension Graph (SLO 6.2.2)

Recall from the Work, Energy, and Power topic that Work Done (\(W\)) is generally calculated as \(W = F \times x\) (where \(x\) is displacement in the direction of the force).

However, when stretching a material, the force \(F\) is not constant (it increases as the extension \(x\) increases).

The standard method for calculating work done when the force is changing is using a graph:

The area under the force-extension (\(F-x\)) graph represents the work done by the force to stretch the material.

2. Elastic Potential Energy (\(E_p\)) (SLO 6.2.3, 6.2.4)

When a material is deformed within its limit of proportionality (meaning Hooke's Law holds true, \(F=kx\)), the work done is stored entirely as Elastic Potential Energy (\(E_p\)). This energy is often called strain energy.

Since the graph is linear up to the limit of proportionality, the shape under the graph is a triangle.


Step-by-Step Derivation of the Energy Formula:

  1. Work done = Area of the triangle = \(\frac{1}{2} \times \text{base} \times \text{height}\)

  2. Base = Extension (\(x\))

  3. Height = Maximum Force (\(F\))

  4. Therefore, Elastic Potential Energy is:

    $$E_p = \frac{1}{2} Fx$$


We can express this solely in terms of the spring constant (\(k\)) and extension (\(x\)) by substituting Hooke's Law (\(F=kx\)):

$$E_p = \frac{1}{2} (kx) x$$

$$E_p = \frac{1}{2} kx^2$$

!!! Important Warning !!!

The simple formula \(E_p = \frac{1}{2} kx^2\) is only valid if the material is stretched within its limit of proportionality (i.e., where the graph is a straight line). If the material is stretched past this limit, you must calculate the energy by finding the area under the curved graph, or you must use the approximation methods required in the question.

Section 3: Visualising the Behaviour (The F-x Graph)

The force-extension graph is the central tool for analyzing material behaviour. Let's look at the key points:

Interpreting the Force-Extension Graph

  • Region O to P (Limit of Proportionality):
    This is the straight-line region. Here, \(F \propto x\). Hooke's Law applies. The material is undergoing elastic deformation. Energy stored is exactly \(E_p = \frac{1}{2} kx^2\).

  • Point P (Limit of Proportionality):
    The point beyond which \(F\) is no longer directly proportional to \(x\). Note that the material is still elastic between P and E, but Hooke's Law fails.

  • Point E (Elastic Limit):
    The absolute maximum extension before the material is permanently deformed. If the force is removed at any point before E, the graph returns to the origin (O).

  • Region E and Beyond (Plastic Deformation):
    Once the force exceeds the elastic limit (E), the material deforms plastically. The structure is permanently altered.

  • Point B (Breaking Stress/Force):
    The point where the material fractures or breaks.

The Hysteresis Loop (Unloading the Material)

When we remove the force, the material typically follows a different path back down the graph.

1. Unloading within the Elastic Region (before E):
If the force is removed (unloaded) before the elastic limit (E), the line follows the exact same path back to the origin O. The stored elastic potential energy is fully returned.

2. Unloading beyond the Elastic Limit (Plastic):
If the material is stretched plastically (e.g., up to point B, then unloaded), the unloading line runs parallel to the original Hooke's Law line (O-P), but it does not return to the origin. Instead, it hits the x-axis at a positive extension, $x_{perm}$.

  • The extension $x_{perm}$ is the permanent extension.

  • The area between the loading curve and the unloading curve (the loop) represents the energy lost (dissipated) as heat due to internal friction during plastic flow.

Did You Know? Materials that exhibit a large region of plastic deformation before breaking are called ductile (e.g., copper). Materials that break shortly after the elastic limit are brittle (e.g., glass).


Key Takeaway for Calculations:

Work Done = Area under the entire \(F-x\) curve.
Elastic Potential Energy (recoverable) = Area up to the point of proportionality, calculated using \(E_p = \frac{1}{2} Fx\).