16.1 Hooke’s Law: Stress and Strain Revisited

Hooke's Law describes how elastic materials respond to forces. When you stretch or compress a spring, it pushes back proportionally to regain its shape. This relationship is central to understanding springs, oscillations, and how materials behave under stress.

Hooke's Law and Elastic Materials

Newton's Third Law in Materials

When you apply a force to a material, you create stress (force per unit area) and deformation (a change in shape or size). The material pushes back with an equal and opposite restoring force, which is Newton's third law in action.

How a material responds after you remove the force determines what category it falls into:

• Elastic materials (like springs) return to their original shape and size once the force is removed.
• Plastic materials (like clay) undergo permanent deformation and stay in their new shape.

Restoring Force and Displacement

For elastic materials, the restoring force is directly proportional to how far the object is displaced from its equilibrium (resting) position. This proportional relationship is Hooke's Law:

$F = -kx$

• $F$: restoring force (N)
• $k$: spring constant, a measure of the material's stiffness (N/m)
• $x$: displacement from the equilibrium position (m)

The negative sign tells you the restoring force always acts in the direction opposite to the displacement. Pull a spring to the right, and it pulls back to the left.

The spring constant $k$ is a property of the specific spring or elastic material:

• A higher $k$ means a stiffer spring. For example, a car suspension spring might have $k = 50{,}000$ N/m, requiring a large force for even a small compression.
• A lower $k$ means a more compliant spring. A soft toy spring might have $k = 5$ N/m, stretching easily under a small force.

This linear relationship only holds within the material's elastic region. Push past that range and the material either deforms permanently or breaks.

Spring Potential Energy

When you compress or stretch a spring, you do work on it, and that energy gets stored as elastic potential energy. To find how much energy is stored, start from the work-energy principle.

1. The force needed to stretch a spring increases linearly from 0 (at equilibrium) to $kx$ (at maximum displacement).
2. The average force over that stretch is $\frac{1}{2}kx$.
3. Work done equals average force times displacement: $W = \frac{1}{2}kx \cdot x$.
4. This gives the elastic potential energy formula:

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

• $U$: elastic potential energy stored in the spring (J)
• $k$: spring constant (N/m)
• $x$: displacement from the equilibrium position (m)

Because energy depends on $x^2$, it increases quadratically with displacement. If you double the displacement, you store four times the energy. Triple it, and you store nine times as much.

Material Properties and Limits

Real materials can only stretch so far before Hooke's Law stops applying. Several terms describe these boundaries:

• Elastic limit: the maximum stress a material can handle and still return to its original shape. Beyond this point, deformation becomes permanent.
• Yield strength: the stress at which a material begins to deform plastically (permanently).
• Tensile strength: the maximum stress a material can withstand before it fractures or fails entirely.

Different types of stress also call for different elastic moduli, each describing how a material responds to a specific kind of deformation:

• Young's modulus: resistance to being stretched or compressed along one axis (tensile/compressive stress).
• Shear modulus: resistance to layers sliding past each other (shear stress).
• Bulk modulus: resistance to uniform compression from all directions, like an object submerged deep underwater.