5.3 Elasticity: Stress and Strain

Elasticity and Hooke's Law are key concepts in understanding how materials respond to forces. They explain why some objects bounce back after being stretched or compressed, while others deform permanently.

These principles are crucial in engineering and everyday life. From designing bridges to choosing the right rubber band, knowing how materials behave under stress helps us make better decisions about their use and safety.

Elasticity and Hooke's Law

Hooke's Law and Elasticity

Hooke's law describes the linear relationship between force and displacement in elastic materials:

$F = kx$

Here, $F$ is the applied force, $x$ is the displacement from the object's natural (unstretched) position, and $k$ is the spring constant. The spring constant measures stiffness: a stiffer spring has a higher $k$ value and requires more force to stretch the same distance.

Elasticity is a material's ability to return to its original shape after the deforming force is removed. Hooke's law only applies within this elastic range. Stretch a rubber band gently and it snaps back; that's elastic behavior, and the process stores elastic potential energy in the material. Pull too hard, though, and the material enters plastic deformation, where Hooke's law no longer holds.

Stress-Strain Curve Interpretation

Two quantities let us describe deformation in a way that doesn't depend on the size of the object:

• Stress ($\sigma$) is the force per unit area applied to a material: $\sigma = F/A$, measured in pascals (Pa). $A$ is the cross-sectional area perpendicular to the force.
• Strain ($\epsilon$) is the fractional change in a dimension: $\epsilon = \Delta L / L_0$, where $\Delta L$ is the change in length and $L_0$ is the original length. Strain is dimensionless (no units).

A stress-strain curve plots stress on the vertical axis against strain on the horizontal axis. The key features to know:

1. Linear (elastic) region — The straight-line portion at the start. Here, Hooke's law applies, and the slope equals the elastic modulus (a measure of stiffness). Remove the stress, and the material returns to its original shape.
2. Yield point — The stress at which the material begins to deform permanently (plastic deformation). Past this point, the material won't fully recover its shape.
3. Ultimate tensile strength — The maximum stress the material can withstand. Beyond this, the material necks (thins out) and heads toward failure.
4. Fracture point — Where the material actually breaks apart.

The area under the curve up to the elastic limit represents the elastic energy stored per unit volume. This is why understanding the full curve matters: it tells you not just how stiff a material is, but how much punishment it can take before it fails.

Types of Material Deformation

Different types of forces produce different types of deformation. There are three main categories:

Tensile deformation occurs when a material is stretched or pulled apart. Think of a cable supporting a hanging weight. Tensile stress is the pulling force per unit area (perpendicular to the cross-section), and tensile strain is the resulting fractional increase in length.

Shear deformation happens when a force is applied parallel to a material's surface, like pushing the top of a book sideways while the bottom stays fixed. Shear stress is the force per unit area applied parallel to the surface, and shear strain is the resulting angular deformation (measured as an angle in radians).

Volumetric (bulk) deformation results from uniform pressure applied from all directions, like the pressure water exerts on a submarine. Volumetric stress is simply the applied pressure, and volumetric strain is the fractional change in volume.

Stress-Strain Relationship and Material Failure

As stress on a material increases, it passes through distinct stages:

1. Elastic deformation — The material stretches proportionally to the force and returns to its original shape when the force is removed.
2. Plastic deformation — The material deforms permanently. It won't return to its original shape even after the force is removed.
3. Fracture — The material breaks. This is the ultimate failure point.

Different materials behave very differently through these stages. Brittle materials like glass fracture with little to no plastic deformation. Ductile materials like copper can stretch significantly before breaking.

Elastic Moduli and Calculations

Moduli Comparisons in the Real World

Each type of deformation has its own elastic modulus, and each modulus is defined as stress divided by strain for that type of deformation:

Young's modulus ($E$) measures stiffness under tension or compression:

$E = \sigma / \epsilon$

• Steel has a Young's modulus of about $200 \times 10^9$ Pa, making it extremely stiff and ideal for construction beams.
• Rubber's Young's modulus is roughly $0.01 \times 10^9$ Pa, which is why it stretches so easily.

Shear modulus ($G$) measures resistance to shear deformation:

$G = \tau / \gamma$

where $\tau$ is shear stress and $\gamma$ is shear strain. Metals have high shear moduli, which is why they're used in gears and bolts. Rubber has a low shear modulus, making it useful for flexible seals and tires.

Bulk modulus ($K$) measures resistance to uniform compression:

$K = -V(\Delta P / \Delta V)$

The negative sign accounts for the fact that an increase in pressure causes a decrease in volume. Water has a high bulk modulus (about $2.2 \times 10^9$ Pa), making it nearly incompressible and ideal for hydraulic systems. Air has a very low bulk modulus, which is why it compresses easily in pneumatic systems.

Dimensional Changes from Forces

These are the working formulas you'll use in problems. Each one is just a rearrangement of the modulus definition solved for the change in dimension:

1. Change in length under tensile or compressive stress:

$\Delta L = \frac{F \cdot L_0}{A \cdot E}$

where $F$ is the applied force, $L_0$ is the original length, $A$ is the cross-sectional area, and $E$ is Young's modulus.

1. Change in volume under uniform pressure:

$\Delta V = -\frac{V_0 \cdot \Delta P}{K}$

where $V_0$ is the original volume, $\Delta P$ is the change in pressure, and $K$ is the bulk modulus. The negative sign means increased pressure shrinks the volume.

1. Shear strain (angular deformation) under shear stress:

$\gamma = \frac{\tau}{G}$

where $\tau$ is the shear stress and $G$ is the shear modulus. Note that $\gamma$ is an angle (in radians), not a length change.

When solving problems, pay attention to units. Stress and moduli are in pascals (Pa), areas in $m^2$, and lengths in meters. A common mistake is mixing up centimeters and meters or forgetting to convert cross-sectional area properly.