11.3 Stress, Strain, and Elastic Moduli

Stress and Strain Fundamentals

Stress and strain describe how materials respond when forces act on them. These concepts explain why a steel beam can support a building, why a rubber band stretches and snaps back, and why a ceramic mug shatters when dropped. Being able to calculate stress and strain lets you predict whether a material will hold up under a given load or deform and fail.

Stress and strain relationship

Stress measures how much force is spread over a given area. If you push on a small area versus a large area with the same force, the stress is very different.

$\sigma = \frac{F}{A}$

where $\sigma$ is stress in pascals (Pa or N/m²), $F$ is the applied force in newtons, and $A$ is the cross-sectional area in m².

Strain measures how much a material deforms relative to its original size. It's a dimensionless ratio (no units):

$\varepsilon = \frac{\Delta L}{L_0}$

where $\Delta L$ is the change in length and $L_0$ is the original length.

Hooke's Law connects the two for materials in their elastic region (meaning they'll return to their original shape when the force is removed):

$\sigma = E\varepsilon$

This only holds up to the elastic limit. Beyond that point, the material deforms permanently.

Types of stress

• Compressive stress pushes inward, causing the material to shorten. Think of the weight of a building pressing down on its foundation.
• Tensile stress pulls outward, stretching the material. An elevator cable supporting a load experiences tensile stress along its length.
• Shear stress acts parallel to a surface, causing layers of the material to slide relative to each other. A bolt holding two metal plates together resists shear stress when the plates are pushed in opposite directions.

Elastic moduli calculations

Each type of deformation has its own elastic modulus, which tells you how stiff a material is against that particular type of stress. All three are measured in pascals (Pa).

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

$E = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L / L_0}$

A high Young's modulus means the material resists stretching or compressing. Steel has $E \approx 200 \times 10^9$ Pa, while rubber is around $E \approx 0.01 \times 10^9$ Pa.

Shear Modulus (G) quantifies resistance to shear deformation:

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

where $\tau$ is shear stress (force parallel to the surface divided by area) and $\gamma$ is shear strain (the angular deformation in radians).

Bulk Modulus (K) describes resistance to uniform compression from all sides, like an object submerged deep underwater:

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

The negative sign is there because an increase in pressure ($\Delta P > 0$) causes a decrease in volume ($\Delta V < 0$), so $K$ comes out positive.

Applying elastic deformation problems

When solving problems with stress, strain, and elastic moduli:

1. Identify what you know. Pick out the given values: force, area, original length, change in length, type of stress.
2. Determine the type of stress. Is the material being stretched, compressed, or sheared? This tells you which modulus to use.
3. Select the right formula. Use $E$, $G$, or $K$ depending on the deformation type.
4. Watch your units. Forces in newtons, areas in m², lengths in meters. A common mistake is leaving area in cm² instead of converting to m².
5. Check that you're in the elastic region. Hooke's law only applies before the yield point. If the problem says the material has permanently deformed, you've gone beyond the elastic limit.

Material behavior under loading

A stress-strain curve maps out how a material responds as you gradually increase the load. The key features, in order:

• Elastic region: Stress and strain are proportional (Hooke's law applies). Remove the load and the material returns to its original shape.
• Yield point: The boundary where permanent (plastic) deformation begins.
• Plastic region: The material deforms permanently. It won't return to its original shape even after the load is removed.
• Ultimate tensile strength: The maximum stress the material can withstand.
• Fracture point: The material breaks.

Ductile materials like copper and steel have a large plastic region. They stretch and neck down before breaking, which gives warning before failure. Brittle materials like glass and ceramics have almost no plastic region. They fracture suddenly with little visible deformation.

Two additional failure modes to know:

• Fatigue occurs when a material is subjected to repeated cyclic loading (like airplane wings flexing during flight). The material can fail at stress levels well below its ultimate strength after enough cycles.
• Creep is slow, time-dependent deformation under constant stress. It becomes more significant at high temperatures, which is why it matters for jet turbine blades operating in extreme heat.