5.1 Stress-Strain Relationships

Elastic Properties

Stress and strain describe how materials respond to applied forces. Stress is the internal force per unit area that particles within a material exert on each other, while strain is the resulting deformation. Together, they form the foundation for analyzing whether a component can handle the loads it'll encounter in service.

Stress and Strain

Stress ($\sigma$) quantifies the internal force distributed over a cross-sectional area:

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

where $F$ is the applied force and $A$ is the cross-sectional area. Units are typically Pascals (Pa) or megapascals (MPa).

Strain ($\epsilon$) measures how much a material deforms relative to its original size:

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

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

These two quantities are linked through the elastic modulus (Young's modulus, $E$):

$E = \frac{\sigma}{\epsilon}$

A higher elastic modulus means the material is stiffer. Steel, for example, has an elastic modulus around 200 GPa, while rubber sits near 0.01 GPa. That's why steel barely deforms under loads that would stretch rubber dramatically.

Hooke's Law and Elastic Limit

Hooke's Law in its spring form states that force is proportional to displacement:

$F = kx$

where $k$ is the spring constant and $x$ is the displacement from equilibrium.

For stress-strain analysis, the equivalent form is:

$\sigma = E\epsilon$

This linear relationship only holds within the elastic region of the stress-strain curve. Within this region, the material returns to its original shape once the load is removed.

The elastic limit (also called the proportional limit) is the maximum stress a material can sustain while still obeying this linear relationship. Beyond it:

• The material enters plastic deformation, meaning it won't fully recover its original shape when unloaded.
• The stress-strain curve deviates from a straight line.
• For design purposes, you generally want to keep working stresses well below this point.

Poisson's Ratio

When you pull a bar in tension, it gets longer in the axial direction but slightly thinner in the transverse (perpendicular) directions. Poisson's ratio ($\nu$) captures this coupling:

$\nu = -\frac{\epsilon_\text{transverse}}{\epsilon_\text{axial}}$

The negative sign makes $\nu$ positive for typical materials, since transverse strain is opposite in sign to axial strain.

• Most engineering materials fall between $\nu = 0.25$ and $\nu = 0.35$. Steel is about 0.30; aluminum is about 0.33.
• A perfectly incompressible material has $\nu = 0.5$. Rubber approaches this value (around 0.49), which is why it deforms in shape but barely changes in volume.
• A Poisson's ratio of 0 means no transverse deformation at all, which is uncommon but can occur in certain foams and auxetic materials.

Poisson's ratio matters in design because real components experience multi-directional strain, even under simple uniaxial loading.

Plastic Deformation and Failure

Yield Strength and Ultimate Strength

Yield strength ($\sigma_y$) is the stress at which a material transitions from elastic to plastic behavior. On a stress-strain curve, it marks the end of the linear region.

• For materials with a clear yield point (like mild steel), this transition is obvious on the curve.
• For materials without a distinct yield point (like aluminum alloys), engineers use the 0.2% offset method: draw a line parallel to the elastic region starting at 0.2% strain, and the intersection with the curve defines $\sigma_y$.
• Yield strength is the primary design criterion for most structural components, because beyond it, permanent deformation begins.

Ultimate tensile strength ($\sigma_u$) is the maximum stress the material reaches before failure. It corresponds to the peak of the engineering stress-strain curve.

• After this peak, the material begins to neck (localize deformation in a narrow region) and the engineering stress drops until fracture.
• While $\sigma_u$ tells you the absolute maximum a material can handle, it's not typically used as a design limit. By the time you reach ultimate strength, the part has already deformed permanently and significantly.

Ductility and Plastic Deformation

Ductility describes how much plastic deformation a material can absorb before it fractures. Two common ways to quantify it:

• Percent elongation: $\frac{L_f - L_0}{L_0} \times 100\%$, where $L_f$ is the length at fracture.
• Percent reduction in area: $\frac{A_0 - A_f}{A_0} \times 100\%$, where $A_f$ is the cross-sectional area at the fracture surface.

Ductile materials (copper, mild steel, aluminum) undergo significant plastic deformation before breaking. This is actually a safety advantage: the visible deformation serves as a warning before catastrophic failure.

Brittle materials (glass, ceramics, cast iron) fracture suddenly with little or no plastic deformation. There's no warning, which makes brittle failure particularly dangerous in structural applications.

At the atomic level, plastic deformation in metals occurs through the movement of dislocations, which are line defects in the crystal lattice. When stress exceeds the yield strength, these dislocations slip along crystallographic planes, causing permanent shape change. This mechanism is why metals can be ductile, while ceramics (which resist dislocation motion) tend to be brittle.