9.1 Stress-strain behavior and mechanical models

Stress-Strain Behavior of Polymers

Polymers don't behave like metals or ceramics under load. Depending on the polymer type, temperature, and how fast you pull on it, the same material can act like a spring, stretch permanently like taffy, or do something in between. This time-dependent mechanical response is what makes polymers both versatile and tricky to predict.

Stress-strain curves capture these behaviors visually, and simple mechanical models (built from springs and dashpots) let you mathematically describe what's happening. Together, they give you the tools to select the right polymer for a given application.

Stress-Strain Behavior

Types of polymer deformation

Elastic deformation is reversible. Stress is proportional to strain following Hooke's law, and the polymer returns to its original shape once you remove the load. Think of a rubber band: stretch it, let go, and it snaps back. The stress-strain curve in this region is a straight line, and the deformation doesn't depend on time.

Plastic deformation is irreversible. Once stress exceeds the yield strength, the polymer undergoes permanent shape change. A plastic grocery bag that gets stretched out and stays stretched is a good example. On the stress-strain curve, this is the region past the yield point where the material keeps deforming without recovering.

Viscoelastic deformation combines both elastic and viscous behavior, and it's time-dependent. The response you get depends on how fast and how long you apply the load. Memory foam is a classic example: push your hand into it and it slowly conforms, then slowly recovers when you release.

Two key viscoelastic phenomena to know:

• Creep: strain gradually increases under constant stress (a loaded polymer shelf sagging over months)
• Stress relaxation: stress gradually decreases under constant strain (a rubber gasket losing its sealing force over time)

Strain rate sensitivity

Polymers behave differently depending on how quickly you deform them. At higher strain rates, polymers tend to be stiffer and more brittle because the polymer chains don't have time to rearrange. At lower strain rates, chains can slide and untangle, so the material appears more ductile and compliant.

This is why the same polymer might shatter under a sudden impact but deform smoothly under slow compression. Strain rate sensitivity is one of the biggest reasons you can't describe polymer mechanics with a single number the way you often can for metals.

Mechanical Models and Properties

Mechanical models for polymer prediction

Since polymers are viscoelastic, you need models that account for both elastic (spring-like) and viscous (fluid-like) responses. Two classic models do this by combining a spring (representing the elastic element, modulus $E$) and a dashpot (representing the viscous element, viscosity $\eta$) in different arrangements.

Maxwell Model (spring + dashpot in series)

This model is best at describing stress relaxation. Because the elements are in series, both experience the same stress, but the total strain is the sum of the strain in each element. Under constant strain, the dashpot slowly flows and the stress decays exponentially:

$E(t) = E_0 \exp(-t/\tau)$

where $E_0$ is the initial modulus and $\tau = \eta / E$ is the relaxation time. A short $\tau$ means stress drops off quickly; a long $\tau$ means the material holds its stress longer.

Kelvin-Voigt Model (spring + dashpot in parallel)

This model is best at describing creep. Because the elements are in parallel, both experience the same strain, but the total stress is shared between them. Under constant stress, the strain gradually increases toward an equilibrium value:

$J(t) = \frac{1}{E}(1 - \exp(-t/\tau))$

where $J(t)$ is the creep compliance, $E$ is the elastic modulus, and $\tau$ is the retardation time. At short times the dashpot resists motion and strain is small; at long times the material approaches a final strain of $1/E$.

Interpretation of stress-strain curves

A stress-strain curve packs a lot of information into one plot. Here are the key properties you should be able to identify and define:

• Young's modulus (elastic modulus): The slope of the initial linear region. It measures stiffness. $E = \sigma / \varepsilon$. A steep slope means a stiff material (like polycarbonate); a shallow slope means a compliant one (like silicone rubber).
• Yield strength: The stress where the curve departs from linearity and plastic deformation begins. Below this point, deformation is recoverable. Above it, permanent shape change occurs.
• Ultimate tensile strength (UTS): The maximum stress the material reaches before failure. This is the highest point on the curve.
• Elongation at break: The strain value where the material fractures. A high elongation at break means the material is ductile (like polyethylene film stretching to several hundred percent strain). A low value means it's brittle (like unmodified polystyrene fracturing at just a few percent strain).

When comparing polymers, plotting their stress-strain curves side by side quickly reveals trade-offs. A stiff, strong polymer often has low elongation at break, while a highly extensible polymer usually has a lower modulus. Choosing the right polymer for an application means deciding which of these properties matters most.