10.2 Viscoelastic models and mechanical analogues

Viscoelastic Behavior in Polymers

Polymers don't behave like simple solids or simple liquids. Because of their long-chain structure and intermolecular interactions, they exhibit both elastic and viscous responses at the same time. This combined behavior is called viscoelasticity, and understanding it is essential for predicting how polymers perform in real applications like car tires, gaskets, and shock absorbers.

To model viscoelasticity, we use mechanical analogues built from springs and dashpots. Two foundational models, Maxwell and Kelvin-Voigt, combine these elements in different ways to capture key phenomena like creep and stress relaxation.

Viscoelasticity in polymers

Viscoelasticity means a material's response to stress has both an elastic component and a viscous component.

• Elastic behavior: the material deforms under stress but returns to its original shape when the stress is removed. Think of stretching a rubber band.
• Viscous behavior: the material deforms under stress but does not return to its original shape. Think of pouring honey.

Polymers show both of these behaviors because their long chains can store energy (elastic) and also slide past one another (viscous), depending on entanglements and intermolecular forces like van der Waals interactions.

Viscoelastic response depends strongly on time and temperature:

• At short time scales or low temperatures, chain motion is restricted and the polymer behaves more like an elastic solid (glassy state).
• At long time scales or high temperatures, chains have time to rearrange and the polymer behaves more like a viscous liquid (rubbery or flow state).

This time-temperature dependence is why the same polymer can feel rigid and brittle in winter but soft and flexible in summer.

Mechanical analogues for viscoelasticity

Before combining elements into full models, you need to understand the two building blocks.

Springs represent purely elastic behavior:

$\sigma = E\varepsilon$

Here $E$ is the elastic modulus (stiffness). A spring deforms instantly under load and recovers completely when the load is removed.

Dashpots represent purely viscous behavior:

$\sigma = \eta \dot{\varepsilon}$

Here $\eta$ is the viscosity and $\dot{\varepsilon}$ is the strain rate. A dashpot resists motion in proportion to how fast you try to deform it, and it never recovers its original shape.

By combining these elements, you can model a range of viscoelastic behaviors:

1. Springs in series: total compliance is the sum of individual compliances (softer overall).
2. Springs in parallel: total modulus is the sum of individual moduli (stiffer overall).
3. Dashpots in series: total strain rate is the sum of individual strain rates (faster deformation).
4. Dashpots in parallel: total viscosity is the sum of individual viscosities (slower deformation).

Maxwell vs. Kelvin-Voigt models

These two models arrange a spring and a dashpot differently, and that arrangement completely changes what behavior each model can capture.

Maxwell Model (spring and dashpot in series)

Because the elements are in series, the stress is the same through both elements:

$\sigma_{spring} = \sigma_{dashpot} = \sigma$

The total strain is the sum of the elastic strain and the viscous strain:

$\varepsilon_{total} = \varepsilon_{elastic} + \varepsilon_{viscous}$

The key consequence: under constant strain, the dashpot allows the stress to gradually transfer and dissipate. This means the Maxwell model predicts stress relaxation well. However, under constant stress, the dashpot keeps flowing forever with no equilibrium, so the model does not predict realistic creep recovery.

Kelvin-Voigt Model (spring and dashpot in parallel)

Because the elements are in parallel, the strain is the same in both elements:

$\varepsilon_{spring} = \varepsilon_{dashpot} = \varepsilon$

The total stress is the sum of the elastic stress and the viscous stress:

$\sigma_{total} = \sigma_{elastic} + \sigma_{viscous}$

The key consequence: under constant stress, the dashpot slows down the deformation while the spring pulls toward an equilibrium strain. This means the Kelvin-Voigt model predicts creep and creep recovery well. However, if you apply a sudden constant strain, the spring instantly bears the full load and the stress never decays, so the model does not predict stress relaxation.

Creep and stress relaxation analysis

These are the two most common experiments for probing viscoelastic properties.

Creep is the gradual increase in deformation when a constant stress is applied.

• In the Maxwell model, strain increases linearly with time and never reaches equilibrium. The material just keeps flowing.
• In the Kelvin-Voigt model, strain rises and asymptotically approaches a constant value. The spring eventually balances the applied stress.
• Creep behavior is characterized by the creep compliance:

$J(t) = \frac{\varepsilon(t)}{\sigma_0}$

where $\sigma_0$ is the constant applied stress and $\varepsilon(t)$ is the time-dependent strain.

Stress relaxation is the gradual decrease in stress when a constant strain is applied.

• In the Maxwell model, stress decays exponentially toward zero. The characteristic time for this decay is the relaxation time $\tau = \eta / E$.
• In the Kelvin-Voigt model, stress remains constant and does not relax at all.
• Stress relaxation behavior is characterized by the relaxation modulus:

$E(t) = \frac{\sigma(t)}{\varepsilon_0}$

where $\varepsilon_0$ is the constant applied strain and $\sigma(t)$ is the time-dependent stress.

Both experiments give you access to fundamental viscoelastic properties: the modulus, the viscosity, and the relaxation time. Creep testing applies a constant stress and tracks strain over time. Stress relaxation testing applies a constant strain and tracks stress over time. Together, they give a fairly complete picture of how a polymer will behave under load.