10.2 Viscoelastic models and mechanical analogues

Polymers are unique materials that exhibit both elastic and viscous properties, known as viscoelasticity. This behavior stems from their long chain structure and intermolecular interactions, making them versatile for various applications like car tires and gaskets.

To understand viscoelasticity, we use mechanical models like Maxwell and Kelvin-Voigt, which combine springs and dashpots. These models help explain phenomena such as creep and stress relaxation, crucial for predicting polymer behavior under different conditions.

Viscoelastic Behavior in Polymers

Viscoelasticity in polymers

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• Viscoelasticity combines elastic and viscous behavior in materials
  • Elastic behavior involves deformation under stress with return to original shape upon stress removal (rubber band)
  • Viscous behavior involves deformation under stress without return to original shape upon stress removal (honey)
• Polymers exhibit viscoelastic behavior due to long chain structure and intermolecular interactions (entanglements, van der Waals forces)
• Viscoelastic behavior depends on time and temperature
  • At short time scales or low temperatures, polymers behave more like elastic solids (glassy state)
  • At long time scales or high temperatures, polymers behave more like viscous liquids (rubbery state)
• Viscoelastic behavior is important for applications such as damping, sealing, and shock absorption (car tires, gaskets)

Maxwell vs Kelvin-Voigt models

• Maxwell model combines elastic and viscous elements in series
  • Elastic element represented by a spring with spring constant $E$
  • Viscous element represented by a dashpot with viscosity $\eta$
  • Total strain is the sum of elastic and viscous strains $\varepsilon_{total} = \varepsilon_{elastic} + \varepsilon_{viscous}$
  • Stress is the same in both elements $\sigma_{spring} = \sigma_{dashpot} = \sigma$
  • Predicts stress relaxation but not creep recovery
• Kelvin-Voigt model combines elastic and viscous elements in parallel
  • Elastic element represented by a spring with spring constant $E$
  • Viscous element represented by a dashpot with viscosity $\eta$
  • Total stress is the sum of elastic and viscous stresses $\sigma_{total} = \sigma_{elastic} + \sigma_{viscous}$
  • Strain is the same in both elements $\varepsilon_{spring} = \varepsilon_{dashpot} = \varepsilon$
  • Predicts creep recovery but not stress relaxation

Mechanical Analogues and Viscoelastic Phenomena

Mechanical analogues for viscoelasticity

• Springs represent elastic behavior
  • Stress is proportional to strain $\sigma = E \varepsilon$
  • $E$ is the spring constant or elastic modulus (stiffness)
• Dashpots represent viscous behavior
  • Stress is proportional to strain rate $\sigma = \eta \dot{\varepsilon}$
  • $\eta$ is the viscosity (resistance to flow)
• Combinations of springs and dashpots model various 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)

Creep and stress relaxation analysis

• Creep is gradual deformation under constant stress
  • In Maxwell model, creep strain increases linearly with time (no equilibrium)
  • In Kelvin-Voigt model, creep strain approaches constant value over time (equilibrium)
  • Creep compliance $J(t) = \varepsilon(t) / \sigma_0$ characterizes creep behavior
• Stress relaxation is gradual stress decrease under constant strain
  • In Maxwell model, stress decays exponentially with time (relaxes to zero)
  • In Kelvin-Voigt model, stress remains constant over time (no relaxation)
  • Relaxation modulus $E(t) = \sigma(t) / \varepsilon_0$ characterizes relaxation behavior
• Creep and stress relaxation experiments probe viscoelastic properties (modulus, viscosity, relaxation time)
  • Creep testing applies constant stress and measures strain vs time
  • Stress relaxation testing applies constant strain and measures stress vs time