Viscoelastic Material ModelsMaterial Testing for CAE & FEA Simulation

Materials respond to applied stress over time through deformation. Metals, for example, undergo plastic deformation or “creep,” where part of the deformation is permanent. Polymers, however, exhibit a more complex behavior called viscoelasticity, where the deformation is time-dependent but recoverable. Unlike metals, polymers don’t recover strain instantaneously upon load removal — instead, recovery is gradual.

The inverse of viscoelastic creep is stress relaxation: when a polymer is held at constant strain, the stress required to maintain that strain decreases over time as the material flows. Both creep and stress relaxation are manifestations of viscoelasticity and form essential elements in understanding polymer performance under long-term loading.

Viscoelasticity is modeled using linear viscoelastic theory, which provides tools for predicting polymer behavior under various time- and temperature-dependent conditions. It includes time-temperature superposition (TTS) and dynamic mechanical analysis (DMA) — methods that allow short-term testing to predict long-term performance. For finite element analysis (FEA) and broader computer-aided engineering (CAE) simulation, viscoelastic characterization data must reflect the relevant time, temperature, frequency, and strain-range conditions of the intended analysis.

Viscoelasticity Assumptions and Fundamentals

A perfectly elastic solid stores all applied energy and returns to its original shape instantly. A viscous fluid flows irreversibly under force. Polymers exhibit both behaviors. The degree to which they behave elastically or viscously depends on material structure and temperature. Thermoplastics can behave like a glassy solid or like a rubber, depending on conditions. Thermosets and rubbers, being crosslinked, don’t melt but shift from rigid to rubbery states with increasing temperature.

Kelvin-Voigt Model

Viscoelasticity is often modeled by combining elastic and viscous elements mathematically. The Kelvin-Voigt model is one such combination:

σ x y = ( σ x y ) E + ( σ x y ) V = G ε x y + η ∂ ε x y ∂ t
where,
σ x y = total stress, decomposed into elastic (E) and viscous (V) components ε x y = total strain G = shear modulus η = viscosity

This model is valid only for small strains. In practice, plastics show linear viscoelastic behavior up to around 1% strain; beyond that, moduli decay and behavior becomes nonlinear.

Creep and Stress Relaxation Relationship

In the linear viscoelastic regime, creep (increasing strain under constant stress) and stress relaxation (decreasing stress under constant strain) are mathematically inverse:

( ε ( t ) ε 0 ) creep = ( σ 0 σ ( t ) ) relax
where,
σ 0 = initial applied stress ε 0 = initial instantaneous strain

This relationship only holds for small deformations within the linear regime.

Dynamic Mechanical Testing

Viscoelasticity is often studied using DMA, where materials are cyclically loaded and unloaded in a sinusoidal pattern. The resulting lag between strain and stress is captured as a phase angle, and two moduli are derived from the complex shear modulus:

G ∗ = G′ + i G″
where,
G ∗ = complex shear modulus G′ = storage modulus (elastic response) G″ = loss modulus (viscous response)

By changing the frequency of testing, the material’s time-dependent behavior can be evaluated: low frequency corresponds to long times; high frequency to short times.

Superposition Principle

Linear viscoelasticity assumes the total response is the sum of responses to individual loads over time (Boltzmann Superposition Principle). This implies that loss and storage moduli must be independent of strain. When this assumption fails (i.e., at higher strains), the behavior is nonlinear viscoelastic.

For rubbers and elastomers, viscoelastic theory is often combined with hyperelasticity, which accommodates large but recoverable strains. Hyper-viscoelastic models allow simulations at large strains by using a family of stress relaxation curves for different strain levels. However, this adds complexity and risk, since behavior may only be locally accurate.

Implementation Strategies

Two primary experimental strategies exist:

  • Time sweeps: A constant strain is applied and held; the stress decay over time is recorded. Common in shear mode.
  • Frequency sweeps: Oscillatory loading is applied at varying frequencies; modulus values are extracted across time scales.

Time-Temperature Superposition (TTS)

Real-world testing for long durations is often impractical. TTS is a technique that extrapolates long-term behavior by conducting tests at various temperatures.

The assumption is that time and temperature are equivalent in affecting polymer behavior. Lower temperatures simulate short-term behavior; higher temperatures simulate long-term behavior. Data at different temperatures are horizontally shifted on a log-time scale to form a single master curve.

The amount of shift is called the shift factor (aT), and the temperature at which the unshifted reference curve was measured is called the reference temperature (TREF). This process produces a sigmoidal modulus vs. time curve in log-log coordinates, showing:

  • Initial modulus plateau
  • Transition (decay) region
  • Long-term modulus plateau

In simulations, the absolute values of the modulus from this curve are typically normalized and used to scale the base modulus from an elastic or hyperelastic model. Because relative values are used, extensometry is not required, and simple specimen geometries are adequate.

One practical limitation of stress relaxation testing is the assumption of an instantaneous step strain, which is not physically realizable. This can introduce error, especially at short timescales.

Testing Methods

Viscoelastic data can be measured using different modes depending on the material:

  • Torsional testing (per ISO 6721-7): Preferred for plastics and materials with moduli from ~10 MPa to ~10 GPa. Commonly uses rectangular bars and offers reliable data.
  • Parallel plate torsion: Used for foams and soft rubbers where torsional bars are impractical.
  • Tensile testing (ISO 6721-5): Suitable for films and sheet rubbers but requires careful system design to avoid compliance issues.
  • Compressive testing: Used for very soft materials (moduli < 10 MPa).

Each mode has trade-offs. For example, tensile testing may induce unidirectional strain that’s hard to reverse, while torsional modes provide fully reversed strain and are thus preferred for linear viscoelasticity.

Linear Viscoelastic Limit

Before valid viscoelastic data can be obtained, the linear viscoelastic region must be identified. This is done via a strain sweep — progressively increasing strain while monitoring the storage and loss moduli. When the moduli begin to change with increasing strain, the limit is exceeded.

Key consequences:

  1. Any strain within the linear region will yield identical viscoelastic curves.
  2. Viscoelastic theory is valid only within this region.

Modeling Viscoelastic Behavior for FEA and CAE Simulation

Finite element analysis (FEA) software commonly uses shear modulus (G) in viscoelastic models, usually derived from torsional tests.

Because the modulus changes dramatically over time/frequency, log-log plots are used to view the data. A well-constructed viscoelastic dataset shows:

  • A high-frequency (short-time) plateau → instantaneous modulus
  • A transition region → modulus decay
  • A low-frequency (long-time) plateau → long-term modulus

In real testing, these ideal plateaus may not fully appear due to experimental limits. When either is missing, engineering judgement is used to estimate a fictive modulus.

Noisy data are common in viscoelastic experiments and are handled by smoothing or discarding outliers. Data are then normalized to the initial modulus and fitted to a Prony series, which is commonly used in FEA for time-dependent material modeling.

Model Extensions and Limitations

Viscoelastic models are usually paired with elastic or hyperelastic models:

  • Elastic + viscoelastic: Modulus from elastic model is scaled by the time-dependent viscoelastic factor.
  • Hyperelastic + viscoelastic: For large-strain, fully recoverable materials. Requires assumption of no plasticity, with time dependence modeled through the viscoelastic scaling.

Limitations:

  • Viscoelastic data assumes linearity with respect to time dependence. Nonlinear behavior (e.g., in large strain or damaged materials) is not well captured.
  • Material must be fully recoverable — i.e., no permanent plastic deformation is allowed for hyper-viscoelastic modeling to be valid.

When used correctly, these models enable accurate prediction of time-dependent mechanical behavior, which is critical in applications like seal deformation, packaging creep, and vibration damping.

Conclusion

Viscoelasticity is a fundamental and complex property of polymers, particularly relevant for long-term or time-dependent applications. Stress relaxation and creep, the two sides of the viscoelastic coin, are captured using linear viscoelastic theory and experimental methods like DMA. Key to this modeling is the recognition of:

  • The linear viscoelastic limit
  • The use of time-temperature superposition for data extrapolation
  • Proper data handling, fitting, and normalization
  • Combining viscoelastic behavior with elastic or hyperelastic base models for FEA and CAE simulation

Correct application of viscoelastic models enables engineers to simulate behavior across time scales that are otherwise inaccessible through direct experimentation — essential for designing polymer components with predictable, reliable performance over time.

To explore the topics discussed on this page further, see Hubert Lobo (Founder, DatapointLabs) and Brian Croop (CEO, DatapointLabs), Determination and Use of Material Properties for Finite Element Analysis (NAFEMS, 2016), Ch.8.

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