Prony Series Viscoelastic Model

Category: Structural Analysis | Integrated 2026-04-06
CAE visualization for viscoelastic prony theory - technical simulation diagram
Prony Series Viscoelastic Model

Prony Series Viscoelastic: Theoretical Foundations

Prony Series Viscoelasticity

🙋

Professor, the Prony series is the standard model for viscoelasticity, right?


🎓

The Prony series is a discretization of the generalized Maxwell model. It expresses the relaxation modulus as a sum of exponential functions:


$$ G(t) = G_\infty + \sum_{i=1}^{N} G_i \exp(-t/\tau_i) $$

$G_i$ is the shear modulus of each Maxwell element, and $\tau_i$ is the relaxation time. It describes the time-dependent behavior of rubber, polymers, and adhesives.


Summary

🎓
  • $G(t) = G_\infty + \sum G_i e^{-t/\tau_i}$ — Sum of exponential functions
  • $N = 5 \sim 15$ terms — Covers a wide time/frequency range
  • Fit $G_i, \tau_i$ from DMA test data
  • Abaqus TEST DATA INPUT — Automatic fitting from measured data

    • Coffee Break Yomoyama Talk

    Historical Origin of the Prony Series

    The name "Prony series" originates from the French mathematician Gaspard de Prony (1755–1839), but its application to viscoelastic relaxation functions was established in the 20th-century field of rheology. In the form E(t)=E∞+Σ Eᵢ exp(-t/τᵢ), it is mathematically equivalent to the parallel model of Maxwell elements (Generalized Maxwell) and can approximate any linear viscoelastic behavior.

    Computational Methods for Prony Series Viscoelastic

    FEM Settings for Prony Series

    🎓

    Abaqus:

    *VISCOELASTIC, TIME=PRONY
g1, k1, tau1
g2, k2, tau2

    Or:

    *VISCOELASTIC, FREQUENCY=PRONY
g1, k1, tau1

    TIME defines time-domain material response; FREQUENCY defines frequency-domain response. In Abaqus/Standard, a viscoelastic definition for the other domain is ignored. Frequency-domain input uses long-term elastic properties. Verify the analysis procedure and elasticity definition. *VISCOELASTIC · *STEADY STATE DYNAMICS


    Summary

    🎓
    • Time domain: *VISCOELASTIC, TIME=PRONY + *VISCO
    • Frequency domain: *VISCOELASTIC, FREQUENCY=PRONY + *STEADY STATE DYNAMICS, DIRECT

      • Coffee Break Yomoyama Talk

      Parameter Identification from Relaxation Curves

      Identify Prony parameters from DMA or stress-relaxation measurements. When combining data from multiple temperatures, construct a master curve only where time–temperature superposition is applicable. Choose the number of terms and relaxation times for the time or frequency range of interest, and assess fit residuals and predictive agreement; no fixed minimum term count guarantees accuracy.

      Prony Series Viscoelastic in Practice

      Practical Checklist

      🎓
      • [ ] Does the DMA data cover the frequency/time range of interest?
      • [ ] Is the Prony series fitting good? (Compare with $G'(\omega), \eta(\omega)$)
      • [ ] Is temperature dependence (WLF law) necessary?
      • [ ] Is the choice of time domain/frequency domain appropriate?

        • Coffee Break Yomoyama Talk

        Application to Automotive Vibration-Isolating Rubber

        Prony series viscoelasticity is standard for vibration damping analysis of engine mount rubber (EPDM-based). The loss factor tanδ in the 10–1000Hz frequency range is adjusted by tuning τᵢ to match target values (typically 0.1–0.3). A workflow combining Abaqus frequency response analysis (*STEADY STATE DYNAMICS) with Prony input for virtual evaluation of NVH (Noise, Vibration, Harshness) performance is widely adopted by Japanese automakers.

        Prony Series Viscoelastic: Software & Solver Comparison

        Tools

        🎓
        • Abaqus *VISCOELASTIC — Prony series. Automatic fitting with TEST DATA INPUT
        • Ansys — Supports Prony series
        • All solvers support it — Basic viscoelastic functionality

          • Coffee Break Yomoyama Talk

          Prony Input Formats by Solver

          Input conventions depend on the solver and material model. For isotropic Abaqus Prony input, each row contains the dimensionless shear relaxation coefficient gᵢ, bulk relaxation coefficient kᵢ, and relaxation time τᵢ. Here gᵢ = Gᵢ/G₀; a Young’s modulus ratio Eᵢ/E₀ cannot generally be substituted directly. Check each solver’s model and normalization definitions. Abaqus documentation: Viscoelastic object

          The Three Most Important Questions for Selection

          • "What are you solving?": Does the necessary physical model/element type for the Prony series viscoelastic model have support? For example, the presence of LES support for fluids, or contact/large deformation capability for structures, makes a difference.
          • "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between an automatic transmission car (GUI) and a manual transmission car (script).
          • "How far will you expand?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.

          Advanced Technologies

          Advanced

          🎓
          • Fractional derivative models — Alternative to Prony series. Wide frequency band with fewer parameters
          • Nonlinear viscoelasticity — Coupling of large deformation and viscoelasticity
          • WLF/TTS — Covers the entire temperature range via temperature-time equivalence

            • Coffee Break Yomoyama Talk

            Combined Model with Hyperelasticity

            For large deformation viscoelasticity like rubber, "visco-hyperelasticity" combining Prony+Mooney-Rivlin is used. In Abaqus, writing *HYPERELASTIC and *VISCOELASTIC in the same *MATERIAL block automatically applies cumulative formulation. The same method is applied in brain tissue surgery simulation (Schroeder 2011), reporting deformation prediction errors within 2mm.

            Prony Series Viscoelastic: Common Issues & Debugging

            Troubles

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            • Poor fitting → Increase the number of terms (5→10→15)
            • Inaccurate temperature dependence → Check WLF parameters
            • Accidentally used structural damping in the time domain → Structural damping is for frequency domain only. Use Prony series for the time domain.

              • Coffee Break Yomoyama Talk

              Time Increment and Prony Accuracy

              Choose time increments that resolve the loading and relaxation times of interest, and refine them to check convergence of the stress history. τmin/10 can be a trial starting point, not a universal accuracy or stability guarantee. For automatic incrementation, the fourth data-line entry of Abaqus *VISCO specifies the maximum time increment. Abaqus documentation: *VISCO

              Related Topics

              StructuralFrequency Response of Viscoelastic Damping MaterialsGlossaryProny Series — CAE GlossaryStructuralStress relaxation and creep relaxationStructuralFrequency response of viscoelastic damping materials — Troubleshooting guideStructuralOgden hyperelastic modelStructuralFrequency Response of Viscoelastic Damping Materials
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              Related fields

              Thermal AnalysisManufacturing Process AnalysisV&V · Quality Assurance
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