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:
$G_i$ 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
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
```
*VISCOELASTIC, TIME=PRONY
g1, k1, tau1
g2, k2, tau2
```
Or:
```
*VISCOELASTIC, FREQUENCY=PRONY
g1, k1, tau1
```
TIME = time domain, FREQUENCY = frequency domain.
Summary
Parameter Identification from Relaxation Curves
DMA (Dynamic Mechanical Analysis) or stress relaxation tests are used to identify Prony parameters. The E'(ω) and E''(ω) obtained from frequency sweeps are converted into a master curve using the time-temperature superposition principle (WLF equation: log aT=−C₁(T−Tref)/(C₂+T−Tref)), and discrete Prony coefficients are fitted via least squares using tools like Pronyseries.py. At least 8–12 τᵢ points are necessary to ensure accuracy.
Prony Series Viscoelastic in Practice
Practical Checklist
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
Prony Input Formats by Solver
Prony series input formats differ by solver. Abaqus inputs normalized relative modulus gᵢ=Eᵢ/E₀ and relaxation time τᵢ pairs, LS-DYNA's MAT_076 inputs E∞ and each Eᵢ as absolute values, MSC Marc uses modulus ratios and time constants, ANSYS uses Prony coefficient tables with shift functions. Conversion errors frequently occur when comparing analyses between different solvers, so always check whether normalization is used.
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
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
Time Increment and Prony Accuracy
For time integration of Prony viscoelasticity, a guideline is time increment Δt < τmin/10 relative to the minimum relaxation time τmin. If Δt is too large, short relaxation components cannot be integrated fully, leading to overestimation of storage modulus E'. In Abaqus, setting the maximum number of increments and DTMAX (maximum Δt) in the *VISCO step is crucial; forgetting these settings frequently causes divergence or inaccurate stress relaxation results.
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