What is a Prony series?

A Prony series is a sum of decaying exponentials that represents the stress relaxation modulus G(t) of a viscoelastic material: G(t) = G_infinity + sum of G_i times exp(-t/tau_i). Each term has a stiffness weight G_i and a relaxation time tau_i. It is the standard input format for viscoelastic materials in FEM software.

How many Prony terms do I need for laminated glass interlayers?

For engineering accuracy, 8 to 12 terms give r-squared above 0.997. Published studies on seven interlayer materials used 12 to 13 terms. The FRACTAN calculator supports 2 to 20 terms.

What is the difference between a Maxwell element and a Voigt element?

A Maxwell element (spring and dashpot in series) describes stress relaxation: exponential stress decay under constant strain. A Voigt element (spring and dashpot in parallel) describes creep: gradual deformation under constant stress. Real polymers need multiple Maxwell elements in parallel (the Generalized Maxwell model) to capture behaviour across many decades of time.

Can I convert Prony series coefficients to dynamic modulus?

Yes. Once G_i and tau_i are known, storage and loss moduli at any frequency omega can be computed exactly: G prime equals G_infinity plus sum of G_i times omega-squared times tau_i-squared divided by (1 plus omega-squared times tau_i-squared), and G double prime equals sum of G_i times omega times tau_i divided by (1 plus omega-squared times tau_i-squared).

Mechanical Models and the Prony Series Explained

Name: Fractan
Author: Fractan

14 min read Updated 2026-03-27 Material Modelling

Springs, Dashpots, and Relaxation Time

Viscoelastic behaviour can be represented by combinations of two idealised mechanical elements:

Spring (perfectly elastic): deforms instantly, recovers completely. Stress proportional to strain: σ = G·γ
Dashpot (perfectly viscous): flows under load, no recovery. Stress proportional to strain rate: σ = η·(dγ/dt)

These are not descriptions of molecular structure — they are mathematical analogies that reproduce the observed macroscopic behaviour. Every spring-dashpot pair introduces a characteristic relaxation time:

τ = η / G

The relaxation time τ has units of seconds. After a time t = τ, the stress in a Maxwell element has decayed to 1/e (≈ 37%) of its initial value.

The two building blocks of viscoelastic models: a spring stores energy; a dashpot dissipates it.

The Maxwell Element

A Maxwell element connects a spring and dashpot in series. Under sudden strain, the spring deforms instantly; over time, the dashpot flows and the stress relaxes exponentially.

Maxwell element: spring and dashpot in series. The stress relaxes as a single exponential with time constant τ_i.

A single Maxwell element gives single-exponential relaxation — far too simple for real polymers, which relax over many decades of time. But it is the building block for the Generalized Maxwell model.

The Voigt Element

A Voigt (Kelvin-Voigt) element connects a spring and dashpot in parallel. Both deform by the same amount, and stresses add. The dashpot retards the spring’s deformation.

creep, not relaxation

Voigt element: spring and dashpot in parallel. The strain approaches equilibrium exponentially (retarded elasticity).

The Maxwell element describes stress relaxation; the Voigt element describes creep. Neither alone captures real polymer behaviour — but combining multiple Maxwell elements does.

The Generalized Maxwell Model

To capture relaxation across many decades of time, multiple Maxwell elements are arranged in parallel, each with its own stiffness G_i and relaxation time τ_i. The total stress is the sum of all individual contributions.

The Generalized Maxwell model: N elements in parallel give the Prony series — the standard representation of viscoelastic relaxation.

This is the mathematical foundation of all modern viscoelastic analysis. Any monotonically decreasing relaxation curve can be fitted with arbitrary accuracy by a Prony series with enough terms.

The Prony Series

The equation

G(t) = G_∞ + ∑_i=1^N G_i · exp(−t / τ_i)

Or equivalently in normalised form: E(t) = E₀ · [1 − ∑ e_i(1 − exp(−t/τ_i))]

Each term has two parameters:

G_i (or e_i = G_i/G₀): the stiffness contribution of the i-th relaxation mechanism [MPa]
τ_i: the relaxation time of the i-th mechanism [s] — the timescale over which that contribution decays

The equilibrium modulus G_∞ is the long-term residual stiffness (at t → ∞). For standard PVB this is near zero (~0.5 MPa); for SentryGlas it is significant (~80 MPa).

What each term means physically

A Prony series with well-spaced relaxation times captures different molecular relaxation mechanisms:

Short τ_i (milliseconds): fast chain-segment motions, glassy relaxation
Medium τ_i (seconds to hours): main glass transition, entanglement slippage
Long τ_i (days to years): slow chain rearrangement, permanent flow

From Prony Series to Dynamic Properties

Once the Prony coefficients {G_i, τ_i} are known, the dynamic moduli at any frequency ω can be computed analytically:

G′(ω) = G_∞ + ∑ G_i · ω²τ_i² / (1 + ω²τ_i²)

G″(ω) = ∑ G_i · ωτ_i / (1 + ω²τ_i²)

This interconversion is exact — not an approximation. It means a single Prony series fitted to stress relaxation data can predict frequency-domain behaviour (storage and loss moduli) at any frequency, and vice versa.

Validation: Centelles et al. (2021) confirmed this by fitting Prony series to relaxation data for seven interlayers and comparing the predicted G′(ω) and G″(ω) against independent DMA frequency sweep measurements. The agreement was excellent, confirming that the interconversion works reliably for glass interlayers.

Fitting in Practice

Step 1: Pre-select relaxation times

The relaxation times τ_i are not fitted — they are pre-selected as logarithmically spaced values spanning the time range of the master curve. For example, with 12 terms covering 10⁻⁴ s to 10⁸ s: one term per decade.

Step 2: Solve for the weights

With τ_i fixed, the problem reduces to linear least squares: find the weights G_i that minimise the squared error between the Prony series and the master curve data.

Step 3: Enforce non-negativity

All weights G_i must be ≥ 0. Negative weights would imply the material stiffens under relaxation at certain time scales — thermodynamically impossible and numerically unstable in FEM solvers.

Step 4: Validate

Check r² (target: > 0.99), visually inspect the fit on a log-log plot, and ideally compare predicted dynamic properties against independent DMA data.

Application	Recommended N	Expected r²
Quick estimate, preliminary design	2–5	> 0.95
Standard engineering (EN 16612 check)	8–12	> 0.99
Research / FEM input	12–20	> 0.999

Precision matters: When entering Prony coefficients into FEM software, use all available decimal places. Rounding the coefficients — even slightly — can cause significant errors because the terms are interdependent and the fit spans many orders of magnitude.

Published Prony Coefficients

Centelles et al. (2021) published complete Prony series (12–13 terms) for seven interlayer materials at a reference temperature of 20°C. Here are the key parameters:

Material	E₀ (MPa)	E_∞ (MPa)	E₀/E_∞	N terms	r²
EVALAM (EVA)	2.75	0.344	8×	13	0.9998
EVASAFE (EVA)	5.48	0.637	9×	13	0.9997
PVB BG-R20	978	0.476	2054×	12	0.9978
Saflex DG-41	1156	1.16	997×	12	0.9987
PVB ES	1193	1.79	667×	12	0.9978
SentryGlas	684	80.5	8.5×	12	0.9996
TPU	13.9	3.38	4.1×	13	0.9994

The E₀/E_∞ ratio reveals the material character: PVB BG-R20 drops 2054× from instantaneous to long-term modulus — near-zero residual stiffness. SentryGlas drops only 8.5×, retaining 80 MPa at equilibrium. This is why SentryGlas is specified for structural applications requiring sustained load capacity.

Generate Your Own Prony Series

Select any interlayer from our multi-vendor database, choose the number of terms, and get fitted Prony coefficients with a downloadable PDF report.

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