Prony Series for Laminated Glass: A Practical Guide

What Are Prony Series?

A Prony series is a mathematical representation of how a viscoelastic material relaxes over time. For a polymer interlayer in laminated glass, the shear relaxation modulus G(t) describes how the material's stiffness decreases under a constant deformation. The Prony series expresses this as a sum of decaying exponentials:

This representation is critical because it is directly compatible with finite element software (ANSYS, Abaqus, Strand7, RFEM) and with the analytical models used in European standards such as EN 16612 and EN 16613.

Why They Matter for Laminated Glass

Laminated glass consists of two or more glass plies bonded by a polymeric interlayer (PVB, ionomer, EVA, or other materials). The interlayer transfers shear between the glass plies, and the degree of shear transfer directly determines the structural performance of the laminate.

The challenge is that polymer interlayers are viscoelastic: their stiffness depends on both time (load duration) and temperature. A PVB interlayer that is stiff under a 3-second wind gust at 20°C may behave almost like a liquid under a permanent load at 60°C. Ignoring this time- and temperature-dependence leads to either unsafe or grossly over-conservative designs.

Prony series allow engineers to capture this behaviour accurately across the full range of design scenarios, from short wind gusts to 50-year permanent loads.

The alternative: simplified assumptions

Without Prony series, engineers typically assume either full shear transfer (monolithic behaviour) or zero shear transfer (layered behaviour). Both are wrong for most real-world conditions:

• Monolithic assumption overestimates stiffness at high temperatures and long load durations, leading to unconservative designs.
• Layered assumption ignores the interlayer contribution entirely, leading to oversized glass and unnecessary cost.

From DMTA Data to Prony Series

The starting point is Dynamic Mechanical Thermal Analysis (DMTA), also called DMA. A small sample of interlayer material is subjected to oscillatory shear deformation at multiple temperatures and frequencies. The test produces:

• Storage modulus G'(ω) — the elastic (in-phase) component
• Loss modulus G''(ω) — the viscous (out-of-phase) component

These are measured across a frequency range (typically 0.1 to 100 Hz) at multiple temperatures (e.g., -20°C to 80°C in 10°C steps).

Building the Master Curve

Raw DMTA data covers a limited frequency window at each temperature. To obtain the relaxation behaviour across the full time range needed for engineering design (seconds to decades), we apply time-temperature superposition (TTS).

TTS exploits the fact that polymers follow a general principle: the same molecular relaxation process that occurs quickly at high temperature occurs slowly at low temperature. By shifting the frequency-domain curves horizontally on a log scale, they collapse onto a single master curve at a chosen reference temperature T_ref.

The shift factors are typically described by the Williams-Landel-Ferry (WLF) equation:

The master curve is then converted from the frequency domain to the time domain to obtain G(t), the relaxation modulus as a function of time. This conversion can be done analytically or numerically.

The Fitting Procedure

Once the master curve G(t) is available, the Prony series coefficients are determined by fitting the series to the data. The procedure involves:

1. Choose the number of Prony terms N. More terms give a better fit but can lead to overfitting. Typically 10 to 20 terms are sufficient for engineering accuracy.
2. Select relaxation times τ_i. These are usually distributed logarithmically across the time range of interest, with one relaxation time per decade of time.
3. Fit the coefficients G_i. A non-negative least squares (NNLS) algorithm minimises the error between the Prony series and the master curve data. The non-negativity constraint ensures physical consistency (negative stiffness contributions are unphysical).
4. Determine G_∞. The equilibrium modulus is the asymptotic value of G(t) at very long times.

Common Pitfalls

1. Insufficient time range

If the Prony series is fitted only to data covering seconds to hours, extrapolating to 50-year permanent loads is unreliable. Ensure your DMTA data, after TTS shifting, covers the full time range relevant to your design scenarios. For EN 16613, this means from at least 1 second to 1.6 billion seconds (50 years).

2. Too many or too few terms

Using 3-5 terms may not capture the transition region accurately. Using 50 terms with noisy data leads to overfitting and oscillatory behaviour. A good rule of thumb: use approximately one Prony term per decade of time in your range of interest.

3. Ignoring temperature dependence

A Prony series fitted at 20°C does not apply at 60°C. You need either separate series at each temperature or a master curve approach with WLF shift factors. The master curve approach is strongly preferred because it provides a single, physically consistent representation across all temperatures.

4. Using Young's modulus instead of shear modulus

DMTA instruments may report E' and E'' (Young's modulus components). For laminated glass analysis, you need the shear modulus G. The conversion uses the Poisson ratio: G = E / 2(1+ν). Typical values are ν = 0.475 for PVB and ν = 0.478 for ionomer.

5. Not validating against experimental data

A Prony series should be validated against independent tests (e.g., creep tests or four-point bending tests on laminated glass specimens). Good DMTA-to-Prony fitting does not guarantee good structural predictions if the TTS assumption breaks down or if the material behaves differently at large strains.

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