# Influence of PVB Interlayer Mechanical Properties on Laminated Glass Elements Design in Dependence of Real Time-Temperature Changes

^{*}

## Abstract

**:**

## 1. Introduction

^{®}), ethylene vinyl acetate (EVA), polyethylene (PE) [10], and thermoplastic polyurethane (TPU) [11].

## 2. Materials and Methods

#### 2.1. Theoretical Background

#### 2.1.1. Viscoelastic Response of the Interlayer

_{T}is the shift factor.

_{g}= T

_{0}(glass transition temperature). When the polymer is close to reaching its glass temperature, it is most affected by time and frequency. For temperatures below T

_{g,}the mobility in polymer chains is low and molecules cannot move freely. On the other hand, when the temperature is above Tg, molecules have significantly more freedom of movement, and as the temperature rises, their free volume does as well. As a result of the change in molecular mobility, that occurs throughout the glass transition interval, mechanical and physical properties of the interlayer change. There are several methods for determining the glass transition temperature (standardized specific volume measurement, DTA, DSC, DMA at a fixed cooling or heating rate), but the one described as superior by Li et al. [25] is the new TTS method. This method complies with time-temperature superposition (TTS) with a dynamic mechanical analyzer (DMA).

_{T}used for the time correction at a given temperature (T

_{1}, T

_{3}, T

_{4}). Equivalent relaxation time is expressed by t/a

_{T}and matched with the reference temperature T

_{ref}(T

_{2}).

_{1}and C

_{2}are the WLF constants that are dependent on the temperature in the exact moment T and reference temperature T

_{0}. The exact determination of T

_{g}is not possible, but according to ASTM D3418-97 [27], it can be concluded from the loss modulus E″(ω). For PVB foil, it is approximately +8 °C. Another value for reference temperature is recommended to use in the WLF approach [14]. Softening temperature Ts = 50 °C was taken as the reference temperature and WLF factors C

_{1}

^{s}= 8.86 and C

_{2}

^{s}= 101.6 in this research. Values for C

_{1}and C

_{2}may be used as general values proposed by William, Landel, and Ferry, or can be evaluated by Expressions (7) and (8):

_{a}(J/mol) represents the activation energy of the observed interlayer and ${R}_{a}$ represents the universal gas constant (${R}_{a}=8.3144621J/molK$). However, the application of this new model becomes essential only when the tested temperature is found to be substantially below T

_{g}.

#### 2.1.2. Effective Thickness Methods

_{i}is the thickness of glass ply (Figure 1a), and d

_{i}is the distance of the mid-plane of the glass ply i, respectively, from the mid-plane of the laminated glass (12).

^{®}technical data [5], it can be noticed only a few load durations with the corresponding temperature that very imprecisely approximate the properties of the interlayer. Trosifol

^{®}technical document includes relaxation shear modulus for different temperatures. The UltraClear foil was considered since it was used in the experiment. In the technical data, the value of shear modulus for a load duration of 1 s decreases drastically with temperature raise (for example: for −20 °C, G = 250 MPa, and for 25 °C, G = 2.7 MPa). Applying double interpolation (temperature and time), obtained more precise values of shear relaxation modulus can be obtained. Trosifol

^{®}also offers datasheets with values of coefficient ω [30], considering the version of European standards [31] used to present composite behavior according to Eurocode standards.

_{i}is the thickness of the glass panes and d

_{i}is the distance from the mid-plane of laminated glass (Figure 5).

_{1}and h

_{2}as individual glass plies thicknesses. Figure 5 illustrates a composite of two glass layers and a polymeric interlayer. The beam length l, width b, and thickness for each layer: h

_{1}and h

_{2}for glass and t for interlayer are the physical values needed for laminated glass definition. Mechanical characteristics of laminated glass are expressed with E for Young’s modulus of glass and G for the Shear modulus of the polymer. The non-dimensional coefficient Γ = 1/(1 + ${\rm K}$) ∈ (0,1) takes into consideration the capability of the PVB foil to transfer shear stress between the glass plies. For the determination of Γ (Equation (12)) a strong approximation is made by using the universal value of β = 9.6 for laminated glass. In the principle of virtual work, another coefficient is found ${\rm K}$ (Equation (13)) where the intermediate layer of thickness t is defined through the shear coefficient of the intermediate layer $\chi $ = t

^{2}/H

^{2}and bending stiffness Bs = EA

^{*}H

^{2}, where A* is the applicable cross-section area A* = A

_{1}A

_{2}/(A

_{1}+ A

_{2}) and H = t + (h

_{1}+ h

_{2})/2.

_{tot}= I

_{1}+ I

_{2}+ A

^{*}H

^{2}, and the value for I

_{S}= A

^{*}H

^{2}/b (Equation (14)). Shear coupling parameter Ψ takes into consideration the boundary and load conditions for the most common cases of the design practice [13]. In the case of simply supported beams under uniform load, the results given by both of these analytical methods match perfectly. This is because the Wölfel–Bennison approach is based on using the universal value β = 9.6 that, according to Wölfel’s theory, applies only to the scenario of simply supported beams under uniformly distributed load. The Wölfel–Bennison method yields insufficiently accurate results when changing boundary and load conditions in comparison with the EET method. Coefficient Ψ can be calculated using Equation (16) for any glass element boundary conditions, subjected to any load condition. Coefficient Ψ can be determined from Table 1 in [13] for glass beams (one-dimensional case) and Table 2.1 and 2.2 in [13] for glass plates (two-dimensional case); in addition to multi-layered laminated glass [33], curved laminated glass [34], and cantilevered laminated glass [35]. For the case of the simply supported beam under uniform load, the expression for Ψ = 168/(17xl

^{2}) was chosen.

#### 2.2. Experimental Research

^{2}). During the installation, several things needed to be secured. It was necessary to avoid (as much as possible) sudden movements of the glass to prevent a change of its initial shape before starting the measurement. The measurement needed to start from the moment the load acts on the glass. It was necessary to ensure that no deflection occurs before the start of the measurement (e.g., with additional pads) (Figure 6a). After the instant deflection was detected, measurements were executed within intervals of 1 min. The glass supports were placed along the shorter sides of the glass panel, freely supported by vertical pads and not clamped anyhow (Figure 6b). Expected deflections and shape changes were dependent on the installed foil. The measured parameters were time, maximum deflection, and glass temperature.

^{®}PVB UltraClear interlayer with a thickness of 0.76 mm. The technical data of Trosifol

^{®}was considered for the mechanical properties of the interlayer [5]. The effective panel span between vertical supports was measured at 2800 mm with an overhang of 100 mm on each side. The load on the panel was only its self-weight. The total test measurement took 2 days with variable indoor temperature conditions. The real-time temperature conditions were observed, during which the inside temperature for 2 days was around 20 °C with a peak of 29.4 °C after the first day of measurement. It is important to notice that most research on laminated glass is performed at a controlled temperature when the only variable is time, and the relaxation properties of shear depend only on the load duration. During this test, two relatively serious rises and falls in temperature occurred, which are expected to happen during the exploitation time as well.

#### 2.3. Numerical Modeling

^{®}technical document was used as a reference and according to the values for Trosifol

^{®}Clear foil, the numerical calculation was provided [5]. Temperature curves for the Shear relaxation modulus (Figure 8) and the Young relaxation modulus (Figure 9) dependent on load duration are presented in Figure 8. The study from Hána et al. [37] provides RFEM analysis of laminated glass that is in good agreement with the enhanced effective thickness (EET) approach and Wölfel–Bennison approach in the case of a simply supported beam under the uniform load [13,38]. Another study from Gwóźdź et al. [39] suggests non-linear analysis to be performed in the RFEM in the case of plates with linear supports on all four edges.

_{1}and C

_{2}[14]. Plate elements of 2800 × 500 mm were only vertically supported on both edges, with meshing element dimensions 0.05 × 0.05 m. The loading was defined as a value of uniform pressure load 0.25 kN/m

^{2}. Finite element modeling analysis in Abaqus SIMULIA was performed only for one point (1500 min and 29.4 °C), while in SCIA Engineer and RFEM software 45 static analyses (points) were observed to draw the deformation curvature over 2800 min.

## 3. Results

#### 3.1. Analytical Calculations

^{®}technical data [30] offer tables with ω coupling values for diverse types of interlayers. Values for UltraClear foil were used for the calculation and display of the deformation curve according to EN 16612 and EN 16613 (Figure 11a).

^{®}UltraClear data sheet for the Shear relaxation modulus) at the peak temperature of 29.4 °C and time point of 1500 min resulted in a gap of 6.51 mm from the measured data which makes 15% error in the result (Figure 11b). Results are sufficiently accurate with the measured deformations when the temperature is close to constant (0–1000 min in Figure 9), but the problem occurs when there is a temperature difference, as shown in Figure 10. The WLF model was used for the correction of the relaxation modulus provided by Trosifol

^{®}. Since the temperature rises from 20.8 °C to 29.4 °C in a relatively short time, foil viscoelastic properties have to be included. The analytical approaches from Galuppi et al. [13] rely on the definition of a secant modulus (time and temperature-dependent). Thois is well presented in Figure 10 (1500 min), where the deformation curvature is in slight growth during the measurement, but from the analytical results (Figure 11), the curve is much steeper. The analytical formulation gives unrealistic high deformation values and for that reason, the WLF method was used to solve the problem of interlayer mechanical properties. Reference temperature was chosen to be Ts = 50 °C and WLF constants C

_{1}= 8.86 and C

_{2}= 101.6 as recommended in [14]. Furthermore, C

_{1}

^{i}and C

_{2}

^{i}were calculated using Equations (7) and (8), where T

_{i}was listed from measuring data for every minute. Finally, the TTS shift factor was calculated (Equation (6)) based on which the Relaxation modulus from Trosifol

^{®}was corrected (Equation (3)). With a new Shear modulus of the interlayer, the Wölfel–Bennison approach proceeded for obtaining deformation. For 2800 points deformation was calculated and the curvature was drawn. In this way, the results have less than a 3% deviation from measured data. The logical input to the code was that the deformation curve can’t reverse under constant stress (which would happen in analytical results if the temperature drops).

^{®}UltraClear (Figure 8). Relaxation modulus values given in the Trosifol

^{®}datasheet were linearly interpolated to determine more precise values for the shear relaxation modulus for the exact time and temperature. Trosifol

^{®}datasheet is determined by dynamic mechanical analysis following EN ISO 6721. The sample storage temperature was 23 °C before measurement. The four-point bending test [40] proved to be an adequate nondestructive method for determining relaxation shear modulus. Values from the Trosifol

^{®}table are in good agreement when the temperature curves are close to each other, which can be seen in Figure 11b.

^{®}is required. Additional time-temperature superposition (TTS) is needed when there is a change in temperature during the time, as happened at the end of the first day of measurement in this study. The shift of modulus from different temperature curves was made to fit the reference temperature. Using the TTS principle on wider time domains as presented in this study, shift factor a

_{T}translates horizontally the Young and shear modulus of foil from the measured temperature in exact time to the reference temperature with the new time t/a

_{T}. When presented graphically a

_{T}(Figure 4 and Figure 12) through the time of the experiment, it can be seen the curvature is mirrored to the temperature curvature, thus reducing the impact of temperature changes on the mechanical properties of foil. For verification of the credibility of the shift factor, a simple check can be conducted when T = T

_{0}–> log a

_{T}= 0 → a

_{T}= 1 and t/a

_{T}= t, which proves the time stays unaltered when the temperature curve matches the reference temperature.

_{1}and C

_{2}were calculated for every minute, based on which the shift factor was obtained. For example, in 1500 min and the temperature peak of 29.4 °C, C

_{1}

^{29.4}= 11,11 and C

_{2}

^{29.4}= 81.00. After the time correction is made using reference temperature T

_{0}= T

_{S}= 50 °C, with the corresponding WLF factors C

_{1}

^{s}= 8.86 and C

_{2}

^{s}= 101.6 as recommended in [14], new shear modulus G (t/a

_{T}, T

_{0}) were calculated. When the mastercurve is obtained and the constants C

_{1}and C

_{2}for each time and temperature point are known, all mastercurves in the range of validity of the WLF model could be obtained. Thus, the time and temperature viscoelastic modulus of the material is completely defined [14]. Additionally, deformations of the laminated glass were calculated using the Wölfel–Bennison with a new shear relaxation modulus. In that way, obtained deformation fits deformation measurement results within less than 3% (Figure 11c).

#### 3.2. Numerical Analysis

^{2}. The performed analysis was static.

^{®}datasheet and manually entered into RF GLASS (Figure 15). Design approaches that are incorporated into the software Dlubal are in good agreement with the analytical Wölfel–Bennison approach [2] proving the accuracy of the Wölfel–Bennison approach with the FEM analysis of solid elements (Figure 16b).

_{1}

^{s}= 8.86 and C

_{2}

^{s}= 101.6. Young modulus and Poisson’s ratio for the elastic properties of foil were selected for the exact time and temperature (1500 min and 39.4 °C). Interpolated elastic relaxation modulus was obtained from the Trosifol

^{®}table E = 0.591 MPa and Poisson’s ratio 0.495. The results are shown in Figure 18. Results fit the measurement data with almost 100% accuracy.

## 4. Conclusions

^{®}catalog E(t) and G(t) with the additional linear interpolation. The WLF model was used for the correction of the relaxation modulus provided by Trosifol. The shear modulus in the producer’s technical data is accurate for the constant temperatures. The problem occurs when there is a change in the temperature. The reason for this lies in the inability of the interlayer to follow the glass due to the memory effect. The analytical approaches based on effective thickness only consider the time and temperature in the current moment. The analytical formulation gives unrealistic high deformation values and for that reason, the WLF method was used to solve the problem of interlayer mechanical properties. For the WLF model constants, C

_{1}

^{WLF}= 8.86, C

_{2}

^{WLF}= 101.6 and the reference temperature Ts = 50 °C most accurate results were obtained in comparison to the measured data. The deformation curvature change that occurs within analytical approaches was well maintained using the shift factor a

_{T}.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Laminated glass composed of two plies under bending: (

**a**) monolithic limit; (

**b**) layered limit; (

**c**) intermediate configuration.

**Figure 2.**Stress and strain diagrams of viscoelastic material: (

**a**) when constant stress is applied; (

**b**) constant strain is applied. Adapted from [23], Elsevier, 2021.

**Figure 3.**Temporal offset of stress and strain diagram of a viscoelastic material under dynamic load. Adapted from [23], Elsevier, 2021.

**Figure 4.**Horizontal shift of temperature curves to match reference temperature curve, T

_{1}< T

_{2}< T

_{3}< T

_{4}.

**Figure 8.**Shear Relaxation curves for PVB Clear from Trosifol

^{®}data sheet at different temperatures.

**Figure 9.**Young Relaxation curves for PVB Clear from Trosifol

^{®}data sheet at different temperatures.

**Figure 11.**Deformation from different glass design analytical approaches: (

**a**) norm EN 16612; (

**b**) Wölfel-Bennison approach; (

**c**) Enhanced effective thickness approach; (

**d**) Effective thickness approach combined with TTS correction of Shear Modulus G.

**Figure 13.**Deformation comparison of different glass design analytical approaches with measured data: EN 16612; Wölfel-Bennison approach; Enhanced effective thickness approach; Effective thickness approach combined with TTS correction of Shear Modulus G.

**Figure 17.**Deformation comparison from different software FEM analysis and measured data from the experiment.

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**MDPI and ACS Style**

Galić, J.; Stepinac, L.; Bošnjak, A.; Zovko, I. Influence of PVB Interlayer Mechanical Properties on Laminated Glass Elements Design in Dependence of Real Time-Temperature Changes. *Polymers* **2022**, *14*, 4402.
https://doi.org/10.3390/polym14204402

**AMA Style**

Galić J, Stepinac L, Bošnjak A, Zovko I. Influence of PVB Interlayer Mechanical Properties on Laminated Glass Elements Design in Dependence of Real Time-Temperature Changes. *Polymers*. 2022; 14(20):4402.
https://doi.org/10.3390/polym14204402

**Chicago/Turabian Style**

Galić, Josip, Lucija Stepinac, Antonia Bošnjak, and Ivana Zovko. 2022. "Influence of PVB Interlayer Mechanical Properties on Laminated Glass Elements Design in Dependence of Real Time-Temperature Changes" *Polymers* 14, no. 20: 4402.
https://doi.org/10.3390/polym14204402