1 Introduction and motivation

It is common nowadays to take into account the mechanical properties of the polymeric interlayer for structural glass calculations according to international standards like EN 16612 (2019) and ASTM E1300 (2016) or in finite element methods. Those methods reflect the coupling effect of the interlayer between the upper and lower glass pane and require modulus data of the polymer material as a function of time and temperature. The great advantage for practical applications can be a reduced glass thickness or an increased span width of the laminate.

The polymer most used as interlayer material is Polyvinyl butyral (PVB). In general, PVB interlayers can be classified in three categories: stiff PVB, conventional PVB, and acoustic PVB. For the latter mono- and multilayer products are commercially available. One of the main differences between these products is the glass transition temperature Tg, resulting predominantly from different plasticizer levels. Table 1 lists the glass transition temperatures of the monolayer products, as determined via dynamical mechanical thermal analysis (DMTA).

Table 1 Glass transition temperature Tg of various PVB Interlayers as determined by DMTA in the linear viscoelastic range with 3 K/min and 1 Hz (Trosifol 2023)
Full size table

The glass transition temperatures cover a range of 25 °C, while conventional PVB is almost centered between stiff PVB and acoustic monolayer PVB. The stiff PVB is designed for structural glass applications and, therefore, requires high modulus values over a wide temperature range. This is obtained by a low plasticizer content and a corresponding high Tg. On the other hand, the high plasticizer content in the acoustic monolayer PVB leads to a reduction of Tg and enables superior damping performance. The Tg of conventional PVB is tailored to ensure good safety properties.

In recent years, interlayer manufacturers have published time and temperature-dependent shear relaxation modulus G(t,T) data in their technical data sheets and brochures, which are used by engineers and designers for glass calculations. Numerous research articles exist, describing best practice on the determination of G(t,T) for monolayer products via DMTA, e.g. (Kuntsche et al. 2015, 2019; Stevels et al. 2016; Haerth et al. 2019; Stevels and D’Haene 2020, 2021). In addition, EN 16613 (2020) includes a methodology for the determination of the interlayer viscoelastic properties. An extraction from a technical data sheet is given in Fig. 1, which shows G(t) as a function of time at a reference temperature of 20 °C for the monolayer products described above as well as the acoustic multilayer product.

Fig. 1
figure 1

Shear Relaxation Modulus as a function of time at 20 °C for various PVB Interlayers (Trosifol 2023)

Full size image

The difference in the glass transition temperature of the various monolayer interlayers is also reflected in the G(t) diagram. As the time–temperature superposition (TTS) principle can be applied to these materials, the change of Tg is basically a shift on the time-axis, while the curve shape remains similar. Depending on the specific load duration, this leads to significant differences in the stiffness of the interlayers, e.g., for 103 s, G(t) is approx. 39 MPa for the stiff PVB and 0.6 MPa for the conventional PVB, while for a load duration of 106 s, this reduces to 0.9 MPa for the stiff PVB and 0.3 MPa for the conventional PVB only.

Beside the monolayer products, Fig. 1 also shows the G(t) data of the multilayer product. Multilayer polymer interlayers for laminated glass are commonly used where a monolayer interlayer cannot provide the desired balance of properties. The most widespread use of a multilayer interlayer is for enhanced acoustic barrier performance. A soft PVB core layer, optimized for acoustic damping, is encapsulated within a standard PVB that is optimized for adhesion to glass, toughness and laminate processing. Obviously, the curve shape of the multilayer is different to the monolayer products. For very short load durations the stiffness of the multilayer PVB is lower compared to the acoustic monolayer PVB, while the reverse behavior can be found between 1 and 100 s. For longer times, both products behave similarly. Note, that the very short load durations become relevant for practical applications at lower temperatures as this is equally to a shift of the curve towards longer times.

The target of this paper is to describe and analyze the differences between monolayer and multilayer products in more detail. In particular, we show how the modulus curve of the multilayer product in Fig. 1 was determined and discuss the temperature dependency of the shift factors. Furthermore, we demonstrate how the behavior of the multilayer can be calculated using the modulus data of the skin and core layers only. A generalized Maxwell approach and a simplified rule-of-mixtures equation are presented. Finally, we compare the modulus data of the multilayer measured via DMTA or calculated via the two approaches with the laminate properties in bending creep tests.

The scientific literature on the dynamic mechanical characterization of PVB multilayers is very limited. In 2019 D'Haene and Stevels (2019) showed a comparison of experimental bending tests with simulated results using Finite Element Methods and found a good agreement. In the present paper the topic is addressed from a more fundamental point of view, using simplified rheological models.

2 Theoretical basics

2.1 Fundamentals on the material behavior of interlayers

Typical load scenarios of laminated safety glass are bending due to wind, snow etc. In those cases, the interlayer provides shear coupling of the individual glass panes of the intact laminated safety glass. The amount of shear coupling is limited by the two limit states, “full shear transfer” and “no shear transfer”, and depends on multiple parameters, as e.g. on the geometry and cross-section of the laminate (height, width, glass thicknesses, interlayer thicknesses), the structural system (e.g. single span beam, plate supported linearly on all sides), the type of load (e.g. concentrated load, uniformly distributed load) and the shear modulus of the interlayer.

Interlayers are polymer-based. Hence, they show a time-dependent material behavior, which means that the shear modulus value (and hence the amount of shear coupling) is dependent on load duration. In the case of intact glass, the distortions of the interlayer are assumed to be small. Hence, the theory of linear viscoelasticity can be used. For interlayers, it has become established to represent the time-dependent material behavior with the generalized Maxwell model (Fig. 2) and to describe the relaxation behavior mathematically with a Prony series. Equation (1) represents the Prony series in shear mode, wherein \({g}_{k}\) (or \({G}_{k}\)), \({\widehat{\uptau }}_{k}\) and \({G}_{0}\) (or \({G}_{\infty }\)) are the so-called Prony-parameters that are to be determined experimentally.

$$ \begin{aligned} & G\left( t \right) = G_{\infty } + \mathop \sum \limits_{k = 1}^{K} G_{k} e^{{ - \frac{t}{{{\hat{\tau }}_{k} }}}} = G_{0} \left( {1 - \mathop \sum \limits_{k = 1}^{K} g_{k} \left( {1 - e^{{ - \frac{t}{{{\hat{\tau }}_{k} }}}} } \right)} \right) \\ & {\text{wherein}}:{\hat{\tau }}_{k} = \eta_{k} /G_{k} \\ \end{aligned} $$
(1)
Fig. 2
figure 2

Generalized Maxwell Model

Full size image

In addition to the time dependence, polymeric interlayers also exhibit a temperature-dependent material behavior. Since the relaxation process is based on thermally activatable molecular movements and rearrangement processes, there is a thermodynamic correlation between the two variables temperature and time. If all relaxation times of a relaxation test exhibit the same temperature dependence, the material behavior is said to be thermorheologically simple (ISO 18437-6 2017; Schwarzl 1990). In this case, a change in temperature results is a horizonal shift of the relaxation curve along the time axis by a temperature dependent shift factor \({a}_{\mathrm{T}}\), see Fig. 3 and Eq. (2). The temperature dependent shift factors can be approximated with time–temperature-superposition (TTS) principles. One of the most frequently used TTS models is the WLF (after Williams, Landel and Ferry (Williams et al. 1955)) approach, Eq. (3).

Fig. 3
figure 3

Time–Temperature Superposition Principle—Shift of the relaxation curve due to temperature change

Full size image

To fully describe the time- and temperature-dependent behavior of thermorheologically simple polymers, the Prony parameters at a given temperature \({T}_{\mathrm{ref}}\) and a time–temperature-superposition (TTS) principle, which can be used to determine the temperature-dependent shift factors, are sufficient.

$${\mathrm{log}}_{10}{a}_{\mathrm{T}}\left(T,{T}_{\mathrm{ref}}\right)={\mathrm{log}}_{10}t-{\mathrm{log}}_{10}{t}_{\mathrm{ref}} ={\mathrm{log}}_{10}\frac{t}{{t}_{\mathrm{ref}}}$$
$$G\left(t,T\right)=G\left({a}_{\mathrm{T}}\cdot {t}_{\mathrm{ref}},{T}_{\mathrm{ref}}\right)$$
(2)
$${\mathrm{log}}_{10}{a}_{\mathrm{T}}=\frac{{-C}_{1}(T-{T}_{\mathrm{ref}})}{{C}_{2}+(T-{T}_{\mathrm{ref}})}$$
(3)

\({C}_{1}\) and \({C}_{2}\) are the so-called WLF parameters, that are material specific and dependent on \({T}_{\mathrm{ref}}\).

DMTA is widely used to experimentally determine the time and temperature dependent mechanical behavior of thermorheologically simple interlayers. DMTA investigations can be performed in different modes: e.g. tension (ISO 6721-4 2019), bending (ISO 6721-5 2019), shear (ISO 6721-6 2019; ISO 6721-10 2015) and torsion (ISO 6721-7 2019). Even though the most common load scenario of laminated glass is bending which results in shear in the interlayer, the current EN 16613 (2020) refers to DMTA in tension mode (which might be changed after revision) and indicates Eq. (4) to calculate the shear modulus.

$$\begin{array}{l}G = \frac{E}{{2\left( {1 + \mu } \right)}} \approx \frac{E}{3}\\{\rm{with}}\;\mu \; = {\rm{ }}0.{\rm{49}}\end{array}$$
(4)

For anisotropic materials, such as multilayer interlayers, Eq. (4) is not valid. Hence, the relaxation behavior must be determined in shear mode. EN 16613 (2020) refers to bending creep tests that can be performed with laminates at different temperatures. However, the bending creep tests performed on laminates have some disadvantages: First, the range of measurable shear modulus values is limited by the limit case “full shear transfer”, which means that shear modulus values of the interlayer can no longer be determined above a certain limit. Second, large climate chambers are necessary to investigate different temperatures and finally, those tests are very time consuming. The more practical alternative is to perform DMTA directly in shear mode.

2.2 Mechanical behavior of multilayers

Multilayers consist of several individual layers. A trilayer consists of two skin layers (skin 1 and skin 2) and a core layer. If the laminated safety glass is subjected to bending, the three layers of the interlayer can be considered as "connected in series" in the model (Fig. 4). If the skin layers of the trilayer are made of the same material, the skin layers can be summarized (Fig. 5). For equilibrium reasons, the stress in all layers is the same, while the total distortion is calculated from the individual distortions (kinematics). Considering the constitutive laws and the initial condition that a sudden distortion is applied and kept constant in the relaxation test, a differential equation system is obtained, which will be discussed in the following section.

Fig. 4
figure 4

Rheological model of trilayer

Full size image
Fig. 5
figure 5

Reduced rheological model: identical skin layers

Full size image

2.2.1 Derivation of the differential equation system

For the sake of clarity, the derivation of the differential equation system and a short study of the material behavior of multilayers in relaxation tests is shown only for the simplified case k = 1 (i.e. one Maxwell element per layer, see Fig. 6).

Fig. 6
figure 6

Simplified rheological model of trilayer to derive differential equations

Full size image

Due to equilibrium:

$$\tau ={\tau }_{\mathrm{S}}={\tau }_{\mathrm{S}\infty }+{\tau }_{\mathrm{S},1}={\tau }_{\mathrm{C}}= {\tau }_{\mathrm{C}\infty }+{\tau }_{\mathrm{C},1}$$
(5)

The kinematics and boundary condition in relaxation tests:

$$\gamma ={\gamma }_{\mathrm{S}}\frac{{d}_{\mathrm{S}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}+{\gamma }_{\mathrm{C}}\frac{{d}_{\mathrm{C}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}=\mathrm{cst}$$
(6)
$$\dot{\gamma }={\dot{\gamma }}_{\mathrm{S}}\frac{{d}_{\mathrm{S}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}+{\dot{\gamma }}_{\mathrm{C}}\frac{{d}_{\mathrm{C}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}=0$$
(7)

And the constitutive models for the individual layers and the multilayer:

$${\gamma }_{\mathrm{S}}= \frac{{\tau }_{\mathrm{S}\infty }}{{G}_{\mathrm{S}\infty }}$$
(8)
$${\dot{\gamma }}_{\mathrm{S}}=\frac{{\dot{\tau }}_{\mathrm{S},1}}{{G}_{\mathrm{S},1}}+\frac{{\tau }_{\mathrm{S},1}}{{\eta }_{\mathrm{S},1}}$$
(9)
$${\gamma }_{\mathrm{C}}= \frac{{\tau }_{\mathrm{C}\infty }}{{G}_{\mathrm{C}\infty }}$$
(10)
$${\dot{\gamma }}_{\mathrm{C}}=\frac{{\dot{\tau }}_{\mathrm{C},1}}{{G}_{\mathrm{C},1}}+\frac{{\tau }_{\mathrm{C},1}}{{\eta }_{\mathrm{C},1}}$$
(11)
$$\gamma =\frac{\tau }{{G}_{\mathrm{multi} }}$$
(12)

The following three differential equations are obtained:

$$\left\{ \begin{gathered} {\text{eq}}{\text{. (5) in (8); derivation ; - eq}}{\text{. (9)} = 0} \hfill \\ {\text{eq}}{\text{. (5) in (10); derivation ; - eq}}{\text{. (11)} = 0} \hfill \\ {\text{eq}}{\text{. (8), (10) and (5) in eq}}{\text{. (7)}} \hfill \\ \end{gathered} \right. $$
$$ \to \left\{\begin{array}{c}\frac{\dot{\tau }-{\dot{\tau }}_{\mathrm{S},1}}{{G}_{\mathrm{S}\infty }}\frac{{d}_{\mathrm{S}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}-\left(\frac{{\dot{\tau }}_{\mathrm{S},1}}{{G}_{\mathrm{S},1}}+\frac{{\tau }_{\mathrm{S},1}}{{\eta }_{\mathrm{S},1}}\right)\frac{{d}_{\mathrm{S}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}=0\\ \\ \\ \frac{\dot{\tau }-{\dot{\tau }}_{\mathrm{C},1}}{{G}_{\mathrm{C}\infty }}\frac{{d}_{\mathrm{C}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}-\left(\frac{{\dot{\tau }}_{\mathrm{C},1}}{{G}_{\mathrm{C},1}}+\frac{{\tau }_{\mathrm{C},1}}{{\eta }_{\mathrm{C},1}}\right)\frac{{d}_{\mathrm{C}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}=0\\ \\ \\ \dot{\gamma }=\frac{\dot{\tau }-{\dot{\tau }}_{\mathrm{S},1}}{{G}_{\mathrm{S}\infty }}\frac{{d}_{\mathrm{S}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}+\frac{\dot{\tau }-{\dot{\tau }}_{\mathrm{C},1}}{{G}_{\mathrm{C}\infty }}\frac{{d}_{\mathrm{C}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}=0\end{array}\right.$$
(13)

The differential equation system in Eq. (13) can be solved with the following initial conditions:

$$\tau (t=0)=\frac{\gamma }{\frac{1}{{G}_{\mathrm{S}\infty }+{G}_{\mathrm{S}.1}}\times \frac{{d}_{\mathrm{S}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}+\frac{1}{{G}_{\mathrm{C}\infty }+{G}_{\mathrm{C}.1}}\times \frac{{d}_{\mathrm{C}}}{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}}$$
(14)
$${\tau }_{\mathrm{S},1}\left(t=0\right)=\frac{\tau (t=0)}{\frac{{G}_{\mathrm{S}\infty }}{{G}_{\mathrm{S}.1}}+1}$$
(15)
$${\tau }_{\mathrm{C},1}\left(t=0\right)=\frac{\tau (t=0)}{\frac{{G}_{\mathrm{C}\infty }}{{G}_{\mathrm{C}.1}}+1}$$
(16)

As a result, \(\tau \) is obtained. Hence, using Eq. (12), the relaxation function of the multilayer is obtained.

2.2.2 Solution of the differential equation system

The system of differential equations shown in Eq. (13) can be solved analytically. However, due to the length of the solution, the representation is omitted. Alternatively, if discrete values of the material parameters for the individual layers (e.g., for skin layer: \({G}_{\mathrm{S}\infty }\), \({G}_{\mathrm{S},1}\), \({\eta }_{\mathrm{S},1}\), \({d}_{\mathrm{S}}\)) and for the applied total strain \(\gamma \) are available, the time step method according to Euler can be used to approximate the solution.

2.2.3 Determination of the multilayer behavior by considering the creep behavior

Setting up and solving the differential equation system becomes very time-consuming as the number of material parameters of the individual layers increases. However, the consideration of the creep behavior is an alternative method to derive multilayer behavior from the individual layer behaviors.

Compared to the relaxation test, the initial conditions change: In the creep test, a stress \(\tau \) is applied at time \(t=0\), which is then held constant (\(\dot{\tau }=0\)). This simplifies the differential equation system to a system of two independent differential equations.

$$\left\{\begin{array}{c}\left({G}_{\mathrm{S}\infty }+{G}_{\mathrm{S},1}\right){\dot{\gamma }}_{\mathrm{S}}+\frac{{G}_{\mathrm{S},1}{G}_{\mathrm{S}\infty }}{{\eta }_{\mathrm{S},1}}{\gamma }_{\mathrm{S}}- \frac{{G}_{\mathrm{S},1}}{{\eta }_{\mathrm{S},1}} \tau =0\\ \\ \left({G}_{\mathrm{C}\infty }+{G}_{\mathrm{C},1}\right){\dot{\gamma }}_{\mathrm{C}}+\frac{{G}_{\mathrm{C},1}{G}_{\mathrm{C}\infty }}{{\eta }_{\mathrm{C},1}}{\gamma }_{\mathrm{C}}- \frac{{G}_{\mathrm{C},1}}{{\eta }_{\mathrm{C},1}} \tau =0\end{array}\right.$$
(17)

With the initial conditions:

$${\gamma }_{\mathrm{S}}(t=0)=\frac{\tau }{{G}_{\mathrm{S}\infty }+{G}_{\mathrm{S},1}}$$
(18)
$${\gamma }_{\mathrm{C}}(t=0)=\frac{\tau }{{G}_{\mathrm{C}\infty }+{G}_{\mathrm{C},1}}$$
(19)

both differential equations can simply be solved independently, and the strain of the multilayer can be calculated using kinematics (see Eq. (5), but \(\ne \mathrm{cst}.\)).

The relationship between the relaxation behavior \(G\left(t\right)\) and the creep behavior \(J\left(t\right)\) can be expressed in Laplace image space (with the space variable \(s\)) as follows (Brinson et al. 2008):

$${s}^{2}\overline{G }\left(s\right)\overline{J }\left(s\right)=1$$
(20)

where \(\overline{G }\left(s\right)\) is the Laplace transform of \(G\left(t\right)\) and \(\overline{J }\left(s\right)\) is the Laplace transform of \(J\left(t\right)\).

Thus, if discrete values of the relaxation material parameters for the individual layers are available, the relaxation functions of the individual layers \({G}_{\mathrm{S}}\left(t\right)\) and \({G}_{\mathrm{C}}\left(t\right)\) can be used to determine the individual creep functions \({J}_{\mathrm{S}}\left(t\right)\) and \({J}_{\mathrm{C}}\left(t\right)\). From the creep functions, the strains \({\gamma }_{\mathrm{S}}\) and \({\gamma }_{\mathrm{C}}\) can be calculated and, using kinematics, the total strain \(\gamma \) is obtained. The creep behavior of the multilayer can be calculated by means of:

$$J(t)=\frac{\gamma }{\tau }$$
(21)

Finally, this creep function is converted with the help of Eq. (20) into the multilayer relaxation function \(G(t)\).

2.2.4 Simplified (rule-of-mixtures) approximation of the multilayer behavior

An approximation method for deriving the multilayer shear modulus from the single layer shear modulus values consists in a purely elastic consideration and in neglecting the mutual influence of the two generalized Maxwell models in the relaxation test. The inverse stiffness at time \(t\) of the multilayer can thus be approximated from the inverse addition of the individual stiffnesses at time \(t\) according to Eq. (22) resp. (23).

$$\frac{{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}}{{G(t)}_{\mathrm{Multi}}}= \frac{{d}_{\mathrm{S}}}{{G(t)}_{\mathrm{S}}}+ \frac{{d}_{\mathrm{C}}}{{G(t)}_{\mathrm{C}}}$$
(22)
$${G(t)}_{\mathrm{Multi}}= \frac{{d}_{\mathrm{S}}+{d}_{\mathrm{C}}}{\frac{{d}_{\mathrm{S}}}{{G(t)}_{{\varvec{S}}}}+ \frac{{d}_{\mathrm{C}}}{{G(t)}_{\mathrm{C}}}}$$
(23)

2.2.5 Specific examples

In this section, numerical examples are considered. Since the combination of two layers with randomly selected material parameters is investigated in the following examples, the terms "skin layer" and "core layer" are no longer used. Instead, it is generally referred to as “Layer 1” and “Layer 2”. The first example aims to compare the different solutions. The randomly selected material parameters from Table 2 apply.

The analytical solution was found with the program Maple. For the time step method according to Euler, a time range of [0:10000] s with a step size of 0.1 s was examined with Excel. The solution via the creep function using Laplace transformation was performed in Matlab.

Figure 7 shows the relaxation behavior of the individual layers (black dotted and dashed lines), as well as that of the multilayer, calculated according to the different methods presented above. It can be seen that the analytical solution, the numerical solution according to Euler and the solution via the creep behavior led to the same result. It is particularly interesting that there is a region where the shear modulus of the multilayer is greater than that of the individual layers. If the result of the simplified solution is compared with the others, good agreement over a large time range can be observed. Deviations only occur at higher times, with the shear modulus values of the simplified method lying below those of the other methods. In addition, the shear modulus values of the simplified variant are always between those of the individual layers.

Fig. 7
figure 7

Comparison of different calculation methods

Full size image

To investigate the influence of the layer thicknesses, the thicknesses of the individual layers are varied in a second example. The other parameters from Table 2 remain unchanged. Figure 8 shows the results. The thicker one layer is compared to the other, the more the behavior of the multilayer approaches that of the thicker layer.

Table 2 Material Model parameters of Layer 1 and 2
Full size table
Fig. 8
figure 8

Influence of different layer thicknesses

Full size image

Last, the influence of different combinations of material parameters for the generalized Maxwell models was investigated. The material parameters investigated are summarized in Table 3.

Table 3 Variation of material model parameters
Full size table

The material parameters of the generalized Maxwell model 2 remained unchanged for the cases A-D. Figure 9 shows the results, including the percentage deviation of the simplified and exact solution. Positive deviations mean that the simplified method overestimates the actual stiffness behavior. Example E shows that the influence of a very soft layer is dominant in the multilayer behavior.

Fig. 9
figure 9

Influence of different material model parameters (Table 3)

Full size image

3 Experimental investigation

3.1 Materials and sample preparation

The interlayer under investigation is a commercially available Trosifol® SC Multilayer with a nominal thickness of 0.76 mm. Beside the commercial product, the skin and core layers were produced as separate monolayers on a lab extruder. The thickness of the monolayers was 0.8 mm and the same resin lots as for the multilayer product were used. The plasticizer content of the individual monolayers was set to the same level as in the multilayer product after storage (equilibration) for 6 weeks.

3.2 DMTA measurements

DMTA measurements were executed with an Anton Paar MCR 302 Rheometer in plate/plate mode under air atmosphere. Test specimens with a diameter of 8 mm were pre-dried for 48 h in a desiccator to reduce moisture content and avoid bubble formation during the measurement. Frequency sweeps were conducted in the linear range of deformation between -20 °C and 105 °C with 5 °C steps.

Master curves were obtained by applying horizontal shift factors to the tan δ results, using IRIS Rheo-Hub software. G(t) data were calculated using the same software and following the Baumgärtel and Winter approach (Baumgärtel and Winter 1989). Origin software was used to describe the temperature dependency of the shift factors by WLF.

3.3 Bending creep tests

To compare the G(t) data from DMTA with the material behavior in a laminate, 4-Point-Bending-Tests with the multilayer product were executed at the University of German Armed Forces Munich (Univ. Munich) and at the test lab Friedmann & Kirchner (F&K). The tests were carried out in accordance with EN 1288-3 (2000) at 20, 30, and 40 °C and for various load durations up to 24 h. The glass thickness used was 6 mm (Univ. Munich) and 4 mm (F&K), respectively, and the interlayer thickness 0.76 mm. The dimension of the test specimens was 1100 × 360 mm. An FEM model was used to extract the modulus from the measured deformation (Botz et al. 2018).

4 Results and discussion

The results are presented as following: Firstly, we show the outcomes of the DMTA measurements with skin, core, and multilayer and discuss the temperature dependency of the shift factors. Secondly, the shear relaxation modulus, measured on the multilayer, is compared with the theoretical predictions described in chapter 3, by using the DMTA results of the skin and core layer. Furthermore, these results are compared with large scale bending creep tests on laminates.

4.1 DMTA results

The shear storage modulus G’ and shear loss modulus G” master curves for the skin, core, and multilayer at a reference temperature of 20 °C is plotted in Fig. 10. Obviously, TTS holds also for the multilayer product, even though it consists of different PVB resins with different plasticizer levels. Over the entire frequency range, the storage modulus is higher for the skin layer compared to the multilayer and for the multilayer compared to the core layer, as expected. In addition, the characteristic relaxation mechanism of the single layers become visible in both modulus curves of the multilayer, particularly indicated by a double-dip in the G” curve.

Fig. 10
figure 10

Master curves of the shear storage modulus G’, and shear loss modulus G” as a function of frequency at 20 °C for skin, core, and multilayer

Full size image

Figure 11 shows the temperature dependency of the horizontal shift factor for the skin, core, and multilayer at a reference temperature of 25 °C. For temperatures above the reference temperature, the shift factors of the multilayer lie between the factors of the core and skin layer, whereas at lower temperatures the datapoints of the core and multilayer overlap. In addition, it is obvious that the curve shape of the skin layer changes from concave in the high temperature range to convex in the low temperature regime, while the core layer follows a convex behavior over almost the whole temperature range investigated.

Fig. 11
figure 11

Shift factors as a function of temperature and fits with the WLF equation

Full size image

A common way to analyze the temperature dependency of the shift factors is to describe the data with the WLF equation (Eq. 3). However, it is evident, that the change in curve shape from convex to concave, as observed for the skin layer, can’t be covered by WLF. Therefore, fitting the shift factors of the skin layer over the entire temperature range, leads to a poor description of the real material behavior, as can be seen by comparing the triangle data points with the black solid line in Fig. 11. This is of particular importance, when recalculating the master curve with the parameters of the WLF equation and not with shift factors only or when evaluating the Prony series at different temperatures. The determined C1 and C2 constants and the correlation coefficient are depicted in Table 4.

Table 4 WLF parameters and correlation coefficients for skin, core and multilayer determined in various temperature ranges
Full size table

This inaccuracy in describing the shift factors over the glass and rubber state by only one set of WLF constants has been found also for other amorphous polymers and is well documented in literature (Sullivan 1990). Therefore, we applied the approach to divide the shift factor diagram into two regions, whereas the temperature at the peak of tan δ, measured at the lowest frequency applied, were chosen as the transition point (representing the glass transition temperature Tg). The grey solid lines, describing the shift factors of the skin layer in Fig. 11, indicate two fits with the WLF equation over a temperature range from −20−25 °C and 25−105 °C, respectively. The corresponding WLF constants and correlation coefficients are given in Table 4. It can be seen that the adaption of the C1 and C2 constants to the data set at −20−25 °C leads to an accurate description of the shift factor dependency, but negative C1 and C2 values. On the other hand, an excellent description could be obtained at high temperatures too, however with positive WLF constants. For the core and multilayer material, the fit was only applied to the higher temperature range, above the respective glass transition temperatures. Interesting to note that the C1 and C2 values of the multilayer lie between the values of the skin and core layer. The reduced correlation coefficient for the multilayer material might be due to the overlap of the two relaxation mechanisms of the skin and core layer.

In conclusion, the fitted WLF parameters, as well known, strongly depend on the temperature range of the considered shift factor data. This is of particular importance, when the data covers temperatures significantly below and above Tg, which is common for PVB interlayers. Care should be taken, when applying the WLF equation to multilayer systems as additional relaxation mechanisms may influence the accuracy of the description of the shift factors. Using for example a polynomial fit, instead of WLF, is purely empirical, but from the engineering point of view, reasonable approach to accurately describe the temperature dependency of the shift factors over a wide temperature range in the glassy and rubber state.

Figure 12 depicts the shear relaxation modulus G(t) as a function of time for the skin, core, and multilayer at 20 °C. G(t) was determined with the G’ and G’’ master curves without using the WLF equation.

Fig. 12
figure 12

Comparison of shear relaxation modulus G(t) as a function of time t at 20 °C, measured and evaluated at Kuraray and at Technical University Darmstadt

Full size image

In order to judge the inter-lab variability for the same materials, Fig. 12 shows the results from TU Darmstadt as well. The main difference regarding the experimental procedure is, that TUD used a profiled plate/plate system (8 mm) and constructed the master curves by horizontally shifting the complex modulus |G*|. Despite those differences a good agreement between the two labs can be concluded, especially for the skin and core layer. For the multilayer product it has to be considered that small variations in the thickness of the individual layers influence the results. Overall, Fig. 12 shows the significant influence of the thin core layer on the behavior of the multilayer product: Even though, the portion of the core layer is below 20% of the multilayer, it significantly reduces the mechanical properties of the final product, particularly in the time range between 10–2 and 102 s. This time range is shifted towards shorter or longer times, depending on the application temperature.

4.2 Comparison of different methods and validation with bending creep tests

In Figs. 13, 14, and 15 the results of the DMTA measurements on the skin, core, and multilayer at 20, 30, and 40 °C, respectively are compared with the theoretical predictions using the simplified approximation (Eq. 23) or the Laplace transformation described in Sect. 3.2.3. In addition, the circular and square data points represent the results of the bending creep tests.

Fig. 13
figure 13

Shear relaxation modulus G(t) as a function of time t at 20 °C. Comparison between DMTA results, bending creep tests, and theoretical predictions

Full size image
Fig. 14
figure 14

Shear relaxation modulus G(t) as a function of time t at 30 °C. Comparison between DMTA results, bending creep tests, and theoretical predictions

Full size image
Fig. 15
figure 15

Shear relaxation modulus G(t) as a function of time t at 40 °C. Comparison between DMTA results, bending creep tests, and theoretical predictions

Full size image

It can be seen that there is a good agreement between the DMTA results, measured directly on the multilayer, and the theoretical predictions at short load durations. Almost no differences can be observed between the simplified approximation (Eq. 23) and the method via Laplace transformation. This is valid for all temperatures investigated. However, at longer times, the strong decrease of the shear relaxation modulus of the core layer significantly influences the predicted modulus data of the multilayer, leading to deviations from the direct DMTA measurements on the multilayer. This phenomenon was already shown by the small parameter study in Sect. 3.2.5 (Fig. 9). Comparing the results of the bending creep tests with the direct DMTA measurements and the theoretical predictions, a better agreement of the two experimental methods is obvious, especially for long load durations. Following this result, the modulus data published in Trosifol (2023) are the ones measured directly on the multilayer.

5 Conclusion and outlook

In the present article the shear relaxation behavior of a PVB acoustic multilayer was investigated from a theoretical and experimental point of view.

Different theoretical methods were introduced, which allow to predict the mechanical behavior of a multilayer in the relaxation test, starting from the material model parameters and the thicknesses of the individual layers. Since multilayers exhibit anisotropic material behavior, it is important to pay attention to the load case to be investigated when combining the rheological models of the individual layers. For multilayers in laminated glass under flexural loading, shear parameters are to be used and the individual rheological models are to be connected in series.

After the rheological models of the individual layers were combined, the relaxation function of the multilayer could be determined by solving a differential equation system. The difficulty arises that although in the relaxation test the total distortion remains the same, the distortions of the individual layers change with time: i.e., the individual layers influence each other.

An alternative with equivalent solution consists of using the mathematical relation between creep and relaxation. In contrast to the relaxation test, the individual models do not influence each other in creep mode and the total creep displacement can be determined via summation of the individual creep displacement.

Finally, a simplified method was introduced to predict the mechanical behavior of the multilayer. This method takes discrete shear modulus values of the individual layers into account, which corresponds to a series connection of elastic springs whereby the shear modulus values vary for each point in time. Hence, this method is particularly remarkable for its speed and ease of use.

The calculation methods presented for the theoretical prediction are generally valid and therefore also transferable to other interlayer combinations.

The theoretical predictions were compared with experimental data determined on the multilayer product via DMTA in shear mode and via bending creep tests. It was shown that master curves can be obtained by applying time temperature superposition to the DMTA data of the multilayer. The temperature dependent shift factors can be described by WLF, whereas the WLF parameters of the multilayer lie between the parameters of the skin and core layer. A good agreement between the DMTA results and the bending creep tests on the laminate was found for all load durations and temperatures investigated. The theoretical predictions indicate lower modulus values compared the experimental results at long load durations, but a good agreement for shorter times. This deviation at elevated load durations needs to be clarified in further investigations.

The described experimental procedure to determine the viscoelastic properties of multilayer systems might also be useful to model the acoustic behavior according to prEN 17940 (2023).