Glass Struct. Eng. (2018) 3:355–371 https://doi.org/10.1007/s40940-018-0067-8

SI: CHALLENGING GLASS PAPER

Sensitivity analysis for the determination of the interlayer shear modulus in laminated glass using a torsional test Martin Botz · Michael. A. Kraus Geralt Siebert

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Received: 29 January 2018 / Accepted: 29 March 2018 / Published online: 18 April 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract Polymeric interlayers used in laminated glass show viscoelastic material behaviour. Therefore, the precise design of laminated glass structures is dependent on temperature and the load duration. For the determination of the above-mentioned material behaviour of the interlayers different small and big scale test setups exist. One of these tests is the torsional test in which the shear modulus of the interlayer can be calculated from measured data during a relaxation test. In this test, a laminated glass plate is conditioned at a certain temperature of interest and then isothermally twisted to a specific angle, thereby the resulting torsional moment at the support is measured over a time span. With this data, it is possible to calculate the corresponding shear modulus of the interlayer. There are a lot of parameters and boundary conditions with potential influence on the test and the results (e.g. accuracy of the thickness of the glass plates and interlayer, accuracy of the twist-angle, clamping of the laminated glass). Based on already conducted torsional tests at the ‘University of German Armed Forces Munich’ a Finite Element Model was implemented. In a sensitivity analysis (Finite-Element-Analysis using ANSYS V17.2) the influences and the interdependencies of the parameters and boundary conditions of the test setup were determined. The results of the analyses can be used to M. Botz (B) · M. A. Kraus · G. Siebert Institute and Laboratory for Structural Engineering, University of German Armed Forces Munich, Neubiberg, Germany e-mail: [email protected]

get an understanding of the significance of the measured and calculated values for the shear modulus of the interlayer using a torsional test. Furthermore, the results can help to optimize the torsional test. Moreover, the torsional test results were compared to small scale test results from a Dynamical Mechanical Thermal Analysis, which exposes small scale test specimen to a steady state oscillation at different frequencies and temperatures. Keywords Laminated glass · Torsional test · Shear modulus · Sensitivity analysis · DMTA

1 Introduction The design of laminated (safety) glass (LG/LSG) is in many cases conservative because the shear transfer between glass and interlayer according to current national design standards (e.d. DIN 18008) cannot be taken into account for the calculation of the stress states in the glass plates. However, in recent years more and more producers of interlayers started to determine the specific material characteristics of their interlayer products to get a technical approval with shear transfer. With these technical approvals, it is possible to design glass structures including the shear transfer between glass and interlayer respecting their time- and temperature dependent product characteristics. For the case of intact laminates, the time dependent material behaviour can be described in the context of linear viscoelasticity

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with the Generalized Maxwell Model. The constitutive model for a Generalized Maxwell Model can be formulated by a so-called Prony-Series. In order to transform a Prony-series for a specific temperature to different temperatures of interest the time-temperature correlation can be taken into account by means of the ‘Time-Temperature-Superposition-Principle’ (TTSP). Polymeric interlayers can be tested in different kind of ways. With the dynamical mechanical thermal analysis (DMTA) it is possible to test only the interlayer using small samples. With this test, the Prony-Series of a polymer can be determined efficiently. In reality, the interlayer is bonded to the glass. Therefore, big scale tests with laminated glass specimen are necessary for the validation of the DMTA tests. One of these big scale tests is the torsional test. This is a relaxation test in which a laminated glass is twisted under isothermal conditions by a defined angle and is hold in this position. By means of the restoring force of the laminated glass, the torsional moment can be measured over time. With a formula developed by (Scarpino et al. 2004) respectively (Kasper 2005) the shear modulus of the interlayer can be calculated from the torsional moment. If the test is carried out for different temperatures, the shear modulus can additionally be determined for the entire construction-relevant temperature range in order to obtain parameters for the Prony-Series. At the University of German Armed Forces Munich, a test setup was developed, in which up to seven specimen can be tested simultaneously in a temperature range from − 30 to + 60 ◦ C. Figure 1 shows the experimental test setup schematically. As will be shown below, small changes in the measured torsional moment can have major influence on the calculated shear modulus. Therefore, the main influencing parameters were numerically investigated and the influence and the interaction of the parameters were determined by means of a sensitivity analysis. The goals are to optimize the experimental setup and to obtain information about the accuracy of the determined shear modulus. In the following, the basics of the time- and temperature-dependent material behavior of polymeric interlayers are described, the torsional test and the implemented numerical model are explained and the results of the sensitivity analysis are presented. Finally, for a stiff type of interlayer the result of a DMTA test is presented and compared to the results of the torsional tests.

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Fig. 1 Torsional test

2 Thermomechanics of polymeric interlayers used in laminated glass Properties of polymeric interlayer show strong time and temperature dependence, thus the theoretical foundations of the thermomechanical description of the phenomena under consideration is laid in this section.

2.1 Time dependent material behaviour For the small strain situation, as it is the case for intact LG/LSG (Schneider et al. 2016), the material behaviour of polymeric interlayers can be described with the theory of linear viscoelasticity (LVE). In the context of this paper, LVE is modeled via the Generalized MaxwellModel (Fig. 2) and an associated Prony-series, (Grellmann and Seidler 2014; Schneider et al. 2016; Schwarzl 1990). In LVE stresses are linearly dependent on strains and both dependent of time, thus the relaxation function G (t) = τ (t)/γ0 is independent of the stress/strain amplitude. The Prony-series consists of K decaying stiffness functions with associated relaxation times τ˜k and an equilibrium stiffness G ∞ . An alternative formulation for the Prony-series uses the standardized modulus g(t) = G k /G 0 , where G 0 corresponds to the shear modulus at time t = 0. The frequency domain representation of the Pronyseries (c.f. Eq. 1) is gathered via a Fourier-transformation real part  of Eq. (1) to result in Eq. (2). The  (ω) and the modulus G e G∗ is called Storage  imaginary part Im G∗ is called Loss modulus G  (ω), (Schwarzl 1990). The ratio of Loss modulus to Storage modulus is defined as the Loss factor tan δ = G  /G  .

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Fig. 2 Generalized Maxwell-Model (left) and Prony-series in the time domain (right)

Fig. 3 Temperature dependent behaviour of selected polymeric interlayers according to (Schneider et al. 2016)

G∗ (ω) = G ∞ +  = G0

K  k=1

K 

Gk

ω2 τk2 1 + ω2 τk2

K 

+i

ω2 τk2

K  k=1

Gk

ωτk 1 + ω2 τk2

K 

ωτk gk + gk +i gk 1− 2τ 2 1 + ω 1 + ω2 τk2 k k=1 k=1 k=1



(2) The Prony parameters gk (or G k ), τ˜k and G 0 (or G ∞ ) have to be determined experimentally. This can either be done via isothermal relaxation or creep experiments repeated at different temperatures or by using the dynamic-mechanical-thermal analysis (DMTA, see Sect. 6). After having obtained the experimental raw data, the unknown Prony-parameters have to be fitted to the test data by means of a suitable identification method (Kuntsche 2015; Kraus and Niederwald 2017) or (Rühl et al. 2017).

2.2 Temperature dependent material behaviour Classification of polymers can be different depending on the aspect of interest. From a technical/engineering

point of view, the classification according to polymeric structure and thus mechanical behaviour is of interest. In dependence of the stiffness temperature behavior, polymers are classified into three main categories: thermoplastics, elastomers and duroplastics, c.f. (DIN 772493 1993) (Schneider et al. 2016). Beyond these three main classes different subclasses such as thermoplastic elastomers exist. Thereby a thermoplastic elastomers contains elastic polymer chains which are embedded in thermoplastic material, see (Schwarzl 1990) and Fig. 3. Each stiffness-temperature graph of the different polymer types in Fig. 3 shows two characteristic regions, which are separated by a significant stiffness drop within a certain temperature interval, which is called the glass transition area, c.f. Fig. 3. The glass transition area is characterized by the glass transition temperature Tg , c.f. (Schwarzl 1990). The temperature range below the glass transition temperature is called energy-elastic range as the polymeric interlayers are in a glass-like state with relatively high stiffness. In contrast to that, at temperatures above the glass tran-

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Fig. 4 Time-temperature-superposition principle (TTSP) for thermorheologically simple materials

sition temperature, the interlayers behave rubberlike and show a relatively low stiffness, which is called entropic elastic area. The stiffness-temperature curve of elastomers is worth a closer inspection. Elastomers are widely meshed and have a slight increase in stiffness with higher temperatures in the entropy-elastic range, c.f. Fig. 3 left. In contrast to that, thermoplastics do not have any molecular crosslinking, thus the stiffness continuously decreases until reaching the melting point, c.f. Fig. 3 right. Thermoplastic elastomers may possess physical cross-linking points (called crystallites), where unordered, amorphous regions are embedded in oriented crystallites. The crystallites dissolve without decomposing the macromolecules with increasing temperature, (Schneider et al. 2016; Schwarzl 1990). 2.3 Thermorheological behavior As already stated in the previous two sections, the viscoelastic / mechanical properties of polymeric materials show strong time and temperature dependent behavior. The time temperature superposition principle (TTSP, Ferry 1980; Schwarzl 1990) is a concept in polymer physics used to determine temperature dependent mechanical properties of (linear) viscoelastic materials at a reference temperature. According to Schwarzl (1990), the TTSP is founded in a thermodynamical correlation between temperature and time as molecular relaxation processes are due to thermally activatable molecular movements and rearrangement processes. Generally speaking, polymers may possess several (distinct) molecular transitions with associated characteristic viscosity-temperature dependencies. It

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was experimentally proven, that the elastic moduli of amorphous polymers increase with loading rate but do decrease with increasing temperature, c.f. (Schwarzl 1990), although the curves of the modulus as a function of time do not change shape as temperature is changed but appear to shift left or right. The TTSP thus implies that a curve at a given temperature can be used as a reference for predicting further stiffness curves at various temperatures by applying a horizontal shift to the reference curve, (Schwarzl 1990; Schneider et al. 2016), where the reference curve is often called Master Curve, c.f. Fig. 4. On the other hand, the construction of Master Curves is due measuring isotherms of modulus at multiple temperatures and subsequent shifting by applying a multiplier (called shift factor) to the time or frequency at which measurement are taken. By this means the individual stiffness isotherms combine to form a single smooth curve (the Master Curve) of stiffness versus frequency or time, c.f. Fig. 5. Analyzing the shift factors used at each particular temperature delivers the shift factor—temperature relationship, which may possess a functional form. In literature different functional forms for the dependency of the shift factors aT on temperature T exist, such as the ‘Arrhenius equation’ (Schwarzl 1990), the ‘William–Landel–Ferry’-equation (WLF) (Williams et al. 1955), linearized forms of both approaches, polynomials (Z-70.3-236 2016), the Arc-tangent function (Woicke et al. 2004), an exponential function (Göhler 2010) or the Narayanaswamy-Tool function (Narayanaswamy 1971). A polymer is said to be thermorheologically simple (Schwarzl 1990; Fesko and

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Fig. 5 Master Curving process (left) and shift factor—temperature relationship (right) in the frequency domain

Fig. 6 Production of poly-vinyl-butyral, according to (Kuraray 2012)

Tschoegl 1971) if all viscosities of the associated Prony-series depend identically on temperature. In this case, the entire stiffness curve can be shifted along the horizontal time/frequency axis by a single specific shift factor aT . The complete thermomechanical description of a thermorheologically simple polymer via a Prony series consists of providing the Prony series at the reference temperature Tr e f and a corresponding timetemperature superposition principle (TTSP).

an autoclave with a pressure of 12–14 bar and a temperature of 140 ◦ C. There is no influence on the chemical structure of the PVB due to the lamination process. The adhesion between glass and the PVB interlayer is mainly generated by the formation of hydrogen bonds between the hydroxyl groups of the materials. 4 Torsional test 4.1 Test set-up

3 Chemical structure and thermomechanical characteristics of poly-vinyl-butyral (PVB) In the context of this study, a structural (stiff) PVB is used, hence in this section a bief introduction on the chemistry and thermomechanical behavior of PVB is given as basis for the interpretation of the results presented in Sect. 6. The main components of PVB-based interlayers are PVB resin (70–75% by weight), plasticizer (25–30% by weight) and additive (less than 1%) (Kuntsche 2015). Additives serve to protect against oxidation and aging of the interlayer and are relevant for the stiffness of the interlayer. Furthermore, they can improve the adhesion and UV transmission. The manufacturing process is shown schematically in Fig. 6. PVB is assigned to armophous thermoplastics and is chemically present as a homopolymer. The lamination process of PVB interlayers is in a first step the precompound using the roller or vacuum process. In a second step, the final permanent bond is accomplished in

The torsion apparatus consists of two steel traverses in which a laminated glass sample can be clamped vertically. Thereby, influence due to the dead load of the sample can be reduced to a minimum. One traverse is rigidly clamped to the supporting structure and the other traverse is rotatable about the x-axis (see Fig. 1). The samples are clamped by concentric jaws, wherein the jaws can be tightened by a counter-rotating thread. Aluminium blocks are fixed to the jaws and are the contact element to the glass surface. The standard test size, according to DIN EN 1288, of 1100 mm × 360 mm (L × w) can be installed and tested in the apparatus. In general, every type of symmetrically glass and interlayer thickness can be tested. The thickness of the glass layers is measured in the factory at 6 points and is indicated on the sample. The interlayer thickness is measured by measuring the total thickness at 4 points and subtract the corresponding glass layer thickness. So far, specimen with a nominal glass layer thickness

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Fig. 7 Pictures of the torsion apparatus within the climate camber, showing the whole clamped and drilled sample (left) and the detail view on the rotated traverse (right)

of 2 × 6 mm and nominal interlayer thickness of 0.76 and 1.52 mm were tested. The test setup is shown in Fig. 7. The load is applied in form of a rotational deformation with an angle of 2◦ (target value). The rotation of the sample is continously recorded over the test time with displacement transducers, the resulting torsional moment is measured with a load cell and associated lever arm to the drilling point. Due to the rigid contact between glass and aluminium the preload force has to be limited to avoid a breakage in the glass. Nevertheless, a certain preload force is necessary to avoid a slippage of the sample due to gravity. The use of common soft glass contact materials such as EPDM is inappropriate in this case because it would possibly impact the measurement due to its viscoelastic behaviour.

4.2 Influencing parameters It can be seen from the test setup that there are different influencing parameters on the results. Assuming that all constructional elements of the torsional apparatus are true to size and the installation of the samples is

GF = −

Table 1 Influencing parameters of the torsional test Torsional apparatus

Sample

Preload force

Thickness of the glass layers

Friction between glass and aluminium

Thickness of the interlayer

Accuracy of the angle of rotation

E-modulus glass Poisson’s ratio glass Poisson’s ratio interlayer

are five influencing parameters left from the samples: (see Table 1) 4.3 Semi-analytic determination of the shear modulus Analogously to Callewaert et al. (2012) the interlayer shear modulus is determined from experimental data by a formula developed in Scarpino et al. (2004) respectively Kasper (2005). In addition to the torsional moment, the geometry of the laminated glazing, the shear modulus of the glazing and the angle of rotation are included in the calculation:

  6 · − 3 · G Glas · h · L · MT · t + 2 · w · G 2Glas · h · t 4 · α   w 2 − 3 · L · MT + 6 · w · G Glas · h 2 · t · α + 12 · w · G Glas · h · t 2 · α + 8 · w · G Glas · t 3 · α

accurate there remain three influencing parameter from the apparatus. Assuming a constant thickness of the glass layers and the interlayer, a parallel lamination and a length and width according to the specification there

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with: GF G Glas h

shear modulus of the interlayer shear modulus of the glass thickness of the interlayer

(4)

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Fig. 8 Influence of the measured torsional moment on the calculated shear modulus

L MT t w α

length of the specimen torsional moment thickness of a single glass panel width of the glass panel angle of rotation

Thus, the shear modulus can be determined for each load duration. As mentioned in part 2, the shear modulus of a viscoelastic material decreases rapidly as a function of temperature at the beginning of a loading. Hence, measurement inaccuracies at the beginning have a greater influence on the calculated shear modulus than at later times. This effect is illustrated in Fig. 8: For the Prony-Series from an approval (Z-70.3230 2016), the corresponding torsional moment were determined for the glass structure shown below (Reference System). These values are varied by 1% and the resulting shear modulus is calculated. Small deviations in the torsional moment have a great influence on the shear modulus determined by Eq. (4). It is worth noticing in Fig. 8 that the deviations of the shear modulus due to the same amount of variation of the torsional moment is not linear, where the effect is more pronounced for a positive variation. A precise investigation of the influence parameters of the torsional test is therefore important in order to be able to provide meaningful results as well as accuracy limits for the results.

5 FEM model and sensitivity analyses 5.1 FEM model The FEM model of the torsional test was created with the software ANSYS (version 17.2). Geometry and material properties The geometric boundary conditions are based on the experimental setup described in Sect. 3. The length and width are set to L = 1100 mm and w = 360 mm. The thickness of the glass layers and the interlayer are parameterized. The traverses and jaws are assigned the material properties of steel. The material properties of the glass layers and the interlayer are also parameterized. Contacts Several contact areas have to be defined: The contact between the interlayer and the glass layer (bonded contact), the contact between the glass and the jaws (frictional contact) and the contact between the jaws and the traverse (bonded contact). For the frictional contact, a coefficient of friction has to be defined. For validation and verification of the model it is assumed to be 0.16 (experimental determined coefficient of friction between glass and aluminium) and in the sensitivity analysis it is parameterized.

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Fig. 9 FEM model, total view (left) and detail view on the clamping side (right)

Type of elements and mesh size The entire model is meshed with volume elements. The traverse and the jaws are meshed automatically with Tet10 elements and the glass layer and the interlayer are meshed with solid Hex20 elements. The mesh size is set to 20 mm based on a previously conducted convergence study. Figure 9 shows the model. Boundary conditions One traverse is fixed by an ‘external displacement’ of the rear surface (translation and rotation in all axes prevented). The other traverse is also fixed by an ‘external displacement’, whereby the rotation around the x-axis remain definable in order to enable the rotation of the traverse. On the threads between the jaws (cylindrical surface) a parametrized preload force is applied. Analyses settings Two load steps are defined. Load step 1: Applying the preload force (clamping the sample) Load step 2: Applying the rotation (within 10 substeps) The calculation is conducted geometrically non-linear and the frictional contact is a non-linear contact. Post processor The torsional moment around the x-axis relative to the rotatable traverse is monitored as output. In addition, the maximum and minimum principal stresses on the glass layer, the deflection of the glass layer and the strain in the interlayer is determined. Verification and validation To verify the model, a comparison is made with the analytical solution according to formula (4). For this purpose, the shear modulus of the interlayer is calculated for different torsional moments and then applied

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in the FEM Model. The thickness of the laminated glass and the angle of rotation are varied. Figure 10 shows an examplary result for the described variant. The deviation between the FEM model and the semianalytical solution according to Kasper (2005) is in similar form and magnitude for other variants. The deviation of the FEM model to the analytical solution is on average 3.5% and in maximum 8%. To validate the model, some samples were provided with strain gauges at different locations and the principal stresses of the FEM model and torsional test were compared: Stresses in the model and test are in good agreement. At this point it can be mentioned that the FEM model can be used as meta-model for the calculation of the shear-modulus using a torsional test.

5.2 Sensitivity analyses In order to determine the influence of the various parameters on the resulting torsional moment and principal stresses in the glass, a sensitivity analysis is carried out with a coupling of ANSYS with OptiSlang (Version 5.2). OptiSlang is used for the automated generation and computation of the experimental design. By means of the sensitivity analysis, the influences of the input parameters on system responses can be examined. Thus, sensitive parameters of the system can be classified. The parameters are varied as discrete values as shown in Table 2. The values results from tolerances in the glass production (glass and interlayer thickness), specifications in literature (material properties of glass and interlayer) and assumptions made (coefficient of friction, preload force, angle of rotation). To receive information on

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Fig. 10 Comparison between FEM and analytical solution

Table 2 Variation of parameter Parameter

Settings

Thickness glass layer 1 [mm]

5.8/6/6.2

Thickness glass layer 2 [mm]

5.8/6/6.2

Thickness interlayer [mm]

0.68/0.76/0.84

E-modulus glass [MPa]

67000/70000/73000

Poisson’s ratio glass [–]

0.2/0.23/0.26

G-Modulus interlayer [MPa]

0.1/10/100

Poisson’s ratio interlayer [–]

0.3/0.45/0.49

Coefficient of friction [–]

0.05/0.16/0.25

Preload force [kN]

1/10/50

Angle of rotation [◦ ]

1.9/1.95/2/2.05/2.1

influences for short and long load duration and different temperatures the shear modulus of the interlayer is varied. The shear modulus is varied from G = 0.1 MPa, corresponding to low shear transfer or equivalently to a long load duration (relaxation process is almost completed to G ∞ ) respectively high temperatures (T > Tg ), G = 10 MPa up to G = 100 MPa, corresponding to full shear transfer equivalent to short load duration respectively low temperature. This results in 32 parameters that can be varied. Different designs of experiments are conceivable. A full-factorial design (i.e. every possible permutation is calculated) achieves the highest statistical significance but results in 98415 designs which is not feasible due to the high calculating time. For this reason, a “Latin Hypercube” design

with a so-called “space filling” is chosen with a design number of 2.000 points. Thus, about 2% of all possible designs are calculated. Nevertheless, a high statistical significance can be achieved by this type of the experimental design (Siebertz et al. 2010). 5.3 Results The Coefficient of Prognosis (CoP) can be used to estimate the quality of the experimental design. The CoP indicates how well unknown points of the regression are estimated. For all parameters, the CoP is between 95– 99%, so the chosen number of design points provides statistically sufficiently accurate results. The evaluation of the results pursues the quantification of the influence of the individual parameters by calculating the mean values, standard deviation and coefficient of variation (CoV) of the system response “torsional moment” and “maximum principal stress” in the glass layer. This procedure is executed for the defined shear moduli of the interlayer. Influence on the torsional moment In Fig. 11, the influence on the torsional moment are presented for the parameters of the torsional apparatus exemplarily for a shear modulus of G = 10 MPa. The results are similar for the other shear moduli, more details are given in Botz et al. (2018). The evaluation of the results focus on the comparison of the mean values of the parameters as the standard deviation and the CoV is similar for all parameters.

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Fig. 11 Influence of the torsional apparatus on the torsional moment

The Coefficient of friction has on average no significant influence on the resulting torsional moment. This means that other materials could be used for clamping (assuming a sufficient rigidity). As the preload force increases, the torsional moment increases up to a preload force of 10 kN. For small preload forces (1 kN), the result is closest to the analytical solution according to Eq. (4). The clamping of the laminated glass should therefore be done with the lowest possible force. The greatest influence on the torsional moment has the angle of rotation. For a deviation of 0.05◦ the torsional moment changes by about 3%. This relationship is linear for all shear moduli. For the torsional test, this means that the angel of rotation must be set and measured very accurately. In Fig. 12, the influence on the torsional moment is presented for the parameters of the samples. The torsional moment increases about 3% per step for an increasing Young’s modulus of the glass. With increasing Poisson‘s ratio of the glass the torsional moment decreases by about 2%. If the total glass thickness increases by 0.2 mm, the torsional moment increases by about 5%. An accurate measurement of the glass thickness is therefore important for accurate results. The influence of the Poisson‘s ratio of the interlayer is only significant for a soft system (G = 0.1 MPa). For high temperatures and/or long load duration, the Poisson‘s ratio of polymers is about 0.49– 0.5. For low temperatures and/or short load duration, values from 0.3 can be found in literature (Grellmann et. al. 2014). Since no influence is recognizable for

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rigid systems (G = 10; 100 MPa), it is sufficient to assume a value 0.49–0.5 for all systems. The influence of the interlayer thickness is dependent on the shear modulus of the interlayer. For a shear modulus of G = 100 MPa (full shear transfer), the torsional moment increases with increasing interlayer thickness as the sectional modulus increases. For a shear modulus of G = 10 MPa, the torsional moment decreases with increasing interlayer thickness, since there is no full shear transfer and the system becomes softer with increasing interlayer thickness. For a shear modulus of G = 0.1 MPa, no influence is noticeable (glass layers are almost decoupled). A precise measurement of the interlayer thickness is therefore also important for a correct result. Influence on the maximum principal stress In Fig. 13, the maximum principal stress on the glass surface is illustrated exemplary. For the sensitivity analysis, the stress was evaluated as the maximum value in the horizontal and vertical path shown in Fig. 13. In the area of the jaws the maximum stress appears. For high preload force and low shear modulus of the interlayer the maximum stress appears between the jaws as shown in Fig. 14. For soft interlayers (e.g. acoustic interlayer) glass cracks appears after long load duration in the described areas of maximum stress (see Fig. 14). It can be seen that the preload force has a major influence on the maximum stress in the glass in dependence of the shear modulus of the interlayer. This aspect

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Fig. 12 Influence of the sample on the torsional moment

is shown in Fig. 15. The displacement in y-direction induced by the preload force is shown in Table 3. For high preload force and a shear modulus of G = 0.1 MPa, the maximum stress on the horizontal path appears at the edge between the jaws. For the other cases, the maximum stress appears in the midspan of the glass layer. As result, the preload force has to be minimized to avoid glass cracks during the test. With respect to the parameters ‘coefficient of friction’, ‘thickness of the interlayer’ and ‘Poisson‘s ratio of the interlayer’, a distinctive influence on the stresses in the glass is not recognizable. For the other parameter (‘Young’s Modulus of the glass’, ‘Poisson‘s ratio of

the glass’, ‘thickness of the glass’, ‘angle of rotation’) the stresses changes linear. This is shown exemplarily in Fig. 16 for the total glass thickness and the angle of rotation.

6 Relaxation and dynamic mechanical thermal analysis (DMTA) experiments on structural PVB In the context of this paper, a comparison of Master Curves obtained from DMTA data against curves from torsional relaxation tests is conducted. The sample material was sent by the manufacturer directly to

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Fig. 13 Maximum principal stress [MPa] on the outer glass surface for G = 10 MPa, preload Force = 1 kN

Fig. 14 a Maximum principal stress for high preload force (50 kN) and G = 0,1 MPa b Glass cracks around the jaws

the test laboratory. The samples have been stored light protected and in the case of the DMTA samples were vacuum packed at approx. 10 ◦ C until testing. The following table summarizes the parameters of the two test settings (Table 4).

6.1 Torsional relaxation test In the torsional test setting, five samples were tested at a temperature of 23 and 50 ◦ C and a load duration of one hour. For a temperature of 0 and 23 ◦ C and a load duration of 30 days three samples were tested respectively. All samples were conditioned in the climate chamber for 24 h before starting the test. The test procedure was already described in Sect. 4. The accuracy of the temperature in the climate chamber was about ± 1 ◦ C around target test temperature.

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6.2 Dynamical mechanical thermal analysis (DMTA) tests The DMTA test was conducted in tensile mode. For the calculation of the shear modulus a Poisson‘s ratio of 0.5 is assumed. The test procedure in the DMTA is a temperature-frequency sweep. The temperature is set to 70 ◦ C and the sample is conditioned. Afterwards the cooling process started to a temperature of − 5 ◦ C. In steps of 5 ◦ C the sample is loaded with a frequency of 0.01–100 Hz in 32 logarithmically equispaced steps and a strain of 1%, where the Storage modulus G  and the Loss modulus G  are measured. The measured raw data can be seen in Fig. 17. With the raw data and the described theory in Sect. 2, the shift factors of a TTSP can be obtained. Finally, a Master Curve was created for a reference temperature TRe f = + 20 ◦ C and the parameters for the corresponding Prony-Series

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Fig. 15 Influence of the preload force on the maximum principal stress in the horizontal path (Smax H) and the vertical path (Smax V)

Table 3 Displacement induced by the preload force Preload force [kN]

1

10

50

Mean displacement [mm]

0.004

0.042

0.208

Standard deviation

0.0015

0.015

0.078

were determined. The Master Curves for the Storage and Loss modulus are depicted in Fig. 17. For further information on the DMTA and the evaluation method, see (Kraus and Niederwald 2017).

6.3 Comparing the results of torsional and DMTA test data Finally the data of both measurements are put together to get insight in the comparability of the results. In Fig. 18 the results are illustrated. For the temperature of 23 ◦ C, the results (Fig. 19a) of the tests are in a good agreement, especially for long load duration. Reason for the deviation for short load duration can be the described sensitivity in chapter 3. Also for the temperature of 50 ◦ C the results (Fig. 19b) are comparable. Reasons for the spread of the results of the torsional test can be on one side due to inaccuracies in the measurement. On the other side, the temperature in the climate chambers varies ± 1 ◦ C. This effects the results

theoretically due to the determined TTSP by log(aT ) = 0.25 in that temperature region. Moreover it was found, that the quality of the laminated glass samples was not always ideal, as e.g. the lamination was not parallel, the thickness was not constant. For the temperature of 0 ◦ C the results (Fig. 19c) between the test methods do not coincide. The result of the DMTA test is much stiffer than the result of the torsional test. As already mentioned in Sect. 4, the calculation of the shear modulus is sensitive for low temperatures as well as for short load durations in a similar manner. Further investigations of the authors indicated limitations in the usage of the formula developed by Kasper in this very context (Botz et al. 2018). From a mechanical prospective, there is a significant difference in the obtainable boundary values of the shear modulus out of the two experiments. For the torsional relaxation test setup, invariance with respect to shear modulus is gained for the full shear transfer case, i.e. at a certain point an increase in the value of the shear modulus does not result in a further growing torsional moment. In the DMTA test setting the raw interlayer material stiffness is measured without being limited to an invariance due to mechanical means. Stevels et al. (2017) conducted a comparison between the results of torsional tests on large scale specimen and DMTA tests. For a temperature of 23 ◦ C the results are quite similar, whereas for lower temperatures of 0 ◦ C

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Fig. 16 Influence of the total glass thickness and angle of rotation on the maximum principal stress Table 4 Parameters settings of the torsional relaxation test and the DMTA Test method

Torsional test

DMTA

Sample geometry [mm]

Laminated glass:

Rectangle interlayer:

Test equipment

L × W × T:

L × W × T:

1100 × 360 × (6-0.76-6)

10 × 30 × 0.76

Torsional apparatus

Eplexor 2000N Netzsch Gabo

Test mode

Torsion

Tension (ν = 0.5)

Temperature program [◦ C]

50; 23; 0

+ 70 : − 5 (cooling)

Temperature steps [◦ C]

Isothermal

−5

Frequency [Hz]

–

0.01 : 100 (32 l.e.s.1 steps)

Incitation

2◦

Load duration

1 h, 30 days

1 l.e.s.

angle of rotation

1% strain –

= logarithmically equispaced

the results do not fully fit well. Stevels is criticizing, that the torsional test is not suitable for short load durations. This is a general problem of a relaxation test. According to the withdrawn DIN 53441 it is advised to set the first evaluation-point t1 equal to 10 times the deformation time ts (i.e. t1 ≥ 10 ts ). This means for example, if in the torsional test the intended angle of rotation is reached within 3 s, the first evaluation-point should be set to 30 s. Moreover, Stevels is criticizing

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the lengthy experimentation of the torsional test. As already mentioned, relaxation tests on the same material are vital for the validation of a Master Curve processed from DMTA tests and a corresponding TTSP. Only by means of the results of long-lasting isothermal relaxation or creep tests, the validity of the applied TTSP to the DMTA data can be proved for long load durations respectively low frequencies or respectively high temperatures.

Sensitivity analysis for the determination of the interlayer shear modulus

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Fig. 17 DMTA raw data: Storage modulus (left), Loss modulus (middle) and TTSP (right)

Fig. 18 Master Curves for Storage modulus (above) and Loss modulus (below)

7 Conclusion The determination of the material properties of polymeric interlayers used in laminated glass is important

for an accurate design of structural glass applications. The complex material behaviour can be examined with different tests. Big scale tests as the described torsional tests are possible in which an entire laminated glass

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Fig. 19 Comparison of the shear modulus Master Curve from DMTA data with and shear modulus curves from torsional relaxation testing at T = 23 ◦ C (a), T = 50 ◦ C (b) and T = 0 ◦ C (c)

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Sensitivity analysis for the determination of the interlayer shear modulus

plate is tested as well as small scale tests as DMTA tests in which only the interlayer is tested. The conducted sensitivity analyses of the torsional test showed that some parameter have major influence on the measured torsional moment and affect thereby the calculated of the shear modulus of the interlayer. The measurement of the the ‘angle of rotation’ and the ‘glass geometry’ has to be done very accurately. Small deviations lead to significant errors. Especially at the beginning of the test, the calculation of the shear modulus is highly sensitive. Moreover, the clamping of the samples has to be taken with minimum preload force to avoid glass cracking during the test. The comparison between conducted torsional and DMTA tests indicates that for long load duration the results are in good agreement. For short load duration and low temperatures, the deviation between the results is significantly greater. Reason for this can be the described sensitivity for short load duration. Furthermore, the evaluation of the torsional test with the formula developed by Kasper might not be invariant for short load duration and low temperatures. In the torsional apparatus, several types of interlayers were tested so far. A comparison to DMTA tests of these interlayers are necessary to confirm the results of this paper. As conclusion it can be said, that the torsional test is a suitable test method for the validation of DMTA tests if the described sensitive parameter are examined precisely. However, further investigations by the authors will be done in this field of research.

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