Unlike glass, interlayers are viscoelastic materials.
Unlike glass, interlayers are viscoelastic materials. This means the stiffness of interlayers varies with temperature and time under conditions encountered in building applications of laminated glass. Interlayer materials are softer at higher temperatures and for longer durations. Another way to put this is that interlayers relax with time and temperature. The fact that common plastic materials get softer with increasing temperatures is intuitive for most people. The effect that materials get softer with time is less intuitive. An easy way to demonstrate the effect is to put a 25 kg weight on a laminated glass specimen and monitor the deflection of the glass panel in time in a four-point bending set-up, as illustrated in Figure 1 (note that the x-axis is logarithmic).
Figure 1. Deflection of a 66.2 Saflex Structural laminate (6 mm glass + 0.76 mm Saflex Structural + 6 mm glass) in time at a temperature of 20°C and 30°C.*
The relaxation modulus has to be known to predict the deflection of laminated glass
To predict the deflections (and stresses) of laminated glass in buildings at different times and temperatures, the relaxation modulus (a measure of interlayer stiffness) of the interlayer must be known. For Saflex interlayers, these can be taken from tables in our product technical sheets which are available on our website. Alternatively, they can be calculated from material models, also available on our website. These can be used to calculate the relaxation modulus for any temperature and duration for the validity range of the model.
The most common way to describe the relaxation behavior of plastic materials with time, including interlayers, is a so-called generalized Maxwell model. Technically, this is a curve fit of the interlayer shear relaxation modulus over a given time or frequency domain. Conceptually, the model consists of n parallel spring-dashpot elements (with each element counted by an index i) and an additional parallel spring. A graphic depiction of such a model is provided in Figure 2.
Figure 2. Spring-dashpot model for linear viscoelasticity
Each spring/combination dashpot has a characteristic time, the ratio between the creep characteristics of the dashpot (a function of time) and the stiffness of the spring. This time is called the relaxation time τ, with τ_{i} = η_{i} /G_{i}. Note that neither η_{i} nor G_{i} are experimentally determined but are merely a construct to conceptualize the notion of multiple characteristic relaxation times in a viscoelastic system.
Spill some data!
The number of elements i that is required depends on the range of the master curve covered by the model and to some extent also the shape of the master curve. For a full description of the master curve typically 20–30 elements are required. To describe the viscoelastic properties of interlayers for structural glass design in the architectural domain, 10–12 elements usually suffice, as typical load durations occur in a limited range. Examples include wind loads (e.g., 3 seconds to 10 minutes), live loads (e.g., 30 seconds to 1 hour), cavity pressure loads (e.g., 6 to 12 hours) and snow loads (e.g., multiple days to one month).
To describe the response of the system, the initial modulus of the interlayer G_{0} must be known, as well as the modulus of the system at “infinite duration” G_{∞}. These effectively serve as model limits and the calculated modulus values cannot be higher than G_{0} or lower than G_{∞}. The relaxation modulus of the interlayer as a function of time(t), G(t) can then be modeled using the following equation in which g_{i} is the normalized modulus G_{i}/G_{0}:
The sum of all g_{i} elements should equal one. Let’s examine an interlayer material model in more detail. In Table 1, all parameters are provided to calculate the shear relaxation modulus of Saflex Structural at 20°C.
Table 1. Model parameters for the evaluation shear relaxation modulus G(t,T) for Saflex Structural
G_{0} = 576 MPa | G_{∞}= 0.23 MPa | |
T_{ref} = 20 °C | ||
i | relaxation time | g_{i} (G_{i}/G_{0}) |
(s) | ||
1 | 1.000E-01 | 0.1713 |
2 | 1.000E+00 | 0.1960 |
3 | 1.000E+01 | 0.2101 |
4 | 1.000E+02 | 0.2054 |
5 | 1.000E+03 | 0.1503 |
6 | 1.000E+04 | 0.0543 |
7 | 1.000E+05 | 0.0101 |
8 | 1.000E+06 | 1.79E-03 |
9 | 1.000E+07 | 4.78E-04 |
10 | 1.000E+10 | 2.93E-04 |
This model contains 10 relaxation times which were chosen in such a way that they shift time a decade from one relaxation time to the next, for user convenience, with exception of going from element 9 to 10. To create insight into the way the model functions, the outcome for elements 2 to 7 was plotted in Figure 3. The other elements have a very low contribution for the time-range plotted. Element 1 is already relaxed at 1 second; the other elements have low contributions at (very) long durations.
As an example, in the 1–10 (s) time range, element 4 has the largest contribution. After 100 seconds (relaxation time) its contribution has significantly reduced, and after 600 seconds (10 minutes) its contribution has become negligible. After 30 seconds, element 5 has the largest contribution and after 1,000 seconds element 6 has the largest contribution, even if the absolute value tracks the g_{i} values in Table 1 and decreases with longer relaxation times.
When all (i = 1–10) elements are added to Figure 3, the results multiplied with G_{0}, and G_{∞} are added and the outcome is the shear relaxation modulus with time as per the previous equation. The result is given in Figure 4 for a period up to 3.5 months. Effectively, Saflex Structural relaxes from 393 MPa to 0.5 MPa over this period. After a single day, the value is 4.1 MPa––still enough to provide effective coupling for laminated glass as indicated by the low deflection increase at 20°C over this time as indicated in Figure 1.
Figure 3. Outcome for individual elements 2 to 7 (gi * EXP(-t/τi) for durations between 1 second and 100.000 seconds (approx. 27 hours)
Figure 4. Shear relaxation modulus Saflex Structural at 20°C– outcome of the Model in Table 1 for 1 to 10.000.000 seconds (approx. 3.5 months)
The model of Table 1 can be extended with a temperature shift function allowing the shear relaxation modulus to be calculated as in Figure 4 for any temperature between, e.g., 10°C and 60°C. The results of this extension are covered in a recent paper that is accessible online (Stevels, W.M & D’Haene P. Determination and verification of PVB interlayer modulus properties, Proceedings to Challenging Glass 7 2020 online: https://journals.open.tudelft.nl/cgc/issue/view/708 ).
Now that you’ve covered the contents of this article, we hope you have some time to relax yourself!
*Dotted lines tracking the points are provided as a guide to the eye only. More details on the experimental setup can be found in our Module 3 of the Saflex Structural series here (insert link).