Limit load and failure mechanisms of soda lime glass foam

Modeling a porous structure of glass foam is important to understand its mechanical properties. There are many ways to describe and model the structure of foam. One of them refers to the microstructure observation and understanding its manufacturing process. During the production of low-density foam, isolated spherical bubbles of gas tend to expand, so the adjacent ones start sticking, while further bubbles grow in polyhedral morphology. Depending on the achieved density and surface tension of the material, the foam is formed with closed or open cells.

Common observation shows that the structure of real foams is neither regular nor periodic, even though it is created by topologically similar cells (dispersion in cell size). Nevertheless, to simplify the problem of theoretical prediction of macroscopic foam behavior we shall focus our attention on idealized periodic microgeometries, created by the repetition of fundamental piece called the unit cell. Study of the physical aspects of foam manufacturing gives us guidelines for unit cell selection.

The latest solution of the minimization problem of unit cell surface energy, where unit cells have planar faces (the bubbles are in contact), was presented by Gabbrielli [19]. Apart from that, and aiming to analyze the foam behavior theoretically, many other idealized structural models were proposed. One of the very first polyhedron, described by Lord Kevin [20] in 1887, was tetrakaidecahedron, which best describes the structure of a single cell. This model has been used in calculation. Figure 1a presents single tetrakaidecahedron, and Fig. 1c shows a model of foam with open cells modes of polyhedrons from Fig. 1a. As it has already been mentioned, the main advantage of tetrakaidecahedron is that it is periodically repetitive. From the structure presented in Fig. 1c, it is possible to cut out other periodical element different from the one presented in Fig. 1a. It is shown in Fig. 1b, and it is made of 6 square frames. Based on this model and frame theory, an equivalent Young’s modulus for the foam can be determined by Eq. 1.where \( E^{*} \) is Young’s modulus for the foam, \( E_{\text{s}} \) is Young’s modulus for the solid material, a is length of the edge of rectangular frame from Fig. 1b, A is an area of beam cross section, \( I_{\text{y}} \) is the second moments of cross section area of beam, \( \psi \) is form factor, and \( \upsilon \) is Poisson’s ratio.

A similar approach can be used for the foam with close cells, but it is necessary to use the model of plate and beam instead of just the frame theory. Gibson and Ashby [21] give the formula of Young’s modulus for the foam with a closed cell in Eq. 2.where \( \frac{{\rho^{*} }}{{\rho_{\text{s}} }} \) is the relative density of the foam, \( \phi \) is a fraction of solid contained in beam elements, a mass of cell is split between plane and beam regions, and \( C_{1} \) and \( C_{1}^{\prime } \) are constants of proportionality.

In the case of a bigger structure, the usage of the plate model causes a significant impediment. Therefore, the finite element (FE) model is used for the analysis of foams with closed cells [22]. Figure 1d shows the FE model of a unitary cell used in this analysis. The model could be prepared basing on solid and shell elements, where the walls of cells are very thin when compared to the rest of their sizes; therefore, for simplification reasons, shell elements are used in Fig. 1d. Typically, the cell wall in the center is thinner than the ones on the line of walls connection. This is caused by the process of foam production, during which the walls of glass bubbles become thinner and thinner when size increases. The density of the glass foam was of 130 kg m⁻³, so we determined the thickness of walls using these data and the density of the solid glass (Table S1).

During the calculation process, we idealized the foam material. We assumed that it had isotropic and linear elastic properties up to the brittle fracture, which determines the limit load of the material. We also assumed that the structure of the foam consists of periodically repetitive cells; therefore, one unitary cell should be enough. This would be true if the boundary on the interface between the foamed glass and the foundation were not considered in the analysis. In that case, the continuity of the structure could be ensured by constraints equations. Those equations relate the movement of free edges of a unitary cell. Based on that model, it is possible to determine stress and strain field far from the boundary of foam specimen and equivalent Young modulus and Poisson ratio for the foam structure. Using those assumption and the FE model, we determined constants from Eq. 1. The FE model is made of shell elements only; therefore, \( \phi \) is equal to 1 and \( C_{1}^{\prime } \) could be omitted. Consequently, for the considered structure, constant \( C_{1} \) is 5.77.

To confirm the results obtained by a theoretical analysis, we performed a series of practical tests.

Figure 5 presents the consecutive stages of the destruction of foam structure S1 before the test (Fig. 5a) and after the destruction of 1/3 (Fig. 5b) and 2/3 (Fig. 5c). These images confirm theoretical calculations (Item 1.1) that the process of destruction begins on the edge and proceeds layer by layer deeper into the structure. The central part of the sample does not seem to be deformed and remains a rigid block. This is evidenced by DIC images presented in Fig. 6 that even shows the deformation of the structure. These images show a vertical displacement of the sample S1 after 1/3 (1st stage; Fig. 61a) and 2/3 (2nd stage; Fig. 62a) of its destruction, deformation in the vertical direction in microstrains (Fig. 61b, 2b) and microshear strain (Fig. 61c, 2c). The process occurs uniformly in the whole extension, and the glass foam layers are destroyed successively one-by-one.

The compression force in the function of displacement and averaged curves with a standard deviation were plotted for all samples (S1–S7), and the results are shown in Fig. 7a. The results reveal a constant increase in a compression force as a function of the displacement up to maximum, at about 10 mm, and after that, a decrease. The visible fluctuation on the curves corresponds to the gradual collapse of the structure of the samples, which occurred layer by layer. The maximum average compression force after 10 mm of the displacement is about 0.8 kN (Fig. 7b).

The same experiments were made for the foam with pores located on the verge and filled with the resin (R samples) that was applied in a way to ensure the flatness of both surfaces of the contact. The pictures of sample R6 before and during the destruction process are shown in Fig. S2. Figure S2 a presents successive stages of the test for the sample R6 before the test, after the detachment of the upper layer of resin (Fig. S2 b) and after the detachment of the lower layer of resin (Fig. S2 c). Comparing this figure with Fig. 6, we can observe that the sample with the resin is more resistant than the one without the resin because there are changes in the first stage of the destruction process (Fig. S2 b). However, in the 2nd stage of the destruction (Fig. S2 c) 1/5 of the sample, R6 is destroyed, while in the same condition only 1/3 of sample S1 (Fig. 6c) remained.

These samples were also analyzed by DIC images. Figure S3 1a presents displacements in the vertical direction in first stage, shown in Fig. S2 b. The deformation in the vertical direction in microstrains and microshear strains is shown in Figs. S3 b and c, respectively. The nature of the destruction is similar to S-type samples (Fig. 6), where the degradation occurs layer by layer and the middle part of the foam remains stiff. The differences between these two samples destruction tests are seen in force versus displacement curves (Fig. 8). In the initial stage of loading, the sample shows a high resistance (Fig. 8a). The force increases to a large value of about 0.52 kN, and then, it decreases to 0.05 kN, which is caused by the difference in the stiffness of two materials: glass and resin. The layer with the resin is detached from the structure. At a later stage, the force increases again to 0.6 kN for displacement of 10 mm (Fig. 8b). The maximum average force of 0.8 kN is the same for S and R samples (Figs. 7b and 8b). However, at the beginning, samples with the resin have greater strength, while the subsequent layers of S-type samples are quickly destroyed without any resistance. At some point, after the destruction of several layers, the increase in the resistance in cells destruction is seen. This is caused by the process of filling the cellular voids with the material formed by crushing the edge of the cells. Owing to this, some force is transferred by plate surfaces as an evenly distributed pressure.

Finally, we analyzed samples with flour (F) and collected pictures of sample F3 before (Fig. S4 a) and during the two stages of the destruction test (Figs. S4 b and c). These images are quite different from previous, because they show cracks of the glass foam structure. A beginning of the crack is almost invisible in Fig. S4 b, but it is clearly visible in Fig. S4 c, and it extends from top to bottom. We confirmed these observations by DIC images, shown in Fig. S5, and plots in Fig. 9. The vertical displacements DIC images confirm the start of the crack (Fig. S5 1a) and its propagation in Fig. S5 2a. The same is observed in the deformation in the vertical direction (Fig. S5 1b and 2b) and microshear strains (Fig. S5 1c and 2c).

In Fig. 9a, we present the force displacement curves for seven samples of the foam with flour and averaged curves with standard deviation in Fig. 9b. From these figures, we can observe that in the initial stage of load application, the behavior of F-type samples is somewhere between S and R types. Sample F is stiffer than S but less stiff than R, because it resists more. When the level of the load is higher, i.e., when it reaches the first stage of the destruction, the flour begins to elbow the structure and tears it apart, which is clearly visible in Fig. 9b and Figs. S5 1a, 1b and 1c, when the sample displacement reached about 4 mm. The maximum of an average compression force for this sample was almost 0.6 kN, and the average displacement is 5 mm (Fig. 9b). After that, a propagation of structure shrinkage and decrease in sample’s thickness occured.

A comparative analysis of all the obtained data is shown in Fig. 10. The glass foam sample and the glass foam covered with the resin sample showed similar results except of the beginning of the force application (Fig. 10a). The sample with the resin was more resistant in the initial stage of force application (elastic deformation until delamination point, Fig. 10b), while the glass foam sample suffered a destruction from the beginning. This is due to the irregularity of surface. An unevenly distributed load shatters the protruding elements. It causes that there is no pure linear range, since some local preliminary structure degradation is present. The maximum force of compression for both samples was about 0.8 kN for the displacement of ~ 10 mm. On the other hand, the sample, covered with flour, due to the lateral expansion of its structure, caused by the very flour, achieved a lower maximal force of 0.6 kN for the displacement of ~ 5 mm. But in the case of flour, the loading could be split into two linear ranges. In the first case, the flour subsides and penetrates the structure. The second case is about the elastic range of foam deformation. The slope of the line for the resin is two times greater than for the flour. (It means that resin is also two times stiffer than flour.) From the numerical results, it can be seen that the average maximum force in S and R specimens was equal to 0.846 kN with a standard deviation of 0.054 kN. Considering the area of the circular sample (diameter 33 mm), the critical stress was of 0.989 MPa, which is similar to the results published by Guo et al. [9] and Attila et al. [10]. Our experimental results also show that the elastic limit load for the resin reinforced foam is 0.68 MPa (0.58 kN), the flour distributed load 0.61 MPa (0.52 kN), and 0.55 MPa (0.47 kN) for the rough material. Our results are also consistent with the properties provided by the manufacturer of foamed glass (Table 2 [23]), while the results of our theoretical model point to a higher material capacity.

The analysis of the presented data allows to single out two significant problems that can be encountered while studying the structure of foamed glass: firstly, the big discrepancy of the material properties and, secondly, the difficulties with modeling the foam structure. In the calculation presented in this article, we used the model with identical and periodical cells. Although there are no two identical cells in the nature, from the macroscopic point of view, this assumption is correct, and it makes it possible to study the boundary effects. The method of load application can change the distribution of stress inside the foam structure. In the case of pure foam (samples S), the cut edges of the open cell do not have any support. The load applied causes in plates tension stress that comes from bending, so the foam does not resist and crashes with force close to zero. The following destruction of several layers of cells produces a dust that fills up the subsequent open cells. The load transferred by the structure is relocated from edges to surfaces of the cells and the foam begins to resist, so the transferred load increases. At some point the equilibrium is set, but then the structure is being destroyed at a constant force. Dynamic and abrupt collapses of subsequent layers of structure cause its degradation. The force transferred by the structure is reduced to zero. In the second case, the free edges of cut cells are supported by resin (samples R), which provides additional stiffness. The foam with the minimal squeeze of structure puts up a significant resistance, which means that the resin protects the structure. It allows a transfer of a bigger load without (or with minimized) the destruction process. As the force grows up, the layer of the resin is detached. The force decreases, and then, it starts to increase again. Then, the process of the destruction of the structure proceeds similarly to the case of pure foam, layer by layer. The samples with the flour (samples F) can be placed somewhere in between the resin and the pure foam. The flour provides a redistribution of the loads and allows the transfer of force through surfaces. Unfortunately, the flour behaves like liquid; it pushes on the structure in all directions. In the initial phase, it helps to transfer the load (a higher low is allowed), but afterward it elbows the structure. The flour becomes the initiator of the crack and bursts the structure. We determined the stress concentration factors and the maximum principal stress for the internal and external regions of the foam by using FE analysis. There is a correlation between the results obtained from the numerical analysis and the experiment. From FE, we observed that the stress in internal regions is lower than for external surfaces and that the maximum stress is higher than the allowed stress for glass. However, our experimental data revealed much lower maximum stress values. Therefore, besides checking the impact of changing the type of resin and relative density on the overall capacity of the structure, it is planned in the future work that other more flexible materials should be investigated as top and bottom coatings of soda lime glass foam. Finally, the effect of the change in the internal pressure should also be examined.