# Primary Seal Deformation in Multipane Glazing Units
## 3 Numerical and Analytical Calculations
### Primary sealant Mechanical Behaviour  
The global FEM model described above gives a general overview of the structural integrity of the product, i.e., the deflection of all panes and their stresses and the strains on the spacers and secondary sealant. It is of major importance for further analysis that this model gives the displacements of all  
Figure 3: Temperature distribution in the structural elements. The recommended upper limit is the hybrid spacer and PIB primary sealant useful upper temperature limit [10]. points of the glass panes and spacers that are in contact with the primary sealant. This provides an opportunity to model the primary sealant separately, where known displacements represent the boundary conditions, as shown in Figure 4.  
The primary sealant could be analysed as a FEM submodel by prescribing displacements as boundary conditions, but it turns out that such an approach is computationally expensive and has several convergence problems due to the PIB material properties where PIB could be treated as an incompressible solid of very low modulus. For practical reasons, an alternative approach that considers the combined effects of separate deformation modes, with each of them representing a separate task based on the type of loading, was chosen. Mode A: Shear strains \(\gamma_{xz}\) occur mainly due to the thermal expansion difference between the glass pane and spacer in the longitudinal (\(z\)) direction. In cases where a unitis double-symmetric, as presented in Figure 1, \(\gamma_{xz}\) is zero on the symmetry planes and is the highest in the corners. In general, however, self-weight affects shear strains too. We assume that \(\gamma_{xz}\) is location-independent at the PIB cross-section and is dependent on the location on the unit's circumference only; therefore, \(\gamma_{xz}(x,y,z)=\gamma_{xz}(z)\) can be calculated from:  
\[\gamma_{xz}^{A}(z)=\frac{\Delta u_{z}(z)}{h} \tag{1}\]  
here \(h\) is the thickness of the PIB layer. Mode B: Shear strains \(\gamma_{xy}\) occur due to the thermal expansion difference between the pane and the spacer in the transverse (\(y\)) direction and due to the increased cavity pressure in the chamber that loads the spacer from the inner side. Simple shear is assumed for convenience, which means that \(\gamma_{xy}\) is the same over the entire PIB cross-section, hence \(\gamma_{xy}(x,y,z)=\gamma_{xy}(z)\). In reality, a spacer made of thin-walled material of low rigidity deforms locally and the spacer wall is no longer precisely parallel with the glass pane; therefore, the average displacements of both contact surfaces with the PIB are needed for the following calculation of the shear strain:  
\[\gamma_{xy}^{B}(z)=\frac{\Delta u_{y}(z)}{h} \tag{2}\]  
Mode C: Glass panes bulge under increased cavity pressure; therefore, the glass pane is no longer parallel with the surface of the spacer. The inclination of the PIB layer can be described by the shear strain \(\gamma_{xy}\) and normal strain \(\varepsilon_{xx}\). The maximal values of both strains on the PIB cross-section can be approximated with:  
\[\varepsilon_{xx}^{C,max}(z)=\pm\frac{L}{h}\tan(\alpha(z)) \tag{3}\]  
\[\gamma_{xy}^{C,max}(z)=\pm\alpha(z) \tag{4}\]  
Mode D: Increased pressure in the chambers moves the glass panes apart and the role of the secondary sealant is to capture this effect. However, the primary sealant follows the deformations of the secondary sealant, causing the thinning of the primary sealant layer. Obviously, when this deformation mode is established, the PIB layer thins, meaning that the effective cross-section for moisture diffusion increases for the duration of the highest summer climatic loadings and is thus of the highest interest. Moreover, the instantaneously deformed state would not be that critical if the material behaviour of  
Figure 4: Deformation modes of the primary sealant. the primary sealant were not viscoplastic.