Simulation of bubble growth during the foaming process and mechanics of the solid foam

Abstract

Elastomeric foams are widely used in different types of applications where different material properties are of interest in each application. All of these properties are governed by the microstructure and the properties of the material matrix. Studying the evolution of the microstructure experimentally is extremely challenging. Thus, here we use direct numerical simulations to gain an insight into the changes that happen from the creation of the gas bubbles in the liquid state, until the solidification into a cellular morphology. Furthermore, the resulting microstructure is then used directly in simulations of solid mechanical testing to determine the mechanical properties of the foam. The matrix fluid is assumed to be Newtonian and incompressible. A linear elastic isotropic material model for the solidified polymer was used to obtain the solid foam properties. The foam was described by a representative volume element (RVE), where a small number of bubbles was randomly distributed. Using this approach, the RVE can describe the bulk behavior of the foam, while remaining computationally tractable. Microstructures with volumes fraction of over 90% (2D) and 45% (3D) are accurately captured. In addition, the influence that the bubble growth rate and the initial bubble distribution of the fluid simulations have on the solid foam properties was studied.

Appendix: Refining in 3D

Mesh refinement is needed to accurately describe the fluid dynamics in the thin layer between bubbles. However, the refinement of the mesh in three dimensions is more cumbersome than in the two-dimensional case, since now the mesh consist of quadratic triangles. Triangles require two different types of refinement, regular and irregular refinement (Yserentant 1991). Regular refinement splits one triangular surface element into four new ones and irregular refinement splits one triangular surface element into two new ones (see, Fig. 22a). Elements generated during irregular refinement are called green elements. An element before refinement is called the parent element. During refinement, new smaller elements are created, which are called siblings. Regular refined elements will therefore have four siblings, while irregular refined elements will have only two. Every element has its own refinement level and by regular refinement the level is increased by one.

Fig. 22

a Regular refinement (top) and irregular refinement (bottom) of an element. The blue circles are the location of the nodes. b Regular refinement of marked element (M). The neighboring elements are not refined and hanging nodes are created (red nodes). To avoid hanging nodes, the neighboring elements are irregularly refined

While refining a single element, unwanted hanging nodes would be generated on its sides, if the neighboring elements were not refined as well (see, Fig. 22b). When an element is marked for regular refinement (M), its neighboring elements are marked to be irregularly refined (G) in order to get rid of hanging nodes (see, Fig. 23a). When a green element is marked for refinement (Fig. 23b), the irregular refinement is first removed and the parent element is regularly refined, as in Fig. 23c.

Fig. 23

a Three elements are marked for regular refinement(M), as a result 5 neighboring elements are marked for irregular refinement(G), to withhold the creation of hanging nodes. b Green elements marked for refinement. c Regular refinement of green elements. d On the left, two neighboring elements of different refinement level are marked, which results in an element that is marked for both regular refinement and for irregular refinement. This cannot be done, however to solve this; refinement is done level by level. The marked elements on the right show that no double green elements can be created. e The final obtained structure

The elements in the mesh are refined level by level, starting with elements with the lowest refinement level, to get rid of hanging node that are generated when two neighboring elements are refined and are of different refinement level (see, Fig. 23d). In Fig. 23, some basic cases of mesh refinement previously mentioned are shown. In every next figure, the marked elements in the previous figure are refined.