Overview of Micromechanical Models

As a first step to modelling the constitutive behavior of foams, the cell morphology and the resulting mode(s) of cell deformation should be taken into account. In general, it is agreed that the cell deformation mechanisms in the elastic regime are elastic bending of the cell struts and axial stretching of the cell walls [11, 73–79]. Furthermore, it is known that in the post-yielding zone, different (but known) collapse modes can occur. This includes plastic bending (yielding of the joints and formation of plastic hinges), elastic buckling as in case of elastomeric foams, plastic buckling, and finally brittle fracture [11, 12, 25, 80–83]. The exact mode of deformation or combinations of different modes are determined by the base polymer material (e.g.

elastomeric, ductile, or brittle material behavior) and the geometrical characteristics of the foam cells such as cell shape, orientation, and wall/strut length and thickness. The cell deformation mechanism is also influenced by foam density, as density has an effect on the cell morphology too. It will be shown in the next chapters that the cell deformation mechanism has a significant impact on the macroscopic foam properties.

Nevertheless, there are sometimes confusing literature data on the micromechanics of foams.

One reason could be due to the morphological complexities, which can vary from case to case. As outlined by Gibson [25], part of the confusion could be related to the understanding about the cell deformation mechanisms, which are then used to derive analytical equations for moduli and strength. Unless the correct deformation mechanisms are identified and applied in the models, the predicted values will deviate from the experimental values. In most literature works, a particular and fixed cell morphology is assumed for the entire foam volume and effort is made to analyze its response to the applied loads [25]. However, there are usually multiple types of cell shapes with varying geometrical features of walls and struts, which result in combinations of different deformation mechanisms. Unless these effects are carefully accounted for, accurate prediction of foam properties at different densities would be almost impossible. The literature review below is first focused on the idealized models capturing the effect of density on properties. Later, models considering higher levels of morphological complexities are briefly mentioned.

Earlier approaches to experimental modeling of foam behavior assumed that the properties depended linearly on the relative foam density. However, later works in the literature shed more light on the relations between properties and foam density. Extensive literature works on different foam types conclude that most of the experimental data can be described by the general power law relation in Eq. 1, which links the relative foam properties to its relative density. The terms f and s could be any mechanical properties of the foam and the solid material it is made of, respectively. The proportionality constant, C, and the power law exponent, n, are usually obtained from experimental tests.

Φ_𝑓

Φ_𝑠 = 𝐶 (𝜌_𝑓

𝜌_𝑠)^𝑛 (1)

However, in the analytical approaches most available models first assume a simplified and idealized definition of the unit cell geometry, such as cubic cells, equiaxed polyhedrals, or tetrakaidecahedron cells. The foam properties are then analytically related to foam density using mathematical relations derived from deformation theories. The idealized cubic models proposed by Gibson and Ashby [11, 12, 84] for open- and closed-cell foams, as well as their extensions to consider anisotropy [84–86], are among the most well-known models. In the cubic model, foam cells are assumed as cubic elements with struts having square cross-sections with constant thickness and length. The cubic model itself and the obtained property-density relations are discussed in chapter 2.3.2.

With the expansion of the computational power, more accurate models based on realistic morphological descriptions are developed. As a first attempt to increasing complexity, some literature models already consider further morphological features such as the ratio of the solid material located in the cell walls and cell struts, respectively (see parameter ∅ in the standard Gibson and Ashby model discussed in chapter 2.3.4). As an example, using FEM models with Kelvin unit cells, Mills [16] suggested that at the same foam density, samples with higher concentration of solid material in the cell walls show stronger mechanical response. In another work, Mills [87] tried to capture the effect of solid material distribution on the mechanical

As a next step to increasing complexity, models based on tetrakaidecahedron cell shapes were proposed [88–92]. These models are more capable of representing a realistic definition of the cellular structure and therefore yielding better results compared to cubic models [15]. For example, Subramaniam [93] looked at the effect of varying strut (edge) thickness on the yielding and plastic behavior of foams (Figure 11). The results showed that assuming constant values for struts cross-sectional areas will result in overestimation of the yield strength, as in reality failure of the weakest (thinnest) strut sections happen at lower stress levels. One main conclusion was that an accurate description of the strut thickness variation along the strut length is essential for precise predictions of the foam’s stiffness and strength.

Figure 11 Demonstration of strut-thickenning effect in foams (reproduced from [93]).

Pierron [94, 95] has also worked on similar local cell deformation and collapse caused by the local variation of the cell morphology. In that work, effort is made to improve the predictions of the compression response of elastic PU foams using optical deformation measurements. By measuring and mapping the local deformation patterns, correlations were made between local cell morphology and the resulting local collapse modes. As advised by Li [96], experimental measurements at different scales (e.g. global and local) should be coupled with the analytical approaches to yield realistic models capable of correctly describing the foam behavior.

According to a more recent work by Ashby [12], cellular structures and lattices can be generally divided to two main categories, based on their principle cell deformation responses: Bending-dominated and stretch-dominated structures. The distinction made between the two types of structures and the understanding about their mechanical responses will be extremely important for processing the data obtained in this thesis. In the results section, it is illustrated that strand PET foams can have a combination of both bending- and stretch-dominated responses, when

loaded in different directions. Therefore, in the following chapters both deformation responses are reviwed in more details.