Mesostructural origins of the anisotropic compressive properties of low-density closed-cell foams: A deeper understanding

TL;DR

This study advances understanding of anisotropic compression in low-density closed-cell foams by linking mesostructure to mechanical response through representative volume elements with Reissner–Mindlin shells and a mixed stress–strain homogenization scheme. It reveals that membrane deformation dominates the initial elastic region, while cell-wall bending becomes critical post-buckling, followed by membrane yielding, and identifies three pathways—load-bearing area fraction, buckling strength, and wall inclination—that translate cell-shape anisotropy into elastic and strength anisotropy. Analytical models for rectangular and Kelvin cells, validated against tessellation-based simulations, show that cell-shape stochasticity, especially in wall orientation, strongly governs strength anisotropy, often outweighing the effects of cell size or wall thickness. The tessellation-based models, incorporating stochastic distributions of cell size, thickness, and shape, reproduce qualitative and quantitative trends in experiments, underscoring the need to move beyond traditional Gibson–Ashby and Sullivan frameworks for high cell-face fraction foams and providing design guidance for tailoring anisotropy in foam-based lattices.

Abstract

Many closed-cell foams exhibit an elongated cell shape in the foam rise direction, resulting in anisotropic compressive properties. Nevertheless, the underlying deformation mechanisms and how cell shape anisotropy induces this mechanical anisotropy are not yet fully understood, in particular for the foams with a high cell face fraction and low relative density. Moreover, the impacts of mesostructural stochastics are often overlooked. This contribution conducts a systematic numerical study on the anisotropic compressive behaviour of low-density closed-cell foams, which accounts for cell shape anisotropy, cell structure and different mesostructural stochastics. Representative volume elements (RVE) of foam mesostructures are modeled, with cell walls described as Reissner-Mindlin shells in a finite rotation setting. A mixed stress-strain driven homogenization scheme is introduced, which allows for enforcing an overall uniaxial stress state. Quantitative analysis of the cell wall deformation behaviour confirms the dominant role of membrane deformation in the initial elastic region, while the bending contribution gets important only after buckling, followed by membrane yielding. Based on the identified deformation mechanisms, analytical models are developed that relates mechanical anisotropy to cell shape anisotropy. It is found that cell shape anisotropy translates into the anisotropy of compressive properties through three pathways, cell load-bearing area fraction, cell wall buckling strength and cell wall inclination angle. Besides, the resulting mechanical anisotropy is strongly affected by the cell shape anisotropy stochastics while almost insensitive to the cell size and cell wall thickness stochastics. The present findings provide deeper insights into the relationships between the anisotropic compressive properties and mesostructures of low-density closed-cell foams.

Figures (27)

• Figure 1: Examples of closed-cell foam mesostructures made from different base materials: (a) aluminium, (b) polyvinylklorid (PVC) and (c) polyisocyanurate (PIR) foams. The red arrow indicates the foam rise direction. Reproduced from Mu2010, Zhou2023a and Andersons2016, respectively, with permission from Elsevier.
• Figure 2: A foam mesostructural RVE with cell walls described as shell continuum: (a) initial to current configurations after imposing the macroscale stress $\hat{\mathbf{P}}$ and deformation gradient $\hat{\mathbf{F}}$ in a mixed manner. $(\bullet)^*$ indicates a quantity with its components partially prescribed. The space-filling volume domain including voids are indicated by the green shadows; decomposition of the mesoscale mid-surface displacement field $\vec{u}_\text{\scriptsize r}$ into the (b) trend field ${ \hat{\newline {\vec{\boldsymbol{u}}}}\,\;}_\text{\scriptsize r}$ and (c) fluctuation field $\vec{w}_\text{\scriptsize r}$. The rotation angle field $\vec{\theta}$ is not visible.
• Figure 3: Sketches of the (a) strain energy partitioning indicator $\mathcal{I}_\text{\scriptsize w}$, (b) strain energy partitioning indicator rate $\dot{\mathcal{I}}_\text{\scriptsize w}$ and (c) membrane plasticity indicator $\mathcal{J}_\text{\scriptsize w}$ versus time, for a probed cell wall. The buckling and membrane yielding events are indicated by the black triangles and circles, respectively.
• Figure 4: Geometrical model configurations of different foam mesostructural RVE: (a-b) idealized cell-based models for two shape anisotropy. All cell walls are assigned with a constant thickness; (c-d) tessellation-based models. "StSt" accounts for the stochastic variations of both cell size and cell wall thickness. "StCt" accounts for the cell size stochastics while assigning a constant thickness to all cell walls. "CtCt" assigns a constant equivalent diameter to all cells and a constant thickness to all cell walls.
• Figure 5: Effective responses of the rectangular parallelepiped cell-based model with $\mathcal{R}=1.5$, under uniaxial compression in the transverse ($\vec{e}_1$) and foam rise ($\vec{e}_3$) directions: (a) stress and (b) strain energy fraction versus applied strain. The yield points are indicated in (a) by the black crosses.