F.E. simulations of closed-cells foams under extreme compression

The mechanical response of isotropic elastomeric closed-cell foams with Classical Voronoi microstructures is examined through three-dimensional finite element simulations. The study focuses on finite-strain uniaxial compression, complemented by infinitesimal-strain shear and hydrostatic loading. Random foam geometries with porosities between 77% and 95% and narrow pore-size distributions were generated by subtracting irregular Voronoi polyhedra, constructed from well-dispersed seed points, from a cubic solid domain. Large-deformation uniaxial compression, reaching nominal strains as high as 95%, was simulated using the Abaqus/Explicit solver.

At small strains, the effective shear modulus of these highly porous foams is accurately predicted by the Differential Hollow Sphere Assemblage (DHSA) model, extending its applicability beyond previously studied porosity ranges. However, the same framework was found to overestimate the effective bulk modulus. To address this limitation, a phenomenological correction was proposed, providing a simple yet reliable estimate of the bulk modulus for highly porous elastomeric foams.

Considering a neo-Hookean polymer matrix, a parametric analysis revealed that the macroscopic compressive response scales with the structural parameter (1−c)^2, where c denotes the porosity. Moreover, when stresses are normalized by ((1-c)^2 x shear modulus of the polymer), all compressive stress–strain curves collapse onto a single master curve.

Normalized compressive stress response with respect to apply strain of closed-cell foams for porosity from 77 to 92%.

At high porosity levels (around 90%), the deformation is primarily accommodated by pore collapse, and the polymer matrix remains close to the small-strain regime. As a result, the foam response is governed almost exclusively by the initial shear modulus of the base polymer, even when considering strain-hardening strain energy for the polymer matrix. In contrast, at lower porosities (around 77%), matrix deformation becomes more pronounced, and the influence of the polymer’s large-strain hyperelastic behavior becomes significant.

The contribution of gas trapped within closed cells was also investigated numerically and compared with experimental observations. The results support the analytical treatment proposed by Gibson and Ashby, confirming that internal gas pressure can substantially stiffen the compressive response of closed-cell foams. Although gas pressure could only be incorporated numerically in the fraction of pores that remained closed, its effect on the overall response was clearly evidenced.

The numerical framework developed in this work provides a robust basis for studying the multiscale mechanics of soft porous materials representative of foams produced by autoclave processing. As a natural extension of this study, and the focus of Part II, additional open- and closed-cell microstructures will be considered to evaluate the generality of the observed scaling laws and to further elucidate the role of microstructural features on macroscopic mechanical behavior.

T. Merlette, J. Diani, 2026. 3D Finite element investigation of hyperelastic foam behavior. I. Voronoi closed-cell microstructures. Mechanics of Materials, 214, 105566, https://doi.org/10.1016/j.mechmat.2025.105566.