M.A. Fortes

Stress and strain in liquid and solid foams

Both liquid and solid foams, together with analogous cellular materials, have distinctive mechanical or rheological properties which find many applications. This review concentrates on the search for a basic understanding of the underlying mechanisms, in terms of specific structural models. Computer simulations play an increasing role and are beginning to be applied to three-dimensional models.

The effect of non-uniformity on the in-plane modulus of honeycombs

An approximate analysis is presented of the Youngs modulus of two-dimensional (isotropic) cellular materials with a distribution of cell dimensions and cell wall thicknesses. Two approaches are used to estimate the modulus: one is based on an energy balance and the other involves a separate calculation of stress and strain. The estimated moduli depend on the moments of the distributions of wall widths and wall thicknesses and are affected by an eventual correlation between the two distributions. Actual values for three-connected foams are calculated for particular distributions.

A general approach to grain growth driven by energy density differences

Abstract A general approach to grain growth driven by energy density differences among the grains, associated with curvature and/or extraneous driving forces (e.g. external fields) is developed. A mean field approximation leads to the definition of a threshold energy density, E ∗ , which depends on moments of the current distributions of grain diameters, a, and grain energy densities, E, such that the rate of change, da / dt , of the diameter of a grain is proportional to ( E ∗ –E ). When curvature effects are negligible, the kinetic equations can be solved analytically to obtain the average grain diameter a > and the diameter distribution as a function of time. A scaling regime is reached for (and only for) power law distributions of E , with a power law kinetics a >∝ t μ . The combined effects of curvature and extraneous driving forces were studied numerically. The rate of growth and the width of the grain diameter distribution change when compared to pure curvature growth, the sign of the change depending on the initial energy density and diameter distributions.

Instabilities in liquid foams

Instabilities play a central role in the physics of foams. Some that change the topology of a dry foam are indicated by the laws promulgated by Plateau in his 1873 book. Their occurrence is less clearly predictable in wet foams. Various other instabilities are related to gravitational loading and gas compressibility. We gather up many examples as a guide to future research and identify problems that remain, including what we call instabilities, which occur before they are expected on the basis of Plateaus laws.

The contact mechanics of cellular solids

A model for the contact of a cellular solid with a compact counter-surface is developed. The cell walls and edges at the surface of the cellular solid deform elastically by compression and bending and, at larger forces, collapse. For open cell materials, a simple general equation is derived relating the fraction f of cell edges in contact to the applied compressive stress σ. The ratio σ/f is proportional to an average strain of the cell struts in contact, which is calculated in particular cases. A geometrical construction is described with which the fraction f can be obtained as a function of the applied stress σ. The analysis of the contact of closed cell materials is also undertaken, leading to relations between the applied stress and the fractions of edges and walls in contact.

Nucleation and glide of dislocations in a monodisperse two-dimensional foam under uniaxial deformation

Abstract Experimental observations of the deformation of a monodisperse two-dimensional (2D) monolayer foam in tension/compression are reported. The foam samples are bound by two parallel walls at a variable separation, w. At critical values of w, dislocations (5/7 pairs) nucleate at the periphery of the foam which then glide along directions at 60° to the walls. The dislocations may suffer reflection at the walls, with a change in Burgers vector. As a result of the glide process, the number of close-packed rows of cells parallel to the walls changes by one and a neck develops. Rearrangements of the bubbles to more stable configurations are also observed, following dislocation glide. A detailed analysis of the topology of nucleation, glide and reflection of dislocation is undertaken. The stress—strain relation is derived and used to calculate the yield stress of the honeycomb foam from the experimentally measured strains at which topological transitions occur. The yield stress is considerably lowered by t...

Bubble size-topology correlations in two-dimensional foams derived from surface energy minimization

We obtain the number fractions xi, the average areas {A}i and the average perimeters {P}i of i-sided bubbles in a two-dimensional foam by minimizing the total surface energy and assuming a simple relation between {P}i and {√A}i. Calculations for linear and Weibull distributions of the square root of the bubble area yield large deviations, particularly at small i, from Lewiss and Deschs laws, which linearly relate {A}i and {P}i, respectively, to i. Nevertheless, we find good agreement with experimental results for xi, {P}i and {A}i in foams.

Surface energy of free clusters of bubbles: an estimation

We propose an approximate equation for the surface energy of two-dimensional free bubble clusters which we compare with exact calculations of the surface energy of symmetrical clusters consisting of a central bubble surrounded by one or two shells of bubbles of two different areas. The accuracy of the equation is good for relatively narrow distributions of the areas and of the number of sides of the bubbles but underestimates the energy for large widths of those distributions. We propose a similar approximate equation for the surface energy of three-dimensional clusters.

Pressures inside bubbles in planar clusters

We provide theoretical estimates and undertake Surface Evolver experiments on the pressures inside bubbles in planar clusters. The equation of equilibrium implies that as the number, N, of unit bubbles become large, the average normalized pressure in an energy-minimizing cluster approaches 2−1/231/4 ≈ 0.9306. An equation is derived for the rigorous theoretical upper and lower bounds on the average pressure in terms of N. Surface Evolver experiments agree with these estimates.

Applicability of Aboav's rule to a three-dimensional Voronoi partition

Abstract Data on mF , the average number of faces in cells adjacent to cells with F faces, obtained by Kumar, Kurtz, Banavas and Sharma in 1992 for a three-dimensional (3D) Poisson-Voronoi partition can be fitted to a linear relation with 1/F. This suggests that Aboavs rule, valid for two-dimensional networks, is likely to be also of general applicability to 3D random tetravalent networks, for example polycrystals and soap froths.