The effective Poisson ratio of random cellular matter having bending dominated architecture

We argue that the effective Poisson ratio of cellular and porous solids is independent of the material of the solid phase, if the mechanism of the cell wall deformation is dominated by beam bending —thus rendering it to be a purely kinematic quantity. Introducing a kinematic simplification and requiring statistical isotropy, we prove a result of remarkable generality that the effective Poisson ratio of irregular planar structures equals 1 for all bending dominated random architectures. We then explore a deeper connection of this behavior with area-preserving deformation of planar closed elastic cells. We show that thin sheets and films made of such microstructured material afford physical realizations of the two-dimensional analogue of incompressible matter. We term such non-stretchable sheet material as well as deformations as isoektasic. Copyright c © EPLA, 2009 Introduction. – Natural and synthetic porous materials such as bone, wood, metal foams [1], biological soft matter [2,3], and optical metamaterial have recently inspired many studies, relating structure to mechanical properties [4,5]. These studies include numerical [6] or laboratory experiments [7], analyses for regular lattices [8,9], the effect of non-homogeneity [10], or theoretical properties bounds [11,12]. Planar architectures are often called “honeycombs” whereas 3D structures are termed as solid “foams”. We assume the cell walls to be made of homogeneous isotropic material, having Young’s modulus E and Poisson’s ratio ν. When remote stress is applied, the cell walls deform, resulting in bulk elastic response of the material which manifests as two effective elastic moduli Ē and ν̄. A majority of solids possess Poisson’s ratio in the range 0.2 to 0.5, the theoretical limit for isotropic continuum being 0.5. Negative values of Poisson’s ratio are not ruled out theoretically; however, physical realizations of negative Poisson’s ratio material had to wait until the discovery of certain microstructural architectures that are usually attributed to “re-entrant corners” present in cells having non-convex shapes [13]. Such materials are also known as auxetic. Complex cellular geometries such as those in fig. 1 are not amenable to exact analysis, experimentation can (a)E-mail: [email protected] Fig. 1: A micrograph of bone uniformly expanded to three different hypothetical characteristic pore size (courtesy Alan Boyde, Queen Mary University, London). provide trends, and detailed computer analyses often obscure general understanding, despite being useful. In contrast, dimensional and scaling arguments are often simple, yet effective [14,15]. We express the effective Young modulus in the general functional form: Ē = f(E, ν, geometry). What role does the microstructural size have in determining the effective elastic properties and, for example, which of the three microstructures in fig. 1 will have the highest Young modulus? If the geometry of the microstructure is characterized by n parameters, the non-dimensional groups in this functional relationship are Ē/E, ν, and (n− 1) nondimensional parameters that describe the shape (e.g. the ratios of the length parameters or angles) because of