analytically calculating effective elastic properties of certain types of polymeric foams. Structural anisotropy in two special microstructures considered is found to ...
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Negative and conventional Poisson’s ratios of polymeric networks with special microstructures Gaoyuan Wei Theory of CondensedMatter, CavendishLaboratov, Madingley Road, Cambridge CB3 OHE, United Kingdom (Received 9 September 199 1; accepted 4 November 199 1) A theoretical model for evaluating effective Poisson’s ratios of polymeric networks with special microstructures has been developed, which takes into account both stretching and bending deformation mechanisms. It is a complete generalization of the formalism first presented by Warren and Kraynik [Mech. Mater. 6,27 (1987) and J. Appl. Mech. 55,341 ( 1988)] for analytically calculating effective elastic properties of certain types of polymeric foams. Structural anisotropy in two special microstructures considered is found to have very significant effects on both magnitude and sign of the effective Poisson’s ratios of the macroscopically isotropic network materials containing them. As two limiting cases, the effective Poisson’s ratios of random assemblies of 2D and 3D bars or rods are shown to be l/3 and l/4, respectively, the latter of which recovers the well-known Poisson’s result.
INTRODUCTION Two recent ca.seslp2of successful fabrications of manmade polymeric materials to exhibit negative Poisson’s ratios have brought about much excitement in the scientific community. 3-5 It is well known that a negative Poisson ratio has, for a long time, been treated as an elastic abnormaly6-27 because of the fact that most naturally occuring materials have positive Poisson ratios 2s-30with rubbers possessing values of the ratio close to l/2-the largest positive number possible for an incompressible, isotropic, 3D object in the linear deformation region. However, all negative Poisson ratio materials, recently termed auxetics or auxetic materialq5 which have been known to mankind so far, have not been well understood at the atomic or molecular level in spite of the fact that the very first appearance of the ratio was in an atomic or molecular model of elastic bars or random assemblies of spheres involving only central forces.31-33 Historically, negative Poisson ratios were first discovered in the calculations of elastic constants of a number of minerals, e.g., iron pyrites, based on experiments on the twisting and bending of rods6 and theories of Voigt34 and Reuss35 (see also Refs. 36 and 37), with the principal Poisson’s ratio being nearly - l/7 for pyrites with cubic symmetry, though recent experiments2’ have shown otherwise. For an aggregate of hexagonal crystals, Li” found that negative values of Poisson’s ratio appear in some directions for cadmium (Cd), while Garber,’ more than a decade earlier, reported that a refractory structural material called pyrolytic graphite in the space technology field exhibits negative Poisson ratios in a certain direction. In 1970, a transient negative Poisson ratio was seen in thin magnetized ferromagnetic films as they age.* More recent findings of negative Poisson ratios are in composite laminates.“*14 polymeric foams,1.12 cancellous or porous bone in the proximal epiphysis of the human tibia, l3 expanded polytetrafluoroethylene in microporous, anisotropic form2 rocks with microcracks, and poly (N-isopropylacrylamide)/water gels near the volume phase transition.22
Formal theoretical investigations of negative Poisson ratios seem to have begun with a 3D structure and its 2D analog of rods, hinges, and springs which exhibit a Poisson ratio of - 1 as proposed and studied by Almgren. l5 Wojciechowski, together with Branka, investigated both theoretically and by Monte Carlo simulation negative Poisson ratios in a two-dimensional molecular system of hard cyclic hexamers in the high-density phase, i.e., with a tilt of the molecular axes relative to the crystalline ones,” while Jaric and Mohanty16 found, based on a density-functional theory, that the nonaffine character of deformation below the unit-cell scale for a hard sphere solid yields a negative Poisson ratioa result that was later partially supported by the Monte Carlo determination performed by Runge and Chester. *’ In 1988, Gibson and Ashby12 used a theoretical model developed for describing mechanics of 2D honeycombs to show that a re-entrant honeycomb (see Fig. 1 in Refs. 15 and 26 for illustrations) exhibits a negative Poisson’s ratio. Further theoretical and computational work along this line can be found in Refs. 5, 23, and 24, and a variety of tensile microstructures involving anisotropic or isotropic particles, tensile springs, and topologically constraining rods or strings, and one structure which contains an internal degree of freedom have been modeled by Evans’ that have negative Poisson’s ratios. Finally, two theoretical attempts at relating effective Poisson ratios of a random isotropic granular system” and a transversely isotropic foam structure2s.26 to their respective microstructures deserve special mention here as they appear to have been the first examples of calculating effective or principal or average Poisson ratios for noncrystalline materials though much of the analysis bear some resemblance to that used in the calculations of elastic constants for single-crystal materials (see, e.g., Ref. 10) and in the theory of transport phenomena associated with spatially periodic media.38‘42 This paper is intended as a complete generalization of the Warren-Kraynik-Warren approach and the associated microstructures25v26 applied to both existing and a novel
J. Chem. Phys. 96 (4), 15 February 1992 3226 0021-9606/92/043226-08$06.00 @ 1992 American Institute of Physics Downloaded 12 Aug 2004 to 129.234.4.76. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Gaoyuan Wei: Poisson’s ratios of polymers ____________________________________________________________________________ 中国科技论文在线 www.paper.edu.cn
class of polymeric networks, the latter of which are now under experimental investigations to show negative Poisson ratio effects.43 It must be emphasized that the theoretical approach adopted here is significally different from those of Gibson and Ashby,” Evans,2 and Evans et al5 in that polymeric networks containing certain microstructures are assumed to be spatially nonperiodic. In other words, these microstructures may be randomly oriented in the networks. In the following, a unified formalism is first presented for two particular planar and three-dimensional network microstructures, which involves first writing down the set of forcebalancing equations for each microstructure; second, identifying the six distinct elements of the symmetric stress tensor in terms of the forces and the surface areas so chosen that each of their normals is parallel to a coordinate axis shown in Fig. 1 or 2; third, averaging six new stress components in a rotated coordinate frame, produced by a uniaxial (principal) strain along one of the three rotated axes, over all possible orientations with respect to the original coordinate system; and finally expressing Poisson ratio in terms of the ratio of the two nonvanishing averaged or effective stress components based on classical elasticity theory for isotropic planar and 3D materials. Simple analytic results relating the so calculated effective Poisson ratios to the architecture and the stiffness of the network materials for several selected cases then follow. Finally, discussions of the results obtained and possible extensions of the theorectical approach presented are given together with a few conclusions. A computation and graphic representation-oriented paper44 will follow this work, which will be devoted entirely to the tailoring of negative Poisson ratios.
FIG. 1. A basic 2D microstructure in polymeric networks-two rodlike segments,each of length L, called arm, and one of length H, backbone, with the ratio L/H denoted by k, meeting at one point calledjunction. The angle between each arm and its projection on the plane perpendicular to the backbone and containing the junction is 10 I
