On the effective properties of foams in the framework of the couple stress theory
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O R I G I NA L A RT I C L E
Andrzej Skrzat
· Victor A. Eremeyev
On the effective properties of foams in the framework of the couple stress theory
Received: 12 November 2019 / Accepted: 12 March 2020 © The Author(s) 2020
Abstract In the framework of the couple stress theory, we discuss the effective elastic properties of a metal open-cell foam. In this theory, we have the couple stress tensor, but the microrotations are fully described by displacements. To this end, we performed calculations for a representative volume element which give the matrices of elastic moduli relating stress and stress tensors with strain and microcurvature tensors. Keywords Foam · Effective properties · Couple stress theory · Anisotropy
1 Introduction Polymer and metal foams have rather perspective properties and widely used in the engineering, see, e.g., [1–5]. From the mechanical point of view, foams may demonstrate quite unusual behavior. It is worth to mention that the re-entrant foams have negative Poisson’s ratio [6] and belong to the class of auxetic materials [7]. As an open-cell foam can be treated as a beam lattice where bending deformations of struts may play a dominant role, it is natural to consider it as a material with couple stresses. In fact, foams are treated as an example of Cosserat-type materials. In order to find the micropolar material parameters, the straightforward experiments on foams were performed in [8–13]. Let us note that for generalized continua such as media with couple stresses, the determination of material parameters used in constitutive equations constitute a rather complex and important problem. Even in the case of classic (Cauchy–type) materials, this problem was stated as a crucial one by Truesdell, see [14,15]. In addition to aforementioned experimental results, we mentioned here the various homogenization techniques. In fact, the homogenization of composite materials with high contrast in properties may lead to generalized models of continuum, such as micropolar and micromorphic media, or strain- and stress-gradient elasticity, Communicated by Andreas Öchsner. Victor A. Eremeyev acknowledges the support of the Government of the Russian Federation (Contract No. 14.Z50.31.0046). A. Skrzat (B) Department of Material Forming and Processing, Faculty of Mechanical Engineering and Aeronautics, Politechnika Rzeszowska im. Ignacego Lukasiewicza, al. Powstanców Warszawy 12, 35-959 Rzeszow, Poland E-mail: [email protected] V. A. Eremeyev Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gda´nsk University of Technology, ul. Gabriela Narutowicza 11/12, 80-233 Gda´nsk, Poland E-mail: [email protected] V. A. Eremeyev Don State Technical University, Gagarina sq., 1, Rostov-on-Don, Russia 344000
A. Skrzat, V. A. Eremeyev
see, e.g., [16–28], and the references therein. Even in the case of isotropic materials undergoing infinitesimal deformations, the constitutive equations contain a lot of elastic parameters. For example, within the linear micropo
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