Accepted Manuscript Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids Holger Mitschke, Fabian Schury, Klaus Mecke, Fabian Wein, Michael Stingl, Gerd E. Schroder-Turk¬ PII: S0020-7683(16)30142-1 DOI: 10.1016/j.ijsolstr.2016.06.027 Reference: SAS 9206 To appear in: International Journal of Solids and Structures Received date: 14 December 2015 Accepted date: 21 June 2016 Please cite this article as: Holger Mitschke, Fabian Schury, Klaus Mecke, Fabian Wein, Michael Stingl, Gerd E. Schroder-Turk,¬ Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.06.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights • Broad comparison of the deformation of cellular solids with mechanisms of frameworks. • Effective Poisson’s ratios agree within the regime of bending-dominated cellular solids. • Geometric framework model emphasizes the importance of morphology for elastic properties. • Determination of infinitesimal framework mechanisms offers versatile microstructure design tool. ACCEPTED MANUSCRIPT 1 ACCEPTED MANUSCRIPT Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids Holger Mitschkea, Fabian Schuryb, Klaus Meckea, Fabian Weinb, Michael Stinglb, Gerd E. Schröder-Turkc aTheoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7B, 91058 Erlangen, Germany bDepartment Mathematik, Applied Mathematics 2, Friedrich-Alexander-Universität Erlangen-Nürnberg cMurdoch University, School of Engineering & Information Technology, Mathematics & Statistics, Murdoch WA 6150, Australia Abstract Control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative Poisson’s ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of bar-and-joint-frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson’s ratio of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending- dominated deformation with negligible dependence of the effective Poisson’s ratio on the underlying material’s Poisson’s ratio and small values of the effective Young’s modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending- dominated deformation mode, the effective Poisson’s ratio is material-dependent and the ˜ Young’s modulus Ecs large. All analysed structures of this type have positive values of ˜ the Poisson’s ratio and large values of Ecs. Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These results clarify the relationship between mechanical properties of a linear- elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young’s modulus. Keywords:ACCEPTEDcellular structures, homogenisation, MANUSCRIPT floppy frameworks, auxeticity, mechanical metamaterial Email address: [email protected] (Holger Mitschke) Preprint submitted to Int. J. Solids Struct. 22nd June 2016 ACCEPTED MANUSCRIPT 1. Introduction Geometry, topology and spatial structure play a crucial role for the deformation behaviour and mechanical properties of cellular materials. This observation underlies the very concept of mechanical meta-materials (Zheng et al., 2014; Florijn et al., 2014; Bückmann et al., 2014; Kadic et al., 2012; Bertoldi et al., 2010; Overvelde et al., 2012), has been recognised by classic work summarised in the book of Gibson and Ashby (1997) or Gibson et al. (2010) and is relevant for the design of actuating or buckling structures (Wicks and Guest, 2004; Guiducci et al., 2015). Cellular solids are two-phase materials, consisting of a solid phase occupying a volume fraction φ of the total space and its complement, the void phase (φ is synonymous for relative density ρ˜ ρs). Many incarnations of cellular materials are possible, including closed-cell foams where the void phase consists of individual hollow cavities, open-cell foams where the solid phase resembles~ a spatial network, etc. Typically, one distinguishes cellular solids from other porous materials by the volume fraction, which is often considered φ 30% (Gibson and Ashby, 1997). While many realisations of cellular solids have a disordered spatial structure, cellular solids with an ordered structure also occur, e.g. honeycombs. The≲ mechanical response of cellular solids to imposed strains depends on a multitude of parameters, including those describing geometric aspects of the spatial structure but also those describing properties of the material constituting the solid phase. This concept is formalised by so-called effective mechanical properties, that is, the mechanical properties of samples of the material with dimension much larger than the microstructure (Torquato, 2002; Milton, 2002). The significant dependence on spatial structure and geometry is recognised in the scaling laws of mechanical properties with volume fraction φ, and their dependence on the type and topology of the spatial structure, see the books by Gibson and Ashby (1997); Gibson et al. (2010); Ashby et al. (2000) or the publication Ashby (2006); note the cross-over behaviour from these scaling laws to the predictions of effective medium theory for low-porosity porous materials (Nachtrab et al., 2011). Similarly, the essential distinction of ‘bending-dominated’ or ‘stretching- dominated’ behaviour reflects differences in spatial structure of the material (Ashby, 2006). Other examples for the dependence on geometry are the relevance on details of the structure, e.g. for low-density open-cell foams the cross-sectional shape (Jang et al., 2008). Despite the significant role of geometry for the effective mechanical properties, the properties of the constituent material also affect the effective mechanical response. We here denote the effective ˜ Poisson’s ratio as ν˜cs and the effective Young’s modulus as Ecs whereas the linear-elastic solid material properties of the solid phase are denoted as νs and Es. The indices ‘s’ and ‘cs’ are the initials of solid and cellular solid. The constituent solid material is considered isotropic and homogeneous and hence, characterised by these two material constants. The effective properties need not to be isotropic with the possibility of different in-plane values 12 ACCEPTED21 1 2 MANUSCRIPT ˜ ˜ 1 ν˜cs , ν˜cs , Ecs , and Ecs for different directions of the applied strain. In general, the ( ) ( ) ( ) ( ) ˜ response of the effective properties ν˜cs and Ecs to changes of the material properties νs and 1Note that for orthotropic symmetry, e.g. honeycomb family in Fig. 3, the number of independent in-plane 12 ˜ 1 21 ˜ 2 moduli is three with the reciprocal relation ν˜cs Ecs ν˜cs Ecs (Gibson and Ashby, 1997). ( ) ( ) ( ) ( ) 3= ACCEPTED MANUSCRIPT Es is not a priori a simple relationship — as we explore below. In particular for biological cellular materials, an additional materials dependence may arise if the constituent material is not isotropic which, for example, is known for wood (Cave, 1968). The structure of cellular materials where the solid phase resembles a network-like structure with edges that are joined at common vertices can be represented by bar-and-joint framework models, also known as bar/strut framework models (Torquato and Stillinger, 2010, Connelly and Whiteley, 1996; therein considered as a subclass of tensegrity frameworks without cables and extendable strut members), pin-jointed frameworks (Pellegrino and Calladine, 1986), (pin-jointed) trusses (Hutchinson and Fleck, 2005) or skeletal structures (Laman, 1970). In this publication bar-and-joint framework models are simply referred to as frameworks. These are composed of stiff edges of fixed length that pivot freely at common vertices, see Fig. 1 (a). All vertex displacements that preserve all edge lengths are allowed deformation modes of these structures. Various aspects of these models have been studied, in particular for periodic and/or symmetric systems (Kangwai et al., 1999; Guest and Hutchinson, 2003; Connelly et al., 2009; Ross et al., 2011; Treacy et al., 2014), and disordered systems (Plischke, 2007; Moukarzel, 2012; Thorpe et al., 2002). Constraint counting rules such as the famous Maxwell (1864) counting rule can give conditions