1 Hierarchical auxetic and isotropic porous medium with extremely negative Poissons ratio

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Hierarchical auxetic and isotropic porous medium with extremely negative Poisson's

ratio

Maryam Morvaridi1, Giorgio Carta2, Federico Bosia1, Antonio S. Gliozzi1, Nicola M. Pugno3,4,

Diego Misseroni3,(*), Michele Brun2,(**)

1Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli

Abruzzi 24, 10129 Torino, Italy

2Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari,

Piazza d’Armi, 09123 Cagliari, Italy

3 Laboratory of Bio-inspired, Bionic, Nano, Meta Materials & Mechanics, Department of Civil,

Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123

Trento, Italy

4 School of Engineering and Materials Science, Queen Mary University of London, Mile End

Road, London E1 4NS, UK

(*) diego.misseroni@unitn.it; (**) mbrun@unica.it

Abstract

We propose a novel two-dimensional hierarchical auxetic structure, consisting of a porous

medium in which a homogeneous matrix includes a rank-two set of cuts characterised by

different scales. The six-fold symmetry of the perforations makes the medium isotropic in the

plane. Remarkably, the mesoscale interaction between the first- and second-level cuts enables

the attainment of a very small value of the Poisson’s ratio, close to the minimum reachable

limit of -1. The effective properties of the hierarchical auxetic structure are determined

numerically, considering both a unit cell with periodic boundary conditions and a finite

structure containing a large number of repeating cells. Further, results of the numerical study

are validated experimentally on a polymeric specimen with appropriately arranged rank-two

cuts, tested under uniaxial tension. We envisage that the proposed hierarchical design can be

useful in numerous engineering applications exploiting an extreme auxetic effect.

Keywords: Auxetic metamaterial, negative Poisson’s ratio, hierarchical structure, porous medium,

experimental validation.

Introduction

Auxetic materials are characterised by the unconventional property of possessing an effective

negative Poisson’s ratio, so that they expand (contract) transversally when stretched

(compressed) longitudinally.

Named “auxetic” after Evans [1], these media owe their special behaviour mainly to their

microstructure rather than to their chemical composition. Hence, auxeticity has been observed

at different scales, from macro- to nano-dimensions. Apart from examples of natural materials

with an intrinsic negative Poisson’s coefficient [2-5], auxetic media are generally artificially-

made systems whose microstructure is designed by exploiting different geometries and

mechanisms: re-entrant unit cells [6-9], star-shaped inclusions [10, 11], chiral configurations

[12-15], double arrowhead honeycombs [16], perforations and cuttings [17-21], rotating rigid

units [22], lattices [23-25] and elastic instabilities [26, 27]. Alternative approaches to design

auxetic media are presented in the reviews [28-32]. Some auxetic systems are also

characterised by a negative value of the coefficient of thermal expansion, implying that they

shrink when subjected to an increase in temperature [33-35]. Additionally, it has been shown

that it is also possible to achieve a smooth transition through a wide range of negative and

positive Poisson’s ratios by using an origami cell that morphs continuously between a Miura

mode and an eggbox mode [36-38].

The increasing interest of the scientific community in auxetic metamaterials is due to their

enhanced mechanical properties with respect to those of conventional materials, including

higher indentation resistance [39] and impact energy absorption abilities [40, 41], improved

fatigue performance [42] and pull-out strength [43], as well as the possibility to bend with

synclastic curvature [44]. For these reasons, auxetic metamaterials have tremendous potential

in many fields, particularly in the aviation industry, e.g. for aircraft design [43, 45, 46], in sports

applications for enhanced comfort and protection [47], in electronics to increase electric power

output [48] and in biomedical engineering for the design of novel types of stents [49-51] and

orthopedic implants [52]. On the other hand, zero Poisson’s ratio materials have been shown

to be useful in other applications, for instance, in the design of morphing aircraft [53-55].

Hierarchical structures are widely exploited in natural materials [56], and in bioinspired

artificial materials [57] to enhance mechanical properties. The simultaneous presence of

multiple length scales, together with material heterogeneity, has been shown to allow the

simultaneous optimization of strength and toughness [58, 59], but also to enable the

improvement of other mechanical characteristics such as adhesion [60, 61], friction [62], or to

achieve band gap engineering in dynamics [63, 64].

In this paper, we investigate the auxetic behaviour of a porous hierarchical medium, consisting

of a homogeneous elastic material containing two classes of perforations, characterised by two

different length scales and exhibiting a six-fold symmetry, which makes the medium isotropic

[65, 66]. First, by numerically studying the periodic elementary cell, we demonstrate that, for

some ratios between the lengths of the two classes of cuts, the effective Poisson’s ratio

approaches the limit of –1. Second, we verify the results of the periodic analysis by determining

the response of a finite model, containing a large number of periodic cells, under uniaxial

loading. Subsequently, we validate the numerical results by experimentally testing a specimen

with hierarchical cuts in uniaxial tension. Finally, we provide some analysis and concluding

remarks.

Numerical model

We introduce a 2-D hexagonal periodic unit cell with oriented cuts, as shown in Fig. 1a. The

cuts are all rotated by the same relative angle ϴ (ϴ-π/6 with respect to the intersected edge of

the hexagonal unit cell of size l). Each cut is of length a and width b, which is also the diameter

of the rounded extremities (in consideration of a practical realization). The proposed design

leads to a periodic pattern with either six-fold or three-fold symmetry, similar to previously

considered designs providing an isotropic auxetic response [19, 20].

Next, we construct a second-rank geometry by adding smaller cuts to the first, arranged

periodically in a hexagonal pattern of size  and in turn oriented at the same angle ϴ (Fig. 1b).

This entails the introduction of a second characteristic size scale in the system, leading to a so-

called “hierarchical” geometry. The transposition of these 2-D geometries in real 3-D structures

is implemented in thin sheets of thickness t, shown in the bottom panels of Fig.1.

Fig. 1: Top and isometric views of the considered unit cells: a) non-hierarchical rank-one hexagonal unit cell;

b) hierarchical rank-two unit cell, with additional second-level cuts.

We numerically evaluate the quasi-static behaviour under uniaxial tension of the two proposed

2-D geometries. For both of them we derive the “effective” properties as functions of the

geometric parameters. This is accomplished by performing a Finite Element (FE) analysis

using the commercial package COMSOL Multiphysics® (version 5.6). A state of plane stress is

assumed in the numerical simulations, and periodic boundary conditions are applied at the

edges of the unit cell. Geometrical parameters are taken as follows: cut length a = 10 mm, cut

width b = 1 mm, lengths of the hexagonal cells l = 9.00 mm and  = 2.25 mm, Young’s

modulus E= 2.285 GPa and Poisson’s coefficient

= 0.370.

The effective elastic properties of the structure are derived by applying macroscopic

longitudinal strains in the horizontal ( = 10) and vertical ( = 10) directions and the

macroscopic shear strain  = 10. Each of these macroscopic strains is applied to the unit

cell using periodic boundary conditions. Specifically, periodic boundary conditions link

displacements on edges having opposite outward normal unit vectors. Referring to Fig. 1, the

periodic boundary conditions satisfy the relations 

() 

()=

() 

(),

where = I, II, III while  are the displacements,  the positions and ,=,. In numerical

computations, additional constraints are imposed to prevent rigid-body motions. The

corresponding macroscopic Cauchy stress components ,  and  are evaluated

numerically as average values of the corresponding local components ,  and  on the

unit cell domains of the models, namely:

 =

||

 = (1 )

()+

()=(1 )

() ( =,,), (1)

where  is the unit cell domain, 

() and 

()= 0 are the average stresses in the solid (S) and

porous (P) phases, respectively, and p is the porosity.

From these values, it is possible to estimate the effective Young's modulus  and Poisson's

ratio  for plane stress conditions. From Hooke’s law, the effective properties of an isotropic

material are given by (see Appendix A)

 =





 , (2)

 =

 , (3)

 =

  =

( ) . (4)

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1HierarchicalauxeticandisotropicporousmediumwithextremelynegativePoisson'sratioMaryamMorvaridi1,GiorgioCarta2,FedericoBosia1,AntonioS.Gliozzi1,NicolaM.Pugno3,4,DiegoMisseroni3,(*),MicheleBrun2,(**)1DepartmentofAppliedScienceandTechnology,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10129Torino,Italy2...

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