548 Advances of Computational Mechanics in Australia

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1 Applied Mechanics and Materials Online: ISSN: , Vol. 846, pp doi: / Trans Tech Publications, Switzerland Geometric bounds for buckling-induced auxetic metamaterials undergoing large deformation Arash Ghaedizadeh 1,a, Jianhu Shen 1,b, Xin Ren 1,2,c and Yi Min Xie 1,d * 1 Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia 2 Key Laboratory of Traffic Safety on Track, School of Traffic & Transportation Engineering, Central South University, Changsha , Hunan Province, China a s @student.rmit.edu.au, b jianhu.shen@rmit.edu.au, c s @student.rmit.edu.au, d, * mike.xie@rmit.edu.au Keywords: Auxetic, Buckling, Elasticity, Densification strain, Energy efficiency method. Abstract. The performance of a metamaterial is dominated by the geometric features and deformation mechanisms of its microstructure. For a certain mechanism, the geometric features have bounds in which the performance of a metamaterial such as negative Poisson s ratio (NPR) can be designed. Previous investigation on buckling-induced auxetic metamaterial revealed that there is a geometric limit for its microstructure to exhibit auxetic behaviour in infinitesimal deformation. However, the limit for auxetic metamaterials undergoing large deformation is different from that under small deformation and has not been reported yet. In this paper, the geometric limit was investigated in an elastic and infinitesimal deformation range using linear buckling analysis. Furthermore, experimentally validated finite element models were used to identify the geometric limits for auxetic metamaterials undergoing large deformation. Depending on the control parameters of the topology, the bounds were represented by a line strip for one control parameter, an area for two control parameters and spatial domain surrounded by a 3D surface for three parameters. The limit was determined by the shape and size of the void of the metamaterials and it was identified through the large deformation analysis as well as the linear buckling analysis. We found that there was a significant difference in the geometric bounds obtained through those two methods. The results from this study can be used to design an auxetic metamaterial for different applications and to control the auxetic performance. Introduction During the past decades several different studies have been carried out on porous metamaterials which exhibit anomalous properties [1]. Nowadays through modeling and experimental studies, there is a better understanding of what cause the rise of these novel features and how they can be applicable to utilize in the large variety of applications including isolation, structural protection, energy absorption and light weight structures [1-8]. It is found that using the elastic instability will take the advantage of the nonlinearly to raise the applications of buckling induced metamaterials with specific properties such as auxetic behaviour, negative compressibility and tunable acoustic and optical reflectivity [5]. Auxetic metamaterial is a typical class of metamaterials and is designed rationally based on experience [2]. The auxetic behaviour is defined as a scale-independent property. Unlike to majority of materials with positive Poisson ratio, auxetic material gets smaller in the transverse direction under a range of compressive strain or exhibits a lateral expansion under axial tension [5, 7, 9, 11]. Previous results on deformation pattern of buckling-induced auxetic metamaterials under compression showed that the NPR behaviour originated from the localization of buckling mode which triggered a global change in the deformation pattern [9]. The buckling in walls of unit building cell led to alternate rotation of rigid parts and suddenly produced a square of ellipses with horizontal and vertical parallel axes [4]. The relation between architecture of microstructure and the nonlinear stress-strain behaviour of cellular metamaterials have been an open All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , RMIT University, Melbourne, Australia-08/07/16,08:59:08)

2 548 Advances of Computational Mechanics in Australia research area for many researchers [4, 5, 11, 2]. Also it has been investigated that the size, arrangement and the shape of void parts had a significant effect on stability and global response of buckling-induced cellular metamaterials [4, 11, 2]. It was found that there was a geometric limit for the microstructure of auxetic metamaterials in the elastic and infinitesimal deformation range. Recently the effect of void shapes on nonlinear response of a planner metamaterial was investigated by Bertoldi et al [4]. More recently a geometrical controllable parameter was defined for a simple 3D auxetic metamaterial to control the volume fraction by Shen et al [2] and through the bulking analysis a line strip for this controllable parameter was defined to exhibit auxetic behaviour in infinitesimal deformation. Here, we have developed an approach to study the bounds for auxetic behaviour of a 2D auxetic metamaterial. The geometric limit was investigated in an infinitesimal deformation through buckling analysis and in large deformation range through post buckling analysis with an explicit algorithm. From a practical view, an effective strain range of NPR behaviour was adopted using an energy efficiency method. This new criterion for auxetic behaviour changed the geometric bounds of auxetic metamaterials. Identification the Shape of Microstructure of Simple 2D Auxetic Metamaterial Investigation into 2D and 3D buckling induced auxetic metamaterials revealed that the auxetic behaviour under large deformation was influenced by the shape and size of void part of microstructure which presented in Fig1c [5, 4, 2]. Inspired by this finding, we used a 2D elastomeric buckling-induced auxetic metamaterials investigated in our previous work [8]. The auxetic mechanism of this new designed metamaterials is similar to the 2D buckling-induced metamaterials which has been designed by Bertoldi et al [9]. In this study, we consider this 2D metamaterial with four geometrical parameters to represent its geometry. These parameters have been defined to control the size of void part of microstructure. Respectively, T A denotes the thickness of thinner part of unit building cell; T B represents the thickness of thicker part of unit building cell, L A shows the length of thicker part of unit building cell and L B is the center to center distance between neighboring units building cells in the unreformed configuration as shown in the Fig 1b. Fig. 1 2D buckling-induced auxetic metamaterial. (a) Unit bundling cell of 2D metamaterial; (b) representative volume element for the metamaterial; (c) topology of the void part of the metamaterial; (d) bulk metamaterial for numerical and experimental investigation (0< T A < T B, 0< L A < L B, 0< T B < L A ). Through changing these geometric parameters, a variety of microstructures with different void sizes can be obtained as shown in Fig 2. For example, L B is fixed as a constant value and T B =0.3T A, by changing L A and T A a variety of microstructures can be produced as shown in Fig 2a. Respectively, Figs 2b and 2c are presenting microstructures when T B =0.6 T A and T B =0.9 T A.

3 Applied Mechanics and Materials Vol Experimental Investigation on Simple 2D Elastomeric Metamaterial Fabrication of bulk material. In order to form the bulk material, nine unit building cells as Fig. 2 Examples of variation of geometric parameters on the microstructure of the designed metamaterial. (a) for T B =0.3 T A ; (b) for T B =0.6 T A; (c) for T B =0.9 T A presented in Fig 1a were patterned along two normal directions as shown in Fig 1a. According to deformation pattern of metamaterial after buckling; four unit building cells are used as the representative volume element (RVE) and shown in Fig 1d. The casting method was used to fabricate the specimen of bulk metamaterial. Firstly a specific mold was created then liquid plastic silicon rubber was casted in the mould. The properties of the silicon-based rubber (TangoPlus) were adopted as those in Ref [2]. The behavior of the base rubber material is accurately linear elastic and the Young s modulus is MPa and the Poisson s ratio was found to be The performance of fabricated bulk material was tested under compression at a strain rate of 5x10-3 S -1. The auxetic response of the 2D bulk metamaterial was recorded by two cameras from two different directions and is presented in Fig 3 for planar view. Fig. 3 Comparison of deformation pattern of 2D auxetic metamaterial between FE results and experimental results. Upper row shows experimental results and lower row shows numerical results ( Scale bar: 20 mm, strain rate: S -1 ) Finite Element Simulation of 2D Metamaterial to Study the Effect of Void Size on Nonlinear Response of Metamaterial Finding the geometric bounds for auxetic metamaterial in an elastic and infinitesimal deformation range. As mentioned before, the NPR behaviour of buckling-induced auxetic metamaterials was originated from the localized buckling mode shape [11]. In this study the linear buckling analysis was employed to find the bounds of the four defined geometric parameters as the effective range of auxetic behaviour. We assumed that if the first buckling mode was similar to the

4 0.8Lb 0.9Lb 1Lb 0.1Lb 0.2Lb 0.3L 0.4Lb 0.5Lb 0.6Lb 0.7Lb 550 Advances of Computational Mechanics in Australia desired local buckling pattern, the bulk material had auxetic behaviour under compression. A systematic study was carried out by varying the defined parameters. Firstly we considered that T B was a constant value (for example T B =0.1 T A ) then the values of L A and T A were decreased to half of its previously used values (Respectively, L A =0.5 L B and T A =0.5 L A ). The first buckling mode shape was checked whether it was similar to the local buckling pattern. If not, we decreased the values of L A and T A in a similar manner and if it did, we increased L A and T A to the half values of the sum of the current value and its upper range bond. By using this method the corresponding limits were calculated and were represented by a surface in the space of the three geometric parameters as shown in Fig. 4. 1La 0.9La Ta 0.8La 0.7La 0.6La 0.5La 0.3La 0.2La 0.1La 0.1Ta 0.2Ta 0.3Ta 0.4Ta 0.5Ta 0.6Ta T b 0.7Ta 0.8Ta 0.9Ta 1Ta La Fig. 4 Geometric bound of three geomtric parameters for auxetic behaviour identifed by buckling analysis Finding geometric bounds for auxetic metamaterial under large deformation. The limitations of the bounds identified by buckling analysis, such as the auxetic behaviour only occured in an infinitesimal deformation range which do not have any practical value to use it in the applications to control the auxetic performance for efficient usage.also, it was found that the results of buckling analysis are influenced by boundary conditions. By changing the applied boundary conditions, the eigenvalues corresponding to different buckling mode shapes can be changed so it is possible that the first buckling mode shape is changed from the desired local buckling to global buckling mode shape. Also It should be noted that for a metamaterials undergoing large deformation, several possible configurations may be happened and these configuration corresponding to different buckling mode shapes. When the dispersion of eigenvalues of first three mode shapes is very low, it is possible that the deformation pattern of metamaterial is changed from a specific configuration to another configuration corresponding to different buckling mode shapes. Here, inspired by these findings the investigation approach of NPR behaviour was developed from buckling analysis to post-buckling analysis and we focused on large deformation of the metamaterial to find the range of geometrical parameters for auxetic behaviour. It should be noted that the effects of shape, size and arrangement void parts on performance of cellular metamaterial were investigated by other researchers before, but to our knowledge the effective strain range for auxetic behaviour has never been employed to find a geometric bound for auxetic behaviour. The starting point of this range is the critical strain at which buckling occurs and the metamaterial begins to exhibit auxetic behaviour and the end point is the densification stain as presented in Fig 5a. Beyond the densification strain, the cellular metamaterial is completely compacted. When the applied strain reaches this strain, the stress value increased sharply as shown in Fig 5b. In this study the energy efficiency method was employed to find the exact value of densification strain.

5 0.1Lb 0.2Lb 0.3Lb 0.5Lb 0.6Lb 0.7Lb 0.1L 0.8Lb 0.2Lb 0.3L 0.4L 0.9Lb 1Lb 0.5Lb 0.6Lb 0.7Lb 0.8Lb 0.9Lb 1Lb Applied Mechanics and Materials Vol (a) Fig. 5 Clarification energy efficiency method to define effective auxetic strain range. (a) Illustration of upper bond an lower bond of effective strain range; (b) Finding densification strain as the upper bound of effective auxetic strain range Under large deformation, the metamaterial will be considered as auxetic when the effective strain range is larger than a prescribed value, i.e or 0.1 which are determined by the intended applications. A systematic investigation was carried out through employing validated FE simulations and the results are presented in Fig. 6. A spatial region surrounded by a 3D surface has been illustrated in Fig. 6a Inside the defined region, the length of effective auxetic strain range is at least The similar surface has been obtained when the effective auxetic strain range is at least 0.1 and the results is presented in Fig. 6b. The results of our investigation revealed that the performance of a buckling-induced metamaterial under large deformation would not exhibit NPR behaviour when the geometric parameters used were within the bound identified by buckling analysis. (b) 1La 0.9La 0.8La 0.7La 0.6La T a 0.5La 0.3La 0.2La 0.1La 0.1Ta 0.2Ta 0.3Ta 0.4Ta 0.5Ta T b 0.6Ta 0.7Ta 0.8Ta 0.9Ta 1Ta La 0.6La T a 0.5La (a) (b) Fig. 6 Geometric bound of three geometric parameters for auxetic behaviour under large deformation. (a) 3D surface for an effective strain range of 0.05; (b) 3D surface for the effective strain range of 0.1 1La 0.9La 0.8La 0.7La 0.3La 0.2La 0.1La 0.1Ta 0.2Ta 0.3Ta 0.4Ta T 0.5Ta b 0.6Ta 0.7Ta 0.8Ta 0.9Ta 1Ta La

6 552 Advances of Computational Mechanics in Australia Fig. 7 Comparison the large deformation analysis result and linear buckling analysis. (a) unreformed metamaterial; (b) First buckling mode shape; (c) Observed deformation pattern under large deformation (strain=0.33) As an example, the observed deformation under large deformation and the first linear buckling mode shape of a metamaterial with specific geometrical features (T B =0.2T A, L A =0.8L B and T A =0.7L A ) are presented in Fig 7. In this case the first buckling mode shape is exactly similar to local buckling pattern for auxetic behaviour; however the result of large deformation analysis revealed that the exhibited deformation pattern was similar to a global buckling without auxetic behaviour. Conclusion and Discussion This study has provided a new method to investigate the effect of pore size of microstructure on the nonlinear response of 2D auxetic metamaterial and finding a geometric bound for auxetic behaviour. The energy efficiency method has been used to find the densification strain as the upper bond of effective auxetic strain range. It is found that using the linear buckling analysis may not be a reliable approach to designing the auxetic metamaterials under large deformation. The geometric bound from this study can be used more effectively to design an auxetic metamaterial for different applications which require an auxetic behaviour occurring in a strain range larger than References: [1] R. Gatt, R. Caruana-Gauci, D. Attard, A.R. Casha, W. Wolak, K. Dudek, L. Mizzi, J.N. Grima, On the properties of real finite-sized planar and tubular stent-like auxetic structures, Phys. Status Solidi B. 251 (2014) [2] J. Shen, S. Zhou, X. Huang, Y.M. Xie, Simple cubic three-dimensional auxetic metamaterials, Phys. Status Solidi B. 251 (2014) [3] X. Ren, J. Shen, A. Ghaedizadeh, H. Tian, Y. Xie, Experiments and parametric studies on 3D metallic auxetic metamaterials with tuneable mechanical properties, Smart Mater. Struct.. 24 (2015) [4] J.T.B. Overvelde, K. Bertoldi, Relating pore shape to the non-linear response of periodic elastomeric structures, J. Mech. Phys. Solids. 64 (2014) [5] T. Mullin, S. Willshaw, F. Box, Pattern switching in soft cellular solids under compression, Soft Matter.. 9 (2013) [6] T.-C. Lim, Constitutive relationship of a material with unconventional Poisson's ratio, J Matter Sci Lett. 22 (2003) [7] R. LAKES, Foam structures with a negative poisson's ratio, Science. 235 (1987) [8] A.Ghaedizadeh, J. Shen, X. Ren, Y. Xie, Tuning the performance of metallic auxetic metamaterials by using buckling and plasticitys, Materials 9 (2016) 54. [9] K. Bertoldi, M.C. Boyce, S. Deschanel, S.M. Prange, T. Mullin, Mechanics of deformationtriggered pattern transformations and superelastic behaviour in periodic elastomeric structures, J. Mech. Phys. Solids.. 56 (2008) [10] K. Bertoldi, P.M. Reis, S. Willshaw, T. Mullin, Negative Poisson's ratio behaviour induced by an elastic instability, Adv. Mater.. 22 (2010) [11] S. Babaee, J. Shim, J.C. Weaver, E.R. Chen, N. Patel, K. Bertoldi, 3D Soft Metamaterials with Negative Poisson's Ratio, Adv. Mater.. 25 (2013)