Elsevier

Materials & Design

Volume 32, Issue 2, February 2011, Pages 512-524
Materials & Design

Compliant hexagonal periodic lattice structures having both high shear strength and high shear strain

https://doi.org/10.1016/j.matdes.2010.08.029Get rights and content

Abstract

Cellular structures having negative Poisson’s ratios can be designed to have high shear flexure properties. In this paper, the elastic limits of hexagonal honeycombs including the ones having negative Poisson’s ratios (NPR) are explored with various cell geometries under simple shear loading. While designing a shear modulus, e.g., G12 of 10 MPa, of hexagonal honeycombs, corresponding meso-structures are designed with three constituent materials; an aluminum alloy (7075-T6), a titanium alloy (Ti–6Al–4V), and a high strength steel (ANSI 4340). The in-plane linear elastic honeycomb model is employed to achieve the shear moduli of the hexagonal honeycombs made of the three constituent materials. The shear strengths, (τpl)12 and shear yield strains, (γpl)12 of hexagonal honeycombs are obtained from finite element analysis using ABAQUS. The titanium alloy honeycombs show a good shear flexure property having both high (τpl)12 and (γpl)12 when it is designed to the target G12. The re-entrant geometry makes honeycombs flexible associated with a high effective bending length.

Introduction

Two-dimensional prismatic cellular materials of periodic meso-structures are called honeycombs. These honeycombs have been primarily used in lightweight sandwich structures for which a high out-of-plane stiffness is desired [1], [2], [3]. Honeycomb structures are also valued for their combination of high strength and low density, and ability to absorb impact energy [4], [5], [6], [7]. In contrast to the highly stiff and strong properties of the out-of-plane direction (longitudinal cell axes) associated with the cell walls’ axial stress, the in-plane properties are two to three orders of magnitude weaker than those of the out-of-plane loading. For this reason, the mechanical properties for the in-plane loading have been thought be the most limiting for design applications. However, some engineers have been motivated to find optimal in-plane meso (or micro) -structures for various requirements; e.g., optimizing thermal expansion [8], optimizing piezoelectric characteristics [9]. In parallel, there were efforts to build a computer aided design (CAD) system for designing heterogeneous cellular materials able to combine multiple materials with geometries [10], [11]. Moreover, there have been efforts to use the lower in-plane stiffness for designing flexible meso-structures in applications that need high deformation under targeted loads [12], [13], [14], [15], [16].

Hexagonal honeycombs are cellular materials employed in various applications in the design of light weight structures. The in-plane moduli of hexagonal honeycombs have been successfully investigated with the cell wall bending model called cellular material theory (CMT) [17]. Other analytical and numerical models describe in-plane properties of honeycombs in the literature; a refined cell wall’s bending model by adding a beam’s stretching and hinging motion [18], a model with the energy method [19], [20], a refined model with round shape at cell edges [21], and a model using the homogenization method [22]. In-plane moduli, yield strengths, and buckling strengths with different cell types –square, hexagonal, triangle, mixed squares and triangles, diamond- were investigated [23], [24], [25].

Triangular, Kagome, and diamond cell honeycombs are known to be the extension dominated cell structures, which is good for high modulus structural design [23]. On the other hand, square and hexagonal cell honeycombs are known to be bending dominated structures, which is good for flexible structural design [23]. Hexagonal cell structures are known to be flexible in both axial and shear directions [26], [27], [28]. Moreover, hexagonal honeycombs can easily be tailored to have targeted in-plane properties by changing cell angles. Therefore, some hexagonal geometries have a potential to compliant structural design. Fig. 1 shows normalized shear strain-shear strength of the triangular honeycomb, the square honeycomb, and the hexagonal honeycombs including the ones with negative Poisson’s ratios (NPR) when they are designed to have the same G12 based on the linear cellular material theory as measured to a constituent material’s yield [17], [23]. Without the re-entrant geometries, the square honeycomb is the best structure satisfying the dual property – a high (τpl)12 and (γpl)12, associated with the shear compliant design. However, the hexagonal structures with high negative cell angles exceed the dual target property of the square honeycomb (Fig. 1).

Our previous study on tailoring a dual target property, e.g., a shear modulus and a shear strain, with cellular structures shows a possibility in building flexible hexagonal honeycomb meso-structures [26]. Motivated by our recent findings on the shear compliant hexagonal honeycombs for the shear band component of a lunar rover wheel, we are seeking more geometric options for the flexible hexagonal honeycomb design. This is applicable in the aerospace morphing wing technology in which some researchers already began to use the in-plane flexibility with honeycombs [13], [14], [15]. The use of re-entrant cellular structures as micro-actuators and displacement amplifiers has also been suggested in the micro-electro-mechanical-system (MEMS) structural design [12].

In ongoing research at Clemson University, we are challenged with developing cellular meso-structures that mimic an example elastomer’s shear property with high deformation recovery and low hysteretic loss [26], [27], [28], [29], [30], [31], [32]. In this study, while pursuing an elastomer’s shear modulus, 10 MPa, we investigate the effect of various hexagonal geometries on shear strengths, (τpl)12 and shear strains, (γpl)12 with an aluminum alloy (7075-T6), a titanium alloy (Ti–6Al–4V), and a high strength steel (ANSI 4340) when yield points of the constituent materials are considered to be the criteria.

Section snippets

Model development

Compliant hexagonal geometries are investigated using CMT because they are easily controlled to have positive to negative Poisson’s ratios by changing cell angles, which is good for this parametric study. Due to the high cost of manufacturing cellular structures with varying geometric parameters, a numerical parametric study of an analytical model and finite element (FE) based numerical test are preferred to physical experiments at the initial stage of design. Therefore, numerical parametric

Maximum shear strength and shear strain of hexagonal meso-structures for a target G12

For the same shear modulus, G12 from Eq. (1), the values of (τpl)12/σys of honeycombs are obtained using Eq. (2) and are plotted in Fig. 6. Cellular structures have high shear strengths at both highly negative and positive cell angles (Fig. 6). If the constitutive relations of the properties of hexagonal honeycombs are linear, the shear strain must have the same behavior as the plots in Fig. 6. Assuming the linear elasticity of hexagonal honeycombs, meso-structures of α = 2 and 4 with θ = −70°

Conclusions

In an effort to tailor the shear compliant hexagonal honeycomb meso-structures having a G12 of 10 MPa with an aluminum alloy (7075-T6), a titanium alloy (Ti–6Al–4V), and a high strength steel (ANSI 4340), honeycomb geometries having high (τpl)12 and high (γpl)12 were sought including ones with NPR. The key findings of the present study are:

  • 7075-T6, Ti–6Al–4V, and ANSI 4340 hexagonal honeycombs can have (τpl)12 and (γpl)12 which are higher than 2.7 MPa and 0.07, respectively when the

Acknowledgement

This work is supported by National Institute of Standards and Technology (NIST) – Advanced Technology Program (ATP).

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