International Journal of Solids and Structures 80 (2016) 284–296 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr Nonlinear shear modulus of re-entrant hexagonal honeycombs under large deformation M.H. Fu a , O.T. Xu a , L.L. Hu a,b,∗ , T.X. Yu c a Department of Applied Mechanics & Engineering, School of Engineering, Sun Yat-sen University, Guangzhou 510275, PR China b State Key Laboratory of Explosion Science and Technology, Beijing 100081, PR China c Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong article info Article history: Received 11 June 2015 Revised 28 October 2015 Available online 2 December 2015 Keywords: Re-entrant hexagonal honeycomb In-plane shear modulus Nonlinear Large deformation abstract The nonlinear, in-plane, shear modulus of re-entrant hexagonal honeycombs under large deformation is an- alytically derived by studying the mechanical behavior of cell structures, which is later verified by numerical simulations. A nonlinear, modified factor is proposed to characterize the difference of the honeycomb’s shear modulus under large and small deformation, revealing its independence from the honeycomb’s relative den- sity. The effects of both strain and cell geometry on the honeycomb’s shear modulus are investigated, exhibit- ing that the honeycomb’s shear modulus increases with shear strain but decreases with the cell-wall-length ratio. For the effect of cell-wall angle, the re-entrant honeycomb’s shear modulus decreases gradually with the cell-wall angle until reaching a minimum and then increases, which is highly different from the mono- tonically increasing relationship of conventional hexagonal honeycombs. When keeping the honeycomb’s relative density constant, the re-entrant honeycomb’s shear modulus monotonously increases with the cell- wall angle and reaches a maximum at h/l ≈ 3.25. Finally, the shear modulus of the re-entrant honeycombs is compared with that of conventional honeycombs. In contrast to the predictions of the classical contin- uum theory, the present study shows that the shear modulus of the re-entrant honeycomb with a negative Poisson’s ratio is not always higher than that of the conventional honeycomb with a positive Poisson’s ratio, which is dominated by the geometry of the cell structure. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction Conventional material shrinks laterally when it is pulled along the axial direction. In contrast, auxetic material with a negative Pois- son’s ratio expands laterally under the axial pull, leading to an in- crease of volume. Auxetic materials have gained increasing atten- tion in recent years with the expectation of several advantages, in- cluding large shear resistance, hardness improvement, and lower fa- tigue crack propagation (Evans and Alderson, 1992; Friis et al., 1988; Alderson and Evans, 1995). Bezazi and Scarpa (2007) studied the mechanical behavior of con- ventional and auxetic foams made of thermoplastic polyurethane. These researchers found that the energy dissipated by the aux- etic foams was significantly higher than that by the conventional foams. Taylor et al. (2011, 2012) numerically studied functionally graded, hierarchical honeycomb with negative Poisson’s ratio and found an enhancement of the equivalent elastic modulus compared to the conventional hexagonal honeycombs. The crash performance ∗ Corresponding author. Tel.: +86 2084110293; fax: +86 2084110293. E-mail address: [email protected], [email protected] (L.L. Hu). (Henderson et al., 2007) and the fracture toughness (Critchley et al., 2013; Yang et al., 2013) of auxetic cellular materials were investigated in the recent years, and multiple advantages were found. When honeycombs are used in structures, they are usually ideal- ized as homogeneous and orthogonally anisotropic in the design of the whole structure for simplicity and efficiency (Saidi et al., 2005; Hu et al., 2012) without considering the cell structure of the hon- eycombs. Thus, the relevant equivalent mechanical properties, such as Young’s modulus, Poisson’s ratio, and shear modulus, need to be known for honeycombs. The relevant research on the conventional hexagonal honeycombs has been examined extensively in recent years (Gibson and Ashby, 1997; Hu and Yu, 2010, 2013; Xu et al., 2012). Based on the classic beam theory, Gibson and Ashby (1997) analyzed the in-plane equiv- alent elastic parameters, such as Young’s modulus, Poisson’s ratio, and shear modulus, of the conventional hexagonal honeycombs un- der small deformation. Zhu and Mills (2000) studied the large de- formation of conventional hexagonal honeycombs, and the in-plane equivalent Young’s modulus was obtained by considering both mate- rial and geometric nonlinearity. Lan and Fu (2009) investigated the in-plane equivalent shear modulus of the conventional hexagonal http://dx.doi.org/10.1016/j.ijsolstr.2015.11.015 0020-7683/© 2015 Elsevier Ltd. All rights reserved.
Nonlinear shear modulus of re-entrant hexagonal honeycombs under large deformation
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