PSI - Issue 71

A. Kumar et al. / Procedia Structural Integrity 71 (2025) 453–460

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(2019); Qi et al. (2020)). Re-entrant honeycomb has a relatively traditional and basic arrangement of cells among all auxetic honeycombs. Therefore, several researchers have utilized both theoretical and numerical methods to compute the elastic modulus of re-entrant honeycombs. Masters and Evans investigated the elastic deformations of re-entrant honeycomb structures (Masters and Evans (1996)). Several research investigations have indicated that the shape and geometry of the representative cell are essential parameters (Kumar et al. (2022); Mukhopadhyay and Adhikari (2016)).

Fig.1. The red dotted line on the re-entrant honeycomb represents the RCE, and the in-plane (loading applied within the 1-2 plane) and out-of plane (loading along the 3 -axis) can be applied in loading conditions

An appropriate representative cell element (RCE) was selected in this study to determine the material properties using FE analysis done using the ABAQUS software. Because the shape is periodic, PBCs were used on the RCE to compute material properties using the micromechanics plugin in Abaqus. After that, the results were compared with the theoretical ones obtained by applying Castigliano's second theorem and other reference results to validate the computed properties. 2. Theoretical Analysis The selected RCE is shown in Fig. 1. The length of vertical and inclined cell walls is h and l , respectively. θ denotes the cell angle, while t denotes the wall thickness. The width of the RCE is w , and Young's modulus is E . 1.1. Out-of-Plane Young's modulus (E 3 )

As illustrated in Fig. 2, the 3 E equation is derived by applying a tensile force 3 F normal to the RCE. The strain energy necessary to deform the RCE in direction- 3 is

Fig. 2. RCE to determine the E 3

0 2 w F

2

= 

d

U

z

3

(1)

AE

In direction- 3 , the deformation, according to the second theorem of Castigliano,