Non-probabilistic models for in-plane elastic properties of cellular hexagonal honeycomb cores involving imprecise parameters

Inherent imprecisions present in the basic parameters of cellular honeycomb cores, such as the cell angle, the material properties, and the geometric parameters, need to be considered in the analysis and design to meet the high-performance requirements. In this paper, imprecisions associated with the basic parameters of honeycomb cores are considered. Non-probabilistic models for the in-plane elastic properties of hexagonal honeycomb cores are developed in which the imprecisely defined input and response parameters are represented by only their mean values and variations without the requirement of knowing the probability density distributions of the imprecise parameters as is required for probabilistic methods. Thus, the proposed models predict not only the nominal values of the in-plane elastic properties but also their variations from the respective mean values. The applicability of the proposed models is demonstrated by considering the analysis of the in-plane elastic properties of a honeycomb core made of aluminum 5052-H-32 in which the core material properties are defined by their mean values and variations. The results show that realistic variations of the in-plane elastic properties are obtained using the proposed non-probabilistic models. The sensitivity of the in-plane elastic properties to the imprecisions present in each basic parameter is also investigated.


Introduction
Cellular materials are widely used as cores in sandwich structures due to their distinct characteristics such as having an extra lightweight and high stiffness and strength values. These characteristics have led designers to use cellular materials in many engineering applications, such as aerospace and automotive structural applications, that require high performance and reliability. Hexagonal honeycomb, balsa, foam, and corrugated cores are typical cellular materials that can be used as cores in sandwich structures. Hexagonal honeycombs, which epitomize two-dimensional cellular materials, are commonly used as cores because of their low weight, high bending strength, and flexural rigidity. In recent decades, many closed-form deterministic analytical models have been developed to determine the mechanical properties of hexagonal honeycomb cores. [1][2][3] The dynamic and failure of honeycomb core sandwich panels with high-velocity impact were studied by Sun et al. 4 using experimental and numerical methods. Zhang et al. 5 conducted an experimental study to investigate the mechanical behavior of hexagonal honeycombs under quasi-static combined compression-shear loading. Ha et al. 6 proposed a new honeycomb, known as bio-inspired honeycomb, and studied the geometrical configurations on the energy absorption of bio-inspired honeycomb sandwich panels. He et al. 7 presented a comprehensive study using numerical, experimental, and theoretical method on the residual flexural properties honeycomb core sandwich panels that are subjected to low-velocity impact. Sun et al. 8 investigated the influences of some structural parameters, such as face sheet thickness, wall and cell geometries, on the behavior of honeycomb cores sandwich panels subjected to lowvelocity impact. However, hexagonal honeycomb cores exhibit disorders in the geometries and properties of materials from which the cores are made. For example, the material properties of the core material, cell angle, and geometry are inherently subjected to irregularities and imprecisions during the formation and manufacturing of cores. Thus, using the available deterministic models for mechanical properties cannot assure the high performance and reliability of hexagonal honeycombs because the inherent scatter in the basic parameters is neglected. The variabilities associated with engineering system responses with stringent design requirements need to be known in order to achieve a desirable performance. Hence, variations must be considered when determining the response of hexagonal honeycomb cores so that improved and more realistic response representations can be obtained.
Several studies have considered the variabilities associated with cellular materials; however, these have often been in terms of their stochastic/probabilistic nature. Zhu et al. 9 studied the effects of irregularities in the cell size and shape of foams on their elastic properties by developing models for the finite element analysis of random Voronoi foams. Hohe and Becker 10 and Hohe and Bechmann 11 presented a numerical analysis using stochastic procedures to investigate the effective mechanical properties of two-dimensional cellular materials taking into account the presence of microstructural irregularities. Flores et al. 12 developed a multi-scale finite element method using the Gaussian process emulator to study the stochastic mechanical response of structures with honeycomb cores. The probabilistic distribution of the residual strength of impacted sandwich structures with honeycomb cores was estimated by Kim et al. 13 in which the mean and variance values of the original properties were used in the probabilistic analysis. Parsi et al. 14 developed a probabilistic model for predicting the dynamic contact force of honeycomb structures subjected to impacts at low velocities. Probabilistic methods have also been used in analysis and design in many engineering fields, such as vibrations, [15][16][17][18] fracture mechanics, [19][20][21][22] computational fluid mechanics, 23,24 and structural stability. 25,26 A stochastic/probabilistic analysis represents each imprecise parameter as a random variable that follows a specific probability distribution. In real-life applications, the full probabilistic information of imprecise quantities is not readily available, which leads to unknown probability density distributions for the imprecisely defined quantities. Thus, an exact stochastic/probabilistic analysis becomes unpractical in the absence of exact probability density distributions of each imprecisely defined parameter, as in most engineering applications. However, due to readily available maximum and minimum values, mean values, and variations are commonly used parameters to represent the imprecisions associated with quantities that are subjected to imprecision. Several nonprobabilistic methods have been proposed to model imprecisions associated with engineering systems where the probability density distributions of imprecise parameters are not needed. [27][28][29][30][31][32][33] Recently, universal grey mathematics has been used in many studies to quantify imprecisions associated with engineering systems. [34][35][36][37][38] These studies have shown that universal grey mathematics has the ability to realistically quantify imprecisions with less computational effort and without overestimating variations in response parameters. Universal grey mathematics only requires the mean values and the corresponding variations of imprecisely defined parameters to model imprecisions in imprecise systems. The superiority of this method lies in the satisfaction of the distributive property, which leads to the realistic quantification of imprecisions in the response parameters without overestimation.
This work presents the modeling and analysis of the in-plane elastic properties of hexagonal honeycomb cores when the basic quantities are defined in terms of mean values and variations. Non-probabilistic models are presented based on universal grey mathematics for the modeling of imprecisions in the response quantities. Unlike probabilistic models, which require the probability distribution density of imprecise parameters to be exactly known, the proposed models can express the in-plane elastic properties of a hexagonal honeycomb core in terms of their mean values and variations with the knowledge of only the mean values and variations of the basic input quantities. Additionally, a sensitivity analysis is performed in which the influence of the variation of each basic variable on the variations of the response quantities is investigated.
In-plane elastic properties of cellular hexagonal honeycomb cores Several approaches are available in the literature to estimate the elastic properties of cellular honeycomb cores, however, without considering variabilities in the imprecise parameters. [1][2][3] The present study is based on the analytical expressions proposed by Gibson and Ashby 2 for the in-plane elastic properties of a cellular honeycomb core, namely, the in-plane Young's moduli (E Ã 1 and E Ã 2 ), the in-plane shear modulus (G Ã 12 ), and the Poisson's ratios (v Ã 12 and v Ã 21 ). The expressions for the in-plane elastic properties of a honeycomb core are derived based on studying the unit cell shown in Figure 1 and treating the cell wall as a beam that deforms due to bending in a linear-elastic manner.
Thus, when the basic parameters are described in their mean values only (deterministic form), the inplane elastic properties of a honeycomb core are given by: However, the aforementioned expressions do not consider the inherent imprecisions in the basic quantities, namely, the wall length l, wall height h, wall thickness t, cell angle u, and cell wall material properties. In reallife applications, during the manufacturing processes, environmental effects and other sources of imprecisions, the basic quantities are subjected to imprecisions which lead to imprecise elastic properties of honeycomb cores. To achieve high performance and reliability, the imprecisions in the basic input quantities need to be considered in the analysis. In the next section, the imprecisions associated with the basic parameters are considered by proposing analytical expressions for the in-plane elastic properties of a honeycomb core using the universal grey mathematics.
Imprecise models for in-plane elastic properties of a honeycomb core Universal grey mathematics is used to predict the inplane elastic properties of a honeycomb core in the presence of imprecise basic parameters. The arithmetic operations associated with the universal grey number theory are presented by Rao and Liu 39 where each imprecise quantity X is represented by its extreme values; the lower bound X and upper bound X . Using and d X = X Àm X m X 3 100, the extreme values can be represented by the mean value and the percent variation of the imprecise quantity as X = m X (1 À 0:01d X ) and X = m X (1 + 0:01d X ). Thus, in this paper, the arithmetic operations are remodeled in which each imprecise quantity X is represented by its mean value m X and the percent variation d X .
The basic input quantities are defined in terms of their mean values and the percent variations. Consequently, the response quantities of a honeycomb core are also represented by their mean values and percent variations. In this work, the imprecise parameter X is represented in the form where m X and d X represent the mean value and the respective percent variation of the imprecise parameter X, respectively. Then, the analytical expressions defined in equations (1)-(5) in conjunction with universal grey mathematics are used to predict the response quantities, namely, the in-plane Young's moduli (E Ã 1 and E Ã 2 ), the in-plane shear modulus (G Ã 12 ), and the Poisson's ratios (v Ã 12 and v Ã 21 ), in terms of their mean values and respective percent variations.
Let X and Y be imprecise parameters that are defined in terms of their mean value m and percent variation d as X = (m X , d X ) and Y = (m Y , d Y ), respectively. Applying universal grey mathematics, the arithmetic operations that are required between any imprecise parameters, X and Y, are given by: The imprecise forms of the models associated with the in-plane elastic properties of a hexagonal honeycomb core can be derived using the basic arithmetic operations defined in equations (7)- (10) in conjunction with the analytical deterministic expressions given in equations (1)- (5). The nature of the arithmetic operations (equations (7)- (10)) and the conditions associated with the subtraction and division operations (equations (8) and (10) To illustrate the process, the computations of the mean value and the percent variation of the Young's modulus in direction 2, , of a hexagonal honeycomb core that is made of aluminum 5052-H-32 is considered. 40 By applying the imprecise parameters of the basic quantities of the honeycomb core, E s = (70.3 3 10 3 MPa, 6 1.5%), l = (3.8 mm, 6 0.5%), h = (5.7 mm, 6 1%), t = (0.08 mm, 6 2%), and u = (30 8 , 6 1:5%), the parameters, the mean value m E Ã 2 and the percent variation d E Ã 2 , which describe the imprecision associated with the Young's modulus in direction 2, E Ã 2 , can be computed as follows Similarly, the mean values and the percent variations of the other in-plane elastic properties of a hexagonal honeycomb core, E Ã 1 , G Ã 12 , v Ã 12 , and v Ã 21 , can be computed by using the basic arithmetic operations defined in equations (7)-(10) in conjunction with their respective deterministic expressions. Also, for comparison, the combinatorial method is used to find the mean values and percent variations in the in-plane elastic properties of honeycomb cores in the presence of imprecise input parameters. The combinatorial method determines the extreme values of the response parameters by, first, considering all combinations of the extreme values of the basic parameter and determining the response parameters for all combinations, then the minimum and maximum of the obtained possible response values are taken as the lower bound X and upper bound X , respectively. However, the combinatorial method is not practical due the large number of analyses needed to find the extreme values of the response parameters. In the following section, the imprecise behavior of the in-plane elastic properties of a hexagonal honeycomb core made of aluminum 5052-H-32 is investigated, where the basic parameters of wall length l, wall height h, wall thickness t, cell angle u, and the cell wall material properties are considered to be in the form of their mean values and percent variations.

Numerical example
The analysis of a hexagonal honeycomb core made of aluminum 5052-H-32 40 is considered to demonstrate the applicability of the proposed imprecise models of the in-plane elastic properties. The analysis is performed using the basic input quantities shown in Table 1 in which three different a = h=l ratios are considered, namely, a = 1, a = 1.5, and a = 2. To investigate the behavior of the response quantities in the presence of imprecisions, five cases are considered in which different variations are introduced in the basic input quantities in each case. The sensitivity of the inplane elastic properties to each imprecise input parameter is studied over a variation range of 0% to 6 5% from the respective mean value. In addition, the influence of the cell angle on the response quantities in the presence of imprecisions is investigated.
Tables 2 to 5 present the results obtained for the inplane elastic properties of a hexagonal honeycomb core considering five cases of imprecisely defined input parameters. The results show that the Young's modulus in direction 2, E Ã 2 , the Poisson's ratio, v Ã 12 , and the shear modulus, G Ã 12 , have acceptable degrees of variation with respect to the variations introduced in the basic input parameters in all of the cases. However, the Young's modulus in direction 1, E Ã 1 , exhibits a realistic variation but approximately doubled the introduced variations in the basic parameters. For example, as shown in Table  2, in the case of h=l = 1, when variations of 6 2% are introduced in all the basic parameters (Case 3), the extreme values of E Ã 2 , v Ã 12 , and G Ã 12 deviate from the respective mean values by 6 3:2084%, 6 1:2094%, and 6 2%, respectively. However, the variation corresponding to E Ã 1 is found to be 6 5:6224%. The results presented in Table 2 are compared with the ones predicted by the combinatorial method which are given in Table 3. Clearly, as shown in Tables 2 and 3, the variations predicted by the present models secure sharper variations than the variations predicted the combinatorial method. Similar behavior is observed for h=l = 1:5 (Table 4) and h=l = 2 (Table 5) in all response quantities. The reason for this behavior is that the expression used to compute the Young's modulus (equation (1)) involved a greater number of imprecisely defined parameters than were used in the computations of the other response quantities (equations (2)-(5)). Additionally, as the ratio of h=l increases, the variations corresponding to E Ã 2 and G Ã 12 also increase. However, the variations predicted for E Ã 1 and v Ã 12 decrease as the ratio of h=l increases. For example, in the case of h=l = 1 (Table 2), when 6 3% is introduced in all the basic quantities (Case 4), the variations corresponding to E Ã 1 , E Ã 2 , v Ã 12 , and G Ã 12 are 6 8:4240%, 6 4:8113%, 6 1:8143%, and 6 3%, respectively. When h=l = 1:5 (Table 4) , and G Ã 12 deviate from their   Table 2. In-plane elastic properties in terms of mean values and variations for a = h=l = 1.

Case
Response parameter respective mean values by 6 8:1988%, 6 5:0375%, 6 1:5876%, and 6 3:2266%, respectively. Figure 2 shows the sensitivities of the in-plane elastic properties, where h=l = 1, to each input parameter, namely, wall length l, wall height h, wall thickness t, cell angle u, and the cell wall elastic modulus E s , when variations of 6 0% to 6 5% are introduced in each input quantity. Compared to the other input quantities, the imprecisions associated with the wall length l and thickness t of the cell highly influence the variations in the elastic moduli, E Ã 1 and E Ã 2 , and the shear modulus G Ã

12
of the honeycomb core. This is because the parameters l and t appear more often in the expressions of E Ã 1 , E Ã 2 , and G Ã 12 compared to the other input parameters. Furthermore, the nonlinearity associated with the parameters l and t in the expressions of the response quantities is a cause of the high sensitivity behavior. For example, when a variation of 6 2% is present only in the wall length l of the cell, the response quantity E Ã 1 varies by 6 7:3213%. By contrast, the elastic modulus E Ã 1 exhibits a variation of 6 2% when only the cell wall elastic modulus E s is imprecisely defined. Figure 2 also shows that the variations associated with the Poisson's ratio v Ã 21 are influenced by only the variations presented in the wall height h and cell angle u. However, the Poisson's ratio is not affected by the variations presented in the other input quantities, namely, wall length l, wall thickness t, and the cell wall elastic modulus E s . Figures 3 to 6 show the behavior of the variations associated with the in-plane elastic properties for different values of cell angle u, considering three cases of imprecisions in all input parameters: 6 1%, 6 2%, and 6 3%. As can be seen in Figure 3, the variations associated with the elastic modulus E Ã 1 decrease as the cell angle increases from 10°to 57.47°, then a sharp increase is observed as the cell angle increases further. This behavior is due to the presence of the trigonometric terms in the respective expressions. Figure 4 shows that the variations corresponding to the elastic modulus E Ã 2 increase with the cell angle for all cases d all basic parameters = 1, 2, and 3. Regarding the variation in the Poisson's ratio v Ã 21 , a behavior similar to that seen in the elastic modulus E Ã 1 is noticed (shown in Figure 5) in which the variations decrease when the cell angle changes from 10°to 39.82°. The variations significantly increase when the cell angle further increases. As shown in Figure 6, the modulus of rigidity G Ã 12 exhibits variations approximately consistent with those introduced in all the basic quantities up to a cell angle of 30°.

Conclusion
This work, for the first time, introduces the modeling and analysis of imprecisions in the in-plane elastic   properties of a hexagonal honeycomb core with imprecisely defined parameters without the requirement of knowing the exact probability distributions of the imprecise parameters as is required for probabilistic approaches. The proposed models are non-probabilistic and can express the in-plane elastic properties in terms of only mean values and percent variations when the basic parameters are available in the form of mean values and percent variations. The variations predicted by the proposed models were found to be realistic and acceptable compared to the imprecisions present in the basic variables. Also, the present results secure sharper and more realistic variations compared to the ones predicted by the combinatorial method. One of the advantages of the proposed models, which cannot be achieved when imprecisions are not considered, is the performance of a sensitivity analysis. This analysis determined the influence of the variations present in each basic variable on the variations of the in-plane elastic properties.

Declaration of conflicting interests
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author received no financial support for the research, authorship, and/or publication of this article.