A method of determining effective elastic properties of honeycomb cores based on equal strain energy

CJA 807 21 February 2017 Chinese Journal of Aeronautics, (2017), xxx(xx): xxx–xxx

No. of Pages 14

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Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

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A method of determining effective elastic properties of honeycomb cores based on equal strain energy

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Qiu Cheng, Guan Zhidong *, Jiang Siyuan, Li Zengshan

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School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China

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Received 25 January 2016; revised 11 August 2016; accepted 29 December 2016

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KEYWORDS

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Elastic properties; Homogenization; Honeycomb; Strain energy; Unit cells

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Abstract A computational homogenization technique (CHT) based on the finite element method (FEM) is discussed to predict the effective elastic properties of honeycomb structures. The need of periodic boundary conditions (BCs) is revealed through the analysis for in-plane and out-of-plane shear moduli of models with different cell numbers. After applying periodic BCs on the representative volume element (RVE), comparison between the volume-average stress method and the boundary stress method is performed, and a new method based on the equality of strain energy to obtain all non-zero components of the stiffness tensor is proposed. Results of finite element (FE) analysis show that the volume-average stress and the boundary stress keep a consistency over different cell geometries and forms. The strain energy method obtains values that differ from those of the volume-average method for non-diagonal terms in the stiffness matrix. Analysis has been done on numerical results for thin-wall honeycombs and different geometries of angles between oblique and vertical walls. The inaccuracy of the volume-average method in terms of the strain energy is shown by numerical benchmarks. Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

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1. Introduction

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For the past several decades, sandwich plates with a honeycomb core have been widely used in the field of aviation. In understanding the behavior of sandwich structures under different types of load, the honeycomb core is often regarded as

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* Corresponding author. E-mail address: [email protected] (Z. Guan). Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

homogeneous solid with orthotropic elastic properties.1 As a result, research on the effective elastic properties of the honeycomb core is of great essence for the calculation and design of honeycomb sandwich structures. A computational homogenization technique (CHT) has been found to be a powerful method to predict the effective properties of structures with periodic media. In order to obtain the effective stiffness tensor, which relates to the equivalent strain and stress, this process is divided into solving six elementary boundary value problems, which refer to uniaxial tensile and shear in three directions.2–5 The equivalent strain is determined after applying the unit displacement boundary conditions (BCs) on the representative volume element (RVE) cell corresponding to one of the six elementary problems. Different

http://dx.doi.org/10.1016/j.cja.2017.02.016 1000-9361 Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Qiu C et al. A method of determining effective elastic properties of honeycomb cores based on equal strain energy, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.02.016

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methods have been used when dealing with the equivalent stress. Volume-average stress is often used to determine effective properties in the literature.2–6 Catapano and Montemurro2 investigated the elastic behavior of a honeycomb with doublethickness vertical walls over a wide range of relative densities and cell geometries. The strain-energy based numerical homogenization technique was also used by Catapano and Jumel3 in determining the elastic properties of particulatepolymer composites. Montemurro et al.4 performed an optimization procedure at both meso and macro scales to obtain a true global optimum configuration of sandwich panels by using the NURBS curves to describe the shape of the unit cell. Malek and Gibson5 got numerical results of a thick-wall honeycomb closer to their analytical solutions by considering nodes at the intersections of vertical and inclined walls. Shi and Tong6 focused on the transverse shear stiffness of honeycomb cores by the two-scale method of homogenization for periodic media. Many researchers also seek the stress on the boundary of the RVE cell. Li et al.7 used the sum of the node force on the boundary of the RVE cell to obtain the equivalent stress. Papka and Kyriakides8 set plates on the top and bottom of the RVE cell to exert BCs. However, regarding honeycomb structures as a combination of cell walls and air, the stress variations on the boundary cause the boundary stress inaccurate to calculate effective properties. Some divergence still exists in numerical results of regular hexagonal honeycomb structures with analytical solutions, especially for the in–plane and outof-plane shear moduli. From the definitions of effective elastic properties expressed by Yu and Tang,9 the equivalent stress is required to make sure that the RVE cell and the corresponding unit volume of the homogeneous solid undergo the same strain energy. Hence, the whole honeycomb structure containing a finite number of RVE cells have the same strain energy as that of the whole volume of the homogeneous solid. The mathematical homogenization theory (MHT) has proven that the strain energy in the RVE can be determined by the volumeaverage stress and strain.10 However, it is not always suitable for the calculation of the volume-average stress method in the CHT. The volume-average method cannot get all precise values in the stiffness matrix, and it is found to get larger strain energy than that obtained from direct analysis in twodimensional porous composites by Hollister and Kikuchi.11 Therefore, we focus on the total strain energy of the RVE cell and propose a new method to determine all the components of the stiffness tensor more accurately in terms of the strain energy. In Section 2, the differences between the proposed energy method and previous methods are analyzed. A process to obtain 9 components of the effective stiffness tensor based on the energy method is introduced. Then, finite element (FE) models are discussed in Section 3. Convergence analysis has been done over material properties, mesh sizes, and BCs applied on the whole model. In addition, two models are proposed to acquire in–plane and out-of-plane shear moduli according to the different deformations of a single RVE cell and a finite number of RVE cells under the same loading. After establishing appropriate models for honeycomb structures, numerical results over a range of cell geometries are compared to analytical solutions in literature in the next section. Finally, Section 5 ends the paper with some conclusions.

C. Qiu et al. 2. Prediction method

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2.1. Introduction of a computational homogeneous technique

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Previous experimental data and theory have proven that a honeycomb core can be classified as an orthotropic material.12 Under this assumption, a honeycomb core conforms to generalized Hook’s law13 as 2 3 2 32 3 r11 0 0 0 e11 C11 C12 C13 6r 7 6C 6 7 0 0 0 76 e22 7 6 22 7 6 12 C22 C23 7 6 7 6 76 7 6 r33 7 6 C13 C23 C33 0 0 0 76 e33 7 6 7 6 76 7 ð1Þ 6r 7 ¼ 6 0 6 7 0 0 C44 0 0 7 6 23 7 6 76 c23 7 6 7 6 76 7 4 r13 5 4 0 0 0 0 C55 0 54 c13 5 0 0 0 0 0 C66 r12 c12

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where r and e are, respectively, the equivalent stress and strain tensors for the whole geometry of an RVE cell. Cij is one of the components of the stiffness tensor C which is symmetric as Cij = Cji. In addition, the shear strain relates the components of the strain tensor as follows, 8 @u e11 ! e11 ¼ @x > > > > e ! e ¼ @v > 22 22 > @y > > > < e33 ! e33 ¼ @w @z ð2Þ > c23 ! 2e23 ¼ @w þ @v > @y @z > > > > þ @u > c13 ! 2e13 ¼ @w > @x @z > : @v c12 ! 2e12 ¼ @x þ @u @y

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where u, v, and w represent the displacements in the x, y, and z directions. To determine the effective stiffness matrix of the RVE cell, six elementary BCs are applied on the RVE cell, which refer to three uniaxial extensions and three shear deformations. For each load case, only one component of the strain tensor is not zero. Then the relative stiffness component is determined by the equivalent stress. Take C11 for example,

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r11 C11 ¼ e11

in load case e11 – 0

ð3Þ

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After obtaining all the 9 independent components in the stiffness matrix, engineering constants can be derived from the compliance matrix which is the inverse of the stiffness matrix.

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2.2. Energy method

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Assuming that an elementary shear boundary displacement is applied on the RVE cell (ckl – 0), Eq. (4) is tenable since the boundary of the RVE cell has an identical displacement. Z 1 ckl dv ¼ ckl ð4Þ V

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where V represents the total volume of the RVE and subscript ‘‘kl” stands for the certain BC. The strain energy of the RVE cell under certain loading can be determined by the FE result as Z 1 U¼ ð5Þ rij eij dv i; j ¼ 1; 2; 3 2

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A method of determining effective elastic properties of honeycomb cores based on equal strain energy 143 144 145 146 148 149 150

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To guarantee the RVE cell and the corresponding volume of the homogeneous material having the same strain energy, the equivalent stress in this method is R rij eij dv U ¼ ð6Þ rkl ¼ ckl V ð1=2Þckl V

where rkl is the corresponding stress component in the stress field obtained from the FE analysis.The related shear modulus can be written as R R R 1 rkl dv 2 rkl dv  ekl dv G1 ¼ V ¼ ð14Þ ckl V2 c2kl

Hence, the effective modulus through this energy method is R rij eij dv i; j ¼ 1; 2; 3 ð7Þ G¼ c2kl V

(2) Boundary stress method The equivalent stress is obtained by summing up node forces on the boundary as P Fkl rkl ¼ ð15Þ S0

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where Fkl is the corresponding node force on the boundary of the RVE cell and S0 means the area of the section on which the displacement is applied. Thus, the effective shear modulus is determined by P Fkl G2 ¼ 0 ð16Þ S ckl

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(3) Energy method The effective property obtained by the energy method is shown in Eq. (7). We review Eqs. (14) and (16) to compare the volumeaverage method and the boundary stress method. Without loss of generality, the volume-average stress in the RVE cell14 is Z Z Z 1 1 1 @xj rij dv ¼ rik dkj dv ¼ rik dv V V V V V V @xk Z Z 1 1 ¼ ½ðrik xj Þ;k  rik ;k xj dv ¼ rik nk xj ds ð17Þ V V V @V

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In Section 2.1, a CHT has been introduced which obtains components of the stiffness tensor by solving six elementary BC problems. However, in this energy method expressed in Eq. (7), only one component of the equivalent strain tensor is non-zero, which means that only one component of the equivalent stress tensor can be calculated through Eq. (6) (i.e., only one component of the equivalent stress rkl contributes to the strain energy). Thus, only diagonal components in the stiffness matrix can be acquired by the six elementary BC problems. In order to get all the 9 independent elastic constants of the corresponding homogeneous solid, a bi-axial strain field is applied to obtain the value of Cij (i – j) in Eq. (1). Taking C12 for example, the BC is set as e11 ¼ e22 ¼ e (two uniaxial tensions applied simultaneously) while the displacements in other directions are zero. According to the equality of the strain energy, 1 1 r11 e11 V þ r22 e22 V ¼ U 2 2

ð8Þ

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2U r11 þ r22 ¼ eV

ð9Þ

From Eq. (1),

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r11 ¼ C11 e11 þ C12 e22

ð10Þ

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r22 ¼ C21 e11 þ C22 e22

ð11Þ

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Adding Eqs. (10) and (11), and considering the symmetry of the stiffness matrix,     1 r11 þ r22 1 2U  C11  C22 ¼  C C12 ¼  C 11 22 2 2 e2 V e ð12Þ

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C13 and C23 can also be acquired by exerting similar BCs on the RVE cell. With the diagonal components obtained by the six elementary load cases, the entire stiffness matrix is determined and 9 engineering constants are then calculated from the compliance matrix.

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2.3. Comparative study

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As mentioned in Section 1, different methods have been applied to determine the equivalent stress. In this section, a comparative study is done among the volume-average method, the boundary method, and the energy method.

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where i; j; k 2 f1; 2; 3g. Eq. (17) shows that the average stress depends uniquely on the surface loading. Here, a further proof is given to show that the average stress only relates to the average stress on the surface where the unit displacement is applied. For a rectangular RVE cell, as shown in Fig. 1, the six boundary surfaces are named A to F respectively. Z Z Z rik nk xj ds ¼ rik nk xj dsA þ    rik nk xj dsF @v Z Z ¼  ðri2 xj ÞA dsA þ ðri2 xj ÞB dsB Z Z þ ðri3 xj ÞC dsC  ðri3 xj ÞD dsD Z Z þ ðri1 xj ÞE dsE  ðri1 xj ÞF dsF ð18Þ

(1) Volume-average method

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The equivalent stress in this method is calculated as follows: Z 1 rkl dv rkl ¼ ð13Þ V

Fig. 1

A rectangular RVE.

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C. Qiu et al. Consider the six elementary displacement BCs:

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(1) e11 – 0

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In this periodic media, for the two points having the same local load on Surfaces E and F, x1E  x1F ¼ ð1 þ e11 ÞuEF

ð19Þ

where uEF means the height of the RVE. As a result of the symmetry of both the honeycomb RVE and the applied BCs, the stress also distributes symmetrically, i.e., ( ðr12 ÞA;B ¼ 0 ð20Þ ðr13 ÞC;D ¼ 0 Considering the correspondence between Surfaces E and F in the periodicity of the RVE, Eq. (18) can be written as Z Z Z rik nk xj ds ¼ ðr11 x1 ÞE dsE  ðr11 x1 ÞF dsF @v Z ¼ r11 ð1 þ e11 ÞuEF dsE ð21Þ Then under this BC, the volume-average stress can be determined from Eqs. (17) and (21) as 1 V

Z

uEF r11 dv ¼ V V

Z

1 r11 ð1 þ e11 ÞdsE  sE

Z r11 dsE

ð22Þ

Similar results can be acquired for uniaxial tensions in other two directions (e22 –0 and e33 –0).

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(2) e12 – 0

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In this periodic media, for the two points having the same local load on Surfaces E and F, x2E  x2F ¼ ð1 þ e12 ÞuEF

V

V

where i; j; k 2 f1; 2; 3g. Eq. (27) indicates an equality between the volumeaverage method and the energy method when calculating the components in the stiffness matrix. However, as mentioned in Section 2.2 for the operating process of the energy method, the equilibrium shown in Eq. (27) only exists for the diagonal elements in the stiffness matrix, because non-diagonal components can’t be calculated directly by Eq. (7). In other words, the volume-average method in the calculation of the equivalent stress is only acceptable for diagonal elements like C11, C22, and so on, while divergence exists at non-diagonal elements. Taking the calculation of C12 for example, in the volumeaverage method, C12 is calculated as Z 1 r011 dv C012 ¼ ð28Þ Ve r011

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applied. Eqs. (22) and (26) show an equality of the volumeaverage stress and the boundary stress, which will be discussed later in Section 4. We review Eqs. (7) and (14) to compare the energy method and theR volume-average method. The difference lies in R R rij dv  eij dv and 2V rkl ekl dv. Without loss of generality, for the energy method, Z Z Z V rij eij dv ¼ V rij ui nj ds ¼ V rij eik xk nj ds V @V @V Z Z ¼ Veik rij xk nj ds ¼ Veik dkj rij dv @V V Z Z ¼ eij dv  rij dv ð27Þ

ð23Þ

As a result of the symmetry of both the honeycomb RVE and the applied BCs, the stress also distributes symmetrically, i.e., ( ðr13 ÞC;D ¼ 0 ð24Þ ðr11 ÞE;F ¼ 0 Considering the correspondence between Surfaces E and F in the periodicity of the RVE, Eq. (18) can be written as Z Z Z rik nk xj ds ¼ ðr12 x2 ÞE dsE  ðr12 x2 ÞF dsF @v Z ¼ r12 ð1 þ e12 ÞuEF dsE ð25Þ Then under this BC, the volume-average stress can be determined from Eqs. (17) and (25) as Z Z Z 1 uEF 1 r12 ð1 þ e12 ÞdsE  r12 dsE r12 dv ¼ ð26Þ V V V sE Similar results can be acquired for shear deformations in other two directions (e13 –0 and e23 –0). The above analysis shows that under both the tensions and shear deformations, the volume-average stress only depends on the average stress on the surface where the unit displacement is

where stands for the local stress when e22 ¼ e is applied. Eq. (12) shows the calculation in the energy method, which considers the total strain energy when applied to e11 ¼ e22 ¼ e. On one hand, by using C012 acquired by the volume-average method as Eq. (28), the total strain energy of the RVE under bi-axial BCs e11 ¼ e22 ¼ e is written as

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2

1 Ve ðC11 þ C22 þ 2C012 Þ U0 ¼ Vðr11 e11 þ r22 e22 Þ ¼ 2 2 Z Ve2 ðC11 þ C22 Þ þ e r011 dv ¼ 2

ð29Þ

r011

where means the stress field inside the RVE when a monoaxial strain e22 ¼ e is employed, which is the process of the volume-average method. On the other hand, similar to the problem for the work and energy under several loads (which is often used for the introduction of Maxwell’s reciprocal theorem15), the total strain energy can be written as Z Ve2 ðC11 þ C22 Þ þ r011 e11 dv U¼ ð30Þ 2 In this equation, r011 also means the stress field inside the RVE when a mono-axial strain e22 ¼ e is employed. Comparison R between Eqs. R (29) and (30) shows that divergence lies in e r011 dv and r011 e11 dv. R R R Set d1 ¼ r011 dv  e11 dv and d2 ¼ V r011 e11 dv. For a homogeneous component R material (with a stiffness R R C12), d1 ¼ C12 e22 dv  e11 dv and d2 ¼ C12 V e11 e22 dv. In FE software ABAQUS, strain and stress fields are present in every integral point inside a single element, which

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form the entire mesh. Therefore, the integrals of d1 and d2 transform into the summation of each element. Assuming that in the FE model, the corresponding strain in each element is e22 : x1 ; x2 ; . . . ; xn and e11 : y1 ; y2 ; . . . ; yn , and the volume in each element as v : z1 ; z2 ; . . . ; zn , then d1 ¼ C12 ðx1 z1 þ x2 z2 þ . . . þ xn zn Þðy1 z1 þ y2 z2 þ . . . þ yn zn Þ 0 1 n X X B C ðxi yj þ xj yi Þzi zj A ¼ C12 @ xi y2i z2i þ ð31Þ i¼1

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i;j¼1;2;...;n i
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d2 ¼ C12 ðx1 y1 z1 þ x2 y2 z2 þ . . . þ xn yn zn Þðz1 þ z2 þ . . . þ zn Þ 0 1 n X X B C ðxi yi þ xj yj Þzi zj A ¼ C12 @ xi y2i z2i þ ð32Þ i¼1

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i;j¼1;2;...;n i
Comparison between Eqs. (31) and (32) shows that the inhomogeneity of the strain distribution is the main reason that causes the difference between the volume-average method and the energy method. Numerical results indicate that a relatively greater strain exists in the air zone than that in the wall, and the impact of the variation of the strain within the wall material is insignificant mainly due to its small volume fraction. According to all the above analysis, the difference of the acquired effective properties between the volume-average method and the energy method, especially for the in-plane moduli, depends on the volume fraction of the air zone and the inhomogeneity of the strain field in the RVE cell. Moreover, all these factors mentioned relate to the wall thickness and cell geometries.

Fig. 2

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3. FE model

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3.1. Convergence

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The RVE cell chosen for a single-wall-thickness honeycomb structure is shown in Fig. 2. As the vertical and oblique walls have the same thickness in the single-wall-thickness honeycomb, the cell geometric parameters in Fig. 2(b) can be t1 ¼ t, t2 ¼ t=2. Three parameters including t=l, h=l and h determine the geometric configuration of the RVE cell. In later analysis, we focus on the most commonly used honeycombs, whose l and h remain the same as l = h = 15. This absolute value is not significant as geometries only provide nondimensional coefficients for the effective properties. Moreover, the core height is set to hc = 50 and it is in the range where the core height has little influence on effective properties according to Ref.2 Regarding the honeycomb structure as a two-phase mixture of the core material and air as shown in Fig. 2(c), aluminum alloy is the common material for honeycombs, whose modulus E ¼ 70 GPa and Poisson’s ratio t ¼ 0:3. The so-called ‘‘elastic air” is endowed to the air zone to get the strain field in the air whose modulus and Poisson’s ratio are set as Eair ¼ 0:001 MPa and tair ¼ 0. In order to evaluate whether the elastic constant of elastic air is appropriate, the in-plane shear modulus is chosen owing to its relatively small value compared to other properties. Fig. 3(a) presents different values of the calculated in-plane shear modulus at different properties of elastic air, in which Gc is the converged value. The deviation of the shear modulus with Eair ¼ 1 MPa from the shear modulus with Eair ¼ 104 MPa is about 0.8%, while the deviation of Eair ¼ 103 MPa from 104 MPa is less than 0.001%. This

382

Single-wall-thickness honeycomb structure.

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C. Qiu et al.

Fig. 3

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Convergence analysis of FE model.

shows that a smaller modulus of elastic air does not lead to a significant change of the numerical result. Therefore, it is appropriate to choose 103 MPa as the elastic property of the air zone. Bending deformation dominates in the honeycomb structure under in-plane loading.16 The contributions of axial and shear deformations are also included for the analysis of the out-of-plane properties,17 so a model composed of 3D solid elements is established to take into account all the threedimensional deformations to get strain and stress fields that are more precise. As a result of the hourglass phenomenon existing in the reduced-integration linear element C3D8R (with enhanced hourglass control) in ABAQUS,18 the number of mesh divisions along the cell wall thickness is studied in the bending problem, as shown in Fig. 3(b). When the number of divisions n ¼ 1, the numerical result of the effective shear modulus shows the hourglass phenomenon as the ‘‘zeroenergy mode” which makes the stress field in the wall material almost zero, thus resulting in a very small effective shear modulus obtained by this mesh size. As the number of divisions increases, hourglass is suppressed and a converged modulus is approached. While the deviation between the results of n ¼ 5 and n ¼ 7 is less than 1%, n ¼ 5 is chosen for the number of divisions along the wall thickness. According to Eq. (3), elementary BCs will be applied to get an equivalent strain for the whole RVE cell. The essence of this computational homogeneous method is that the stiffness of the honeycomb structure is replaced by the stiffness of equivalent solids. Therefore, it is of great significance to simulate the deformation of the whole structure accurately. Hence, nonlinear effects brought by the possible large deformation of the wall are taken into account. Based on the above analysis, numerical results considering nonlinear effects at varying strain are shown in Fig. 3(c). The results indicate that a nonlinear effect happens at a strain of 101, whose effective modulus is above 50% higher than that at a 103 strain. The difference between results at 102 and 103 strains is within 1% as well as the difference between those at 103 and 104 strains. Thus, a strain of 103 is the chosen value for the elementary BCs. In the meanwhile, the maximum stress of the honeycomb wall under a 103 strain BC is 21.1 MPa from the FE results, which is lower than the yield stress of 5052 aluminum alloy commonly used in commercial honeycomb structures.19,20 Therefore, the honeycomb wall stays in the linear elastic stage under this loading, and it is appropriate to use this model to calculate the elastic properties.

3.2. Boundary conditions

457

Catapano and Montemurro2 gave detailed BCs when taking into account the symmetries of the unit cell. As discussed by Hori and Nemat-Nasser,21 the homogeneous stress BC and the homogeneous strain BC only provide the lower and upper bounds of effective moduli. The plane-remains-plane homogeneous BCs (or unit-displacement BCs) not only over-constrain the boundaries, but also violate the stress periodicity conditions. In theoretical analysis of the FE model proposed by Catapano and Montemurro2 in the calculation for the in-plane shear modulus G12 and the out-of-plane shear modulus G13, we found that the calculated value differs from different cell numbers.

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3.2.1. G12 under unit-displacement BCs

470

In this part, the accuracy of the RVE cell selected in Fig. 2(b) to simulate the entire honeycomb structure under in-plane shear loading is discussed. The effect of cell numbers on the effective shear modulus is analyzed, based on which a new model (The new model is called Model 2, as the model shown in Fig. 2(c) is called Model 1) is proposed to get more precise properties of in-plane shear under unit-displacement BCs. When the honeycomb structure is subject to a shear loading in the 1-2 direction as shown in Fig. 4, as mentioned before, bending deformation of the walls dominates in the honeycomb core. The deflection of the vertical walls and the rotation of the oblique walls together cause a deformation for the whole honeycomb core under 1-2 shear loading. From the previous theory in literature,22 the proportion of the deflection of the vertical walls in the whole shear deformation on the boundary is

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3

F1 h 24EI F1 h ðhþlÞ F1 hl2 þ 64EI 24EI 2

¼ 84:2%

ð33Þ

sin h

Therefore, the deflection of the vertical walls plays an important role in determining the effective in-plane shear modulus. Fig. 4(a) and (b) illustrate the difference between a single RVE cell and a finite number of RVE cells under the same in-plane shear loading, which show that the local load on the vertical walls is not the same, thus causing different deflections. For a single RVE cell, the deflection of the vertical Wall AE (CF) is

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A method of determining effective elastic properties of honeycomb cores based on equal strain energy

Fig. 4

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w1 ¼

ðF=2Þh3 2Ebt32

Honeycomb structure under shear loading along 1-2 direction.

ð34Þ

While for the honeycomb core consisting of a finite number of RVE cells, the deflection of Wall AE (CF) is 3

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533

w2 ¼

Fh

2Ebð2t2 Þ

ð35Þ

3

Load conditions on other parts of the RVE cell are the same, so there is no difference in the caused shear deformation. From the above analysis, it can be seen that the discrepancy in the deflection of the vertical Wall AE (CF) will result in a difference of the calculated effective shear modulus, and the result of a single RVE cell will be lower than that of the honeycomb core containing a finite number of RVE cells. The essence of this phenomenon is that the process of the homogeneous technique in this loading condition is replacing the shearing stiffness of the honeycomb core by the bending stiffness of the wall. The vertical Wall AE (CF) in the chosen RVE cell has a half-wall thickness, which leads to a considerable reduction in the bending stiffness. However, the halved wall causes little section change of the RVE cell due to the very small volume fraction of the wall in the whole honeycomb core. For these reasons, the effective shear modulus of a single RVE cell is different from that of the whole honeycomb core. To make a single RVE cell precisely simulate the deformation of a real honeycomb core with respect to in-plane shear, We introduce a new model named Model 2 on the basis of the FE model (Model 1) shown in Fig. 2. Only a geometric change is done in Model 2 as t2 = 0.795t from t2 = t/2 in Model 1. For Model 2, the deflection of Wall AE (CF) under the same in-plane loading is w01 ¼

ðF=2Þh

3

2Ebð0:795tÞ

3

¼ w2

7

ð36Þ

This result has the same value of the deflection as in a finite number of RVE cells. FE models containing different numbers of cells as respectively n = 1, 2, 4, 8, 14, 16 are established to evaluate the effect of the cell number on the effective in-plane shear modulus, and these models have the same BCs as those in Ref.2 Calculations have been conducted on honeycombs of three geometries: (1) t ¼ 0:2; h ¼ 30o ; l ¼ h ¼ 15 (2) t ¼ 0:4; h ¼ 30o ; l ¼ h ¼ 15 (3) t ¼ 0:2; h ¼ 45o ; l ¼ h ¼ 15

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The FE results in Fig. 5 indicate that for all the geometric situations, the effective in-plane shear modulus increases as more cells are included in the FE models, which is consistent with the previous analysis. Moreover, all the curves have a tendency of approaching a converged value that is closer to the modulus obtained by Model 2. Therefore, the results presented in Fig. 5 can be a validation for the accuracy of Model 2 to simulate a real honeycomb core under in-plane shear loading.

546

3.2.2. G13 under unit-displacement BCs

554

Similar to Section 3.2.1, the effect of the cell number on the effective out-of-plane shear modulus is analyzed, on the basis of which another new model called Model 3 is proposed to get more precise properties of the out-of-plane shear modulus in the 1-3 direction for the whole honeycomb core. When the honeycomb structure is subject to a shear loading in the 1-3 direction as shown in Fig. 6, the overall deformation of the honeycomb core is governed by the shear deformations of the vertical and oblique walls. We conduct an analysis on the shear flows inside each wall in a similar way used by Kelsey et al.23

555

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C. Qiu et al. From equilibrium conditions, 8 qb ¼ qc þ qd > > > < qa þ qc ¼ qb > sðh þ l sin hÞ  2l cos h ¼ ðqb þ qc Þl cos h > > : qa  h2 þ ðqb  qc Þl sin h þ ðqd þ qe Þ  h2 ¼ 0 

Fig. 5 Effective shear modulus vs number of RVE cells (Model 1 t2 = t/2, Model 2 t2 = 0.795t).

566 567 568 569

For the single RVE cell presented in Fig. 6(a) under 1-3 shear loading (with a shear stress s), the following equation is acquired by equilibrium conditions: 8 q ¼ qd > > > b > > > > qc ¼ qe > > > < qa þ qc ¼ qb > > > > > sðh þ l sin hÞ  2l cos h ¼ ðqb þ qc Þl cos h > > > > > : qa  h2 þ ðqb  qc Þl sin h þ ðqd  qe Þ  h2 ¼ 0

ð37Þ

571 572 573

575 576 577 578 580

From Eq. (37), 

qa ¼ 0 qb ¼ qc ¼ qd ¼ qe ¼ sðh þ l sin hÞ

ð38Þ

For the real honeycomb core consisting of a finite number of RVE cells, under the consideration of periodicity, qa ¼ qd ¼ qe

ð39Þ

Fig. 6

581 582

ð40Þ 584

From Eqs. (39) and (40), we get qa ¼ qd ¼ qe ¼ 0 qb ¼ qc ¼ sðh þ l sin hÞ

585 586

ð41Þ

Comparison between Eqs. (38) and (41) shows that under the same shear loading in the 1-3 direction, the shear flow in Wall AE as well as CF varies in a single RVE cell and a finite number of RVE cells. In a single RVE cell, similar to the situation in the simulation of in-plane shear loading, Walls AE and CF suffer higher shear flows than those in a finite number of RVE cells, leading to differences in the deformation for the entire honeycomb structure. The higher shear flows qd and qe cause the in-plane bending deformation of the oblique walls as shown in Fig. 7 from the FE analysis of a single RVE cell. It can be seen that the extra bending deformation of the oblique walls is in a direction contributing to the 1-3 shear deformation, and FE results illustrate that this bending deformation is weakening as the cell number increases. For these reasons, a single RVE cell has a larger deformation than that of a finite number of RVE cells under the same loading owing to the bending deformation of the oblique walls, thus having a lower effective shear modulus in the 1-3 direction than that of a finite number of RVE cells. In order to find an appropriate RVE model to simulate the deformation of the real honeycomb structure under the 1-3 direction shearing precisely, the thickness of the oblique walls is adjusted to suppress the extra bending deformation. The geometry of the proposed new model called Model 3 is determined according to the FE results shown in Fig. 8. Fig. 8 shows the effective shear modulus along the 1-3 direction of a RVE cell with a geometry of t = 0.2, h ¼ 30 , and l = h = 15 at different thicknesses of the oblique walls.

Honeycomb structure under shear loading along 1-3 direction.

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A method of determining effective elastic properties of honeycomb cores based on equal strain energy

9

Fig. 7 In-plane bending deformation of oblique walls under shear loading along 1-3 direction.

635

The horizontal dotted line in Fig. 8 represents the converged value of the finite number of RVE cells. It can be seen that when the thickness of the oblique walls t1 = 0.243 = 1.215t, we get a value very close to the converged value for G13. According to these results, t1 = 1.215t is chosen as the geometric change of Model 3 from Model 1, and the accuracy of Model 3 is going to be evaluated later in Fig. 9. Similar to the in-plane shear loading, FE models containing different numbers of cells as respectively n = 1, 2, 4, 8, 14, 16 are also used to evaluate the effect of the cell number on the effective out-of-plane shear modulus along the 1-3 direction, and honeycombs of three geometries are also taken into consideration, as shown in Fig. 9. With the number of cells increasing, the effective shear modulus G13 grows and approaches a converged value, as analyzed before. In addition, the converged value is very close to the result of Model 3 for all the three cell geometries. Therefore, this new model can provide a relatively more accurate value of G13 than that of Model 1 under this shear loading.

636

3.2.3. Periodic boundary conditions

637

As stated before, a single RVE cell cannot provide accurate G12 and G13 as those of the whole honeycomb core with unit-displacement BCs. However, when applied to periodic BCs, the effective properties remain constant with the cell number changing. Moreover, periodic BCs are required in determining the elastic properties of periodic media to ensure the periodicity of displacements and tractions on the boundary of the RVE. In order to generate a symmetrical mesh for the convenience of prescribing the periodic BCs, a whole unit cell is remained for FE analysis.

617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634

638 639 640 641 642 643 644 645 646

Fig. 9 Effective shear modulus vs number of RVE cells (Model 1 t1 = t, Model 3 t1 = 1.215t).

Mathematical expressions can be found in the literature by Whitcomb,24 Xia,25 Li and Wongsto,26 which have been used in periodic media like unidirectional composites and plane and satin weave composites. Constraint equations (CEs) are utilized for periodic BCs of the rectangular solid RVE shown in Fig. 1. These equations can be sorted into three categories, i.e., equations for surfaces, edges, and vertices. (1) Equations for surfaces

647 648 649 650 651 652 653

654 655

Under three uniaxial tensions and shear deformations (e011 ; e022 ; e033 ; c012 ; c013 ; c023 ), CEs can be applied to three pairs of surfaces. For surfaces perpendicular to 1-axis, i.e., Surfaces E and F: 8 0 > < ux¼W1  ux¼0 ¼ e11 W1 ð42Þ vx¼W1  vx¼0 ¼ 0 > : wx¼W1  wx¼0 ¼ 0

656

For surfaces perpendicular to 2-axis, i.e., Surfaces A and B: 8 0 > < uy¼W2  uy¼0 ¼ c12 W2 ð43Þ vy¼W2  vy¼0 ¼ e022 W2 > : wy¼W2  wy¼0 ¼ 0

663 664

For surfaces perpendicular to 3-axis, i.e., Surfaces C and D: 8 0 > < uz¼W3  uz¼0 ¼ c13 W3 0 ð44Þ vz¼W3  vz¼0 ¼ c23 W3 > : wz¼W3  wz¼0 ¼ e033 W3

667 668

657 658 659 660

662

666

670 671

(2) Equations for edges

672 674 673

Fig. 8 Effective shear modulus along 1-3 direction vs thickness of oblique wall AB (BC).

Two or three in Eqs. (42)(44) are satisfied for nodes on the edges of the RVE. As these constraints are not independent, FE analysis cannot function properly if the CEs for edges are not considered separately as well as the CEs for vertices. For edges parallel to the 1-axis, i.e., Lines hd, ed, fb, and gc: 8 0 > < uea  uhd ¼ e11 W1 ð45Þ vea  vhd ¼ 0 > : wea  whd ¼ 0

675

8 0 0 > < ufb  uhd ¼ e11 W1 þ c12 W2 0 vfb  vhd ¼ e22 W2 > : wfb  whd ¼ 0

683

676 677 678 679 680

682

ð46Þ

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No. of Pages 14

10 8 0 > < ugc  uhd ¼ c12 W2 0 vgc  vhd ¼ e22 W2 > : wgc  whd ¼ 0

C. Qiu et al.

ð47Þ

Similar equations can be given for edges parallel to the 2-axis and 3-axis. (3) Equations for vertices

691

cases. Therefore, it can be concluded that the volumeaverage stress is equal to the boundary stress regardless of changes in cell geometries and forms. As discussed in Section 2.3, Eqs. (22) and (24) show an equality of the volume-average stress and the boundary stress, which is validated by the results in Table 1. Understanding such an equality, only the energy method and the volumeaverage method are operated in later FE analysis.

727

4.2. Results by energy method

735

In this part, we have compared the numerical results by the volume-average method and the energy method. Different cell geometries such as the wall thickness and the angle between vertical and oblique walls are taken into consideration to evaluate the supposed discrepancy between the two methods. Fig. 10 shows the numerical results of the energy method and the volume-average method at different wall thicknesses which range from 0.2 to 1.0 as the RVE cell has a dimension of l = h = 15. It can be seen in Fig. 10 that the three shear moduli G12, G13, and G23 calculated by the volume-average method and the energy method are all nearly the same with a maximum difference of less than 3%. Since the shear moduli only relate to the diagonal elements in the stiffness matrix while calculated from the compliance matrix, this equilibrium of these three shear moduli can be a validation for Eq. (27). Nevertheless, for the elastic properties relate to non-diagonal components (i.e. E1, E2, E3, t12, t13, and t23), divergences exist between the volume-average method and the energy method. Fig. 10(d) shows that the volume-average method gets the same results as those of Malek & Gibson’s model, which validates our proper use of the volume-average method. As stated in Section 2.3, the main motivation that we put forward this energy method is the supposed discrepancy in the calculation of C12. It has been proven in Section 2.3 that an inaccurate calculation of C12 in the volume-average method leads to an inaccurate strain energy under bi-axial BCs. The discrepancy of the strain energy in each model of unit cells is presented in Fig. 11. It can be seen that the discrepancy remains 1.3–1.6% within our computing range. Although the relative error remains nearly constant, the absolute value increases as the wall thickness increases. For the in-plane elastic properties E1 and E2, within the range of calculation, the absolute value of the discrepancy between the two methods varies with the cell wall thickness increasing. However, in these geometries, as the wall thickness increases, the relative error between these two methods is slightly changed from nearly 8% to 6%. This follows from

736

728 729 730 731 732 733 734

692 693 694 695 696 697

699 700

702

Point d is fixed to avoid rigid-body motions of the RVE. Then seven CEs are defined for other vertices with the reference of Point d. Equations for Points f and g are given here as examples: 8 0 0 0 > < uf  ud ¼ e11 W1 þ c12 W2 þ c13 W3 0 0 ð48Þ vf  vd ¼ e22 W2 þ c23 W3 > : 0 wf  wd ¼ e33 W3 8 0 0 > < ug  ud ¼ c12 W2 þ c13 W3 vg  vd ¼ e022 W2 þ c023 W3 > : wg  wd ¼ e033 W3

ð49Þ

710

Periodic BCs are prescribed in the ABAQUS software combined with Python scripts. Python scripts are used to find nodes on surfaces, edges, and vertices, generate CEs for corresponding nodes in the mesh according to Eqs. (42)(49), and submit jobs in the ABAQUS environment. After jobs are finished, post-processing Python scripts are executed to calculate effective moduli by both the volume-average method and the energy method.

711

4. Results

712

4.1. Volume-average stress and boundary stress

713

A total of six cases as listed in Table 1 are analyzed. Controlling parameters include the form of the cell (single-wall thickness and double-wall thickness), the thickness of the wall (t = 0.2 and t = 0.4), and the angle between vertical and oblique walls (h ¼ 30 and h ¼ 45 ). The reason for choosing these parameters is that later analysis has shown that the energy method and the volume-average method have larger divergences under the chosen geometries. For each case, the volume-average stress and the boundary stress are both used to get the effective modulus. E1 =E01 in Table 1 means the ratio of the modulus obtained by the boundary stress to that by the volume-average stress. An inspection of Table 1 shows that the values of E=E0 , G=G0 , and t=t0 in all directions are very close to unity for all

703 704 705 706 707 708 709

714 715 716 717 718 719 720 721 722 723 724 725 726

Table 1

Comparison between boundary stress method and volume-average stress method.

t

h (°)

Feature

E1 E01

E2 E02

E3 E03

t12 t012

t13 t013

t23 t023

G12 G012

G13 G013

G23 G023

0.2 0.4 0.2 0.2 0.4 0.2

30 30 45 30 30 45

Single-wall Single-wall Single-wall Double-wall Double-wall Double-wall

1 0.998 1.001 1 1.001 1

1.001 1.001 0.999 1.001 1 1.001

1 1 0.998 1 1 0.999

0.999 0.997 0.996 1.002 1.001 1.002

1 0.999 1 0.999 0.999 1

1 0.999 1 1 0.999 1.002

0.999 0.999 1.002 1 0.999 1.001

0.998 1.002 0.998 0.998 1.002 0.997

0.998 1 1 0.999 1.001 0.997

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A method of determining effective elastic properties of honeycomb cores based on equal strain energy

Fig. 10

Numerical results of strain energy method and volume-average stress method vs wall thickness.

Fig. 11 Discrepancy of strain energy of honeycomb cores under bi-axial BCs.

774 775 776 777 778 779 780

11

the fact that the discrepancy of the strain energy under bi-axial BCs remains almost the same as shown in Fig. 11, representing the difference of non-diagonal elements in the stiffness matrix. The numerical results for E3 have little discrepancy due to the homogeneous strain and stress fields of the whole honeycomb structure under uniaxial tensions in the 3 directions. As shown in Eqs. (31) and (32), uniform strain and stress fields do not

lead to a difference between d1 and d2. The curves of t13 and t23 show a similar tendency to those of the in-plane moduli E1 and E2, which conforms to the hypothesis proposed by Malek and Gibson as t31 = t32 = t where t is the Poisson’s ratio of the cell wall material. Fig. 12 shows the numerical results of the energy method and the volume-average method with analytical solutions at different angles between vertical and oblique walls. Cell geometries of three angles h ¼ 30 ; 45 ; 60 and three cell wall thicknesses t = 0.2, 0.3, 0.4 are conducted in calculation. Similar conclusions can be drawn from results illustrated in Fig. 12 with those in Fig. 10. On one hand, approximately the same values for three shear moduli are acquired by the volume-average and energy methods. On the other hand, divergence exists for other properties related to non-diagonal components of the stiffness matrix and this divergence varies with different cell geometries. Results in Fig. 12 show that the deviation of effective properties obtained by both the volume-average method and the energy method varies as the angle changes from 30° to 60°. Nevertheless, effective elastic properties obtained using the energy method remain at values that are lower than those using the volume-average method. According to the experiment data of a regular honeycomb (h/l = 1) provided in

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No. of Pages 14

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Fig. 12

C. Qiu et al.

Numerical results of strain energy method and volume-average stress method vs angle between vertical and oblique walls.

820

Ref.2 which are also slightly lower than their numerical results (based on the volume-average method), results by the energy method may provide more accurate values. In a consistency with Fig. 10, numerical results at angles of 45° and 60° also indicate a slight increase of the discrepancy between the two methods as the wall thickness increases. From the results in Figs. 10–12, it can be concluded that the energy method gets different values of effective properties of a thin-wall thickness honeycomb from those of the volumeaverage method regardless of the variation of the angle between vertical and oblique walls. It can be proven that the discrepancy of the strain energy under bi-axial BCs can be considered as a symbol for the difference of the two methods in determining effective properties. The inaccuracy in calculating the strain energy shows the inaccurate moduli acquired by the volume-average method.

821

4.3. Comments on discrepancy

822

As shown in Fig. 11, the discrepancy of the strain energy under bi-axial BCs between the volume-average method and the energy method is very small. Here, three 4-element models are established. On one hand, the 4-element models show a possible big deviation of the strain energy between the two methods, which represents the need of performing numerical bi-axial tests. On

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823 824 825 826 827 828

the other hand, different models may have different inhomogeneity of the strain distribution, which we think results in the discrepancy between the two methods. The models are shown in Fig. 13. Four C3D8I elements are utilized to form the models. Each element is assigned specific material properties that are preset in Table 2. The material properties are set to present the supposed discrepancy between d1 and d2 according to their expressions analyzed in Eqs. (31) and (32). The results of these 4-element models are shown in Table 3. From the results, it can be found that the deviation between d1 and d2 can be larger than 50% in Model 3. However, since C12 is about one sixth of C11 and C12, the deviation of the strain energy reduces to 7.28%. The 4-element models can prove the discrepancy of the strain energy under bi-axial BCs.

Fig. 13

Four-element model composed of 4 materials.

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A method of determining effective elastic properties of honeycomb cores based on equal strain energy Table 2

Material properties used in 4-element models.

Material

E1 (GPa)

E2 (GPa)

E3 (GPa)

t12

t13

t23

G12 (MPa)

G13 (MPa)

G23 (MPa)

1-1 1-2 1-3 1-4

5 10 10 5

10 5 5 10

10 10 10 10

0.3 0.3 0.3 0.3

0 0 0 0

0 0 0 0

100 100 100 100

100 100 100 100

100 100 100 100

2-1 2-2 2-3 2-4

2 10 10 2

10 2 2 10

10 10 10 10

0.01 0.3 0.3 0.01

0 0 0 0

0 0 0 0

100 100 100 100

100 100 100 100

100 100 100 100

3-1 3-2 3-3 3-4

103 10 10 103

10 103 103 10

10 10 10 10

0.001 0.3 0.3 0.001

0 0 0 0

0 0 0 0

100 100 100 100

100 100 100 100

100 100 100 100

Table 3

Results of 4-element models.

Model Model-1 Model-2 Model-3

844 845 846 847 848 849 850

851 852 853 854

Maximum e11 3

1.3  10 1.7  103 1.8  103

Minimum e11 3

0.7  10 0.3  103 7.9  107

Deviation between d1 and d2 (%)

Deviation of strain energy (%)

8.42 24.17 57.14

1.27 3.41 7.28

It can also be seen that the deviation between d1 and d2 varies in the three models as well as the deviation of the strain energy. The maximum and minimum strains are also listed in Table 3. It can be seen that in the 4-element models, the deviation between d1 and d2 increases with the inhomogeneity of strain increasing. From these results, we get: (1) The discrepancy between the two methods varies with different problems. (2) The inhomogeneity of the strain distribution influences that discrepancy.

855

856

858 857 859 860 861 862 863 864 865 866 867 868 869

5. Conclusions (1) An equality for the volume-average method and the boundary stress method previously used in predicting the effective moduli is validated, and this equality remains constant as cell geometries and cell forms vary. (2) An energy method is proposed based on the equality of the strain energy of the RVE cell and the corresponding homogeneous solid. Analysis has been done on the discrepancy between the energy method and the volume-average method. Numerical results show that the energy method obtains values of effective properties closer to experimental results for in-plane moduli.

870

871 872 873 874

13

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2. Catapano A, Montemurro M. A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I: Homogenization of core properties. Compos Struct 2014;118:664–76. 3. Catapano A, Jumel J. A numerical approach for determining the effective elastic symmetries of particulate-polymer composites. Compos Part B Eng 2015;78:227–43. 4. Montemurro M, Catapano A, Doroszewski D. A multi-scale for the simultaneous shape and material optimization of sandwich panels with cellular core. Compos Part B Eng 2016;91:458–72. 5. Malek S, Gibson L. Effective elastic properties of periodic hexagonal honeycombs. Mech Mater 2015;91(1):226–40. 6. Shi GY, Tong P. Equivalent transverse shear stiffness of honeycomb cores. Int J Solids Struct 1995;32(10):1383–93. 7. Li K, Gao XL, Subhash G. Effects of cell shape and cell wall thickness variations on the elastic properties of two-dimensional cellular solids. Int J Solids Struct 2005;42(5–6):1777–95. 8. Papka SD, Kyriakides S. Experiments and full-scale numerical simulations of in-pane crushing of a honeycomb. Acta Mater 1998;46(8):2765–76. 9. Yu W, Tang T. Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials. Int J Solids Struct 2007;44(11–12):3738–55. 10. Sun CT, Vaidya RS. Prediction of composite properties from a representative volume element. Compos Sci Technol 1996;56 (2):171–9. 11. Hollister SJ, Kikuchi N. A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput Mech 1992;10(2):73–95. 12. Karakoc A, Freund J. Experimental studies on mechanical properties of cellular structures using NomexÒ honeycomb cores. Compos Struct 2012;94(6):2017–24. 13. Shen GL, Hu GK, Liu B. Mechanics of composite materials. 2nd ed. Beijing: Tsinghua University Press; 2013, p. 45–6. 14. Hill R. Elastic properties of reinforced solids: Some theoretical principles. J Mech Phys Solids 1963;11(5):357–72. 15. Beer FP, Johnston Jr ER, DeWolf JT, Mazurek DF. Mechanics of materials. 6th ed. Columbus, OH: McGraw-Hill Education; 2012, p. 732–4.

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16. Gibson LJ, Ashby MF, Schajer GS, Robertson CI. The mechanics of two-dimensional cellular materials. Proceed Roy Soc Lond Ser A, Math Phys Sci 1982;382(1782):25–42. 17. Burton WS, Noor AK. Assessment of continuum models for sandwich panel honeycomb cores. Comput Method Appl Mech 1997;145(3):341–60. 18. Belytschko T, Liu WK, Kennedy JM. Hourglass control in linear and nonlinear problems. Comput Method Appl Mech 1984;43 (3):251, p. 251–276. 19. Hexcel Corporation. HexWebTM honeycomb attributes and properties—A comprehensive guide to standard Hexcel honeycomb materials, configurations, and mechanical properties[Internet]; 1999. Available from [updated 2016, cited 2016 Jan 4]. 20. Kutz M. Handbook of materials selection. New York: John Wiley & Sons, Inc.; 2002, p. 100–3.

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C. Qiu et al. 21. Hori M, Nemat-Nasser S. On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mech Mater 1999;31(10):667–82. 22. Wang YJ. Note on the in-plane shear modulus of honeycomb structures. Chin J Appl Mech 1993;10(1):93–7 [Chinese]. 23. Kelsey S, Gellatly RA, Clark BW. The shear modulus of foil honeycomb cores. Aircr Eng Aerosp Tecnol 1958;30(10):294–302. 24. Whitcomb JD, Chapman CD, Tang X. Derivation of boundary conditions for micromechanics analyses of plain and satin weave composites. Surf Sci 2000;34(9):724–47. 25. Xia Z, Zhang Y, Ellyin F. A unified periodical boundary conditions for representative volume elements of composites and applications. Int J Solids Struct 2003;40(8):1907–21. 26. Li S, Wongsto A. Unit cells for micromechanical analyses of particle-reinforced composites. Mech Mater 2004;36(7):543–72.

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