As shown in Fig. 5(a-b), the equilibrium equation of the BHC subjected to forces can be expressed as:

## y and z direction

In the same way as the effective elastic modulus analysis of the cell in the *x* direction, the force sketch of the core, when the cell is subjected to *P**y* load in the *y* direction, is shown in Fig. 5(c-d), and the equilibrium equation of the BHC subjected to forces can be expressed as:

$${\text{P}}_{\text{y}}^{\text{*}}\text{=}{\text{σ}}_{\text{y}}\text{l}\text{cos}\text{θ}\text{∙}\text{T}\text{ }\text{, }{\text{M}}_{\text{y}}\text{=}\frac{\text{1}}{\text{2}}{\text{P}}_{\text{y}}^{\text{*}}\text{l}\text{cos}\text{θ}$$

29

where \({\text{P}}_{\text{y}}^{\text{*}}\) is the load on the cell in the *y* direction, *σ**y* is the stress generated when the force is applied, and *M**y* is the moment.

The modulus of elasticity \({\text{E}}_{\text{y}}^{\text{*}}\) of the sandwich-inclined cell wall under stress is calculated as:

$$\left\{\begin{array}{c}{\text{E}}_{\text{y}}^{\text{*}}\text{=}\frac{{\text{σ}}_{\text{y}}}{{\text{ε}}_{\text{yy}}} \text{,}\text{ }{\text{ε}}_{\text{yy}}\text{=}\frac{{\text{w}}_{\text{y}}}{\text{h}\text{+}\text{l}\text{sin}\text{θ}}\\ {\text{w}}_{\text{y}}\text{=}{\text{-}\text{w}}_{\text{ay}}\text{sin}\text{θ}\text{+}{\text{w}}_{\text{sy}}\text{cos}\text{θ}\text{+}{\text{w}}_{\text{by}}\text{cos}\text{θ}\text{+}{\text{2}\text{w}}_{\text{h}\text{y}}\end{array}\right.$$

30

where *ε**yy* is the strain when the force is applied, *w**y* is the total deformation when the force is applied, *w**ay*, *w**sy*, and *w**by* are the axial compression, shear deformation, and bending deformation of the inclined wall, respectively, when the force is applied (Fig. 5(c)), and *w**hy* is the compression produced by the vertical cell wall in the *y* direction (Fig. 5(d)), and their calculated expressions are as follows:

$$\left\{\begin{array}{c}{\text{w}}_{\text{ay}}\text{=}\frac{{\text{P}}_{\text{y}}{\text{l}}_{\text{e}}\text{sin}\text{θ}}{{\text{E}}_{\text{y}}\text{T}{\text{∙}\text{t}}_{\text{l}}}\text{,}{\text{w}}_{\text{sy}}\text{=}\frac{{\text{P}}_{\text{y}}{{\text{l}}_{\text{e}}}^{\text{3}}\text{cos}\text{θ}}{\text{12}{\text{E}}_{\text{y}}{\text{I}}_{\text{l}}}\left(\text{2.4+1.5}{\text{ν}}_{\text{yx}}\right){\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{2}}\text{=}\frac{{\text{P}}_{\text{y}}{\text{l}}_{\text{e}}\text{cos}\text{θ}}{{\text{E}}_{\text{y}}\text{T}{\text{∙}\text{t}}_{\text{l}}}\left(\text{2.4+1.5}{\text{ν}}_{\text{yx}}\right)\\ {\text{w}}_{\text{by}}\text{=}\frac{{\text{P}}_{\text{y}}{{\text{l}}_{\text{e}}}^{\text{3}}\text{cos}\text{θ}}{\text{12}{\text{E}}_{\text{y}}{\text{I}}_{\text{l}}}\text{=}\frac{{\text{P}}_{\text{y}}{{\text{l}}_{\text{e}}}^{\text{3}}\text{cos}\text{θ}}{{\text{E}}_{\text{y}}\text{T}\text{∙}{{\text{t}}_{\text{l}}}^{\text{3}}}\text{,}\text{ }{\text{w}}_{\text{h}\text{y}}\text{=}\frac{{\text{P}}_{\text{y}}{\text{h}}_{\text{e}}}{{\text{E}}_{\text{y}}\text{T}\text{∙}{\text{t}}_{\text{h}}}\end{array}\right.$$

31

where *E**y* and *ν**yx* are the modulus of elasticity and Poisson’s ratio in the *y* direction of the substrate, respectively, *h**e* is the effective bending length of the vertical cell wall of the sandwich, and *t**h* is the minimum thickness value of the inclined cell wall.

The equivalent elastic modulus parameter \({\text{E}}_{\text{y}}^{\text{*}}\) expression of the BHC in the *y*-axis direction can be obtained using the above equation:

$${\text{E}}_{\text{y}}^{\text{*}}\text{=}{\text{E}}_{\text{y}}{\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{3}}\frac{\left(\text{h}\text{+}\text{l}\text{sin}\text{θ}\right)}{\text{l}{\text{cos}}^{\text{3}}\text{θ}}\left[\frac{\text{1}}{\text{1+}\left(\text{2.4+1.5}{\text{ν}}_{\text{y}\text{x}}\text{+}{\text{tan}}^{\text{2}}\text{θ}\text{+2}\frac{{\text{h}}_{\text{e}}}{{\text{l}}_{\text{e}}}\frac{{\text{t}}_{\text{l}}}{{\text{t}}_{\text{h}}}{\text{sec}}^{\text{2}}\text{θ}\right){\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{2}}}\right]$$

32

The equivalent effect variation \({\text{ε}}_{\text{y}\text{x}}^{\text{*}}\) produced by the BHC when the force is applied in the *y* direction is expressed as follows:

$${\text{ε}}_{\text{y}\text{x}}^{\text{*}}\text{=}\frac{{\text{w}}_{\text{y}\text{x}}}{\text{l}\text{cos}\text{θ}}\text{=}\frac{{\text{w}}_{\text{a}\text{y}}\text{cos}\text{θ}\text{+}{\text{w}}_{\text{s}\text{y}}\text{sin}\text{θ}\text{-}{\text{w}}_{\text{b}\text{y}}\text{cos}\text{θ}}{\text{l}\text{cos}\text{θ}}\text{=}{\text{E}}_{\text{y}}{\left(\frac{{\text{l}}_{\text{e}}}{{\text{t}}_{\text{l}}}\right)}^{\text{3}}\frac{\text{sin}\text{θ}}{\text{l}}\left[\frac{\text{1}}{\left(\text{1.4+1.5}{\text{ν}}_{\text{y}\text{x}}\right){\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{2}}\text{-}\text{cot}\text{θ}}\right]$$

33

where *w**yx* is the deformation in the *x* direction when the cell is subjected to a force in the *y* direction.

From Eqs. 31 and 33, the Poisson’s ratio \({\text{υ}}_{\text{y}\text{x}}^{\text{*}}\) of the BHC in the *y* direction can be expressed as:

$${\text{υ}}_{\text{y}\text{x}}^{\text{*}}\text{=}\left|\frac{{\text{ε}}_{\text{y}\text{x}}^{\text{*}}}{{\text{ε}}_{\text{yy}}}\right|\text{=}\left|\frac{\text{sin}\text{θ}\left(\text{h}\text{+}\text{l}\text{sin}\text{θ}\right)}{\text{l}{\text{cos}}^{\text{2}}\text{θ}}\left[\frac{\text{1+}\left(\text{1.4+1.5}{\text{ν}}_{\text{y}\text{x}}\right){\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{2}}}{\text{1+}\left(\text{2.4+1.5}{\text{ν}}_{\text{y}\text{x}}\text{+}{\text{tan}}^{\text{2}}\text{θ}\text{+2}\frac{{\text{h}}_{\text{e}}}{{\text{l}}_{\text{e}}}\frac{{\text{t}}_{\text{l}}}{{\text{t}}_{\text{h}}}{\text{sec}}^{\text{2}}\text{θ}\right){\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{2}}}\right]\right|$$

34

When the cell is subjected to axial load in the out-of-plane *z* direction, the honeycomb cell wall is mainly subjected to compressive force. Therefore, the stress and outof-plane Young’s modulus of the BHC in the *z* direction can be calculated from the deformation generated by the load (Sorohan et al. 2018a):

$$\left\{\begin{array}{c}{\text{P}}_{\text{z}}^{\text{*}}\text{=}{\text{σ}}_{\text{z}}\left({\text{l}}^{\text{2}}\text{sin}\text{2}\text{θ}\text{+2}\text{hl}\text{cos}\text{θ}\text{-}\text{πab}\right)\\ {\text{E}}_{\text{z}}^{\text{*}}\text{=}{\text{E}}_{\text{z}}\text{∙}\frac{{\text{ρ}}^{\text{*}}}{{\text{ρ}}_{\text{s}}}\end{array}\right.$$

35

where \({\text{P}}_{\text{z}}^{\text{*}}\) and \({\text{E}}_{\text{z}}^{\text{*}}\) are the damage load and equivalent modulus of elasticity of the core in the *z* direction, respectively, and *E**z* is the modulus of elasticity of the base material in the *z* direction.

The effective Poisson’s ratio out-of-plane of the face of the material is equal to the Poisson’s ratio of the base of the material. The finite element analysis method and the correction formula of Poisson’s ratio for anisotropic materials can be obtained from the equivalent Young’s modulus of elasticity and Poisson’s ratio of the BHC in the *x*, *y* and *z* directions as follows (Malek and Gibson 2015; Zhao et al. 2020):

$$\left\{\begin{array}{c}{\text{υ}}_{\text{z}\text{x}}^{\text{*}}\text{=}{\text{υ}}_{\text{z}\text{x}}\text{ }\text{,}\text{ }{\text{υ}}_{\text{zy}}^{\text{*}}\text{=}{\text{υ}}_{\text{zy}}\\ {\text{υ}}_{\text{x}\text{z}}^{\text{*}}\text{=}\frac{{\text{E}}_{\text{x}}^{\text{*}}}{{\text{E}}_{\text{z}}^{\text{*}}}{\text{υ}}_{\text{z}\text{x}}^{\text{*}}\text{ }\text{,}\text{ }{\text{υ}}_{\text{yz}}^{\text{*}}\text{=}\frac{{\text{E}}_{\text{y}}^{\text{*}}}{{\text{E}}_{\text{z}}^{\text{*}}}{\text{υ}}_{\text{zy}}^{\text{*}}\end{array}\right.$$

36

**Calculation of effective shear modulus**

The calculation of the effective shear modulus of BHC includes two cases of in-plane shear modulus \({\text{G}}_{\text{xy}}^{\text{*}}\) and out-plane shear modulus \({\text{G}}_{\text{xz}}^{\text{*}}\) and \({\text{G}}_{\text{yz}}^{\text{*}}\), which are analyzed as follows.

**Shear modulus** \({\text{G}}_{\text{xy}}^{\text{*}}\)

The shear diagram of the adjacent BHC wall subjected to a shear load *P* is shown in Fig. 6.

As shown in Fig. 6(b and d), the shear stress *τ**xy* and the combined moment *M**xy* in the *xy* direction on the vertical cell wall of a single sandwich core cell wall are calculated as:

$${\text{τ}}_{\text{xy}}\text{=}\frac{\text{P}}{\text{4}\text{l}\text{cos}\text{θ}\text{∙}\text{T}}\text{, }{\text{M}}_{\text{xy}}\text{=}\frac{\text{Ph}}{\text{8}}$$

37

where *τ**xy* is the vertical cell wall shear stress in the *xy* direction, *P* is the shear force, and *M**xy* is the combined moment.

As shown in Fig. 6(c), the shear force *S* on the inclined cell wall of a single cell is calculated as follows:

$$\text{S}\text{=}\frac{\text{P}\text{(}\text{h}\text{+}\text{l}\text{sin}\text{θ}\text{)}}{\text{4}\text{l}\text{cos}\text{θ}}$$

38

As shown in Fig. 6(e), the deflection angle *ψ* generated by the deflection when the vertical cell wall of a single sandwich core is stressed is calculated below:

$$\text{ψ}\text{=}\frac{\text{P}{\text{h}}_{\text{e}}\text{l}}{\text{4}{\text{8}\text{E}}_{\text{x}}{\text{I}}_{\text{h}}}\text{ }\text{, }{\text{I}}_{\text{h}}\text{=}\frac{\text{1}}{\text{12}\text{T}\text{∙}{{\text{t}}_{\text{h}}}^{\text{3}}}$$

39

where *I**h* is the moment of inertia generated by force along the *y* direction of the vertical cell wall.

The deflection *µ**h* and shear strain *γ**h* of the vertical cell wall of the BHC due to the rotation angle *ψ* in the *x* direction are calculated as follows:

$$\left\{\begin{array}{c}{\text{μ}}_{\text{h}}\text{=}\frac{{\text{h}}_{\text{e}}}{\text{2}}\text{ψ}\text{+}\frac{\text{P}}{\text{6}{\text{E}}_{\text{x}}{\text{I}}_{\text{h}}}{\left(\frac{{\text{h}}_{\text{e}}}{\text{2}}\right)}^{\text{3}}\text{+}\frac{\text{P}{\text{h}}_{\text{e}}^{\text{3}}}{\text{4}{\text{8}\text{E}}_{\text{x}}{\text{I}}_{\text{h}}}\left(\text{2.4+1.5}{\text{ν}}_{\text{xy}}\right){\left(\frac{{\text{t}}_{\text{h}}}{{\text{h}}_{\text{e}}}\right)}^{\text{2}}\\ {\text{γ}}_{\text{h}}\text{=}\frac{\text{2}{\text{μ}}_{\text{h}}}{\text{h}\text{+}\text{l}\text{sin}\text{θ}}\text{=}\frac{\text{P}}{\text{4}{\text{E}}_{\text{x}}\text{T}\text{∙}{{\text{t}}_{\text{h}}}^{\text{3}}}\frac{{{\text{h}}_{\text{e}}}^{\text{2}}{\text{l}}_{\text{e}}}{\text{h}\text{+}\text{l}\text{sin}\text{θ}}\left[{\text{l}}_{\text{e}}\text{+2}{\text{h}}_{\text{e}}\text{+2}{\text{h}}_{\text{e}}\left(\text{2.4+1.5}{\text{ν}}_{\text{xy}}\right){\left(\frac{{\text{t}}_{\text{h}}}{{\text{h}}_{\text{e}}}\right)}^{\text{2}}\right]\end{array}\right.$$

40

where *µ**h* is the deflection generated by the rotational force of the vertical cell wall along the *x* direction, and *γ**h* is the shear strain.

Similarly, the axial compression *µ**la* and shear deformation *µ**ls* of the BHC inclined cell wall in the *y* direction caused by the turning angle are calculated below:

$$\left\{\begin{array}{c}{\text{μ}}_{\text{la}}\text{=}\frac{\text{P}{\text{l}}_{\text{e}}\text{cos}\text{θ}}{\text{8}{\text{E}}_{\text{y}}\text{T}\text{∙}{\text{t}}_{\text{l}}}\text{+}\frac{\text{S}{\text{l}}_{\text{e}}\text{sin}\text{θ}}{\text{2}{\text{E}}_{\text{y}}\text{T}\text{∙}{\text{t}}_{\text{l}}}\text{=}\frac{\text{P}}{\text{8}{\text{E}}_{\text{y}}\text{T}\text{∙}{\text{t}}_{\text{l}}}\frac{{\text{l}}_{\text{e}}}{\text{l}}\left[\text{l}\text{+}\text{tan}\text{θ}\left(\text{h+l}\text{sin}\text{θ}\right)\right]\\ {\text{μ}}_{\text{ls}}\text{=}\frac{\text{P}{\text{h}}_{\text{e}}^{\text{2}}}{\text{96}{\text{E}}_{\text{y}}{\text{I}}_{\text{l}}}\frac{{\text{l}}_{\text{e}}}{{{\text{h}}_{\text{e}}\text{/}\text{l}}_{\text{e}}}\left(\text{2.4+1.5}{\text{ν}}_{\text{y}\text{x}}\right){\left(\frac{{\text{t}}_{\text{l}}}{{\text{l}}_{\text{e}}}\right)}^{\text{2}}\text{=}\frac{\text{P}{\text{h}}_{\text{e}}}{\text{8}{\text{E}}_{\text{y}}\text{T}\text{∙}{\text{t}}_{\text{l}}}\left(\text{2.4+1.5}{\text{ν}}_{\text{y}\text{x}}\right)\end{array}\right.$$

41

Thus, the total shear strain *γ**l* generated by the inclined cell wall of the BHC is calculated as follows:

$${\text{γ}}_{\text{l}}\text{=}\frac{\text{2}\left[{\text{μ}}_{\text{la}}\text{sin}\text{θ}\text{+}{\text{μ}}_{\text{ls}}\text{cos}\text{θ}\right]}{\text{lcos}\text{θ}}\text{=}\frac{\text{P}}{\text{4}{\text{E}}_{\text{y}}\text{T}\text{∙}{\text{t}}_{\text{l}}}\frac{\text{1}}{{\text{l}}^{\text{2}}}\left[{\text{l}}_{\text{e}}\text{tan}\text{θ}\text{[}\text{l}\text{+}\text{tan}\text{θ}\left(\text{h+l}\text{sin}\text{θ}\right)\text{]+}{\text{h}}_{\text{e}}\text{l}\left(\text{2.4+1.5}{\text{ν}}_{\text{y}\text{x}}\right)\right]$$

42

In summary, the expression for the shear modulus \({\text{G}}_{\text{xy}}^{\text{*}}\) of the BHC is below:

$${\text{G}}_{\text{xy}}^{\text{*}}\text{=}\frac{{\text{τ}}_{\text{xy}}}{{\text{γ}}_{\text{xy}}}\text{=}\frac{{\text{τ}}_{\text{xy}}}{{\text{γ}}_{\text{h}}\text{+}{\text{γ}}_{\text{l}}}\text{=}\frac{{{\text{E}}_{\text{x}}\text{E}}_{\text{y}}\text{∙}{\text{t}}_{\text{l}}\text{∙}{{\text{t}}_{\text{h}}}^{\text{3}}}{{\text{A}\text{∙}\text{E}}_{\text{y}}\text{∙}{\text{t}}_{\text{l}}\text{+}\text{B}{\text{E}}_{\text{x}}\text{∙}{{\text{t}}_{\text{h}}}^{\text{3}}}$$

43

where:

$$\left\{\begin{array}{c}\text{A}\text{=}\frac{{{\text{h}}_{\text{e}}}^{\text{2}}}{\text{cos}\text{θ}\text{(}\text{h}\text{+}\text{l}\text{sin}\text{θ}\text{)}}\frac{{\text{l}}_{\text{e}}}{\text{l}}\left[{\text{l}}_{\text{e}}\text{+2}{\text{h}}_{\text{e}}\text{+2}{\text{h}}_{\text{e}}\left(\text{2.4+1.5}{\text{ν}}_{\text{x}\text{y}}\right){\left(\frac{{\text{t}}_{\text{h}}}{{\text{h}}_{\text{e}}}\right)}^{\text{2}}\right]\\ \text{B}\text{=}\frac{\text{1}}{{\text{l}}^{\text{3}}\text{cos}\text{θ}}\left[{\text{l}}_{\text{e}}\text{tan}\text{θ}\left[\text{l}\text{+}\text{tan}\text{θ}\left(\text{h+l}\text{sin}\text{θ}\right)\right]\text{+}{\text{h}}_{\text{e}}\text{l}\left(\text{2.4+1.5}{\text{ν}}_{\text{y}\text{x}}\right)\right]\end{array}\right.$$

44

$$\mathbf{S}\mathbf{h}\mathbf{e}\mathbf{a}\mathbf{r} \mathbf{m}\mathbf{o}\mathbf{d}\mathbf{u}\mathbf{l}\mathbf{u}\mathbf{s} {\text{G}}_{\text{xz}}^{\text{*}} \mathbf{a}\mathbf{n}\mathbf{d} {\text{G}}_{\text{yz}}^{\text{*}}$$

Numerous scholars have reported the upper and lower limits of the out-of-plane shear modulus of HSC with equal wall thickness(Sorohan et al. 2018a, b). The closed-form equations of the out-of-plane shear modulus of HSC can be obtained by analyzing the out-of-plane equivalent elastic constant shear modulus of HSC using the method reported by Sardar M (Malek and Gibson 2015) and Tanmoy M et al. (Mukhopadhyay and Adhikari 2017). The shear force distribution of BHC is shown in Fig. 6(f-h), considering the effect of displacement and interaction forces of BHC at the nodes of contact locations and the equilibrium of forces.

As shown in Fig. 6(f-h), the shear stresses on the adjacent cell wall modules of the cell are *τ**a*, *τ**b*, *τ**c*, *τ**d*, *τ**e* with their shear strains *γ**b*, *γ**c*, *γ**d*, *γ**e*, respectively. When the stress fields in regions *d* and *e* are uniform when they are stressed and the contact point area modules are of the same material properties, according to the equilibrium condition of the forces, there are:

$$\left\{\begin{array}{c}{\text{τ}}_{\text{a}}\text{=}\text{0}\\ {\text{τ}}_{\text{b}}\text{=}{\text{τ}}_{\text{c}}\text{=}{\text{τ}}_{\text{d}}\text{cos}\text{θ}\text{=}{\text{τ}}_{\text{e}}\text{cos}\text{θ}\\ {\text{γ}}_{\text{b}}\text{=}{\text{γ}}_{\text{c}}\text{=}{\text{γ}}_{\text{d}}\text{cos}\text{θ}\text{=}{\text{γ}}_{\text{e}}\text{cos}\text{θ}\end{array}\right.$$

45

The total shear strain of the BHC in the plane caused by shear force is *γ**xz*, and the displacement generated by its inclined cell wall is consistent with the total displacement generated by the core; thus, the following relationship equations exists:

$${\text{γ}}_{\text{b}}{\text{l}}_{\text{e}}\text{=}{\text{γ}}_{\text{xz}}{\text{l}}_{\text{e}}\text{cos}\text{θ}$$

46

According to the force equilibrium condition, the shear force *P**xz* of the core in the *x* direction is equal to the sum of the shear forces at the modules in the node area; thus, the following relationship exists:

$${\text{τ}}_{\text{b}}\text{cos}\text{θ}\left({\text{t}}_{\text{l}}{\text{l}}_{\text{e}}\text{-2}{\text{S}}_{\text{e}}\right)\text{+}{\text{S}}_{\text{d}}{\text{τ}}_{\text{d}}\text{+2}{\text{S}}_{\text{e}}{\text{τ}}_{\text{e}}\text{=}{\text{τ}}_{\text{xz}}{\text{l}}_{\text{e}}\text{cos}\text{θ}\left(\text{h+l}\text{sin}\text{θ}\right)$$

47

where *S**e* is the shear force of module *e*, and *S**d* is the shear force of module *d*. The shear modulus \({\text{G}}_{\text{xz}}^{\text{*}}\) of the sandwich core in the *xoz* plane is calculated as follows:

$${\text{G}}_{\text{xz}}^{\text{*}}\text{=}{\text{G}}_{\text{xz}}\left(\frac{{\text{t}}_{\text{l}}}{\text{cos}\text{θ}\text{(}\text{h+l}\text{sin}\text{θ}\text{)}}\right)\left[\frac{{\text{l}}_{\text{e}}}{\text{l}}{\text{cos}}^{\text{2}}\text{θ}\text{+}\frac{{\text{t}}_{\text{l}}}{\text{l}}\left[\frac{\text{3}}{\text{4}}\text{tan}\text{θ}\text{-}\frac{\text{cos}\text{θ}}{\text{2}}\left(\text{2}\text{sin}\text{θ}\text{-1}\right)\right]\right]$$

48

Similarly, according to the equilibrium condition of forces, the relationship between the shear stresses in the *y* direction generated by the shear load in the *yoz* plane for the nodal region modules *a*, *b*, *d*, and *e* is obtained as:

$${\text{τ}}_{\text{b}}\text{sin}\text{θ}\left({\text{t}}_{\text{l}}{\text{l}}_{\text{e}}\text{-2}{\text{S}}_{\text{e}}\right)\text{+}{\text{S}}_{\text{d}}{\text{τ}}_{\text{d}}\text{+2}{\text{S}}_{\text{e}}{\text{τ}}_{\text{e}}\text{+}\frac{{\text{S}}_{\text{a}}{\text{τ}}_{\text{a}}}{\text{2}}\text{=}{\text{τ}}_{\text{yz}}{\text{l}}_{\text{e}}\text{cos}\text{θ}\left(\text{h+l}\text{sin}\text{θ}\right)$$

49

In modules *d* and *e*, the existence of the relationship between the transferability of the force and the equilibrium condition can be expressed as:

$$\left\{\begin{array}{c}{\text{τ}}_{\text{b}}\text{=}{\text{τ}}_{\text{c}}\text{=}{\text{τ}}_{\text{d}}\text{sin}\text{θ}\text{=}{\text{τ}}_{\text{e}}\text{sin}\text{θ}\\ {\text{γ}}_{\text{b}}\text{=}{\text{γ}}_{\text{c}}\text{=}{\text{γ}}_{\text{d}}\text{sin}\text{θ}\text{=}{\text{γ}}_{\text{e}}\text{sin}\text{θ}\\ {\text{γ}}_{\text{b}}{\text{l}}_{\text{e}}\text{+}{\text{γ}}_{\text{a}}{\text{h}}_{\text{e}}\text{=}{\text{γ}}_{\text{yz}}\left({\text{h}}_{\text{e}}\text{+}{\text{l}}_{\text{e}}\text{sin}\text{θ}\right)\end{array}\right.$$

50

Hence, the expression for the calculated shear modulus \({\text{G}}_{\text{yz}}^{\text{*}}\) of the BHC in the *yoz* plane can be obtained using the above equation, and the derived equation can be expressed below:

$${\text{G}}_{\text{yz}}^{\text{*}}\text{=}{\text{G}}_{\text{yz}}\left(\frac{{\text{t}}_{\text{l}}}{\text{cos}\text{θ}\text{(}\text{h+l}\text{sin}\text{θ}\text{)}}\right)\left[\frac{{\text{h}}_{\text{e}}}{\text{2}\text{l}}{\text{+}\text{sin}}^{\text{2}}\text{θ}\frac{{\text{l}}_{\text{e}}}{\text{l}}\text{-}\frac{{\text{t}}_{\text{h}}}{\text{2}\text{l}}\text{tan}\text{θ}\left({\text{2}\text{sin}}^{\text{2}}\text{θ}\text{-2}\text{sin}\text{θ}\text{-}\frac{\text{3}}{\text{2}}\right)\right]$$

51

**Numerical approach**

**Boundary conditions**

Recently, the computational homogenization technique (CHT) has been widely used to estimate the effective properties of non-homogeneous short fiber composite materials with complex morphological characteristics and to determine the effective stiffness tensor corresponding to three uniaxial tensile and three simple shear loads by solving six elementary edge value problems. Additionally, the stiffness tensor for a given volume of material depends on the type of boundary conditions applied, and periodic boundary conditions are well suited for periodic and random media boundary conditions, which can be used to obtain effective elastic properties more accurately than other boundary conditions (Malek and Gibson 2015; Ongaro 2018; Zhao et al. 2020). Therefore, FEM full-field microstructure simulation-based CHT can be used as a tool to provide reference solutions and to evaluate the correlation of different micromechanical models, under which the 3D finite element solid model can be accurately simulated when the BHC is subjected to forces through discretization (Catapano and Montemurro 2014a, b) (Fig. 7(a)).

The arrangement of BHC is similar to that of other periodic structures and materials (Fig. 2), and the numerical simulation of its grouping may result in different effective elastic properties due to different boundary conditions. To ensure the accuracy of the force analysis of BHC and simulate the real deformation and load of the HSC under stress, setting the boundary condition can effectively prevent the model from reducing the analysis accuracy due to the over-constrained or under-constrained state. The expression for the stiffness \(\stackrel{\text{-}}{\text{C}}\) of a composite material with linear elastic properties is given as (Zhao et al. 2020):

$$\stackrel{\text{-}}{\text{σ}}\text{=}\stackrel{\text{-}}{\text{C}}\text{:}\stackrel{\text{-}}{\text{ε}}$$

52

where \(\stackrel{\text{-}}{\text{σ}}\) and \(\stackrel{\text{-}}{\text{ε}}\) represent the average value of the stress and strain tensors of the periodic material or structure, respectively.

For a given structure or material *Ω* with boundary conditions \(\text{∂Ω}\) subjected to an arbitrary load and a displacement field generated by force *u*(*x*), have two corresponding points *M* and *N* on the boundary conditions \(\text{∂Ω}\) of the structure or material satisfying the following relations (Malek and Gibson 2015):

$$\stackrel{\text{-}}{\text{ε}}\text{=}\frac{\text{u}\left({\text{x}}_{\text{M}}\right)\text{-}\text{u}\left({\text{x}}_{\text{N}}\right)}{{\text{x}}_{\text{M}}\text{-}{\text{x}}_{\text{N}}}\text{, }\text{∀}\text{x}\text{∈}\text{∂}\text{Ω}$$

53

Therefore, the relationship between the stress field *ε*(*x*) generated when the material is stressed for the periodic boundary condition existing with its total volume *V**Ω* is given by (Harper et al. 2012):

$$\stackrel{\text{-}}{\text{ε}}\text{=}⟨\text{ε}\left(\text{x}\right)⟩\text{=}\frac{\text{1}}{{\text{V}}_{\text{Ω}}}\underset{\text{>0}}{\overset{\text{Ω}}{\int }}\text{ε}\left(\text{x}\right)\text{dV}$$

54

**Failure criteria**

BHC, under the action of out-of-plane shear, cell wall deformation, and stress distribution, may be affected by the adjacent honeycomb cell wall and boundary conditions. The deformation compatibility control of HSC is mainly at the intersecting surfaces of the two adjacent sandwich core faces, and the force equilibrium conditions determine the deformation close to the middle plane of the core and away from the face. Meanwhile, when the warping behavior of the unit wall at the boundary is neglected, the maximum shear and damage stresses of the sandwich panel are usually located in the center of the core stack, with their stresses distributed in the middle of the unit wall. In this analysis section, two possible damage mechanisms of the honeycomb under out-of-plane shear loading are considered: elastic buckling of the cell wall and shear yielding of the cell wall material. In addition, the HSC is a thin-walled structure. The shear stress distribution can be considered in terms of the thickness of the honeycomb cell wall, whose material is considered to be homogeneous, isotropic homogeneous, and linearly elastic, thus allowing the selection of a cell for analysis after considering the periodicity and symmetry of the honeycomb structure (Namvar and Vosoughi 2020) (Fig. 7(b)).

As shown in Fig. Figure 7(b), the *z*-axis is perpendicular to the *xoy* plane, and the periodically arranged BHC with two two cell walls comprising modules *a*, *b*, *c*, and d are subjected to shear stresses in the *xoy* plane as *τ**a*, *τ**b*, *τ**c*, and *τ**d*. In the force analysis of the BHC, the force in the cell wall module d is neglected, and the load is applied in the *y* direction of the honeycomb, whose equilibrium conditions for the forces in the *xoz* plane are given below(Qiao et al. 2008):

$$\left\{\begin{array}{c}{\text{τ}}_{\text{b}}\text{=}{\text{τ}}_{\text{c}}\\ {\text{τ}}_{\text{a}}{\text{t}}_{\text{h}}\text{=}\text{2}{\text{τ}}_{\text{b}}{\text{t}}_{\text{l}}\end{array}\right.$$

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When the BHC is subjected to shear load in the *y* direction, the relationship between its internal and external shear stresses \({\text{τ}}_{\text{yz}}^{\text{*}}\) is obtained by the equilibrium condition of the force as follows:

$$\text{2}{\text{τ}}_{\text{yz}}^{\text{*}}\left(\text{h+l}\text{sin}\text{θ}\right)\text{l}\text{cos}\text{θ}\text{=}{\text{τ}}_{\text{a}}{\text{t}}_{\text{h}}\text{h+}\text{2}{\text{τ}}_{\text{b}}{\text{t}}_{\text{l}}\text{sin}\text{θ}$$

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In addition, when the BHC is subjected to shear load in the *x* direction because the core cell wall of *a* module is perpendicular to the loading direction, *τ**a* = 0, the equilibrium condition of the force yields:

$$\left\{\begin{array}{c}{\text{τ}}_{\text{b}}\text{=}{\text{τ}}_{\text{c}}\\ {\text{τ}}_{\text{xz}}^{\text{*}}\left(\text{h+l}\text{sin}\text{θ}\right)\text{l}\text{cos}\text{θ}\text{=}{\text{τ}}_{\text{b}}{\text{t}}_{\text{l}}\text{cos}\text{θ}\end{array}\right.$$

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In calculating the elastic buckling of the cell wall of the BHC under shear load, the *Eulerian* relationship for the buckling load of the thin plate under shear can be obtained for the critical buckling load of the vertical cell wall and the inclined cell wall of the BHC using the following formula:

$$\left\{\begin{array}{c}{\text{P}}_{\text{cr}\text{-}\text{h}}\text{=}\frac{\text{k}{\text{π}}^{\text{2}}{\text{E}}_{\text{x}}}{\text{12}\left(\text{1-}{\text{v}}_{\text{xz}}^{\text{2}}\right)}\frac{{{\text{t}}_{\text{h}}}^{\text{3}}}{\text{h}}\\ {\text{P}}_{\text{cr}\text{-}\text{l}}\text{=}\frac{\text{k}{\text{π}}^{\text{2}}{\text{E}}_{\text{y}}}{\text{12}\left(\text{1-}{\text{v}}_{\text{yz}}^{\text{2}}\right)}\frac{{{\text{t}}_{\text{l}}}^{\text{3}}}{\text{l}}\end{array}\right.$$

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where *P**cr*−*h* and *P**cr*−*l* represent the critical buckling loads of the vertical and inclined cell walls of the BHC, respectively, and *k* is the endpoint restraint factor at the intersection of the core cell walls, which is related to the ratio of \(\frac{{\text{t}}_{\text{h}}}{\text{h}}\) and \(\frac{{\text{t}}_{\text{l}}}{\text{l}}\).

In addition, when the core is subjected to shear stress in the *x* and *y* directions to reach its yield stress of the damage load, *τ**max* = *σ**x* or *τ**max* = *σ**y*, indicating a core is subjected to shear failure model as follows:

$$\left\{\begin{array}{c}{\text{τ}}_{\text{xz}}^{\text{*}}\text{=}\frac{{\text{τ}}_{\text{max}}}{\text{h+l}\text{sin}\text{θ}}\frac{{\text{t}}_{\text{l}}}{\text{l}}\\ {\text{τ}}_{\text{yz}}^{\text{*}}\text{=}\frac{{\text{τ}}_{\text{max}}}{\text{cos}\text{θ}}\frac{{\text{t}}_{\text{l}}}{\text{l}}\end{array}\right.$$

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