*2.2. VAM-Based Reduction Analysis of OSFP*

## 2.2.1. Zeroth-Order Approximation

Plugging Equation (8) into Equation (17) gives the total potential energy density as

2Π = (*<sup>ε</sup>* <sup>+</sup> *<sup>x</sup>*3*κ*) <sup>T</sup>*De*(*ε* + *x*3*κ*) + 2(*ε* + *x*3*κ*) <sup>T</sup>*De∂ew*||*<sup>α</sup>* <sup>+</sup>2(*∂ew*||*α*)T*De∂ew*|| <sup>+</sup> <sup>2</sup>(*<sup>ε</sup>* <sup>+</sup> *<sup>x</sup>*3*κ*) <sup>T</sup>*Desw*||,3 <sup>+</sup> <sup>2</sup>(*<sup>ε</sup>* <sup>+</sup> *<sup>x</sup>*3*κ*) <sup>T</sup>*Des∂tw*3*<sup>α</sup>* +2 - *<sup>∂</sup>ew*||*<sup>α</sup>* T *<sup>D</sup>es*- *<sup>w</sup>*||,3 <sup>+</sup> *<sup>∂</sup>tw*3*<sup>α</sup>* + 2(*ε* + *x*3*κ*) <sup>T</sup>*Detw*3,3 +2 - *∂ew*||*<sup>α</sup>* T *Detw*3,3 + *w*<sup>T</sup> ||,3*Dsw*||,3 <sup>+</sup> <sup>2</sup>*w*<sup>T</sup> ||,3*Ds∂tw*3*<sup>α</sup>* <sup>+</sup> <sup>2</sup>(*∂tw*3,*α*) <sup>T</sup>*Ds∂tw*3,*<sup>α</sup>* +2*w*<sup>T</sup> ||,3*Dstw*3,3 <sup>+</sup> <sup>2</sup>(*∂tw*3,*α*) <sup>T</sup>*Dstw*3,3 + *Dtw*<sup>2</sup> 3,3 −2 '' *f* <sup>T</sup> *<sup>i</sup> wi* (( + *τ*<sup>T</sup> *<sup>i</sup> <sup>w</sup>*<sup>T</sup> *<sup>i</sup>* + *<sup>β</sup>*<sup>T</sup> *<sup>i</sup> <sup>w</sup>*<sup>T</sup> *i* (18)

where the underlined items and the double-underlined item can be ignored according to VAM.

To impose the constraints on the fluctuating function, we introduce the Lagrange multipliers *λi*, such as

$$
\delta(\Pi + \lambda\_i \langle w\_i \rangle) = 0 \tag{19}
$$

The zeroth-order approximate variational expression is

$$
\begin{pmatrix}
\left[\left(\boldsymbol{\varepsilon} + \boldsymbol{x}\_{3}\boldsymbol{\kappa}\right)^{\mathrm{T}}\boldsymbol{D}\_{\mathrm{c}\boldsymbol{s}} + \boldsymbol{w}\_{||\boldsymbol{\beta}}^{\mathrm{T}}\boldsymbol{D}\_{\mathrm{s}} + \boldsymbol{w}\_{3,3}^{\mathrm{T}}\boldsymbol{D}\_{\mathrm{s}t}^{\mathrm{T}}\right] \delta\boldsymbol{w}\_{||\boldsymbol{\beta}} \\
+ \lambda\_{i} \delta\boldsymbol{w}\_{i} + \left[\left(\boldsymbol{\varepsilon} + \boldsymbol{x}\_{3}\boldsymbol{\kappa}\right)^{\mathrm{T}}\boldsymbol{D}\_{\mathrm{c}t} + \boldsymbol{w}\_{||\boldsymbol{\beta}}^{\mathrm{T}}\boldsymbol{D}\_{\mathrm{s}t} + \boldsymbol{w}\_{3,3}^{\mathrm{T}}\boldsymbol{D}\_{t}\right] \delta\boldsymbol{w}\_{3,3}
\end{pmatrix} = \boldsymbol{0} \tag{20}
$$

The corresponding Euler–Lagrange equations are

$$\begin{aligned} \left[ \left( \boldsymbol{\varepsilon} + \mathbf{x}\_3 \boldsymbol{\kappa} \right)^{\mathrm{T}} \mathbf{D}\_{\mathrm{cs}} + \boldsymbol{w}\_{||.\boldsymbol{\beta}}^{\mathrm{T}} \mathbf{D}\_{\mathrm{s}} + \boldsymbol{w}\_{3,3}^{\mathrm{T}} \mathbf{D}\_{\mathrm{st}}^{\mathrm{T}} \right]\_{,3} &= \boldsymbol{\lambda}\_{||} \\ \left[ \left( \boldsymbol{\varepsilon} + \mathbf{x}\_3 \boldsymbol{\kappa} \right)^{\mathrm{T}} \mathbf{D}\_{\mathrm{ct}} + \boldsymbol{w}\_{||.\boldsymbol{\beta}}^{\mathrm{T}} \mathbf{D}\_{\mathrm{st}} + \boldsymbol{w}\_{3,3}^{\mathrm{T}} \mathbf{D}\_{\mathrm{t}} \right]\_{,3} &= \boldsymbol{\lambda}\_3 \end{aligned} \tag{21}$$

where *<sup>λ</sup>*|| = [*λ*<sup>1</sup> *<sup>λ</sup>*2] T.

The boundary conditions of the top and bottom of the panel can be defined as

$$\begin{cases} \left[ \left( \mathfrak{e} + \mathfrak{x}\_{3} \mathfrak{kappa} \right)^{\mathrm{T}} \mathbf{D}\_{\mathrm{cs}} + w\_{\parallel,\mathcal{3}}^{\mathrm{T}} \mathbf{D}\_{\mathrm{s}} + w\_{\mathrm{3},\mathcal{3}} \mathbf{D}\_{\mathrm{st}}^{\mathrm{T}} \right]^{+/-} = \mathbf{0} \\ \left[ \left( \mathfrak{e} + \mathfrak{x}\_{3} \mathfrak{kappa} \right)^{\mathrm{T}} \mathbf{D}\_{\mathrm{ct}} + w\_{\parallel,\mathcal{3}}^{\mathrm{T}} \mathbf{D}\_{\mathrm{st}} + w\_{\mathrm{3},\mathcal{3}} \mathbf{D}\_{\mathrm{t}} \right]^{+/-} = \mathbf{0} \end{cases} \tag{22}$$

where the superscript "+ / −" indicates the items at the top and bottom of the panel.

From these conditions, we can solve *<sup>w</sup>*|| and *<sup>w</sup>*<sup>3</sup> as

$$\boldsymbol{w}\_{||} = \left\langle - (\boldsymbol{\varepsilon} + \boldsymbol{x}\_{3} \boldsymbol{\kappa}) \overline{\boldsymbol{D}}\_{t^{3}} \boldsymbol{D}\_{s}^{-1} \right\rangle^{\mathrm{T}}, \boldsymbol{w}\_{3} = \left\langle - (\boldsymbol{\varepsilon} + \boldsymbol{x}\_{3} \boldsymbol{\kappa}) \overline{\boldsymbol{D}}\_{t^{3}} \boldsymbol{D}\_{t}^{-1} \right\rangle \tag{23}$$

where

$$
\overline{D}\_{\rm tS} = D\_{\rm tS} - \overline{D}\_{\rm tt} \mathbf{D}\_{\rm st}^T \overline{D}\_{\rm t}^{-1}, \\
\overline{D}\_{\rm tt} = D\_{\rm tI} - D\_{\rm tS} \mathbf{D}\_{\rm s}^{-1} \mathbf{D}\_{\rm tt}, \\
\overline{D}\_{\rm t} = \mathbf{D}\_{\rm l} - \mathbf{D}\_{\rm sl}^T \mathbf{D}\_{\rm s}^{-1} \mathbf{D}\_{\rm tl} \tag{24}
$$

Plugging Equation (23) into Equation (18) gives the zeroth-order strain energy as

$$\mathcal{U}\_{2D} = \frac{1}{2} \left\langle \left(\boldsymbol{\varepsilon} + \mathbf{x}\_3 \boldsymbol{\kappa}\right)^{\mathrm{T}} \bar{\mathbf{D}}\_{\boldsymbol{\ell}} (\boldsymbol{\varepsilon} + \mathbf{x}\_3 \boldsymbol{\kappa}) \right\rangle = \frac{1}{2} \left\{ \begin{array}{cc} \boldsymbol{\varepsilon} \\ \boldsymbol{\kappa} \end{array} \right\}^{\mathrm{T}} \left[ \begin{array}{cc} \boldsymbol{A} & \mathbf{B} \\ \mathbf{B}^{\mathrm{T}} & \mathbf{D} \end{array} \right] \left\{ \begin{array}{cc} \boldsymbol{\varepsilon} \\ \boldsymbol{\kappa} \end{array} \right\} \tag{25}$$

where *A*, *D* and *B* are tensile, bending, and coupling stiffness sub-matrix, respectively, and can be expressed as

$$\begin{array}{l} \mathbf{A} = \langle \langle \vec{D}\_{\mathbf{c}} \rangle \rangle, \mathbf{B} = \langle \langle \mathbf{x}\_{3} \vec{D}\_{\mathbf{c}} \rangle \rangle, \mathbf{D} = \langle \langle \mathbf{x}\_{3}^{2} \vec{D}\_{\mathbf{c}} \rangle \rangle, \\ \mathbf{D}\_{\mathbf{c}} = \mathbf{D}\_{\mathbf{c}} - \mathbf{D}\_{\mathbf{c}s} \mathbf{D}\_{\mathbf{s}}^{-1} \mathbf{D}\_{\mathbf{c}s}^{\mathrm{T}} - \mathbf{D}\_{\mathbf{c}t} \mathbf{D}\_{\mathbf{c}t}^{\mathrm{T}} / \mathbf{D}\_{\mathbf{t}} \end{array} \tag{26}$$

#### 2.2.2. Transforming into Reissner–Mindlin Model

There are two additional transverse shear strains *<sup>γ</sup>* <sup>=</sup> 2*γ*<sup>13</sup> <sup>2</sup>*γ*23*<sup>T</sup>* in the Reissner– Mindlin model. To transform Equation (25) into the Reissner–Mindlin model, we must eliminate the coupled stiffness terms between  and *γ* as follows:

$$\begin{split} 2\Pi\_{\Omega} &= \Gamma^{T}\mathcal{D}\_{t}\Gamma = \mathcal{R}^{T}A\mathcal{R} + 2\mathcal{R}^{T}\mathcal{B}\gamma + \gamma^{T}\mathcal{C}\gamma\\ &= \mathcal{R}^{T}\left(A - \mathcal{B}\mathcal{C}^{-1}\mathcal{B}^{T}\right)\mathcal{R} + \left(\gamma + \mathcal{C}^{-1}\mathcal{B}^{T}\mathcal{R}\right)^{T}\mathcal{C}\left(\gamma + \mathcal{C}^{-1}\mathcal{B}^{T}\mathcal{R}\right) \end{split} \tag{27}$$

where R is Reissner–Mindlin generalized strains.

The final form of the total energy can be expressed as

$$2\Pi\_{\mathcal{R}} = \mathcal{R}^T X \mathcal{R} + \gamma^T \mathbf{G} \gamma + 2\mathcal{R}^T \mathbf{F} \tag{28}$$

where *F* is a load-related term and

$$\begin{cases} X = A - B\mathcal{C}^{-1}\mathcal{B}^T\\ G = \mathcal{C} \end{cases} \tag{29}$$

The resultant stress of the panel can be expressed as

$$\begin{array}{llll}\frac{\partial \Pi\_{\mathcal{R}}}{\partial \boldsymbol{\varepsilon}\_{11}} = N\_{11\prime} & \frac{\partial \Pi\_{\mathcal{R}}}{\partial 2\underline{\varepsilon}\_{12}} = N\_{12\prime} & \frac{\partial \Pi\_{\mathcal{R}}}{\partial \underline{\varepsilon}\_{22}} = N\_{22} \\\frac{\partial \Pi\_{\mathcal{R}}}{\partial \underline{\kappa}\_{11}} = M\_{11\prime} & \frac{\partial \Pi\_{\mathcal{R}}}{\partial 2\underline{\kappa}\_{12}} = M\_{12\prime} & \frac{\partial \Pi\_{\mathcal{R}}}{\partial \underline{\kappa}\_{22}} = M\_{22} \\\frac{\partial \Pi\_{\mathcal{R}}}{\partial 2\gamma\_{13}} = Q\_{1\prime} & \frac{\partial \Pi\_{\mathcal{R}}}{\partial 2\gamma\_{23}} = Q\_{2} \end{array} \tag{30}$$

Due to the symmetry of the axis and plane, some stiffness components disappear, and the constitutive relation of the OSFP can be obtained as


The original 3D geometric nonlinear problem in Equation (17) is mathematically decomposed into constitutive modeling over the unit cell in Equation (31) and geometric nonlinear plate analysis. That is to say, as an alternative to the direct numerical simulation using 3D nonlinear finite element analysis, the global analysis of the OSFP can be reduced to 2D plate analysis using the linear solver in ABAQUS, with the constitutive relation obtained from the constitutive modeling of the unit cell.
