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Modelling architected plate using a non-local derivative-free shear deformable plate theory

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Abstract

The internal length scale relating to the cell size plays a crucial role in predicting the response of architected structures when subjected to external stimuli. A Volterra derivative-based approach for arriving at the non-local derivative-free continuum laws for architected structures is proposed. A mainstay of the work is the derivative-free directionality term, which recovers its classical counterpart in the infinitesimal limit. Using this approach, we derive the non-local integro-differential governing equations of a shear deformable plate. We also suggest a physical basis for the consideration of energy for nonaffine deformations and accurately estimate it by performing buckling analysis. This discards the requirement of the additional energy to be incorporated in an arbitrary manner for suppressing the unwanted spurious oscillations induced from zero energy modes. The numerical results demonstrate the efficacy of the proposed framework in precisely capturing the mechanical response of web-core shear deformable plate, thereby, manifesting the supremacy of the reduced model in shrinking the cost and computational time. To bolster our claim, various numerical models with different loading conditions have been analysed and compared against the three-dimensional FEM results.

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Acknowledgements

The authors acknowledge Professor Arun Srinivasa (Department of Mechanical Engineering, Texas A& M University, College Station, TX 77845, Unites States) for sharing his valuable comments and suggestions, which greatly improved the quality of the results included here.

Funding

MS and SS acknowledge SERB (ECR/2018/001672) for supporting this work. JNR acknowledges the National Science Foundation (CMMI grant No. 1952873) for supporting this work.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Technology Indore, Indore, Madhya Pradesh, 453552, India

    Mukul Saxena & Saikat Sarkar

  2. Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77845, USA

    J. N. Reddy

Authors
  1. Mukul Saxena
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  2. Saikat Sarkar
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Correspondence to Saikat Sarkar.

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Appendices

Appendix 1: Localization of derivative-free deformation gradient G

The proposed derivative-free directionality term approaches its classical deformation gradient counterpart in the infinitesimal limit. For demonstration, let us assume sufficient smoothness of the field such that the displacement (\({u_3}\)) at a material point \(\varvec{Y}\), in the neighbourhood of \(\varvec{X}\), can be approximated using a truncated Taylor expansion as:

$$\begin{aligned} \begin{aligned} {(u_3)_Y\approx (u_3)_X+\triangledown (u_3,X_d)(Y_d-X_d)}, \end{aligned} \end{aligned}$$
(34)

where \({\triangledown }\) is the classical gradient operator. The average stretch around \(\varvec{X}\) may also be approximated in a similar way.

$$\begin{aligned} \begin{aligned} {(\bar{u}_3)_Y \approx (u_3)_X+\triangledown (u_3,X_d)(\overline{Y_d-X_d})}. \end{aligned} \end{aligned}$$
(35)

The nonlocal derivative-free deformation gradient is expressed as:

$$\begin{aligned} \begin{aligned}&G(u_3,X_d)=I+\hat{G}(u_3,X_d)=I+\left[ \int _{\varOmega _x}\left( (u_3)_Y-(\bar{u}_3)_Y\right) \left( Y_d-\bar{Y}_d\right) ^T dY\right] \\&\quad \left[ \int _{\varOmega _x}\left( Y_d-\bar{Y}_d\right) \left( Y_d-\bar{Y}_d\right) ^T dY\right] ^{-1}, \end{aligned} \end{aligned}$$
(36)

where I is the identity tensor. Replacing the terms in Eq. (36) with those given in Eqs. (34) and (35), we get,

$$\begin{aligned} \begin{aligned} {G} \approx&{I+\left[ \int _{\varOmega _x}\left( (u_3)_X+\triangledown (u_3,X_d)(Y_d-X_d)-((u_3)_X+ \triangledown (u_3,X_d)(\overline{Y_d-X_d}))\right) \left( Y_d-\bar{Y}_d\right) ^TdY\right] }.\\&{I+\left[ \int _{\varOmega _X} (Y_d-\bar{Y}_d)(Y_d-\bar{Y}_d)^T dY\right] ^{-1}}\\ =&{I+\left[ \int _{\varOmega _x}\triangledown (u_3,X_d)(Y_d-X_d-\overline{Y_d-X_d}) \left( Y_d-\bar{Y}_d\right) ^TdY\right] } {\left[ \int _{\varOmega _X} (Y_d-\bar{Y}_d)(Y_d-\bar{Y}_d)^TdY\right] ^{-1}}\\ =&{I+\triangledown (u_3,X_d)\left[ \int _{\varOmega _x}(Y_d-X_d-\bar{Y}_d-\bar{X}_d) \left( Y_d-\bar{Y}_d\right) ^TdY\right] } {\left[ \int _{\varOmega _X} (Y_d-\bar{Y}_d)(Y_d-\bar{Y}_d)^T dY\right] ^{-1}}\\ =&{I+\triangledown (u_3,X_d)\left[ \int _{\varOmega _x}(Y_d-\bar{Y}_d)\left( Y_d-\bar{Y}_d\right) ^TdY\right] } {\left[ \int _{\varOmega _X} (Y_d-\bar{Y}_d)(Y_d-\bar{Y}_d)^T dY\right] ^{-1}}\\ =&{I+\triangledown (u_3,X_d)}\\ =&{F} \end{aligned} \end{aligned}$$
(37)

where F is the classical deformation gradient.

Appendix 2: Comparison of DFCT based non-local deformation gradient and PD based counterpart

The following non-local gradient term in the DFCT is actually rooted in measure theory and has been derived via a stochastic projection technique [33]. A similar expression may also be found in stochastic filtering [42, 43].

$$\begin{aligned} \begin{aligned}&{ {\hat{G}_{DFCT}(u_3,X_d)=\left[ \int _{\varOmega _x}\left( (u_3)_Y-(\bar{u}_3)_Y\right) \left( Y_d-\bar{Y}_d\right) ^T dY\right] \left[ \int _{\varOmega _x}\left( Y_d-\bar{Y}_d\right) \left( Y_d-\bar{Y}_d\right) ^T dY\right] ^{-1}}} \end{aligned} \end{aligned}$$
(38)

On the other hand, the non-local gradient term for the PD correspondence may be written as [44,45,46]:

$$\begin{aligned} \begin{aligned}&{ {\hat{G}_{PD}(u_3,X_d)=\left[ \int _{\varOmega _x}\left( (u_3)_Y-({u_3})_X\right) \left( Y_d-X_d\right) ^T dY\right] \left[ \int _{\varOmega _x}\left( Y_d-X_d\right) \left( Y_d-X_d\right) ^T dY\right] ^{-1}}} \end{aligned} \end{aligned}$$
(39)

The above two expressions become identically same when \({(u_3)_X}\) and \({{X}_d}\), which is perhaps the case when there are detectable symmetries (e.g. through material homogeneity and/or symmetries in applied loading configurations). This is however not true in general and accordingly the two expressions differ. To numerically assess the performances of the two expressions, we have considered a shear deformable plate model, arrived at via constitutive correspondences using PD [47,48,49] and DFCT gradient terms. For simplicity, the material properties of the plate have been kept uniform and boundary conditions in the form of simple supports are considered at all the four sides of the plate. Under a uniform distribution of particles, the two approaches give the same solution. However, for random distribution, the PD variant exhibits unphysical oscillations, whereas the DFCT appears to work fine (see Fig 18).

Fig. 18
figure 18

Comparison of zero energy oscillations in simply supported shear deformable plate subjected to uniformly distributed load when analysed via (a) PD and (b) proposed method

Full size image

Appendix 3: Equivalence of shear deformable plate theory and the shear-rigid plate theory in static case

Here we demonstrate that the derivative-free shear deformable plate theory is equivalent to shear-rigid (Kirchhoff) plate theory for a thin plate. The non-local governing equations for the shear deformable plate subjected to transverse load q/unit area can be written as:

$$\begin{aligned}&{ {D\hat{G}(\hat{G}(\varPhi _1,X_1),X_1)+D\mu \hat{G}(\hat{G}(\varPhi _{2},X_1),X_2)}}\nonumber \\ {}&{ {+\frac{\mathcal {S} h^3}{12}\left( \hat{G}(\hat{G}(\varPhi _1,X_2),X_2)+\hat{G}(\hat{G}(\varPhi _{2},X_2),X_1)\right) -k_s\mathcal {S} h\left( \hat{G}(u_3,X_1)+\varPhi _1\right) =0}}\nonumber \\ \end{aligned}$$
(40)
$$\begin{aligned}&{ {D\hat{G}(\hat{G}(\varPhi _{2},X_2),X_2)+D\mu \hat{G}(\hat{G}(\varPhi _{1},X_1),X_2)}}\nonumber \\ {}&{ {+\frac{\mathcal {S} h^3}{12}\left( \hat{G}(\hat{G}(\varPhi _{2},X_1),X_1)+\hat{G}(\hat{G}(\varPhi _{1},X_2),X_1)\right) -k_s\mathcal {S} h\left( \hat{G}(u_3,X_2)+\varPhi _{2}\right) =0}} \end{aligned}$$
(41)
$$\begin{aligned}&{ {k_s\mathcal {S} h\left( \hat{G}(\hat{G}(u_3,X_1),X_1)+\hat{G}(\varPhi _1,X_1)\right) +k_s\mathcal {S} h\left( \hat{G}(\hat{G}(u_3,X_2),X_2)+\hat{G}(\varPhi _{2},X_2)\right) }}\nonumber \\ {}&{ {+\vartheta _1\hat{G}(\hat{G}(u_3,X_1),X_1)+\vartheta _2\hat{G}(\hat{G}(u_3,X_2),X_2)-q=0}} \end{aligned}$$
(42)

Equations 40 and 41 may be written as:

$$\begin{aligned}&{ {D\hat{G}(\hat{G}(\hat{G}(\varPhi _1,X_1),X_1),X_1)+D\mu \hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_1),X_2),X_1)}}\nonumber \\&+\frac{\mathcal {S} h^3}{12}\left( \hat{G}(\hat{G}(\hat{G}(\varPhi _1,X_2),X_2),X_1)+\hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_2),X_1),X_1) \right) \nonumber \\&-k_s\mathcal {S} h\left( \hat{G}(\hat{G}(u_3,X_1),X_1)+\hat{G}(\varPhi _1,X_1)\right) =0\nonumber \\ \end{aligned}$$
(43)
$$\begin{aligned}&{ {D\hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_2),X_2),X_2)+D\mu \hat{G}(\hat{G}(\hat{G}(\varPhi _{1},X_1),X_2),X_2)}}\nonumber \\&+\frac{\mathcal {S} h^3}{12}\left( \hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_1),X_1),X_2)+\hat{G}(\hat{G}(\hat{G}(\varPhi _{1} ,X_2),X_1),X_2)\right) \nonumber \\&-k_s\mathcal {S} h\left( \hat{G}(\hat{G}(u_3,X_2),X_2)+\hat{G}(\varPhi _{2},X_2)\right) =0 \end{aligned}$$
(44)

Upon adding the Eqs. 43 and 44, we get:

$$\begin{aligned} \begin{aligned}&{ {D\hat{G}(\hat{G}(\hat{G}(\varPhi _1,X_1),X_1),X_1)+D\hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_2),X_2),X_2)}}\\ {}&{ {D\hat{G}(\hat{G}(\hat{G}(\varPhi _1,X_1),X_2),X_2)\left( \mu +\frac{2\mathcal {S}h^3}{12D}\right) +D\hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_1),X_1),X_2)\left( \mu +\frac{2\mathcal {S}h^3}{12D}\right) }}\\ {}&{ {-{k_s\mathcal {S} h}\left( \hat{G}(\hat{G}(u_3,X_1),X_1)+\hat{G}(\varPhi _1,X_1)\right) -{k_s\mathcal {S} h}\left( \hat{G}(\hat{G}(u_3,X_2),X_2)+\hat{G}(\varPhi _{2},X_2)\right) =0}} \end{aligned} \end{aligned}$$
(45)

Using the fact that, \({\left( \mu +\frac{2\mathcal {S}h^3}{12D}\right) =1}\), the above Eqn takes the form:

$$\begin{aligned} \begin{aligned}&{ {D\hat{G}(\hat{G}(\hat{G}(\varPhi _1,X_1),X_1),X_1)+D\hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_2),X_2),X_2)}}\\ {}&{ {D\hat{G}(\hat{G}(\hat{G}(\varPhi _1,X_1),X_2),X_2)+D\hat{G}(\hat{G}(\hat{G}(\varPhi _{2},X_1),X_1),X_2)}}\\ {}&{ {-{k_s\mathcal {S} h}\left( \hat{G}(\hat{G}(u_3,X_1),X_1)+\hat{G}(\varPhi _1,X_1)\right) -{k_s\mathcal {S} h}\left( \hat{G}(\hat{G}(u_3,X_2),X_2)+\hat{G}(\varPhi _{2},X_2)\right) =0}} \end{aligned} \end{aligned}$$
(46)

However, Eq. 42 may be rewritten as:

$$\begin{aligned} \begin{aligned}&{ {k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+\hat{G}(\hat{G}(\hat{G} (\varPhi _1,X_1),X_1),X_1)\right) }}\\ {}&{ {+k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_1),X_1)+\hat{G}(\hat{G}(\hat{G} (\varPhi _{2},X_2),X_1),X_1)\right) }}\\ {}&+\vartheta _1\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+\vartheta _2\hat{G} (\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_1),X_1)\\&-\hat{G}(\hat{G}(q,X_1),X_1)=0 \end{aligned} \end{aligned}$$
(47)

and,

$$\begin{aligned} \begin{aligned}&{ {k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_2),X_2)+\hat{G}(\hat{G}(\hat{G} (\varPhi _1,X_1),X_2),X_2)\right) }}\\ {}&{ {+k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_2),X_2)+\hat{G}(\hat{G}(\hat{G} (\varPhi _{2},X_2),X_2),X_2)\right) }}\\ {}&+\vartheta _1\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_2),X_2)+\vartheta _2\hat{G} (\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_2),X_2)\\&-\hat{G}(\hat{G}(q,X_2),X_2)=0 \end{aligned} \end{aligned}$$
(48)

Adding the Eqs. 47 and 48, we get:

$$\begin{aligned} \begin{aligned}&{ {k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+\hat{G}(\hat{G}(\hat{G} (\varPhi _1,X_1),X_1),X_1)\right) }}\\ {}&{ {+k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+\hat{G}(\hat{G}(\hat{G} (\varPhi _1,X_1),X_2),X_2)\right) }}\\ {}&{ {+k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_1),X_1)+\hat{G}(\hat{G}(\hat{G} (\varPhi _{2},X_2),X_1),X_1)\right) }}\\ {}&+\vartheta _1\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+\vartheta _2\hat{G} (\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_1),X_1)\\&-\hat{G}(\hat{G}(q,X_1),X_1)\\&{ {+k_s\mathcal {S} h\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_1),X_1)+\hat{G}(\hat{G}(\hat{G} (\varPhi _{2},X_2),X_1),X_1)\right) }}\\ {}&+\vartheta _1\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+\vartheta _2\hat{G} (\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_1),X_1)\\&-\hat{G}(\hat{G}(q,X_1),X_1) =0 \end{aligned} \end{aligned}$$
(49)

Substituting Eqs. 44 in Eq. 49 and rearranging the terms leads to:

$$\begin{aligned} \begin{aligned}&D\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+2\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1) ,X_1),X_2),X_2)\right. \\&\left. +\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_2),X_2)\right) \\&+\vartheta _1\left( \frac{D}{k_s\mathcal {S}h}\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1) \right. \right. \\&\left. \left. +\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_2),X_2)\right) +\hat{G}(\hat{G}(u_3,X_1),X_1)\right) \\&+\vartheta _2\left( \frac{D}{k_s\mathcal {S}h}\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_2),X_2) +\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_2),X_2)\right) \right. \\&\left. +\hat{G}(\hat{G}(u_3,X_2),X_2)\right) \\&{ {-q-\frac{D}{k_s\mathcal {S}h}\left( \hat{G}(\hat{G}(q,X_1),X_1)+\hat{G}(\hat{G}(q,X_2),X_2)\right) =0}} \end{aligned} \end{aligned}$$
(50)

With the assumption that for a shear rigid (Kirchhoff) plate:

$$\begin{aligned} \begin{aligned}&{ {\frac{D}{k_s\mathcal {S}h}<<1}} \end{aligned} \end{aligned}$$
(51)

we arrive at the governing equation for a shear rigid plate in the static case.

$$\begin{aligned} \begin{aligned}&D\left( \hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_1),X_1),X_1),X_1)+2\hat{G}(\hat{G} (\hat{G}(\hat{G}(u_3,X_1),X_1),X_2),X_2)\right. \\&\left. +\hat{G}(\hat{G}(\hat{G}(\hat{G}(u_3,X_2),X_2),X_2),X_2)\right) \\ {}&{ {+\vartheta _1\left( \hat{G}(\hat{G}(u_3,X_1),X_1)\right) } {+\vartheta _2\left( \hat{G}(\hat{G}(u_3,X_2),X_2)\right) -q=0}} \end{aligned} \end{aligned}$$
(52)

Therefore, with the assumption that \({\frac{D}{k_s\mathcal {S}h}<<1}\), the shear deformable plate theory becomes equivalent to shear-rigid (Kirchhoff) plate theory for a thin plate.

Appendix 4: Computation of analytical expressions for \({\hat{G}(\hat{G}(\xi ,X_d),X_d)}\)

Let us assume a field variable \(\xi\) defined by the far-off interactions of any material point \(\varvec{X}\) such that,

$$\begin{aligned} \begin{aligned}&{\xi }= {\xi ^0\,\,\text{sin}(\alpha X_1)\,\,\text{sin}(\beta X_2)} \end{aligned} \end{aligned}$$
(53)

The derivative-free directionality term \({\hat{G}(\xi ,X_1)}\) and \({\hat{G}(\xi ,X_2)}\) can be computed as:

$$\begin{aligned} \begin{aligned}&{\hat{G}(\xi ,X_1)= \left[ \int _{\varOmega _X}( {\xi -\overline{\xi }})( {(Y_1-\overline{Y}_1})^TdY_1\right] \left[ \int _{\varOmega _X}( {Y_1-\overline{Y}_1})( {(Y_1-\overline{Y}_1})^TdY_1\right] ^{-1}}\\&{\hat{G}(\xi ,X_2)= \left[ \int _{\varOmega _X}( {\xi -\overline{\xi }})( {(Y_2-\overline{Y}_2})^TdY_2\right] \left[ \int _{\varOmega _X}( {Y_2-\overline{Y}_2})( {Y_2-\overline{Y}_2})^TdY_2\right] ^{-1}} \end{aligned} \end{aligned}$$
(54)

For an influence domain of definite length \({r_c}\), the above expressions take the form,

$$\begin{aligned} \begin{aligned}&{\hat{G}(\xi ,X_1)}=\\ {}&{ \left[ \int _{X_1-r_c}^{X_1+r_c}( {\xi -\overline{\xi }})( {Y_1-\overline{Y}_1})^TdY_1\right] \left[ \int _{X_1-r_c}^{X_1+r_c}( {Y_1-\overline{Y}_1})( {Y_1-\overline{Y}_1})^TdY_1\right] ^{-1}}\\ {}&{\hat{G}(\xi ,X_2)}=\\ {}&{ \left[ \int _{X_2-r_c}^{X_2+r_c}( {\xi -\overline{\xi }})( {Y_2-\overline{Y}_2})^TdY_2\right] \left[ \int _{X_2-r_c}^{X_2+r_c}( {Y_2-\overline{Y}_2})( {Y_2-\overline{Y}_2})^TdY_2\right] ^{-1}} \end{aligned} \end{aligned}$$
(55)

where,

$$\begin{aligned} \begin{aligned}&{ \left[ \int _{X_1-r_c}^{X_1+r_c}( {Y_1-\overline{Y}_1})( {Y_1-\overline{Y}_1})^TdY_1\right] =\frac{2r_c^3}{3}}\\&{ \left[ \int _{X_2-r_c}^{X_2+r_c}( {Y_2-\overline{Y}_2})( {Y_2-\overline{Y}_2})^TdY_2\right] =\frac{2r_c^3}{3}}\\ \end{aligned} \end{aligned}$$
(56)

Substitution of Eq. (56) in the the Eq. (55) yields:

$$\begin{aligned} \begin{aligned} {\hat{G}(\xi ,X_1)}=&{-C_1\xi ^0\text{cos}(\alpha X_1)\,\text{sin}(\beta X_2)}\\ {\hat{G}(\xi ,X_2)}=&{-C_2\xi ^0\text{sin}(\alpha X_1)\,\text{cos}(\beta X_2)}\\ \end{aligned} \end{aligned}$$
(57)

where,

$$\begin{aligned} {} \begin{aligned}&{C_1=\frac{3}{\alpha ^2r_c^3}\left( \alpha r_c \text{cos}\left( {\alpha r_c}\right) - \text{sin}\left( {\alpha r_c}\right) \right) }\\&{C_2=\frac{3}{\beta ^2r_c^3}\left( \beta r_c \text{cos}\left( {\beta r_c}\right) - \text{sin}\left( {\beta r_c}\right) \right) } \end{aligned} \end{aligned}$$
(58)

The expression for \({\hat{G}\left( \hat{G}\left( \xi ,X_1\right) ,X_1\right) }\) can be computed as:

$$\begin{aligned} \begin{aligned} {\hat{G}\left( \hat{G}\left( \xi ,X_1\right) ,X_1\right) }=&{ \left[ \int _{X_1-r_c}^{X_1+r_c}( {\hat{G}(\xi ,X_1)-\overline{\hat{G}(\xi ,X_1)}})( {Y_1-\overline{Y}_1})^TdY_1\right] \frac{3}{2rc^3}}\\ =&{C_1^2\xi ^0\text{sin}(\alpha X_1)\,\text{sin}(\beta X_2)} \end{aligned} \end{aligned}$$
(59)

Similarly,

$$\begin{aligned} \begin{aligned} {\hat{G}\left( \hat{G}\left( \xi ,X_2\right) ,X_2\right) }=&{ \left[ \int _{X_2-r_c}^{X_2+r_c}( {\hat{G}(\xi ,X_2)-\overline{\hat{G}(\xi ,X_2)}})( {Y_2-\overline{Y}_2})^TdY_2\right] \frac{3}{2rc^3}}\\ =&{C_2^2\xi ^0\text{sin}(\alpha X_1)\,\text{sin}(\beta X_2)} \end{aligned} \end{aligned}$$
(60)

and,

$$\begin{aligned} \begin{aligned} {\hat{G}\left( \hat{G}\left( \xi ,X_1\right) ,X_2\right) }=&{ \left[ \int _{X_2-r_c}^{X_2+r_c}( {\hat{G}(\xi ,X_1)-\overline{\hat{G}(\xi ,X_1)}})( {Y_2-\overline{Y}_2})^TdY_2\right] \frac{3}{2rc^3}}\\ =&{C_1C_2\xi ^0\text{sin}(\alpha X_1)\,\text{sin}(\beta X_2)} \end{aligned} \end{aligned}$$
(61)

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Saxena, M., Sarkar, S. & Reddy, J.N. Modelling architected plate using a non-local derivative-free shear deformable plate theory. Meccanica 58, 1671–1692 (2023). https://doi.org/10.1007/s11012-023-01677-y

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