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Microstructural properties of novel nanocomposite material based on hydroxyapatite and carbon nanotubes: fabrication and nonlinear instability simulation

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Abstract

In this work, a novel porous bio-nanocomposite material containing the commercial nanocrystalline hydroxyapatite (n-HA) composed using single-walled carbon nanotubes (SWCNTs) with different weight fractions (0, 2.5, 5, and 7.5 wt%) is fabricated via a space holder technique for bone tissue engineering applications. The hydroxyapatite (HA) presents a weak mechanical performance which the addition of the SWCNT can enhance the mechanical strength of the matrix. The manufactured n-HA-SWCNT bio-nanocomposite samples are then coated by gelatin-ibuprofen (GN-IBO) bio-polymer with a dip-coating method. The X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy-dispersive spectroscopy (EDS) are employed for the phase characterization and surface morphology analysis of the fabricated bio-nanocomposite specimens. The mechanical properties, the rate of drug release, and biological characteristics of the specimens with and without SWCNTs are investigated. After that, the nonlinear instability and vibration responses of an axially loaded plate-form bone implant made of the manufactured n-HA-SWCNT bio-nanocomposite samples coated with GN-IBO thin layers are simulated through a sandwich-plate model and based upon the experimentally extracted mechanical properties. The obtained results indicate the presence of SWCNT and n-HA in the composition due to the fuzzy metamorphism or degradation. It is found that by increasing the amount of SWCNTs, the mass density of the fabricated bio-nanocomposite samples reduces, but the porosity and strength of them are higher than those of the pure n-HA. Therefore, it is expected that these newly manufactured bio-nanocomposites have an excellent capability for bone formation with better bonds with surrounding tissues.

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Authors and Affiliations

  1. School of Science and Technology, The University of Georgia, 0171, Tbilisi, Georgia

    S. Sahmani

  2. New Technologies Research Center, Amirkabir University of Technology, 15875-4413, Tehran, Iran

    S. Saber-Samandari & A. Khandan

  3. Mechanical Engineering Department, Amirkabir University of Technology, 15875-4413, Tehran, Iran

    M. M. Aghdam

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  1. S. Sahmani
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  2. S. Saber-Samandari
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  3. M. M. Aghdam
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  4. A. Khandan
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Appendices

Appendix A

$${u}_{x}\left(x,y,z\right)=u\left(x,y\right)-z\frac{\partial w(x,y)}{\partial x}+\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]{\psi }_{x}\left(x,y\right)$$
(3)
$${u}_{y}\left(x,y,z\right)=v\left(x,y\right)-z\frac{\partial w(x,y)}{\partial y}+\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]{\psi }_{y}\left(x,y\right)$$
(4)
$${u}_{z}\left(x,y,z\right)=w\left(x,y\right)$$
(5)

where \(u,v\) and \(w\) denote the displacement components of the plate type implant for x, y, and z-axis, respectively. Furthermore, \({\psi }_{x}\) and \({\psi }_{y}\) represent, the rotations relevant to the cross section of the implant at neutral-plane normal, respectively, for the y and x-axis.

According to the von Karman nonlinear strain displacement formulation, this yields

$$ \begin{aligned} \begin{aligned}\left\{\begin{array}{c}{\varepsilon }_{xx}\\ {\varepsilon }_{yy}\\ {\gamma }_{xy}\end{array}\right\}&=\left\{\begin{array}{c}{\varepsilon }_{xx}^{0} \\ {\varepsilon }_{yy}^{0}\\ {\gamma }_{xy}^{0}\end{array}\right\}+z\left\{\begin{array}{c}{\kappa }_{xx}^{(1)}\\ {\kappa }_{yy}^{(1)}\\ {\kappa }_{xy}^{(1)}\end{array}\right\}+\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(2)}\\ {\kappa }_{yy}^{(2)}\\ {\kappa }_{xy}^{(2)}\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial u}{\partial x}+\frac{1}{2}{\left(\frac{\partial w}{\partial x}\right)}^{2}\\ \frac{\partial v}{\partial y}+\frac{1}{2}{\left(\frac{\partial w}{\partial y}\right)}^{2}\\ \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}+\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\end{array}\right\}-z\left\{\begin{array}{c}\frac{{\partial }^{2}w}{\partial {x}^{2}}\\ \frac{{\partial }^{2}w}{\partial {y}^{2}}\\ 2\frac{{\partial }^{2}w}{\partial x\partial y}\end{array}\right\}\\&\quad+\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]\left\{\begin{array}{c}\frac{\partial {\psi }_{x}}{\partial x}\\ \frac{\partial {\psi }_{y}}{\partial y}\\ \frac{\partial {\psi }_{x}}{\partial y}+\frac{\partial {\psi }_{y}}{\partial x}\end{array}\right\}\end{aligned} \end{aligned} $$
(6)
$$\left\{\begin{array}{c}{\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right\}=\left[\mathrm{cosh}\left(\frac{1}{2}\right)-\mathrm{cosh}\left(\frac{z}{h}\right)\right]\left\{\begin{array}{c}{\psi }_{x}\\ {\psi }_{y}\end{array}\right\}$$
(7)

where \({\varepsilon }_{ij}^{0}\), \({\kappa }_{ij}^{(1)}\), \({\kappa }_{ij}^{(2)}\) (\(i,j=x,y\)) are respectively related to the midplane strain components, the first and higher-order curvature components.

Accordingly, the constitutive relationship for a sandwich plate type implant will be like the following:

$$\left\{\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\\ {\sigma }_{xz}\\ {\sigma }_{yz}\end{array}\right\}=\left(\left[{Q}_{f1}\right]+\left[{Q}_{b}\right]+\left[{Q}_{f2}\right]\right)\left\{\begin{array}{c}{\varepsilon }_{xx}\\ {\varepsilon }_{yy}\\ {\gamma }_{xy}\\ {\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right\}$$
(8)

where

$$\left[{Q}^{f1}\right]=\left[{Q}^{f2}\right]=\left[\begin{array}{ccccc}\frac{{E}_{f}}{1-{\nu }_{f}^{2}}& \frac{{\nu }_{f}{E}_{f}}{1-{\nu }_{f}^{2}}& 0& 0& 0\\ \frac{{\nu }_{f}{E}_{f}}{1-{\nu }_{f}^{2}}& \frac{{E}_{f}}{1-{\nu }_{f}^{2}}& 0& 0& 0\\ 0& 0& \frac{{E}_{f}}{\left(1-{\nu }_{f}\right)\left(1+2{\nu }_{f}\right)}& 0& 0\\ 0& 0& 0& \frac{{E}_{f}}{\left(1-{\nu }_{f}\right)\left(1+2{\nu }_{f}\right)}& 0\\ 0& 0& 0& 0& \frac{{E}_{f}}{\left(1-{\nu }_{f}\right)\left(1+2{\nu }_{f}\right)}\end{array}\right]$$
(9)
$$\left[{Q}^{b}\right]=\left[\begin{array}{ccccc}\frac{{E}_{b}}{1-{\nu }_{b}^{2}}& \frac{{\nu }_{b}{E}_{b}}{1-{\nu }_{b}^{2}}& 0& 0& 0\\ \frac{{\nu }_{b}{E}_{b}}{1-{\nu }_{b}^{2}}& \frac{{E}_{b}}{1-{\nu }_{b}^{2}}& 0& 0& 0\\ 0& 0& \frac{{E}_{b}}{\left(1-{\nu }_{b}\right)\left(1+2{\nu }_{b}\right)}& 0& 0\\ 0& 0& 0& \frac{{E}_{b}}{\left(1-{\nu }_{b}\right)\left(1+2{\nu }_{b}\right)}& 0\\ 0& 0& 0& 0& \frac{{E}_{b}}{\left(1-{\nu }_{b}\right)\left(1+2{\nu }_{b}\right)}\end{array}\right]$$
(10)

in which \({E}_{b}\) and \({E}_{f}\) represent Young's modulus of the bulk and coating film, \({\nu }_{b}\) and \({\nu }_{f}\) are their Poisson's ratio for the sandwich plate type implant, that is extracted investigationally due to the fabricated n-HA-SWCNT bio-nanocomposites dip coated by GN-IBO thin layers as has been shown in Table 1.

Considering the continuum elasticity, the overall strain energy of the sandwich plate type implant could be stated as

$$ \begin{aligned}{\Pi }_{s}&=\frac{1}{2}{\int }_{S}{\int }_{-\frac{h}{2}}^\frac{h}{2}{\sigma }_{ij}{\varepsilon }_{ij}dzdS=\frac{1}{2}{\int }_{S}\left\{{N}_{xx}{\varepsilon }_{xx}^{0}+{N}_{yy}{\varepsilon }_{yy}^{0}+{N}_{xy}{\gamma }_{xy}^{0}+{M}_{xx}{\kappa }_{xx}^{(1)}+{M}_{yy}{\kappa }_{yy}^{(1)}\right.\\&\quad\left.+{M}_{xy}{\kappa }_{xy}^{(1)}+{R}_{xx}{\kappa }_{xx}^{(2)}+{R}_{yy}{\kappa }_{yy}^{(2)}+{R}_{xy}{\kappa }_{xy}^{(2)}+{Q}_{x}{\gamma }_{xz}+{Q}_{y}{\gamma }_{yz}\right\}dS\end{aligned} $$
(11)

where the stress resultants could be stated as

$$\left\{\begin{array}{c}{N}_{xx}\\ {N}_{yy}\\ {N}_{xy}\end{array}\right\}=\left[\begin{array}{ccc}{A}_{11}^{*}& {A}_{12}^{*}& 0\\ {A}_{12}^{*}& {A}_{22}^{*}& 0\\ 0& 0& {A}_{66}^{*}\end{array}\right]\left\{\begin{array}{c}{\varepsilon }_{xx}^{0}\\ {\varepsilon }_{yy}^{0}\\ {\gamma }_{xy}^{0}\end{array}\right\}+\left[\begin{array}{ccc}{B}_{11}^{*}& {B}_{12}^{*}& 0\\ {B}_{12}^{*}& {B}_{22}^{*}& 0\\ 0& 0& {B}_{66}^{*}\end{array}\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(1)}\\ {\kappa }_{yy}^{(1)}\\ {\kappa }_{xy}^{(1)}\end{array}\right\}+\left[\begin{array}{ccc}{B}_{11}^{**}& {B}_{12}^{**}& 0\\ {B}_{12}^{**}& {B}_{22}^{**}& 0\\ 0& 0& {B}_{66}^{**}\end{array}\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(2)}\\ {\kappa }_{yy}^{(2)}\\ {\kappa }_{xy}^{(2)}\end{array}\right\}$$
(12)
$$\left\{\begin{array}{c}{M}_{xx}\\ {M}_{yy}\\ {M}_{xy}\end{array}\right\}=\left[\begin{array}{ccc}{B}_{11}^{*}& {B}_{12}^{*}& 0\\ {B}_{12}^{*}& {B}_{22}^{*}& 0\\ 0& 0& {B}_{66}^{*}\end{array}\right]\left\{\begin{array}{c}{\varepsilon }_{xx}^{0}\\ {\varepsilon }_{yy}^{0}\\ {\gamma }_{xy}^{0}\end{array}\right\}+\left[\begin{array}{ccc}{D}_{11}^{*}& {D}_{12}^{*}& 0\\ {D}_{12}^{*}& {D}_{22}^{*}& 0\\ 0& 0& {D}_{66}^{*}\end{array}\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(1)}\\ {\kappa }_{yy}^{(1)}\\ {\kappa }_{xy}^{(1)}\end{array}\right\}+\left[\begin{array}{ccc}{D}_{11}^{**}& {D}_{12}^{**}& 0\\ {D}_{12}^{**}& {D}_{22}^{**}& 0\\ 0& 0& {D}_{66}^{**}\end{array}\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(2)}\\ {\kappa }_{yy}^{(2)}\\ {\kappa }_{xy}^{(2)}\end{array}\right\}(13)$$
(13)
$$\left\{\begin{array}{c}{R}_{xx}\\ {R}_{yy}\\ {R}_{xy}\end{array}\right\}=\left[\begin{array}{ccc}{B}_{11}^{**}& {B}_{12}^{**}& 0\\ {B}_{12}^{**}& {B}_{22}^{**}& 0\\ 0& 0& {B}_{66}^{**}\end{array}\right]\left\{\begin{array}{c}{\varepsilon }_{xx}^{0}\\ {\varepsilon }_{yy}^{0}\\ {\gamma }_{xy}^{0}\end{array}\right\}+\left[\begin{array}{ccc}{D}_{11}^{**}& {D}_{12}^{**}& 0\\ {D}_{12}^{**}& {D}_{22}^{**}& 0\\ 0& 0& {D}_{66}^{**}\end{array}\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(1)}\\ {\kappa }_{yy}^{(1)}\\ {\kappa }_{xy}^{(1)}\end{array}\right\}+\left[\begin{array}{ccc}{G}_{11}^{*}& {G}_{12}^{*}& 0\\ {G}_{12}^{*}& {G}_{22}^{*}& 0\\ 0& 0& {G}_{66}^{*}\end{array}\right]\left\{\begin{array}{c}{\kappa }_{xx}^{(2)}\\ {\kappa }_{yy}^{(2)}\\ {\kappa }_{xy}^{(2)}\end{array}\right\}$$
(14)
$$\left\{\begin{array}{c}{Q}_{x}\\ {Q}_{y}\end{array}\right\}=\left[\begin{array}{cc}{A}_{44}^{*}& 0\\ 0& {A}_{55}^{*}\end{array}\right]\left\{\begin{array}{c}{\psi }_{x}\\ {\psi }_{y}\end{array}\right\}$$
(15)

in which the stiffness parameters are given in Appendix A.

Besides, the sandwich plate type implant’s kinetic energy of the manufactured n-HA-SWCNT bio-nanocomposites dip coated considering GN-IBO thin layers based upon the hyperbolic shear deformable plate model may be stated as

$${\Pi }_{T}=\frac{1}{2}{\int }_{S}{\int }_{-\frac{h}{2}}^\frac{h}{2}\rho \left\{{\left(\frac{\partial u}{\partial t}-z\frac{{\partial }^{2}w}{\partial x\partial t}+\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]\frac{\partial {\psi }_{x}}{\partial t}\right)}^{2}+{\left(\frac{\partial v}{\partial t}-z\frac{{\partial }^{2}w}{\partial y\partial t}+\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]\frac{\partial {\psi }_{y}}{\partial t}\right)}^{2}+{\left(\frac{\partial w}{\partial t}\right)}^{2}\right\}dzdS=\frac{1}{2}{\int }_{S}\left\{{I}_{0}{\left(\frac{\partial u}{\partial t}\right)}^{2}-2{I}_{1}\frac{\partial u}{\partial t}\frac{{\partial }^{2}w}{\partial x\partial t}+2{I}_{3}\frac{\partial u}{\partial t}\frac{\partial {\psi }_{x}}{\partial t}+{I}_{2}{\left(\frac{{\partial }^{2}w}{\partial x\partial t}\right)}^{2}-2{I}_{4}\frac{{\partial }^{2}w}{\partial x\partial t}\frac{\partial {\psi }_{x}}{\partial t}+{I}_{5}{\left(\frac{\partial {\psi }_{x}}{\partial t}\right)}^{2}+{I}_{0}{\left(\frac{\partial v}{\partial t}\right)}^{2}-2{I}_{1}\frac{\partial v}{\partial t}\frac{{\partial }^{2}w}{\partial y\partial t}+2{I}_{3}\frac{\partial v}{\partial t}\frac{\partial {\psi }_{y}}{\partial t}+{I}_{2}{\left(\frac{{\partial }^{2}w}{\partial y\partial t}\right)}^{2}-2{I}_{4}\frac{{\partial }^{2}w}{\partial y\partial t}\frac{\partial {\psi }_{y}}{\partial t}+{I}_{5}{\left(\frac{\partial {\psi }_{y}}{\partial t}\right)}^{2}+{I}_{0}{\left(\frac{\partial w}{\partial t}\right)}^{2}\right\}dS$$
(16)

where \({I}_{0}, {I}_{1},\dots ,{I}_{5}\) are given in Appendix A.

Additionally, \({\Pi }_{P}\) achieved by the external distributed load \(\mathcal{q}\) may be expressed as

$${\Pi }_{P}={\int }_{S}\mathcal{q}wdS$$
(17)

So, after utilizing Hamilton's principle, the non-linear governing differential equations of motion considering the stress resultants might be written as

$$\frac{\partial {N}_{xx}}{\partial x}+\frac{\partial {N}_{xy}}{\partial y}={I}_{0}\frac{{\partial }^{2}u}{\partial {t}^{2}}-{I}_{1}\frac{{\partial }^{3}w}{\partial x\partial {t}^{2}}+{I}_{3}\frac{{\partial }^{2}{\psi }_{x}}{\partial {t}^{2}}$$
(18)
$$\frac{\partial {N}_{xy}}{\partial x}+\frac{\partial {N}_{yy}}{\partial y}={I}_{0}\frac{{\partial }^{2}v}{\partial {t}^{2}}-{I}_{1}\frac{{\partial }^{3}w}{\partial y\partial {t}^{2}}+{I}_{3}\frac{{\partial }^{2}{\psi }_{y}}{\partial {t}^{2}}$$
(19)
$$\frac{{\partial }^{2}{M}_{xx}}{\partial {x}^{2}}+2\frac{{\partial }^{2}{M}_{xy}}{\partial x\partial y}+\frac{{\partial }^{2}{M}_{yy}}{\partial {y}^{2}}+{N}_{xx}\frac{{\partial }^{2}w}{\partial {x}^{2}}+2{N}_{xy}\frac{{\partial }^{2}w}{\partial x\partial y}+{N}_{yy}\frac{{\partial }^{2}w}{\partial {y}^{2}}=-{I}_{1}\left(\frac{{\partial }^{3}u}{\partial x\partial {t}^{2}}+\frac{{\partial }^{3}v}{\partial y\partial {t}^{2}}\right)+{I}_{0}\frac{{\partial }^{2}w}{\partial {t}^{2}}-{I}_{2}\left(\frac{{\partial }^{4}w}{\partial {x}^{2}\partial {t}^{2}}+\frac{{\partial }^{4}w}{\partial {y}^{2}\partial {t}^{2}}\right)-{I}_{4}\left(\frac{{\partial }^{3}{\psi }_{x}}{\partial x\partial {t}^{2}}+\frac{{\partial }^{3}{\psi }_{y}}{\partial y\partial {t}^{2}}\right)+\mathcal{q}$$
(20)
$$\frac{\partial {R}_{xx}}{\partial x}+\frac{\partial {R}_{xy}}{\partial y}-{Q}_{x}={I}_{3}\frac{{\partial }^{2}u}{\partial {t}^{2}}-{I}_{4}\frac{{\partial }^{3}w}{\partial x\partial {t}^{2}}+{I}_{5}\frac{{\partial }^{2}{\psi }_{x}}{\partial {t}^{2}}$$
(21)
$$\frac{\partial {R}_{xy}}{\partial x}+\frac{\partial {R}_{yy}}{\partial y}-{Q}_{y}={I}_{3}\frac{{\partial }^{2}v}{\partial {t}^{2}}-{I}_{4}\frac{{\partial }^{3}w}{\partial y\partial {t}^{2}}+{I}_{5}\frac{{\partial }^{2}{\psi }_{y}}{\partial {t}^{2}}$$
(22)

In that way, the Airy stress function \(f\left(x,y\right)\) is stated as

$${N}_{xx}=\frac{\partial f(x,y)}{\partial {y}^{2}} , {N}_{yy}=\frac{\partial f(x,y)}{\partial {x}^{2}} , {N}_{xy}=-\frac{\partial f\left(x,y\right)}{\partial x\partial y}$$
(23)

Moreover, considering the perfect geometric assumption for the plate type implant, the compatibility equation relevant to the mid-plane strain components yields

$$\frac{{\partial }^{2}{\varepsilon }_{xx}^{0}}{\partial {x}^{2}}+\frac{{\partial }^{2}{\varepsilon }_{yy}^{0}}{\partial {y}^{2}}-\frac{{\partial }^{2}{\gamma }_{xy}^{0}}{\partial x\partial y}={\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^{2}-\frac{{\partial }^{2}w}{\partial {x}^{2}}\frac{{\partial }^{2}w}{\partial {y}^{2}}$$
(24)

Using formulations (6) and (11), the non-linear governing differential equations of motion may be stated as the displacement components like the following,

$$\left(\frac{{A}_{11}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}f}{\partial {x}^{4}}+\left(\frac{1}{{A}_{66}^{*}}-\frac{2{A}_{12}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}f}{\partial {x}^{2}\partial {y}^{2}}+\left(\frac{{A}_{22}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}f}{\partial {y}^{4}}+\left(\frac{{A}_{11}^{*}{B}_{12}^{*}-{A}_{12}^{*}{B}_{11}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}w}{\partial {x}^{4}}+\left(\frac{{A}_{11}^{*}{B}_{11}^{*}+{A}_{22}^{*}{B}_{22}^{*}-2{A}_{12}^{*}{B}_{12}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{2{B}_{66}^{*}}{{A}_{66}^{*}}\right)\frac{{\partial }^{4}w}{\partial {x}^{2}\partial {y}^{2}}+\left(\frac{{A}_{22}^{*}{B}_{12}^{*}-{A}_{12}^{*}{B}_{22}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}w}{\partial {y}^{4}}-\left(\frac{{A}_{11}^{*}{B}_{12}^{**}-{A}_{12}^{*}{B}_{11}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{3}{\psi }_{x}}{\partial {x}^{3}}-\left(\frac{{A}_{11}^{*}{B}_{11}^{**}-{A}_{12}^{*}{B}_{12}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{{B}_{66}^{**}}{{A}_{66}^{*}}\right)\frac{{\partial }^{3}{\psi }_{x}}{\partial x\partial {y}^{2}}-\left(\frac{{A}_{22}^{*}{B}_{12}^{**}-{A}_{12}^{*}{B}_{22}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{3}{\psi }_{y}}{\partial {y}^{3}}-\left(\frac{{A}_{22}^{*}{B}_{22}^{**}-{A}_{12}^{*}{B}_{12}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{{B}_{66}^{**}}{{A}_{66}^{*}}\right)\frac{{\partial }^{3}{\psi }_{y}}{\partial {x}^{2}\partial y}={\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^{2}-\frac{{\partial }^{2}w}{\partial {x}^{2}}\frac{{\partial }^{2}w}{\partial {y}^{2}}$$
(25)
$$\left({D}_{11}^{*}-\frac{{A}_{11}^{*}\left({\left({B}_{11}^{*}\right)}^{2}+{\left({B}_{12}^{*}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}w}{\partial {x}^{4}}+\left(2\left({D}_{12}^{*}+2{D}_{66}^{*}\right)-\frac{{A}_{11}^{*}\left({\left({B}_{11}^{*}\right)}^{2}+{\left({B}_{12}^{*}\right)}^{2}\right)+{A}_{22}^{*}\left({\left({B}_{22}^{*}\right)}^{2}+{\left({B}_{12}^{*}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-4\frac{{\left({B}_{66}^{*}\right)}^{2}}{{A}_{66}^{*}}\right)\frac{{\partial }^{4}w}{\partial {x}^{2}\partial {y}^{2}}+\left({D}_{22}^{*}-\frac{{A}_{22}^{*}\left({\left({B}_{22}^{*}\right)}^{2}+{\left({B}_{12}^{*}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{4}w}{\partial {y}^{4}}+\left(\frac{{A}_{11}^{*}\left({\left({B}_{11}^{**}\right)}^{2}+{\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-{D}_{11}^{**}\right)\frac{{\partial }^{3}{\psi }_{x}}{\partial {x}^{3}}+\left(\frac{2{A}_{11}^{*}{B}_{11}^{**}{B}_{12}^{**}-{A}_{12}^{*}\left({{\left({B}_{11}^{**}\right)}^{2}+\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}+2\frac{{\left({B}_{66}^{**}\right)}^{2}}{{A}_{66}^{*}}-\left({D}_{12}^{**}+2{D}_{66}^{**}\right)\right)\frac{{\partial }^{3}{\psi }_{x}}{\partial x\partial {y}^{2}}+\left(\frac{{A}_{22}^{*}\left({\left({B}_{22}^{**}\right)}^{2}+{\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-{D}_{22}^{**}\right)\frac{{\partial }^{3}{\psi }_{y}}{\partial {y}^{3}}+\left(\frac{2{A}_{22}^{*}{B}_{22}^{**}{B}_{12}^{**}-{A}_{12}^{*}\left({{\left({B}_{22}^{**}\right)}^{2}+\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}+2\frac{{\left({B}_{66}^{**}\right)}^{2}}{{A}_{66}^{*}}-\left({D}_{12}^{**}+2{D}_{66}^{**}\right)\right)\frac{{\partial }^{3}{\psi }_{y}}{\partial {x}^{2}\partial y}=\frac{{\partial }^{2}w}{\partial {x}^{2}}\frac{{\partial }^{2}f}{\partial {y}^{2}}-2\frac{{\partial }^{2}w}{\partial x\partial y}\frac{{\partial }^{2}f}{\partial x\partial y}+\frac{{\partial }^{2}w}{\partial {y}^{2}}\frac{{\partial }^{2}f}{\partial {x}^{2}}-{I}_{0}\frac{{\partial }^{2}w}{\partial {t}^{2}}+\left({I}_{2}-\frac{{I}_{1}{I}_{3}}{{I}_{0}}\right)\left(\frac{{\partial }^{4}w}{\partial {x}^{2}\partial {t}^{2}}+\frac{{\partial }^{4}w}{\partial {y}^{2}\partial {t}^{2}}\right)+\left({I}_{4}-\frac{{I}_{1}{I}_{3}}{{I}_{0}}\right)\left(\frac{{\partial }^{3}{\psi }_{x}}{\partial x\partial {t}^{2}}+\frac{{\partial }^{3}{\psi }_{y}}{\partial y\partial {t}^{2}}\right)+\mathcal{q}$$
(26)
$$\left(\frac{{A}_{11}^{*}{B}_{12}^{**}-{A}_{12}^{*}{B}_{11}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{3}f}{\partial {x}^{3}}+\left(\frac{{A}_{11}^{*}{B}_{11}^{**}-{A}_{12}^{*}{B}_{12}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{{B}_{66}^{**}}{{A}_{66}^{*}}\right)\frac{{\partial }^{3}f}{\partial x\partial {y}^{2}}+\left(\frac{{A}_{11}^{*}\left({B}_{11}^{*}{B}_{11}^{**}+{B}_{12}^{*}{B}_{12}^{**}\right)-{A}_{12}^{*}\left({B}_{12}^{*}{B}_{11}^{**}+{B}_{11}^{*}{B}_{12}^{**}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-{D}_{11}^{**}\right)\frac{{\partial }^{3}w}{\partial {x}^{3}}+\left(\frac{{A}_{11}^{*}\left({B}_{11}^{*}{B}_{12}^{**}+{B}_{12}^{*}{B}_{11}^{**}\right)-{A}_{12}^{*}\left({B}_{12}^{*}{B}_{12}^{**}+{B}_{11}^{*}{B}_{11}^{**}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}+2\frac{{B}_{66}^{*}{B}_{66}^{**}}{{A}_{66}^{*}}-\left({D}_{12}^{**}+2{D}_{66}^{**}\right)\right)\frac{{\partial }^{3}w}{\partial x\partial {y}^{2}}+\left({G}_{11}^{*}-\frac{{A}_{11}^{*}\left({\left({B}_{11}^{**}\right)}^{2}+{\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}{\psi }_{x}}{\partial {x}^{2}}+\left({G}_{66}^{*}-\frac{{\left({B}_{66}^{**}\right)}^{2}}{{A}_{66}^{*}}\right)\frac{{\partial }^{2}{\psi }_{x}}{\partial {y}^{2}}+\left({G}_{12}^{*}+{G}_{66}^{*}+\frac{{A}_{12}^{*}\left({\left({B}_{11}^{**}\right)}^{2}+{\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{{\left({B}_{66}^{**}\right)}^{2}}{{A}_{66}^{*}}\right)\frac{{\partial }^{2}{\psi }_{y}}{\partial x\partial y}-{A}_{44}^{*}{\psi }_{x}=\left(\frac{{I}_{3}^{2}}{{I}_{0}}-{I}_{4}\right)\frac{{\partial }^{3}w}{\partial x\partial {t}^{2}}+\left({I}_{5}-\frac{{I}_{3}^{2}}{{I}_{0}}\right)\frac{{\partial }^{2}{\psi }_{x}}{\partial {t}^{2}}$$
(27)
$$\left(\frac{{A}_{22}^{*}{B}_{12}^{**}-{A}_{12}^{*}{B}_{22}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{3}f}{\partial {y}^{3}}+\left(\frac{{A}_{22}^{*}{B}_{22}^{**}-{A}_{12}^{*}{B}_{12}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{{B}_{66}^{**}}{{A}_{66}^{*}}\right)\frac{{\partial }^{3}f}{\partial {x}^{2}\partial y}+\left(\frac{{A}_{22}^{*}\left({B}_{22}^{*}{B}_{22}^{**}+{B}_{12}^{*}{B}_{12}^{**}\right)-{A}_{12}^{*}\left({B}_{12}^{*}{B}_{22}^{**}+{B}_{22}^{*}{B}_{12}^{**}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-{D}_{22}^{**}\right)\frac{{\partial }^{3}w}{\partial {y}^{3}}+\left(\frac{{A}_{22}^{*}\left({B}_{22}^{*}{B}_{12}^{**}+{B}_{12}^{*}{B}_{22}^{**}\right)-{A}_{12}^{*}\left({B}_{12}^{*}{B}_{12}^{**}+{B}_{22}^{*}{B}_{22}^{**}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}+2\frac{{B}_{66}^{*}{B}_{66}^{**}}{{A}_{66}^{*}}-\left({D}_{12}^{**}+2{D}_{66}^{**}\right)\right)\frac{{\partial }^{3}w}{\partial {x}^{2}\partial y}+\left({G}_{22}^{*}-\frac{{A}_{22}^{*}\left({\left({B}_{22}^{**}\right)}^{2}+{\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}{\psi }_{y}}{\partial {y}^{2}}+\left({G}_{66}^{*}-\frac{{\left({B}_{66}^{**}\right)}^{2}}{{A}_{66}^{*}}\right)\frac{{\partial }^{2}{\psi }_{y}}{\partial {x}^{2}}+\left({G}_{12}^{*}+{G}_{66}^{*}+\frac{{A}_{12}^{*}\left({\left({B}_{22}^{**}\right)}^{2}+{\left({B}_{12}^{**}\right)}^{2}\right)}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}-\frac{{\left({B}_{66}^{**}\right)}^{2}}{{A}_{66}^{*}}\right)\frac{{\partial }^{2}{\psi }_{x}}{\partial x\partial y}-{A}_{55}^{*}{\psi }_{y}=\left(\frac{{I}_{3}^{2}}{{I}_{0}}-{I}_{4}\right)\frac{{\partial }^{3}w}{\partial y\partial {t}^{2}}+\left({I}_{5}-\frac{{I}_{3}^{2}}{{I}_{0}}\right)\frac{{\partial }^{2}{\psi }_{y}}{\partial {t}^{2}}$$
(28)

Likewise, the equilibrium condition of the applied loads in the x-axis direction could be written as

$${\int }_{0}^{{L}_{2}}{N}_{xx}dy+{L}_{2}h{\sigma }_{xx}=0$$
(29)

Furthermore, the shortenings of the sandwich plate type implant along the x and y-axis are formulated as follows

$$\frac{{\Delta }_{x}}{{L}_{1}}=-\frac{1}{{L}_{1}{L}_{2}}{\int }_{0}^{{L}_{1}}{\int }_{0}^{{L}_{2}}\frac{\partial u}{\partial x}dxdy=-\frac{1}{{L}_{1}{L}_{2}}{\int }_{0}^{{L}_{1}}{\int }_{0}^{{L}_{2}}\left\{\left(\frac{{A}_{11}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}f}{\partial {y}^{2}}-\left(\frac{{A}_{12}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}f}{\partial {x}^{2}}+\left(\frac{{A}_{11}^{*}{B}_{11}^{*}-{A}_{12}^{*}{B}_{12}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}w}{\partial {x}^{2}}+\left(\frac{{A}_{11}^{*}{B}_{12}^{*}-{A}_{12}^{*}{B}_{11}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}w}{\partial {y}^{2}}-\left(\frac{{A}_{11}^{*}{B}_{11}^{**}-{A}_{12}^{*}{B}_{12}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{\partial {\psi }_{x}}{\partial x}-\left(\frac{{A}_{11}^{*}{B}_{12}^{**}-{A}_{12}^{*}{B}_{11}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{\partial {\psi }_{y}}{\partial y}-\frac{1}{2}{\left(\frac{\partial w}{\partial x}\right)}^{2}\right\}dxdy$$
(30)
$$\frac{{\Delta }_{y}}{{L}_{2}}=-\frac{1}{{L}_{1}{L}_{2}}{\int }_{0}^{{L}_{1}}{\int }_{0}^{{L}_{2}}\frac{\partial v}{\partial y}dxdy=-\frac{1}{{L}_{1}{L}_{2}}{\int }_{0}^{{L}_{1}}{\int }_{0}^{{L}_{2}}\left\{\left(\frac{{A}_{22}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}f}{\partial {x}^{2}}-\left(\frac{{A}_{12}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}f}{\partial {y}^{2}}+\left(\frac{{A}_{22}^{*}{B}_{12}^{*}-{A}_{12}^{*}{B}_{22}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}w}{\partial {x}^{2}}+\left(\frac{{A}_{22}^{*}{B}_{22}^{*}-{A}_{12}^{*}{B}_{12}^{*}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{{\partial }^{2}w}{\partial {y}^{2}}-\left(\frac{{A}_{22}^{*}{B}_{12}^{**}-{A}_{12}^{*}{B}_{22}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{\partial {\psi }_{x}}{\partial x}-\left(\frac{{A}_{22}^{*}{B}_{22}^{**}-{A}_{12}^{*}{B}_{12}^{**}}{{{A}_{11}^{*}{A}_{22}^{*}-\left({A}_{12}^{*}\right)}^{2}}\right)\frac{\partial {\psi }_{y}}{\partial y}-\frac{1}{2}{\left(\frac{\partial w}{\partial y}\right)}^{2}\right\}dxdy$$
(31)
$${\mathcal{P}}_{x}={\mathcal{P}}_{x}^{\left(0\right)}+{\mathcal{P}}_{x}^{\left(2\right)}{\left({\mathcal{W}}_{m}\right)}^{2}+{\mathcal{P}}_{x}^{\left(4\right)}{\left({\mathcal{W}}_{m}\right)}^{4}+\dots $$
(32)
$${\delta }_{x}={\delta }_{x}^{(0)}+{\delta }_{x}^{(2)}{\left({\mathcal{W}}_{m}\right)}^{2}+{\delta }_{x}^{(4)}{\left({\mathcal{W}}_{m}\right)}^{4}+\dots $$
(33)
$${\omega }_{L}=\sqrt{{K}_{1}/{K}_{0}} (18)$$
(34)
$${\omega }_{NL}={\omega }_{L}\sqrt{1+\frac{9{K}_{1}{K}_{3}-10{K}_{2}^{2}}{12{K}_{1}^{2}}{\left({\mathcal{W}}_{m}\right)}^{2}}$$
(35)

Appendix B

$$ \left\{ {\begin{array}{*{20}c} {A_{{11}}^{*} } \\ {A_{{22}}^{*} } \\ {A_{{12}}^{*} } \\ {A_{{66}}^{*} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{{f1}} } \\ {Q_{{22}}^{{f1}} } \\ {Q_{{12}}^{{f1}} } \\ {Q_{{66}}^{{f1}} } \\ \end{array} } \right\}\int_{{z_{0} }}^{{z_{1} }} {dz + \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{b} } \\ {Q_{{22}}^{b} } \\ {Q_{{12}}^{b} } \\ {Q_{{66}}^{b} } \\ \end{array} } \right\}} + \int_{{z_{1} }}^{{z_{2} }} {dz + \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{{f2}} } \\ {Q_{{22}}^{{f2}} } \\ {Q_{{12}}^{{f2}} } \\ {Q_{{66}}^{{f2}} } \\ \end{array} } \right\} + \int_{{z_{2} }}^{{z_{3} }} {dz} } $$
(36)
$$ \left\{ {\begin{array}{*{20}c} {B_{{11}}^{*} } \\ {B_{{22}}^{*} } \\ {B_{{12}}^{*} } \\ {B_{{66}}^{*} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{{f1}} } \\ {Q_{{22}}^{{f1}} } \\ {Q_{{12}}^{{f1}} } \\ {Q_{{66}}^{{f1}} } \\ \end{array} } \right\}\int_{{z_{0} }}^{{z_{1} }} {zdz + \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{b} } \\ {Q_{{22}}^{b} } \\ {Q_{{12}}^{b} } \\ {Q_{{66}}^{b} } \\ \end{array} } \right\}} \int_{{z_{1} }}^{{z_{2} }} {zdz + \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{{f2}} } \\ {Q_{{22}}^{{f2}} } \\ {Q_{{12}}^{{f2}} } \\ {Q_{{66}}^{{f2}} } \\ \end{array} } \right\}} \int_{{z_{2} }}^{{z_{3} }} {zdz} $$
(37)
$$ \left\{ {\begin{array}{*{20}c} {D_{{11}}^{*} } \\ {D_{{22}}^{*} } \\ {D_{{12}}^{*} } \\ {D_{{66}}^{*} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{{f1}} } \\ {Q_{{22}}^{{f1}} } \\ {Q_{{12}}^{{f1}} } \\ {Q_{{66}}^{{f1}} } \\ \end{array} } \right\}\int_{{z_{0} }}^{{z_{1} }} {zdz + \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{b} } \\ {Q_{{22}}^{b} } \\ {Q_{{12}}^{b} } \\ {Q_{{66}}^{b} } \\ \end{array} } \right\}} \int_{{z_{1} }}^{{z_{2} }} {zdz + \left\{ {\begin{array}{*{20}c} {Q_{{11}}^{{f2}} } \\ {Q_{{22}}^{{f2}} } \\ {Q_{{12}}^{{f2}} } \\ {Q_{{66}}^{{f2}} } \\ \end{array} } \right\}} \int_{{z_{2} }}^{{z_{3} }} {zdz} $$
(38)
$$\left\{\begin{array}{c}{B}_{11}^{**}\\ {B}_{22}^{**}\\ {B}_{12}^{**}\\ {B}_{66}^{**}\end{array}\right\}=\left\{\begin{array}{c}{Q}_{11}^{f1}\\ {Q}_{22}^{f1}\\ {Q}_{12}^{f1}\\ {Q}_{66}^{f1}\end{array}\right\}{\int }_{{z}_{0}}^{{z}_{1}}\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+\left\{\begin{array}{c}{Q}_{11}^{b}\\ {Q}_{22}^{b}\\ {Q}_{12}^{b}\\ {Q}_{66}^{b}\end{array}\right\}{\int }_{{z}_{1}}^{{z}_{2}}\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+\left\{\begin{array}{c}{Q}_{11}^{f2}\\ {Q}_{22}^{f2}\\ {Q}_{12}^{f2}\\ {Q}_{66}^{f2}\end{array}\right\}{\int }_{{z}_{2}}^{{z}_{3}}\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz$$
(39)
$$\left\{\begin{array}{c}{D}_{11}^{**}\\ {D}_{22}^{**}\\ {D}_{12}^{**}\\ {D}_{66}^{**}\end{array}\right\}=\left\{\begin{array}{c}{Q}_{11}^{f1}\\ {Q}_{22}^{f1}\\ {Q}_{12}^{f1}\\ {Q}_{66}^{f1}\end{array}\right\}{\int }_{{z}_{0}}^{{z}_{1}}\left[{z}^{2}\mathrm{cosh}\left(\frac{1}{2}\right)-zh\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+\left\{\begin{array}{c}{Q}_{11}^{b}\\ {Q}_{22}^{b}\\ {Q}_{12}^{b}\\ {Q}_{66}^{b}\end{array}\right\}{\int }_{{z}_{1}}^{{z}_{2}}\left[{z}^{2}\mathrm{cosh}\left(\frac{1}{2}\right)-zh\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+\left\{\begin{array}{c}{Q}_{11}^{f2}\\ {Q}_{22}^{f2}\\ {Q}_{12}^{f2}\\ {Q}_{66}^{f2}\end{array}\right\}{\int }_{{z}_{2}}^{{z}_{3}}\left[{z}^{2}\mathrm{cosh}\left(\frac{1}{2}\right)-zh\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz$$
(40)
$$\left\{\begin{array}{c}{G}_{11}^{*}\\ {G}_{22}^{*}\\ {G}_{12}^{*}\\ {G}_{66}^{*}\end{array}\right\}=\left\{\begin{array}{c}{Q}_{11}^{f1}\\ {Q}_{22}^{f1}\\ {Q}_{12}^{f1}\\ {Q}_{66}^{f1}\end{array}\right\}{\int }_{{z}_{0}}^{{z}_{1}}\left[{\left(z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right)}^{2}\right]dz+\left\{\begin{array}{c}{Q}_{11}^{b}\\ {Q}_{22}^{b}\\ {Q}_{12}^{b}\\ {Q}_{66}^{b}\end{array}\right\}{\int }_{{z}_{1}}^{{z}_{2}}\left[{\left(z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right)}^{2}\right]dz+\left\{\begin{array}{c}{Q}_{11}^{f2}\\ {Q}_{22}^{f2}\\ {Q}_{12}^{f2}\\ {Q}_{66}^{f2}\end{array}\right\}{\int }_{{z}_{2}}^{{z}_{3}}\left[{\left(z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right)}^{2}\right]dz$$
(41)
$$\left\{\begin{array}{c}{A}_{44}^{*}\\ {A}_{55}^{*}\end{array}\right\}=\left\{\begin{array}{c}{Q}_{44}^{f1}\\ {Q}_{55}^{f1}\end{array}\right\}{\int }_{{z}_{0}}^{{z}_{1}}\left[\mathrm{cosh}\left(\frac{1}{2}\right)-\mathrm{cosh}\left(\frac{z}{h}\right)\right]dz+\left\{\begin{array}{c}{Q}_{44}^{b}\\ {Q}_{55}^{b}\end{array}\right\}{\int }_{{z}_{1}}^{{z}_{2}}\left[\mathrm{cosh}\left(\frac{1}{2}\right)-\mathrm{cosh}\left(\frac{z}{h}\right)\right]dz+\left\{\begin{array}{c}{Q}_{44}^{f2}\\ {Q}_{55}^{f2}\end{array}\right\}{\int }_{{z}_{2}}^{{z}_{3}}\left[\mathrm{cosh}\left(\frac{1}{2}\right)-\mathrm{cosh}\left(\frac{z}{h}\right)\right]dz$$
(42)
$${I}_{0}={\rho }_{f1}{\int }_{{z}_{0}}^{{z}_{1}}dz+{\rho }_{b}{\int }_{{z}_{1}}^{{z}_{2}}dz+{\rho }_{f2}{\int }_{{z}_{2}}^{{z}_{3}}dz$$
(43)
$${I}_{1}={\rho }_{f1}{\int }_{{z}_{0}}^{{z}_{1}}zdz+{\rho }_{b}{\int }_{{z}_{1}}^{{z}_{2}}zdz+{\rho }_{f2}{\int }_{{z}_{2}}^{{z}_{3}}zdz$$
(44)
$${I}_{2}={\rho }_{f1}{\int }_{{z}_{0}}^{{z}_{1}}{z}^{2}dz+{\rho }_{b}{\int }_{{z}_{1}}^{{z}_{2}}{z}^{2}dz+{\rho }_{f2}{\int }_{{z}_{2}}^{{z}_{3}}{z}^{2}dz$$
(45)
$${I}_{3}={\rho }_{f1}{\int }_{{z}_{0}}^{{z}_{1}}\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+{\rho }_{b}{\int }_{{z}_{1}}^{{z}_{2}}\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+{\rho }_{f2}{\int }_{{z}_{2}}^{{z}_{3}}\left[z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz$$
(46)
$${I}_{4}={\rho }_{f1}{\int }_{{z}_{0}}^{{z}_{1}}\left[{z}^{2}\mathrm{cosh}\left(\frac{1}{2}\right)-zh\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+{\rho }_{b}{\int }_{{z}_{1}}^{{z}_{2}}\left[{z}^{2}\mathrm{cosh}\left(\frac{1}{2}\right)-zh\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz+{\rho }_{f2}{\int }_{{z}_{2}}^{{z}_{3}}\left[{z}^{2}\mathrm{cosh}\left(\frac{1}{2}\right)-zh\mathrm{sinh}\left(\frac{z}{h}\right)\right]dz$$
(47)
$${I}_{5}={\rho }_{f1}{\int }_{{z}_{0}}^{{z}_{1}}\left[{\left(z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right)}^{2}\right]dz+{\rho }_{b}{\int }_{{z}_{1}}^{{z}_{2}}\left[{\left(z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right)}^{2}\right]dz+{\rho }_{f2}{\int }_{{z}_{2}}^{{z}_{3}}\left[{\left(z\mathrm{cosh}\left(\frac{1}{2}\right)-h\mathrm{sinh}\left(\frac{z}{h}\right)\right)}^{2}\right]dz$$
(48)

where \({\rho }_{f1}, {\rho }_{f2}, {\rho }_{b}\) stand for the mass density of the upper and lower coating film, and the bulk of the sandwich plate type implant, respectively.

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Sahmani, S., Saber-Samandari, S., Aghdam, M.M. et al. Microstructural properties of novel nanocomposite material based on hydroxyapatite and carbon nanotubes: fabrication and nonlinear instability simulation. J Nanostruct Chem 12, 1–22 (2022). https://doi.org/10.1007/s40097-021-00395-9

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  • DOI: https://doi.org/10.1007/s40097-021-00395-9

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