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Nonlinear instability of a thin laminated composite circular plate subjected to a tensile periodic load

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Abstract

In this work, we have investigated the nonlinear instability behavior of a thin laminated composite circular plate subjected to a tensile periodic load. A nonlinear system with quadratic and cubic nonlinear terms, referred to the effect of large deflection, is obtained. The method of multiple time scales is used to obtain periodic solutions for the second- and third-order systems, which directly leads to the solvability conditions of the two orders including the nonlinearity terms. Different cases of resonance with the instability criteria are taken into account. For the numerical application, the effect of the tensile periodic load was studied on a plate of graphite fabric-carbon material.

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Correspondence to Khaled M. Elmorabie.

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Appendices

Appendix A1

The elements of the transformed reduced stiffness matrix \([{\bar{Q}}_{ij}]^{(k)}\) are

$$\begin{aligned}{}[{\bar{Q}}]=[T]^{-1}[Q][T]^{-T} \end{aligned}$$

where [T] is the transformation matrix between the principal material coordinates and the plate’s coordinates given by

$$\begin{aligned} T=\left[ \begin{array}{ccc} \cos ^{2}\vartheta &{}\quad \sin ^{2}\vartheta &{}\quad 2 \sin \vartheta \cos \vartheta \\ \sin ^{2}\vartheta &{}\quad \cos ^{2}\vartheta &{}\quad -2 \sin \vartheta \cos \vartheta \\ \sin \vartheta \cos \vartheta &{}\quad -\sin \vartheta \cos \vartheta &{}\quad \cos ^{2}\vartheta -\sin ^{2}\vartheta \end{array} \right] , \end{aligned}$$
Fig. 14
figure 14

a Composite plate design of multi-shape with a plane symmetry \(S=0\), b the diagram explains the transversal cut of a multi-shape composite plate divided into 4 panels with filled holes

and \(\vartheta \) is the orientation angle.

Appendix A2

In general, we considered unidirectional high-strength fiber-reinforced laminae of graphite/carbon materials, so the composite laminate corresponds to a collection of laminae which are orientated in different directions. This sequence of different orientations is termed the lamination scheme or stacking sequence. The stacking sequence and material properties of individual lamina provide additional flexibility to the designers to tailor the stiffness and strength of the laminate. It should be noticed that the material parameters of a composite beam or plate are varying in three cases:

  • In a circular plate with bi-dimensional varying thickness, i.e., \(h=h(r)\),

  • In a plate made of a functionally graded polar orthotropic material,

  • In a plate made of anisotropic layers.

    To clarify why the physical properties of an orthotropic composite plate are invariant and moreover independent of the radial and circumferential coordinates, we will give an example of how a plate of anisotropic layers is fabricated

  • Let \(\Omega \) be a symmetric laminated composite structure composed of the superposition of 2N anisotropic layers, each one of constant thickness \(\varepsilon >0\), but this time each one with a variable shape \({\mathcal {M}}_i \subset {\mathcal {M}}\), where \({\mathcal {M}}\) is a regular sub-domain of \(R^2\). By abuse of notation, we denote \(\Omega \) as:

    $$\begin{aligned} \Omega = \{{\mathcal {M}}_{i}\},~~~i= -N,\ldots ,-1,1,\ldots .N. \end{aligned}$$

    Since we suppose \(\Omega \) symmetric, i.e., \({\mathcal {M}}_{-i} = {\mathcal {M}}_{i}\), we consider only N layers, so we rather write \(\Omega = \{{\mathcal {M}}_{i}\},~~~i=1,2,\ldots ,N\). The index i grows from the inside to the outside of the laminated composite structure, see Fig. 2.

We considered each laminate to be an orthotropic material and unidirectionally reinforced. For the purpose of this study, each layer is a non-homogeneous two-phase material, where each “hole” is filled with another “weak” material with different physical properties (weight, electric or heat conductivity, etc.). We will denote this weak material as \(A_{0}\) (in brown color). Let \(\tau _{i}\) be the characteristic function of the i-layer. According to the classical theory of plates, the composite structure \(\Omega \) is characterized by the superposition of the elastic properties of each layer. Thus, the extensional stiffness tensor A reads [23]:

$$\begin{aligned} A(x)=2 \varepsilon \sum _{i=1}^{N} \big (\tau _{i}(x) A_i + (1- \tau _{i}(x))A_0 \big ),~~~~x=(x_1, x_2) \end{aligned}$$

where \(A_i\) is the extensional stiffness of the i-layer, meanwhile the bending stiffness tensor D reads

$$\begin{aligned} D(x)=\frac{2 \varepsilon ^3}{3} \sum _{i=1}^{N} \big ((i^3 -(i-1)^3 )(\tau _{i}(x) A_i + (1- \tau _{i}(x))A_0) \big ). \end{aligned}$$

Now, to fabricate a laminated composite plate with invariant material properties, we assumed each ply of an orthotropic material and unidirectionally reinforced. Each hole will inject with the same fiber material, which leads to the characteristic function \(\tau _{i}\) for each i-layer will be invariant and will not be changing at any point of the injected ply and so for all-composite layers. Further, we use the following parameterized expression of the extensional stiffness tensor \(A_i\) for a laminate with two orientations [23]:

$$\begin{aligned} A_i= \xi _{i1} [{\bar{Q}}_{0^\circ }] + \xi _{i2} [{\bar{Q}}_{90^\circ }] = \sum _{j=1}^{2} \xi _{ij} [{\bar{Q}}_{j}] \end{aligned}$$

where \(\xi = (\xi _{ij})\in \{0,1\},~i=1,2,\ldots ,N,~j=1,2\) is a binary matrix of orientations and satisfies the following identity:

$$\begin{aligned} \xi _{ij}=\left\{ \begin{array}{l} \displaystyle 1,\quad \text { if the layer in position } i \text { has fiber orientation } j,\\ \displaystyle 0,\quad 1~~if~not. \end{array} \right. \end{aligned}$$

Besides the definition of the proportion of plies of each fiber orientation within the composite laminate [23]:

$$\begin{aligned} p_j = \frac{1}{N}\sum _{i=1}^{N} \xi _{ij}~,~~~~\mathrm {and}~~~~~~p_j = \frac{1}{N^3}\sum _{i=1}^{N} \big (i^3 -(i-1)^3 \big ) \xi _{ij}. \end{aligned}$$

By inserting the parameterized expression of \(A_i\) into Eq. (2), with \(A_0 = A_i\), we get

$$\begin{aligned} A= & {} 2 \varepsilon \sum _{i=1}^{N} \big (A_i) = 2 \varepsilon N \sum _{i=1}^{N} \left( \frac{1}{N} \sum _{j=1}^{2} \xi _{ij} [{\bar{Q}}_{j}]\right) = 2 \varepsilon N \sum _{j=1}^{2} \big (p_j [{\bar{Q}}_{j}])\\= & {} h \big ( p_1 [{\bar{Q}}_{0^\circ }] + p_2 [{\bar{Q}}_{90^\circ }] \big ),~~~~~~h=2\varepsilon N~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$

where h is the total thickness of the composite plate. In our model, the proportion of plies \(p_j\) is constant, since we have only two plies. Then we can put \(p_j=1/2,~~j=1,2\). Thus,

$$\begin{aligned} A= \frac{1}{2} h \big ([{\bar{Q}}_{0^\circ }]+ [{\bar{Q}}_{90^\circ }] \big )= h {\bar{A}}. \end{aligned}$$

Similarly, the bending stiffness tensor D can be represented as

$$\begin{aligned} D=\frac{2 \varepsilon ^3}{3} \sum _{i=1}^{N} \big ((i^3 -(i-1)^3 )( A_i) \big )= \frac{ h^3}{12} \left[ \frac{1}{2}\big ([{\bar{Q}}_{0^\circ }]+ [{\bar{Q}}_{90^\circ }] \big )\right] =\frac{ h^3}{12} {\bar{A}}. \end{aligned}$$

The above results prove that the physical properties of an orthotropic composite plate are invariant and independent of the space coordinates.

Appendix A3

The constants \(a_{ij}, b_{i,j}\), and \(d_{ij},~~i,j=1,2,6,\) are

$$\begin{aligned}&\left( \begin{array}{cc} a_{11} &{}\quad a_{12} \\ a_{21} &{}\quad a_{22} \\ \end{array} \right) =\left( \begin{array}{cc} A_{22}/\Delta &{}\quad -A_{12}/\Delta \\ -A_{12}/\Delta &{}\quad A_{11}/\Delta \end{array} \right) ~,\\&\left( \begin{array}{cc} b_{11} &{}\quad b_{12} \\ b_{21} &{}\quad b_{22} \\ \end{array} \right) =\left( \begin{array}{cc} (A_{22}B_{11}-A_{12}B_{12})/\Delta &{}\quad (A_{22}B_{12}-A_{12}B_{22})/\Delta \\ (A_{11}B_{12}-A_{12}B_{11})/\Delta &{}\quad (A_{11}B_{22}-A_{12}B_{12})/\Delta \\ \end{array} \right) ,~ \end{aligned}$$

\([d_{ij}]=-[B_{ij}][b_{ij}]+[D_{ij}]\),     \(i,j=1,2\) where \(a_{66}=1/A_{66}\),

\(b_{66}=B_{66}/A_{66}\),

\(d_{66}=(A_{66}D_{66}-B_{66}^{2})/A_{66}\),

\(\Delta = A_{11}A_{22}-A_{12}^{2}\).

Appendix A4

The first-order approximation solutions of Eq. (55) take the following forms:

$$\begin{aligned} \displaystyle q_{12}= & {} -\frac{A_1\Gamma _1 f \omega _1 e^{i T_0 \omega _1+i T_0 \Omega }}{\Omega (2 \omega _1+\omega _2)(2 \omega _1+\Omega )}-\displaystyle \frac{A_1 \Gamma _1 f \omega _2 e^{i T_0 \omega _1+i T_0 \Omega }}{2\Omega (2 \omega _1+\omega _2)(2 \omega _1+\Omega )}-\frac{4 A_1^2 \Gamma _{11} e^{2 i T_0 \omega _1}}{3 (2 \omega _1+\omega _2)(2 \omega _1+\Omega )}\\&\displaystyle -\,\frac{2 A_1^2 \Omega \Gamma _{11} e^{2 i T_0 \omega _1}}{3\omega _1(2 \omega _1+\omega _2)(2 \omega _1+\Omega )}-\displaystyle \frac{A_1^2 \Omega \omega _2 \Gamma _{11} e^{2 i T_0 \omega _1}}{3\omega _1^2(2 \omega _1+\omega _2)(2 \omega _1+\Omega )}-\frac{2 A_1^2 \omega _2 \Gamma _{11} e^{2 i T_0 \omega _1}}{3\omega _1(2 \omega _1+\omega _2)(2 \omega _1+\Omega )}\\&\displaystyle -\frac{A_1 A_2 \Omega \Gamma _{12} e^{i T_0 \omega _1+i T_0 \omega _2} }{\omega _2(2 \omega _1+\omega _2) (2 \omega _1+\Omega )}- \frac{2 A_1 A_2 \omega _1 \Gamma _{12} e^{i T_0 \omega _1+i T_0 \omega _2}}{\omega _2(2 \omega _1+\omega _2)(2 \omega _1+\Omega )} - \frac{\bar{A_1} \Gamma _1 f \omega _1 e^{-i T_0 \omega _1-i T_0 \Omega }}{\Omega (2\omega _1+\omega _2) (2\omega _1+\Omega )}\\&\displaystyle -\frac{\bar{A_1} \Gamma _1 f \omega _2 e^{-i T_0 \omega _1-i T_0 \Omega }}{2\Omega (2 \omega _1+\omega _2) (2\omega _1+\Omega )}+ \frac{8 A_1 \bar{A_1} \Gamma _{11}}{(2 \omega _1+\omega _2) (2 \omega _1+\Omega )}+ \frac{4 \Omega A_1 \bar{A_1} \Gamma _{11}}{\omega _1(2 \omega _1+\omega _2) (2 \omega _1+\Omega )}\\&\displaystyle +\frac{2 \Omega A_1 \bar{A_1} \omega _2 \Gamma _{11}}{\omega _1^2(2 \omega _1+\omega _2) (2 \omega _1+\Omega )}+ \frac{4 A_1 \bar{A_1} \omega _2 \Gamma _{11}}{\omega _1(2 \omega _1+\omega _2) (2 \omega _1+\Omega )} - \frac{4 \bar{A_1}^2 \Gamma _{11} e^{-2i T_0 \omega _1}}{3 (2\omega _1+\omega _2) (2 \omega _1+\Omega )}\\&\displaystyle -\frac{2 \bar{A_1}^2 \Omega \Gamma _{11} e^{-2i T_0 \omega _1}}{3 \omega _1 (2 \omega _1+\omega _2) (2 \omega _1+\Omega )}- \frac{ \bar{A_1}^2 \omega _2 \Omega \Gamma _{11} e^{-2i T_0 \omega _1}}{3 \omega _1^2 (2 \omega _1+\omega _2) (2 \omega _1+\Omega )} - \frac{2 \bar{A_1}^2 \omega _2 \Gamma _{11} e^{-2i T_0 \omega _1}}{3 \omega _1 (2 \omega _1+\omega _2) (2 \omega _1+\Omega )}\\&\displaystyle -\frac{\bar{A_1} \bar{A_2} \Omega \Gamma _{12} e^{-i T_0 \omega _1-i T_0 \omega _2} }{\omega _2 (2 \omega _1+\omega _2) (2 \omega _1+\Omega )} - \frac{2 \bar{A_1} \bar{A_2} \omega _1 \Gamma _{12} e^{-i T_0 \omega _1-i T_0 \omega _2}}{\omega _2 (2 \omega _1+\omega _2) (2 \omega _1+\Omega )}~ \end{aligned}$$

Similarly, the reader can deduce the factor \( q_{22}\).

Appendix A5

$$\begin{aligned} \displaystyle \xi _1= & {} \displaystyle \frac{-1}{8 \omega _1^2}\epsilon \Big (a_{20} (\Gamma _{12}(\mu \epsilon \cos (\Phi _{10})- 4 \omega _1 \sin (\Phi _{10}))+ \displaystyle 8 \alpha _{12} \omega _1^2 \epsilon \cos (\Phi _{10}))+ 4 \mu \omega _1^2)\Big )~,~\\ \displaystyle \xi _2= & {} \displaystyle \frac{-1}{8 \omega _1^2} a_{10} \epsilon \Big (\Gamma _{12} (\mu \epsilon \cos (\Phi _{10})- 4 \omega _1 \sin (\Phi _{10}))+ 8 \alpha _{12} \omega _1^2 \epsilon \cos (\Phi _{10})\Big )~,\\ \displaystyle \xi _3= & {} \displaystyle \frac{1}{8 \omega _1^2} a_{10} a_{20} \epsilon \Big (\Gamma _{12} (4 \omega _1 \cos (\Phi _{10})+ \mu \epsilon \sin (\Phi _{10}))+ 8 \alpha _{12} \omega _1^2 \epsilon \sin (\Phi _{10})\Big ),\\ \xi _4= & {} -\frac{\mu \epsilon }{2}~,\\ \xi _5= & {} a_{10} a_{20} \alpha _{24} \epsilon ^2~,\\ \displaystyle \xi _6= & {} \displaystyle \frac{1}{12 \omega _1^3}\Big (4 \omega _1^3 (3 (\epsilon ^2 (6 a_{20} (a_{20} \alpha _{15}+ \alpha _{12} \sin (\Phi _{10})+ 18\alpha _{14} a_{10}^2- 2 \alpha _{22} a_{10}+ 6 \alpha _{11}- \displaystyle \alpha _{24})+ \sigma _1)\\&+ 4 \sigma _2)+ 9 a_{20} \omega _1 \epsilon \Gamma _{12}(4 \omega _1 \cos (\Phi _{10})+ \mu \epsilon \sin (\Phi _{10}))- 9 a_{20}^2 \epsilon ^2 \Gamma _{12}^2)\Big ),\\ \displaystyle \xi _7= & {} \displaystyle \frac{1}{4 \omega _1^3}\epsilon \Big (3 a_{10} (-2 a_{20} \epsilon \Gamma _{2,1}^2+\omega _1 \Gamma _{2,1}(4 \omega _1 \cos (\Phi _{10})+\mu _1 \epsilon \sin (\Phi _{10}))\\&+ 8 \omega _1^3 \epsilon ( \displaystyle 2 a_{20} \alpha _{15}+\alpha _{12} \sin (\Phi _{10})+\alpha _{13} \cos (\Phi _{10})))-8 a_{20} \alpha _{23} \omega _1^3 \epsilon \Big ),\\ \displaystyle \xi _8= & {} \displaystyle \frac{1}{4 \omega _1^2} 3 a_{10} a_{20} \epsilon \Big (\epsilon \cos (\Phi _{10})(\mu \Gamma _{12}+ 8 \alpha _{12} \omega _1^2)- 4 \omega _1 \sin (\Phi _{10})\Gamma _{12}\Big )~, \\ \displaystyle \xi _9= & {} \displaystyle \frac{1}{8 \omega _1^3} \Big (-3 a_{20}^2 \epsilon ^2 \Gamma _{12}^2+ 3 a_{20} \omega _1 \epsilon \Gamma _{12} (4 \omega _1 \cos (\Phi _{10})+\mu \epsilon \sin (\Phi _{10}))+ 8 \omega _1^3 (3 \epsilon ^2 ( \\&\displaystyle a_{20}(a_{20} \alpha _{15}+\alpha _{12} \sin (\Phi _{10})+ 3 \alpha _{14} a_{10}^2+\alpha _{11})+\sigma _2))\Big )~,\\ \displaystyle \xi _{10}= & {} \displaystyle \frac{1}{8 \omega _1^3} 3 a_{10} \epsilon \Big (-2 a_{20} \epsilon \Gamma _{12}^2+ \omega _1 \Gamma _{12}(4 \omega _1 \cos (\Phi _{10})+\mu \epsilon \sin (\Phi _{10}))+ 8 \omega _1^3 \epsilon (\displaystyle 2 a_{20} \alpha _{15}+\alpha _{12} \sin (\Phi _{10}))\Big )~,\\ \displaystyle \xi _{11}= & {} \displaystyle \frac{1}{8 \omega _1^2} 3 a_{10} a_{20} \epsilon \Big (\epsilon \cos (\Phi _{10})(\mu \Gamma _{12}+ 8 \alpha _{12} \omega _1^2)- 4 \omega _1 \sin (\Phi _{10})(\Gamma _{12})\Big ). \end{aligned}$$

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Elmorabie, K.M., Zakaria, K. & Sirwah, M.A. Nonlinear instability of a thin laminated composite circular plate subjected to a tensile periodic load. Acta Mech 231, 5213–5238 (2020). https://doi.org/10.1007/s00707-020-02783-8

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