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Nonlinear thermal postbuckling of functionally graded graphene-reinforced composite laminated plates with circular or elliptical delamination

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Abstract

This research investigates the nonlinear thermal stability responses of functionally graded graphene-reinforced composite (FG-GRC) laminated plates with embedded circular and elliptical delamination as well as edge delamination, subjected to a uniform temperature rise and a variety of mechanical boundary conditions. The thermomechanical properties of the GRCs are estimated using the extended Halpin–Tsai micromechanical model that incorporates efficiency parameters to take into account nanoscale size and surface effects of the graphene reinforcement. The von Karman geometrical nonlinearity is adopted in a solution based on the third-order shear deformation theory. The nonlinear equilibrium equations derived by the minimum total potential energy principle are solved using the Ritz method in conjunction with the Newton–Raphson iterative procedure. Parametric studies reveal that the types of graphene distribution pattern and geometry of delamination zones have a substantial effect on the thermal equilibrium paths and buckling temperature of the GRC delaminated plates. FG-X graphene sheet pattern raises the critical buckling temperature and compressive strength of the baselaminate and reduces the nonlinear thermal postbuckling deflection; however, it causes a significant increase in normal stress distribution at the top and the bottom surfaces of the delaminated plates.

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Funding

The authors gratefully acknowledge the financial support to the present work provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) (Grant No. RGPIN-2022-03462) and Alberta Innovates (Grant No. 222301535 C).

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, University of Alberta, Edmonton, AB, Canada

    S. F. Nikrad & Z. T. Chen

  2. Bioresource Engineering Department, McGill University, Montreal, QC, Canada

    A. H. Akbarzadeh

  3. Mechanical Engineering Department, McGill University, Montreal, QC, Canada

    A. H. Akbarzadeh

Authors
  1. S. F. Nikrad
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  2. Z. T. Chen
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  3. A. H. Akbarzadeh
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Correspondence to Z. T. Chen or A. H. Akbarzadeh.

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Appendices

Appendix A

According to the TSDT, the normal and transverse shear strain tensors of the FG-GRC laminated plate can be written as [61]:

$$\begin{aligned}\left\{\begin{array}{c}{\varepsilon }_{xx}\\ {\varepsilon }_{yy}\\ {\gamma }_{xy}\end{array}\right\}&=\left\{\begin{array}{c}{\varepsilon }_{xx}^{\left(0\right)}\\ {\varepsilon }_{yy}^{\left(0\right)}\\ {\gamma }_{xy}^{\left(0\right)}\end{array}\right\}+z\left\{\begin{array}{c}{\varepsilon }_{xx}^{\left(1\right)}\\ {\varepsilon }_{yy}^{\left(1\right)}\\ {\gamma }_{xy}^{\left(1\right)}\end{array}\right\}+{z}^{3}\left\{\begin{array}{c}{\varepsilon }_{xx}^{\left(3\right)}\\ {\varepsilon }_{yy}^{\left(3\right)}\\ {\gamma }_{xy}^{\left(3\right)}\end{array}\right\}\\ \left\{\begin{array}{c}{\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right\}&=\left\{\begin{array}{c}{\gamma }_{yz}^{\left(0\right)}\\ {\gamma }_{xz}^{\left(0\right)}\end{array}\right\}+{z}^{2}\left\{\begin{array}{c}{\gamma }_{yz}^{\left(2\right)}\\ {\gamma }_{xz}^{\left(2\right)}\end{array}\right\}\end{aligned}$$
(A.1)

where

$$\begin{aligned}\left\{\begin{array}{c}{\varepsilon }_{xx}^{\left(0\right)}\\ {\varepsilon }_{yy}^{\left(0\right)}\\ {\gamma }_{xy}^{\left(0\right)}\end{array}\right\}&=\left\{\begin{array}{c}\frac{\partial {u}_{0}}{\partial x}+\left(\frac{1}{2}\right){\left(\frac{\partial {w}_{0}}{\partial x}\right)}^{2}\\ \frac{\partial {v}_{0}}{\partial y}+\left(\frac{1}{2}\right){\left(\frac{\partial {w}_{0}}{\partial y}\right)}^{2}\\ \frac{\partial {u}_{0}}{\partial y}+\frac{\partial {v}_{0}}{\partial x}+\frac{\partial {w}_{0}}{\partial x}\frac{\partial {w}_{0}}{\partial y}\end{array}\right\}\\ \left\{\begin{array}{c}{\varepsilon }_{xx}^{\left(1\right)}\\ {\varepsilon }_{yy}^{\left(1\right)}\\ {\gamma }_{xy}^{\left(1\right)}\end{array}\right\}&=\left\{\begin{array}{c}\frac{\partial {\varphi }_{x}}{\partial x}\\ \frac{\partial {\varphi }_{y}}{\partial y}\\ \frac{\partial {\varphi }_{x}}{\partial y}+\frac{\partial {\varphi }_{y}}{\partial x}\end{array}\right\}\\ \left\{\begin{array}{c}{\varepsilon }_{xx}^{\left(3\right)}\\ {\varepsilon }_{yy}^{\left(3\right)}\\ {\gamma }_{xy}^{\left(3\right)}\end{array}\right\}&=\left\{\begin{array}{c}\left(\frac{-4}{3{h}^{2}}\right)\left(\frac{\partial {\varphi }_{x}}{\partial x}+\frac{{\partial }^{2}{w}_{0}}{\partial {x}^{2}}\right)\\ \left(\frac{-4}{3{h}^{2}}\right)\left(\frac{\partial {\varphi }_{y}}{\partial y}+\frac{{\partial }^{2}{w}_{0}}{\partial {y}^{2}}\right)\\ \left(\frac{-4}{3{h}^{2}}\right)\left(\frac{\partial {\varphi }_{x}}{\partial y}+\frac{\partial {\varphi }_{y}}{\partial x}+2\frac{{\partial }^{2}{w}_{0}}{\partial x\partial y}\right)\end{array}\right\}\\ \left\{\begin{array}{c}{\gamma }_{yz}^{\left(0\right)}\\ {\gamma }_{xz}^{\left(0\right)}\end{array}\right\}&=\left\{\begin{array}{c}{\varphi }_{y}+\frac{\partial {w}_{0}}{\partial y}\\ {\varphi }_{x}+\frac{\partial {w}_{0}}{\partial x}\end{array}\right\} , \left\{\begin{array}{c}{\gamma }_{yz}^{\left(2\right)}\\ {\gamma }_{xz}^{\left(2\right)}\end{array}\right\}=\left(\frac{-4}{{h}^{2}}\right)\left\{\begin{array}{c}{\gamma }_{yz}^{\left(0\right)}\\ {\gamma }_{xz}^{\left(0\right)}\end{array}\right\}\end{aligned}$$
(A.2)

Appendix B

Displacement and rotational functions of FG-GRC plate with elliptical embedded delamination (SSSS).

2.1 Out-of-plane displacement

$$\begin{aligned}{W}^{\left(1\right)}&=\sum_{m=0}^{{M}_{1}}\sum_{n=0}^{{N}_{1}}\left(x-\frac{{L}_{1}}{2}\right)\left(x+\frac{{L}_{1}}{2}\right)\left(y-\frac{{b}_{1}}{2}\right)\left(y+\frac{{b}_{1}}{2}\right){W}_{mn}^{\left(1\right)}{x}^{m}{y}^{n}\\ {W}^{\left(2\right)}&={W}^{\left(1\right)}+\sum_{m=0}^{{M}_{2}}\sum_{n=0}^{{N}_{2}}{\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right)}^{2}{W}_{mn}^{\left(2\right)}{x}^{m}{y}^{n}\\ {W}^{\left(3\right)}&={W}^{\left(1\right)}+\sum_{m=0}^{{M}_{3}}\sum_{n=0}^{{N}_{3}}{\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right)}^{2}{W}_{mn}^{\left(3\right)}{x}^{m}{y}^{n}\end{aligned}$$
(B.1)

2.2 In-plane displacements

$$\begin{aligned}{U}^{\left(1\right)}&=\sum_{p=0}^{{P}_{1}}\sum_{q=0}^{{Q}_{1}}\left(x-\frac{{L}_{1}}{2}\right)\left(x+\frac{{L}_{1}}{2}\right){U}_{pq}^{\left(1\right)}{x}^{p}{y}^{q}\\ {U}^{\left(2\right)}&={U}^{\left(1\right)}-\frac{4{h}_{2}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+{h}_{2}{\varphi }_{x}^{\left(1\right)}+\sum_{p=0}^{{P}_{2}}\sum_{q=0}^{{Q}_{2}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){U}_{pq}^{\left(2\right)}{x}^{p}{y}^{q}\\ {U}^{\left(3\right)}&={U}^{\left(1\right)}-\frac{4{h}_{3}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+{h}_{3}{\varphi }_{x}^{\left(1\right)}+\sum_{p=0}^{{P}_{3}}\sum_{q=0}^{{Q}_{3}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){U}_{pq}^{\left(3\right)}{x}^{p}{y}^{q}\\ {V}^{\left(1\right)}&=\sum_{s=0}^{{S}_{1}}\sum_{r=0}^{{R}_{1}}\left(y-\frac{{b}_{1}}{2}\right){V}_{sr}^{\left(1\right)}{x}^{s}{y}^{r}\\ {V}^{\left(2\right)}&={V}^{\left(1\right)}-\frac{4{h}_{2}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+{h}_{2}{\varphi }_{y}^{\left(1\right)}+\sum_{s=0}^{{S}_{2}}\sum_{r=0}^{{R}_{2}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){V}_{sr}^{\left(2\right)}{x}^{s}{y}^{r}\\ {V}^{\left(3\right)}&={V}^{\left(1\right)}-\frac{4{h}_{3}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+{h}_{3}{\varphi }_{y}^{\left(1\right)}+\sum_{s=0}^{{S}_{3}}\sum_{r=0}^{{R}_{3}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){V}_{sr}^{\left(3\right)}{x}^{s}{y}^{r}\end{aligned}$$
(B.2)

2.3 Rotational functions

$$\begin{aligned}{\varphi }_{x}^{\left(1\right)}&=\sum_{j=0}^{{J}_{1}}\sum_{k=0}^{{K}_{1}}{\varphi }_{{x}_{jk}}^{\left(1\right)}{x}^{j}{y}^{k}\\ {\varphi }_{x}^{\left(2\right)}&={\varphi }_{x}^{\left(1\right)}-\frac{4{h}_{2}^{2}}{{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+\sum_{j=0}^{{J}_{2}}\sum_{k=0}^{{K}_{2}}{\varphi }_{{x}_{jk}}^{\left(2\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{j}{y}^{k}\\ {\varphi }_{x}^{\left(3\right)}&={\varphi }_{x}^{\left(1\right)}-\frac{4{h}_{3}^{2}}{{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+\sum_{j=0}^{{J}_{3}}\sum_{k=0}^{{K}_{3}}{\varphi }_{{x}_{jk}}^{\left(3\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{j}{y}^{k}\\ {\varphi }_{y}^{\left(1\right)}&=\sum_{i=0}^{{I}_{1}}\sum_{t=0}^{{T}_{1}}{\varphi }_{{y}_{it}}^{\left(1\right)}{x}^{i}{y}^{t}\\ {\varphi }_{y}^{\left(2\right)}&={\varphi }_{y}^{\left(1\right)}-\frac{4{h}_{2}^{2}}{{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+\sum_{i=0}^{{I}_{2}}\sum_{t=0}^{{T}_{2}}{\varphi }_{{y}_{it}}^{\left(2\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{i}{y}^{t}\\ {\varphi }_{y}^{\left(3\right)}&={\varphi }_{y}^{\left(1\right)}-\frac{4{h}_{3}^{2}}{{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+\sum_{i=0}^{{I}_{3}}\sum_{t=0}^{{T}_{3}}{\varphi }_{{y}_{it}}^{\left(3\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{i}{y}^{t}\end{aligned}$$
(B.3)

Displacement and rotational functions of FG-GRC plate with elliptical embedded delamination (CCCC).

2.4 Out-of-plane displacement

$$\begin{aligned}{W}^{\left(1\right)}&=\sum_{m=0}^{{M}_{1}}\sum_{n=0}^{{N}_{1}}{\left(x-\frac{{L}_{1}}{2}\right)}^{2}{\left(x+\frac{{L}_{1}}{2}\right)}^{2}{\left(y-\frac{{b}_{1}}{2}\right)}^{2}{\left(y+\frac{{b}_{1}}{2}\right)}^{2}{W}_{mn}^{\left(1\right)}{x}^{m}{y}^{n}\\ {W}^{\left(2\right)}&={W}^{\left(1\right)}+\sum_{m=0}^{{M}_{2}}\sum_{n=0}^{{N}_{2}}{\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right)}^{2}{W}_{mn}^{\left(2\right)}{x}^{m}{y}^{n}\\ {W}^{\left(3\right)}&={W}^{\left(1\right)}+\sum_{m=0}^{{M}_{3}}\sum_{n=0}^{{N}_{3}}{\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right)}^{2}{W}_{mn}^{\left(3\right)}{x}^{m}{y}^{n}\end{aligned}$$
(B.4)

2.5 In-plane displacements

$$\begin{aligned}{U}^{\left(1\right)}&=\sum_{p=0}^{{P}_{1}}\sum_{q=0}^{{Q}_{1}}\left(x-\frac{{L}_{1}}{2}\right)\left(x+\frac{{L}_{1}}{2}\right){U}_{pq}^{\left(1\right)}{x}^{p}{y}^{q}\\ {U}^{\left(2\right)}&={U}^{\left(1\right)}-\frac{4{h}_{2}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+{h}_{2}{\varphi }_{x}^{\left(1\right)}+\sum_{p=0}^{{P}_{2}}\sum_{q=0}^{{Q}_{2}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){U}_{pq}^{\left(2\right)}{x}^{p}{y}^{q}\\ {U}^{\left(3\right)}&={U}^{\left(1\right)}-\frac{4{h}_{3}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+{h}_{3}{\varphi }_{x}^{\left(1\right)}+\sum_{p=0}^{{P}_{3}}\sum_{q=0}^{{Q}_{3}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){U}_{pq}^{\left(3\right)}{x}^{p}{y}^{q}\\ {V}^{\left(1\right)}&=\sum_{s=0}^{{S}_{1}}\sum_{r=0}^{{R}_{1}}\left(y-\frac{{b}_{1}}{2}\right){V}_{sr}^{\left(1\right)}{x}^{s}{y}^{r}\\ {V}^{\left(2\right)}&={V}^{\left(1\right)}-\frac{4{h}_{2}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+{h}_{2}{\varphi }_{y}^{\left(1\right)}+\sum_{s=0}^{{S}_{2}}\sum_{r=0}^{{R}_{2}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){V}_{sr}^{\left(2\right)}{x}^{s}{y}^{r}\\ {V}^{\left(3\right)}&={V}^{\left(1\right)}-\frac{4{h}_{3}^{3}}{3{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+{h}_{3}{\varphi }_{y}^{\left(1\right)}+\sum_{s=0}^{{S}_{3}}\sum_{r=0}^{{R}_{3}}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){V}_{sr}^{\left(3\right)}{x}^{s}{y}^{r}\end{aligned}$$
(B.5)

2.6 Rotational functions

$$\begin{aligned}{\varphi }_{x}^{\left(1\right)}&=\sum_{j=0}^{{J}_{1}}\sum_{k=0}^{{K}_{1}}\left(x+\frac{{L}_{1}}{2}\right)\left(x-\frac{{L}_{1}}{2}\right){\varphi }_{{x}_{jk}}^{\left(1\right)}{x}^{j}{y}^{k}\\ {\varphi }_{x}^{\left(2\right)}&={\varphi }_{x}^{\left(1\right)}-\frac{4{h}_{2}^{2}}{{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+\sum_{j=0}^{{J}_{2}}\sum_{k=0}^{{K}_{2}}{\varphi }_{{x}_{jk}}^{\left(2\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{j}{y}^{k}\\ {\varphi }_{x}^{\left(3\right)}&={\varphi }_{x}^{\left(1\right)}-\frac{4{h}_{3}^{2}}{{h}_{1}^{2}}\left({\varphi }_{x}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial x}\right)+\sum_{j=0}^{{J}_{3}}\sum_{k=0}^{{K}_{3}}{\varphi }_{{x}_{jk}}^{\left(3\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{j}{y}^{k}\\ {\varphi }_{y}^{\left(1\right)}&=\sum_{i=0}^{{I}_{1}}\sum_{t=0}^{{T}_{1}}\left(y+\frac{{b}_{1}}{2}\right)\left(y-\frac{{b}_{1}}{2}\right){\varphi }_{{y}_{it}}^{\left(1\right)}{x}^{i}{y}^{t}\\ {\varphi }_{y}^{\left(2\right)}&={\varphi }_{y}^{\left(1\right)}-\frac{4{h}_{2}^{2}}{{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+\sum_{i=0}^{{I}_{2}}\sum_{t=0}^{{T}_{2}}{\varphi }_{{y}_{it}}^{\left(2\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{i}{y}^{t}\\ {\varphi }_{y}^{\left(3\right)}&={\varphi }_{y}^{\left(1\right)}-\frac{4{h}_{3}^{2}}{{h}_{1}^{2}}\left({\varphi }_{y}^{\left(1\right)}+\frac{{\partial W}^{\left(1\right)}}{\partial y}\right)+\sum_{i=0}^{{I}_{3}}\sum_{t=0}^{{T}_{3}}{\varphi }_{{y}_{it}}^{\left(3\right)}\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-1\right){x}^{i}{y}^{t}\end{aligned}$$
(B.6)

Appendix C

$$\begin{aligned}{W}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(1\right)}&={W}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(2\right)}={W}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(3\right)},\quad{\frac{{\partial W}^{\left(1\right)}}{\partial x}}_{\left|x=\frac{{-L}_{2}}{2}\right.}= {\frac{{\partial W}^{\left(2\right)}}{\partial x}}_{\left|x=\frac{{-L}_{2}}{2}\right.}={\frac{{\partial W}^{\left(3\right)}}{\partial x}}_{\left|x=\frac{{-L}_{2}}{2}\right.}\\ {W}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(1\right)}&={W}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(2\right)}={W}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(3\right)},\quad{\frac{{\partial W}^{\left(1\right)}}{\partial x}}_{\left|x=\frac{{L}_{2}}{2}\right.}= {\frac{{\partial W}^{\left(2\right)}}{\partial x}}_{\left|x=\frac{{L}_{2}}{2}\right.}={\frac{{\partial W}^{\left(3\right)}}{\partial x}}_{\left|x=\frac{{L}_{2}}{2}\right.}\\ {W}_{\left|y={b}_{2}\right.}^{\left(1\right)}&={W}_{\left|y={b}_{2}\right.}^{\left(2\right)}={W}_{\left|y={b}_{2}\right.}^{\left(3\right)},\quad{\frac{{\partial W}^{\left(1\right)}}{\partial y}}_{\left|y={b}_{2}\right.}= {\frac{{\partial W}^{\left(2\right)}}{\partial y}}_{\left|y={b}_{2}\right.}={\frac{{\partial W}^{\left(3\right)}}{\partial y}}_{\left|y={b}_{2}\right.}\\ {U}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(2\right)}&={U}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{2}{\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{2}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {U}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(2\right)}&={U}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{2}{\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{2}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {U}_{\left|y={b}_{2}\right.}^{\left(2\right)}&={U}_{\left|y={b}_{2}\right.}^{\left(1\right)}+{h}_{2}{\varphi }_{{x}_{\left|y={b}_{2}\right.}}^{\left(1\right)}-{h}_{2}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|y={b}_{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|y={b}_{2}\right.}}^{\left(1\right)}\right)\\ {U}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(3\right)}&={U}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{3}{\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{3}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {U}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(3\right)}&={U}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{3}{\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{3}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {U}_{\left|y={b}_{2}\right.}^{\left(3\right)}&={U}_{\left|y={b}_{2}\right.}^{\left(1\right)}+{h}_{3}{\varphi }_{{x}_{\left|y={b}_{2}\right.}}^{\left(1\right)}-{h}_{3}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|y={b}_{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|y={b}_{2}\right.}}^{\left(1\right)}\right)\\ {V}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(2\right)}&={V}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{2}{\varphi }_{{y}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{2}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {V}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(2\right)}&={V}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{2}{\varphi }_{{y}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{2}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {V}_{\left|y={b}_{2}\right.}^{\left(2\right)}&={V}_{\left|y={b}_{2}\right.}^{\left(1\right)}+{h}_{2}{\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}-{h}_{2}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}\right)\\ {V}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(3\right)}&={V}_{\left|x=\frac{{-L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{3}{\varphi }_{{y}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{3}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {V}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(3\right)}&={V}_{\left|x=\frac{{L}_{2}}{2}\right.}^{\left(1\right)}+{h}_{3}{\varphi }_{{y}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}-{h}_{3}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {V}_{\left|y={b}_{2}\right.}^{\left(3\right)}&={V}_{\left|y={b}_{2}\right.}^{\left(1\right)}+{h}_{3}{\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}-{h}_{3}^{3}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}\right)\\ {\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(2\right)}&={\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}-3{h}_{2}^{2}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(2\right)}&={\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}-3{h}_{2}^{2}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(2\right)}&={\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}-3{h}_{2}^{2}{C}_{1}^{\left(1\right)}\left({\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}+{W}_{{,y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}\right)\\ {\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(3\right)}&={\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}-3{h}_{3}^{2}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{-L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(3\right)}&={\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}-3{h}_{3}^{2}{C}_{1}^{\left(1\right)}\left({\varphi }_{{x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}+{W}_{{,x}_{\left|x=\frac{{L}_{2}}{2}\right.}}^{\left(1\right)}\right)\\ {\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(3\right)}&={\varphi }_{{y}_{\left|y={b}_{2}\right.}}^{\left(1\right)}-3{h}_{3}^{2}{C}_{1}^{\left(1\right)}\end{aligned}$$
(C.1)

Appendix D

See Figs. 26, 27 and 28.

Fig. 26
figure 26

Influence of clamped and simply-supported boundary conditions on the thermal equilibrium paths of FG-A laminated plates with embedded elliptical delamination, AD/A = 0.25, \(\frac{{h}_{s}}{h}=0.2\)

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Fig. 27
figure 27

Comparison of deflection through the sublaminate and baselaminate of FG-X plate with embedded elliptical delamination at three different temperatures AD/A = 0.25, SSSS. a Sublaminate. b Baselaminate

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Fig. 28
figure 28

Comparison of normal Stress (\({\sigma }_{11}\)) through the sublaminate and baselaminate of FG-X plate with embedded elliptical delamination at three different temperatures AD/A = 0.25, SSSS. a Sublaminate. b Baselaminate

Full size image

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Nikrad, S.F., Chen, Z.T. & Akbarzadeh, A.H. Nonlinear thermal postbuckling of functionally graded graphene-reinforced composite laminated plates with circular or elliptical delamination. Acta Mech 234, 5999–6039 (2023). https://doi.org/10.1007/s00707-023-03694-0

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