Combined analytical and numerical approach for auxetic FG-CNTRC plate subjected to a sudden load

Abstract

In the current work, the dynamic behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate with negative Poisson’s ratio (NPR) is investigated by combining higher-order shear deformation theory and large deflection theory. First, explicit solutions are proposed to predict the effective Poisson’s ratio (EPR) of the laminates. Taking carbon nanotube-reinforced composite (CNTRC) as an example, the maximum NPR is obtained for $\left( { \pm \theta } \right)_{{3{\text{T}}}}$ laminate as well. Results show that the EPR ($v_{13}^{\text{e}}$,$v_{23}^{\text{e}}$) can range from a positive value of 0.311 to a negative value of 0.63. For the dynamic response problem, the asymptotic solutions with a two-step perturbation approach are derived for FG-CNTRC plates to capture the relationship between the center deflection and time. Several key factors such as functionally graded distribution, variations in the elastic foundation, and thermal stress produced by changing the temperature field are considered in the subsequent analysis. Numerical simulations are carried out to examine the corresponding dynamic behavior of FG-CNTRC plates when these factors are taken into account.

Appendices

Appendix 1

In Eqs. (9)–(12), the thermal resultants $\bar{N}_{i}^{T}$, $\bar{M}_{i}^{T}$ and $\bar{P}_{i}^{T}$ are given by

$$\left[ {\begin{array}{*{20}c} {\bar{N}_{x}^{T} } \\ {\bar{N}_{y}^{T} } \\ {\bar{N}_{xy}^{T} } \\ \end{array} \begin{array}{*{20}c} {\bar{M}_{x}^{T} } \\ {\bar{M}_{y}^{T} } \\ {\bar{M}_{xy}^{T} } \\ \end{array} \begin{array}{*{20}c} {\bar{P}_{x}^{T} } \\ {\bar{P}_{y}^{T} } \\ {\bar{P}_{xy}^{T} } \\ \end{array} } \right] = \sum\limits_{k = 1}^{N} {\int\limits_{{h_{k - 1} }}^{{h_{k} }} {\left[ {\begin{array}{*{20}c} {A_{x} } \\ {A_{y} } \\ {A_{xy} } \\ \end{array} } \right]} }_{k} (1,Z,Z^{3} )\Delta T\,{\text{d}}Z,$$

(37a)

and $\bar{S}_{i}^{T} ,\left( {i = x,y,xy} \right)$ are given as

$$\left[ {\begin{array}{*{20}c} {\bar{S}_{x}^{T} } \\ {\bar{S}_{y}^{T} } \\ {\bar{S}_{xy}^{T} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\bar{M}_{x}^{T} } \\ {\bar{M}_{y}^{T} } \\ {\bar{M}_{xy}^{T} } \\ \end{array} } \right] - \frac{4}{{3h^{2} }}\left[ {\begin{array}{*{20}c} {\bar{P}_{x}^{T} } \\ {\bar{P}_{y}^{T} } \\ {\bar{P}_{xy}^{T} } \\ \end{array} } \right],$$

(37b)

in which $\Delta T$ is temperature increment from an initial state (T₀), ΔT = T−T₀, and

$$\left[ \begin{aligned} A_{x} \hfill \\ A_{y} \hfill \\ A_{xy} \hfill \\ \end{aligned} \right] = - \left[ {\begin{array}{*{20}c} {\bar{Q}_{11} } & {\bar{Q}_{12} } & {\bar{Q}_{16} } \\ {\bar{Q}_{12} } & {\bar{Q}_{22} } & {\bar{Q}_{26} } \\ {\bar{Q}_{16} } & {\bar{Q}_{26} } & {\bar{Q}_{66} } \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {c^{2} } & {s^{2} } \\ {s^{2} } & {c^{2} } \\ {2cs} & { - 2cs} \\ \end{array} } \right]\,\left[ \begin{aligned} \alpha_{11} \hfill \\ \alpha_{22} \hfill \\ \end{aligned} \right],$$

(38)

where $\bar{Q}_{ij}$ are the component of the transformed lamina stiffness matrix which are evaluated as follows:

$$\left[ \begin{aligned} \bar{Q}_{11} \hfill \\ \bar{Q}_{12} \hfill \\ \bar{Q}_{22} \hfill \\ \bar{Q}_{16} \hfill \\ \bar{Q}_{26} \hfill \\ \bar{Q}_{66} \hfill \\ \end{aligned} \right] = \left[ {\begin{array}{*{20}c} {c^{4} } & {2c^{2} s^{2} } & {s^{4} } & {4c^{2} s^{2} } \\ {c^{2} s^{2} } & {c^{4} + s^{4} } & {c^{2} s^{2} } & { - 4c^{2} s^{2} } \\ {s^{4} } & {2c^{2} s^{2} } & {c^{4} } & {4c^{2} s^{2} } \\ {c^{3} s} & {cs^{3} - c^{3} s} & { - cs^{3} } & { - 2cs(c^{2} - s^{2} )} \\ {cs^{3} } & {c^{3} s - cs^{3} } & { - c^{3} s} & {2cs(c^{2} - s^{2} )} \\ {c^{2} s^{2} } & { - 2c^{2} s^{2} } & {c^{2} s^{2} } & {(c^{2} - s^{2} )^{2} } \\ \end{array} } \right]{\kern 1pt} {\kern 1pt} \left[ {\begin{array}{*{20}c} {Q_{11} } \\ {Q_{12} } \\ {Q_{22} } \\ {Q_{66} } \\ \end{array} } \right],$$

(39a)

$$\left[ {\begin{array}{*{20}c} {\bar{Q}_{44} } \\ {\bar{Q}_{45} } \\ {\bar{Q}_{55} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {c^{2} } \\ { - cs} \\ {s^{2} } \\ \end{array} } & {\begin{array}{*{20}c} {s^{2} } \\ {cs} \\ {c^{2} } \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {Q_{44} } \\ {Q_{55} } \\ \end{array} } \right],$$

(39b)

where

$$\begin{aligned} Q_{11} = E_{11} (1 - \nu_{12} \nu_{21} )^{ - 1} ,Q_{22} = E_{22} (1 - \nu_{12} \nu_{21} )^{ - 1} ,Q_{12} = \nu_{21} E_{11} (1 - \nu_{12} \nu_{21} )^{ - 1} \hfill \\ Q_{16} = Q_{26} = 0,Q_{66} = G_{12} ,Q_{44} = G_{23} ,Q_{55} = G_{13} . \hfill \\ \end{aligned}$$

(40)

The coefficient $I_{i}$ can be calculated as:

$$(I_{1} ,I_{2} ,I_{3} ,I_{4} ,I_{5} ,I_{7} ) = \sum\limits_{k = 1}^{N} {} \int_{{h_{k - 1} }}^{{h_{k} }} {\rho_{k} \left( {1,Z,Z^{2} ,Z^{3} ,Z^{4} ,Z^{6} } \right)} \text{d}Z,$$

(41a)

and

$$\begin{aligned} \bar{I}_{2} & = I_{2} - \frac{{4I_{4} }}{{3h^{2} }},\bar{I}_{5} = I_{5} - \frac{{4I_{7} }}{{3h^{2} }},\bar{I}_{3} = I_{3} - \frac{{8I_{5} }}{{3h^{2} }} + \frac{{16I_{7} }}{{9h^{4} }}, \\ I_{8} & = \frac{{I_{2} \bar{I}_{2} }}{{I_{1} }} - \bar{I}_{3} - \frac{4}{{3h^{2} }}\bar{I}_{5} ,I_{9} = \frac{4}{{3h^{2} }}\left( {\bar{I}_{5} - \frac{{\bar{I}_{2} I_{4} }}{{I_{1} }}} \right),I_{{10}} = \frac{{\bar{I}_{2} \bar{I}_{2} }}{{I_{1} }} - \bar{I}_{3} . \\ \end{aligned}$$

(41b)

Appendix 2

The matrices in the Eq. (17) are derived in Shen [49]

$$\left[ {\begin{array}{*{20}c} {A_{ij}^{ * } } & {B_{ij}^{ * } } & {D_{ij}^{ * } } \\ {E_{ij}^{ * } } & {F_{ij}^{ * } } & {H_{ij}^{ * } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_{ij}^{ - 1} } & { - A_{ij}^{ - 1} B_{ij} } & {D_{ij} - B_{ij} A_{ij}^{ - 1} B_{ij} } \\ { - A_{ij}^{ - 1} E_{ij} } & {F_{ij} - E_{ij} A_{ij}^{ - 1} B_{ij} } & {H_{ij} - E_{ij} A_{ij}^{ - 1} E_{ij} } \\ \end{array} } \right],\left( {i,j = 1,2,6} \right),$$

(42)

in which A_ij, B_ij, D_ij, etc. refer to the plate stiffnesses, which are functions of ($\overline{Q}_{ij}$)_k

$$\left( {A_{ij} ,B_{ij} ,D_{ij} ,E_{ij} ,F_{ij} ,H_{ij} } \right) = \sum\limits_{k = 1}^{N} {\int\limits_{{h_{k - 1} }}^{{h_{k} }} {(\bar{Q}_{ij} } } )_{k} \left( {1,Z,Z^{2} ,Z^{3} ,Z^{4} ,Z^{6} } \right){\text{d}}Z\;\left( {i,j = { 1}, 2, 6} \right),$$

(43a)

$$\left( {A_{ij} ,D_{ij} ,F_{ij} } \right) = \sum\limits_{k = 1}^{N} {\int\limits_{{h_{k - 1} }}^{{h_{k} }} {(\bar{Q}_{ij} } } )_{k} (1,Z^{2} ,Z^{4} ){\text{d}}Z\;\left( {i,j = { 4}, 5} \right)\left( {i,j = { 4}, 5} \right),$$

(43b)

In the general case, the matrices ${\text{A}}_{ij}^{*}$,${\text{D}}_{ij}^{*}$ and ${\text{H}}_{ij}^{*}$ are symmetric although the matrices ${\text{B}}_{ij}^{*}$,${\text{E}}_{ij}^{*}$ and ${\text{F}}_{ij}^{*}$ might be not.

In the case of (± θ)_3T laminated plate:

$$\begin{gathered} \left\{ {\begin{array}{*{20}c} {A_{{45}} = D_{{45}} = F_{{45}} = 0} \\ {F_{{61}}^{*} = F_{{62}}^{*} = B_{{66}}^{*} = E_{{66}}^{*} = 0} \\ \end{array} } \right. \hfill \\ \left\{ {\begin{array}{*{20}c} {A_{{ij}}^{*} = D_{{ij}}^{*} = F_{{ij}}^{*} = H_{{ij}}^{*} = 0{\mkern 1mu} \left( {i = 1,2;j = 6} \right)} \\ {B_{{ij}}^{*} = E_{{ij}}^{*} = 0\quad \quad \quad \quad {\mkern 1mu} \left( {i,j = 1,2} \right)} \\ \end{array} } \right. \hfill \\ \end{gathered}$$

(44)

Appendix 3

In Eq. (35),

$$\begin{aligned} g_{40} & = - \left[ {\gamma_{170} - \gamma_{171} (m^{2} + n^{2} \beta^{2} )} \right] - g_{08}^{*} - \gamma_{14} \gamma_{24} m^{2} n^{2} \beta^{2} \frac{{g_{05}^{*} g_{07} }}{{g_{06} }} \\ & + \gamma_{80} \left( {\gamma_{14} \gamma_{24} \frac{{m^{2} g_{02} + n^{2} \beta^{2} g_{01} }}{{g_{00} }}\frac{{g_{05} }}{{g_{06} }} - \frac{{m^{2} g_{04} + n^{2} \beta^{2} g_{03} }}{{g_{00} }}} \right), \\ \end{aligned}$$

(45)

$$g_{41} = \left\{ {\begin{array}{*{20}c} {Q_{11} - \gamma_{14} (\gamma_{T1} m^{2} + \gamma_{T2} n^{2} \beta^{2} )\Delta T + 3g_{43} \varPhi^{2} (T)\;(immovable\,)} \\ {Q_{11} \left[ {1 - \frac{P}{{P_{cr} }}\frac{{(m^{2} + \eta n^{2} \beta^{2} )}}{{m^{2} }}} \right]\quad \quad \quad \quad \quad \;\;\left( {\text{movable}} \right),} \\ \end{array} } \right.$$

(46)

$$g_{ 42} = 3g_{ 43} \varPhi (T),g_{43} = \frac{{\gamma_{14} \gamma_{24} }}{16}\left( {\frac{{m^{4} }}{{\gamma_{7} }} + \frac{{n^{4} \beta^{4} }}{{\gamma_{6} }} + C_{33} } \right),$$

(47)

$$C_{33} = \left\{ {\begin{array}{*{20}c} {2\frac{{m^{4} + \gamma_{24}^{2} n^{4} \beta^{4} + 2\gamma_{5} m^{2} n^{2} \beta^{2} }}{{\gamma_{24}^{2} - \gamma_{5}^{2} }}\;(\text{immovable})} \\ {0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left( {\text{movable}} \right)} \\ \end{array} ,} \right.$$

(48)

where the other symbols are given in Shen [45]

$$\begin{aligned} Q_{11} = g_{08} + \gamma_{14} \gamma_{24} m^{2} n^{2} \beta^{2} \frac{{g_{05} g_{07} }}{{g_{06} }} + \left[ {K_{1} + K_{2} (m^{2} + n^{2} \beta^{2} )} \right], \hfill \\ \gamma_{6} = 1 + \frac{{4m^{2} \gamma_{14} \gamma_{24} \gamma_{230}^{2} }}{{\gamma_{42} + \gamma_{430} 4m^{2} }},\gamma_{7} = \gamma_{24}^{2} + \frac{{4n^{2} \beta^{2} \gamma_{14} \gamma_{24} \gamma_{223}^{2} }}{{\gamma_{31} + \gamma_{322} 4n^{2} \beta^{2} }}, \hfill \\ \end{aligned}$$

(49)

$$\varPhi (T) = \lambda + \varTheta_{3} (\lambda )^{3} + \cdot \cdot \cdot$$

(50)

The coefficients $\lambda$ and $\varTheta_{3}$ can be obtained as follow for $m = n = 1$

$$\begin{aligned} \lambda & = \frac{16}{{\pi^{2} G_{08} }}\left( {(\gamma_{T3} m^{2} + \gamma_{T4} n^{2} \beta^{2} ) - \frac{{(\gamma_{T3} - \gamma_{T6} )m^{2} g_{102} + (\gamma_{T4} - \gamma_{T7} )n^{2} \beta^{2} g_{101} }}{{g_{00} }}} \right)\Delta T \\ & \times \frac{h}{{\left[ {D_{11}^{*} D_{22}^{*} A_{11}^{*} A_{22}^{*} } \right]^{1/4} }}, \\ \varTheta_{3} & = - \frac{{\gamma_{14} \gamma_{24} }}{{16G_{08} }}\left( {\frac{{m^{4} }}{{\gamma_{7} }} + \frac{{n^{4} \beta^{4} }}{{\gamma_{6} }} + C_{33} } \right), \\ \left[ {\begin{array}{*{20}c} {g_{ 101} } \\ {g_{102} } \\ \end{array} } \right] & = \left[ {\begin{array}{*{20}c} {\left( {\gamma_{31} + \gamma_{320} m^{2} + \gamma_{322} n^{2} \beta^{2} } \right)\left( {\gamma_{230} m^{2} + \gamma_{232} n^{2} \beta^{2} } \right) - \gamma_{331} n^{2} \beta^{2} \left( {\gamma_{221} m^{2} + \gamma_{223} n^{2} \beta^{2} } \right)} \\ {\left( {\gamma_{42} + \gamma_{430} m^{2} + \gamma_{432} n^{2} \beta^{2} } \right)\left( {\gamma_{221} m^{2} + \gamma_{223} n^{2} \beta^{2} } \right) - \gamma_{331} m^{2} \left( {\gamma_{230} m^{2} + \gamma_{232} n^{2} \beta^{2} } \right)} \\ \end{array} } \right], \\ G_{08} & = Q_{11} - \gamma_{14} \left( {\gamma_{T1} m^{2} + \gamma_{T2} n^{2} \beta^{2} } \right)\Delta T. \\ \end{aligned}$$

(51)