Hygrothermal analysis of antisymmetric cross-ply laminates using a refined plate theory

Abstract

The effect of hygrothermal conditions on the antisymmetric cross-ply laminates has been investigated using a unified shear deformation plate theory. The present plate theory enables the trial and testing of different through-the-thickness transverse shear-deformation distributions and, among them, strain distributions do not involve the undesirable implications of the transverse shear correction factors. The differential equations of laminated plates whose deformations are governed by either the shear deformation theories or the classical one are derived. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. A wide variety of results is presented for the static response of simply supported rectangular plates under non-uniform sinusoidal hygrothermal/thermal loadings. The influence of material anisotropy, aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the hygrothermally induced response is studied.

Appendices

Appendix 1

The elements of the symmetric matrix [L], for RPT, are given by:

$$ \begin{aligned} L_{11} & = A_{11} d_{11} + 2A_{16} d_{12} + A_{66} d_{22} , \\ L_{12} & = A_{16} d_{11} + (A_{12} + A_{66} )d_{12} + A_{26} d_{22} , \\ L_{13} & = - B_{11} d_{111} - 3B_{16} d_{112} - (B_{12} + 2B_{66} )d_{122} - B_{26} d_{222} , \\ L_{14} & = B_{11}^{a} d_{11} + 2B_{16}^{a} d_{12} + B_{66}^{a} d_{22} , \\ L_{15} & = B_{16}^{a} d_{11} + (B_{12}^{a} + B_{66}^{a} )d_{12} + B_{26}^{a} d_{22} , \\ L_{16} & = L_{13}^{a} d_{1} + L_{63}^{a} d_{2} , \\ L_{22} & = A_{66} d_{11} + 2A_{26} d_{12} + A_{22} d_{22} , \\ L_{23} & = - B_{16} d_{111} - (B_{12} + 2B_{66} )d_{112} - 3B_{26} d_{122} - B_{22} d_{222} , \\ L_{24} & = L_{15} , \\ L_{25} & = B_{66}^{a} d_{11} + 2B_{26}^{a} d_{12} + B_{22}^{a} d_{22} , \\ L_{26} & = L_{63}^{a} d_{1} + L_{23}^{a} d_{2} , \\ L_{33} & = D_{11} d_{1111} + 4D_{16} d_{1112} + 2(D_{12} + 2D_{66} )d_{1122} + 4D_{26} d_{1222} + D_{22} d_{2222} , \\ L_{34} & = - D_{11}^{a} d_{111} - 3D_{16}^{a} d_{112} - (D_{12}^{a} + 2D_{66}^{a} )d_{122} - D_{26}^{a} d_{222} , \\ L_{35} & = - D_{16}^{a} d_{111} - (D_{12}^{a} + 2D_{66}^{a} )d_{112} - 3D_{26}^{a} d_{122} - D_{22}^{a} d_{222} , \\ L_{36} & = - (\bar{L}_{13} d_{11} + 2\bar{L}_{63} d_{12} + \bar{L}_{23} d_{22} ), \\ L_{44} & = F_{11}^{a} d_{11} + 2F_{16}^{a} d_{12} + F_{66}^{a} d_{22} - A_{55}^{a} , \\ L_{45} & = F_{16}^{a} d_{11} + (F_{12}^{a} + F_{66}^{a} )d_{12} + F_{26}^{a} d_{22} - A_{45}^{a} , \\ L_{46} & = (\hat{L}_{13} - A_{55}^{a} )d_{1} + (\hat{L}_{63} - A_{45}^{a} )d_{2} , \\ L_{55} & = F_{66}^{a} d_{11} + 2F_{26}^{a} d_{12} + F_{22}^{a} d_{22} - A_{44}^{a} , \\ L_{56} & = (\hat{L}_{63} - A_{45}^{a} )d_{1} + (\hat{L}_{23} - A_{44}^{a} )d_{2} , \\ L_{66} & = - (A_{55}^{a} d_{1} + 2A_{45}^{a} d_{12} + A_{44}^{a} d_{22} - \tilde{L}_{33} ). \\ \end{aligned} $$

For the FPT, HPT and SPT, the components of [L] are the same as given above for the RPT except L _i6 = 0(i = 1, 2, …, 6). However, for the CPT, the components of [L] are reduced to be L _ij (i, j = 1, 2, 3).

Appendix 2

The transformation formulae for the stiffness c ^(k) _ij are

$$ \begin{aligned} \left\{ \begin{gathered} c_{11} \hfill \\ c_{12} \hfill \\ c_{22} \hfill \\ c_{16} \hfill \\ c_{26} \hfill \\ c_{66} \hfill \\ \end{gathered} \right\}^{(k)} & = \left[ {\begin{array}{*{20}c} {c^{4} } & {2c^{2} s^{2} } & {s^{4} } & {4c^{2} s^{2} } \\ {c^{2} s^{2} } & {c^{4} + s^{4} } & {c^{2} s^{2} } & { - 4c^{2} s^{2} } \\ {s^{4} } & {2c^{2} s^{2} } & {c^{4} } & {4c^{2} s^{2} } \\ {c^{3} s} & {cs^{3} - c^{3} s} & { - cs^{3} } & { - 2cs\left( {c^{2} - s^{2} } \right)} \\ {cs^{3} } & {c^{3} s - cs^{3} } & { - c^{3} s} & {2cs\left( {c^{2} - s^{2} } \right)} \\ {c^{2} s^{2} } & { - 2c^{2} s^{2} } & {c^{2} s^{2} } & {\left( {c^{2} - s^{2} } \right)^{2} } \\ \end{array} } \right]\left\{ \begin{gathered} c_{11} \hfill \\ c_{12} \hfill \\ c_{22} \hfill \\ c_{66} \hfill \\ \end{gathered} \right\}, \\ \left\{ \begin{gathered} c_{44} \hfill \\ c_{45} \hfill \\ c_{55} \hfill \\ \end{gathered} \right\}^{(k)} & = \left[ {\begin{array}{*{20}c} {c^{2} } & {s^{2} } \\ { - cs} & {cs} \\ {s^{2} } & {c^{2} } \\ \end{array} } \right]\left\{ \begin{gathered} c_{44} \hfill \\ c_{55} \hfill \\ \end{gathered} \right\},\quad \left\{ \begin{gathered} c_{13} \hfill \\ c_{23} \hfill \\ c_{63} \hfill \\ \end{gathered} \right\}^{(k)} = \left[ {\begin{array}{*{20}c} {c^{2} } & {s^{2} } \\ {s^{2} } & {c^{2} } \\ {sc} & { - sc} \\ \end{array} } \right]\left\{ \begin{gathered} c_{13} \hfill \\ c_{23} \hfill \\ \end{gathered} \right\},\quad c_{33}^{(k)} = c_{33} , \\ \end{aligned} $$

where c = cos θ _k , s = sin θ _k and c _ij are the material stiffness of the lamina. For RPT one has

$$ \begin{aligned} c_{11} & = \frac{{E_{x} (1 - \nu_{yz} \nu_{zy} )}}{\varDelta },\quad c_{12} = \frac{{E_{x} (\nu_{yx} + \nu_{yz} \nu_{zx} )}}{\varDelta },\quad c_{13} = \frac{{E_{x} (\nu_{zx} + \nu_{yx} \nu_{zy} )}}{\varDelta }, \\ c_{22} & = \frac{{E_{y} (1 - \nu_{xz} \nu_{zx} )}}{\varDelta },\quad c_{23} = \frac{{E_{y} (\nu_{zy} + \nu_{xy} \nu_{zx} )}}{\varDelta },\quad c_{33} = \frac{{E_{z} (1 - \nu_{xy} \nu_{yx} )}}{\varDelta }, \\ c_{44} & = G_{yz} ,\quad c_{55} = G_{xz} ,\quad c_{66} = G_{xy} . \\ \end{aligned} $$

in which Δ = 1 − ν _xy ν _yx  − ν _yz ν _zy  − ν _zx ν _xz  − 2ν _yx ν _xz ν _zy , E _i are Young’s moduli in the material principal directions, ν _ij are Poisson’s ratios and G _ij are shear moduli. The material stiffness for the CPT and other shear deformation plate theories may be reduced to:

$$ \begin{aligned} c_{11} & = \frac{{E_{x} }}{{1 - \nu_{xy} \nu_{yx} }},\quad c_{12} = \frac{{\nu_{xy} E_{y} }}{{1 - \nu_{xy} \nu_{yx} }} = \frac{{\nu_{yx} E_{x} }}{{1 - \nu_{xy} \nu_{yx} }},\quad c_{22} = \frac{{E_{y} }}{{1 - \nu_{xy} \nu_{yx} }},\quad c_{13} = \frac{{\nu_{xz} E_{z} }}{{1 - \nu_{xy} \nu_{yx} }}, \\ c_{23} & = \frac{{\nu_{yz} E_{z} }}{{1 - \nu_{xy} \nu_{yx} }},\quad c_{33} = \frac{{E_{z} }}{{1 - \nu_{xy} \nu_{yx} }},\quad c_{44} = G_{yz} ,\quad c_{55} = G_{xz} ,\quad c_{66} = G_{xy} . \\ \end{aligned} $$

Appendix 3

The components of the generalized force vector {F} are given by

$$ \begin{aligned} F_{1}^{ij} & = \lambda \left( {A_{1}^{T} \bar{T}_{1} + B_{1}^{T} \bar{T}_{2} + {}^{a}B_{1}^{T} \bar{T}_{3} + a_{1}^{T} \bar{C}_{1} + b_{1}^{T} \bar{C}_{2} + {}^{a}b_{1}^{T} \bar{C}_{3} } \right), \\ F_{2}^{ij} & = \mu \left( {A_{2}^{T} \bar{T}_{1} + B_{2}^{T} \bar{T}_{2} + {}^{a}B_{2}^{T} \bar{T}_{3} + a_{2}^{T} \bar{C}_{1} + b_{2}^{T} \bar{C}_{2} + {}^{a}b_{2}^{T} \bar{C}_{3} } \right), \\ F_{3}^{ij} & = - q_{0} - h\left[ {\left( {B_{1}^{T} \lambda^{2} + B_{2}^{T} \mu^{2} } \right)\bar{T}_{1} + \left( {D_{1}^{T} \lambda^{2} + D_{2}^{T} \mu^{2} } \right)\bar{T}_{2} + \left( {{}^{a}D_{1}^{T} \lambda^{2} + {}^{a}D_{2}^{T} \mu^{2} } \right)\bar{T}_{3} } \right. \\ & \left. { + \left( {b_{1}^{T} \lambda^{2} + b_{2}^{T} \mu^{2} } \right)\bar{C}_{1} + \left( {d_{1}^{T} \lambda^{2} + d_{2}^{T} \mu^{2} } \right)\bar{C}_{2} + \left( {{}^{a}d_{1}^{T} \lambda^{2} + {}^{a}d_{2}^{T} \mu^{2} } \right)\bar{C}_{3} } \right], \\ F_{4}^{ij} & = h\lambda \left( {{}^{a}B_{1}^{T} \bar{T}_{1} + {}^{a}D_{1}^{T} \bar{T}_{2} + {}^{a}F_{1}^{T} \bar{T}_{3} + {}^{a}b_{1}^{T} \bar{C}_{1} + {}^{a}d_{1}^{T} \bar{C}_{2} + {}^{a}f_{1}^{T} \bar{C}_{3} } \right), \\ F_{5}^{ij} & = h\mu \left( {{}^{a}B_{2}^{T} \bar{T}_{1} + {}^{a}D_{2}^{T} \bar{T}_{2} + {}^{a}F_{2}^{T} \bar{T}_{3} + {}^{a}B_{2}^{T} \bar{C}_{1} + {}^{a}D_{2}^{T} \bar{C}_{2} + {}^{a}F_{2}^{T} \bar{C}_{3} } \right), \\ F_{6}^{ij} & = - h\left( {L^{T} \bar{T}_{1} + {}^{a}L^{T} \bar{T}_{2} + {}^{b}L^{T} \bar{T}_{3} + l^{T} \bar{C}_{1} + {}^{a}l^{T} \bar{C}_{2} + {}^{b}l^{T} \bar{C}_{3} } \right), \\ \end{aligned} $$

where

$$ \begin{aligned} \{ A_{i}^{T} ,B_{i}^{T} ,D_{i}^{T} \} & = \sum\limits_{k = 1}^{n} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {(c_{1i}^{(k)} \alpha_{x} + c_{i2}^{(k)} \alpha_{y} )\{ 1,\bar{z},\bar{z}^{2} \} {\text{d}}z,} } \quad (i = 1,2), \\ \{ a_{i}^{T} ,b_{i}^{T} ,d_{i}^{T} \} & = \sum\limits_{k = 1}^{n} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {(c_{1i}^{(k)} \beta_{x} + c_{i2}^{(k)} \beta_{y} )\{ 1,\bar{z},\bar{z}^{2} \} {\text{d}}z,} } \quad (i = 1,2), \\ \left\{ {{}^{a}B_{i}^{T} ,{}^{a}D_{i}^{T} ,{}^{a}F_{i}^{T} } \right\} & = \sum\limits_{k = 1}^{n} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {\left( {c_{1i}^{(k)} \alpha_{x} + c_{i2}^{(k)} \alpha_{y} } \right)\bar{\varPsi }(z)\{ 1,\bar{z},\bar{\varPsi }(z)\} {\text{d}}z,} } \quad (i = 1,2), \\ \left\{ {{}^{a}b_{i}^{T} ,{}^{a}d_{i}^{T} ,{}^{a}f_{i}^{T} } \right\} & = \sum\limits_{k = 1}^{n} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {\left( {c_{1i}^{(k)} \beta_{x} + c_{i2}^{(k)} \beta_{y} } \right)\bar{\varPsi }(z)\{ 1,\bar{z},\bar{\varPsi }(z)\} {\text{d}}z,} } \quad (i = 1,2), \\ \left\{ {L^{T} ,{}^{a}L^{T} ,{}^{b}L^{T} } \right\} & = \sum\limits_{k = 1}^{n} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {\left( {c_{13}^{(k)} \alpha_{x} + c_{23}^{(k)} \alpha_{y} } \right)\bar{\varPsi }^{\prime\prime}(z)\{ 1,\bar{z},\bar{\varPsi }(z)\} {\text{d}}z,} } \\ \left\{ {l^{T} ,{}^{a}l^{T} ,{}^{b}l^{T} } \right\} & = \sum\limits_{k = 1}^{n} {\int\limits_{{z_{k} }}^{{z_{k + 1} }} {\left( {c_{13}^{(k)} \beta_{x} + c_{23}^{(k)} \beta_{y} } \right)\bar{\varPsi }^{\prime\prime}(z)\{ 1,\bar{z},\bar{\varPsi }(z)\} {\rm dz},} } \\ \end{aligned} $$

in which $\bar{z} = z/h,\quad \bar{\varPsi }(z) = \varPsi (z)/h$, and $\bar{\varPsi }^{\prime\prime}(z) = \varPsi^{\prime\prime}(z)/h$.

The elements of the symmetric matrix [C], for RPT, are given by:

$$ \begin{aligned} C_{11} & = - A_{11} \lambda^{2} - A_{66} \mu^{2} , \\ C_{12} & = - (A_{12} + A_{66} )\lambda \mu , \\ C_{13} & = \lambda [B_{11} \lambda^{2} + (B_{12} + 2B_{66} )\mu^{2} ], \\ C_{14} & = - B_{11}^{a} \lambda^{2} - B_{66}^{a} \mu^{2} , \\ C_{15} & = - (B_{12}^{a} + B_{66}^{a} )\lambda \mu , \\ C_{16} & = L_{13}^{a} \lambda , \\ C_{22} & = - A_{66} \lambda^{2} - A_{22} \mu^{2} , \\ C_{23} & = \mu [(B_{12} + 2B_{66} )\lambda^{2} + B_{22} \mu^{2} ], \\ C_{24} & = C_{15} , \\ C_{25} & = - B_{66}^{a} \lambda^{2} - B_{22}^{a} \mu^{2} , \\ C_{26} & = L_{23}^{a} \mu , \\ C_{33} & = - D_{11} \lambda^{4} - 2(D_{12} + 2D_{66} )\lambda^{2} \mu^{2} - D_{22} \mu^{4} , \\ C_{34} & = \lambda [D_{11}^{a} \lambda^{2} + (D_{12}^{a} + 2D_{66}^{a} )\mu^{2} ], \\ C_{35} & = \mu [(D_{12}^{a} + 2D_{66}^{a} )\lambda^{2} + D_{22}^{a} \mu^{2} ], \\ C_{36} & = - (\bar{L}_{13} \lambda^{2} + \bar{L}_{23} \mu^{2} ), \\ C_{44} & = - (F_{11}^{a} \lambda^{2} + F_{66}^{a} \mu^{2} + A_{55}^{a} ), \\ C_{45} & = - (F_{12}^{a} + F_{66}^{a} )\lambda \mu , \\ C_{46} & = (\hat{L}_{13} - A_{55}^{a} )\lambda , \\ C_{55} & = - F_{66}^{a} \lambda^{2} - F_{22}^{a} \mu^{2} - A_{44}^{a} , \\ C_{56} & = (\hat{L}_{23} - A_{44}^{a} )\mu , \\ C_{66} & = - (A_{55}^{a} \lambda^{2} + A_{44}^{a} \mu^{2} + \tilde{L}_{33} ). \\ \end{aligned} $$

For the FPT, HPT and SPT, the components of [C] are the same as given above for the RPT except C _i6 = 0(i = 1, 2, …, 6). However, for the CPT, the components of [C] are reduced to be C _ij (i, j = 1, 2, 3).