Investigation of interlaminar stresses surrounding circular hole in composite laminates under uniform heat flux

Abstract

In this paper, interlaminar stresses in a symmetrical laminated composite plate with a circular hole under uniform heat flux were examined. The analytical solution was achieved based on the boundary-layer theory by Lekhnitskii’s solution. The stress relations related out-of-plane stress components were obtained by variational principle states, the minimum principle of complementary energy, zero-order equilibrium relations and boundary conditions for each layer. The unknown factors in the stress relations were obtained by the equilibrium equation in integral form. This solution prepared a calculation method via adaptability for examining the 3D thermal stresses in perforated laminated with curved boundaries. The results were obtained for $[45/-45]$s, [0/90]s, $[30/-30]$s lay-ups made of graphite/epoxy and glass/epoxy materials.

Appendix

$\overline{S}$ is a stiffness matrix is obtained as follows:

$$\begin{aligned}&\overline{S}_{11}=S_{11} \cos ^4 \gamma + \left( 2S_{12}+S_{66} \right) \sin ^2\gamma \cos ^2 \gamma + S_{22} \sin ^4 \gamma \nonumber \\&\overline{S}_{12}=\left( S_{11}+S_{22}-S_{66} \right) \sin ^2 \gamma \cos ^2 \gamma + S_{12} \left( \cos ^4 \gamma +\sin ^4 \gamma \right) \nonumber \\&\overline{S}_{13}= S_{13} \cos ^2 \gamma + S_{23}\sin ^2 \gamma \nonumber \\&\overline{S}_{16}= \left( 2S_{11}-2S_{12}-S_{66} \right) \sin \gamma - \left( 2S_{22}-2S_{12}-S_{66} \right) \sin ^3 \gamma \cos \gamma \nonumber \\&\overline{S}_{22}= S_{11} \sin ^4 \gamma + \left( 2S_{12}+s_{66} \right) \sin ^2 \gamma \cos ^2 \gamma + S_{22} \cos ^4 \gamma \nonumber \\&\overline{S}_{23}= S_{13} \sin ^2 \gamma + S_{23} \cos ^2 \gamma \nonumber \\&\overline{S}_{26}= \left( 2S_{11}-2S_{12}-S_{66} \right) \sin ^3 \gamma \cos \gamma - \left( 2S_{22}-2S_{12}-S_{66} \right) \sin \gamma \cos ^3 \gamma \nonumber \\&\overline{S}_{33}= S_{33} \nonumber \\&\overline{S}_{36}= 2 \sin \gamma \cos \gamma \left( S_{13}-S_{23} \right) \nonumber \\&\overline{S}_{44}= S_{44} \cos ^2 \gamma + S_{55} \sin ^2 \gamma \nonumber \\&\overline{S}_{46}= \sin \gamma \cos \gamma \left( S_{55}-S_{44} \right) \nonumber \\&\overline{S}_{55}= S_{44} \sin ^2 \gamma + S_{55} \cos ^2 \gamma \nonumber \\&\overline{S}_{66}= 2 \left( 2S_{11}+2S_{22}-4S_{12}-S_{66} \right) \sin ^2 \gamma \cos ^2 \gamma + S_{66} \left( \cos ^4 \gamma + \sin ^4 \gamma \right) . \end{aligned}$$

(A.1)

From the engineering constants, the compliance coefficients along the principal material directions are obtained using

$$\begin{aligned}&S_{11} = \frac{1}{E_1}&S_{12} = - \frac{\eta _{21}}{E_2}&S_{13} = \frac{\eta _{31}}{E_3}&\nonumber \\&S_{22} = \frac{1}{E_2}&S_{23} = - \frac{\eta _{32}}{E_3}&S_{33} = \frac{1}{E_3}&\nonumber \\&S_{44} = \frac{1}{G_{23}}&S_{55} = \frac{1}{G_{13}}&S_{66} = \frac{1}{G_{12}}.&\end{aligned}$$

(A.2)

The $P_{ij}$ coefficients in Eqs. (37) and (38):

(A.3)

where $o_1 = c_{12}^{(\ell )}$ and $o_2 = c_{32}^{(\ell )}$ and $\left. \varvec{\varTheta } = \sigma _{\theta \theta }^{0 \;\, 2} \right| _{\ell =0}$. The energy equation can also be rewritten as follows:

$$\begin{aligned} \delta {\varvec{\varPi }}_c =&\delta \left\{ \frac{-t^2}{4 \lambda _1^2 \left( 1 + \lambda _2^2 \right) \lambda _2^2} \left( \lambda _1^5 R \left( \lambda _2^5 P_{33} + \lambda _2^4 P_{33} \right) +\lambda _1^4 \lambda _2^4 P_{33} t + \lambda _1^3 R \left( \lambda _2^4 \left( -2P_{36}-2P_{13} + P_{44} + P_{55} \right) \right. \right. \right. \nonumber \\&\left. \left. \left. + 2 \lambda _2^3 \left( P_{45} + P_{44} - P_{36} + P_{55} - 2P_{13} \right) +\lambda _2 P_{44} \right) + \lambda _1^2 \left( \lambda _2^4 t \left( \frac{1}{2}P_{44}+P_{13}+P_{36}+\frac{1}{2}P_{55}+P_{45} \right) \right. \right. \right. \nonumber \\&\left. \left. \left. + \lambda _2^3 t \left( 3 P_{45} - P_{36} + P_{44} + 2P_{55} \right) + \lambda _2^2 t \left( \frac{1}{2} P_{44} + \frac{1}{2} P_{55} + P_{13} \right) \right) + \lambda _1 \left( \lambda _2^4 R \left( 2 P_{16} + P_{11} + P_{66} \right) \right. \right. \right. \nonumber \\&\left. \left. \left. + \lambda _2^3 R \left( 2 P_{66} + 6 P_{16} + 4 P_{11} \right) +\lambda _2 R \left( 4 P_{16} + 4 P_{11} + P_{66} \right) + \lambda _1 \lambda _2 R P_{11} \right) + \lambda _2^4 t \left( \frac{1}{2} P_{11} + P_{16} + \frac{1}{2} P_{66} \right) \right. \right. \nonumber \\&\left. \left. + \lambda _2^3 t \left( 2 P_{11} + 3 P_{16} + P_{66} \right) + \lambda _2^2 t \left( 4 P_{11} + 4 P_{16} + \frac{1}{2} P_{66} \right) + 2 \lambda _2 P_{11} t + \frac{1}{2} P_{11} t \right) \right\} = 0. \end{aligned}$$

(A.4)