# Buckling of Composite Structures with
## 2 Kinematic Relations  
The 3D displacements are defined in the following way:  
\[ \begin{split}\widetilde{U}_{1}(\xi_{1},\xi_{2},\xi_{3})& =u_{1}(\xi_{1},\xi_{2})\Big{(}1+\frac{z}{R_{1}}\Big{)}+z\phi_{1}+z^ {2}\psi_{1}+z^{3}\gamma_{1}+z^{4}\theta_{1}\\ \widetilde{U}_{2}(\xi_{1},\xi_{2},\xi_{3})&=u_{2}( \xi_{1},\xi_{2})\Big{(}1+\frac{z}{R_{2}}\Big{)}+z\phi_{2}+z^{2}\psi_{2}+z^{3} \gamma_{2}+z^{4}\theta_{2}\\ \widetilde{U}_{3}(\xi_{1},\xi_{2},\xi_{3})&=u_{3}( \xi_{1},\xi_{2})+z\chi_{1}+z^{2}\chi_{2}+z^{3}\chi_{3}\end{split}  \tag{1}\]  
The non-linear strain components are expressed as follows [51; 52; 53]:  
\[ \begin{split}\varepsilon_{11}&=e_{11}+\frac{1}{2} \bigg{[}\varepsilon_{11}^{2}+\Big{(}\frac{1}{2}\varepsilon_{12}+\omega_{2} \Big{)}^{2}+\Big{(}\frac{1}{2}\varepsilon_{13}-\omega_{2}\Big{)}^{2}\bigg{]},\\ \varepsilon_{12}&=e_{12}+e_{11}\Big{(}\frac{2}{2} \varepsilon_{12}-\omega_{3}\Big{)}+e_{22}\Big{(}\frac{1}{2}\varepsilon_{12}+ \omega_{3}\Big{)}+\Big{(}\frac{1}{2}\varepsilon_{13}-\omega_{2}\Big{)}\Big{(} \frac{1}{2}\varepsilon_{23}+\omega_{1}\Big{)},\\ e_{11}&=\frac{1}{H_{1}}\Big{(}\frac{\partial \widetilde{U}_{1}}{\partial\xi_{1}}+\frac{1}{H_{2}}\frac{\partial H_{1}}{ \partial\xi_{2}}\widetilde{U}_{2}+\frac{1}{H_{3}}\frac{\partial H_{1}}{ \partial\xi_{3}}\widetilde{U}_{3}\Big{)},\\ 2\omega_{1}&=\frac{1}{H_{1}H_{2}}\Big{[}\frac{ \partial\left(H_{3}\widetilde{U}_{3}\right)}{\partial\xi_{2}}-\frac{\partial \left(H_{2}\widetilde{U}_{2}\right)}{\partial\xi_{3}}\Bigg{]},\\ e_{12}&=e_{21}=\frac{H_{2}}{H_{1}}\frac{\partial \left(H_{2}\right)}{\partial\xi_{1}}+\frac{H_{1}}{H_{2}}\frac{\partial}{ \partial\xi_{2}}\bigg{(}\frac{\widetilde{U}_{1}}{H_{1}}\bigg{)},\ \ \ \ \ H_{\gamma}=A_{\gamma}(1+\frac{z}{R_{ \gamma}}),\ \ \ \ \ H_{3}=1,\ \ \ \ \ \gamma=1,2.\end{split}  \tag{2}\]  
The strains \(\varepsilon_{13}\), \(\varepsilon_{22}\), \(\varepsilon_{23}\), and \(\varepsilon_{33}\) are derived by the classical cyclic permutation of the subscripts 1, 2, and 3, whereas the non-linear terms are introduced in the following way:  
\[\varepsilon_{11}^{nont}=\frac{1}{2}\bigg{(}\frac{\partial u_{3}}{A_{1}\partial \xi_{1}}\bigg{)}^{2},\varepsilon_{22}^{nont}=\frac{1}{2}\bigg{(}\frac{\partial u _{3}}{A_{2}\partial\xi_{2}}\bigg{)}^{2},\varepsilon_{12}^{nont}=\frac{ \partial^{2}u_{3}}{A_{1}A_{2}\partial\xi_{1}^{2}\partial\xi_{2}} \tag{3}\]  
The strain components are functions of the parameters \(u\), \(v\), \(w\), \(\phi_{1}\), \(\phi_{2}\), \(\chi_{1}\), \(\chi_{2}\), \(\chi_{3}\)--see Ref. [50]. We shall consider composite structures containing an embedded delamination. The global system of coordinates is defined in the mid-surface of the structure, as illustrated in Figure 4. As is normal for laminates, the material system of coordinates coincides with the fiber direction. The applied formulation of variational principles characterized in the third section is based on the following geometrical and material assumptions: (1) the displacement discontinuities can occur in the delaminated regions only; (2) both for laminates and FGMs, material is linearly elastic--see the Section 3; (3) the geometrical nonlinearities are taken into account in the form of large deformations, as in Equation (3), and not strains; (4) the stretching of the normal is considered in the form written in Equation (1). The description of fracture, buckling and post-buckling behaviour of composite structures with delaminations can be carried out in two general ways: (1) a 3D approach--usually the finite element method; and (2) 2D analytical approximations. In the latter case, the form and the correctness of the analysis depend mainly on the form of the prescribed 2D approximations of 3D displacements. The model of the delamination is usually employed in the analysis of the delamination sensitivity where the imperfection is added to the normal displacements, i.e.