# Buckling of Composite Structures with
## Appendix A Mean Stresses and Mechanics of Fracture Criteria  
### Mean Stress Criterion  
The experimental results demonstrate evidently that transverse shear stress components \(\sigma_{13}\), \(\sigma_{23}\), and \(\sigma_{33}\) play the fundamental role in the description of the delamination failure process. Referring to the form of failure, the above stress components are usually called the inter-laminar stress components. The stress criterion is usually formulated as follows:  
\[\left(\frac{\overline{\sigma}_{13}}{Z_{1}}\right)^{2}+\left(\frac{\overline{ \sigma}_{23}}{Z_{2}}\right)^{2}+\left(\frac{\overline{\sigma}_{33}}{Z_{3}} \right)^{2}=1, \tag{11}\]  
where:  
delamination (both marginal and internal). The fact of occurrence of delamination is determined by the size of the \(G\) coefficient, which is defined as the amount of energy required to create a delamination with a unit area. The value of the total \(G\) coefficient is defined as the sum of the components from individual delamination (\(l=1,2,\dots\) ):  
\[G=\sum_{l=1}^{L}G_{l}=\sum_{l=1}^{L}\left(G_{I}+G_{II}+G_{III}\right)_{l} \tag{10}\]  
Each of the partial fracture opening works \(G_{l}\) is the sum of three components, defining three typical forms of fracture: \(I\)--due to stretching, \(II\)--shear in the direction parallel to the area of the fracture, and \(III\)--shearing in the direction perpendicular to the area of the fracture--see Figure 11. Theoretical and experimental analysis shows that there are many cases where the partial coefficients are not a sum but some unknown function of the components \(G_{I}\), \(G_{II}\), and \(G_{III}\), i.e. :  
\[G_{I}=G_{I}(G_{I},G_{II},G_{III}) \tag{11}\]  
Under the linear theory of fracture mechanics, the stress field in the vicinity of the fracture tip is assessed by calculating the stress intensity factor (SIF). These values are also used as a criterion for starting the fatigue crack growth process. The values of the stress intensity factors corresponding to each of the forms of failure are determined from the following relations ([PERSON] et al. [73]):  
\[K_{I}=\sqrt{\frac{G_{I}}{R^{11}}},\quad K_{II}=\sqrt{\frac{G_{II}}{R^{2}}}, \quad K_{III}=\sqrt{\frac{G_{III}}{R^{3}}}, \tag{12}\]  
where:  
\[R^{11}=\frac{\tau}{\sqrt{S_{11}}},\quad R^{22}=\frac{\tau}{\sqrt{S_{2}}},\quad R ^{33}=\frac{\tau}{\sqrt{S_{33}}},\tau=\sqrt{\left[\sqrt{S_{11}S_{22}}+S_{12}+ S_{66}/2\right]S_{11}S_{22}/2}, \tag{13}\]  
and \(S_{ij}=[C_{ij}]^{-1}\). The theoretical determination of the \(G\) coefficient value is based on the principle of energy conservation, i.e. :  
\[G=\frac{\partial(W_{ext}-W_{lost})}{\partial a}, \tag{14}\]  
where \(W_{ext}\) is the work done by external forces and \(W_{lost}\) is the deformation energy lost due to fracture elongation from an initial value of a to length \(a+\Delta a\). A different definition was proposed by [PERSON] [74] when investigating the phenomenon of failure of an infinite plate with a slot of length \(a\). If the surface forces described by the \(s\) vector are applied to the edges of the slot, the work needed to close the slot is equal to the value of the \(G\) coefficient and is expressed by the formula:\[G=\lim_{\Delta a\to 0}\frac{1}{2\Delta a}\int\limits_{0}^{\Delta a}s\cdot \Delta\widetilde{U}da. \tag{10}\]  
\(\Delta\widetilde{U}\) determines the increment of the displacement vector on the crack surface. In the case of cracks closed inside the laminate, it very often turns out that the correct size for determining the fracture growth is not its length dimension, but its surface area. For linearly elastic bodies, the coefficient \(G\) is equivalent to [PERSON]'s integral \(RJ\) defined as follows:  
\[RJ=\int_{\Gamma}\left(Jdy-T_{i}\widetilde{U}_{i,x}ds\right), \tag{11}\]  
where \(J\) (see Equation (10)) is the strain energy density, T is the force vector normal to the contour \(\Gamma\), \(\widetilde{U}\) the displacement vector, and \(ds\) is the contour length element \(\Gamma\).