# Buckling of Composite Structures with
###### Abstract  
In the present paper, buckling problems of constructions with single delamination are examined. Structures were made of unidirectional laminates and functionally graded materials (FGM). Two types of delaminations (closed and opened) were both investigated in experiments both in rectangular plates and axi-symmetric shells. The first part of the work is devoted to the formulation of contact problems (embedded, closed delaminations) with the aid of various functional inequalities. Then, computational models are discussed. To study the influence of the variable material configuration of FGMs, the fourth-order plate/shell relations were adopted. Finally, three particular problems examined are the buckling of flat rectangular plates, spherical shells, and compressed rectangular plates with elliptical delaminations. The experiments were conducted using imperfection sensitivity analysis and post-buckling non-linear analysis. The results demonstrate that the unsymmetric configurations of FGM structures lead to the reduction of buckling loads for structures with delaminations. For FG structures, those effects are described by the simple coefficient. Linear fracture mechanics were employed to distinguish the form of unilateral boundary problems (closed or opened). In the first case, the stable variations of the strain energy release rate \(G_{l}\) with the delamination length variations were observed, whereas in the second case the unstable variations were observed. contact (unilateral) problems; laminates; functionally graded materials; variational inequalities; buckling; delaminations; experimental results; fracture mechanics +
Footnote †: journal: Applied Sciences  
Acticle Buckling of Composite Structures with  
Delaminations--Laminates and Functionally Graded Materials. Figure 1: FGM categories. Figure 2: A square compressed plate with the embedded circular closed delamination \(a/L_{x}=a/L_{y}=0.3\), \(h/L_{x}=0.012\)—Ref. [11] (not to scale). The delamination associated with the buckling of structures can be classified into types of strips: rectangular, circular, and elliptical. The effect of contact between separated layers (sub-laminates) is also considered. Several papers investigate the growth of cracks and delaminations in flat plates and cylindrical shells with axi-symmetric delaminations [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. The problems of delamination failure are considered both under static, impact and fatigue loads. The research analysis can be divided into the following subjects:  
* For predicting the initiation of delamination failure, it is possible to apply analytical or numerical methods (finite element modeling); see, e.g., Ref. [11] combined with the formulation of appropriate strength criteria presented in Appendix A--see Refs. [12; 13];
* Buckling analysis--the length and the form of the delaminated region are prescribed in advance and then the carrying capacity of structures (mainly plates), understood in the sense of local buckling, is studied, where both single [14] and multiple [15] delaminations are considered;
* Contact problems: during the loading or unloading process, the initially opened region of delamination can both grow and/or shrink; however, in the latter case, the strict and concise contact (even frictionless) analysis is required--see, for instance, 1D analysis, 2D analysis and 3D analysis Refs. [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30];
* Different problems in conjunction with the impact/fatigue delamination growth (laminates and FGMs) are also investigated. Several authors use the finite element method and cohesive elements approach to describe various modes of impact/fatigue loading [31; 32]. Modeling of impact/fatigue problems and damage propagation in delaminated composite structures are also discussed in Refs. [33; 34; 35; 36; 37; 38; 39; 40; 41]. However, it should be pointed out that the correct and accurate solution to the above problems requires a different approach due to the existence of unilateral boundary conditions.