Buckling of Composite Structures with Delaminations—Laminates and Functionally Graded Materials

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_I with the delamination length variations were observed, whereas in the second case the unstable variations were observed.

1. Introduction

The invention, conception and manufacturing of functionally graded materials (FGMs) introduce a new class of composite materials that should be considered in the investigations of structural behaviour and their response to loading and boundary conditions.

The possible configurations of functionally graded materials are presented in Figure 1.

For discontinuous gradients, three types of gradients are introduced: (1) gradient composition, (2) gradient microstructure and (3) gradient porosity.

For FGMs, the review of possible compositions of materials and their applications is discussed in Refs. [1,2,3,4,5,6,7,8,9,10].

Multi-layered composite structures (laminates and those made of functionally graded materials or nanostructures) may be subject to various forms of local (matrix, fiber cracking, fiber separation, or delamination) or global (buckling or free vibration) failure. In the case of the simultaneous occurrence of global and local forms of damage, the mathematical and numerical description of the deformation and final failure of the analyzed structures is drastically changed and becomes difficult—see the experimental results shown in Figure 2 and Figure 3. They demonstrate the differences between deformations for closed and opened delaminations. The detailed description of experimental results both for static and fatigue loads is presented in Refs. [11,12]—see also Appendix A.

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. The mathematical formulation of such problems is carried out with the use of variational inequalities; see Panagiotopoulos [42], Muc [43].

The importance and complexity of the numerical approach are underlined in different papers [44,45,46,47,48,49] where various numerical methods have been studied characterizing the application of dual methods, nonlinear programming methods, asymptotic methods, and the Bubnov–Galerkin and Rayleigh–Ritz methods.

The aim of the present paper is two-fold:

• To demonstrate and discuss the formulation of the buckling of laminated and functionally graded structures with 1D and 2D delaminations;
• To compare the results for composite structures made of laminates or functionally graded materials.

The analysis was conducted with the aid of 2D higher-order plate/shell theories, formulated in Ref. [50], and of 3D FE formulations. Linear fracture mechanics are adopted (the strain energy release factor G_I–see Appendix A) to distinguish the form of unilateral boundary conditions.

2. Kinematic Relations

The 3D displacements are defined in the following way:

$$\begin{gathered} {\tilde{U}}_1({\xi }_1, {\xi }_2,{\xi }_3)=u_1({\xi }_1, {\xi }_2)\left(1+\frac{z}{R_1}\right)+z{\varphi }_1+z^2{\psi }_1+z^3{\gamma }_1+z^4{\theta }_1 \\ {\tilde{U}}_2({\xi }_1, {\xi }_2,{\xi }_3)=u_2({\xi }_1, {\xi }_2)\left(1+\frac{z}{R_2}\right)+z{\varphi }_2+z^2{\psi }_2+z^3{\gamma }_2+z^4{\theta }_2 \\ {\tilde{U}}_3({\xi }_1, {\xi }_2,{\xi }_3)=u_3\left({\xi }_1, {\xi }_2\right)+z{\chi }_1+z^2{\chi }_2+z^3{\chi }_3\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \end{gathered}$$

(1)

The non-linear strain components are expressed as follows [51,52,53]:

$$\begin{array}{l}{\epsilon }_{11}=e_{11}+\frac{1}{2}\left[e_{11}^2+{\left(\frac{1}{2}{e}_{12}^{}+{\omega }_2\right)}^2+{\left(\frac{1}{2}{e}_{13}-{\omega }_2\right)}^2\right],\\ {\epsilon }_{12}=e_{12}+e_{11}\left(\frac{1}{2}{e}_{12}-{\omega }_3\right)+e_{22}\left(\frac{1}{2}{e}_{12}+{\omega }_3\right)+\left(\frac{1}{2}{e}_{13}-{\omega }_2\right)\left(\frac{1}{2}{e}_{23}+{\omega }_1\right),\\ e_{11}=\frac{1}{H_1}\left(\frac{\partial {\tilde{U}}_1}{\partial {\xi }_1}+\frac{1}{H_2}\frac{\partial H_1}{\partial {\xi }_2}{\tilde{U}}_2+\frac{1}{H_3}\frac{\partial H_1}{\partial {\xi }_3}{\tilde{U}}_3\right),\\ 2{\omega }_1=\frac{1}{H_1{H}_2}\left[\frac{\partial \left(H_3{\tilde{U}}_3\right)}{\partial {\xi }_2}-\frac{\partial \left(H_2{\tilde{U}}_2\right)}{\partial {\xi }_3}\right],\\ e_{12}=e_{21}=\frac{H_2}{H_1}\frac{\partial }{\partial {\xi }_1}\left(\frac{{\tilde{U}}_2}{H_2}\right)+\frac{H_1}{H_2}\frac{\partial }{\partial {\xi }_2}\left(\frac{{\tilde{U}}_1}{H_1}\right),\begin{array}{cc}& H_{\gamma }=A_{\gamma }(1+\frac{z}{R_{\gamma }}),\begin{array}{cc}& H_3=1,\begin{array}{cc}& \gamma =1,2.\end{array}\end{array}\end{array}\end{array}$$

(2)

The strains ε₁₃, ε₂₂, ε₂₃, and ε₃₃ 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:

$${\epsilon }_{11}^{nonl}=\frac{1}{2}{\left(\frac{\partial u_3}{A_1\partial {\xi }_1}\right)}^2,{\epsilon }_{22}^{nonl}=\frac{1}{2}{\left(\frac{\partial u_3}{A_2\partial {\xi }_2}\right)}^2,{\epsilon }_{12}^{nonl}=\frac{{\partial }^2{u}_3}{A_1{A}_2\partial {\xi }_1\partial {\xi }_2}$$

(3)

The strain components are functions of the parameters u, v, w, ϕ₁, ϕ_2, χ₁, χ₂, χ₃—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.:

$${\tilde{U}}_3({\xi }_1, {\xi }_2,{\xi }_3)=u_3\left({\xi }_1, {\xi }_2\right)+z{\chi }_1+z^2{\chi }_2+z^3{\chi }_3+w^{*}({\xi }_1,{\xi }_2)$$

(4)

$w^{*}({\xi }_1,{\xi }_2)$ represents a small deviation of the plate/shell middle plane from a flat shape. The imperfections of the structure, considering the simply supported boundary conditions, are assumed as:

$$w^{\ast }\left({\xi }_1,{\xi }_2\right)=\delta \sin \left(\pi {\xi }_1/a\right)\sin \left(\pi {\xi }_2/b\right)$$

(5)

where the coefficient δ varies between 0 and 1, and δ represents the amplitude of the imperfection. The above representation of the delamination form is commonly used in the literature—see, e.g., [17,54,55,56].

The proposed formulation is applied to the analysis of:

‒

One-dimensional axisymmetric spherical shells, R₁ = R₂ = R, where the coordinate ${\xi }_2$ variable is eliminated;

‒

Two-dimensional flat rectangular plates, Cartesian rectangular coordinates—R₁ $\to \infty$, R₂ $\to \infty$.

3. A Brief Description of Contact Problems Associated with Local Buckling

The description of contact problems for composite structures can be divided into two classes:

• One-dimensional contact problems; the delamination is represented by a single line that separates (or not) the sub-laminates—see Figure 5;
• Two-dimensional contact problems; the delamination is described by a surface—see Figure 6.

The first two modes plotted in Figure 5a,b occur as the upper sub-laminate is thin and the area of the imperfection (delamination) is large. The third mode (Figure 5c) is observed as the delamination has a small area and the imperfection is located deeper in the structure. In the latter case, the delamination can be closed and its effects are connected with contact reactions between upper and lower sub-laminates—see, e.g., Refs. [57,58].

The failure mode and loads are functions of the delamination size and form— see Equation (5).

For 1D delaminations, Figure 7 shows two characteristic cases: when the delamination length a < a_crit is not the critical length and does not cause local sub-laminate stability loss, and the second, when sub-laminate stability loss above the critical length occurs. Therefore, it is obvious that the construction of models characterizing the local loss of stability by the sub-laminate should allow for the determination of both the critical length of a_crit delamination as a function of geometrical and material parameters, as well as the assessment (after local buckling) of the degree of decrease in the value of the critical load as a function of the length of the gap a (or the surface area A for two-dimensional delamination)—see Figure 7.

To evaluate the curve plotted in Figure 8, the imperfection sensitivity method was adopted. In Ref. [59] it is demonstrated that buckling loads can be derived with the aid of the lower bound method, where the form of the imperfection is described by Equation (5). Varying the parameters characterizing the imperfection function, it is possible to find the lower bound corresponding to the curve plotted in Figure 8.

4. Variational Formulations of Contact Problems with Local Buckling

The composite structure represents a 3D body (laminates or functionally graded material, FGM) that is a space occupied by layered structures ${\displaystyle {\sum }_{k=1}^N{V}_k}$, where V_k is a space occupied by the individual k-th layer description of the layered structures.

The functional characterizing composite structures J can be divided into three parts:

$$J=J_{en}+J_{bbc}+J_{ubc}$$

(6)

The first part, J_en, corresponds to the strain (internal) energy. The classical approaches are presented below:

‒

Hu–Washizu

$$\begin{array}{l}{J}_{en}\left(\sigma ,e,u\right)={\displaystyle \sum _{k=1}^N[{\displaystyle \underset{V^{(k)}}{\int }(\frac{1}{2}{\left[e^{(k)}\right]}^{Tr}\left[Q^{(k)}\right]\left[e^{(k)}\right]-{\left[{\sigma }^{(k)}\right]}^{Tr}\left[e^{(k)}\right]}}\\ +{\left[{\sigma }^{(k)}\right]}^{Tr}{\left[e^{(k)}\right]}^{Tr}\left[\left[e^{(k)}\right]\left(u_{\alpha }^{(k)}\right)\right])]dV\end{array}$$

(7)

‒

Hellinger–Reissner

$$\begin{array}{l}{J}_{en}\left(\sigma ,e\right)={\displaystyle \sum _{k=1}^N[{\displaystyle \underset{V^{(k)}}{\int }(-\frac{1}{2}{\left[{\sigma }^{(k)}\right]}^{Tr}{\left({\left[Q^{(k)}\right]}^{-1}\right)}^{Tr}\left[{\sigma }^{(k)}\right]}}\\ +{\left[{\sigma }^{(k)}\right]}^{Tr}\left[e^{(k)}\left(u_{\alpha }^{(k)}\right)\right])]dV\\ \end{array}$$

(8)

‒

Lagrange

$$J_{en}\left(u\right)={\displaystyle \sum _{k=1}^N\left[{\displaystyle \underset{V^{(k)}}{\int }\left(\frac{1}{2}{\left[\left[e^{(k)}\right]\left(u_{\alpha }^{(k)}\right)\right]}^{Tr}\left[Q^{(k)}\right]\left[e^{(k)}\left(u_{\alpha }^{(k)}\right)\right]\right)}\right]}dV$$

(9)

where σ is the stress tensor, e is the strain tensor and u denotes the components of the displacements. The relation between stresses and strains is described by the linear elastic relation. For laminated composites, the traditional relation takes the classical form presented, e.g., by Jones [60]. For FGMs, the physical relation takes the following form:

$$\left[Q(z)\right]=\frac{E(z)}{(1+\nu )}\left[\begin{array}{cccccc}\frac{1-\nu }{1-2\nu }& \frac{\nu }{1-2\nu }& \frac{\nu }{1-2\nu }& 0& 0& 0\\ \frac{\nu }{1-2\nu }& \frac{1-\nu }{1-2\nu }& \frac{\nu }{1-2\nu }& 0& 0& 0\\ \frac{\nu }{1-2\nu }& \frac{\nu }{1-2\nu }& \frac{1-\nu }{1-2\nu }& 0& 0& 0\\ 0& 0& 0& 0.5& 0& 0\\ 0& 0& 0& 0& 0.5& 0\\ 0& 0& 0& 0& 0& 0.5\end{array}\right]$$

(10)

where:

$$E(z)/E_b=[(E_t/E_b-1)find(z)+1]find(z)={\left(\frac{z}{h}+\frac{1}{2}\right)}^n$$

(11)

The second term in the functional (6) corresponds to the bilateral boundary conditions formulated in the equality form:

$${\sigma }_{(\alpha \beta )}^{(k)}{n}_{\beta }^{(k)}=s_{\alpha },u_{j\alpha }^{(k)}=W_{j\alpha } \mathrm{on} S,k=1 \mathrm{or} \mathrm{k}=\mathrm{N}$$

(12)

They determine values of the displacement field W or the distributed external loads s.

The third component in the functional J (6) represents the unilateral boundary conditions, i.e., the kinematic boundary conditions between layers:

$$u_{\alpha }^{(k)}\left(x,y,z_l\right)=u_{\alpha }^{(k-1)}\left(x,y,z_l\right)$$

(13)

and transverse shear and normal stress continuity conditions at the contact interfaces:

$${\sigma }_{\alpha 3}^{(k)}\left(x,y,z_l\right)={\sigma }_{\alpha 3}^{(k-1)}\left(x,y,z_l\right), \alpha =1,2,3$$

(14)

They take the following forms:

‒

the Hu–Washizu approach

$$J_{ubc}=-{\displaystyle \sum _{l=1}^{N-1}\left({\displaystyle \underset{S^{(l,l+1)}}{\overset{}{\int }}\left[u^{(l)}-u^{(l+1)}\right]\left[{\sigma }_3^{\left(l,l+1\right)}\right]dS}\right)}$$

(15)

‒

the Hellinger–Reissner approach

$$J_{ubc}={\displaystyle \sum _{l=1}^{N-1}\left({\displaystyle \underset{S^{(l,l+1)}}{\overset{}{\int }}\left[u^{(l)}-u^{(l+1)}\right]\left[{\sigma }_3^{\left(l,l+1\right)}\right]dS}\right)}$$

(16)

‒

the Lagrange approach

$$J_{ubc}=-{\displaystyle \sum _{l=1}^{N-1}\left({\displaystyle \underset{S^{(l,l+1)}}{\overset{}{\int }}\left[u^{(l)}-u^{(l+1)}\right]\left[\lambda \right]dS}\right)}$$

(17)

where λ denotes the Lagrange multiplier.

5. Computational Models

The construction of composite structures with delamination is presented in Figure 8. The total area V is divided into three (Figure 5a) or four parts (Figure 5b,c). Two of the areas (1) and (4) represent the domains without delaminations, and the domains (2) and (3) correspond to the division of the thickness along the line of the delamination (unilateral boundary condition). For each of the areas (2) and (3), the independent sets of kinematical relations is formulated, whereas in the areas (1) and (4), the global system of coordinates is used. A similar approach is introduced in Refs. [17,61].

Correct numerical modeling of the problem of development of delamination and subsequent loss of stability by sub-laminates requires taking into account the following factors in the analysis:

‒

The application of the large displacement (or deformation) option to determine the bifurcation point;

‒

Structure modeling using 2D or 3D elements is necessary for the analysis of the sub-laminate buckling state, because the classic FEM packages contain only shell elements based on the 3D or first-order transverse shear theory, and as stated previously, in delamination problems, the theories of higher-order shells should be used;

‒

The area where delamination occurs and its surroundings should be discretized using 2D or 3D elements with a triangular base to better approximate, especially at the edge of delamination, the values of the G energy release factors;

‒

Along the thickness of FGMs, the division should include 15 to 20 FE since the material properties vary significantly along the coordinate z—see Equation (10).

The mechanical properties of materials used in computations are written in

Table 1. Mechanical properties.

Unidirectional Composites—See Jones [60]		FGMs—See Equations (10) and (11)
E1/E2 = E1/E3 = 20, E1/G13 = 34, E1/G23 = 20	ν12 = 0.28	Et/Eb = 5	ν = 0.3

Here, 1 corresponds to the direction parallel to fibres. n = 0 in Equation (11) characterizes isotropic (quasi-isotropic) properties.

.

6. Three Parametrical Shell Theories

The simplest Love–Kirchhoff hypothesis (the three parametrical shell theory) can be applied to observe and consider the unilateral constraint problems for 2D laminated structures. The form of the displacement distributions (1) is reduced to:

$$\begin{gathered} {\tilde{U}}_1({\xi }_1, {\xi }_2,{\xi }_3)=u_1({\xi }_1, {\xi }_2)\left(1+\frac{z}{R_1}\right)+z\frac{\partial u_3}{\partial {\xi }_1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ {\tilde{U}}_2({\xi }_1, {\xi }_2,{\xi }_3)=u_2({\xi }_1, {\xi }_2)\left(1+\frac{z}{R_2}\right)+z\frac{\partial u_3}{\partial {\xi }_2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ \hspace{1em}{\tilde{U}}_3({\xi }_1, {\xi }_2,{\xi }_3)=u_3\left({\xi }_1, {\xi }_2\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \end{gathered}$$

(18)

The strain and stress components can be derived from the Equations (2), (3) and (10).

6.1. The Flat Rectangular Plate with An Infinite Width

Let us analyse the simplest case of the unilateral constraint problem for the flat rectangular plate. The problem is reduced to the 1D problem as explained in Ref. [59]. In the plate shown in Figure 9, the length of the delamination is equal to a.

At the edge of the delaminated zone, one can write the continuity conditions for membrane forces N:

N₍₁₎ = N₍₂₎ + N₍₂₎

(19)

bending moments M:

M₍₁₎ = M₍₂₎ + M₍₃₎ − 0.5 N₍₂₎t₍₂₎ − 0.5 N₍₃₎t₍₃₎

(20)

and axial displacements u:

u₍₃₎(b + a,0) − u₍₃₎(b,0.5 t₍₂₎)= u₍₂₎(b + a,0) − u₍₂₎(b, − 0.5 t₍₃₎)

(21)

For the L-K hypothesis, the stress–strain relations take the following form:

$$M_{(i)}=-{\left(\frac{B_{xx}}{A_{xx}}\right)}_{(i)}{N}_{(i)}-{\left(D_{xx}-\frac{B_{xx}^2}{A_{xx}}\right)}_{(i)}{w}_{(i),xx}, i=1, 2, 3$$

(22)

$$A_{xx(i)}={\displaystyle \int E_{xx(i)}}d{z}_{(i)},B_{xx(i)}={\displaystyle \int E_{xx(i)}{z}_{(i)}}d{z}_{(i)},D_{xx(i)}={\displaystyle \int E_{xx(i)}{z}_{(i)}^2}d{z}_{(i)}$$

(23)

Assuming that the buckling mode takes the local buckling form (see Figure 5a) and the symmetry concerning the plate coordinate x the fundamental differential equations (in the (1), (2), (3) and (4) domains (Figure 9)) can be written in the classical way valid for 1D beams:

$${\left(N_{xx}{}_{(i)}\right)}_{,x}=0,{\left(M_{xx}{}_{(i)}\right)}_{,xx}=P_{(i)}{u}_{3(i),xx},(i)=1,2,3,4$$

(24)

Figure 10 shows values of the critical pressure with a symmetric delamination for several delamination length parameters. The buckling load drops drastically as a is higher than the dimensionless thickness of the delaminated zone h₍₃₎/h₍₁₎. The decrease of buckling loads is much more significant for plates made of FGMs.

For the analysed case, the strain energy release rate G_I can be derived in an analytical or numerical way—see Appendix A. As may be seen in Figure 10, the small value of the strain energy release rate G_I is associated with a catastrophic growth of the delamination length. For higher values of G_I, the delamination zone is also unstable; however, it becomes stable for small values of the a/L_x ratio. It will be discussed later for spherical shells.

6.2. Axi-Symmetric Spherical Shells under External Pressure

The buckling of delaminated spherical shells and segments has also been studied in the literature [17,62,63,64,65,66,67,68]. The investigations deal with the analysis of shallow shells [17,62,63,64,65,66,67] or spherical shell segments [68].

The cross-section of spherical shells is presented in Figure 11. The shell geometry is characterized by the shallowness parameter ρ defined below.

$$\rho =\sqrt[4]{12(1-{\nu }_{12}{\nu }_{21})}\frac{a}{\sqrt{Rh}}.$$

(25)

For isotropic (quasi-isotropic) material properties, the dimensionless value of the external buckling pressure is described by the following relation:

$$p_{\dim }=\frac{2E_{quasi}}{\sqrt{3\left(1-{\nu }^2\right)}}{\left(\frac{h}{R}\right)}^2$$

(26)

The forms of the spherical shells with delaminations are plotted in Figure 12.

For shallow shells, the fundamental relations are presented and discussed in Refs. [50,59,69].

$$\begin{array}{l}M{\Delta }^2{u}_3-\frac{1}{R}\Delta f-\frac{1}{{\xi }_1}\left(\frac{df}{d{\xi }_1}-\frac{n^2}{{\xi }_1}f\right)\frac{d^2{u}_3}{d{\xi }_1^2}-\frac{1}{{\xi }_1}\left(\frac{d{u}_3}{d{\xi }_1}-\frac{n^2}{{\xi }_1}{u}_3\right)\frac{d^{2}f}{d{\xi }_1^2}\\ +2\frac{n^2}{{\xi }_1^2}\left(\frac{df}{d{\xi }_1}-\frac{n^2}{{\xi }_1}f\right)\left(\frac{d{u}_3}{d{\xi }_1}-\frac{n^2}{{\xi }_1}{u}_3\right)-p_{zn}=0,M=\frac{\tilde{D}-{\tilde{B}}^2}{\left(1-{\nu }^2\right)\tilde{A}}\end{array}$$

(27)

$$\frac{1}{\tilde{A}}{\Delta }^{2}f+\frac{1}{R}\Delta u_3-\frac{n^2}{{\xi }_1^2}{\left(\frac{d{u}_3}{d{\xi }_1}-\frac{1}{{\xi }_1}{u}_3\right)}^2+\left(\frac{d{u}_3}{d{\xi }_1}-\frac{n^2}{{\xi }_1}{u}_3\right)\frac{d^2{u}_3}{d{\xi }_1^2}=0$$

(28)

$\Delta$ is the Laplace operator, and f denotes the Airy stress function. The above relations are reduced to the eight-order differential equation with respect to the normal displacement u₃(ξ₁).

The main goal of the present section is to characterize the influence of the form of delaminations on the reduction of buckling pressures occurring for ideal/perfect spherical shells. The analysis was carried out via imperfection sensitivity analysis to determine the behaviour analogous to the curve plotted in Figure 7, i.e., to find the distributions of buckling pressures with respect to the values of parameters q_k describing the shapes of geometrical imperfections plotted in Figure 13:

Min p_cr(q_k) k = 1, 2, …, K

(29)

The buckling pressures of spherical shells were derived with the use of the Bubnov–Galerkin method. The detailed form of the series expansions is discussed in Refs. [59,69]. Buckling pressures are much more sensitive to increased radius polar imperfections—see Figure 14. For cross-ply and FGMs, the values of buckling loads are almost identical due to the value of thickness ratio h/R.

7. Numerical Examples for Thicker Structures; 2-D Higher-Order or 3-D Approaches

For thicker structures using the above (higher-order or 3D) formulations, it is possible to solve various issues of delaminated composite structures with local buckling. The problems shown below deal with single delamination.

7.1. Axi-Symmetric Shells under External Pressure

For shallow spherical shells (the small ρ), the buckling mode corresponds to the axisymmetric collapse. As the value ρ increases, the buckling mode becomes bifurcation buckling—see Refs. [50,59]. It is necessary to verify the possible buckling modes for structures subjected to unilateral constraints. We propose herein to adopt the criteria discussed in Appendix A. However, this is a complicated task due to the existing scatter of material properties characterizing failure stresses in Equation (A1) or to postulate the explicit form of the functions (A2), (A3) or (A11). Therefore, we propose to evaluate the strain energy rates with the aid of the finite element package Nisa II Endure. Since the shear effects are negligibly small (see Figure A1) only the tension component G_I was computed. Figure 15 represents the variations of the strain energy release rates G_I with the single delamination length a.

The distributions of the values demonstrate the influence of the shallowness parameter ρ. For small values, stable behaviour is observed, i.e., the growth and the decrease. For higher values, stable deformations are observed. Increasing the value of ρ, stable behaviour is again observed that corresponds to the state plotted in Figure 15. The analysis conducted in Ref. [17] illustrates the influence of the parameter ρ on the shell deformations.

As may be observed in Figure 16 for the cases plotted in Figure 16b,c, the unilateral boundary conditions of Equations (15)–(17) should be taken into account in the numerical analysis.

The variations of buckling pressures with the shallowness parameter are plotted in Figure 17 for laminated shells. The single delamination reduces values of buckling pressures. The buckling forms of such structures are plotted in Figure 5—see also Ref. [50].

The unsymmetric properties of the FGM structures decrease buckling pressures (Figure 18), since they reduce shell-bending stiffnesses, being the most significant aspect in the evaluation of buckling loads—see Equation (19). The values of buckling pressures (Figure 18) vary with the change of the E_t/E_b ratio and of the n coefficient—see Equation (10). The distributions of buckling pressures are similar for both symmetric laminates and unsymmetric configurations of structures made of functionally graded materials. The broader discussion of the results is presented in Ref. [50].

7.2. Compressed Rectangular Plates with a Single Delamination

Now, let us consider the buckling problem of compressed plates with delamination with the form plotted in Figure 6. Delamination results in the both pre- and post-buckling behaviour of structures—see Figure 19. For perfect plates without delamination the rapid change between pre- and post-buckling deformations is observed. For structures with delaminations, the initiation and then the development of a delaminated area leads to nonlinear behaviour (Figure 19). Similarly, for spherical shells the effect of unsymmetry of FGM configuration is lower than for composite laminates.

The value p_dim corresponds to buckling loads of compressed isotropic (quasi-isotropic) plates and is equal to:

$$p_{\dim }=\frac{{\pi }^2{E}_{quasi}}{12\left(1-{\nu }^2\right)}\frac{h^2}{L_x^2}{\left(\frac{m{L}_x}{L_y}+\frac{L_y}{m{L}_x}\right)}^2$$

(30)

where the symbols L_x, L_y denote the length of the plate along the x and y directions.

The post-buckling form of the plate is shown in Figure 19 and Figure 20. The elliptical delamination is located at z = h/3. Such a construction of the delamination is assumed to simplify computation and derivation of buckling pressures.

8. Concluding Remarks and Further Recommendations

For laminated composite constructions the detailed information concerning with the theoretical and experimental analysis of delaminated structures can be found in Refs [70,71].

In this section, primary conclusions will be made from reviewing the important results described in previous sections, and the contribution of the present study to the field of mechanics.

Recommendations for further research will also be described.

• A 2D higher-order method of analysis based on the stretching of the normal to the shell mid-surface is presented herein for obtaining the local buckling loads of structures with unilateral constraints of rectangular plates and axisymmetric spherical shells.
• Applying the general theory of elasticity with tensor analysis, six field equations, which consist of six displacement relations, have been derived and written.
• The energy functionals (variational inequalities) in terms of the physical components of the displacements, Hu–Washizu, Hellinger–Reissner, and Lagrange strain energy J, have been derived.
• Delaminations are represented in the form of geometrical imperfections added to normal displacements. They are characterized by a set of geometrical parameters. The parameters are derived as the lower bound of functions of critical load and the parameter of delaminations.
• The numerical solutions can be obtained with the use of analytical solutions (the Rayleigh–Ritz method) or FE analysis.
• The analysis demonstrates that buckling loads with unilateral constraints are reduced for structures made of FGM materials in comparison to laminated structures due to the reduction of bending stiffnesses.
• With increasing computational capabilities, analysts are gradually becoming more able to deal with three-dimensional problems made of unsymmetrical laminated and FG materials.
• The presented analysis can be extended for investigations of plated/shell structures with multiple delaminations.
• The present analysis of buckling characteristics of structures subjected to unilateral constraints shows the necessity of further studies to define the safe lower bounds in the design of laminated and FGM-composed constructions.