3D Chinese lantern-type stability loss of a cylinder composed of functionally graded materials under axial compression

The instability of constructions subjected to external forces is a significant focus of engineering research. Many researches have studied on these problems and investigated to determine the influencing parameters on instability of construction, such as material characteristics, loading conditions, and geometric parameters. Some of these studies, like the subject of this study, focus on the stability problem of column-type structures. For example, in Ref. [1], multilayered composite cylindrical shells subjected to axial compression demonstrate varying critical forces based on layer stacking configurations. The Leibenzon–Ishlinsky method has been employed to ascertain critical loads and examine the effects of anisotropy and pillar size on the stability of initially anisotropic cylindrical pillars [2]. Investigations into hollow cylinders composed of Murnaghan material under tension, compression, and inflation have identified potential instability regions, even under tensile stresses, with stability analysis performed via a bifurcation method [3]. Cylindrical shells exhibit axisymmetric stability loss at the onset of bulging in longitudinal impact, with theoretical wavelengths correlating closely with experimental results [4]. The deformation and stability loss of closed cylindrical shells filled with sand under bending loads were experimentally examined by Gonik et al. [5]. Kostka et al. examined the stability loss of thin-walled cylindrical shells featuring elliptical cross-sections by experimental and numerical approaches [6]. The hydrostatic buckling characteristics of moderately thick composite cylindrical shells was established by a theoretical model founded on the first-order shear deformation theory [7]. The stability of a shotcrete-supported crown was evaluated utilizing Discontinuity Layout Optimization method in [8]. The effect of hybridization fiber composites and thermoset polymer as reinforcement for energy absorption tube was reviewed by a wide range of methodology and particular parameter [9].

A novel methodology for assessing critical loads related to the stability loss of cylindrical shells, plates, and rods was introduced in [10]. The formulae and examples of three-dimensional non-axisymmetric stability in viscoelastic anisotropic cylindrical shells were delineated in [11]. These studies collectively illustrate the intricate nature of cylinder stability, affected by variables like material qualities, thermal conditions, electrical charges, and cross-sectional shape. Croll reinterprets and expands upon the classical theory of axial load buckling in circular cylinders and underscores the necessity of situating numerical studies within classical theory and its modified stiffness extension [12]. Hunt et al. examine concealed symmetries in buckling modes, employing Donnell equations and creating computer programs for large deflection analysis [13]. Hoff and Rehfield provide closed-form solutions to linear Donnell equations, analyzing different simple support conditions that result in buckling stresses approximately half of the classical critical value [14]. Grabovsky and Harutyunyan employ a stringent “near-flip” buckling theory for axially compressed cylindrical shells, validating the classical buckling load formula while forecasting scaling instability due to load imperfections [15]. Their findings indicate that buckling may occur at stresses proportional to thickness raised to the powers of 1.5 or 1.25, consistent with historical experimental data.

The “Chinese lantern” failure mode is a distinct type of structural instability seen in cylindrical constructions subjected to axial compression. This phenomenon is especially pertinent in materials exhibiting certain mechanical properties, such as boroaluminum tubes [16]. The “Chinese lantern” failure mode arises when the material exhibits a high ratio of axial to transverse stress thresholds. This failure mode is defined by a loss of the shell’s stability, resulting in a characteristic buckling pattern [6]. For materials demonstrating the “Chinese lantern” failure, the strength surface derived from the maximum strain criterion intersects the axial stress axis [17]. The elastic modulus along the reinforcing direction might differ markedly among several production groups, influencing stability and failure attributes. The “Chinese lantern” failure mode in cylindrical structures, such as boroaluminum tubes, is affected by the material’s stress ratio and manufacturing quality. Improving manufacturing techniques to enhance the transverse strain limit can augment structural stability and avert this type of failure. Experimental and theoretical evaluations demonstrate a robust association, substantiating the application of stability theory and strain requirements in forecasting this failure mechanism.

The stability of functionally graded material (FGM) cylinders is a significant research focus owing to their applications across diverse engineering domains. These materials have features that progressively change, improving their performance under varying loading circumstances. Research attention on FGM mechanics has consistently increased across both academic and engineering sectors. Multiple reviews addressing various facets of FGMs have been published over the years [18‐20].

Jabbari et al. [21] presented an analytical approach for cylinder geometry and boundary conditions based on the idea that Young’s modulus and thermal conductivity adhere to a univariate power law. Research has also been conducted on more intricate cylindrical structures utilizing functionally graded materials (FGM). The failure characteristics of active–passive damping in the functionally graded piezoelectric sandwich structure were examined under a normal magnetic field and unbounded conditions in [22]. The issue of vibrations in sandwich cylindrical shells with a metal foam core was investigated using the Chebyshev collocation method in [23]. The problems concerning free and steady-state oscillations of an elastic cylinder due to the intricacy of FGM production, both radial and longitudinal variable properties, were investigated in [24]. This synthesis examines the stability loss of FGM cylinders under diverse situations, informed by recent research findings. The stability of FGM cylinders is profoundly affected by the gradation of material properties over the thickness, the volume percentage of constituents, and geometric parameters including length-to-radius and thickness-to-radius ratios [25‐29]. The modified Donnell type stability of cylindrical shells made of ceramic, functionally graded materials, and metal layers under axial load and supported by Winkler-Pasternak foundations was examined using Galerkin’s method in [30]. The issue of global stability loss in a circular cylinder was analytically examined in [31] for the case in which the cylinder is composed of viscoelastic material. Analytical and numerical methods yield dependable forecasts of stability loss, crucial for the design and implementation of new materials in engineering structures.

Based on the above aspects, a special stability loss problem of cylindrical structural elements under axial pressure is investigated in this work. The similar problem is studied in [31] for cylinder made from time-dependent material. This study gives some developments on previous researches by introducing a novel mathematical framework for analyzing the Chinese lanterns-type stability loss of functionally graded material cylinders, subjected to axial compression. The classical buckling theory of a cylinder predicts that a cylindrical shell buckles under uniform axial pressure and that the classical critical stress can be determined by an empirical formula depending on the modulus of elasticity, the radius of the cylinder, and the Poisson’s ratio [12]. It is clear that the experimental results, especially for cylindrical shells, do not agree with the numerical results obtained within the framework of classical theory. Therefore, this is a topic that is being worked on a lot. In this study, buckling problems of cylindrical shells under the action of an axial force that does not agree with the results of classical theory were developed and the buckling problems of solid/hollow bodies of FGM cylinder type were investigated within the framework of the 3D geometrically nonlinear exact theory of Elasticity.

Because the phenomenon of Chinese Lantern-type buckling is crucial for comprehending abrupt stability failures in thin-walled and lightweight engineering structures. This research, in contrast to previous studies, incorporates material grading that substantially influences the stability characteristics in lantern-type stability loss problems. In this study, a comprehensive methodology is used to explain the mechanism of Chinese lantern-type stability loss problem of a fixed-supported circular hollow and solid cylinder made of functionally graded materials through the combination of analytical and numerical techniques (the finite element method). Within the framework of the initial imperfection criterion, it is assumed that the cylinder has a structurally very small sinusoidal initial imperfection before loading. The influence of some parameters on the structural durability of the investigated problems is examined in detail. The contributions of this study to the field in question can be characterized as theoretical and methodological contributions as well as scientific results that can be used in practice.

The cylinder is made of functionally graded material (FGM) with a constant elastic modulus along the axial direction. Poisson’s ratio is assumed to be invariant across the material. The cylinder is fixed-supported at both ends, and initial imperfections have a sinusoidal distribution. The analysis focuses on axisymmetric stability loss and uniformly distributed axial compressive stress. The initial imperfection criterion assesses the loss of stability, and the governing equations are formulated using three-dimensional geometrically nonlinear equations of elasticity theory. By applying the linearization method [32], the solution of this nonlinear boundary value problem can be reduced to the solution of a series of linear boundary value problems (approximations). According to [31, 32], the consecutive solutions to the first two approximations are adequate for determining the critical force.

A considered structural elements as cylinder of length \(\ell \), with an outer radius a and an inner radius b \((b<a)\), is depicted in Fig. 1, subjected to a uniformly distributed normal compressive stress at its ends with an intensity p. A cylindrical coordinate system \((r,\theta ,z)\) is utilized, with the Oz axis aligned with the cylinder’s axis of symmetry and its origin located at the cylinder’s base.

The cylinder is presumed to be composed of a functionally graded material, indicating that the material qualities continuously changed with the coordinates. In the majority of investigations concerning such structural elements with FGM, the influence of Poisson’s ratio-dependent on coordinates-on stress and displacement distributions is disregarded; hence, Poisson’s ratio is considered constant in this study. The modulus of elasticity is presumed to vary along the z-axis in accordance with a power law distribution, represented by the following function.

\(E_0 ~(E_1)\) is the modulus of elasticity at \(z=0 ~(z=\ell )\) and n represents a real number that serves as the exponent in the power law. Two cases are possible: if \(E_0<E_1\), the FGM exhibits gradual stiffening; conversely, if \(E_1<E_0\), the FGM demonstrates progressive softening in the designated direction. The material property gradient for the solution domain is illustrated for the problem in Fig. 2.

In the context of the initial imperfection criterion method, it is posited that the cylinder possesses a structurally minimal initial imperfection before loading (Fig. 1b), with these initial structural distortions being axisymmetric and resembling the mode of a Chinese lantern. The external force that causes the amplitudes of these imperfections to escalate toward infinity is termed the critical force. In this study, the initial defect of the cylinder is defined as a sinusoidal shape, represented by the following equation for its lateral surface [31, 32].where \(t_3 \in (0,\ell )\) is a parameter and L denotes the amplitude of the initial imperfection. It is assumed that \(L \ll \ell \). The dimensionless tiny parameter \(\varepsilon \) is introduced to quantify the extent of the initial imperfection [31, 32].We consider the cross-section of the cylinders perpendicular to their middle line tangent vector to be a circle with constant radius. Given the assumptions, we examine the evolution of the infinitesimal initial imperfections of the cylinder under the influence of normal compression forces of intensity p acting on its ends along the z-axis. According to the considered problem symmetry, this study is conducted in the framework of the axisymmetrical state so solution domain can be selected as for solid cylinder \(\{-a \le r \le 0\) and \(0 \le z \le \ell \}\) or for hollow cylinder \(\{-a \le r \le -b\) and \(0 \le z \le \ell \}\) and it is assumed that the subsequent set of geometrically nonlinear 3D exact field equations of Elasticity Theory is provided in the solution domains:whereThe variables \(t_{rr},~t_{\theta \theta },~t_{zz},~t_{zr}\) and \(t_{rz}\) are components of the non-symmetric Kirchhoff stress tensor, whereas \(\sigma _{rr},~\sigma _{\theta \theta },~\sigma _{zz},~\sigma _{zr}\) and \(\sigma _{rz}\) \((\varepsilon _{rr},~\varepsilon _{\theta \theta },~\varepsilon _{zz},~\varepsilon _{zr}\) and \(\varepsilon _{rz})\) denote the components of the Green stress (strain) tensor. Additionally, \(u_r\) and \(u_z\) signify the displacement vector components in the r and z directions, respectively. The Eq. (4) represents the governing equation concerning the non-symmetric Kirchhoff stress tensor components. Equation (5) delineates the relationship between the components of the non-symmetric Kirchhoff stress tensor and the components of the conventional stress tensor within a cylindrical coordinate system. Furthermore, Eq. (6) exemplifies the nonlinear relationship between the components of the Green strain tensor and the components of the displacement vector.

The constitutive equations for the material of the cylinder in the cylindrical coordinate system are as follows:where \(C_{ij}~~(ij=11,12,13,33,66)\) denotes the function that characterizes the material qualities and is contingent upon location. The relations in (7) pertain to a cylinder composed of transversely isotropic material, with its symmetry axis aligned along the Oz axis. In the case where the material of the cylinder is isotropic FGM, the material constants \(C_{ij}\) (in (7)) are expressed as follows:E(z) represents the modulus of elasticity and \(\nu \) denotes the Poisson’s ratio. The boundary conditions for the problem are given as follows:In this context, p denotes the intensity of the normal pressure applied to the ends of the cylinder, whereas the term \(n_r~(n_z)\) refers to the radial (vertical) component of the unit outer normal of the specified surface. To address the boundary value problem (4)–(9), we express the desired values as a power series in relation to the tiny parameter \(\varepsilon \) as indicated in Eq. (3) [31].where \((ij)=rr,\theta \theta ,zz,rz\) and \((i)=r,z\). By substituting the expression (10) into the relations (4) to (9) and organizing them according to the degree of the parameter \(\varepsilon \), we obtain a series-boundary value problem. Each problem is arranged based on the degree of the parameter \(\varepsilon \) and according to the degree of \(\varepsilon \), labeling them as zeroth \((\varepsilon ^0)\), first \((\varepsilon ^1)\) and subsequent approximations.

Each approximation incorporates elements from all preceding approximations. The zeroth and the first approximations are adequate to ascertain the critical force for the current problem, as outlined in [31, 32]. Consequently, the equations, relationships, and boundary conditions for the zeroth approximation are delineated below:The values pertaining to the zeroth approximation are ascertained as follows according to (11)–(14):From (15), it can be inferred that in the zeroth approximation, the components of the displacement vector can be expressed as follows [32]:where \(A_i\) and \(B_i\) are constant coefficients whose values are determined from the zeroth approximation (i.e., from the boundary value problem in (11)–(14). Now we ascertain the values pertaining to the first approximation. Considering the expression (15), the equations, relationships, and boundary conditions for the first approximation are enumerated below:In Eqs. (11)–(20), previously defined known quantities are used. The zeroth approximation’s solution is determined analytically. On the other hand, the finite element approach is used to determine the numerical solution to the boundary value problem of the first approximation. To this end, in accordance with [33], we present the subsequent functional:whereThe quantities in the first (zeroth) approximation are denoted by the upper indices (1) ((0)) in Eqs. (21) and (22). It is observed that in the specified load case, only the value of \(\sigma _{zz}^{(0)}\) is non-zero, while the other stress quantities with upper indices (0), namely \(\sigma _{rr}^{(0)}\), \(\sigma _{zr}^{(0)}\) and \(\sigma _{rz}^{(0)}\) are equal to zero. Prior to establishing the FEM modeling, it is essential to clarify that functional (21) corresponds to the boundary value problem (17)–(20). By taking the variation of the functional (21) with respect to the displacements \(u_r^{(1)}\) and \(u_z^{(1)}\) and equating it to zero, one can derive the two differential equations presented in (17) as well as the boundary conditions corresponding to the stresses outlined in (20) as follows:The variations in the quantities of (25) are treated as in (24), and after numerous mathematical manipulations and grouping for \(\delta u_r^{(1)}\) and \(\delta u_z^{(1)}\), the subsequent equation is derived.It should be noted that in (26), the boundary condition pertaining to \(u_r^{(1)}\) and \(u_z^{(1)}\) must be fulfilled by \(\delta u_r^{(1)}\) and \(\delta u_z^{(1)}\), specifically, \(\delta u_r^{(1)}\Big |_{z=0}^{\ell }=0\) and \(\delta u_z^{(1)}\Big |_{z=0}^{\ell }=0\). Each term in (26) is equal to zero individually. Since \(\delta u_r^{(1)}\) and \(\delta u_z^{(1)}\) are arbitrary, their coefficients must be zero. Consequently, the governing equations in (17) and the boundary conditions in (20) are adopted, demonstrating that the functional (21) is applicable for finite element method modeling of the boundary value problem (17)–(20). Consequently, by employing functional (21) and the Ritz method, together with established FEM processes [34], the FEM modeling is developed, resulting in the following system of linear algebraic equations.where F is the load vector, K is the Stiffness matrix, and \(u^{(1)}\) is the nodal displacement vector.The element stiffness matrix \(K^{(e)}\) iswhereand \(W_e ~(W=\cup _{e=1}^{M} W_e)\) is the domain of e-th finite element, T indicates the transpose, M is the total number of finite elements, N is the shape function, and \(C^{(e)}\) is an Elasticity matrix whose constituents are \(C_{ij}(z)\) in Eq. (7). Nodal displacements are derived by resolving Eq. (27). In each finite element, Gauss quadrature method with 10 sample points is employed to derive the numerical values of definite integrals. The solution domain W is divided into nine-noded quadrilateral (Q9) elements [34]. In summary, the solution technique outlined above involves solving the zeroth (Eqs. 11–14) and first (Eqs. 17–20) approximations for different values of p in (14), following which the critical parameter \(p_{cr}\) is ascertained using the initial imperfection criterion, i.e.,

The critical external force \(p_{cr}\), which induces the loss of stability in the Chinese lantern-type FGM circular cylinder, is calculated by utilizing a specific value of p in Eq. (9), along with the solutions of the zeroth approximation and the first approximation, while applying the initial imperfection criterion given in Eq. (28). The impacts of various mechanical and geometrical parameters on the numerical results of the stability loss problem are provided in the tables and graphs below.

FEM model in the half cross-section is employed due to the symmetry of the problem concerning the radial axis. The solution domain is partitioned into 25 and 10 rectangular elements along the Oz \((Ox_3 )\) and Or \((Ox_1)\) axes, respectively. The FEM modeling utilizes 250 rectangular elements with 9 nodes each, resulting in a total of 1071 nodes and 1951 NDOFs. Initially, Tables 1 and 2 are provided for the assessment of mesh sensitivity. The parameters \(N_1\) and \(N_2\) represent the number of rectangular elements along the Oz \((Ox_3)\) and Or \((Ox_1)\) axes, respectively. Tables 1 and 2 display the critical axial compressive force values, i.e., \(p_{cr}/E_0\) which cause stability loss for variations in \(N_1\) (Table 1) and \(N_2\) (Table 2) in axially symmetric circular cross-sectioned cylinders made of isotropic homogeneous material, i.e., \(E_1/E_0 =1\) for \(b/\ell =0\) and \(a/\ell =0.5\).

The numerical results in Tables 1 and 2 indicate that the critical axial compressive force values diminish as the parameters \(N_1\) and \(N_2\) increase. Nevertheless, beyond a specified value of the parameters \(N_1\) and \(N_2\), the critical axial compressive force values remain invariant and converge toward a particular limit. These results validate the accuracy of the mesh sensitivity employed to ascertain the numerical solution. Consequently, the parameter values \(N_1=25\) and \(N_2=10\) are presumed for the numerical computations.

Tables 3 and 4 present the critical axial compressive force values, i.e., \(p_{cr}/E_0\), that induce stability loss for solid (\(b/\ell =0\)) (Table 3) and hollow (\(b/\ell \ne 0\)) (Table 4) axially symmetric circular cross-sectioned cylinders composed of isotropic homogeneous material, i.e., \(E_1/E_0 = 1\). These tables indicate that increasing the radius or decreasing the length of the cylinder in the Oz direction results in a decrease in the values of critical compressive force. These results deviate from the well-known mechanical and physical aspects that increase the cylinder length reduce the force that causes loss of stability. This unusual finding – specifically, that an increase in length leads to an increase in the critical force – can be explained as an increased force necessary to expand the lantern-like folds as the cylinder’s length increases. Moreover, when the dimensions of the voids expand, the values of the critical force diminish (Table 4). So the existence of voids in the cylinder lowers the stability limit even more, hence stressing the need of careful design issues in hollow cylindrical constructions.

Tables 5 and 6 present the values of the critical axial compressive force that induces stability loss in FGM cylinders.

The effect of material inhomogeneity parameter i.e., \(E_1/E_0\), on the values of \(p_{cr}/E_0\) is presented in Table 5 for \(\nu =0.3\) and power law exponent \(n=2\), for both solid (\(b/\ell =0\)) and hollow (\(b/\ell = 0.2\)) FGM cylinders. As dimensionless values are employed in the numerical computation, the ratio of \(E_1/E_0\) signifies the maximum Young’s modulus in the cylinder. \(E_1/E_0=1\) pertains to the case in which the material is homogeneous. The results indicate that as the \(E_1/E_0\) value deviates from 1, the critical compressive force values escalate and the critical compressive force values obtained for the case \(E_1<E_0\) remain smaller than the corresponding value for the case \(E_0<E_1\). The critical values for the hollow cylinder are consistently lower than those for the solid cylinder. Particularly as the value of \(E_1/E_0\) escalates, this reduction amplifies further. One important discovery is that a higher material heterogeneity can improve the critical axial load capacity, which is especially relevant for load-bearing construction design with best performance.

The effect of the power law exponent (i.e., n in Eq. (1)) on the values of \(p_{cr}/E_0\) is presented in Table 6 for \(\nu =0.3\) and \(E_1/E_0=5\), for both solid (\(b/\ell =0\)) and hollow (\(b/\ell =0.2\)) FGM cylinders. Based on the numerical data presented in Table 6, the values of \(p_{cr}/E_0\) increase as the power law exponent, i.e., n increases. Despite the critical axial compressive force values for the hollow cylinder in the n variation being lower than those for the solid cylinder, the relative difference in the n variation is more pronounced.

Geometric and material parameters including the power law exponent and the ratio of elastic moduli are found to be rather important in determining the stability loss mechanism. Because of their gradual variation in mechanical characteristics, FGM cylinders showed better resistance against stability loss than homogeneous materials. The numerical results also show that material property gradation along the axial direction is quite important for stability performance and provides a special means of reducing instability-related failures. This result directly affects engineering uses where more stability is needed, including pipeline construction under compressive loads, bridge supports, and aircraft components exposed to such forces [9].

The effect of the variation in \(b/\ell \) on the values of the critical axial compressive force, i.e., \(p_{cr}/E_0\) for the case where the cylinder material is functionally graded, is determined in Table 7. According to the numerical results in Table 7, the critical force values decrease as the dimensions of the voids increase.

The graph depicts the influence of the variation in \(b/\ell \) on the values of the critical axial compressive force \(p_{cr}/E_0\) for both isotropic (\(E_1/E_0=1\)) and functionally graded (\(E_1/E_0=5, n=2\)) cylinders in Fig. 3. The obtained \(p_{cr}/E_0\) values are higher when the cylinder material is assumed to be FGM than when it is assumed to be isotropic homogeneous; and in this case, as \(b/\ell \) increases, the values of \(p_{cr}/E_0\) decrease further.

Figure 4 compares the variations in a) n and b) \(E_1/E_0\) for both a solid (\(b/\ell =0\)) and hollow (\(b/\ell =0.2\)) cylinder composed of FGM material for \(a/\ell =0.5, \nu =0.3\). The graphs indicate that the critical axial compressive force values for the solid cylinder exceed those for the hollow cylinder in all cases. The critical force values in a cylinder composed of functionally graded material represented as a power law are significantly more influenced by variations in \(E_1/E_0\) than by alterations in n.

The stability modes on the outer surface of the solid cylinder for specific “p” values, where \(p/E_0\) (\(<p_{cr}/E_0\)), for the dimensionless displacements \(\tilde{u}_z\) and \(\tilde{u}_r\) are illustrated in Fig. 5 for the case \(E_1/E_0=5\), \(a/\ell =0.5\), \(b/\ell =0\) \(\nu =0.3\) and \(n=2\). The critical force value for this problem is \(p_{cr}/E_0=0.7245\). These plots indicate that when ‘p’ approaches \(p \rightarrow p_{cr}\), the conditions \(\{|u_r|;|u_z|\} \rightarrow \infty \) are fulfilled.

Figure 6 illustrates the dimensionless surface plots for the cylinder’s cross-section defined by \(\{-a \le r \le 0 ~~\text {and}~~ 0 \le z \le \ell \}\) or \(\{0 \le x_1 \le a ~~\text {and}~~ 0 \le x_3 \le \ell \} \) for \(p/E_0=0.30\) \((p_{cr}/E_0=0.7245)\) regarding a) \(\tilde{u}_r\) and b) \(\tilde{u}_z\) when \(a/\ell =0.5\) and \(b/\ell =0\), illustrating the stability loss form of a solid cylinder.

Figure 7 presents the dimensionless surface plots for the cylinder’s cross-section defined by \(\{-a \le r \le 0 ~~\text {and}~~ 0 \le z \le \ell \}\) or \(\{0 \le x_1 \le a-b ~~\text {and}~~ 0 \le x_3 \le \ell \} \) for \(p/E_0=0.30\) \((p_{cr}/E_0=0.6135)\), illustrating a) \(\tilde{u}_r\) and b) \(\tilde{u}_z\), when \(a/\ell =0.5\) and \(b/\ell =0.2\), indicating the stability loss form for a hollow cylinder.

It is important to observe that in Figs. 6 and 7, the transformation \(x=a+r\) has been employed to connect the components between \(O_r\) in polar coordinates and \(O'_x\) in Cartesian coordinates. According to the graphs, both displacement values obtained for the hollow cylinder are greater than the values obtained for the solid cylinder.

The physical consequences of the obtained results show that the interaction between material heterogeneity and geometric restrictions controls the stability behavior of FGM cylinders. The study shows that an adaptive response to compressive load is made possible by a slow change of material properties along the axial direction, so postponing the beginning of buckling. Applications where stability failure might cause catastrophic structural collapse benefit especially from this effect.

The results demonstrate that voids within the cylinder reduce the critical axial force required for stability loss. This results from the redistribution of stress concentrations around the voids, so compromising the structural load-bearing capacity generally. Nevertheless, the deliberate choice of the power law exponent in FGMs helps to offset this effect and enables better structural resilience in spite of internal cavities.

Engineer-wise, the research emphasizes the need of adjusting material gradation to improve structural performance. Engineers can create lightweight, high-strength cylindrical components with outstanding resistance to compressive loads by precisely optimizing the material distribution. These results highlight the benefits of FGMs over conventional isotropic materials in uses needing both mechanical strength and adaptability to changing load conditions.

This work examines the axisymmetric stability loss of fixed supported circular solid and hollow cylinders in the Chinese lantern mode subjected to axial pressure. The cylinder is thought to be constructed using functionally graded material (FGM). The cylinder’s material is graded along the axis of the cylinder. The side surface of the cylinder is presumed to possess an infinitesimal internal imperfections formed as sinusoidal in the Chinese lantern configuration. The relevant boundary value problem about the evolution of this flaw under the axial pressure of the cylinder is articulated within the context of the exact equations of geometric nonlinear elasticity theory. The diminutive nature of the initial imperfections is defined by a small parameter, with all desired values expressed as a power series in relation to this parameter. Only the zeroth and first approximations in this series are utilized to ascertain the critical pressure for stability loss based on the initial imperfection criterion. The values of solution to the zeroth approximation are determined analytically, whereas the numerical FEM method is employed to ascertain the values of solution to the first approximation. Consequently, the relevant function is presented, and its validation for the finite element method modeling of the boundary value problem is illustrated within the framework of the first approximation. The authors have coded all the algorithms and programs required to obtain numerical results in FTN77, and the numerical results obtained are presented in tables and figures. The numerical results derived from the aforementioned procedure are presented. Consequently, the following inferences are summarized based on these results: