The Effect of Porosity on Elastic Stability of Toroidal Shell Segments Made of Saturated Porous Functionally Graded Materials

Abstract

This research deals with the stability analysis of shallow segments of the toroidal shell made of saturated porous functionally graded (FG) material. The nonhomogeneous material properties of porous shell are assumed to be functionally graded as a function of the thickness and porosity parameters. The porous toroidal shell segments with positive and negative Gaussian curvatures and nonuniform distributed porosity are considered. The nonlinear equilibrium equations of the porous shell are derived via the total potential energy of the system. The governing equations are obtained on the basis of classical thin shell theory and the assumptions of Biot's poroelasticity theory. The equations are a set of the coupled partial differential equations. The analytical method including the Airy stress function is used to solve the stability equations of porous shell under mechanical loads in three cases. Porous toroidal shell segments subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. Closed-form solutions are expressed for the elastic buckling behavior of the convex and concave porous toroidal shell segments. The effects of porosity distribution and geometrical parameters of the shell on the critical buckling loads of porous toroidal shell segments are studied.

1 Introduction

Thin-walled structures such as elastic shells, curved/flat plates, arches, and beams are widely used in the mechanical engineering and design problems. Other applications of thin-walled structures are in the fields of architectural, marine, and civil engineering. On the other hand, an important case of double curvature shells, namely the toroidal shells, may be used in pressure vessels.

During the last several years, functionally graded (FG) porous material structures with uniform and nonuniform distributed porosity are widely used in engineering problems. The FG porous materials are made of two different parts; one of which is solid and another part may be either liquid or gas. Biot introduced the constitutive equations (stress–strain relations) for an elastic porous element. In Biot's poroelasticity theory [1], the constitutive equations can be expressed in the terms of elastic constants of the fluid and solid phases.

Static and dynamic behaviors of the toroidal shell segments as the type of double curvature shells are well studied in the past years, and the buckling analysis for such shells can be an important design problem.

Stein and McElman [2] are the pioneers to investigate the linear buckling analysis of the toroidal shell segments. They researched the stability problem for the shallow segments of homogeneous isotropic toroidal shells under various mechanical loads. Toroidal shell segments with both positive and negative Gaussian curvatures are analyzed in this study, employing an analytical approach. Moreover, the initial post-buckling behavior of homogeneous isotropic toroidal shell segments is studied by Hutchinson [3]. He presented the closed-form solutions for the nonlinear post-buckling responses of toroidal shell segments with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loads. Fluegge and Sobel [4] presented a study on the buckling behavior of isotropic toroidal shells under symmetric external pressure load using the analytical method. Jordan [5] investigated the stability analysis of isotropic toroidal shells under uniform hydrostatic pressure load based on the classical shell theory. Moreover, an asymmetric buckling response of toroidal shells subjected to axial tension loads is researched by Radhamohan and Prasad [6].

Many theoretical investigations on the bending, vibration, and buckling problems of the FG porous thin-walled structures have been developed by authors. For example, Magnucki and Stasiewicz [7] researched the buckling behavior of porous beams with variable material properties and simply supported boundary conditions on the basis of shear deformation theory. The linear axisymmetric deflection and buckling responses of saturated porous circular plates with varying material properties and simply supported edges are studied by Magnucka-Blandzi [8] based on the classical plate theory. Chen et al. [9] analyzed the elastic buckling and static bending problems of shear deformable FG porous beams with simply supported end conditions. Chen et al. [10] presented a study on the free and forced vibration responses of the shear deformable FG porous beams based on the Biot poroelasticity theory. Moreover, Chen et al. [11] studied the free vibration analysis of shear deformable sandwich beams with a functionally graded porous core using an approximate method. They investigated the effect of porosity distribution parameter on the static and dynamic responses of the FG porous beams. Also, thermal post-buckling responses of geometrically imperfect FG porous beams with temperature dependent properties are analyzed by Babaei et al. [12] based on the higher order shear deformation beam theory.

Bich et al. [13] investigated the linear buckling analysis of eccentrically stiffened FG toroidal shell segments under external mechanical loads based on classical thin shell theory. Ninh et al. [14] presented a study on the torsional buckling behavior of eccentrically stiffened FG toroidal shell segments surrounded by two-parameter elastic foundation. Bich and Ninh [15] researched the post-buckling behavior of sigmoid FG toroidal shell segment surrounded by elastic medium subjected to thermo-mechanical loads. Ninh and Bich [16] studied the nonlinear buckling analysis of eccentrically stiffened FG toroidal shell segments under torsional load in thermal environment. Thang and Thoi [17] presented a study to analyze the nonlinear dynamic buckling of sigmoid functionally graded material toroidal shell segments with axial and circumferential stiffeners. Bich et al. [18] worked on the nonlinear buckling behavior of FG toroidal shell segments filled inside by elastic foundation under external pressure load including temperature effects. Bich et al. [19] reported the nonlinear dynamic responses of eccentrically stiffened FG toroidal shell segments surrounded by elastic medium in thermal environment. Ninh and Bich [20] researched the nonlinear thermal vibration of eccentrically stiffened ceramic–functionally graded material (FGM)–metal layer toroidal shell segments surrounded by elastic foundation. Tung and Hieu [21] investigated the nonlinear buckling analysis of carbon nanotube-reinforced composite toroidal shell segment surrounded by elastic medium under uniform external pressure load. Vuong and Duc [22] presented a study on the nonlinear response of buckling behavior of eccentrically stiffened FGM toroidal shell segments in thermal environment. Hieu and Tung [23] studied the thermo-mechanical nonlinear buckling analysis of pressure-loaded carbon nanotube reinforced composite toroidal shell segment surrounded by elastic medium. Ali and Hasan [24] researched the nonlinear dynamic buckling behavior of the imperfect shear deformable orthotropic FG toroidal shell segments under the longitudinal constant velocity. Phuong et al. [25] worked on the thermo-mechanical post-buckling behavior of FG graphene-reinforced composite laminated toroidal shell segments surrounded by the Pasternak elastic foundation. Vuong and Duc [26] analyzed the nonlinear vibration of FG moderately thick toroidal shell segment within the framework of the third order shear deformation theory. Vuong and Duc [27] reported the nonlinear buckling and post-buckling responses of an FG toroidal shell segment under a torsional load in a thermal environment. Finally, Zhang et al. [28] presented a study on the experimental and numerical buckling analyses of toroidal shell segments under uniform external pressure load. For more works on the instability analysis of FGM cylindrical shells (a simple case of the toroidal shells), one may refer to Refs. [29–35].

For the first time, toroidal shell segments made of saturated porous FG materials are modeled using the Biot poroelasticity theory. The nonhomogeneous material properties of the porous toroidal shell segments are taken by a specific distribution function through the shell thickness. The aim of the present research is to obtain the elastic stability of porous toroidal shell segments under various mechanical loads. The porous toroidal shell segments with both positive and negative Gaussian curvatures under lateral pressure, axial compression, and hydrostatic pressure loads are analyzed. The nonlinear equilibrium equations are obtained using the minimum potential energy principle. The stability equations are derived from the equilibrium equations by employing the adjacent equilibrium criterion. Closed-form solutions of the shallow segments of porous toroidal shells with simply supported end conditions are presented. The obtained results show the effects of porosity variation and geometrical property of toroidal shells on the critical buckling loads of these structures.

2 Kinematic Equations

The shallow segments of toroidal shells with thickness $h$, equator radius $b$, length $l$, and radius of curvature $r$ are considered in Fig. 1. It is assumed that the toroidal shell segments are made of saturated porous FG materials. The nonhomogeneous properties of the shells with nonuniform distributed porosity are graded in the shell thickness. The geometry and schematic of the shallow toroidal shell segments are described in the Cartesian coordinates system. The radius of curvature $r$ can be positive for a convex toroidal shell segment (positive Gaussian curvature) or negative for a concave toroidal shell segment (negative Gaussian curvature).

Fig. 1.

Schematic and geometric characteristic of the toroidal shell segments with positive/negative Gaussian curvature [2]

On the basis of classical thin shell theory, the nonzero components of strain tensor across the thickness of the shallow toroidal shell at a distance $z$ from the middle surface of the shell are expressed as follows [36]:

$${\epsilon }_x\hspace{0.17em}=\hspace{0.17em}\overline{{\epsilon }_x}+z\hspace{0.17em}{K}_x,\hspace{0.17em}{\epsilon }_y\hspace{0.17em}=\hspace{0.17em}\overline{{\epsilon }_y}+z\hspace{0.17em}{K}_y,\hspace{0.17em}{\gamma }_{xy}={\overline{\gamma }}_{xy}+2z\hspace{0.17em}{K}_{xy}$$ (1)

In the above relations, ($\overline{{\epsilon }_x},\hspace{0.17em}\overline{{\epsilon }_y},{\overline{\gamma }}_{xy}$) are the nonlinear components of the strain tensor at the middle surface of the porous toroidal shell with shallow curvatures, and ($K_x,K_y,\hspace{0.17em}{K}_{xy}$) are the linear curvature components.

The nonlinear strain-displacement equations are presented for the shallow toroidal shell segments using the von Karman type of geometric nonlinearity. The nonlinear strain components at the middle surface of shallow shells are related to the displacement components $u,\hspace{0.17em}v,$ and $w$, respectively, in the $x,\hspace{0.17em}y,$ and $z$ directions as [2,3]

$$\overline{{\epsilon }_x}={u,}_x+\frac{1}{2}{\left({w,}_x\right)}^2\pm \frac{w}{r},\hspace{0.17em}\overline{{\epsilon }_y}={v,}_y+\frac{1}{2}{\left({w,}_y\right)}^2+\frac{w}{b},\hspace{0.17em}{\overline{\gamma }}_{xy}={u,}_y+{v,}_{x\hspace{0.17em}}+{w,}_x{w,}_y$$ (2)

Also, the curvatures are related to the lateral displacement component $w$ of the shell as [36]

$$K_x={-w,}_{xx},\hspace{0.17em}{K}_y={-w,}_{yy},\hspace{0.17em}{K}_{xy}=-{w,}_{xy}$$ (3)

in which a comma in subscript indicates partial differentiation. Also, a convention is used in the first of Eq. (2) and in the next equations. According to the present convention, when there is a double sign, the upper sign applies to the convex toroidal shell segment and the lower sign applies to the concave toroidal shell segment.

3 Constitutive Equations

Based on Biot's poroelasticity theory [1], the constitutive equations are presented for toroidal shells made of saturated porous FG material. According to this theory, the porosity in the shell thickness can be distributed as uniform, symmetric, and nonsymmetric. It is assumed that the nonhomogeneous properties of the porous toroidal shell segments with nonsymmetric distributed porosity in the thickness direction are presented as [37]

$$E\left(z\right)=-E_0\left\{\beta \text{cos}\left[\frac{\pi }{2h}\left(z+\frac{h}{2}\right)\right]-1\right\},\hspace{0.17em}G\left(z\right)=-G_0\left\{\beta \text{cos}\left[\frac{\pi }{2h}\left(z+\frac{h}{2}\right)\right]-1\right\}$$ (4)

in which

$$\beta =1-\frac{E_1}{E_0}=1-\frac{G_1}{G_0}$$ (5)

Here, $E_1$ and $E_0$, respectively, are the elasticity modulus at $z=-h/2$ and $z=h/2$, and $\beta$ is the porosity distribution parameter which is variable as $0<\beta <1$. Also,$\hspace{0.17em}{G}_1$ and $G_0$, respectively, are the shear modulus at $z=-h/2$ and $z=h/2$, where the shear modulus depends on Young's modulus as $E=2G\left(1+\nu \right)$.

The linear constitutive equations (stress–strain relations) for the porous elastic functionally graded materials can be presented using Biot's model of the poroelasticity theory as follows [37]:

$${\mathrm{\sigma }}_{ij}=2\hspace{0.17em}G\left(z\right){\epsilon }_{ij}+\frac{2\hspace{0.17em}G\left(z\right)v_u}{1-2v_u}{\epsilon }_{kk}{\delta }_{ij}-\alpha \hspace{0.17em}{p}_0{\delta }_{ij}$$ (6)

in which

$$\begin{array}{c}{p}_0=M\hspace{0.17em}\left(\zeta -\mathrm{\alpha }\right),\hspace{0.17em}M=\frac{2G\left(z\right)\left(v_u-v\right)}{{\alpha }^2\left(1-2v_u\right)\left(1-2v\right)}\hspace{0.17em}\\ v_u=\frac{v+\alpha B(1-2v)/3}{1-\alpha B(1-2v)/3},\hspace{0.17em}\alpha =\frac{3\left(v_u-v\right)}{\left(v_u+1\right)\left(1-2v\right)}\end{array}$$ (7)

In the above equations, $p_0$ is the porous fluid pressure, $M$ is Biot's modulus, $\zeta$ is variation of fluid volume, and $\alpha$ is the coefficient of effective stress $0<\alpha <1$. Also, ${\epsilon }_{kk}$ is the volumetric strain and $v_u$ is undrained Poisson's ratio as $v<v_u<0.5$. Besides, $B$ is the Skempton coefficient and ${\delta }_{ij}$ is Kronecker's delta.

On the basis of undrained conditions ($\zeta =0$) and plane stress elasticity theory, two-dimensional constitutive equations of the saturated porous FG toroidal shell segments can be rewritten as [37]

$${\mathrm{\sigma }}_x=A_1{\epsilon }_x+B_1{\epsilon }_y,\hspace{0.17em}{\mathrm{\sigma }}_y=A_1{\epsilon }_y+B_1{\epsilon }_x,\hspace{0.17em}{\mathrm{\sigma }}_{xy}=G{\left(z\right)\hspace{0.17em}\gamma }_{xy}$$ (8)

in which

$$\begin{array}{c}{A}_1=2\hspace{0.17em}G\left(z\right)\left[1+\left(\frac{v_u}{1-2v_u}+\frac{v_u-v}{\left(1-2v_u\right)\left(1-2v\right)}\right)\left(1-\frac{c_2}{c_1}\right)\right]\\ B_1=2\hspace{0.17em}G\left(z\right)\left[\left(\frac{v_u}{1-2v_u}+\frac{v_u-v}{\left(1-2v_u\right)\left(1-2v\right)}\right)\left(1-\frac{c_2}{c_1}\right)\right]\hspace{0.17em}\end{array}$$ (9)

In the above equations, two known parameters $c_1$ and $c_2$ are defined as

$$\begin{array}{c}{c}_1=2\hspace{0.17em}\mathrm{G}\left(\mathrm{z}\right)\left[1+\frac{v_u}{1-2v_u}+\frac{v_u-v}{\left(1-2v_u\right)\left(1-2v\right)}\right]\hspace{0.17em}\\ c_2=2\hspace{0.17em}G\left(z\right)\left[\frac{v_u}{1-2v_u}+\frac{v_u-v}{\left(1-2v_u\right)\left(1-2v\right)}\right]\hspace{0.17em}\end{array}$$ (10)

The force resultants are written by integrating the stress–strain relations through the thickness as follows:

$$\begin{array}{c}{N}_x={\int }_{-h/2}^{h/2}{\sigma }_x\hspace{0.17em}dz=2A_{2\hspace{0.17em}}{\overline{\epsilon }}_x+B_2{\overline{\epsilon }}_y+A_3{K}_x+B_3{K}_y\hspace{0.17em}\\ N_y={\int }_{-h/2}^{h/2}{\sigma }_y\hspace{0.17em}dz=2A_2{\overline{\epsilon }}_y+B_2{\overline{\epsilon }}_x+A_3{K}_y+B_3{K}_x\hspace{0.17em}\\ N_{xy}={\int }_{-\frac{h}{2}}^{\frac{h}{2}}{\sigma }_{xy}\hspace{0.17em}dz=2C_2{\overline{\gamma }}_{xy}+C_3{K}_{xy}\hspace{0.17em}\end{array}$$ (11)

The moment resultants are obtained by integrating the moment of stresses through the thickness as

$$\begin{array}{c}{M}_x={\int }_{-h/2}^{h/2}{\sigma }_x\hspace{0.17em}z\hspace{0.17em}dz=A_3{\overline{\epsilon }}_x+B_3{\overline{\epsilon }}_y+2A_4{K}_x+B_4{K}_y\hspace{0.17em}\\ M_y={\int }_{-h/2}^{h/2}{\sigma }_y\hspace{0.17em}z\hspace{0.17em}dz=A_3{\overline{\epsilon }}_y+B_3{\overline{\epsilon }}_x+2A_4{\hspace{0.17em}K}_y+B_4{K}_x\hspace{0.17em}\\ M_{xy}={\int }_{-h/2}^{h/2}{\sigma }_{xy}\hspace{0.17em}z\hspace{0.17em}dz=\frac{C_3}{2}{\overline{\gamma }}_{xy}+C_4{K}_{xy}\hspace{0.17em}\end{array}$$ (12)

where in the above equations the following coefficients are used:

$$\begin{array}{c}{A}_2=\frac{1}{2}{\int }_{-h/2}^{h/2}{A}_1\hspace{0.17em}dz,\hspace{0.17em}{A}_3=\frac{1}{2}{\int }_{-h/2}^{h/2}2A_1\hspace{0.17em}z\hspace{0.17em}dz,\hspace{0.17em}{A}_4=\frac{1}{2}{\int }_{-h/2}^{h/2}{A}_1{z}^{2}dz\hspace{0.17em}\\ B_2=\frac{1}{2}{\int }_{-h/2}^{h/2}2B_1\hspace{0.17em}dz,\hspace{0.17em}{B}_3=\frac{1}{2}{\int }_{-h/2}^{h/2}2\hspace{0.17em}{B}_{1}z\hspace{0.17em}dz,\hspace{0.17em}{B}_4=\frac{1}{2}{\int }_{-h/2}^{h/2}2B_1{z}^{2}dz\hspace{0.17em}\\ C_2=\frac{1}{2}{\int }_{-h/2}^{h/2}G\left(z\right)dz,\hspace{0.17em}{C}_3=\frac{1}{2}{\int }_{-h/2}^{h/2}4G\left(z\right)z\hspace{0.17em}dz,\hspace{0.17em}{C}_4=\frac{1}{2}{\int }_{-h/2}^{h/2}4G\left(z\right)\hspace{0.17em}{z}^{2}dz\hspace{0.17em}\end{array}$$ (13)

4 Governing Equations

The nonlinear equilibrium and linear stability equations of the saturated porous FG toroidal shell segments are presented in this section. The governing equations of the porous shells under mechanical loads are derived by employing the energy methods [38]. At first, the nonlinear equations for an equilibrium position of the porous toroidal shell segments are obtained using the minimum potential energy principle. Then, from the equilibrium equations, the stability equations of the porous shells are derived using the adjacent equilibrium criterion.

The total potential energy $V$ of the porous toroidal shell segments under lateral pressure and axial compression loads is the sum of the strain energy $U$ and the potential energy $\Omega$ of the applied external loads.

The expression for the potential energy of the lateral pressure $p$ and axial compression $q$ is written as [38]

$$\mathrm{\Omega }=-\iint p\hspace{0.17em}w\hspace{0.17em}dx\hspace{0.17em}dy+\iint q\hspace{0.17em}u\hspace{0.17em}\mathit{dydz}$$ (14)

The strain energy of the toroidal shell made of saturated porous functionally graded material is given by

$$U=\frac{1}{2}\iint \int \left({\sigma }_x{\epsilon }_x+{\sigma }_y{\epsilon }_y+{\sigma }_{xy}{\gamma }_{xy}\right)\hspace{0.17em}dx\hspace{0.17em}dy\hspace{0.17em}dz$$ (15)

The expression for the total potential energy $V$ of the system is obtained from Eqs. (14) and (15) as follows:

$$\begin{array}{c}V=U+\mathrm{\Omega }=\iint \left[A_2\left({\overline{{\epsilon }_x}}^2+{\overline{{\epsilon }_y}}^2\right)+B_2\overline{{\epsilon }_x}\overline{{\epsilon }_y}+C_2{{\overline{\gamma }}_{xy}}^2\right]\hspace{0.17em}dx\hspace{0.17em}dy\\ +\iint \left[A_3\left(K_{x\hspace{0.17em}}\overline{{\epsilon }_x}+K_y\overline{{\epsilon }_y}\right)+B_3\left(K_y\overline{{\epsilon }_x}+K_x\overline{{\epsilon }_y}\right)+C_3{K}_{xy}{\overline{\gamma }}_{xy}\right]dx\hspace{0.17em}dy\\ +\iint \left[A_4\left({K_x}^2+{K_y}^2\right)+B_4{K}_{y\hspace{0.17em}}{K}_x+C_4{K_{xy}}^2\right]dx\hspace{0.17em}dy-\iint \left(p\hspace{0.17em}w+qh\hspace{0.17em}{u,}_x\right)\hspace{0.17em}dx\hspace{0.17em}dy\end{array}\hspace{0.17em}$$ (16)

The nonlinear equilibrium equations of the porous toroidal shell segments can be obtained using the minimum potential energy principle. Using the stress resultants introduced in Eqs. (11) and (12), and applying the variational method to Eq. (16), the equilibrium equations of the porous toroidal shell are established as

$$\begin{array}{c}{N}_{x,x}+N_{xy,y}=0\hspace{0.17em}\\ N_{y,y}+N_{xy,x}=0\hspace{0.17em}\\ M_{x,xx}+2M_{xy,xy}+M_{y,yy}\mp \frac{N_x}{r}-\frac{N_y}{b}+N_x{w}_{,xx}+2N_{xy}{w}_{,xy}+N_y{w}_{,yy}+p=0\hspace{0.17em}\end{array}$$ (17)

The linear stability equations of the porous toroidal shell segments under applied mechanical loads can be derived using the well-known adjacent equilibrium criterion. It is assumed that the primary equilibrium position of the porous toroidal shells is rewritten using the displacement components $u^0,v^0$, and $w^0$, and the arbitrary small increments of the displacement components are presented by $u^1,v^1$, and $w^1$. Therefore, the total displacement components of the neighboring configuration can be defined as follows [39]:

$$u=u^0+u^1,\hspace{0.17em}v=v^0+v^1,\hspace{0.17em}w=w^0+w^1$$ (18)

Also, the forces and moment resultants of porous toroidal shell segments can be rewritten as

$$\begin{array}{c}{N}_x=N_x^0+N_x^1,\hspace{0.17em}{M}_x=M_x^0+M_x^1\hspace{0.17em}\\ N_y=N_y^0+N_y^1,\hspace{0.17em}{M}_y=M_y^0+M_y^1\hspace{0.17em}\\ N_{xy}=N_{xy}^0+N_{xy}^1,\hspace{0.17em}{M}_{xy}=M_{xy}^0+M_{xy}^1\end{array}$$ (19)

in which the terms with superscripts $0$ are related to the displacements $u^0,v^0$, and $w^0$, and the terms with superscripts $1$ denote the increments of the force and moment resultants.

By substituting Eqs. (18) and (19) into Eq. (17) and neglecting the nonlinear higher-order terms, the linear stability equations of the porous toroidal shell segments are established as

$$\begin{array}{c}{N}_{x,x}^1+N_{xy,y}^1=0\hspace{0.17em}\\ N_{y,y}^1+N_{xy,x}^1=0\hspace{0.17em}\\ M_{x,xx}^1+2M_{xy,xy}^1+M_{y,yy}^1\mp \frac{N_x^1}{r}-\frac{N_y^1}{b}+N_x^0{w}_{,xx}^1+2N_{xy}^0{w}_{,xy}^1+N_y^0{w}_{,yy}^1=0\hspace{0.17em}\end{array}$$ (20)

The prebuckling force and moment resultants for the symmetric configuration of porous shells are defined as

$$\begin{array}{c}{N}_x^0=2A_2\left(u_{,x}^0\pm \frac{w^0}{r}\right)+B_2\left(\frac{w^0}{b}\right),\hspace{0.17em}{M}_x^0=A_3\left(u_{,x}^0\pm \frac{w^0}{r}\right)+B_3\left(\frac{w^0}{b}\right)\\ N_y^0=2A_2\left(\frac{w^0}{b}\right)+B_2\left(u_{,x}^0\pm \frac{w^0}{r}\right),\hspace{0.17em}{M}_y^0=A_3\left(\frac{w^0}{b}\right)+B_3\left(u_{,x}^0\pm \frac{w^0}{r}\right)\\ N_{xy}^0=2C_2{v}_{,x}^0,\hspace{0.17em}{M}_{xy}^0=\frac{1}{2}{C}_3{v}_{,x}^0\end{array}$$ (21)

The increments of force resultants of the porous toroidal shell segments can be written as follows:

$$\begin{array}{c}{N}_x^1=2A_2\left(u_{,x}^1\pm \frac{w^1}{r}\right)+B_2\left(v_{,y}^1+\frac{w^1}{b}\right)-A_3{w}_{,xx}^1-B_3{w}_{,yy}^1\hspace{0.17em}\\ N_y^1=2A_2\left(v_{,y}^1+\frac{w^1}{b}\right)+B_2\left(u_{,x}^1\pm \frac{w^1}{r}\right)-A_3{w}_{,yy}^1-B_3{w}_{,xx}^1\\ N_{xy}^1=2C_2\left(u_{,y}^1+v_{,x}^1\right)-C_3{w}_{,xy}^1\end{array}$$ (22)

The increments of moment resultants of the porous toroidal shells can be presented as

$$\begin{array}{c}{M}_x^1=A_3\left(u_{,x}^1\pm \frac{w^1}{r}\right)+B_3\left(v_{,y}^1+\frac{w^1}{b}\right)-2A_4{w}_{,xx}^1-B_4{w}_{,yy}^1\hspace{0.17em}\\ M_y^1=A_3\left(v_{,y}^1+\frac{w^1}{b}\right)+B_3\left(u_{,x}^1\pm \frac{w^1}{r}\right)-2A_4{w}_{,yy}^1-B_4{w}_{,xx}^1\\ M_{xy}^1=\frac{1}{2}{C}_3\left({u_{,y}^1+v}_{,x}^1\right)-C_4{w}_{,xy}^1\end{array}$$ (23)

The increments of linear strain components and curvatures at the middle surface of shells are defined as

$$\begin{array}{c}{\epsilon }_x^1=u_{,x}^1\pm \frac{w^1}{r},\hspace{0.17em}{\epsilon }_y^1=v_{,y}^1+\frac{w^1}{b},\hspace{0.17em}{\gamma }_{xy}^1=u_{,y}^1+v_{,x}^1\hspace{0.17em}\\ K_x^1=-w_{,xx}^1,\hspace{0.17em}{K}_y^1=-w_{,yy}^1,\hspace{0.17em}{K}_{xy}^1=-w_{,xy}^1\hspace{0.17em}\end{array}$$ (24)

The linear strain components at the middle surface of the porous toroidal shells introduced in Eq. (24) must satisfy the following compatibility equation [3]:

$${\epsilon }_{x,yy}^1-{\gamma }_{xy,xy}^1+{\epsilon }_{y,xx}^1=\pm \frac{1}{r}{w}_{,yy}^1+\frac{1}{b}{w}_{,xx}^1$$ (25)

By substituting Eq. (24) into Eq. (22), the increments of the force resultants can be rewritten as

$$\begin{array}{c}{N}_x^1=2A_2{\epsilon }_x^1+B_2{\epsilon }_y^1+A_3{K}_x^1+B_3{K}_y^1\hspace{0.17em}\\ N_y^1=2A_2{\epsilon }_y^1+B_2{\epsilon }_x^1+A_3{K}_y^1+B_3{K}_x^1\hspace{0.17em}\\ N_{xy}^1=2C_2{\gamma }_{xy}^1+C_3{K}_{xy}^1\end{array}$$ (26)

The reverse relations of the increments of force resultants presented in Eq. (26) can be derived as follows:

$$\begin{array}{c}{\epsilon }_x^1=\frac{1}{\Delta }\left\{2A_2{N}_x^1-B_2{N}_y^1-\left(2A_2{A}_3-B_2{B}_{3\hspace{0.17em}}\right){\hspace{0.17em}K}_x^1-\left(2A_2{B}_3-B_2{A}_{3\hspace{0.17em}}\right)K_y^1\right\}\hspace{0.17em}\\ {\epsilon }_y^1=\frac{1}{\Delta }\left\{2A_2{N}_y^1-B_2{N}_x^1-\left(2A_2{A}_3-B_2{B}_{3\hspace{0.17em}}\right)K_y^1{-\hspace{0.17em}\left(2A_2{B}_3-B_2{A}_{3\hspace{0.17em}}\right)\hspace{0.17em}K}_x^1\right\}\hspace{0.17em}\\ {\gamma }_{xy}^1=\frac{1}{2C_2}\left(N_{xy}^1-C_{3\hspace{0.17em}}{K}_{xy}^1\right)\end{array}$$ (27)

in which $\Delta \hspace{0.17em}={\left(2A_2\right)}^2-{\left(B_2\right)}^2$.

By substituting Eq. (27) into Eq. (23), the increments of the moment resultants are rewritten as

$$\begin{array}{c}{M}_x^1=\frac{1}{\Delta }\left\{\left(2A_2{A}_3-B_2{B}_3\right)N_x^1+(2A_2{B_3-B_2{A}_3)\hspace{0.17em}N}_y^1+2[{B_{2}A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta \hspace{0.17em}{A}_4]K_x^1+{\left[B_2\left({A_3}^2+{B_3}^2\right)-4A_2{A}_3{B}_3+\Delta \hspace{0.17em}{B}_4\right]\hspace{0.17em}K}_y^1\right\}\hspace{0.17em}\\ M_x^1=\frac{1}{\Delta }\left\{\left(2A_2{A}_3-B_2{B}_3\right)N_y^1+(2A_2{B_3-B_2{A}_3)\hspace{0.17em}N}_x^1+2[{B_{2}A}_3{B}_3-A_2({A_3}^2+{B_3}^2)+\Delta \hspace{0.17em}{A}_4]K_y^1+{\left[B_2({A_3}^2+{B_3}^2)-4A_2{A}_3{B}_3+\Delta \hspace{0.17em}{B}_4\right]\hspace{0.17em}K}_x^1\right\}\hspace{0.17em}\\ M_{xy}^1=\frac{C_3}{4C_2}{\hspace{0.17em}N}_{xy}^1+\left(C_4-\frac{{C_3}^2}{4C_2}\right)K_{xy}^1\hspace{0.17em}\end{array}$$ (28)

To simplify the stability equations, the first two of Eq. (20) can be satisfied when the increments of force resultants are represented using an Airy stress function $f(x,y)$ as follows [3]:

$$N_x^1={f,}_{yy},\hspace{0.17em}{N}_y^1={f,}_{xx},\hspace{0.17em}{N}_{xy}^1=-{f,}_{xy}$$ (29)

By substituting Eq. (29) into Eqs. (28) and (27), substituting Eq. (28) into the third of Eq. (20), and substituting Eq. (27) into Eq. (25), a new system of two partial differential equations can be obtained as

$${2A_{2}f}_{,\mathit{xxxx}}+\left(\frac{\Delta }{2C_2}-2B_2\right)f_{,\mathit{xxyy}}+2A_2{f}_{,\mathit{yyyy}}+\left(2A_2{B}_3-B_2{A}_3\right)w_{,\mathit{xxxx}}^1+\left(2A_2{B}_3-B_2{A}_3\right)w_{,\mathit{yyyy}}^1+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)w_{,\mathit{xxyy}}^1+\Delta \hspace{0.17em}\left(\pm \frac{w_{,yy}^1}{r}-\frac{w_{,xx}^1}{b}\right)=0\hspace{0.17em}$$ (30)

$$\left(2A_2{B}_3-B_2{A}_3\right)f_{,\mathit{xxxx}}+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)f_{,\mathit{xxyy}}+\left(2A_2{B}_3-B_2{A}_3\right)f_{,\mathit{yyyy}}-2\left[B_2{A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta {\hspace{0.17em}A}_4\right]w_{,\mathit{xxxx}}^1-2\hspace{0.17em}\left[B_2\left({A_3}^2+{B_3}^2\right)-4A_2{A}_3{B}_3+\Delta \left(B_4+C_4-\frac{{C_3}^2}{4C_2}\right)\right]w_{,\mathit{xxyy}}^1-{2\left[B_2{A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta {\hspace{0.17em}A}_4\right]\hspace{0.17em}w}_{,\mathit{yyyy}}+\Delta \left(\pm \frac{f_{,yy}}{r}-\frac{f_{,xx}}{b}+N_x^0{w}_{,xx}^1+2N_{xy}^0{w}_{,xy}^1+N_y^0{w}_{,yy}^1\right)=0\hspace{0.17em}$$ (31)

in which Eqs. (30) and (31) are two linear equations in terms of two dependent unknown functions $w^1$ and $f$. These are a set of coupled partial differential equations to study the stability of porous toroidal shell segments.

5 Analytical Solutions

Closed-form solutions of the stability equations can be obtained in this section using an analytical approach. The toroidal shell segments with both positive and negative Gaussian curvatures made of saturated porous material are considered. The cases of porous shells with free and simply supported two end edges subjected to three cases of mechanical loads are analyzed. In case 5.1, the lateral uniform pressure $\mathit{p}$ is distributed on the external surface of the shells. In case 5.2, an axial compressive load $\mathit{q}$ is distributed on the two end edges of the shell. In case 5.3, the hydrostatic pressure $\mathit{P}$ is uniformly distributed on the external surface and two edges of the shells.

The assumed boundary conditions for shallow segments of porous toroidal shells can be written as [3]

$$at\hspace{0.17em}:\hspace{0.17em}x=0,l\hspace{0.17em}:\hspace{0.17em}{w}^1=0,\hspace{0.17em}{w,}_{xx}^1=0,\hspace{0.17em}f=0,\hspace{0.17em}{f,}_{xx}=0$$ (32)

Boundary conditions Eq. (32) are satisfied when the buckling mode shape $w^1$ and stress function $f$ are represented by employing the following double Fourier-series as:

$$w^1=\sum \sum W_{mn}\text{sin}\frac{m\pi x}{l}\sin \frac{ny}{b},\hspace{0.17em}f=\sum \sum F_{mn}\text{sin}\frac{m\pi x}{l}\text{sin}\frac{ny}{b}$$ (33)

in which $m$ and $n$, respectively, are the numbers of half waves in the axial and circumferential directions.

Substituting Eq. (33) into Eqs. (30) and (31), and solving Eq. (30), one has

$$F_{mn}=\frac{\left(2A_2{B}_3-B_2{A}_3\right)\left(m^4+{\lambda }^4\right)+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)m^2{\lambda }^2+\frac{\Delta }{b}\frac{l^2}{{\pi }^2}(m^2\pm \frac{b}{r}{\lambda }^2)}{-2A_2\left(m^4+{\lambda }^4\right)-\left(\frac{\Delta }{2C_2}-2B_2\right)m^2{\lambda }^2}{W}_{mn}$$ (34)

in which $\lambda =(nl/\pi b)$.

Finally, substituting Eq. (34) into Eq. (31) yields

$$\left[\left(2A_2{B}_3-B_2{A}_3\right)\left(m^4+{\lambda }^4\right)+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)m^2{\lambda }^2+\frac{\Delta }{b}\frac{l^2}{{\pi }^2}\left(m^2\pm \frac{b}{r}{\lambda }^2\right)\right]\times \frac{\left(2A_2{B}_3-B_2{A}_3\right)\left(m^4+{\lambda }^4\right)+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)m^2{\lambda }^2+\frac{\Delta }{b}\frac{l^2}{{\pi }^2}\left(m^2\pm \frac{b}{r}{\lambda }^2\right)}{-2A_2\left(m^4+{\lambda }^4\right)-\left(\frac{\Delta }{2C_2}-2B_2\right)m^2{\lambda }^2}\hspace{0.17em}-2\left[B_2\left({A_3}^2+{B_3}^2\right)-4A_2{A}_3{B}_3+\Delta \left(B_4+C_4-\frac{{C_3}^2}{4C_2}\right)\right]m^2{\lambda }^2-\hspace{0.17em}\frac{\Delta l^2}{{\pi }^2}\left(N_x^0{m}^2-2N_{xy\hspace{0.17em}}^0\lambda m+N_y^0{\lambda }^2\right)-2\left[B_2{A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta {\hspace{0.17em}A}_4\right]\left(m^4+{\lambda }^4\right)=0$$ (35)

Equation (35) is an analytical solution for the system of partial differential Eqs. (30) and (31). In the next steps, the stability of porous toroidal shell segments is analyzed for three different types of mechanical loads.

Case 5.1: Buckling under lateral pressure $\mathit{p}$

The prebuckling force resultants of the porous toroidal shell segments under lateral uniform pressure $p$ are obtained by solving the membrane form of the equilibrium equations as follows:

$$N_x^0=N_{xy}^0=0,\hspace{0.17em}{N}_y^0=-pb$$ (36)

By substituting Eq. (36) into Eq. (35), the lateral pressure load of buckling is obtained as

$$\mathit{p}=\frac{{\left[\left(2A_2{B}_3-B_2{A}_3\right)\left(m^4+{\lambda }^4\right)+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)m^2{\lambda }^2+\frac{\Delta }{b}\frac{l^2}{{\pi }^2}\left(m^2\pm \frac{b}{r}{\lambda }^2\right)\right]}^2}{\mathrm{\Delta }\hspace{0.17em}b\frac{l^2}{{\pi }^2}\left[{2A}_2\left(m^4+{\lambda }^4\right){\lambda }^2+\left(\frac{\Delta }{2C_2}-2B_2\right)m^2{\lambda }^4\right]}+\frac{2{\pi }^2}{\Delta \hspace{0.17em}b{\hspace{0.17em}l}^2}\left\{\left[B_2{A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta {\hspace{0.17em}A}_4\right]\left(\frac{m^4+{\lambda }^4}{{\lambda }^2}\right)+\left[B_2\left({A_3}^2+{B_3}^2\right)-4A_2{A}_3{B}_3+\Delta \left(B_4+C_4-\frac{{C_3}^2}{4C_2}\right)\right]m^2\right\}$$ (37)

The critical lateral pressure load ($p_{\mathrm{cr}}$) of the porous toroidal shell segments can be derived from Eq. (37) when the equation is minimized with respect to allowable changes in the buckling mode parameters ($m,\hspace{0.17em}\lambda$).

Case 5.2: Buckling under axial compression $\mathit{q}$

The prebuckling force resultants of the porous toroidal shell segments under axial compression load $q$ are obtained by solving the membrane form of the equilibrium equations as follows:

$$N_y^0=N_{xy}^0=0,\hspace{0.17em}{N}_x^0=-qh$$ (38)

By substituting Eq. (38) into Eq. (35), the axial compression load of buckling is derived as follows:

$$\mathit{q}=\frac{{\left[\left(2A_2{B}_3-B_2{A}_3\right)\left(m^4+{\lambda }^4\right)+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{4C_2}\right)m^2{\lambda }^2+\frac{\Delta }{b}\frac{l^2}{{\pi }^2}\left(m^2\pm \frac{b}{r}{\lambda }^2\right)\right]}^2}{\mathrm{\Delta }h\frac{l^2}{{\pi }^2}\left[2A_2\left(m^4+{\lambda }^4\right)m^2+\left(\frac{\Delta }{2C_2}-2B_2\right)m^4{\lambda }^2\right]}\hspace{0.17em}+\frac{2{\pi }^2}{\Delta \hspace{0.17em}h\hspace{0.17em}{l}^2}\left\{\left[B_2{A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta {\hspace{0.17em}A}_4\right]\left(\frac{m^4+{\lambda }^4}{m^2}\right)+\left[B_2({A_3}^2+{B_3}^2)-4A_2{A}_3{B}_3+\Delta \left(B_4+C_4-\frac{{C_3}^2}{4C_2}\right)\right]{\lambda }^2\right\}$$ (39)

The critical axial compression load ($q_{\mathrm{cr}}$) of the porous toroidal shell segments can be derived from Eq. (39) when the equation is minimized with respect to allowable changes in the buckling mode parameters ($m,\hspace{0.17em}\lambda$).

Case 5.3: Buckling under hydrostatic pressure $\mathit{P}$

The prebuckling force resultants of the porous toroidal shell segments under uniform hydrostatic pressure $P$ are obtained by solving the membrane form of the equilibrium equations as follows:

$$N_{xy}^0=0,\hspace{0.17em}{N}_x^0=-\frac{Pb}{2},\hspace{0.17em}{N}_y^0=-Pb\left(1\mp \frac{b}{2r}\right)$$ (40)

Similarly, by substituting Eq. (40) into Eq. (35), the hydrostatic pressure load of buckling is derived as follows:

$$\mathit{P}=\frac{{\left[\left(2A_2{B}_3-B_2{A}_3\right)\left(m^4+{\lambda }^4\right)+2\left(2A_2{A}_3-B_2{B}_3-\frac{\Delta C_3}{{4C}_2}\right)m^2{\lambda }^2+\frac{\Delta }{b}\frac{l^2}{{\pi }^2}\left(m^2\pm \frac{b}{r}{\lambda }^2\right)\right]}^2}{\mathrm{\Delta }\hspace{0.17em}b\frac{l^2}{{\pi }^2}\left[\frac{m^2}{2}-{\lambda }^2\left(\pm \frac{b}{2r}-1\right)\right]\left[2A_2\left(m^4+{\lambda }^4\right)+\left(\frac{\Delta }{{2C}_2}-2B_2\right)m^2{\lambda }^2\right]}+\frac{2{\pi }^2}{\Delta \hspace{0.17em}b{l}^2\left[\frac{m^2}{2}+{\lambda }^2\left(1\mp \frac{b}{2r}\right)\right]}\left\{\left[B_2{A}_3{B}_3-A_2\left({A_3}^2+{B_3}^2\right)+\Delta \hspace{0.17em}{A}_4\right]\left(m^4+{\lambda }^4\right)+\left[B_2({A_3}^2+{B_3}^2)-4A_2{A}_3{B}_3+\Delta \left(B_4+C_4-\frac{{C_3}^2}{4C_2}\right)\right]m^2{\lambda }^2\right\}$$ (41)

The critical hydrostatic pressure load ($P_{\mathrm{cr}}$) of the porous toroidal shell segments can be derived from Eq. (41) when the equation is minimized with respect to allowable changes in the buckling mode parameters ($m,\hspace{0.17em}\lambda$).

6 Results and Discussions

In this section, the buckling behavior of the shallow segments of a toroidal shell made of saturated porous functionally graded material is studied. The critical buckling loads of the porous toroidal shell segments with both positive and negative Gaussian curvatures are investigated. The case of porous toroidal shells under lateral pressure, axial compression, and hydrostatic pressure loads is analyzed. The effects of shell thickness, radius of curvature, and porosity distribution parameter on the critical buckling loads of the shells are presented.

6.1 Comparison Study.

To demonstrate the present formulation and solution method, a comparison study is performed in this section. To this end, the analytical closed-form solutions of the homogeneous isotropic shells are compared with the results of Stein and McElman [2] and Hutchinson [3]. The case of isotropic toroidal shell segments with free and simply supported boundary conditions and positive/negative Gaussian curvature is studied. Three different types of mechanical loads such as lateral pressure, axial compression, and hydrostatic pressure are considered.

Using the obtained results from Stein and McElman [2] and Hutchinson [3], the closed-form solutions for the buckling loads of isotropic toroidal shell segments are expressed as follows [2,3]:

$$\mathit{p}=\frac{Eh}{b}\left\{\frac{{\pi }^2{h}^2{\left(m^2+{\lambda }^2\right)}^2}{12l^2\left(1-{\nu }^2\right){\lambda }^2}+\frac{l^2{\left[m^2\pm {\lambda }^2\left(b/r\right)\right]}^2}{{\pi }^2{b}^2{\lambda }^2{\left(m^2+{\lambda }^2\right)}^2}\right\}$$ (42)

$$\mathit{q}=E\left\{\frac{{\pi }^2{h}^2{\left(m^2+{\lambda }^2\right)}^2}{12l^2\left(1-{\nu }^2\right)m^2}+\frac{l^2{\left[m^2\pm {\lambda }^2\left(b/r\right)\right]}^2}{{\pi }^2{b}^2{m}^2{\left(m^2+{\lambda }^2\right)}^2}\right\}$$ (43)

$$\mathit{P}=\frac{2\hspace{0.17em}Eh}{b{m}^2+b{\lambda }^2\left[2\mp \left(b/r\right)\right]}\left\{\frac{{\pi }^2{h}^2{\left(m^2+{\lambda }^2\right)}^2}{12\hspace{0.17em}{l}^2\left(1-{\nu }^2\right)}+\frac{{l^2\left[m^2\pm {\lambda }^2\left(b/r\right)\right]}^2}{{\pi }^2{b}^2{\left(m^2+{\lambda }^2\right)}^2}\right\}$$ (44)

The above closed-form solutions can be derived from our results presented in Eqs. (37), (39), and (41), respectively, for the lateral pressure, axial compression, and hydrostatic pressure loads. By substituting ($\beta =0$) and (${\nu }_u=\mathrm{\nu }$) into Eqs. (37), (39), and (41), the explicit solutions (42), (43) and (44), respectively, are obtained.

Also, a comparison study is displayed in Fig. 2 to assure the validity and accuracy of the closed-form solutions obtained in this investigation. The critical buckling load curves of the present solutions are compared with those reported by Stein and McElman [2]. The isotropic toroidal shell segment with positive/negative Gaussian curvature under lateral pressure load is analyzed. For the sake of comparison, the dimensionless parameters $K=b{l}^2{p}_{cr}/D{\pi }^2$, $Z=l^2\sqrt{1-{\upsilon }^2}/bh$ and $D=E{h}^3/12(1-{\upsilon }^2)$ are used. The results depicted in Fig. 2 indicate that the current results are in excellent agreement with those of Stein and McElman [2].

Fig. 2.

Comparison between results of this study and those reported in Ref. [2] for isotropic shells under lateral pressure load

6.2 Parametric Studies.

The novel parametric studies are presented in this section for the buckling analysis of the shallow segments of the porous toroidal shell. The case of porous shells with nonuniform and nonsymmetric distributed porosity in the shell thickness is considered. Porous toroidal shells with both positive and negative Gaussian curvatures and simply supported end conditions are analyzed. Novel numerical results are obtained as the critical buckling load for the porous shells under various mechanical loads. The effects of the shell thickness, radius of curvature, and porosity parameter on the buckling behavior of the porous toroidal shell segments are investigated.

It is assumed that the saturated porous functionally graded materials (Tennessee marble) are used in this study as the constituents of the shells. The following numerical values are presented for the material properties [40]:

$$G_0=24\hspace{0.17em}\mathrm{Gpa},\hspace{0.17em}\mathrm{\nu }=0.25,\hspace{0.17em}{\nu }_u=0.27$$ (45)

Also, the relationship between the geometrical parameters of the toroidal shell segments is ($l=b=100h$).

The numerical results including the critical buckling loads of porous toroidal shell segments are presented here. Buckling loads of the porous shells subjected to lateral pressure, axial compression, and hydrostatic pressure are presented in Eqs. (37), (39), and (41), respectively. The critical buckling loads are obtained when these equations are minimized with respect to allowable changes in the buckling mode parameters ($m,\hspace{0.17em}n$).

The critical buckling load of the porous cylindrical shell (a simple case of toroidal shells) can be obtained from the results presented in this study. Toroidal shell segments with both positive and negative Gaussian curvatures can be reduced to a porous cylindrical shell when the radius of curvature $r$ is set equal to $\mathrm{\infty }$ or ($h/r=0$). Also, by decreasing the porosity parameter $\beta$, the porous toroidal shell segments will be close to the isotropic shells.

The critical buckling loads of the porous toroidal shell segments with both positive and negative Gaussian curvatures under three types of applied mechanical loads are depicted in Tables 1–3. The effects of porosity parameter $\beta$ and $h/r$ ratio on the critical buckling load of porous toroidal shell segments are investigated. The buckling mode parameters ($m,\hspace{0.17em}n$) associated with the critical buckling load of porous shells are also presented. The convex/concave porous toroidal shell segment with $h/r=0,\hspace{0.17em}0.002,\hspace{0.17em}0.004,\hspace{0.17em}0.006,\hspace{0.17em}0.008,\hspace{0.17em}0.01$ is considered. Also, different porosity parameters are assumed for the porous shells which are $\beta =0,\hspace{0.17em}0.2,\hspace{0.17em}0.4,\hspace{0.17em}0.6,\hspace{0.17em}0.8,\hspace{0.17em}1$.

Table 1.

The critical buckling load $p_{\mathrm{cr}}\left(\mathrm{Mpa}\right)$ of porous toroidal shell segments under lateral pressure load

	$h\hspace{0.17em}/\hspace{0.17em}r$	$(m,\hspace{0.17em}n)$	$\beta =0.0$	$\beta =0.2$	$\beta =0.4$	$\beta =0.6$	$\beta =0.8$	$\beta =1.0$
$r>0$	$0.000$	$(1,\hspace{0.17em}8)$	$0.6461$	$0.5660$	$0.4804$	$0.3860$	$0.2759$	$0.1334$
	$0.002$	$(1,\hspace{0.17em}10)$	$1.1233$	$0.9834$	$0.8358$	$0.6758$	$0.4936$	$0.2661$
	$0.004$	$(2,\hspace{0.17em}10)$	$1.3218$	$1.1597$	$0.9849$	$0.7895$	$0.5576$	$0.2706$
	$0.006$	$(2,\hspace{0.17em}10)$	$1.4523$	$1.2733$	$1.0816$	$0.8693$	$0.6204$	$0.2965$
	$0.008$	$(2,\hspace{0.17em}10)$	$1.6120$	$1.4123$	$1.2000$	$0.9671$	$0.6976$	$0.3530$
	$0.010$	$(2,\hspace{0.17em}10)$	$1.8008$	$1.5768$	$1.3401$	$1.0829$	$0.7890$	$0.4204$
$r<0$	$0.000$	$(1,\hspace{0.17em}8)$	$0.6461$	$0.5660$	$0.4804$	$0.3860$	$0.2759$	$0.1334$
	$0.002$	$(1,\hspace{0.17em}6)$	$0.3849$	$0.3374$	$0.2860$	$0.2285$	$0.1602$	$0.0694$
	$0.004$	$(1,\hspace{0.17em}5)$	$0.2732$	$0.2403$	$0.2043$	$0.1632$	$0.1129$	$0.0439$
	$0.006$	$(1,\hspace{0.17em}4)$	$0.2391$	$0.2100$	$0.1781$	$0.1418$	$0.0976$	$0.0373$
	$0.008$	$(1,\hspace{0.17em}4)$	$0.7582$	$0.6662$	$0.5715$	$0.4723$	$0.3653$	$0.2421$
	$0.010$	$(1,\hspace{0.17em}3)$	$0.4045$	$0.3526$	$0.2980$	$0.2393$	$0.1731$	$0.0916$

Table 3.

The critical buckling load $P_{\mathrm{cr}}\left(\mathrm{Mpa}\right)$ of porous toroidal shell segments under hydrostatic pressure load

	$h\hspace{0.17em}/\hspace{0.17em}r$	$(m,\hspace{0.17em}n)$	$\beta =0.0$	$\beta =0.2$	$\beta =0.4$	$\beta =0.6$	$\beta =0.8$	$\beta =1.0$
$r>0$	$0.000$	$(1,\hspace{0.17em}8)$	$0.5999$	$0.5255$	$0.4461$	$0.3584$	$0.2561$	$0.1239$
	$0.002$	$(2,\hspace{0.17em}9)$	$1.0624$	$0.9322$	$0.7915$	$0.6341$	$0.4472$	$0.1996$
	$0.004$	$(2,\hspace{0.17em}9)$	$1.2985$	$1.1382$	$0.9665$	$0.7767$	$0.5544$	$0.2657$
	$0.006$	$(2,\hspace{0.17em}10)$	$1.6184$	$1.4189$	$1.2052$	$0.9687$	$0.6914$	$0.3304$
	$0.008$	$(2,\hspace{0.17em}10)$	$2.0216$	$1.7712$	$1.5049$	$1.2128$	$0.8748$	$0.4427$
	$0.010$	$(3,\hspace{0.17em}10)$	$2.3995$	$2.1069$	$1.7893$	$1.4315$	$1.0020$	$0.4254$
$r<0$	$0.000$	$(1,\hspace{0.17em}8)$	$0.5999$	$0.5255$	$0.4461$	$0.3584$	$0.2561$	$0.1239$
	$0.002$	$(1,\hspace{0.17em}6)$	$0.3112$	$0.2727$	$0.2312$	$0.1847$	$0.1259$	$0.0561$
	$0.004$	$(1,\hspace{0.17em}5)$	$0.1955$	$0.1720$	$0.1462$	$0.1168$	$0.0808$	$0.0314$
	$0.006$	$(1,\hspace{0.17em}4)$	$0.1487$	$0.1306$	$0.1107$	$0.0882$	$0.0607$	$0.0232$
	$0.008$	$(1,\hspace{0.17em}4)$	$0.4438$	$0.3900$	$0.3345$	$0.2765$	$0.2138$	$0.1417$
	$0.010$	$(1,\hspace{0.17em}3)$	$0.1974$	$0.1721$	$0.1455$	$0.1168$	$0.0845$	$0.0447$

Table 1 displays the critical buckling loads for the convex segment $r>0$ and concave segment $r<0$ of the porous toroidal shell under uniform lateral pressure load. From the numerical results of this study, it is seen that the critical buckling load for both segments of the porous toroidal shell decreases as the porosity parameter $\beta$ increases. But, the effect of $h/r$ ratio on the buckling behavior of the toroidal shell segments with positive and negative Gaussian curvatures is not the same. As seen from the results of Table 1, the critical buckling load of the convex segment of the porous toroidal shell increases with increasing the $h/r$ ratio. But, the critical buckling load of the concave toroidal shell segment has no stable tendency under variations of the $h/r$ ratio.

Table 2 shows the effects of porosity parameter $\beta$ and thickness-to-radius ratio $h/r$ on the critical buckling load of the porous toroidal shell segments under axial compression load. The case of porous toroidal shell segments with both positive and negative Gaussian curvatures is considered in this investigation. As seen from the numerical results of Table 2, the critical buckling load for both convex and concave segments of the porous toroidal shell decreases when the porosity distribution parameter $\beta$ increases.

Table 2.

The critical buckling load $q_{\mathrm{cr}}\left(\mathrm{Mpa}\right)$ of porous toroidal shell segments under axial compression load

	$h\hspace{0.17em}/\hspace{0.17em}r$	$(m,\hspace{0.17em}n)$	$\beta =0.0$	$\beta =0.2$	$\beta =0.4$	$\beta =0.6$	$\beta =0.8$	$\beta =1.0$
$r>0$	$0.000$	$(3,\hspace{0.17em}7)$	$146.7650$	$128.7519$	$109.3340$	$87.6431$	$61.9096$	$27.8844$
	$0.002$	$(3,\hspace{0.17em}7)$	$153.0015$	$134.1793$	$113.9522$	$91.4519$	$64.9092$	$30.0748$
	$0.004$	$(3,\hspace{0.17em}7)$	$159.8902$	$140.1757$	$119.0563$	$95.6638$	$68.2289$	$32.5022$
	$0.006$	$(3,\hspace{0.17em}7)$	$167.4309$	$146.7411$	$124.6464$	$100.2787$	$71.8685$	$35.1665$
	$0.008$	$(4,\hspace{0.17em}6)$	$173.0863$	$151.8348$	$129.0463$	$103.7613$	$74.0458$	$35.2483$
	$0.010$	$(4,\hspace{0.17em}6)$	$176.1906$	$154.5390$	$131.3478$	$105.6653$	$75.5524$	$36.3521$
$r<0$	$0.000$	$(3,\hspace{0.17em}7)$	$146.7650$	$128.7519$	$109.3340$	$87.6431$	$61.9096$	$27.8844$
	$0.002$	$(3,\hspace{0.17em}6)$	$139.5646$	$122.2933$	$103.8754$	$83.6021$	$60.0292$	$29.6883$
	$0.004$	$(2,\hspace{0.17em}6)$	$121.7472$	$106.5499$	$90.4013$	$72.7136$	$52.2886$	$26.2490$
	$0.006$	$(2,\hspace{0.17em}6)$	$101.6905$	$89.1164$	$75.5911$	$60.5266$	$42.7248$	$19.3085$
	$0.008$	$(2,\hspace{0.17em}6)$	$88.3577$	$77.5508$	$65.7926$	$52.4952$	$36.4605$	$14.8113$
	$0.010$	$(2,\hspace{0.17em}6)$	$81.7489$	$71.8529$	$61.0057$	$48.6193$	$33.4957$	$12.7575$

The effects of porosity parameter $\beta$ and $h/r$ ratio on the critical buckling load of the porous toroidal shell segments under hydrostatic pressure load are shown in Table 3. The numerical results of this investigation are presented for the porous toroidal shell segments with both positive and negative Gaussian curvatures. It is seen that the critical buckling load of the porous shells decreases with an increase in the porosity parameter $\beta$. Also, the critical buckling loads of a convex porous shell are higher than those of the concave porous shell.

Figure 3 shows the effect of porosity parameter $\beta$ on Young's modulus variations through the thickness direction of the porous toroidal shell. The numerical results are presented in Fig. 3 as the dimensionless Young's modulus $E\left(z\right)/E_0$ versus the thickness parameter $z/h$. It is seen that the Young's modulus variations through the shell thickness can be uniform when the porosity distribution parameter is set equal to $\beta =0$.

Fig. 3.

The effect of porosity distribution parameter $\beta$ on the Young's modulus variations through the shell thickness

To better understand the effects of $h/r$ ratio and porosity parameter $\beta$ on the critical buckling load of the saturated porous toroidal shell segments, the novel parametric studies are shown in Figs. 4–9. The numerical results are displayed in Figs. 4–9 as the critical buckling loads versus porosity distribution change. The porous toroidal shell segments with both positive and negative Gaussian curvatures subjected to lateral pressure, axial compression, and hydrostatic pressure loads are analyzed in these figures.

Fig. 4.

The critical buckling load curves of a convex porous toroidal shell segment $r>0$ under lateral pressure load

Fig. 9.

The critical buckling load curves of a concave porous toroidal shell segment $r<0$ under hydrostatic pressure load

Figure 4 shows the critical buckling load curves of a porous convex toroidal shell segment $r>0$ subjected to uniform lateral pressure load. The results of this investigation are displayed in Fig. 4 as the critical buckling loads verses porosity distribution parameter $\beta$ for different $h/r$ ratios. Moreover, in Fig. 5 the critical buckling load curves for a porous concave toroidal shell segment $r<0$ under lateral pressure are presented. Figure 6 illustrates the critical buckling load curves of the porous convex toroidal shell segment $r>0$ under axial compression load. For different values of $h/r$ ratio, the critical buckling loads verses porosity distribution changes are shown in this figure. Moreover, the critical buckling load curves are depicted in Fig. 7 for the porous concave toroidal shell segment $r<0$ under axial compression load. Figure 8 shows the effects of $h/r$ ratio and porosity distribution parameter $\beta$ on the critical buckling load curves of a porous convex toroidal shell segment $r>0$ under hydrostatic pressure load. Finally, the critical buckling load curves of a porous concave toroidal shell segment $r<0$ under hydrostatic pressure load are presented in Fig. 9.

Fig. 5.

The critical buckling load curves of a concave porous toroidal shell segment $r<0$ under lateral pressure load

Fig. 6.

The critical buckling load curves of a convex porous toroidal shell segment $r>0$ under axial compression load

Fig. 7.

The critical buckling load curves of a concave porous toroidal shell segment $r<0$ under axial compression load

Fig. 8.

The critical buckling load curves of a convex porous toroidal shell segment $r>0$ under hydrostatic pressure load

Figures 4–9 show the effects of porosity parameter $\beta$ and thickness-to-radius ratio $h/r$ on the critical buckling load curves of porous toroidal shell segments under lateral pressure, axial compression, and hydrostatic pressure. Both positive and negative Gaussian curvatures for the shallow segments of the porous toroidal shell are considered in these figures. As seen from these investigations, with increasing the coefficient of porosity $\beta$ the critical buckling load curves of the porous convex/concave toroidal shell segment decrease significantly. Also, it is seen that the critical buckling load of porous toroidal shell segments will be close to the homogeneous isotropic toroidal shells by decreasing the porosity distribution parameter $\beta$.

The critical buckling loads and stability of the porous toroidal shell segment with positive Gaussian curvature increase with increasing the thickness-to-radius ratio $h/r$. But, a porous toroidal shell segment with negative Gaussian curvature has no stable tendency under variations of the $h/r$ ratio. In particular in Fig. 5 when the $h/r$ ratio increases from $0$ to $0.006$, critical buckling loads of the porous toroidal shell segment with negative Gaussian curvature decrease, but it increases at the $h/r=0.008.$ Also, it is seen that the critical buckling loads of a porous convex toroidal shell segment are higher than those of a concave porous toroidal shell segment.

7 Conclusion

In this research, the effects of porosity distribution parameter $\beta$ and thickness-to-radius ratio $h/r$ on the critical buckling load of the porous toroidal shell segments are studied. The shallow segments of the toroidal shell made of saturated porous FG material with nonuniform distributed porosity in the shell thickness are considered. The porous toroidal shell segments with both positive and negative Gaussian curvatures and simply supported boundary conditions are considered in this study. Porous shells under lateral pressure, axial compression, and hydrostatic pressure loads are analytically analyzed. The energy method, as the minimum potential energy principle, is used to derive the governing equations of the porous toroidal shell segments. The nonlinear equilibrium equations of the system are obtained on the basis of classical thin shell theory, Biot's model of the poroelasticity, and the von Karman type of kinematic assumptions. The linear stability equations of porous toroidal shell segments are derived from the nonlinear equilibrium equations by using the adjacent equilibrium criterion. The stability equations are established as a system of the coupled partial differential equations. These equations are solved using an analytical method including the Airy stress function. The simply supported boundary conditions are satisfied by employing the general solutions in the form of double Fourier-series. The resulting closed-form solutions in the cases of lateral pressure, axial compression, and hydrostatic pressure loads are presented in this research to study the elastic stability of porous toroidal shell segments.

From the numerical results of this investigation, it is concluded that:

1. The critical buckling loads and stability of the porous toroidal shell segments with both positive and negative Gaussian curvatures decrease with increasing the porosity parameter $\beta$.

2. The critical buckling load of the porous toroidal shell segment with positive Gaussian curvature $r>0$ is higher than that of the negative Gaussian curvature $r<0$.

3. For a porous toroidal shell segment with positive Gaussian curvature $r>0$, the critical buckling load and stability increase with increasing the thickness-to-radius ratio $h/r$.

4. For a porous toroidal shell segment with negative Gaussian curvature $r<0$, the critical buckling load has no stable tendency under variations of the $h/r$ ratio.

5. By decreasing the porosity distribution parameter $\beta$, the critical buckling load of porous toroidal shell segments will be close to the homogeneous isotropic shells.