Mechanical Behavior of Hollow Corrugated Sandwich Cylinders Under Inner Pressure Loading

Abstract

Taking pressure-bearing equipment as a prototype, the mechanical behavior of hollow corrugated sandwich cylinders under inner pressure loading was investigated. Considering the hollow cylindrical shell with finite length, the elastic closed-form solutions of hollow corrugated sandwich cylinders under inner pressure loading were presented. The optimization for minimum weight was performed. The stress distribution characteristics of the structures with different topological parameters were discussed. It can be found that an inflection point of the structure loading efficiency exists during an increase in the pressure, providing that under lower inner pressure loading, the loading efficiency of hollow corrugated sandwich structures is not invariably superior to homogeneous structures, yet while subjected to higher inner pressures, the sandwich structures exhibit a significant advantage in load-bearing efficiency.

1. Introduction

Pressure-bearing equipment, including pressured vessels, pressured pipes, pressured chambers, et al., are widely applied in aviation, national defense, the chemical industry, foodstuff, and transportation. Pressure-bearing equipment are key devices for bearing inner pressure as a basic load, facing the threat of leaking and exploding [1]. Lightweight, security, and reliability are key requirements for the design of pressure-bearing equipment. In recent years, with the development of material manufacturing and processing technology, lightweight porous metal sandwich plates have been widely studied in structure-bearing, active cooling, sound absorption, noise lowering, explosion and ballistic impact resistance, and structure actuation, due to their excellent structural efficiency and multifunctional application prospects [2,3,4]. Due to the high void ratio and good designability, hollow corrugated sandwich cylinder structures can be used as pressure-bearing equipment instead of homogeneous metal structures [5]. It is promising to reduce the weight while achieving multifunctional effects such as structural support, crashworthiness, energy absorption, et al. [6,7].

Investigation of the behavior of long hollow cylinders with infinite length under inner or outer pressure loading is one of the classical problems in engineering mechanics. A thick-walled hollow cylinder with an isotropic base material was treated comprehensively in a purely elastic stress state by Timoshenko [8,9], in a fully plastic stress state by Boressi et al. [10] and Mendelson [11], and in an elasto-plastic stress state by Parker [12] and Perry and Aboudi [13]. For pressurized hollow cylinders with anisotropic materials, Lekhniëtìskiæi [14] and Luo and Li [15] presented the elastic closed-form solutions. Khalili et al. [16] analyzed the response of multilayer composite circular cylindrical shells under different loads based on first-order shear deformation theory. Rizzetto et al. [17] numerically investigated the nonlinear dynamic stability of isotropic and composite cylindrical shells under axial loading. Semenyuk et al. [18] proposed a generalized design model to analyze the stability and initial post-buckling equilibrium trajectory of sandwich cylindrical shells with elastic cores. For cylinder shells with fiber-reinforced composite materials, Kardomateas [19] presented the benchmark solutions to the problem of buckling of orthotropic cylindrical shells under external pressure or axial compression loading. On that basis, Kardomateas [20,21] constructed the elasticity solution for a cylindrical sandwich shell under different loadings. Liu et al. [22] performed the optimum weight design of hollow sandwich cylinders with ultralightweight cellular cores, considering five different core topologies, i.e., Kagome truss, single-layered pyramidal truss, double-layered pyramidal truss, single-layered corrugated core, and double-layered corrugated core, respectively. Compared to hollow cylinders having solid walls, truss-core sandwich cylinders and single-layer corrugated core sandwich cylinders exhibit superior lightweight advantages, particularly in scenarios involving heavier loads. Taking the engine combustor of an aerospace vehicle as a prototype, Liu et al. [23] conducted further research on the bi-functional optimization of actively cooled, pressurized hollow sandwich cylinders with prismatic cores.

In conclusion, research on the structural efficiency of sandwich structures with cellular material cores mainly focuses on the sandwich plate under different loading. Few research studies were related to the optimization of sandwich cylinders under inner pressure loading. Existing research only analyzed the situation of that structure subjected to high-level inner pressure. However, during the investigation of the lightweight design of structures with cellular materials, it can be found that under lower inner pressure, the loading efficiency of a hollow corrugated sandwich cylinder does not surpass that of a homogeneous plate, and its inner diameter and thickness are limited. Consequently, we presume that the loading efficiency of a pressurized hollow corrugated sandwich cylinder may have an inflection when the pressure loading increases. This hypothesis motivates the research presented in this paper.

In this research, the mechanical behavior of hollow corrugated sandwich cylinders under inner pressure loading was investigated. Taking small inner pressure load-bearing equipment as a prototype, the mechanics behavior of hollow corrugated sandwich cylinder shells under inner pressure loading was analyzed. In Section 2, the representative structure and its topological parameters of the corrugated cell were introduced, and the expression of structure mass was given. In Section 3, the elastic closed-form solution of the pressurized hollow corrugated sandwich cylinder shell was constructed, which was verified by finite element analysis (FEA). In Section 4, the optimum design model for minimum weight is set up. In Section 5, the stress distribution and the physical mechanism of the occurrence of the inflection were discussed. The results of this research can guide the application and optimization of sandwich structures in pressure-bearing equipment.

2. Description of Problems

2.1. Structure Model

Considering the reduction in weight, a hollow corrugated sandwich cylinder shell was proposed, of which the topological configuration is shown in Figure 1.

As illustrated in Figure 1b, L is the axial length of the hollow corrugated sandwich cylinder, p is half the width of the cell; $t_0$ and $t_i$ are the thickness of the top and nether panels, respectively; $h_c$ is the thickness of the corrugated sandwich structure; $t_w$ is the wall thickness of the trapezoid corrugated sandwich; $\alpha$ is the incline angle of the trapezoid sandwich, and f is the trapezoid width. It was assumed that the corrugated panel and the facesheets are welded together with no slip between them.

2.2. Loading and Mass Expression

Considering a long, hollow, corrugated sandwich cylinder subjected to a uniform internal pressure $P_i$, of which the cross profile is shown in Figure 2a. Assume that the geometry axis is coincident with the physics axis, i.e., the Z-axis of the cylindrical coordinate system.

Figure 2b is the topological configuration of the hollow corrugated sandwich cylinder. $R_0$ and $R_i$ are the inner radius of the inner and outer facesheets, respectively; $t_0$ and $t_i$ are the thicknesses of the inner and outer facesheets, respectively; $t_w$ is the thickness of the trapezoidal corrugated core; and $l_w$ is the length of the incline cell. α_i, α_o, β, and θ are central angles related to the structure of the corrugation. In contrast to the flat panel construction, the description of the topological parameters of the corrugated unit undergoes minor alterations due to the curvature. In a flat plate, the trapezoid width f (shown in Figure 1b) is constant, while in a cylinder shell, the trapezoid widths of the inside and outside facesheets are difficult to remain constant, which are expressed by f_o and f_i in Figure 2b. The mass of the outer facesheet, the inner facesheet, and the core can be calculated as follows:

$$\left\{\begin{array}{l}{m}_{outer\_face}={\rho }_s\pi \left(2R_o+t_o\right)t_o\hfill \\ m_{inner\_face}={\rho }_s\pi \left(2R_i+t_i\right)t_i\hfill \\ m_{core}={\rho }_s\left(2l_w{t}_w+f_o{t}_w+f_i{t}_w\right)n_{core}\hfill \end{array}\right.,$$

(1)

where ρ_s is the density of steel, n_core is the number of the corrugations, and

$$l_w=\sqrt{R_o^2+{\left(R_i+t_i\right)}^2-2R_o\left(R_i+t_i\right)\cos \beta },$$

(2)

$$\beta ={\displaystyle \frac{\theta }{2}}-{\alpha }_o-{\alpha }_i,$$

(3)

$${\alpha }_o={\displaystyle \frac{f_o}{2R_o}},$$

(4)

$${\alpha }_i={\displaystyle \frac{f_i}{2\left(R_i+t_i\right)}},$$

(5)

$$\theta ={\displaystyle \frac{2\pi }{n_{core}}}.$$

(6)

Therefore, the mass per unit length of the hollow corrugated sandwich cylinder can be expressed as

$$\begin{array}{ll}m& =m_{outer\_face}+m_{inner\_face}+m_{core}\\ & ={\rho }_s\pi \left(2R_o+t_o\right)t_o+{\rho }_s\pi \left(2R_i+t_i\right)t_i+{\rho }_s\left(2l_w{t}_w+f_o{t}_w+f_i{t}_w\right)n_{core}\\ & ={\rho }_s\left\{\pi \left[\left(2R_o+t_o\right)t_o+\left(2R_i+t_i\right)t_i\right]+n_{core}{t}_w\left(2l_w+f_o+f_i\right)\right\}\end{array}.$$

(7)

3. Structural Analysis

As shown in Figure 3, assuming $P_a$ and $P_b$ are the inner and outer pressure on the sandwich exerted through the facesheets. Thus, the corrugated panel is synchronously subjected to the inner pressure $P_a$ and the outer pressure $P_b$; the force on the outer facesheet is only inner pressure $P_b$, while the force on the inner facesheet is only outer pressure $P_a$.

3.1. Closed-Form Solution for Corrugated Core Panel

For a long, hollow sandwich cylinder with no column caps, the problem can be treated as plane axisymmetric. Based on the elastic solution of the anisotropic hollow cylinder under inner and outer pressure, the closed-form stress solution of the sandwich can be obtained [15].

$$\left\{\begin{array}{l}\begin{array}{ll}{\sigma }_r^c& ={\displaystyle \frac{P_a{c}^{k+1}-P_b}{1-c^{2k}}}{\rho }^{k-1}+{\displaystyle \frac{P_b{c}^{k-1}-P_a}{1-c^{2k}}}{c}^{k+1}{\rho }^{-k-1}\\ & +B_3{\eta }_1\left(1+{\displaystyle \frac{1-c^{k+1}}{c^{2k}}}{\rho }^{k-1}+{\displaystyle \frac{1-c^{k+1}}{1-c^{2k}}}{c}^{k+1}{\rho }^{-k-1}\right)\\ & +\vartheta {\mu }_1{R}_0\left(2\rho -{\displaystyle \frac{1-c^{k+1}}{1-c^{2k}}}{\rho }^{k-1}+{\displaystyle \frac{1-c^{k-2}}{1-c^{2k}}}{c}^{k+1}{\rho }^{-k-1}\right)\end{array}\hfill \\ \begin{array}{ll}{\sigma }_{\theta }^c& ={\displaystyle \frac{P_a{c}^{k+1}-P_b}{1-c^{2k}}}k{\rho }^{k-1}-{\displaystyle \frac{P_b{c}^{k-1}-P_a}{1-c^{2k}}}k{c}^{k+1}{\rho }^{-k-1}\\ & +B_3{\eta }_1\left(1+{\displaystyle \frac{1-c^{k+1}}{c^{2k}}}k{\rho }^{k-1}+{\displaystyle \frac{1-c^{k+1}}{1-c^{2k}}}{c}^{k+1}{\rho }^{-k-1}\right)\\ & +\vartheta {\mu }_1{R}_0\left(2\rho -{\displaystyle \frac{1-c^{k+2}}{1-c^{2k}}}k{\rho }^{k-1}+{\displaystyle \frac{1-c^{k-2}}{1-c^{2k}}}{c}^{k+2}{\rho }^{-k-1}\right)\end{array}\hfill \\ \begin{array}{lll}{\sigma }_z^c& =& -{\displaystyle \frac{1}{s_{33}}}\left(s_{13}{\sigma }_{RR}^c+s_{23}{\sigma }_{\theta \theta }^c+s_{34}{\tau }_{\theta Z}^c\right)\\ & =& -{\displaystyle \frac{1}{s_{33}}}\left[{\displaystyle \frac{P_a{c}^{k+1}-P_b}{1-c^{2k}}}\left(s_{13}+k{s}_{23}\right){\rho }^{k-1}+{\displaystyle \frac{P_b{c}^{k-1}-P_a}{1-c^{2k}}}\left(s_{13}-k{s}_{23}\right)c^{k+1}{\rho }^{-k-1}\right]\\ & & +B_3\left[-{\eta }_2+{\eta }_1\left(1+{\displaystyle \frac{1-c^{k+1}}{c^{2k}}}{g}_{k1}{\rho }^{k-1}+{\displaystyle \frac{1-c^{k-1}}{1-c^{2k}}}{g}_{k2}{c}^{k+1}{\rho }^{-k-1}\right)\right]\\ & & +\vartheta R_0\left({\mu }_2\rho +{\mu }_1{\displaystyle \frac{1-c^{k+2}}{1-c^{2k}}}{g}_{k1}{\rho }^{k-1}+{\displaystyle \frac{1-c^{k-2}}{1-c^{2k}}}{g}_{k2}{c}^{k+2}{\rho }^{-k-1}\right)\end{array}\hfill \end{array}\right.,$$

(8)

where

$${\mu }_1={\displaystyle \frac{{\beta }_{14}-2{\beta }_{24}}{4({\beta }_{22}{\beta }_{44}-{\beta }_{24}^2)-({\beta }_{11}{\beta }_{44}-{\beta }_{14}^2)}},$$

(9)

$${\mu }_2={\displaystyle \frac{{\beta }_{11}-4{\beta }_{22}}{4({\beta }_{22}{\beta }_{44}-{\beta }_{24}^2)-({\beta }_{11}{\beta }_{44}-{\beta }_{14}^2)}},$$

(10)

$${\eta }_1={\displaystyle \frac{(s_{13}-s_{23}){\beta }_{44}-s_{34}({\beta }_{14}-{\beta }_{24})}{{\beta }_{22}{\beta }_{44}-{\beta }_{24}^2-({\beta }_{11}{\beta }_{44}-{\beta }_{14}^2)}},$$

(11)

$${\eta }_2={\displaystyle \frac{(s_{13}-s_{23})({\beta }_{14}+{\beta }_{24})-s_{34}({\beta }_{11}-{\beta }_{22})}{{\beta }_{22}{\beta }_{44}-{\beta }_{24}^2-({\beta }_{11}{\beta }_{44}-{\beta }_{14}^2)}},$$

(12)

$$k=\sqrt{{\displaystyle \frac{{\beta }_{44}{\beta }_{11}-{\beta }_{14}^2}{{\beta }_{22}{\beta }_{44}-{\beta }_{24}^2}}},$$

(13)

$$g_{k1}={\displaystyle \frac{{\beta }_{14}+k{\beta }_{24}}{{\beta }_{44}}},$$

(14)

$$g_{k2}={\displaystyle \frac{{\beta }_{14}-k{\beta }_{24}}{{\beta }_{44}}},$$

(15)

$$c={\displaystyle \frac{R_i+T_i}{R_o}},$$

(16)

$${\beta }_{ij}=s_{ij}-{\displaystyle \frac{s_{i3}{s}_{j3}}{s_{33}}},(i,j=1,\dots ,6),$$

(17)

$$\rho ={\displaystyle \frac{R}{R_o-T_o/2}},R\in [R_i+T_i,R_o],$$

(18)

where s₁₁, s₁₂, s₂₂, s₁₃, and s₂₃ are the homogenized effective compliance tensors of the core. ${\sigma }_{RR}^c$, ${\sigma }_{\theta \theta }^c$, and ${\tau }_{\theta Z}^c$ are stresses in an isotropic core. c, k, and ρ are constants.

The stress and displacement are independent of the coordinate Z, thus the boundary condition of both ends can be treated as fixed [20], which is as follows:

$$B_3=\vartheta =0.$$

(19)

Thus, the closed-form stress solution of the sandwich can be written as

$$\left\{\begin{array}{l}{\sigma }_r^c={\displaystyle \frac{P_a{c}^{k+1}-P_b}{1-c^{2k}}}{\rho }^{k-1}+{\displaystyle \frac{P_b{c}^{k-1}-P_a}{1-c^{2k}}}{c}^{k+1}{\rho }^{-k-1}\hfill \\ {\sigma }_{\theta }^c={\displaystyle \frac{P_a{c}^{k+1}-P_b}{1-c^{2k}}}k{\rho }^{k-1}-{\displaystyle \frac{P_b{c}^{k-1}-P_a}{1-c^{2k}}}k{c}^{k+1}{\rho }^{-k-1}\hfill \\ {\sigma }_z^c=-{\displaystyle \frac{1}{s_{33}}}\left(s_{13}{\sigma }_{RR}^c+s_{23}{\sigma }_{\theta \theta }^c+s_{34}{\tau }_{\theta Z}^c\right)=-{\displaystyle \frac{1}{s_{33}}}\left[{\displaystyle \frac{P_a{c}^{k+1}-P_b}{1-c^{2k}}}\left(s_{13}+k{s}_{23}\right){\rho }^{k-1}+{\displaystyle \frac{P_b{c}^{k-1}-P_a}{1-c^{2k}}}\left(s_{13}-k{s}_{23}\right)c^{k+1}{\rho }^{-k-1}\right]\hfill \end{array}\right.$$

(20)

3.2. Closed-Form Solutions of the Inner and Outer Facesheets

For an infinitely long sandwich hollow cylinder without end caps, the generalized plane deformation assumption is adopted. Hence, both the stresses and the displacements are independent of Z. Based on Lame’s equations [4], the planar elastic stresses of the inner facesheet can be described as

$$\left\{\begin{array}{l}{\sigma }_r^i={\displaystyle \frac{1-\frac{{\left(R_i+T_i\right)}^2}{R^2}}{\frac{{(R_i+T_i)}^2}{R_i^2}-1}}{P}_i+{\displaystyle \frac{\frac{{\left(R_i+T_i\right)}^2}{R^2}-\frac{{(R_i+T_i)}^2}{R_i^2}}{\frac{{(R_i+T_i)}^2}{R_i^2}-1}}{P}_a\hfill \\ {\sigma }_{\theta }^i={\displaystyle \frac{1+\frac{{\left(R_i+T_i\right)}^2}{R^2}}{\frac{{(R_i+T_i)}^2}{R_i^2}-1}}{P}_i-{\displaystyle \frac{\frac{{\left(R_i+T_i\right)}^2}{R^2}+\frac{{(R_i+T_i)}^2}{R_i^2}}{\frac{{(R_i+T_i)}^2}{R_i^2}-1}}{P}_a\hfill \\ u^i={\displaystyle \frac{\frac{1-\mu }{E}+\frac{1+\mu }{E}\frac{{\left(R_i+T_i\right)}^2}{R^2}}{\frac{{(R_i+T_i)}^2}{R_i^2}-1}}{P}_{i}R-{\displaystyle \frac{\frac{1+\mu }{E}\frac{{\left(R_i+T_i\right)}^2}{R^2}+\frac{1-\mu }{E}\frac{{(R_i+T_i)}^2}{R_i^2}}{\frac{{(R_i+T_i)}^2}{R_i^2}-1}}{P}_{a}R\hfill \end{array}\right.,$$

(21)

where

$$R\in \left[R_i,R_i+T_i\right],$$

(22)

and, for the outer facesheet, the planar elastic stresses are

$$\left\{\begin{array}{l}{\sigma }_r^o={\displaystyle \frac{1-\frac{{\left(R_o+T_o\right)}^2}{R^2}}{\frac{{(R_o+T_o)}^2}{R_o^2}-1}}{P}_b\hfill \\ {\sigma }_{\theta }^o={\displaystyle \frac{1+\frac{{\left(R_o+T_o\right)}^2}{R^2}}{\frac{{(R_o+T_o)}^2}{R_o^2}-1}}{P}_b\hfill \\ u^o={\displaystyle \frac{\frac{1+\mu }{E}\frac{{\left(R_o+T_o\right)}^2}{R^2}+\frac{1-\mu }{E}}{\frac{{(R_o+T_o)}^2}{R_o^2}-1}}{P}_{b}R\hfill \end{array}\right.,$$

(23)

where

$$R\in \left[R_i,R_i+T_i\right].$$

(24)

3.3. Displacement Compatibility Conditions

For there is no slip between the facesheets and the corrugated core, the displacement compatibility conditions are as follows:

$$u_R^i(R_i+T_i)=u_R^c(R_i+T_i),$$

(25)

$$u_R^o(R_o)=u_R^c(R_o),$$

(26)

$$u_R^c(R)=\left({\beta }_{12}{\sigma }_{RR}^c+{\beta }_{22}{\sigma }_{\theta \theta }^c\right)R,R\in \left[R_i+T_i,R_o\right].$$

(27)

3.4. Model Validation of the Closed-Form Solution

The basis material of the facesheets and the corrugated core is 304 stainless steel [15], of which the main physical parameters are $E=206 \mathrm{GPa}$, $\mu =0.3$, and $\rho =7900{ \mathrm{kg}/\mathrm{m}}^3$, respectively. The structural parameters are shown in

Table 1. Structural parameters of hollow corrugated sandwich cylinder shell.

Case	Ro (m)	p/Ro	hc/Ro	f/Ro	To/Ro	Ti/Ro	Tc/Ro	Pi/E
1	0.15	0.0837	0.333	0.0533	0.00367	0.00867	0.0026	15 × 10−5

.

Table 2. Stress results of hollow corrugated sandwich cylinder shell.

Case			Results in [23]	Closed-Form Solution in This Paper	Error (%)
1	Inner Facesheet	${\sigma }_r^i$ (Pa)	−3.01 × 107	−3.09 × 107	2.66
		${\sigma }_{\theta }^i$ (Pa)	1.89 × 109	1.81 × 109	4.23
		${\sigma }_z^i$ (Pa)	5.54 × 108	5.32 × 108	3.97
		${\sigma }_{von}^i$ (Pa)	1.69 × 109	1.63 × 109	3.55
2	Outer Facesheet	${\sigma }_r^o$ (Pa)	−4.3 × 106	−4.22 × 106	1.86
		${\sigma }_{\theta }^o$ (Pa)	1.17 × 109	1.15 × 109	1.71
		${\sigma }_z^o$ (Pa)	3.5 × 108	3.45 × 108	1.43
		${\sigma }_{von}^o$ (Pa)	1.05 × 109	1.03 × 109	1.90
3	Core	${\sigma }_y^c$ (Pa)	−6.23 × 107	−6.58 × 107	5.62
		${\sigma }_x^c$ (Pa)	−2.08 × 108	−1.82 × 108	12.50
		${\sigma }_{von}^c$ (Pa)	1.84 × 108	1.60 × 108	13.04

shows the calculated stress of the hollow corrugated sandwich cylinder shell; it can be seen that the error of ${\sigma }_x^c$ is 12.5%, resulting in the enlargement of the cumulative error for ${\sigma }_{von}^c$. The errors of other physical quantities are all less than 5%, providing that the calculation is accurate.

4. The Minimum Weight Design Model

4.1. Dimensionless Expression of Mass

Compared to a solid cylinder shell, the dimensionless mass can be expressed as

$$\begin{array}{ll}{\displaystyle \frac{m}{{\rho }_s{R}_o^2}}& ={\displaystyle \frac{\pi \left[\left(2R_o+t_o\right)t_o+\left(2R_i+t_i\right)t_i\right]+n_{core}{t}_w\left(2l_w+f_o+f_i\right)}{R_o^2}}\\ & =\pi \left[\left(2+{\displaystyle \frac{t_o}{R_o}}\right){\displaystyle \frac{t_o}{R_o}}+\left(2{\displaystyle \frac{R_i}{R_o}}+{\displaystyle \frac{t_i}{R_o}}\right){\displaystyle \frac{t_i}{R_o}}\right]+n_{core}{\displaystyle \frac{t_w}{R_o}}\left(2{\displaystyle \frac{l_w}{R_o}}+{\displaystyle \frac{f_o}{R_o}}+{\displaystyle \frac{f_i}{R_o}}\right)\end{array}.$$

(28)

4.2. Failure Mode

The deformation of the incline core panel of the hollow corrugated sandwich cylinder is along the axial direction; thus, the maximum membrane stress can be described as follows:

$$\left\{\begin{array}{l}{\sigma }_y^c={\sigma }_r^c(s_{11}{\sin }^2\alpha +s_{12}{\cos }^2\alpha )+{\sigma }_{\theta }^c(s_{12}{\sin }^2\alpha +s_{22}{\cos }^2\alpha )+{\sigma }_z^c(s_{13}{\sin }^2\alpha +s_{23}{\cos }^2\alpha )\hfill \\ {\sigma }_x^c={\displaystyle \frac{{\sigma }_z^{c}p\left(2R_o-3h_c\right)\sin \alpha }{2(R_o-2h_c)T_c}}\hfill \end{array}\right.$$

(29)

Considering three failure modes, i.e., facesheet yielding, core yielding, and core buckling,

$$\left\{\begin{array}{ll}{\displaystyle \frac{\sqrt{2}}{2}}{\left.\left\{{\left({\displaystyle \frac{{\sigma }_{\theta }^{\lambda }}{E}}-{\displaystyle \frac{{\sigma }_r^{\lambda }}{E}}\right)}^2+{\left({\displaystyle \frac{{\sigma }_z^{\lambda }}{E}}-{\displaystyle \frac{{\sigma }_r^{\lambda }}{E}}\right)}^2\right.+{\left({\displaystyle \frac{{\sigma }_z^{\lambda }}{E}}-{\displaystyle \frac{{\sigma }_{\theta }^{\lambda }}{E}}\right)}^2\right\}}^{1/2}\le {\displaystyle \frac{{\sigma }_Y}{E}}\hfill & \mathrm{facesheet} \mathrm{yielding}\\ {\displaystyle \frac{\sqrt{2}}{2}}{\left.\left\{{\left({\displaystyle \frac{{\sigma }_x^c}{E}}-{\displaystyle \frac{{\sigma }_y^c}{E}}\right)}^2+{\left({\displaystyle \frac{{\sigma }_x^c}{E}}\right)}^2\right.+{\left({\displaystyle \frac{{\sigma }_y^c}{E}}\right)}^2\right\}}^{1/2}\le {\displaystyle \frac{{\sigma }_Y}{E}}\hfill & \mathrm{core} \mathrm{yielding}\\ {\displaystyle \frac{{\sigma }_x^c}{E}}\le {\displaystyle \frac{{\pi }^2{T}_c^2{\sin }^2\alpha }{12(1-{\mu }^2)h_c^2}}\hfill & \mathrm{core} \mathrm{buckling}\end{array}\right.$$

(30)

where

$$\lambda =i,o.$$

(31)

The length of the incline cell is far greater than its width; thus, the effect of the axial normal stress ${\sigma }_{x^{*}}$ on the core member buckling can be neglected. Because the horizontal component of the corrugate core cell has been merged with the facesheets, the failure mode of the cell is considered to be the same as that of the facesheets.

When failure occurs in the facessheet or the corrugate core, the relevant limit inner pressure load can be obtained as follows:

$$P_{\lim }=P_{ik}, if\hspace{1em}{\sigma }_k=\left[\sigma \right], k\in \left(0,\infty \right).$$

(32)

4.3. Minimum Weight Design Model

According to the structural model, and combining the dimensionless mass expression and the failure mode, a minimum weight design model of the hollow corrugated sandwich cylinder under inner pressure load is presented. The calculating flow chart is shown in Figure 4.

5. Results and Discussion

5.1. Stress Distribution of Typical Corrugated Cylinder Shell

Figure 5 shows the stress distribution of the facesheets and the corrugated core varying with inner pressure load. When only considering the elastic deformation, with the increase in the inner pressure loading, the stresses of the facesheets and the corrugated core panel increase. The inboard stress of the inside facesheet is the highest, followed by the outside facesheet, while the corrugated core exhibits the lowest stress with a relatively gradual change.

5.2. Minimum Weight Analysis of Hollow Corrugated Cylinder Shell

Figure 6 shows the dimensionless minimum weight as a function of the dimensionless loading parameter $P_i/E$ for the hollow corrugated sandwich cylinder and the homogeneous panel cylinder. By analyzing the data in Figure 6, the following findings can be obtained:

(1)

The minimum weight of the hollow homogeneous panel cylinder exhibits approximately linear growth with respect to the limit load, whereas the hollow corrugated sandwich cylinder undergoes a gradual change in weight;

(2)

Compared to the homogeneous panel cylinder, when the inner pressure is relatively low (P_i < $1.57\times {10}^{-5}\cdot E$), a higher minimum weight is required for the hollow corrugated sandwich cylinder to withstand the same inner pressure load. However, as the limit inner pressure load exceeds this threshold, there is a notable improvement in structural efficiency, resulting in the hollow corrugated sandwich cylinder having a lighter minimum weight than the homogeneous panel cylinder for the same inner pressure load;

(3)

Furthermore, when the thickness of the hollow corrugated sandwich cylinder h_c is varied, it is observed that the parameter h_c significantly influences the emergence of the inflection point. At relatively small thicknesses ($t_i/h_c>0.026$), the structural efficiency of the hollow corrugated sandwich cylinder is minimal. However, as the thickness increases to a certain value ($t_i/h_c<0.026$), the structural efficiency of the hollow corrugated sandwich cylinder becomes notably superior to that of the homogeneous panel structure. Moreover, as the thickness of the corrugated core increases, the structural efficiency also increases.

Consequently, it can be concluded that the structural efficiency of the hollow corrugated sandwich cylinder is significantly higher when the loading pressure is relatively high, and there may be an inflection point with the increase in the loading. Additionally, the optimal structural efficiency is closely related to the topological parameters of the corrugated sandwich structure. Subsequently, the effect of every topological parameter on the occurrence of the inflection will be discussed, respectively.

5.3. Minimum Weight Comparison of Different Hollow Corrugated Sandwich Cylinders

Figure 7, Figure 8, Figure 9 and Figure 10 present the relations of the minimum weight and the limit load of the hollow corrugated sandwich cylinder with different topological parameters.

As shown in Figure 7, it can be found that when the thickness of the inner facesheet increases, the compression strength of the hollow corrugated sandwich cylinder increases accordingly to bear higher inner pressure; thus, the inflection point is relatively carry-forward. Furthermore, when the thickness of the inside facesheet is $t_i/h_c=0.01$, the limit load is less than $1.0\times {10}^{-5}\cdot E$, thus its structural efficiency is inferior to that of the homogeneous panel cylinder, and there is no inflection. Similarly, there is no inflection when the thickness is $t_i/h_c=0.02$. Additionally, it is noteworthy that the structural efficiency of the hollow corrugated cylinder varies with different inner pressure loads. When the pressure is low, the minimum weight decreases as the thickness of the inside facesheet increases. However, when the pressure is high, the results are reversed: the minimum weight increases with the increasing thickness of the inside facesheet, and the inflection point of this transformation occurs relatively earlier.

It can be seen from Figure 8 that with the increase in the thickness of the outer facesheet, the minimum weight of the hollow corrugated sandwich cylinder increases, and the inflection point occurs relatively earlier. Furthermore, when the thickness of the outer facesheet $t_o/h_c$ exceeds 0.02, the structural efficiency is inferior to that of the homogeneous panel cylinder, and there is no inflection observed. Additionally, it is worth noting that when the outer facesheet is removed, the structural efficiency significantly improves under low inner pressure.

As depicted in Figure 9, as the thickness of the corrugated core panel increases, the minimum weight of the hollow corrugated sandwich cylinder rises, accompanied by a relatively advanced inflection point. Furthermore, when the thickness of the corrugated core panel $t_w/h_c$ exceeds 0.0033, the limit load is less than $1.76\times {10}^{-5}\cdot E$; the structural efficiency is inferior to that of the homogeneous panel cylinder, and there is no inflection point observed.

It can be seen from Figure 10 that when the inclining angle of the corrugated core panel increases, the number of the corrugated core adds, and the minimum weight of the hollow corrugated sandwich cylinder increases accordingly. Particularly, when the inclining angle of the corrugated core panel α exceeds 75°, the limit load is less than $1.7\times {10}^{-5}\cdot E$; the structural efficiency is inferior to that of the homogeneous panel cylinder, and there is no inflection point.

5.4. Sensitivity Analysis of Structural Parameters

To investigate the sensitivity of all structural parameters, Figure 11 and Figure 12 illustrate the impact of size design variables related to the layer thickness and shape design variables associated with the corrugated core layer. As depicted in Figure 11, as the inner pressure increases, the influence of the inner facesheet thickness (t_i) becomes most pronounced. Analogously, Figure 12 demonstrates that with the increase in the inner pressure, the effect of the core thickness (h_c) stands out the most. Additionally, the trapezoid width exhibits minimal sensitivity to structural efficiency. In summary, when designing a hollow corrugated sandwich cylinder to withstand inner pressure loads, designers should prioritize careful consideration of the thicknesses of both the facesheets and the core.

In order to further quantitatively analyze the sensitivity of structural parameters on the minimum weight, Gray Relational Analysis (GRA) was used to calculate the relevance of different parameters [24]. Detailed results were listed in

Table 3. Relevance of structural parameters based on GRA.

Parameters		Relevance
Size design variables	ti	0.6133
	to	0.6087
	tw	0.5720
Shape design variables	hc	0.6954
	ρ	0.6873
	f	0.6690

. It can be seen that for the size design variables, the relevance of t_i is the largest while the relevance of t_w is the smallest. For the shape design variables, the relevance of h_c is the largest while the relevance of f is the smallest. Furthermore, the relevance of shape design variables is larger than that of size design variables, indicating that the shape design variables are more sensitive for the optimal weight design of structures.

6. Conclusions

Due to the high structural efficiency and growing trend of multifunctional application, the hollow corrugated sandwich cylinder has been widely used in aviation, high-speed trains, packaging, and other fields. In this paper, the mechanical behaviors of the hollow corrugated sandwich cylinder under inner pressure loading were investigated, yielding the following key conclusions:

(1)

Concerning the long, hollow sandwich cylinder without column caps, an elastic closed-form solution and the minimum weight design model of the hollow corrugated sandwich cylinder were proposed with the homogenization method.

(2)

The stress distribution of the hollow corrugated sandwich cylinder with different topological structural parameters was compared and analyzed. It is found that there is an inflection point of the structural efficiency during the increase in the pressure. Notably, the hollow corrugated sandwich cylinder demonstrates a relatively pronounced structural advantage only when subjected to significant internal pressure.

(3)

The optimal structural efficiency of the hollow corrugated sandwich cylinder is closely related to its topological parameters. With the increase in the topological parameters, including core thickness, inner facesheet thickness, outer facesheet thickness, core thickness, and core inclining angle, both the minimum weight and the limit load increase accordingly, leading to a relative advancement of the inflection point in the structural efficiency curve.

(4)

The model did not consider the influence of the failure between the core and facesheets, which should be further investigated to improve the accuracy and practicability of the model.