1. Introduction
To exploit underwater resources and develop deep-ocean technology, deep-sea submersibles are significant mobile platforms that require multiple physical capabilities including load-bearing, buckling, fatigue, corrosion-resistant performance, etc. These key properties are affected by various internal and external factors such as constituent materials, structural styles, loads, and deep-sea environment, and the constituent materials have played a significant role and have been the research focus [1]. The high-performance materials currently used within the cylindrical, spherical, or ellipsoidal-shaped submersibles can mainly be divided into three types: metal materials such as high-strength titanium alloy, high-strength steel, and aluminum alloy; non-metallic materials like composite materials, ceramics, and organic glass; and fiber-reinforced resin-based composites including glass fiber reinforced resin and carbon fiber reinforced resin [2], [3], [4], [5]. However, other functionally physical properties such as thermal, dynamic, or acoustic behaviors have not been paid considerable concern. The design and fabricating of innovative functional materials and their corresponding composite structures have become an inevitable trend thanks to the rapid industrialization of current high-end manufacturing technologies.
Composite structures sandwiched with lightweight cellular materials have been extensively used in aeronautics, automotive, marine, and civil engineering applications, while the low-density lightweight cellular core contributes to their multiple functionalities [6]. Honeycomb materials are the main class of artificial cellular materials whose geometries can be readily parameterized, and they are usually characterized by a tessellation of an array of periodic unit cells to fill a planar area or 3D space as a sandwiched core [7]. The physical characteristics of honeycomb materials are governed by the geometry of their unit cells and can thus be flexibly changed, such as Poisson's ratio, elastic modulus, bulk modulus, and coefficient of thermal expansion [8], etc. Among them, the Poisson's ratio can be tuned to possess not merely the positive Poisson's ratio (PPR) that widely exists in nature but negative or zero values, and the honeycombs that exhibit negative or zero Poisson's ratios (NPR and ZPR) typically belong to mechanical metamaterials. In accord with the definition of the Poisson's ratio, the NPR or "auxetic" metamaterial exhibits lateral expansion instead of contraction when stretched under uniaxial loading, while the ZPR one yields no transverse deformation under unidirectional tensile or compressive loading. Furthermore, the former possesses superior mechanical properties such as enhanced shear resistance, indentation resistance, and fracture toughness [9], [10], [11], while the latter has been specially recommended on aircraft morphing skins without double curvatures under out-of-plane bending moments [12]. On the one hand, three well-established auxetic microstructures have been identified to distinguish NPR metamaterials based on their deformation mechanisms [13], including re-entrant, chiral [14], and rotating-rigid [15] ones. The most frequently reported re-entrant structures among these configurations are characterized as "direct inward" or featuring a negative angle, and their deformations are dominated by the realignment of their hinged cell ligaments. A variety of re-entrant honeycomb configurations have been proposed and investigated, such as re-entrant hexagons [8], arrowheads [16], lozenge and square grids [17], and star systems [18]. On the other hand, the ZPR metamaterials have been paid increasing attention in recent years and can be categorized into two main types: compliance in one direction and two directions [19]. The former contains the accordion design with solid strips connected by bending hexagonal ligaments [20], accordion topology with chevron bending ligaments [21], and modified semi re-entrant design [22], while the latter includes semi re-entrant [23], [24], [25], [26], AuxHex [27], SILICOMB [28], star-shaped [29] and cross-circular ZPR honeycombs [30]. Among these ZPR configurations, the semi-re-entrant honeycomb is a classic type that has drawn considerable concern since it is directly constituted by combining halves of the regular hexagonal PPR honeycomb and the re-entrant hexagonal NPR honeycomb.
Inspired by the deformation mechanism of the NPR and ZPR mechanical metamaterials, their corresponding material cores are expected to be applied to sandwich cylindrical shell meta-structures as protective covers. Compared with conventional PPR honeycombs, the circularly adjacent NPR and ZPR honeycomb unit cells can deform with less circumferential extrusion when subjected to radial forced compression, which potentially benefits the energy-absorption performance of the cellular core, as the conceptional comparison depicted in Fig. 1. Therefore, the sandwich cylindrical shells with NPR and ZPR metamaterials are projected to bear huge hydrostatic pressure for deep-sea submersibles, devices, or equipment. Regarding the recent research on sandwich cylindrical shells with cellular cores, Cong et al. [31] investigated the nonlinear dynamic response of eccentrically stiffened sandwich cylindrical shells with auxetic re-entrant hexagonal honeycomb cores. Lan et al. [32] compared the blast resistance of sandwich cylindrical shells with auxetic re-entrant hexagonal honeycomb cores and aluminum foams. Guo et al. [33] simulated the deformation behaviors and energy absorption of cylindrical structures with auxetic re-entrant hexagonal lattices under axial crushing loads. Gao et al. [34] carried out the theoretical prediction of dynamic responses under impact loading for sandwich cylindrical shells with an auxetic 3D arrowhead honeycomb core. Li et al. [35] studied the mechanical and vibroacoustic performance of sandwich cylindrical shell meta-structures with functionally graded auxetic re-entrant hexagonal honeycomb cores and carried out an optimization design of the sound transmission, and a pressure-resistant cylindrical shell meta-structure sandwiched with a ZPR AuxHex honeycomb core was also designed with balanced band gap characteristics [36]. Ren et al. [37] also studied the quasi-static and sound insulation performance of a cylindrical cellular shell with a bidirectional negative-stiffness metamaterial core. In addition, by curling the planar metamaterials, Wei et al. [38] designed a series of multi-functional cylindrical meta-structure that can simultaneously program both thermal expansion and Poisson's ratio. Ling et al. [39] analyzed the large magnitudes of programmable Poisson's ratio within a series of lightweight cylindrical meatstructures. Qin et al. [40] have recently introduced fish-cell ZPR metamaterials into cylindrical meta-structures and studied their structural mechanics. Note that the profile of the honeycomb unit cells is usually distorted to fit the circular shape of the cylindrical shell, which might cause distinct structural behaviors compared with those in the scenario of sandwich panels. However, cylindrical shell meta-structures with a ZPR cellular core have still rarely been investigated, while the comparative study on the physical properties of sandwich cylindrical shells with PPR, ZPR, and NPR cellular cores has not been reported before, which covers a full characteristic range of Poisson's ratios of the sandwiched cellular materials.
Fig.1 Conceptional comparison of pressure-resistant sandwich cylindrical shells with (a) the PPR; (b) the NPR; and (c) the ZPR cellular cores. |
Therefore, a novel cylindrical meta-structure sandwiched with the semi-re-entrant ZPR metamaterial has been designed in this study, while its similarly-shaped sandwich cylindrical shells with PPR and NPR honeycombs have also been presented. The main contribution of the presented work is comparing the mechanical and acoustic performance of sandwich cylindrical shells with cellular materials featuring a varied range of Poisson's ratios to guide potential applications on submerged pressure-resistant covers. The representative honeycomb unit cells are chosen to be the regular hexagonal, re-entrant hexagonal, and semi-re-entrant hexagonal honeycomb as the most classic topologies, respectively, while the sandwich cylindrical shell with a semi-re-entrant ZPR cellular core has not been investigated yet. In the following contents, Section 2 presents the model design regarding the three respective unit cells and the sandwich cylindrical shell, and it also introduces the computational methods of the periodic boundary condition (PBC), the applied loads, and the boundary condition for the sandwich structure. Section 3 chooses three representative honeycomb unit cells with equivalent load-bearing capacities and shows the results of pressure-resistant, buckling, and sound insulation performance of the sandwich cylindrical shells with the three honeycomb cellular cores, while comparison and tradeoffs have been discussed then in detail. Section 4 draws a conclusion.
2. Model design and computational methods
2.1. Geometries and mechanical behavior of the PPR, NPR, and ZPR honeycombs
The respective configurations of the conventional hexagonal PPR, re-entrant hexagonal NPR, and semi-re-entrant ZPR honeycombs are illustrated in Fig. 2. The ZPR honeycomb comprises halves of PPR and NPR honeycombs (separated by blue dash lines) whose internal angles are opposite to each other [12], which has been explained with plus and equal signs.
Fig.2 Geometries and physical parameters of (a) the conventional hexagonal PPR honeycomb; (b) re-entrant hexagonal NPR honeycomb; (c) the semi-re-entrant ZPR honeycomb; and (d) the bearing condition of each inclined ligament. |
Four parameters define the cell geometries of the three honeycombs as depicted in Fig. 2 (a-c), where θ is the internal angle; h is the length of the horizontal ligaments; l is the length of the inclined ligaments; t is the ligament thicknesses, while their values are identical for all of the three honeycombs. Lx and Ly are the horizontal and vertical lengths for a honeycomb inscribed in a rectangle measuring Lx × Ly, and their subscripts p, n, and z represent that the lengths belong to the PPR, NPR, and ZPR honeycombs, respectively. The following geometrical relationships are satisfied for both the PPR and NPR honeycombs
Here, the cell aspect ratio α= Lx/ Ly is defined. For the combined ZPR honeycomb, the following equation set can be derived
When all the three honeycombs are loaded with the applied stress as σy in the y-direction, the inclined ligament yield mechanical deflection as depicted in Fig. 2 (d). From standard beam theory [8], the flexural deflection δf, the shear deflection δs and the axial deflection δa can be respectively written as
where P represents the internal force of the inclined ligament; I=zt3/12 denotes the second moment of inertia of the planar beam with b being the out-of-plane thickness of the cellular material network; νs and Es represent Poisson's ratio and Young's modulus of the base material. Therefore, as marked in Fig. 2 (a-c) with corresponding directions, the horizontal and vertical deflections , , , and of each unit cell can be derived as
When the honeycomb systems are subjected to the applied stress in the y-direction, the Poisson's ratios νyx of the PPR, NPR, and ZPR honeycombs can be obtained based on the definition as
where
Here, the coefficient η denotes the effects of beams' axial and shear deformations. The Poisson's ratio of the ZPR honeycomb is finally proved to be zero as expressed
2.2. Load-bearing capabilities of the honeycombs within a periodic lattice
Specifically, the pressure-resistant properties of the sandwich cylindrical shells can be preliminarily predicted by the load-bearing capabilities of their honeycombs with appropriate boundary conditions. Here, the finite element homogenization method based on representative volume elements (RVEs) is employed and the periodic boundary condition (PBC) is adopted. The macroscopic mechanical behavior of a metamaterial structure can be reflected by their corresponding micro-architectures via the PBC that satisfies Hill's energy law, which offers a strong alternative to investigating the mechanical performance of a homogeneous material whose micro-architectures feature arbitrary geometries of even complicated topology [41]. Therefore, the displacement periodic boundary condition is selected and implemented on the three honeycombs. The periodic boundary in the two-dimensional case can be expressed in the FE form by [42]
where u, v, and θ are the vector components of nodal displacement vectors in x, y, and z directions while the subscripts l, r, t, and b represent nodes on left, right, bottom, and top boundaries, respectively. εx and εy are nominal strains that are normalized dimensionless parameters along x and y directions. The parameters and the boundary condition are shown in Fig. 3 by taking the combined ZPR honeycomb as an example.
Fig.3 (a) The cellular honeycomb lattice under distributed pressure; and (b) the periodic boundary condition of a unit cell. |
To specifically obtain the mechanical properties in the y-direction, the left equation set on Px in Eq. (11) will be omitted and only an initial εy should be given for loading. The PBC can be substituted into the FEM equation by Lagrange's multiplier method. The vertical component of nodal reaction forces can be attained by solving the FEM equation, while the Von Mises stress σMises can be derived as well as the axial compressive stress σc of the beam elements as marked in Fig. 3. The equivalent external pressure σy can be theoretically derived by the ratio of the resultant force along Py and the length of Py
Here, σy is treated as a distributed line load for the 2D case where the out-of-plane width of the structure is treated as 1. To connect the mechanical field under the displacement PBC with the actual hydrostatic pressure σ⁎ y, a virtually-slight deformation εy is given with the PBC in advance, and its corresponding virtual external pressure σy can be derived by Eq. (12). Then, the stress field σ Mises can also be obtained. Within the range of linear elasticity of the base material, the actual Von Mises stress σ⁎Mises can be derived by σ⁎ Mises=( σ⁎ y/σy) σ Mises. Here, for the displacement and stress states of the unit cells, an FE mathematical model of general static analysis is employed using in-house developed FE codes.
2.3. Geometries, mechanical and vibroacoustic modeling of the sandwich cylindrical shell
Since the studied sandwich cylindrical shell is assumed to be infinite out of plane in this scenario, the three-dimensional modeling can thus be simplified into a planar strain problem. As depicted in Fig. 4 that uses the ZPR honeycomb as an example, the two-dimensional configuration of the sandwich cylindrical shell contains two inner and outer face sheets and a honeycomb core, and the geometry of honeycomb cells is slightly distorted to adapt to geometrical compliance due to the circular tessellation. Here, Nc and Nr are the numbers of honeycombs along the circumferential and radial directions, respectively; R0 and RNr are the inner and outer radius; tf is the thickness of the face sheets; i represents the element sequence number in the radial direction; ti is the feature thickness of the element in the ith layer. Note that the ligament thickness ti+1= βti complies with a similar condition and keeps the relative density of the core constant as well, where β denotes the ratio of similitude between two adjacent honeycomb cells along the radial direction.
Fig.4 Geometrical parameters, load, and boundary conditions of the sandwich cylindrical shell for (a) mechanical; and (b) vibroacoustic modeling via the FEA. |
In this work, the Finite Element Analysis (FEA) is utilized for both mechanical and vibroacoustic modeling, and the Timoshenko beam theory is adopted to catch the mechanical behavior of beam elements more exactly. Each unit cell comprises a rigid-jointed network of beams discretized into multiple Timoshenko beam elements. For mechanical modeling, the considered cylindrical shell is loaded with hydrostatic pressure ps distributed on the outer face sheet, and the structure is treated as a free constrained body with an inertia-relief constraint represented by green arrows as depicted in Fig. 4 (a). Here, with the tool of ABAQUS® FE packages, a mathematical model of general static analysis is adopted for pressure resistance via a Static general step, while a mathematical model of eigenvalue buckling prediction is utilized for structural stability via a Buckle step. Here, their corresponding mathematical formulas are omitted due to fundamentality.
Regarding vibroacoustic modeling, the considered sandwich structure is exposed to a cylindrical incident wave of acoustic pressure pa on the inner face sheet. The vibration of the inner face sheet is transmitted through the cellular core to the outer face sheet, and then the sandwich cylindrical shell radiates sound in the fluid medium that is discretized by acoustic finite elements as illustrated in Fig. 4 (b). The interface of the fluid and the structure interacts with a tie constraint while the outer surface of the acoustic field is set as a non-reflecting boundary with a radius of 2 RNr. The sound transmission loss can be expressed as [36]
where W represents sound power; p means sound pressure of each acoustic particle; ρ and c are density and sound velocity of the acoustic medium, respectively; and the subscripts i and t denote the incident and transmitted sides where these physical quantities locate. Since the contribution of airborne transmission to the total radiated sound power can be neglected due to the high structural stiffness, only structure-borne transmission is considered [43]. Here, the fluid and solid interactions are considered, especially when the acoustic medium is selected as water, and an FE mathematical model of vibroacoustic coupling analysis is employed for natural frequencies and sound transmission performance, which is solved by the ABAQUS® steps of Frequency and Steady-state dynamics, respectively. The mathematical equation of vibroacoustic coupling can be written as [44]
where Q is the vibroacoustic coupling matrix; , , and are the structural stiffness, damping, and mass matrices, respectively; , , and are the fluid stiffness, damping, and mass matrices, respectively; u is the nodal displacement vector of the structure; p is the dynamic nodal pressure vector of the fluid; and are the external excitation vector of the structure and fluid, respectively; is the density of the fluid; ω is the circular frequency while .
3. Results and discussions
3.1. Verification of load-bearing capabilities of three selected honeycombs
The horizontal dimension, cell aspect ratio, and featured angle of the selected three honeycomb unit cells are set as Ly=50 mm, α=1.732, and θ=±30 deg, which is a classic combination of geometric parameters in the open literature. The wall thickness of the selected honeycombs is set as 6 mm. The geometric configurations of the selected three honeycomb unit cells are depicted in Fig. 5 (a-c). The base material is selected as titanium alloy, whose Young's modulus Es=1.15 × 105 MPa; density ρs=4.440 × 103 kg/m3; Poisson's ratio vs=0.33; the proportional limit σp,s=860 MPa; and the coefficient for structural safety cs=1.2. Here, the equivalent Young's modulus becomes Es/(1- ν2) because of the planar strain assumption.
Fig.5 Geometric configurations of (a) the PPR; (b) the NPR; and (c) the ZPR honeycomb unit cells; and displacement deformation and stress distribution of (d) the PPR; (e) the NPR; and (f) the ZPR honeycomb unit cells. |
The PPR, NPR, and ZPR honeycomb unit cells are all constrained by the PBC as expressed in Eq. (11) and applied with pressure on both sides as depicted in Fig. 5 (a-c). Due to the uncertain annular arrangement and the uncertain stiffness of the inner face sheet seen as elastic support, the practical boundary condition of the honeycomb unit cells within the sandwich cylindrical shell is hard to be directly and accurately extracted. This part aims to verify whether the load-bearing capabilities of the three selected honeycombs are theoretically equivalent under uniaxial compression so that the comparison in the following contents can be more convincible. Therefore, the geometric configurations, displacement deformation, and stress distribution of the three respective honeycomb unit cells are programming calculated by FEA via MATLAB® codes as shown in Fig. 5 (d-e). Furthermore, the theoretical Poisson's ratio can be derived by
where ur, ul, vt1, vt2, vb1, and vb2 represent the nodal displacement on boundaries, which are marked in Fig. 5 (d-e).
The inclined ligaments of each unit cell play a major load-bearing role, and the intersections of the inclined and horizontal ligaments are the locations that yield the maximum beam stress. Note that the maximum stresses are all 5.778 × 102 MPa for the three selected honeycombs and are both less than σp,s/ cs=7.167 × 102 MPa, which ensures structural safety of the unit cells within the cellular cores. The nominal strains εy are all 4.581 × 10−3 and the relative densities ρ⁎/ρs are 0.24, 0.32, and 0.28 for the PPR, NPR, and ZPR honeycomb unit cells, respectively, which indicates that the PPR honeycomb yields the best lightweight property. In addition, Poisson's ratios that are derived from the theoretical formulas in Eqs. (7) to (10) are compared with those calculated via the FEA as illustrated in Fig. 5 (d-e), which have been listed in Table 1. Therefore, the comparative results significantly benchmark the effectiveness of the FEA simulation as introduced in Section 2.2 and thus validate the theoretically equivalent load-bearing capabilities of the three selected honeycombs under axial compression.
Table 1 The Poisson's ratio values derived from theoretical formulas and simulated by FEA codes. |
| Poisson's ratio values | PPR honeycomb | NPR honeycomb | ZPR honeycomb |
|---|---|---|---|
| From theoretical formulas | 8.627 × 10−1 | -8.627 × 10−1 | 0 |
| From FEA codes | 8.591 × 10−1 | -8.591 × 10−1 | -2.051 × 10−14 |
3.2. Mechanical performance of the sandwich cylindrical shells with the three honeycomb cores
The pressure-resistant capability of the sandwich cylindrical shell with all the respective cellular cores is simulated and analyzed here. The inner radius R0 is primitively selected as 1000 mm (20 Ly), while the number of unit cells along the circumferential direction Nc is derived to be 72 as an integer value so that the inner radius R0 is then slightly modified as 992.4 mm. The similarity ratios between radially adjacent unit cells are all 1.050, and the number of unit cells along the radius direction Nr is 9. The outer radius R10 of the cylindrical structure is calculated to be 1544.8 mm. The thicknesses tf of the inner and outer face sheets are all 30 mm. To fit the circular shape, the rectangular cell frame should be modified into a trapezoidal one, which results in slight geometric distortion. Nonetheless, the weight-bulk ratios for the sandwich cylindrical shell with PPR, NPR, and ZPR cellular cores are 738 kg/m3, 895 kg/m3, and 815 kg/m3 (all less than the density of water), respectively, which can provide buoyancy reserves for the engineering applications on submersible equipment or structures.
In this scenario, the hydrostatic distributed pressure σ⁎ y=10 MPa in the water depth of approximately 1000 m is applied on the outer face sheet of the sandwich cylindrical shell with an inertia-relief constraint as depicted in Fig. 4 (a), and the studied structures are all discretized by B22-typed Timoshenko beam elements with an overall mesh size of 2.5 mm. The entire structure is stress-free in accord with the practical loading condition, which differs from the virtual loading condition under axial compression in Section 3.1 for previous verification. The practical boundary condition with the inner face sheet stress-free is theoretically weaker than that with both sides under pressure, which is more conservative for structural safety. The information on the structural FE models for the sandwich cylindrical shells with three respective cellular cores is presented in Table 2.
Table 2 The information on the structural FE models of the sandwich cylindrical shells with three respective cellular cores. |
| Empty Cell | With the PPR cellular core | With the NPR cellular core | With the ZPR cellular core |
|---|---|---|---|
| Number of nodes | 59546 | 76824 | 68328 |
| Number of elements | 30457 | 39096 | 34848 |
| Element types | B22 | B22 | B22 |
In the macroscopic view, the displacement distributions of the entire sandwich cylindrical shells with the PPR, NPR, and ZPR cellular cores are illustrated and compared in Fig. 6. The outer face sheets of the sandwich cylindrical shells with the NPR and ZPR cellular cores feature larger deformation than their inner face sheets while the situation is the opposite for the sandwich structure with the PPR cellular core, which implies that the former two types of cellular cores can protect the inner space better than the latter one. In addition, the stress distributions of the entire sandwich cylindrical shells with the PPR, NPR, and ZPR cellular cores are also depicted in Fig. 7. Thus, it can be noted that the order of structural strength is PPR>NPR>ZPR for the honeycomb cores by comparing the maximum Von Mises stress of the entire sandwich cylindrical shells.
Fig.6 Displacement distributions of the cylindrical shell structures with (a) PPR; (b) NPR; and (c) ZPR cellular cores. |
Fig.7 Stress distributions of the cylindrical shell structures with (a) PPR; (b) NPR; and (c) ZPR cellular cores. |
As the dominant reason, the PPR cellular core is expanding whereas the NPR and ZPR cores are oppositely contracting under the applied pressure. The largest stress value of 2.376 × 102 MPa distributes on the inner face sheet of the sandwich structure with the PPR cellular core while the stress of 1.097 × 102 MPa on the outer face sheet has generally been amplified to the inner face sheet. For the sandwich structure with the NPR cellular core, both the inner and outer face sheets yield relatively large stress values of 2.865 × 102 MPa and 2.433 × 102 MPa, which indicates that the stress on the outer face sheet has been greatly transmitted to the inner face sheet through the NPR metamaterials. Regarding the sandwich structure with the NPR cellular core, the stress value of 1.206 × 102 MPa on the inner face sheet is less than half of the stress of 2.544 × 102 MPa on the outer face sheet, but the ZPR metamaterials yield the largest stress value of 3.868 × 102 MPa. It demonstrates the best energy-absorption capability of the ZPR metamaterial core than the other two cellular cores, whereas it places a higher demand on the load-bearing behavior of the ZPR honeycomb. By contrast, the PPR honeycombs feature relatively low-stress states, while the maximum stress value of 3.426 × 102 MPa for the NPR honeycomb is even close to the value for the ZPR honeycomb.
In the microscopic view, the stress distributions of representative unit cells for the three selected honeycombs that locate on the central 6th layer (marked with red lines in Fig. 7) are also extracted in Fig. 8 to reveal the deep deformation mechanism. To compare with the theoretical stress states in Fig. 5, the practical stress states are depicted in Fig. 8 (a-c) for the three embedded honeycombs, while the axial principal stress states are additionally illustrated to identify the stress direction. Here, a positive sign denotes tensile stress while a negative sign represents compressive stress. In general, the stress states of all honeycomb unit cells differ from the theoretical situation under axial compression as illustrated in Fig. 5. The maximum stress values of the three studied honeycombs indicate that the PPR honeycomb withstands the lowest stress state while the ZPR honeycomb absorbs the most energy under the external hydrostatic pressure.
Fig.8 Von Mises stress state and specific nodal deformation of the (a) PPR; (b) NPR; and (c) ZPR honeycomb unit cells; and axial principal stress state of the (d) PPR; (e) NPR; and (f) ZPR honeycomb unit cells. |
Based on the basic deformation mechanism, the PPR honeycomb unit cell is stretched along the radial direction since the radial compressive stress of the inclined ligaments is decreased by the macroscopic expansion of the PPR cellular core as seen in Fig. 7 (a). As a result, the horizontal ligaments should yield considerable compressive stress since the adjacent unit cells approach each other because of the inherent PPR characteristic, which can be demonstrated by the stress state as illustrated in Fig. 8 (a) and (d).
For the NPR honeycomb unit cell, its Von Mises stress distribution just resembles the theoretical situation as shown in Fig. 5 (b), but the maximum Von Mises has also dropped certainly due to the stress-free inner face sheet. It must be noted that the horizontal ligaments feature tensile stress since the adjacent unit cells move away from each other due to the inherent "auxetic" effect, which can also be reflected in Fig. 8 (b) and (e).
In addition, the Von Mises stress state of the ZPR honeycomb unit cell is depicted in Fig. 8 (c) and (f), which is similar to that depicted in Fig. 5 (c). However, a slight asymmetry should be noted in the stress distribution, which is mainly caused by geometric transformation or distortion of the unit cells due to the annular arrangement. As a result, the middle parts of the left inclined ligaments of the unit cell feature tensile stress, and compressive stress also distributes on the horizontal ligaments but yields a relatively small value of stress. Likewise, the maximum Von Mises stress of the ZPR unit cell has also declined because the inner face sheet is freely constrained.
The practical ratios between horizontal and vertical deformation for the three honeycombs are calculated via the same formula expression as Eq. (15) and listed in Table 3, and they are derived from the specific nodal deformation marked in Fig. 8 (a-c) where the black local coordinates are also depicted. It can be seen that the practical ratios between horizontal and vertical deformation are greatly inconsistent with the theoretical Poisson's ratios, while the PPR honeycomb unit cell yields a positive value while the NPR and ZPR honeycomb unit cells just feature negative values. Note that the practical ratio of the NPR honeycomb unit cell is closest to 0 rather than the ZPR one, which implies that the adjacent NPR honeycomb unit cells feature the least circumferential deformation interference. The results are just opposite to the ideal expectation about their microscopic deformation as depicted in Fig. 1. As one of the main causes, the circular arrangement just constrains the globally-circumferential or locally-horizontal deformation of the adjacent unit cells compared with the horizontal-freely vertical compression that is applied in Fig. 5. Furthermore, the stiffness of the inner and outer face sheets also greatly affects the macroscopic deformation (expansion or compression) of the sandwiched cellular core.
Table 3 The practical ratios between horizontal and vertical deformation and the theoretical Poisson's ratios for comparison. |
| Physical indexes | PPR honeycomb | NPR honeycomb | ZPR honeycomb |
|---|---|---|---|
| Practical ratios between horizontal and vertical deformation | 6.679 | -8.541 × 10−3 | -5.932 × 10−3 |
| Theoretical Poisson's ratios | 8.627 × 10−1 | -8.627 × 10−1 | 0 |
In consideration of the engineering application of deep-sea protective covers, the buckling performance of the researched sandwich cylindrical shells under the applied hydrostatic pressure also requires verification here as illustrated in Fig. 9. Considering the elastic buckling, the mode shapes illustrated in Fig. 9 (a)-(c) correspond to the second Fourier order of a theoretical cylindrical shell under an external pressure of 10 MPa, as depicted in Fig. 4 (a). The first-order buckling factors of the entire structures with the three selected honeycombs are all greater than 1.0, which demonstrates their sufficient structural stability. Crucially, the buckling factors of the sandwich structure with the PPR cellular core are the largest while those of the sandwich structure with the NPR cellular core yield the lowest values, which directly shows the order of their buckling performance (PPR>ZPR>NPR).
Fig.9 The first entire modes and buckling factors of the sandwich structures with the (a) PPR; (b) NPR; and (c) ZPR cellular cores under an external pressure of 10 MPa with no extra constraint; the first local modes and buckling factors of the (d) PPR; (e) NPR; (f) ZPR cellular cores under an external pressure of 10 MPa with the inner face sheets fixed. |
Note that Fig. 9 (d)-(f) show the first local modes of their corresponding cellular cores under an external pressure of 10 MPa with their inner face sheets fixed, which aims to verify the local stability of the cellular core with a virtual boundary condition. Concerning the different patterns of the first local modes for the three cellular cores, cell distortion is generated within several honeycomb layers for the PPR cellular core while the NPR porous structure undergoes an overall collapse to the inner face sheet. For the ZPR cellular core, periodic contraction and expansion along with the circumferential direction are produced and the outer face sheet features a wave shape.
Concerning nonlinear elastic-plastic or plastic buckling leading to collapse [45], studies on cylindrical and spherical shells containing initial flaws have been carried out mainly based on local and modal imperfections [46]. In terms of the sandwich cylindrical structures in this study, their nonlinear instability deserves further investigation when considering initial geometric imperfections or the plasticity of the porous materials with even larger deformation. Additive manufacturing techniques such as 3D printing would be helpful in greatly eliminating manufacturing errors of the investigated cylindrical sandwich structures.
3.3. Vibro-acoustic performance of the sandwich cylindrical shells with the three honeycomb cores
In this section, the sound insulation performance of the sandwich cylindrical shells with the three selected honeycombs is simulated in accord with the boundary condition as illustrated in Fig. 4 (b). The sandwich cylindrical shells are still discretized by B22-typed Timoshenko beam elements with an overall mesh size of 2.5 mm, and then the acoustic medium is discretized by both AC2D3 and AC2D4 acoustic elements with an overall mesh size of 5 mm in ABAQUS® to ensure a vibroacoustic discrete criterion based on the structural and acoustical wavelengths that correspond to the maximum frequency of interest (10000 Hz) in this scenario. Here, the structural mesh size is less than one-twentieth of the bending wavelength of 3.8 mm, while the acoustic mesh size is no more than one-sixth of the minimum acoustic wavelengths of 5.7 mm and 24.7 mm for air and water, respectively. In addition, the annulus-shaped acoustic FE field of the three respective models identically features an inner radius of R10=1544.8 mm and an outer radius of 2 R10, which comprises 156087 nodes, 4195 AC2D3 elements, and 152825 AC2D4 elements.
A harmonic incident wave whose amplitude is 1 Pa is applied as the acoustic load as shown in Fig. 4 (b), where the inner acoustic medium is air. The outer acoustic medium is set as both air and seawater with ρ=1.024 × 103 kg/m3 and c=1.480 × 103 m/s to investigate the effect of fluid and structure interaction. The bulk modulus of air and seawater are 1.420 × 10−1 MPa and 2.242 × 103 MPa, respectively. The sound transmission loss can then be derived by substituting the incident sound pressure on the inner face sheet (1 Pa) and the transmitted sound pressure on the inner surface of the acoustic field into Eq. (13).
The STL curves of the sandwich cylindrical shells with the three selected cellular cores from 1-10000 Hz are plotted in Fig. 10. The researched frequency region of the STL generally covers the stiffness-controlled, resonance-controlled, and part of mass-controlled regions, while the coincidence-controlled region starting at the critical frequency are not involved where the STL curve would feature an obvious drop. Prominently, the averaged STLs within the researched frequency range are also listed in Table 4, which shows that the order of sound insulation capability is ZPR>NPR>PPR for the sandwiched honeycomb cores. The trends of the STL curves are generally consistent despite the different acoustic mediums. However, the STL curves in the seawater environment have slightly shifted to lower frequencies compared with those corresponding to the air environment due to the fluid-structure interaction. The sound insulation capability below 4000 Hz is better in the air than that in the seawater, since the acoustic medium of a high density exhibits more obvious effects on vibroacoustic attenuation in lower frequencies. Furthermore, the STL curves in Fig. 10 (a) vary less sharply than those in Fig. 10 (b), which is also caused by the non-negligible dynamic damping effect of the seawater.
Fig.10 The STL curves of the sandwich cylindrical shells with the three selected honeycombs surrounded by the acoustic medium of (a) air; and (b) seawater. |
Table 4 The averaged STLs of the respective sandwich cylindrical shells enveloped by air and seawater. |
| Outer acoustic medium | PPR cellular core | NPR cellular core | ZPR cellular core |
|---|---|---|---|
| Air | 6.108 × 101 dB | 7.002 × 101 dB | 8.072 × 101 dB |
| Seawater | 4.533 × 101 dB | 5.402 × 101 dB | 6.535 × 101 dB |
Here, relevant characteristic natural frequencies of the three sandwich cylindrical shells have also been given to explain some key phenomena. Based on vibroacoustic theory, it is known that the breathing resonant modes of a cylindrical shell would cause a sharp decrease of STLs since the radial inflation and compression of the cylindrical structure can extremely radiate the vibration energy to the acoustic medium. Thus, the first and second orders of the breathing modes of the investigated cylindrical sandwich shells have been illustrated in Fig. 11. The frequency values exactly match the first and second dips of the STL curves of the sandwich cylindrical shells with the three respective cellular cores in Fig. 11 (a), respectively. As plotted in Fig. 11 (b), the breathing modes of the sandwich cylindrical shells surrounded by seawater also weaken the sound insulation performance for the same reason, but the modes have been omitted here for simplicity. Note that the vibroacoustic coupling strongly affects the frequency values compared with the former scenario surrounded by air.
Fig.11 The first breathing modes of the sandwich structures with the (a) PPR; (b) NPR; and (c) ZPR cellular cores; the second breathing modes of the sandwich structures with the (d) PPR; (e) NPR; and (f) ZPR cellular cores. |
In addition, the acoustic pressure distribution of the respective sandwich cylindrical shells surrounded by air at the first and second breathing resonant frequencies has been illustrated in Fig. 12, respectively. As a theoretical vibroacoustic phenomenon, the cylindrical shell structures directly inflate and compress along the radial direction, and thus an ideal cylindrical wave is generated in the acoustic medium. Note that it is the most efficient pattern to radiate the vibration energy to acoustic power, and the STL dramatically drops as a result. Measurements of vibroacoustic control close to the breathing resonances are required to increase the sound insulation performance.
Fig.12 The acoustic pressure distribution of the sandwich cylindrical shells at the first breathing resonant frequencies with the (a) PPR; (b) NPR; and (c) ZPR cellular cores; at the second breathing resonant frequencies with the (a) PPR; (b) NPR; and (c) ZPR cellular cores. |
4. Conclusion
This work studies the mechanical and acoustic performance of sandwich cylindrical shells with PPR, NPR, and ZPR cellular cores whose unit cells feature equivalent load-bearing capacities verified by using the tool of PBC. The pressure-resistant, buckling, and vibroacoustic performance have been simulated, while the mechanical behaviors of the respective PPR, NPR, and ZPR honeycomb unit cells have been extracted and compared in a specific microscopic view. The following conclusions can be drawn:
For the pressure-resistant property, the order of structural strength is PPR>NPR>ZPR for the embedded cellular cores by comparing the maximum Von Mises stress of the entire sandwich cylindrical shell. Additionally, the microscopic deformation of the respective sandwiched honeycomb unit cells differs sharply from the theoretical uniaxial-compressed situation due to the circular arrangement and stiffness of face sheets, while the adjacent NPR honeycomb unit cells feature the least circumferential deformation interference. Furthermore, the order of the energy-absorption capacity is ZPR>NPR>PPR for the three respective honeycomb unit cells by comparing their microscopic stress states, which requires a tradeoff when the pressure-resistant property is simultaneously considered. Regarding the buckling performance, the order of structural stability is PPR>ZPR>NPR for the sandwiched honeycomb cores by comparing their first buckling factors. The entire mode shapes of the entire sandwich cylindrical shell are semblable while those of local cellular cores are distinctly dependent on honeycomb types. For the sound insulation performance, the order is ZPR>NPR>PPR for the sandwiched honeycomb cores in both air and seawater, while the STL curves in the seawater slightly shift to lower frequencies compared with those in the air due to the fluid-structure interaction. The sandwich cylindrical shells with respective cellular cores radiate the most vibration energy to the acoustic medium with the breathing modes, and thus the STLs decrease sharply at the corresponding breathing resonant frequencies within the resonant-controlled region.
To summarize, the PPR cellular core sandwiched in sandwich cylindrical shells takes advantage of high structural strength and stability, while the ZPR honeycomb advances the energy-absorption capability and sound insulation performance of the corresponding meta-structure. The NPR honeycomb yields moderate properties and also has the potential for lower circumferential deformation between adjacent unit cells to better adapt to the specific circular arrangement. Nonlinear instability of elastic-plastic or plastic buckling deserves further investigation when considering initial geometric imperfections or the plasticity of the porous materials with even larger deformation, while the current additive manufacturing techniques can help to greatly reduce manufacturing errors.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
