Regulation Mechanism of the Shock Waves in a Pentamode Lattice-Ring Structure Subjected to Collision

Abstract

This paper hopes to explore the application potential of pentamode materials in the field of shock protection. Hammer percussion tests revealed that the peak strain of the inner-ring front shock surface of the pentamode lattice-ring structure is 103.9% of that of the inner-ring rear shock surface. According to the simulation results, for a solid ring of equal mass made of the same base material, the ratio mentioned above reaches 3385.7%. Compared with the solid ring of equal mass made of the same base material, the pentamode lattice-ring structure saw a decline of 65.5% in the peak strain of its inner-ring front shock surface. The distribution laws of the group velocity characterizing energy-flow characteristics were discovered by calculating cell dispersion curves in various layers of the pentamode lattice-ring structure. The laws governing the effects of cellular structure parameters on group velocity anisotropy and pentamode characteristic parameters were also revealed. It was found that the deflection angle of the energy-flow vector is positively correlated with group velocity anisotropy and that the effects of pentamode characteristic parameters π and μ on the deflection angle of the energy-flow vector vary greatly in different value ranges. The deflection angle of the energy-flow vector has a decisive effect on the protection performance of the pentamode lattice-ring structure. The conclusions of this study can provide some theoretical support for the shock protection of submarine structures.

1. Introduction

As a type of underwater vehicle, submarines mostly work in deep seas with complex topography. Since any collision accident may likely cause serious consequences, protecting submarines from collision shock is an important research task. The protection against collision shock is largely based on the energy dissipation theory, i.e., the energy generated by collision shock is absorbed through material deformation. However, conventional materials can no longer meet this requirement [1,2]. The concept of pentamode materials was first proposed by Milton and Cherkaev in 1995 [3]. Among the six eigenvalues of the elastic stiffness matrix of pentamode materials, only a single one was non-zero, i.e., it is a metamaterial with a “fluid” property. As a type of emerging metamaterial, pentamode materials can be reasonably used to design a cellular structure capable of guiding shock stress waves to propagate in the predetermined direction, offering a new solution to the regulation of shock stress waves.

At present, the research on pentamode materials has been widely carried out in the field of acoustics. Milton et al. [4] studied the coordinate transformation of elastodynamic equations based on electromagnetic coordinate transformation [5]. In theory, the modulus parameters of a structure can be designed to produce an elastic wave stealth performance, except that coordinate transformation requires the material to have strong anisotropy. In this regard, pentamode materials can satisfactorily meet design requirements. In 2008, Norris [6] proposed the concept of an invisible acoustic cloak based on the coordinate transformation of elastodynamic equations and the special properties of pentamode materials. Scandrett et al. [7,8] demonstrated the feasibility of constructing acoustic cloaks using layered pentamode materials, making it possible to design acoustically invisible cloaks. For the first time, Kadic et al. [9] fabricated a micron-scale 3D pentamode material model based on lithography. However, there is still a long way to go before practical application. LI et al. [10] designed a pentamode spherical acoustic invisible cloak composed of double-cone cells. They found that the structural parameters of pentamode materials significantly affected stealth performance and that stronger material anisotropy improved stealth performance. Nie [11] studied the problem of acoustic scattering with a 3D spherical cloak composed of non-ideal pentamode materials, analyzed the effects of its scattering characteristics, material defects, and inner-boundary constraints, and proposed a corresponding optimization strategy. The optimized cloak produced a desirable stealth performance in the target frequency band. Layman et al. [12] proposed a 2D pentamode material of honeycomb structure, thus effectively simplifying the 3D structure of pentamode materials into a 2D one and greatly reducing the fabrication difficulty while ensuring the acoustic performance. Chen et al. [13] designed a 2D pentamode acoustic cloak made of single-phase solid materials and numerically verified its acoustic wave diffraction. However, due to the resonance caused by a weak shear modulus, the cloak produced a stealth performance only in some frequency ranges. Chen et al. [14] also designed a theoretical scattering model for a pentamode cylindrical acoustic cloak with shear stiffness and systematically investigated the effects of material parameters, damping, and inner-boundary constraints on scattering. They found that the shear stiffness of pentamode materials introduced strong resonance in the low-frequency range, weakening the broadband stealth performance. Moreover, the authors also found that the best broadband stealth performance was achieved when radial fixed inner-boundary constraints were imposed.

With the increasing attention of pentamode materials in the field of acoustics, the research on overcoming the defects of pentamode materials has been gradually carried out. Due to the lack of a thorough understanding of the formation mechanism of pentamode materials, most existing pentamode materials have very similar topological characteristics, making it impossible to obtain design parameters for ideal or specified properties. To solve this problem, Wu [15] proposed a new pentamode microstructure using the topology optimization method to achieve pentamode characteristics through the overall elastic deformation of the microstructure and endow the microstructure with large and achievable quality factors. Finally, the author verified the effectiveness of the proposed topology design method based on the laser melting technology. Dong [16] employed a bottom-up topology optimization approach to propose a series of new isotropic or anisotropic pentamode microstructure configurations. Moreover, Li [17] also proposed an effective topology optimization method to discover pentamode microstructures with new material properties. Apart from optimizing the structural form of pentamode materials, some researchers further optimized the regulation effect using composite materials. Guo et al. [18] found that adding phononic crystals into a 2D pentamode structure can achieve protected propagation of elastic waves and turn the propagation path. Compared with previous pentamode materials, Wang et al. [19,20] discovered that locally resonant pentamode materials retained the characteristics of pentamode materials and possessed phononic band gaps, offering a new idea for regulating broadband acoustic waves. Krushynska et al. [21] proposed a new design idea for locally resonant pentamode materials and regulated broadband elastic waves by arranging pentamode materials and phononic crystals in a staggered pattern, thereby allowing them to work together. Cai et al. [22] proposed a method to adjust the first phononic band gaps of locally resonant pentamode materials. By studying the equivalent relationship between the local resonance mode at the lower edge of the first phononic band gaps and the spring-mass system, the authors found that the first phononic band gaps could be adjusted via microstructure design between 52 Hz and 548 Hz.

Researchers have extensively investigated the application of pentamode materials in acoustic cloaks due to their special mechanical behavior. However, no adequate attention has been paid to their application potential in other engineering fields, such as static stealth, vibration isolation devices, seismic wave regulation, and shock protection. Existing literature about pentamode materials mainly focuses on the mechanical properties of compression and shear and the protection against seismic waves, but rarely touches upon the application in shock protection. Zhang [23] predicted the mechanical properties of a pentamode material by changing the thickness of thin walls and the number of layers, aiming to endow it with an excellent load-bearing capacity. As the thickness of thin walls increased from 0.15 mm to 0.45 mm, the pentamode material showed an increase in compression modulus and a decrease in Poisson’s ratio. Skripnyak [24] assessed the dissipation effect of pentamode materials under shock loading and obtained the numerical simulation results of the mechanical behaviors of titanium-alloy pentamode materials. He used the inelastic deformation model and the ductile damage criterion to describe the mechanical properties of metamaterial cells at different strain rates and temperatures. Zhang [25] predicted the stress distribution and deformation mechanism of pentamode materials using a modified finite element model and verified the structural parameters obtained with dispersion curves. Inspired by invisible core-shell nanoparticles in optics, Bückmann [26] designed an imperceptible cloak with an elastomechanical unfeelability cloak core-shell using a pentamode material. The cloak’s calculated and measured displacement fields testified to its sound static stealth performance. Amendola’s team [27,28] investigated the mechanical responses of a pentamode lattice-sandwich structure under large elastic strains and its potential uses in vibration isolation and shock protection. It was found that the pentamode lattice-sandwich structure could be turned into a new shock protection device and vibration isolation device through a reasonable design of mesh geometry, node stiffness, and lamination schemes. Fraternali [29] pointed out that pentamode materials with hinged or semirigid connections could serve as effective shock protectors under shear shock loading.

In view of the fact that the application potential of pentamode materials in the field of shock protection has not been effectively developed, this paper studies the regulation mechanism of a pentamode lattice-ring structure against shock stress waves, and establishes the relationship between cell structure and protection performance. First, based on the coordinate transformation theory and additive manufacturing technology, a pentamode lattice-ring structure is designed and manufactured. Secondly, the hammer percussion tests and simulations of the pentamode lattice-ring structure are performed, and the dynamic responses of the pentamode lattice-ring structure under collision shock is explored. Through the hammer percussion tests and simulations, it is found that the pentamode lattice-ring structure has a good shock protection effect. Finally, the dispersion characteristics and energy-flow characteristics of the pentamode lattice-ring structure cells are studied, and the regulation mechanism of the pentamode lattice-ring structure against shock stress waves is explored. Moreover, the effects of group velocity anisotropy and pentamode characteristic parameters on the deflection angle of the energy-flow vector are considered, and the effect of the energy-flow vector’s deflection angle on the ring structure’s protection performance is accounted for. The conclusions of this study can provide some theoretical support for the applications of pentamode materials in the field of shock protection.

2. Hammer Percussion Tests on Pentamode Lattice-Ring Structure

2.1. Coordinate Transformation Theory

In an elastodynamic problem, the physical field variables are stress $\mathit{\sigma }$ and displacement $\mathit{u}$; the material properties are elastic tensor $\mathit{C}$ and density $\rho$; and the governing equations are the Navier equation and the constitutive equation:

$$\{\begin{array}{c}\mathsf{\nabla }·\mathit{\sigma }=\rho \frac{{\partial }^2\mathit{u}}{\partial t^2}\\ \mathit{\sigma }=\mathit{C}:\mathsf{\nabla }\mathit{u}\end{array}$$

(1)

After converting the virtual space (initial space) into the physical space (transformation space) with the aid of mapping $\mathit{A}$ [30]:

$$\{\begin{array}{c}{\sigma }_{ij}^{\prime }={\alpha }_I{\alpha }_J{\sigma }_{ij}\\ u_i^{\prime }={\beta }_I{u}_i\\ C_{ijkl}^{\prime }={\chi }_I{\chi }_J{\chi }_K{\chi }_L{C}_{ijkl}\\ {\rho }_{ij}^{\prime }={\phi }_I{\delta }_{ij}\rho \end{array}$$

(2)

where ${\sigma }_{ij}$, $u_j$, and $C_{ijkl}$ denote the components of stress $\mathit{\sigma }$, displacement $\mathit{u}$, and elastic tensor $\mathit{C}$, respectively; ${\alpha }_i$, ${\beta }_i$, ${\chi }_i$, and ${\phi }_i$ (i = I, J, K, L) denote the proportion factors of stress, displacement, stiffness, and density, respectively; and ${\delta }_{ij}$ denotes the Kronecker function. Since the governing equations do not change before and after coordinate transformation and the kinetic energy and potential energy before and after mapping are conserved, the following expressions can be written:

$$\{\begin{array}{c}{\phi }_i=\frac{{\lambda }_i^2}{det\mathit{A}}\\ {\chi }_i{\chi }_j{\chi }_k{\chi }_l=\frac{{\lambda }_i{\lambda }_j{\lambda }_k{\lambda }_l}{det\mathit{A}}\\ {\alpha }_i{\alpha }_j=\frac{{\lambda }_i{\lambda }_j}{det\mathit{A}}\\ {\beta }_i=1/{\lambda }_i\end{array}$$

(3)

where ${\lambda }_i$ denotes the eigenvalue of mapping $\mathit{A}$. The final mapping transformation relationship is [30]:

$$\{\begin{array}{c}{\sigma }_{ij}^{\prime }=\frac{{\lambda }_I{\lambda }_J}{det\mathit{A}}{\sigma }_{ij}\\ u_i^{\prime }=\frac{1}{{\lambda }_I}{u}_i\\ C_{ijkl}^{\prime }=\frac{{\lambda }_I{\lambda }_J{\lambda }_K{\lambda }_L}{det\mathit{A}}{C}_{ijkl}\\ {\rho }_{ij}^{\prime }=\frac{{\lambda }_I^2}{det\mathit{A}}{\delta }_{ij}\rho \end{array}$$

(4)

In a 2D polar coordinate system, the mapping relationship can be expressed as: $r^{\prime }=f\left(r\right)$ and ${\theta }^{\prime }=\theta$, where r and $\theta$ denote the polar coordinates in the virtual space, and $r^{\prime }$ and ${\theta }^{\prime }$ denote the polar coordinates in the physical space, respectively. Hence:

$$\{\begin{array}{c}{\lambda }_r=\frac{d{r}^{\prime }}{dr}\\ {\lambda }_{\theta }=\frac{r^{\prime }}{r}\frac{d{\theta }^{\prime }}{d\theta }\\ det\mathit{A}=\frac{f^{\prime }\left(r\right)f\left(r\right)}{r}\end{array}$$

(5)

By setting $f\left(r\right)=\frac{R_1}{R_2}r+R_1$ and substituting Equation (5) into Equation (4), the material density and elastic matrix can be determined [30]:

$$\{\begin{array}{c}{K}_{rrrr}^{\prime }=\frac{r{f}^{\prime }{\left(r\right)}^3}{f\left(r\right)}E\\ K_{\theta \theta \theta \theta }^{\prime }=\frac{f{\left(r\right)}^3}{r^3{f}^{\prime }\left(r\right)}E\\ K_{\theta \theta rr}^{\prime }=K_{rr\theta \theta }^{\prime }=\frac{f^{\prime }\left(r\right)f\left(r\right)}{r}E\\ {\rho }^{\prime }=\frac{{\lambda }_r^2}{det\mathit{A}}\rho \end{array}$$

(6)

where $\rho$ and $E$ denote material density and elastic modulus in the virtual space, respectively, and $K_{iIjJ}^{\prime }$ denotes the entry of material elastic matrix in the physical space. Figure 1 shows the change curves of material density and elastic matrix in the physical space.

According to the variation trends of density and elastic matrix, the material at the inner ring of the pentamode lattice-ring structure must have extremely high tangential stiffness and extremely low radial stiffness and density. Considering that existing material properties and fabrication processes cannot meet design requirements, the design and fabrication of the pentamode lattice-ring structure should prioritize $K_{\theta \theta \theta \theta }^{\prime }$ before giving due consideration to other parameters. In this study, a test model was fabricated for the pentamode lattice-ring structure based on additive manufacturing technologies. Stainless steel with the following material parameters was adopted as the base material: elastic modulus $E$ = 200 GPa; density $\rho$ = 7800 kg/m³; and Poisson’s ratio $\nu$ = 0.3. Parameters of the test model were as follows: inner radius $R_1$ = 52 mm; outer radius $R_2$ = 104 mm; cell wall thickness $t$ = 1 mm; and model height $h$ = 10 mm. The test model was composed of 13 layers, each with 50 different cells. Figure 2 and

Table 1. Structural dimensions and parameters of the test model.

Number of Layers	a (mm)	b (mm)	θ (°)	Kθθθθ (Pa)	Krrrr (Pa)	Kθθrr (Pa)	G (Pa)	ρ (kg/m3)	π	μ
1	0.96	3.58	76	9.73 × 1010	2.17 × 109	1.23 × 1010	1.71 × 1010	4435.60	0.850	1.178
2	1.19	3.73	77.5	9.07 × 1010	1.81 × 109	1.07 × 1010	1.12 × 1010	4148.31	0.835	0.877
3	1.32	3.94	72.5	6.56 × 1010	3.32 × 109	1.34 × 1010	1.04 × 1010	3443.00	0.907	0.707
4	1.47	4.18	71.5	5.72 × 1010	3.55 × 109	1.31 × 1010	8.39 × 109	3146.46	0.922	0.589
5	1.62	4.42	70.5	5.03 × 1010	3.79 × 109	1.29 × 1010	6.86 × 109	2895.80	0.935	0.497
6	1.79	4.69	69.5	4.40 × 1010	4.01 × 109	1.26 × 1010	5.54 × 109	2661.12	0.945	0.417
7	1.97	4.98	68.5	3.85 × 1010	4.23 × 109	1.22 × 1010	4.49 × 109	2451.11	0.954	0.352
8	2.16	5.31	67.5	3.38 × 1010	4.40 × 109	1.17 × 1010	3.64 × 109	2257.64	0.962	0.298
9	2.38	5.66	67	3.02 × 1010	4.37 × 109	1.11 × 1010	2.88 × 109	2091.02	0.967	0.250
10	2.6	6.05	66.5	2.72 × 1010	4.31 × 109	1.05 × 1010	2.30 × 109	1939.06	0.972	0.213
11	2.85	6.49	65.5	2.37 × 1010	4.43 × 109	1.00 × 1010	1.84 × 109	1802.35	0.976	0.180
12	3.12	6.96	65	2.12 × 1010	4.34 × 109	9.41 × 109	1.46 × 109	1649.35	0.980	0.152
13	3.39	7.3	65	1.98 × 1010	4.18 × 109	8.93 × 109	1.19 × 109	1558.56	0.982	0.130

a—vertical member length of cell; b—diagonal member length of cell; θ—topological angle of cell. Layers 1–13 are marked starting from the inside to the outside of the pentamode lattice-ring structure.

show the structural and cellular dimensions of the test model, respectively.

2.2. Equivalent Mechanical Parameters and Pentamode Characteristic Parameters of Pentamode Lattice-Ring Structure

A honeycomb hexagonal periodic structure’s equivalent mechanical parameters are mainly affected by member length, member width, and topological angle. Its equivalent elastic matrix is as follows:

$$\mathit{C}=\left[\begin{array}{ccc}{K}_{\theta \theta \theta \theta }& K_{\theta \theta rr}& 0\\ K_{\theta \theta rr}& K_{rrrr}& 0\\ 0& 0& G\end{array}\right]$$

(7)

$$K_{\theta \theta \theta \theta }=\frac{E_S\delta \left(4{\sin }^2\theta +{\delta }^2{\cos }^2\theta +2\xi {\delta }^2\right)\sin \theta }{2\left(\xi +\cos \theta \right)\left(2+4\xi {\cos }^2\theta +\xi {\delta }^2{\sin }^2\theta \right)}$$

(8)

$$K_{rrrr}=\frac{E_S\delta \left(\xi +\cos \theta \right)\left(4{\cos }^2\theta +{\delta }^2{\sin }^2\theta \right)}{2\left(2+4\xi {\cos }^2\theta +\xi {\delta }^2{\sin }^2\theta \right)\sin \theta }$$

(9)

$$K_{\theta \theta rr}=\frac{E_S\delta \left(4-{\delta }^2\right)\sin \theta \cos \theta }{2\left(2+4\xi {\cos }^2\theta +\xi {\delta }^2{\sin }^2\theta \right)}$$

(10)

$$G=\frac{4E_S{\delta }^3\left(\xi +\cos \theta \right)\sin \theta }{{\delta }^2+4\left(1+2\xi \right){\xi }^2{\sin }^2\theta +\left(2+\xi \cos \theta \right){\delta }^2\xi \cos \theta }$$

(11)

where $K_{\theta \theta \theta \theta }$ denotes the equivalent elastic modulus in the $\theta$-axis direction; $K_{rrrr}$ denotes the equivalent elastic modulus in the r-axis direction; $K_{\theta \theta rr}$ denotes the coupling elastic modulus between two principal directions; $G$ denotes the equivalent shear modulus; $E_S$ denotes the elastic modulus of the base material; $\xi$ denotes member length ratio; $\xi =a/b$; $\delta$ denotes member slenderness ratio; $\delta =t/b$; and $\theta$ denotes the topological angle of the cell. The ideality of pentamode materials is mainly characterized by the following parameters:

$$\pi =\frac{\left|K_{\theta \theta rr}\right|}{\sqrt{K_{\theta \theta \theta \theta }{K}_{rrrr}}}$$

(12)

$$\mu =\frac{G}{\sqrt{K_{\theta \theta \theta \theta }{K}_{rrrr}}}$$

(13)

when $\pi =1$ and $\mu =0$, the material is an ideal pentamode material. However, restricted by fabrication process and use requirements,$\pi$ and $\mu$ can only tend towards 1 and 0, respectively. Table 1 provides the equivalent mechanical and pentamode characteristic parameters of cells in various layers of the pentamode lattice-ring structure. It can be seen that the equivalent elastic matrix of cells in various layers presents strong anisotropy and that the value of $K_{\theta \theta \theta \theta }$ is higher than $K_{rrrr}$. In addition, the pentamode characteristics of inner-layer cells are inferior to those of outer-layer cells. Restricted by the fabrication process, inner-layer cells had an excessively large member slenderness ratio, which led to poor pentamode characteristics.

3. Hammer Percussion Tests on Pentamode Lattice-Ring Structure

3.1. Working Conditions of Hammer Percussion Tests

Five measuring points were selected on the inner ring of the model, located at its upper part (point A), upper-middle part (point B), middle part (point C), lower-middle part (point D), and lower part (point E). A strain gauge was attached to each measuring point along the circumferential direction to collect and record the strain values at these points. Two supports were made of polycarbonate (PC) as the base material based on additive manufacturing technologies. The shape of the supports was an isosceles right triangle with an indented hypotenuses 25 mm in length. Material parameters of PC: elastic modulus $E$ = 2000 MPa; density $\rho$ = 1200 kg/m³; and Poisson’s ratio $\nu$ = 0.4. The supports were bonded to the outer layer of the pentamode lattice-ring structure using super glue. Then, the supports and the test model were fixed by bolts onto an iron frame, and the test model was struck with a hammer. The force signals of hammer percussion were collected. The diagram of the test device is shown in Figure 3.

3.2. Test Results

After the removal of accidental errors through multiple repeated tests, the test data in the peak-strain stage of the five measuring points were selected to draw their strain time-history curves, as shown in Figure 4.

Table 2. Peak strains and protection performance parameters.

Data Source	$\mathbf{Peak}\mathbf{Strain}(\mathsf{\mu }\mathsf{\epsilon })$						$\mathbf{Peak}\mathbf{Strain}\mathbf{Moment}\left(\mathrm{ms}\right)$					γ
	${\mathit{\epsilon }}_A$	${\mathit{\epsilon }}_{\mathit{B}}$	${\mathit{\epsilon }}_{\mathit{C}}$	${\mathit{\epsilon }}_{\mathit{D}}$	${\mathit{\epsilon }}_{\mathit{E}}$	${\mathit{\epsilon }}_A/{\mathit{\epsilon }}_{\mathit{E}}$	${\mathit{t}}_{\mathit{A}}$	${\mathit{t}}_{\mathit{B}}$	${\mathit{t}}_{\mathit{C}}$	${\mathit{t}}_{\mathit{D}}$	${\mathit{t}}_{\mathit{E}}$
Pentamode lattice-ring structure (test)	101.3	−56.8	−123.2	29.6	97.5	103.9%	0.22	0.21	0.21	0.23	0.22	31.4
Pentamode lattice-ring structure (simulation)	91.5	−48.3	−120.4	26.2	106.4	86.0%	0.23	0.175	0.18	0.255	0.22	38.0
Solid ring of equal mass (simulation)	237.0	−125.4	−105.6	31.6	7.0	3385.7%	0.205	0.21	0.215	0.22	0.22	0.9

provides the tested peak strains and peak strain moments of the five measuring points. According to the strain time-history curves, measuring point C had the largest peak strain; measuring point A, located on the front shock surface, and measuring point E, located on the rear shock surface, had almost the same peak strain (i.e., ${\epsilon }_A/{\epsilon }_E=103.9\%$); the peak strain of measuring point E was higher than that of measuring point D, which was a very abnormal phenomenon. In addition, the peak strain moments of measuring points B and C were earlier than those of other measuring points, suggesting that shock stress waves first arrived at measuring points B and C instead of measuring point A.

The above observation constituted an obvious difference between the pentamode lattice-ring structure and conventional rings. If a conventional ring were to be impacted, the peak strain of the front shock surface would be far greater than that of the rear shock surface. The peak strains of the five measuring points could be ranked in descending order of A, B, C, D, and E. The order in which shock stress waves arrived at various measuring points would be A, B, C, D, and E. The test results of the pentamode lattice-ring structure significantly differed from those of a conventional ring, suggesting that shock stress waves followed a different propagation path in the pentamode lattice-ring structure, which could effectively weaken the shock on the front shock surface.

4. Shock Simulation of Pentamode Lattice-Ring Structure

4.1. Simulation Modeling

The force signals collected were processed using MATLAB software, followed by the fitting of the analytical formula of the percussion force function $F\left(T\right)$, $F\left(T\right)=7.686\times {10}^{36}\times T^{10}-3.605\times {10}^{34}\times T^9+7.320\times {10}^{31}\times T^8-8.263\times {10}^{28}\times T^7+5.550\times {10}^{25}\times T^6-2.20\times {10}^{22}\times T^5+4814\times T^4-4928\times T^3+1.811\times {10}^{10}\times T^2-42633.244\times T-1.824$. The spectral analysis shown in Figure 5 was performed on the percussion force. The main frequencies of the force signals were distributed below 5000 Hz.

A model was built for the pentamode lattice-ring structure using COMSOL finite element simulation software. Free triangular mesh division was adopted, and a load of $F\left(T\right)$ was applied where the hammer struck, as shown in Figure 6a. A model was set up for a solid steel ring of equal mass to increase the contrast. The solid ring had an outer-ring radius of $R_3$ = 74.5 mm and an inner-ring radius of $R_1$. The other working conditions were the same as those for the pentamode lattice-ring structure, as shown in Figure 6b.

4.2. Simulation Results

The model simulation was performed using finite element simulation software and the strain values at the five measuring points of the pentamode lattice-ring structure were output. A comparison between simulated and test results is shown in Figure 4. Table 2 provides the simulated peak strains and peak strain moments of the five measuring points. The errors between tested and simulated peak strains all fell within 15% for the same measuring point. The order of arrival of the peak strain moments for various measuring points in the tests was identical to that of the simulation. Hence, a mutual verification between test and simulation results was observed. According to Figure 4, the peak strain of the measuring point on the front shock surface (point A) in the test data is slightly higher than that of the measuring point on the rear shock surface (point E). Moreover, in the simulation data, the peak strain of the measuring point on the front shock surface (point A) is slightly lower than that of the measuring point on the rear shock surface (point E). This is probably because the supports could not be entirely regarded as fixed constraints under existing test conditions, and there were certain errors in the hammer striking points.

In Table 2, ${\epsilon }_i$ denotes the peak strain of measurement point i$,\mathrm{and} t_i$ denotes the peak strain moment of measurement point i. The simulation results of the pentamode lattice-ring structure and the solid ring of equal mass were compared in Figure 7. Compared with the solid ring of equal mass, the pentamode lattice-ring structure showed a significant decline in the peak strain of the front shock surface and a significant rise in the peak strain of the rear shock surface. Specifically, measuring points A, B, and D showed declines of 65.5%, 61.4%, and 17.1%, while measuring points C and E experienced an increase of 14.0% and 3958.3%, respectively. Moreover, the order of arrival of the peak strain moments of various measuring points on the solid ring of equal mass was from measuring point A to measuring point E, which distinguished the solid ring of equal mass from the pentamode lattice-ring structure. Therefore, the propagation mode of shock stress waves in the solid ring of equal mass differed from that in the pentamode lattice-ring structure. A protection performance parameter $\gamma$, proposed to characterize the protective effect of the ring structure against shock stress waves, is defined as follows:

$$\gamma =\frac{{\rho }_S{\epsilon }_E{R}_1^3}{{\rho }_0{\epsilon }_A{R}_n^3}\times 100$$

(14)

where ${\rho }_0$ denotes the effective density of the ring structure, ${\rho }_0=\frac{M}{V}$, kg/m³; $M$ denotes the total weight of the ring structure, $V=h\left(\pi R_n^2-\pi R_1^2\right)$; $h$ denotes the height of the ring structure; ${\rho }_S$ denotes the density of the base material of the ring structure; $R_1$ and $R_n$ denote the inner radius and outer radius of the ring structure, respectively, m; and ${\epsilon }_A$ and ${\epsilon }_E$ denote the peak strains of the ring structure’s inner-ring front shock surface and rear shock surface, respectively. A higher $\gamma$ value means a smaller thickness, smaller mass, and better protection effect of the ring structure. Table 2 provides the $\gamma$ values of the pentamode lattice-ring structure and the solid ring of equal mass. The $\gamma$ value of the pentamode lattice-ring structure exceeded 30, while that of the solid ring of equal mass was only 0.9. This indicates that the protective effect of the pentamode lattice-ring structure against shock stress waves is far stronger than that of the solid ring of equal mass.

The stress field distributions of the pentamode lattice-ring structure and the solid ring of equal mass at various moments were output using COMSOL finite element simulation software, as shown in Figure 8. As can be seen from the simulation results, after the pentamode lattice-ring structure was impacted at 0.05 ms, stress was mainly concentrated at the top of the model and presented a bifurcation trend. At 0.10 ms, the stress produced an obvious bifurcation effect and mainly transferred downward along the bifurcation direction, while the stress at measuring point A was very low. At 0.15 ms, stress first transferred to measuring points B and C, which explains why the peak strain moments of measuring points B and C were earlier than those of other measuring points. At 0.20 ms, measuring point C showed a stress concentration significantly higher than the one at other measuring points, while the stress continued to transfer downward and gradually converged to the supports. At 0.25–0.30 ms, stress converged to the two supports, and high stress appeared at measuring point E.

On the other hand, after the solid ring of equal mass was impacted at 0.05–0.10 ms, stress presented no bifurcation trend and first transferred to measuring point A. At 0.15–0.30 ms, stress successfully transferred to measuring points B, C, and D, finally converging at the supports, while measuring point E was almost stress-free. Compared with the solid ring of equal mass, the pentamode lattice-ring structure showed a significant decline in the peak strain of measuring point A and a significant rise in the peak strain of measuring point E due to the difference in stress transfer paths between the pentamode lattice-ring structure and the solid ring of equal mass. Moreover, a change in the order of arrival of the peak strain moments for various measuring points was also observed.

5. Analysis of Dispersion Characteristics and Energy-Flow Characteristics

Dispersion characteristics constitute an inherent property of the material, reflecting the propagation characteristics of waves in the material. An analysis of dispersion characteristics helps clarify the shock stress wave’s energy-flow characteristics. Energy-flow characteristics, on the other hand, determine the protective effect of the pentamode lattice-ring structure. Therefore, establishing the relationships among dispersion characteristics, energy-flow characteristics, and protection are of great significance for the applications of pentamode materials in the field of shock protection.

5.1. Dispersion Curves and Group Velocity

Pentamode materials are periodic structures composed of cells as the smallest unit. By performing frequency sweeping along the boundary of the irreducible Brillouin zone of cells, we can obtain the dispersion curve of periodic structures. The tangent slope of a point on the dispersion curve is the magnitude of group velocity at the frequency. Group velocity is defined as the equal-amplitude plane’s propagation velocity, i.e., energy’s propagation velocity. The dispersion curves of cells in 13 layers can be solved using COMSOL finite element simulation software, as shown in Figure 9.

According to Figure 9, below 10,000 Hz, the group velocities of cells are hardly affected by frequency, and the s-wave group velocities of the two principal directions have an equal magnitude.

Table 3. Group velocities of cells for the pentamode lattice-ring structure.

Group Velocity	Number of Layers
	1	2	3	4	5	6	7	8	9	10	11	12	13
$C_{\mathrm{P}\theta }$ (m/s)	5123.7	5098.4	4821.1	4709.0	4598.7	4484.9	4370.5	4110.4	4175.3	4101.1	4064.0	3913.8	3877.7
$C_{\mathrm{P}r}$ (m/s)	989.2	967.8	1285.5	1350.3	1417.1	1487.4	1560.5	1575.4	1669.0	1701.5	1693.0	1814.6	1824.3
$C_{\mathrm{S}}$ (m/s)	1767.2	1599.4	1670.7	1597.4	1527.4	1450.1	1374.5	1253.9	1210.8	1132.0	1045.3	986.0	918.0
$C_{\mathrm{P}\theta }$$/C_{\mathrm{P}r}$	5.180	5.268	3.750	3.487	3.245	3.015	2.801	2.609	2.502	2.410	2.400	2.157	2.126
$\alpha$ (°)	70.13	70.66	64.82	63.60	62.78	61.83	61.07	60.27	59.50	57.89	57.36	55.32	54.10

Legend: $C_{\mathrm{P}\theta }$ denotes the magnitude of the p-wave group velocity in the ΓX direction (tangential direction); $C_{\mathrm{P}r}$ denotes the magnitude of the p-wave group velocity in the ΓY direction (radial direction); $C_{\mathrm{S}}$ denotes the magnitude of the s-wave group velocity; and $\alpha$ denotes the deflection angle of the energy-flow vector, i.e., the included angle between the energy-flow vector in the ring and the radial direction of the ring.

provides the group velocities of cells in various layers. For the same excitation, the energy of the p-wave and the s-wave is directly proportional to the square of the p-wave velocity and the square of the s-wave velocity, respectively. By contrast, below 10,000 Hz, the p-wave group velocity of cells in each layer is more than four times the s-wave group velocity. The energy contained by the s-wave is less than 1/16 of that of the p-wave. Hence, the group velocity of the p-wave is the main research object.

5.2. Effects of Cellular Structure Parameters on Group Velocity Anisotropy and Pentamode Characteristic Parameters

Cellular structure parameters include topological angle $\theta$, vertical member length $a$, diagonal member length $b$, and wall thickness $t$. A certain cellular structure parameter was changed based on the cells in the 13th layer to explore its effects on group velocity anisotropy and pentamode characteristic parameters, as shown in Figure 10 and Figure 11.

When topological angle $\theta$ increased, $C_{\mathrm{P}\theta }$ and $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ increased with it, while $C_{\mathrm{P}r}$ and $C_{\mathrm{S}}$ decreased. When vertical member length $a$ increased, $C_{\mathrm{P}\theta }$, $C_{\mathrm{S}}$, and $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ increased with it, while $C_{\mathrm{P}r}$ decreased. When diagonal member length $b$ increased, $C_{\mathrm{P}r}$ increased with it, while $C_{\mathrm{P}\theta }$, $C_{\mathrm{S}}$, and $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ decreased. When wall thickness $t$ increased, $C_{\mathrm{P}\theta }$, $C_{\mathrm{P}r}$, and $C_{\mathrm{S}}$ increased with it, while $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ first increased and then decreased. Therefore, adopting a larger topological angle $\theta$, larger vertical member length $a$, smaller diagonal member length $b$, and suitable wall thickness $t$ will produce a greater $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value. In addition, according to the variation range and rate of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, adjusting topological angle $\theta$ can help to more effectively change the $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value. Notably, the s-wave group velocity $C_{\mathrm{S}}$ should not be overly large. In cell design, $C_{\mathrm{S}}$ should be properly controlled to weaken the effect of the s-wave on the regulation effect.

When topological angle $\theta$ increased, $K_{\theta \theta \theta \theta }$ increased with it, while $K_{rrrr}$, $K_{\theta \theta rr}$, and $G$ decreased. Moreover, $\pi$ first increased and then decreased, and $\mu$ first decreased and then increased. When vertical member length $a$ increased, $K_{rrrr}$ increased with it, while $K_{\theta \theta \theta \theta }$, $K_{\theta \theta rr}$, $G$, $\pi$, and $\mu$ decreased; the amplitude of variation in $\pi$ could almost be ignored. When diagonal member length $b$ increased, $\pi$ increased with it, while $K_{\theta \theta \theta \theta }$, $K_{rrrr}$, $K_{\theta \theta rr}$, $G$, and $\mu$ decreased. When wall thickness $t$ increased,$K_{\theta \theta \theta \theta }$, $K_{rrrr}$, $K_{\theta \theta rr}$, $G$, and $\mu$ increased with it, while $\pi$ decreased. Therefore, adopting a suitable topological angle $\theta$, larger vertical member length $a$, smaller diagonal member length $b$, and smaller wall thickness $t$ will improve the pentamode characteristic parameters.

5.3. Effect of Group Velocity Anisotropy on Energy-Flow Direction

Since group velocity determines energy-flow direction, group velocity anisotropy will affect energy-flow direction. Based on its equivalent mechanical parameters, the pentamode lattice-ring structure was equated to a homogeneous ring of equal size. Various layers of the equivalent homogeneous ring were made of anisotropic materials with inconsistent radial and tangential stiffness. A contrast ring of equal size, made of isotropic materials, was also set up. A point load was applied at the vertex of each ring in the magnitude of the force signal fitting function $F\left(T\right)$ shown in Section 3.1. The energy field distributions of the pentamode lattice-ring structure, the equivalent homogeneous ring, and the contrast ring at different moments were output, as shown in Figure 12. According to Figure 12, the overall energy-flow trajectory of the equivalent homogeneous ring was consistent with that of the pentamode lattice-ring structure, showing an obvious directionality. The energy of shock stress waves diverged to both sides of the ring, effectively reducing energy concentration on the front shock surface. However, cavities in the pentamode lattice-ring structure caused the energy of shock stress waves to transfer along the members, resulting in certain energy divergence. As a result, the width of the energy-flow trajectory of the pentamode lattice-ring structure was significantly larger than that of the equivalent homogeneous ring. The energy-flow trajectory of the contrast ring significantly differed from that of the pentamode lattice-ring structure and the equivalent homogeneous ring. The energy of shock stress waves propagated and spread spherically, and obvious energy concentration occurred on the front shock surface.

The direction of the energy-flow vector in the same layer of material of the equivalent homogeneous ring is gradually changing in a Cartesian coordinate system, while the energy-flow trajectory is shaped as an arc. By contrast, in a polar coordinate system, the direction of the energy-flow vector in the same layer of material almost remained unchanged, mainly because various layers of the equivalent homogeneous ring were made of anisotropic materials with inconsistent radial and tangential stiffness. The deflection angle of the energy-flow vector was read in a polar coordinate system. The deflection angle $\alpha$ of the energy-flow vector for each of the 13 material layers is provided in Table 3. By analyzing the variation laws, it can be seen that the variation trend was positively correlated with $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, and that a higher $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value implied stronger group velocity anisotropy.

A 200 mm × 50 mm rectangular material model was built. A shock load was applied at the midpoint of the upper edge of the model, and low-reflection boundaries were set up on the four sides. An anisotropic material with $\pi =1$ and $\mu =0.01$ was adopted. Changing the $K_{\theta \theta \theta \theta }$ and $K_{rrrr}$ of the material also modifies its $C_{\mathrm{P}\theta }$ and $C_{\mathrm{P}r}$. The energy-flow vector in the model was output to identify the effect of group velocity anisotropy $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ on the deflection angle α of the energy-flow vector, as shown in Figure 13. According to Figure 13, stronger group velocity anisotropy $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ meant a larger deflection angle α of the energy-flow vector, and $\Delta \alpha$/$\Delta \left(C_{\mathrm{P}\theta }/C_{\mathrm{P}r}\right)$ decreased with an increase in $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ and gradually approached 0. In other words, with an increase in group velocity anisotropy $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, the role of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ in increasing the deflection angle $\alpha$ of the energy-flow vector can be gradually assumed as negligible. In practical applications, it is impossible for $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ to be infinite, and the deflection angle $\alpha$ of the energy-flow vector has an upper limit, i.e., $\alpha \le 90°$. Moreover, pursuing an excessively large $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value will reduce the cost performance of the material. Therefore, the design of pentamode cells should give comprehensive consideration to the fabrication process and cost performance, thus choosing a reasonable $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value.

5.4. Effects of Pentamode Characteristic Parameters on Energy-Flow Direction

Pentamode characteristic parameters determine the ideality of pentamode materials, i.e., the similarity between the material and the fluid. Their energy-flow characteristics differ greatly for materials with different pentamode characteristic parameters. A 200 mm × 50 mm rectangular material model with $K_{\theta \theta \theta \theta }=9\times {10}^{10}\mathrm{Pa}$ was built. A shock load was applied at the midpoint of the upper edge of the model, and low-reflection boundaries were set up on the four sides. An anisotropic material with $K_{\theta \theta \theta \theta }=9\times {10}^{10}\mathrm{Pa}$ and $K_{rrrr}=1\times {10}^{10}\mathrm{Pa}$ was adopted. Changing the $K_{\theta \theta rr}$ and $G$ of the material also modifies the values of $\pi$ and $\mu$. The energy field distribution in the model at 0.15 ms was output to identify the effects of pentamode characteristic parameters $\pi$ and $\mu$ on the deflection angle $\alpha$ of the energy-flow vector, as shown in Figure 14.

According to Figure 14, pentamode characteristic parameters $\pi$ and $\mu$ affect the deflection angle $\alpha$ of the energy-flow vector. When $\pi =1$, regardless of the value of $\mu$, the deflection angle $\alpha$ of the energy-flow vector always stabilizes at 55°, in which case the deflection angle $\alpha$ of the energy-flow vector is independent of $\mu$. When $\pi \ne 1$, pentamode characteristic parameters $\pi$ and $\mu$ jointly affect the deflection angle $\alpha$ of the energy-flow vector. With a decrease in the value of $\pi$ or $\mu$, the deflection angle $\alpha$ of the energy-flow vector gradually decreases, especially when $\mu \le 0.1$. In addition, when $\pi \le 0.9$ and $\mu$ approaches 0, the material loses its energy-flow vector deflection effect. Thus, to ensure that a pentamode material has a large deflection angle $\alpha$ of the energy-flow vector, the pentamode material should have a certain $\mu$ value, i.e., a certain shear strength, under the premise of $\pi$ tending towards one.

5.5. Effects of Energy-Flow Direction on Protection Performance Parameters

Based on the model in Figure 6b, the inner and outer radii of the ring were set as $R_1$ and $R_2$, respectively. The deflection angle $\alpha$ of the energy-flow vector was changed to identify its effect on protection performance parameter $\gamma$, as shown in Figure 15.

With an increase in the deflection angle $\alpha$ of the energy-flow vector, the peak strain of measuring point A gradually decreases, while that of measuring point E gradually increases. Protection performance parameter $\gamma$ was positively correlated with the deflection angle $\alpha$ of the energy-flow vector, and $\Delta \gamma /\Delta \alpha$ increased with an increase in the deflection angle $\alpha$ of the energy-flow vector. Increasing the deflection angle $\alpha$ of the energy-flow vector can quickly increase the protection performance parameter $\gamma$, while the deflection angle $\alpha$ of the energy-flow vector is positively correlated with the values of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$. Thus, increasing the values of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$ will improve the effect of pentamode materials in regulating shock stress waves. When designing the deflection angle $\alpha$ of the energy-flow vector, the values of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$ can be increased within a reasonable design range to achieve a better shock protection effect.

6. Conclusions

In the study this paper, a test model was designed and fabricated for the pentamode lattice-ring structure based on the coordinate transformation theory and additive manufacturing technologies. Hammer percussion tests, shock simulation, and dispersion characteristic analysis revealed the dynamic response characteristics of the pentamode lattice-ring structure under collision shock loading. The specific conclusions of this study are as follows:

(1) Hammer percussion tests revealed that the peak strain of the inner-ring front shock surface of the pentamode lattice-ring structure was 103.9% of that of the inner-ring rear shock surface. Simulation results showed that, for a solid ring of equal mass made of the same base material, this ratio reached as high as 3385.7%. Moreover, compared with the solid ring, the pentamode lattice-ring structure saw a decline of 65.5% in the peak strain of its inner-ring front shock surface. The protection performance parameter $\gamma$ of the pentamode lattice-ring structure exceeded 30, while that of the solid ring of equal mass was only 0.9. In addition, after the pentamode lattice-ring structure was impacted, stress transfer presented a bifurcation trend. Stress first transferred to measuring points B and C, significantly lowering the stress at measuring point A. Thus, the pentamode lattice-ring structure can effectively weaken the shock from shock waves on the front shock surface, avoiding the early failure of the front shock surface.

(2) By investigating the effects of cellular structure parameters on group velocity anisotropy and pentamode characteristic parameters, it was found that $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ was positively correlated with topological angle $\theta$ and vertical member length $a$ and negatively correlated with diagonal member length $b$. In addition, $\pi$ was positively correlated with $b$ and negatively correlated with $a$ and wall thickness $t$. Lastly, $\mu$ was positively correlated with wall thickness $t$ and negatively correlated with $a$ and diagonal member length $b$. In addition, when wall thickness $t$ increased, $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ first increased and then decreased. When topological angle $\theta$ increased,$\pi$ first increased and then decreased, while $\mu$ first decreased and then increased. This means that, given other parameters, wall thickness $t$ has an optimal value that maximizes $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$. In addition, topological angle $\theta$ has an optimal value that optimizes pentamode characteristic parameters. According to the variation range and rate of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, adjusting topological angle $\theta$ can help to more effectively increase group velocity anisotropy $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$. The design of pentamode cells can be optimized based on the above conclusion to obtain ideal values for $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$.

(3) By analyzing the effects of group velocity anisotropy $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ on the deflection angle $\alpha$ of the energy-flow vector, it was found that stronger group velocity anisotropy $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ meant a larger deflection angle $\alpha$ of the energy-flow vector, and that $\Delta \alpha /\Delta \left(C_{P\theta }/C_{Pr}\right)$ decreased with increasing $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ and gradually tended towards 0. In practical applications, it is impossible for $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ to be infinite, and the deflection angle of the energy-flow vector has an upper limit. Moreover, pursuing an excessively large $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value will reduce the cost performance of the material. Therefore, designing the pentamode cells should give comprehensive consideration to the fabrication process and cost performance by choosing a reasonable $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$ value.

(4) By analyzing the effects of pentamode characteristic parameters $\pi$ and $\mu$ on the deflection angle $\alpha$ of the energy-flow vector, it was discovered that pentamode characteristic parameters $\pi$ and $\mu$ significantly affected the deflection angle $\alpha$ of the energy-flow vector. When $\pi =1$, the deflection angle $\alpha$ of the energy-flow vector was independent of $\mu$. When $\pi \ne 1$, the deflection angle $\alpha$ of the energy-flow vector was positively correlated with pentamode characteristic parameters $\pi$ and $\mu$, especially when $\mu \le 0.1$. In addition, when $\pi \le 0.9$ and $\mu$ approached 0, the material lost its energy-flow vector deflection effect. Thus, to ensure that a pentamode material has a large deflection angle α of the energy-flow vector, the pentamode material should have a large $\mu$ value, i.e., a certain shear strength, under the premise of $\pi$ approaching 1.

(5) By analyzing the effects of the deflection angle $\alpha$ of the energy-flow vector on protection performance parameter $\gamma$, it was found that protection performance parameter $\gamma$ was positively correlated with the deflection angle $\alpha$ of the energy-flow vector, and that $\Delta \gamma /\Delta \alpha$ increased with the deflection angle $\alpha$ of the energy-flow vector. Increasing the deflection angle $\alpha$ of the energy-flow vector could quickly increase the protection performance parameter $\gamma$, while the deflection angle α of the energy-flow vector is positively correlated with the values of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$. Thus, increasing the values of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$ will improve the effects of pentamode materials in regulating shock stress waves.

(6) When designing the pentamode lattice-ring structure, the mechanical parameter distribution of the pentamode lattice-ring structure should first be obtained based on the coordinate transformation theory. Then, the designing of the pentamode cells should make the equivalent mechanical parameters for each layer in which cells meet the requirements of coordinate transformation as much as possible. Due to the limitations of material performance and manufacturing process, the design and fabrication of the pentamode lattice-ring structure should prioritize $K_{\theta \theta \theta \theta }^{\prime }$ before giving due consideration to other parameters. In addition, when designing the cell structure, the values of $C_{\mathrm{P}\theta }$/$C_{\mathrm{P}r}$, $\pi$, and $\mu$ should be increased within a reasonable design range to achieve a better shock protection effect.