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Auxetic mechanical metamaterials that exhibit a negative Poisson’s ratio (NPR) can be artificially designed to exhibit a unique range of physical and mechanical properties. Novel sandwich structures composed of uniform and gradient auxetic double arrowhead honeycomb (DAH) cores were investigated in terms of their vibration and sound transmission performance stimulated by nonhomogeneous metamaterials with nonperiodic cell geometries. The spectral element method (SEM) was employed to accurately evaluate the natural frequencies and dynamic responses with a limited number of elements at high frequencies. The results indicated that the vibrating mode shapes and deformations of the DAH sandwich models were strongly affected by the patterned gradient metamaterials. In addition, the sound insulation performance of the considered DAH sandwich models was investigated regarding the sound transmission loss (STL) from 1 Hz to 1500 Hz under a normal incident planar wave, and this performance was compared with that for hexagonal honeycomb sandwich panels. A programmable structural-acoustic optimization was implemented to maximize the STL while maintaining a constant weight and high strength. The results showed that the uniform DAH sandwich models with larger NPRs generally exhibited better vibration and acoustic attenuation behaviors and that the optimized gradient increasing NPR models yielded higher STL values than the optimized gradient decreasing NPR models for two specified frequency cases, with improvements of 6.52 dB and 2.52 dB and a higher bending stiffness but a lower overall STL. Thus, sandwich panels consisting of auxetic DAHs can achieve desirable vibroacoustic performance with a higher bending stiffness than conventional hexagonal honeycomb sandwich structures, and the design of gradient DAHs can be extended to obtain optimized vibration and noise-control capabilities.

Metamaterials are artificial structures engineered to achieve unusual properties, and metamaterials that behave mechanically and have a negative Poisson’s ratio (NPR) are called “auxetic” mechanical metamaterials [

In recent decades, cellular solids have been progressively employed [

The aforementioned sandwiched cellular structures are usually characterized by repeating unit cells with a fixed geometry. However, most current metamaterials are generally classified into two types based on their structural arrangements: homogenous (with uniform periodic structures) and inhomogeneous (with nonuniform nonperiodic structures) [

Accordingly, a novel sandwich panel with cellular cores of auxetic DAHs is proposed and investigated with respect to its vibration and acoustic characteristics. To the best of our knowledge, such structures have not been previously investigated. In addition, gradient double arrowhead cellular cores are designed and implemented in the direction of the panel thickness based on the features of the nonuniform metamaterials. Programmable structural-acoustic optimization processes are then implemented to design a modified-gradient DAH for an optimized sandwich panel that radiates less sound at different frequency regions. To precisely predict the structural vibroacoustic performance, the spectral element method (SEM) [

In the remainder of the paper, Section

As shown in Figure _{1} and _{2} denote the respective internal angles between the two inclined cell ribs and the vertical axis, _{x} and _{y} are the cell dimensions, and each unit cell is associated with a rectangle that measures _{x} × _{y}, where _{x} = 2_{y}/_{x} is the cell aspect ratio. The cell aspect ratio _{1} and _{2} define the cell shape. For loading in the _{2} is given, and _{1} can then be derived through equation (_{y}/_{x}. In addition, the feasible constraint of

(a) Schematic diagrams of an array of auxetic DAHs and (b) a unit cell.

The considered sandwich panels with DAH cores are depicted in Figure _{x} is calculated to be 50 mm and _{y} is 10 mm with

Sandwich panels with DAH cores from models 1 to 7: (a) the uniform models; (b) the gradient DNPR models; (c) the gradient INPR models.

To maintain a constant weight, the thickness of the two face sheets is constrained by a constant value of _{s} = 2 mm. In addition, the thickness of the cellular cores is varied to maintain the weight equal to that of a baseline cellular sandwich panel with dimensions of 2 m × 50 mm and a relative density of 0.1. To this point in the study, both the overall length and weight have been held constant to compare the various properties of the sandwich models. The base material for the sandwich panel is aluminum with a Young’s modulus _{s} = 71.9 GPa, Poisson’s ratio _{s} = 2700 kg/m^{3}. The total weights of the two face sheets and core are 21.6 kg/m and 27.0 kg/m, respectively.

The first three models shown in Figure _{2} increases from 45 to 65°. Both of these values monotonously decrease as _{2} increases. Here, we stipulate that a large NPR signifies a Poisson’s ratio with a large absolute value. Thus, the DAHs with larger NPRs yield large effective elastic modulus values. In contrast, as illustrated in Figures _{2} can generate sandwich models with DAHs of a decreasing NPR (DNPR) and an increasing NPR (INPR), respectively, particularly in this scenario. Regarding the gradient sandwich panels, the DAHs with large NPRs mainly yield large

Mechanical indices: (a)

The considered sandwich panel is assumed to be infinite along the

Global and local coordinate systems and DOFs, where (^{(e)} denotes the element length.

The partial differential equations of motion that describe the longitudinal and transverse vibrations of a uniform Bernoulli–Euler beam are represented in [_{s}/(1 ‒ ^{2}) here under the planar strain assumption. Equation (

The homogeneous solution to equation (

In this case, _{i} (_{L} and _{F} are the wavenumbers of the longitudinal and transverse elastic waves, respectively. The spectral nodal displacement vector ^{(e)}), equation (_{i} with respect to

Sign convention for a Bernoulli–Euler beam element.

The spectral nodal longitudinal tensile force, transverse shear force, and bending moment can be related to the corresponding force and moment matrix _{c}(_{d}(

Substituting equation (

Finally, the spectral equation for each element is assembled, and the spectral equation of the entire system is given by

Then, the eigenvalue problem for a spectral element structural system can be reduced from equation (_{i} can be determined using the condition that_{i}. Here,

For a significantly minimized system scale, the total numbers of nodes and elements are 447 and 881 for each sandwich model, respectively. Since the SEM can provide exact frequency-domain solutions without refined mesh discretization as the frequency increases [

The considered sandwich model with DAH cores is exposed to a planar incident wave of acoustic pressure on the bottom face sheet as shown in Figure

Acoustic loading and boundary conditions for a baffled, simply supported sandwich panel with a DAH core.

The acoustic incident wave is harmonic with a unit amplitude _{B} represents the bottom face sheet. The spectral nodal force can be obtained by equation (_{a} is the acoustic wavenumber, _{U} with a normal velocity _{t} can be derived via integration over the upper face sheet [

The STL characteristics of a panel can be generally divided into four distinct regions: stiffness, resonance, mass, and coincidence regions from low to high frequencies [_{p} is the mass per unit surface area of the panel and _{p} = 48.6 kg/m for the considered models that are assumed as single panels in this scenario.

The considered models are treated as undamped linear elastic systems because the effects of damping on natural frequency searching are generally small [

Series of the natural frequencies of the representative uniform and gradient DAH sandwich models via SEM (Hz).

Model | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
---|---|---|---|---|---|---|---|---|---|---|

Model 1 | 88.8 | 210.5 | 357.8 | 516.1 | 677.5 | 784.2 | 836.4 | 910.0 | 923.1 | 928.0 |

Model 2 | 86.6 | 210.6 | 364.3 | 532.4 | 706.7 | 856.6 | 881.2 | 1049.6 | 1129.2 | 1153.1 |

Model 3 | 83.7 | 207.3 | 363.5 | 536.9 | 718.7 | 885.3 | 904.1 | 1081.8 | 1255.4 | 1333.3 |

Model 4 | 80.6 | 201.9 | 357.0 | 530.6 | 713.6 | 882.0 | 902.6 | 1080.9 | 1257.4 | 1326.9 |

Model 5 | 77.8 | 194.6 | 342.3 | 503.4 | 581.2 | 615.7 | 666.1 | 667.3 | 719.9 | 816.7 |

Model 6 | 74.4 | 182.5 | 317.9 | 467.5 | 623.9 | 781.5 | 870.3 | 937.1 | 1087.5 | 1179.9 |

Model 7 | 94.5 | 230.2 | 395.0 | 565.8 | 729.2 | 823.6 | 878.1 | 969.4 | 1003.0 | 1023.0 |

The first ten natural frequencies of the considered (a) uniform and (b) gradient DAH sandwich models.

For the uniform models 1 to 5, the fundamental frequency that represents the structural bending stiffness decreases as the feature angle _{2} of the DAHs increases. This result is associated with the low _{2} values are larger than those of the models with small _{2} values until the feature angle exceeds 60°. This result implies that the static mechanical indices cannot completely reflect the structural dynamic behavior at high-order natural frequencies. Concerning the gradient models, the natural frequencies of model 7 are always higher than the frequencies of model 6, mainly due to the large differences in the panel thickness.

The vibrating modes of the corresponding natural frequencies for the representative models are depicted in Figure

Vibrating mode shapes of the corresponding natural frequencies for the representative models: (a) the third bending global modes and (b) the foremost local modes.

For dynamic responses, damping can be introduced by a complex modulus ^{∗} = _{s}(1 +

Vibrating deformations of the considered DAH sandwich models: (a) model 3 and (b) model 6 at 500, 1000, and 1400 Hz, where _{m} denotes the displacement magnitude of the middle bottom face sheet.

To visualize the sound radiation distributions after vibrating deformations as shown in Figure _{0} is 20

SPL distributions of the considered models: (a) model 3 and (b) model 5 at 500, 1000, and 1400 Hz.

In this study, the STL responses of the considered models with varying tailored DAH cores were primarily investigated in the region from 1 to 1500 Hz, which covers the stiffness and resonance regions based on the natural frequencies discussed in Section _{p} = 48.6 kg/m for the baseline panel introduced in Section

The compared sandwich models with uniform hexagonal honeycomb cores: cell feature angles of (a) 10° and (b) −10°.

To verify the effectiveness and accuracy of the vibroacoustic calculation method in this paper, the numeric results compared with those of [

The calculated STLs of the sandwich panels for verification by comparing (a) Figure 11 in [

STLs of the representative sandwich models: (a) models 1 to 5; (b) models 6 and 7; (c) models 8 and 9. (d) The overall average STLs for the constant-weight sandwich models from 1 to 1500 Hz.

The overall average STL (STL_{o}) of the considered models is shown in Figure _{o}. It can be concluded that a uniform sandwich model with a small _{2} or large NPR yields the best sound insulation performance, as shown in Figures _{o} of model 6 is larger than the STL of model 7, as shown in Figure _{2} for the bottom layer, respectively. As plotted in Figure _{o} of model 9 with auxetic hexagonal honeycombs is larger than that of model 8, which is in agreement with the findings in [

The structural-acoustic optimization problem is defined as_{2}, respectively; STL_{a}(_{2}) is the average STL between the specified frequencies of _{1} and _{2}; _{0}(_{2}) is the fundamental frequency of the optimized model; _{0} is the minimum allowable natural frequency; and _{2} sequences, where the positive sign represents a decreasing gradient _{2} (a gradient INPR) and the negative sign represents an increasing gradient _{2} (a gradient DNPR). The baseline model was employed with model 3 due to its moderate mechanical and acoustic performance, with values of _{0} = 83.7 Hz and STL_{o} = 39.67 dB.

The programmable design was achieved via a global optimization method based on the MultiStart algorithm. As a workflow [^{2} random start points, and the iteration tolerance for the local solver was 1 × 10^{−3} to maintain computational efficiency and reasonability. The optimization procedures were executed for specified tonal and frequency band cases.

Here, a tonal excitation at 1400 Hz, which was close to an STL dip in the baseline model, was chosen for the tonal case. The STLs of the baseline and optimized gradient DAH sandwich models are compared in Figure _{a} increments are approximately 9.72 dB for the optimized gradient DNPR model and 16.24 dB for the optimized gradient INPR model. The SPL distributions and dynamic deformations at 1400 Hz are illustrated in Figure

Optimal design at 1400 Hz: (a) STLs of the baseline and optimized models; (b) iteration steps; (c) configurations of the optimized gradient NPR models; (d) mechanical properties of the DAHs in the optimized gradient NPR models.

Characteristic parameters of the baseline and optimized models for optimal design at 1400 Hz.

Model |
_{2} (deg) |
STL_{a} (dB) |
_{0} (Hz) |
STL_{o} (dB) |
---|---|---|---|---|

Baseline | [55.00, 55.00, 55.00, 55.00, 55.00] | 39.37 | 83.72 | 39.67 |

Optimized gradient DNPR | [45.98, 46.53, 46.53, 49.00, 51.03] | 49.09 | 83.99 | 40.24 |

Optimized gradient INPR | [64.91, 64.91, 56.90, 54.96, 45.09] | 55.61 | 95.27 | 37.68 |

SPL distributions and dynamic deformations of the optimized models at 1400 Hz: (a) the optimized gradient DNPR model and (b) the optimized gradient INPR model.

Here, the chosen computational frequency interval was 20 Hz for the frequency band from 1000 to 1500 Hz. The optimized information is depicted in Figure _{a} of the baseline model. More optimized solutions could be obtained by increasing the number of start points, decreasing the iteration tolerance, or using larger integral orders.

Optimal design from 1000 to 1500 Hz: (a) STLs of the baseline and optimized models; (b) iteration steps; (c) configurations of the optimized gradient NPR models; (d) mechanical properties of the DAHs within the optimized gradient NPR models.

Characteristic parameters of the baseline and optimized models for optimal design from 1000 to 1500 Hz.

Model |
_{2} (deg) |
STL_{a} (dB) |
_{0} (Hz) |
STL_{o} (dB) |
---|---|---|---|---|

Baseline | [55.00, 55.00, 55.00, 55.00, 55.00] | 45.36 | 83.72 | 39.67 |

Optimized gradient DNPR | [46.71, 47.14, 47.14, 48.31, 51.53] | 47.87 | 83.73 | 40.16 |

Optimized gradient INPR | [65.00, 65.00, 55.51, 46.78, 46.25] | 50.39 | 92.91 | 38.31 |

The optimized gradient INPR models can achieve better insulation of radiated sound power than can the optimized gradient DNPR models, with improvements of 6.52 dB and 2.52 dB for the tonal and frequency band cases in this scenario, respectively. As plotted in Figures _{a} that even exceeds the mass law curve values and yield a high bending stiffness; however, the use of gradient INPRs generally sacrifices overall sound insulation, with a lower STL_{o}. It can be concluded that the DAHs with large NPRs in upper layers improve the potential for obtaining an optimized STL within a specified frequency region, following the phenomenon that a uniform DAH sandwich model with a large NPR yields high sound insulation performance. Referring to the STL curves in Figures

Sandwich structures composed of uniform and gradient DAH cores across the panel thickness were presented, and the vibration and sound transmission properties of the proposed structures were evaluated by comparing uniform and gradient DNPR and INPR models, as well as uniform hexagonal honeycomb sandwich panels. The following conclusions were drawn from the study:

Regarding the vibration properties, the various core configurations of the gradient sandwich DAH panels yield more diverse mode shapes and provide more flexible vibrating deformation mechanisms than do the uniform models.

With respect to the sound transmission performance with normal incidence, the uniform DAH sandwich models with large NPRs generally yield better sound insulation behaviors. Moreover, the considered DAH sandwich models insulate less sound but are far stiffer than the conventional hexagonal honeycomb sandwich models.

The programmable optimized gradient INPR models yield higher STL values than the optimized DNPR models, with improvements of 6.52 dB and 2.52 dB for the specified frequency cases and a large bending stiffness; however, a lower overall sound insulation was observed. For the gradient sandwich models, the DAHs with large NPRs in the upper layers improve the potential for obtaining an optimized target STL, and the DAHs in the bottom layers affect mainly the STL curve shapes and govern the overall noise insulation properties.

Further consideration could be given to the vibroacoustic performance and design of the proposed structures at high frequencies and under other excitation conditions. Furthermore, the design variables can also be extended to consider more than a single feature angle. Moreover, further studies could be conducted to investigate gradient auxetic metamaterials with varying topologies.

The programming data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The support for this work provided by the National Natural Science Foundation of China (51479115), High-Tech Ship Research Projects by MIIT ([2014]148 and [2016]548), and Opening Project by the State Key Laboratory of Ocean Engineering (GKZD010071) is gratefully acknowledged.