Design of resonant structures in resin matrix to mitigate the blast wave with a very wide frequency range

In this work, resonant structures (RSs) are embedded in the resin matrix to form the micro-scale artificial composite materials to mitigate the blast wave with a very wide frequency range (BWR). The propagation of stress waves in the resin and composite materials is described, and the composite materials exhibit stronger blast wave attenuation characteristic compared with the pure resin material. The attenuation mechanism of the composite materials is explained in detail through the absorption, storage and conversion of impact energy. In addition, the influences of materials of the RSs on the performances of the composite materials are analyzed, and the RS is redesigned to further improve the attenuation effect of the composite material. Equivalent model of the composite material is first proposed and established based on the weakly nonlinear lattice system (WNLS). At the same time, artificial tree algorithm is applied to design its spring stiffness parameters. Based on the WNLS, a three-dimensional composite material plate structure is built to mitigate the overpressure of blast wave at the macro-scale. Compared with traditional materials, the composite material exhibits superior attenuation effect and greater lightweight.


Introduction
Blast wave generated by explosion causes great damage to the human body [1,2]. Therefore, it is very significant to study to effectively reduce the overpressure of blast wave in order to protect the human body. Previous scholars have studied ways to use different forms of material and structure [3][4][5][6][7] to mitigate blast waves. Nesterenko [8] studied the attenuation effect of the 'soft' condensed matter by using a simplified approach, and some anomalous effects were also analyzed. Christou et al [9] presented a computational continuum model of fluid-structure interaction to study the mitigation effects of foam structure, foam structure with filler SiO 2 aerogel and foam structure with filler soda-lime glass beads, respectively. Wadley et al [10] proposed the concept of cellular materials to achieve active and passive attenuation of blast wave overpressure. Through pre-compress the cellular material and release prior to the arrival of the explosion blast wave, a momentum opposite to the explosion was produced. It can achieve the offset of the momentum of the blast wave. Su et al [11] studied one novel device consisting of a piston-cylinder Smart Materials and Structures Smart Mater. Struct. 29 (2020) 045042 (27pp) https://doi.org/10.1088/1361-665X/ab75a4 4 Authors to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. to mitigate the blast wave. When the device is subjected to the blast wave, the blast wave is generated inside and repeatedly reflected. Numerical simulations show that the device can significantly reduce the peak pressure of the blast wave.
Vivek and Sitharam [12] used a proportional test equipment to study the attenuation effect of a sealed sand barrier system on the blast wave. The test results show that the sand barrier system can effectively attenuate the shock wave, and the geotextile layer plays an important role in providing a gradual upward pressure on the downstream end of the barrier. Dan and Ozer [13] studied the propagation characteristics of blast waves through a set of plate barriers. Numerical results show that these plate barriers can effectively mitigate the blast wave.
In short, studies on blast wave protection mainly focuses on using material deformation to absorb blast wave energy, using complex structure to alleviate blast wave propagation or using the inverse potential energy principle to counter incoming blast wave. These measures require large material damage or complex and large structure which mean these measures cannot be reused or require a large volume and cost. Fortunately, scholars have used RSs to overcome the shortcomings of these traditional methods. RS can be reused many times and can efficiently attenuate blast waves with a small and lightweight structure [14,15]. RSs exhibit extraordinary physical properties [16][17][18][19][20][21] through the local resonance of oscillators, and these physical properties can effectively attenuate the blast wave.
Li et al [14] proposed a multi-resonator RS. Theory studies demonstrate that the multi-resonator RS has more band gaps compared with traditional RSs. Numerical results prove that the multi-resonator RS can achieve a great attenuation effect of blast wave. In addition, sensitivity analyses give effects of parameters of the RS on blast wave attenuation. Tang et al [15] applied the negative effective mass property of RSs to mitigate the impact load and blast wave. The influences of number and mass of resonator on the frequency spectrum and amplitude were also studied. Wang et al [22] proposed a novel method to study the attenuation effect of polymer matrix with particles on blast wave. The three-dimensional models of two kinds of RSs were set up, and the energy absorption and particle motion were studied. Mitchell et al [23,24] designed the metaconcrete to attenuate the elastic waves. The resonant motions of inclusions can significantly absorb the impact energy and reduce the structure stress. Sagr et al [16] presented a dissipative RS with multiple Maxwell oscillators. The vibration and dissipation characteristics were studied by the damped single-resonator and dual-resonator microstructure models. The dissipative RS was applied to mitigate the blast wave. The band gaps of blast wave were expanded significantly by reasonably designing the resonant motions and viscoelastic damping effects of oscillators [25,26]. Although scholars have conducted studies on blast wave protection based on RSs, few works have mentioned the design of dual-scale RSs for the attenuation of BWR. The attenuation of BWR involves two key factors. The first one is how to design the RS to enhance their attenuation effects. The second factor is how to achieve macro-scale verification after completing the micro-scale design of RSs.
Resin material is a kind of common engineering material, which has the edges of light weight, low cost, and easy manufacturing. In addition, resin materials are a good choice for the matrix of composite materials although the mechanical and physical properties of pure resin are not outstanding. Composite materials produced by the resin matrix and other highperformance filler materials can bring a leap in material properties. For example, the composite material of carbon fiber and resin matrix is characterized with high strength and light weight [27,28], the composite material of waste paper/jute fabric and resin matrix has high mechanical performance and environmental protection property [29], and the combination of graphene sheet and resin matrix can make paints own extremely strong conductive property [30].
Starting from the design difficulties of RSs and the excellent properties of resin, we embed the RSs into the resin matrix to form a new composite material at the micro-scale. The new composite material has excellent mechanical and physical properties, which can effectively alleviate the BWR and absorb the impact energy. Sensitivity analyses of materials of the RSs are carried out to enhance the attenuation effect of composite material, and the RS is redesigned to further improve the performance of the composite material. In addition, the energy absorption characteristics of the composite materials are presented to explain the attenuation mechanism of blast waves. Furthermore, an equivalent model is set up based on the WNLS to model the performance of the composite material. Finally, a macro-scale composite material plate structure is built based on the equivalent model to attenuate the blast wave overpressure and verify the attenuation effect of RSs.
This work is organized as follows. In section 2, the micro-scale composite material model with internal RSs is established, and sensitivity analyses of the materials and studies on an improved RS model are presented. Section 3 describes the theory of the WNLS and the inverse of spring stiffness parameters. The attenuation effects of the macroscale composite material model are studied in section 4. A preliminary experimental scheme is provided in section 5. Finally, section 6 gives the conclusions.

Model of composite material with RSs
Resin is a common material in engineering, with the advantages of low price and mature technology. In this work, a micro-scale composite material structure consists of RSs and resin matrix is applied to study the propagation characteristics of stress waves. Figure 1 shows the composite material model which length, width, and height are 2 mm, 1 mm, and 1 mm, respectively. The RSs that are embedded in the resin matrix consist of frame, coating and oscillator. Seven layer resin materials and six layer RSs are periodically arranged in the entire structure. The cylindrical holes are periodically arranged on the frame structure, and the coating and oscillator structures are installed in these holes. Figure 2 shows the detailed size of one RS. Oscillator is a cylindrical structure, and its thickness and circular diameter are 60 and 100 μm. The wall thickness of the coating structure is 10 μm. The materials of frame, coating and oscillator are PP resin, PP resin and steel. Table 3 shows the parameters of these two materials.  As shown in figure 2(a), the RS in the composite material can be equivalently viewed as the single-resonator microstructure. Its outer and inner masses are m 1 and m 2 , and their displacements are u 1 and u 2 . The stiffness of the internal spring between m 1 and m 2 is k 2 . The outer force acting on mass m 1 is F. To simplify the derivation of the governing equations of the RS, the mass of spring is ignored here. Then, equation (1) can be acquired based on the Newton's second law Since the motion of effective mass m eff of the microstructure is the same as the external mass m 1 , the following equation (2) is always true Assuming the outer force and mass motions are in a harmonic form, m eff of the microstructure can be obtained as follows based on equations (1) and (2) 2 is resonant frequency of m 2 . Then, the dimensionless effective mass m eff /m st of the microstructure can be obtained where m st is the static mass of the microstructure and θ = m 2 /m 1 is the ratio of m 2 and m 1 . Figure 3 shows the plots of dimensionless effective mass m eff /m st and dimensionless frequency ω/ω 2 with different θ. Obviously, dimensionless effective mass m eff /m st is negative over a narrow frequency range, and this negative mass band gap can be applied to mitigate the waves [14,15,17]. Furthermore, the band gap becomes wider as θ increases. The increase of θ means the increase of m 1 or the decrease of m 2 . Therefore, a heavy oscillator is beneficial to the attenuation effect of RSs.  From figure 4(a), the blast wave is applied to the left side of the composite material. Periodic boundary condition is imposed on the four sides of the structure. The commercial software LSDYNA is applied to perform the impact analysis, and the analysis time is 3 μs. Materials resin and steel are modeled by the piecewise linear plasticity material (No. 24) in the material library. In order to avoid stress wave reflection during the analysis, a 2 mm PP resin material is added to the back end of the composite material. The entire model consists of 1983 096 elements and 2040 755 nodes. The pressures at sections 1 and 2 are selected as the output response. The resin material with the same geometry is applied to study the attenuation effect of the composite material. Figure 4(b) shows the time-domain curve of the blast wave, and its overpressure is 5.3×10 −3 GPa. Figure 4(c) is the spectrum curve of the blast wave obtained by fast Fourier transform, and the frequency range is 0-2 MHz. Figure 5 illustrates the propagation of stress waves in the resin material and the composite material at different times (0.18, 0.29, 0.5, 0.99, 1.4, and 2.1 μs). In figure 5(a), the stress waves are first generated in the impact region on the left side and then propagated in the resin material. The stress waves propagate to 0.2 mm at 0.18 μs and propagate to around 0.5 mm at 0.29 μs. In resin material, the propagation speed of stress waves is uniform, and the wave front is substantially in one plane. In addition, the stress amplitude in the propagation region is different. This indicates that there is periodic compression and stretching during the propagation of stress waves. When t=0.99 μs, the stress waves propagate through half of the whole structure. At this time, the stress in the material is mainly concentrated in the middle of the structure, and the propagation of stress waves begins to appear uneven. When t=1.4 μs, the wave front of stress waves reaches around 3 mm, and the stress waves in the material exhibit a significant wave behavior symmetrically along the X axis. There are two peak stresses at the left and right boundaries, and the wave front exhibits a slight convex shape. At t=2.1 μs, the wave front reaches the right border of the structure. Since the right boundary of the material is a free boundary, the stress is zero at the boundary. The stress waves exhibit a wavy behavior symmetrically along the Xaxis, and there are three areas of stress concentration in the structure. Figure 6(a) shows the internal energy and kinetic energy curves of the pure resin model. In figure 6(a), both the internal energy and kinetic energy increase rapidly after the blast wave acts on the structure, and they have similar growth rates. At about 2.45 μs, there is a sharp peak for the curve of kinetic energy and a marked valley for the curve of internal energy. This indicates that the stress waves have propagated to the end of the resin material, and these stress waves are reflecting. At this time, the internal energy of elasticity stored inside the material is converted into the kinetic energy. After 2.45 μs, the internal energy increases rapidly and the kinetic energy decreases quickly until they reach the inflection points. This means that the kinetic energy is further converted into the internal energy of elasticity of the material, after the reflection of the stress waves. Figure 6(b) illustrates the energy absorption of the front and rear components in the pure resin model, and the energy in this figure is a combination of internal energy and kinetic energy. From figure 6(b), it can be seen that the energy stored in the front component increases rapidly first and then decreases quickly to a low level. The energy stored in the rear component begins to rise rapidly as the energy of the front component decreases, and it remains at a high level. This indicates that the stress waves propagate from front to back, and a high value of energy is generated in the front component. As the stress waves propagate to the rear component, the internal energy of elasticity and kinetic energy originally stored in the front component are released, and a large value of energy is generated in the rear component.
Unlike the propagation characteristics of stress waves in homogeneous resin material, the propagation of stress waves in composite material exhibits more complexity. As shown in figure 5(b), the stress waves propagate to the first layer of RSs at 0.18 μs. At t=0.29 μs, the stress concentration is obviously exhibited in the RSs and the connection area of the RSs and matrix structures, the peak stress of these regions is significantly greater than the matrix structure. In addition, the stress waves exhibit a periodic wave arc before and after the RSs. At t=0.5 μs, the stress waves pass through the second layer of RSs, and its wave front reaches the third layer of RSs. The stress waves between the second and third layers of RSs exhibits a regular fan shape. When t=0.99 μs, the stress waves just propagate to the last layer of RSs. At this point, the stress concentration is still mainly in the first five layer of RSs. Compared with the pure resin material, the obvious stress concentration phenomenon in composite material indicates that stress waves are constantly oscillated and reflected in the RSs, and the blast wave energy is significantly absorbed. When t=1.4 μs, the stress waves pass through the middle region of the whole structure. At this time, the stress is mainly concentrated at the first, fourth, fifth and sixth layers of RSs, and the blast wave energy is still absorbed, stored and reflected by the sandwich structures. Finally, at t=2.1 μs, the wave front approaches the right boundary of the structure, while the structural stress is still concentrated at the RSs. Table 1 shows the numerical results of these two models. From figure 5 and table 1, the peak stresses in the resin and composite material models are 9.24×10 −3 and 36.52×10 −3 GPa. The structural stress is significantly concentrated in the interlayer regions of the RS, which indicates that the composite material can better absorb the blast wave energy compared with the pure resin material. In addition, it can be seen that the propagation velocity of stress waves in the pure resin material is greater than that of the composite material. The main reason is that the stress waves are continuously absorbed and reflected in these RSs.

Figures 7(a) and (b)
show the curves of internal energy and kinetic energy of all parts in the composite material. In figure 7(a), it is obvious that the internal energy of the parts is matrix, frame structure, coating and oscillator from large to small orderly. From figure 7(b), the kinetic energy of the parts from large to small is oscillator, matrix, frame structure and coating orderly. Obviously, internal energy is mainly absorbed by matrix and frame structure, while kinetic energy is mainly stored in oscillator and matrix. The maximum kinetic energy of the oscillators (MEO) is 1.05×10 −6 J. This means that a large part of the blast energy is converted into the kinetic energy of the oscillators. This indicates that the RSs can manipulate the impact energy through the resonant motions of the oscillators and alleviate the impact damage.  energy are constantly transformed into each other. The main reason is that part of the energy carried by the stress waves is converted into the kinetic energy of the resonant motions of oscillators with the propagation of stress waves, and this part of kinetic energy are continuously released and absorbed, resulting in continuous conversion of internal energy and kinetic energy. In addition, the reflection time of the stress waves is about 2.8 μs, which is later than that of the pure resin model. This indicates that the propagation speed of stress waves in the composite material is slower than that of the resin. Figure 7(d) shows the energy absorbed by the composite material and the rear component. The energy absorbed by the composite material increases first and then decreases, and the energy absorbed by the rear component begins to increase as the energy of the composite material decreases. The phenomenon is caused by the reason that is same as the pure resin material. The internal energy of elasticity and kinetic energy stored in the composite material are released as the stress waves propagate to the rear component, and the internal energy of plasticity is retained inside the composite material. From figures 6(b), 7(d) and table 1, time of the intersection point of two energy curves of the front assembly and the rear component (TIE) in the composite material model is 1.87 μs, which is significantly larger than 1.19 μs of the pure resin model. This indicates that the storage time of the impact energy in the composite material is longer than that of the pure resin material, which is mainly because the energy absorption and conversion process caused by the resonant motions of oscillators reduces the propagation speed of the stress waves. The maximum energy absorbed by the front assembly (MEF) of the composite material and pure resin models are 3.68×10 −6 J and 3.07×10 −6 J, respectively. Their maximum energy absorbed by the rear component (MER) are 3.01×10 −6 and 2.98×10 −6 J. The ratios of MER and MEF (RMM) of the composite material and pure resin models are 81.79% and 97.07%. A smaller value of RMM indicates a better energy absorption of the front assembly and a smaller proportion of the blast energy acting on the rear component. Obviously, the composite material model has a smaller value of RMM, which can absorb the impact energy better. In addition, from figures 6(b), 7(d) and table 1, the internal energy of plasticity remaining in the front assembly (IPF) of the composite material and pure resin models are 0.67×10 −6 and 0.22×10 −6 J. The ratios of IPF and MEF (RIM) of these two models are 81.79% and 97.07%, respectively. A larger RIM implies that the structure has a better energy absorption effect. Obviously, the value of RIM of the composite material is larger than that of the pure resin material, which also proves that the composite material can better absorb the impact energy, thereby protecting the rear structure. The energy absorption of the composite material also implies its attenuation effect of the pressure.
Figures 7(e) and (f) are the pressure curves at sections 1 and 2, and table 1 shows the peak pressure of these two sections. From figures 7(e), (f) and table 1, the peak pressure at section 1 (PP1) of the resin and composite material models are 6.02×10 −3 , 6.78×10 −3 GPa, and the peak pressure at section 2 (PP2) of these two models are 4.73×10 −3 and 3.69×10 −3 GPa. It is obvious that the PP1 of the composite material model is larger than that of the resin model, while its PP2 is smaller. Furthermore, it can be seen from figure 7 that Table 1. Dynamic responses of the resin and composite material models. Note. PP1 is the peak pressures at section 1, PP2 indicates the peak pressures at section 2, MEF stands for the maximum energy absorbed by the front assembly, MER denotes the maximum energy absorbed by the rear component, IPF represents the internal energy of plasticity remaining in the front assembly, MEO is the maximum kinetic energy of the oscillators, RMM is the ratio of MER and MEC, RIM denotes the ratio of IPF and MEF, and TIE means time of the intersection point of two energy curves of the front assembly and the rear component.

Analyses of materials of RSs on the performance of composite material
To better study the attenuation effect of RSs, the matching analyses of materials are conducted in this section. The combination of materials of RSs is shown in tables 2, and 3 is the parameters of each material.   a1) and (b1) illustrate the kinetic energy curves of all components in the composite material of groups 2 and 3, respectively. For group 2, the kinetic energy of all components is matrix, frame structure, oscillator and coating in order from large to small. It is clear that the kinetic energy is mainly stored in the matrix. The kinetic energy of oscillators starts to increase at about 1.0 μs, and the rate of growth as well as the moment of onset are significantly slower than other three parts. The main reason is that the propagation speed of stress wave in rubber is significantly slower than other materials. Regarding group 3, the order of these components is matrix, oscillator, frame structure and coating. Obviously, the kinetic energy is mainly stored in the matrix and oscillator. A large amount of impact energy is converted into the kinetic energy of oscillators, which facilitates the absorption of impact energy and the attenuation of impact load. Combining the results of group 1, it is clear that group 1 has the largest values of MEO, RIM and TIE, and the smallest value of RMM. Therefore, group 1 has the best manipulation effect of impact energy.
From figures 8(c) and (d) as well as table 5, the PP1 of groups 1, 2 and 3 are 6.78×10 −3 GPa, 6.06×10 −3 GPa and 7.48×10 −3 GPa, respectively, and the PP2 are 3.67×10 −3 GPa, 4.76×10 −3 GPa and 3.78×10 −3 GPa, respectively. The peak stress of structure is 36.53× 10 −3 GPa, 10.96×10 −3 GPa and 50.44×10 −3 GPa, respectively. It is obvious that the peak pressure and maximum structural stress of group 2 are comparable to those of the resin model. The peak pressures and the maximum structural stress of group 1 are all smaller than group 3. From figure 8(a), the pressure curve of groups 1-3 at the section 1 vibrates up and down along the pressure curve of the resin model. The peak pressure of group 3 is the largest, the peak pressure of group 1 is slightly larger than that of the resin model, and the peak pressure of group 2 is comparable to the resin model. From figure 8(b), the pressure curve of group 2 at section 2 is similar to the resin model. Their pressure amplitudes are close, while the time of peak pressure is relatively delayed. The main reason is that the structural stiffness of the rubber coating is weak, and the locally resonant frequency of the oscillator is low which cannot effectively attenuate the BWR. However, the low-frequency local resonance generated by the rubber coating can still reflect a small amount of stress waves, so the pressure curve at section 1 still fluctuates, and the PP2 is delayed slightly. Furthermore, the pressure curves of groups 1 and 3 at section 2 are close. Both of them lag behind the resin model significantly. The peak pressure of group 1 is slightly smaller than that of group 3. A smaller PP2 means a better attenuation effect. In addition, the resin coating is lighter than the aluminum coating. Therefore, the performance of group 1 is slightly better than that of group 3, and both groups 1 and 3 are significantly better than group 2. Based on the above analyses, reasonable coating stiffness has a positive effect on improving the performance of the composite materials.
Figures 9(a) and (b) show the structural stress nephograms of the rubber and aluminum models at 0.29, 0.99, and 2.1 μs. In figure 9(a1), the stress waves propagate to the middle of the first interlayer and the second interlayer, causing stress concentration in the first interlayer region. In figure 9(a2), the stress waves propagate to the sixth sandwich structure and the structural stress concentration is mainly concentrated at the fourth and fifth sandwich structures. From figure 9(a3), the stress wave front propagates to the rightmost region of the structure at 2.1 μs. The stress wave propagation characteristic is symmetric along the X direction. At this time, there is no significant stress concentration in the sandwich structure. In figure 9(b1), the stress waves pass through the first sandwich structure, and the stress concentration phenomenon in the first interlayer region is obvious. In figure 9(b2), the stress waves propagate to the sixth sandwich structure, and the stress is mainly concentrated at the second, third and fourth layers. From figure 9(b3), the stress wave front propagates to the rightmost area of the structure at 2.1 μs. At this time, a symmetrical wave phenomenon along the X direction appears in the resin material region, and the obvious stress concentration appears at the first, fourth and sixth layers. Compared with the structural stress nephograms of group 1, the stress concentration region of group 2 at the sandwich structure is smaller and the stress amplitude is lower. The stress concentration phenomenon of group 3 is similar to group 1. Considering the above results, the blast wave attenuation effect of group 2 is the worst.      Note. PP1 is the peak pressure at section 1, PP2 indicates the peak pressure at section 2, MEF stands for the maximum energy absorbed by the front assembly, MER denotes the maximum energy absorbed by the rear component, IPF represents the internal energy of plasticity remaining in the front assembly, MEO is the maximum kinetic energy of the oscillators, RMM is the ratio of MER and MEC, RIM denotes the ratio of IPF and MEF, and TIE means time of the intersection point of two energy curves of the front assembly and the rear component. increase of oscillator density, the PP1 increases, the PP2 decreases, and the maximum structural stress increases. The results imply that the heavy oscillator is favorable for blast wave attenuation. In addition, from figure 10(b), the time of PP2 of these composite material models is delayed compared with the resin model. As the density of the oscillator increases, the delay is more pronounced. The above analysis results indicate that a better performance of the composite material requires a heavier oscillator, which is consistent with the conclusion of theoretical analysis in section 2.1.1 where increasing θ can widen the band gap of the negative effective mass. Figures 11(a)-(e) are the stress nephograms of the composite material models with resin, aluminum, copper, lead, and gold oscillator, respectively. It is clear that as the density of the oscillator increases, the concentration of stress becomes more pronounced, while the propagation distance of the stress wave decreases. The results mean that the heavy oscillators can absorb and reflect the stress wave energy more effectively.

Attenuation effect of the composite material with improved RSs
In this section, redesign of the frame, oscillator and coating structures is applied to further enhance the attenuation effect of the RSs. Figure 12 shows the improved structure. Both sides of the structure are the resin matrix. The frame structure is in the middle of the matrix, and cylindrical holes are periodically arranged in the frame structure. Six rows of oscillators and coatings are installed inside the cylindrical holes, and the pink area of the frame structure is the mounting region of the oscillator and coating. Compared with the original RSs, the changes of the improved RSs include: volume of the oscillator is increased, volume of the coating is reduced, layout of the coating is adjusted and five layers of matrix in the middle of the RSs is removed. These changes are beneficial to obtain a more reasonable coating stiffness and a heavier oscillator, which are conducive to improve the performance of the composite material. Meanwhile, deletion of excess matrix can make room for the RSs.
The detailed sizes of the improved oscillator and coating structures are shown in figure 13. From figure 13, the outer surface of the cylindrical oscillator is increased by four protruding structures to increase the structure mass. Diameter and length of the improved oscillator are 100 and 166.7 μm. The  coating structure is changed to eight arcuate columnar structures which are arranged in the middle of the cylindrical convex structure. The length, width and height of one coating structure are 60 μm, 19.5 μm and 20 μm, respectively.
Taking into account the performance of the RSs under different combination of material in the previous section as well as the weight of the structure, steel and resin are selected as the materials of oscillator and coating, respectively. The analysis time of this model is 3.5 μs, and a 2 mm resin material is also added to the back end of the RSs to avoid the reflection of stress wave. The entire model contains a total of 1775 424 elements and 1864 127 nodes. Figure 14 shows the stress nephograms of the improved RS model at different times (0.18, 0.29, 0.5, 0.99, 1.4 and 2.1 μs). When t=0.18 μs, the stress waves propagate to the sandwich structure. When t=0.29 μs, the stress waves just pass through the first layer sandwich structure and produces a significant stress concentration phenomenon. When t= 0.5 μs, the stress waves propagate to the third layer, and the stress concentration region is significantly increased. When t=0.99 μs, the stress wave front reaches the rightmost end of the frame structure and begins to propagate toward the resin matrix structure. The stress is mainly concentrated on the regions of the first, second, third and fourth layers of RSs. When t=1.4 μs, the stress wave front reaches the structural region of about 2.5 mm, and stress fluctuation occurs in the resin structure. The stress concentration in the frame structure is still obvious. When t=2.1 μs, the stress wave front reaches the position of about 3.2 mm, and the wave phenomenon which is symmetric along the X axis is generated in the resin structure. At this point, the stress amplitude of the resin structure is small, and the stress waves are still reflected back and forth within the frame structure. Throughout the analysis process, the maximum stress of the improved RS model is 92.14×10 −3 GPa, which is significantly higher than that of the other composite material models in section 2.2. This demonstrates that the blast wave attenuation effect of the improved RS model is enhanced. Figure 15 and table 6 show the plots of energy and pressure as well as their numerical results of the improved RS model. From figure 15(a), the internal energy is mainly absorbed by the frame structure and the coating, which is different from the composite material model in group 1. The main reason is the frame structure of the improved RS is different from the previous model. Figure 15(b) shows the kinetic energy of all components in the composite material. The value of MEO is 1.82×10 −6 J, which is significantly greater than that of the composite material in group 1. It is clear that the kinetic energy is mainly stored in the oscillators, which implies that the improved RS can efficiently manipulate the impact energy and alleviate the impact load. In figure 15(c), the internal energy and kinetic energy curves continuously oscillate, and the amplitude of the oscillation is larger than that of group 1. At about 3.4 μs, there is a sharp peak in kinetic energy and a significant trough in internal energy. At this point, the stress waves have propagated to one end of the resin material, and these stress waves are reflecting. The reflection time of stress wave of the improved RS model is significantly later than the pure resin model and the group 1. This means that the stress wave energy is constantly converted inside the improved RS model, and the propagation speed of the stress waves is significantly reduced compared with the pure resin model and group 1. In figure 15(d) and table 6, the values of MEF, MER, IPF, RMM, RIM and TIE     similar as the resin model. Its PP2 is 2.45×10 −3 GPa, which is significantly less than the resin and other composite material models. In addition, the results in figure 15(f) also show that the delay of the peak pressure moment of the improved RS model is more pronounced compared with the resin and other composite material models. This 'delayed release' feature can be further used for other engineering problems. The overall analysis results demonstrate that the performance of the composite material is significantly improved with the redesign of the RS.

Parameter reverse of the WNLS
The improved RS model in section 2.3 is selected as the original model to study the attenuation effect of blast wave at the macro-scale. Since this composite material contain too much element and node, it is very difficult to construct the macro-scale model by directly combining its micro-scale information. It is necessary to simplify the three-dimensional composite material model. Scholars had studied the linear and nonlinear one-dimensional lattice systems [31][32][33][34] which has the advantages of simple principle, less parameters and high computational efficiency. Since the analysis model is a nonlinear impact model, a one-dimensional WNLS is applied to approximate the attenuation effect of the composite material to reduce the computational cost.

The mathematical model of the WNLS
As shown in figure 16, an equivalent model of the improved RS model is built based on the WNLS. In figure 16, the light blue parts on both sides are the matrix, and the middle part is the WNLS. There are six single-resonator microstructures in this WNLS. Each microstructure is linked by two external nonlinear springs which stiffness is controlled by two parameters, and the nonlinearity of spring is described by a polynomial. For this weak nonlinearity model, a small parameter ε is introduced to simulate the restoring force of the outer spring. The spring stiffness k 1 and restoring force F 1 can be written as equations (5) and (6) where u j 2 ( ) and u j 1 ( ) are the displacements of the inner and outer masses of jth cell. Then, equations (7) and (8) can be ( ) where δ 2 =k 2 /b 1 and δ 1 =b 2 /b 1 is the ratio of stiffness parameters. Next, equations (9) and (10) can be rewritten as equations (11) and (12) through the defining of dimensionless time τ=ωt and dimensionless frequency w w W = The perturbation approach [33,35] is applied to acquire the dispersion equation of the WNLS. As shown in equation (13), the disturbances of u , Equation (13) is substituted into equations (11) and (12), and the coefficients of ε 0 and ε 1 are extracted separately. Then, the governing equations equations (11) and (12) can be rewritten as follows where subscript a = 0, 1 corresponds to ε 0 and ε 1 respectively. O α and P α are written as follows The governing equation of the linear lattice system is recovered by the lowest order. With nonlinear terms serving as the forcing function, the higher order expansion is a heterogeneous form of the linear system. Through the requirement of non-long-term solutions, the correction of frequency for each order nonlinearity is acquired. Assume that the solutions of the linear lattice system are as follows where a and q is the lattice constant and wavenumber, c represents the complex conjugate. H 1 and H 2 denote the amplitudes of the external and internal masses, respectively. The following equations can be obtained by substituting equations (19) and (20) into equations (14) and ( 2 ( ( ) ) ω 0am is the smaller root which corresponds to the acoustic mode, and ω 0om is the larger root that corresponds to the optical mode [33]. For α=1, the governing equations of the WNLS can be updated as follows In the equation of c 1 , the asterisk indicates the complex conjugate. The forced term e inqa on the right side the equal  sign of equation (26) is long-term and must be removed.
The dispersion curves of WNLS and linear lattice system obtained by the perturbation approach [33,35] are shown in figure 17. The dimensionless parameters are set as θ 2 =1 and δ 2 =1. The nonlinear parameters are selected as ed H | | = 0, respectively, to describe the WNLS and the linear lattice system. From figure 17, it is seen that both the WNLS and the linear lattice system have a band gap. The shaded area in figure 17 is the band gap of ed H 1 1 2 | | = 0.02. The band gap of the linear lattice system is slightly different from the WNLS, and these band gaps can be applied to manipulate the propagation of waves. As the nonlinear parameter ed H 1 1 2 | | increases, the band gap width of the WNLS decreases. In this work, we design the spring stiffness parameters of the WNLS to adjust its band gap and its attenuation effect, so that the performance of the WNLS can approximates the improved RS model well.

Parameters reverse of the WNLS based on AT algorithm
In figure 16, total mass of this equivalent model is 3.92×10 −6 kg which is equal to the improved RS model. The masses of the matrix and the WNLS are 0.416×10 −6 kg and 3.51×10 −6 kg, respectively. The outer and internal masses of one microstructure are m 1 =m 2 = 0.2925× 10 −6 kg. The pressures of the equivalent and improved RS models at sections 1 and 2 are the output responses. The fit of the response curves between these two models is defined as the target. The IGD metric [36] is applied to evaluate the fit. A small IGD value means a better fit. The design variables of the equivalent model are the inner and outer spring stiffness parameters b 1 , b 2 and k 2 . Table 7 shows the design space of these parameters. Since the calculation results are the pressure curves at sections 1 and 2, there are two IGD values (IGD 1 and IGD 2 ) that corresponding to the responses at sections 1 and 2. Based on AT algorithm [37,38], the inverse of the spring stiffness parameters is carried out with the goal of minimizing the IGD= IGD 1 +IGD 2 .
AT is an efficient heuristic algorithm that was proposed by Li et al in (2017). Figure 18 shows the optimization flow of AT. In AT, the positions of the branches represent the design variables, and the branches themselves are the solutions of the optimization problem. A thicker branch means a better solution. Two evolutionary operators of branches are applied to update the solutions which are the crossover and self-evolution operators. Through the execution of all operations in the algorithm, AT can efficiently solve a variety of optimization problems. Figure 19 illustrates the pressure curves of the improved RS and equivalent models. In figure 19(a), at section 1, the peak pressure, peak pressure moment and curve form of the equivalent model are consistent with the improved RS model before 1 μs. From figure 19(b), at section 2, the rise and fall speed as well as the amplitude and amplitude moment of the pressure curves between these two models are very consistent. Table 8

Attenuation effect of the macro-scale composite material plate structure
The macro-scale composite material plate structure is established based on the equivalent model. Figure 20 shows the geometry of the macro-scale composite material model. This model is a square plate structure with a side length of 192 mm, a plate thickness of 2 mm, and a structural mass of 0.145 kg. The model contains 847 892 elements and 960 794 nodes. Dynamic response of a three-dimensional block rubber material under the blast wave is studied to test the attenuation effect of the composite material. Figure 21(a) shows the block rubber material which has a rectangular structure with length, width and height of 190 mm, 190 mm and 100 mm, respectively. In addition, traditional materials of resin, aluminum alloy and steel are also applied to evaluate the performance of the macro-scale composite material plate. Structures of the traditional materials are also the square plate with a side length of 192 mm. Three cases are applied. The first case is to directly apply the blast wave to the rubber material. The second one is that the blast wave is applied to the traditional material plate structure while placing the rubber material behind the traditional material. The third case is to apply the blast wave to the composite material plate structure while placing the rubber material behind the composite material. Two contrasting methods are applied. The first is to set the same mass for traditional material and composite material models. The second one is that plate thicknesses of traditional material and composite material models are the same. The thicknesses of traditional material models can be acquired based on the above two contrasting. Figures 21(b) and (c) show the rubber material with traditional material and composite material plate structures, respectively. Figures 21(d) is a schematic diagram of how the blast wave applied to the surface of composite material plate. The overpressure of the incident blast wave is 1.33× 10 −3 GPa, and its spectrum range is 0-2 MHz. The analysis time is 200 μs.
When the mass of traditional material plate structure is the same as composite material model, the plate thicknesses of resin, aluminum and steel materials are 3.77 mm, 1.45 mm and 0.5 mm, respectively. Table 9 shows the maximum stress of rubber material at different times. The second column of this table is the thickness of the plate structure, and the last column is the maximum structural stress for the entire analysis time. It can be seen that the peak stresses of the rubber material are 18.05×10 .90×10 −6 and 9.39×10 −6 GPa, and its maximum structural stress is 15.65×10 −6 GPa. Peak structural stresses of the aluminum plate model at different times are 5.70×10 −6 , 9.67×10 −6 , 14.25×10 −6 , 12.55×10 −6 , 10.96×10 −6 , 10.08×10 −6 and 9.55×10 −6 GPa, and its maximum value is 15.84×10 −6 GPa. Peak stresses of the steel plate model are 6.68×10 −6 , 11.95×10 −6 , 18.22×10 −6 , 15.32×10 −6 , 12.53×10 −6 , 11.21×10 −6 and 10.54× 10 −6 GPa, and its maximum value is 19.90×10 −6 GPa. Peak stresses of the composite material plate model at different times are 3.21×10 −6 , 5.27×10 −6 , 7.48×10 −6 , 5.09×10 −6 , 4.73×10 −6 , 4.67×10 −6 and 4.59× 10 −6 GPa, and its maximum value is 7.48×10 −6 GPa. Therefore, the peak structural stress of the composite material model is significantly lower than the original, resin, aluminum and steel models. Figure 22(a) is the history curve of maximum stress and time of the rubber material corresponding to different models with the same structure mass. It can be seen that the peak stress of all models have a sharp rise at about 0-25 μs, and then gradually decrease with time. In addition, the maximum stress of the rubber material of the composite  Therefore, peak stress of the composite material model is still the smallest, which is significantly lower than the resin and aluminum models, and slightly lower than the steel model. Meanwhile, structural mass of the composite material structure is significantly lower than that of the aluminum and steel models. From figure 22(b), the peak stress curves of the resin and aluminum models are close, which are significantly higher than the composite material model. The peak stress curve of the steel model is significantly lower than that of the resin and aluminum models, but slightly higher than the composite material model.
The above analysis results show that: (1) the composite material with RSs can effectively mitigate the BWR, (2) with the same structure mass or plate thickness, composite material achieves better attenuation effect compared with traditional material models, (3) composite material model has higher light weight potential than other models. Figures 23 and 24 are the stress nephograms of the pure rubber model and the composite material model at different times. From figure 23, the structural stress is first created at the four edges of the rubber material and increases over time.  At about 20 ms, the amplitude of the stress reaches its maximum value. At about 50 ms, the stress waves are transmitted about 13 mm along the side of the rubber material, and the phenomenon of stress concentration occurs in the central area of the rubber. Over time, stress waves continue to propagate along the side of the rubber material. At about 100 ms and 150 ms, the stress waves propagate to around 22 mm and 32 mm, respectively, and the stress concentration mainly occurs at the front of the stress waves. In addition, the stress in the center of the rubber appears wavy. In Figure 24(a), the structural stress appears around rubber material first. At approximately 20 ms, the magnitude of the stress reaches its maximum value. After 50 ms, the stress starts to concentrate on the central area of the rubber material. Compared with the pure rubber model, the stress is more evenly distributed in the central area of the composite material model. In short, from figures 23 and 24, the stress amplitude of the composite material model is significantly smaller than that of the pure rubber model at each time. Therefore, the composite material model can effectively attenuate the blast wave and alleviate the damage of the rubber material.

Description of the experimental scheme
Compared with the process of simulation analyses, we provide a preliminary experimental scheme to verify the attenuation effect of the composite material. Computer-aided design technique is applied to construct the three dimensional model of the composite material, and 3D printing technology can be applied for the manufacture of the composite material.
A shock tube [39,40] can be applied to generate and propagate the blast waves.
The experimental process is summarized as follows. (1) The ideal blast wave is obtained by changing the parameters of the shock tube. (2) Impact tests are carried out in five cases. The first case is that the rubber material is impacted directly by the blast wave generated by the shock tube, and the damage of the rubber material is recorded. The second case is to combine the rubber and composite materials, and the composite material plate is placed before the rubber material. The blast wave impacts the composite material directly, and the stress wave generated by the impact will propagate to the rubber material and cause damage to it. The damage of the rubber material in this case is recorded. The third, fourth and fifth cases are that experiments are performed by replacing the composite material plate in the second case with the steel, aluminum and resin plates, respectively. The results of these three cases are also recorded. (3) The attenuation effect of the composite material is verified by comparing the damage of the rubber material in all cases.
Although this is a preliminary experimental scheme and many details need to be refined, this strategy provides a clear and feasible idea for experimental verification. In addition, the focus of this work is to study the propagation mechanism of stress waves, explore the energy absorption mechanism of RSs, demonstrate the attenuation effect of composite materials and propose the equivalent model of the composite materials with RSs through simulated analyses. Therefore, the authors expect to exhibit the experimental process of this work in detail and share these verification results with the academic community in the subsequent work.

Conclusion
In this work, a three-dimensional micro-scale artificial composite materials consist of resonant structures (RSs) and resin matrix are designed to mitigate the blast wave with a very wide frequency range (BWR). Analyses of propagation characteristic of stress waves, energy absorption and material sensitivity are carried out to study the attenuation effect of the composite materials. Computational results show that: (1) RSs can significantly attenuate the blast wave, (2) the better attenuation effect of blast wave requires heavy oscillator and reasonable coating stiffness. An improved RS is presented to further enhance the performance of the composite material. In addition, an equivalent modeling method for constructing the highprecision macro-scale composite material is presented to study blast wave protection. A weakly nonlinear lattice system (WNLS) is applied first to equal the attenuation effect of the composite material, and the spring stiffness parameters is obtained based on the AT (artificial tree) algorithm. Then, a macro-scale composite material plate structure is established based on the WNLS. Finally, the composite material model is applied to attenuate the BWR and protect the rubber material at the macro-scale. Analysis results indicate that: (1) the composite material model realizes great attenuation effect of the BWR, (2) compared with traditional materials, composite material achieves better attenuation effect and exhibits superior lightweight potential. This work shows the broad application prospect of artificial composite material with RSs and resin matrix in the field of blast protection.