- Email: [email protected]
Study on the mechanism of band gap and directional wave propagation of the auxetic chiral lattices
Journal Pre-proofs Study on the mechanism of band gap and directional wave propagation of the auxetic chiral lattices Kai Zhang, Pengcheng Zhao, Cheng Zhao, Fang Hong, Zichen Deng PII: DOI: Reference:
S0263-8223(19)34229-1 https://doi.org/10.1016/j.compstruct.2020.111952 COST 111952
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
7 November 2019 8 January 2020 17 January 2020
Please cite this article as: Zhang, K., Zhao, P., Zhao, C., Hong, F., Deng, Z., Study on the mechanism of band gap and directional wave propagation of the auxetic chiral lattices, Composite Structures (2020), doi: https://doi.org/ 10.1016/j.compstruct.2020.111952
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Ltd.
Study on the mechanism of band gap and directional wave propagation of the auxetic chiral lattices Kai Zhang1,2*, Pengcheng Zhao1,2, Cheng Zhao1,2, Fang Hong1,2, Zichen Deng1,2 1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, P. R. China 2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072, P. R. China
Abstract In this study, the wave propagation properties in terms of the band gap and directions of wave propagation of the auxetic chiral structure are analyzed. The mechanism of generation of the band gap are carefully investigated. The auxetic chiral structure are assembled with the repeat unit cells and the unit cell contains a number of rigidly connected beams. The dynamic model of the unit cell are established by the principle of finite element method. The wave behaviors of the lattices are calculated by solving the dynamic model with the help of the Bloch’s theorem. The band structure are obtained and the effects of the chiral angles on the width and position of the band gap distributions are carefully studied. Especially, the mechanism of formation of the band gap are also analyzed by investigating the vibrational mode calculated by the commercial finite element software. The group velocities are calculated to analyze the effects of the geometrical parameters on the directional frequency-dependent energy flows in the structures. We also use the commercial finite element software to simulate the directional wave behaviors in the structure. We find that the first mode of the elastic wave spread only along certain specific directions in the auxetic chiral structure. The *
Corresponding author, Email: [email protected] (Kai Zhang) 1 / 34
speed of wave propagation will be reduced, and the direction of wave propagation rotates counterclockwise when the chiral angle increases. Keywords: Wave propagation, Auxetic chiral structure, Band gap, Group velocity
2 / 34
1 Introduction Mechanical metamaterials are a kind of artificial archiechtured materials have broad engineering applications like the energy absorption foams, sensors and actuators, smart composites [1] in aerospace, automobile, civil industries, etc. Their unique mechanical properties can be achieved via innovative geometric design [2-4]. In the past few years, scholars had used rigid and flexible components to carefully design microstructures with special configurations. A number of unique mechanical properties was found, such as negative Poisson's ratio, vibration attenuation, impact resistance, etc [5, 6]. Additionally, the fast growing additive manufacturing technology enables rapid production of the complex geometry, which provides a new way to design and manufacture the innovative metamaterials [7-9]. The auxetic metamaterial is one important category of the metamaterials, and has the light weight while retaining high rigidity and strength [10]. It also has excellent energy absorption capability, and special acoustic properties [11-13]. Furthermore, auxeticity can increase fracture toughness, and shear stiffness [14-16] of the structure and let the structure to undergo dramatic shape changes [17-19]. The above properties highlight the potential applications of auxetic metamaterials in developing novel lattice metal [20-22], corrugated panels [23], and foam metal [24]. Recently, chiral structures with auxetic properties have attracted great interests around the academic world as its advantages for potential applications in flexible and smart structures, and components. Normally, the existing auxetic mechanical metamaterials have two basic deformation mechanisms. One is the deformations of the symmetric units with re-entrant angles (also be called as instability-induced auxetic effects) [2527], and the other one is the rotations of the chiral units when the structure deformed (chirality-induced auxetic effects) [28-30]. Although the deformations of the auxetic chiral lattice have been extensively studied, the study of the dynamic properties of the auxetic chiral lattice from the viewpoint of wave propagation has not been fully studied. In recent years, the study of wave propagation in periodic metamaterial lattice structures 3 / 34
with tunable wave properties and directional waveguides has aroused considerable interests, and the progress largely depends on determining the wave propagation characteristics in the periodic systems [31-33]. In these wave propagation properties, the band structure is one the important properties and it has deep relation with topologies and geometries of structures [34]. The group velocity is also another important properties to analyze the anisotropic characteristics and the directional waveguide of the structure, which is the elastic wave propagates along a specific direction in the structure [35, 36]. There are also many efforts were carried out to tune the wave propagation in the lattices by applying the external effects like the temperature fields [37] and magneto-elastic effects [38]. However, the wave behavior is still highly determined by the topology and geometry of the structures [39-41]. Therefore, the relations between the wave behaviors and geometrical parameters become significantly important topics for the analysis of wave propagation and design of the innovational structure. Because of the special mechanical properties, the chiral lattices have the unique vibrational filtering [42] and wave steering capabilities [43] which make the lattice to be one of the best choice for the sound isolation [44]. We also found that the chirality-induced rotational efficiency can deduce the effective auxetic effect, which can significantly amplify the effects of chiral metamaterial [28]. Additionally, by utilizing the two basic deformation mechanisms of re-entrant angle and chirality [28], new auxetic mechanical metamaterials are developed. Furthermore, the study on the wave propagation behaviors of lattice structure with zigzag arms [29, 30, 45] shows that lower-frequencies complete band gaps can be generated and good directional properties are produced in such structure compared with the traditional lattices with straight arms. These results infer that the rotating auxetic chiral structures can exhibit unique wave propagation behaviors. Therefore, we have great interests to carry out the study on the wave propagation analysis to find the relationship between the structural deformation and wave behaviors. To study the wave propagation characteristics of the auxetic chiral lattices, we establish 4 / 34
the dynamic model of the structure and analyze the effects of configurations and geometrical parameters of the structure on the band structures. The relationship between the band gap and the vibration mode is carefully investigated. We compare the analytical results with the results obtained by using the commercial finite element software COMSOL for verifying the band structures and transmission spectra of the structures. The vibration modes at the band gap edges are calculated to analyze the mechanism of the generation of band gap. The directional characteristics of the wave propagation in terms of the group velocities in the auxetic chiral lattice are analyzed. The transient propagation of elastic waves in finite structures, are calculated to show the directional properties of the considered systems. The paper is organized in four sections, including the Introduction section above. In Section 2, brief descriptions of the two kinds of auxetic chiral lattices are given, and analytical models for the dynamical behaviors of the structures are established. Section 3 investigates the performance of various types of structures in terms of band gaps and directionality of the wave propagation. Additionally, the finite element (FE) simulations are carried out to verify the theoretical calculations and to discuss the mechanism of formation of band gaps. Finally, Section 4 summarizes the important conclusions drawn by this study.
2 The dynamical model of the auxetic chiral lattices 2.1 Geometries of the auxetic chiral lattices In this study, the characteristics of wave propagation in two types of auxetic chiral structures are investigated. The periodic auxetic chiral lattices which are Type I lattice and Type II lattice are shown in Figs. 1(a) and (b). The unit cell of chiral auxetic lattices are four-fold symmetry, and two different types of unit cells are designed respectively as shown in Figs. 1(c) and (d). The dimensions of the unit cells are a a , and the unit cells are assembled with a series of rigid connections of beams. For simplicity, we name
5 / 34
the two beams as Beam I and Beam II and the lengths of Beam I and II are
l1 and l2,
respectively. Furthermore, the slenderness ratio of the beam is defined as . The crosssection of the beam is supposed as the square and the width of the beam is d a / (2 ) . The chiral angle between the Beam I and the X-axis is . Type I lattice is established by the orthogonally connected Beams I and II and their lengths have the relationship: l12 l22
a2 . Furthermore, the relationships between the 4
lengths of the Beam I and II and the chiral angle are l1 a cos and l 2 a sin . 2
2
For the Type II lattice, it is assembled with Beam I and Beam II which have the same lengths. The relationship between the length and chiral angle is l1 l 2
a . 4 cos
Therefore, the independent geometrical parameters of two types lattices are chiral angle
and slenderness ratio . The maximum chiral angle of Type II lattice is
max arctan2=63 . At the same time, material parameters of the structures are assumed as: Young’s modulus E=200 GPa, Poisson’s ratio
=0.3 and mass density
=7850 kg·m-3.
(a)
(b)
6 / 34
d
d l1
l1
l2 BeamІ
l2
BeamІ BeamІІ
BeamІІ
(c)
(d)
Fig. 1 Two types of auxetic chiral lattices: (a) Type I lattice, (b) Type II lattice, (c) Unit cell of Type I lattice, and (d) Unit cell of Type II lattice
2.2 The Bloch’s theorem Wave propagation in periodic systems is traditionally studied through the application of Bloch’s theorem, which is a relevant concept that originated in solid state physics. By applying the Bloch's theorem, we can study the wave characteristics in the whole lattice by analyzing the wave motion within a single unit cell, which greatly improves the calculation efficiency and saves the calculation cost. The joints of any lattice structure can be regarded as a collection of points, called lattice points, and these could be represented by a set of basis vectors. By selecting a suitable unit cell, the entire lattice can be obtained by tessellating the unit cell along the basis vectors. We also should define a reciprocal lattice in the wave vector space and the basis vectors of the direct and reciprocal lattice satisfy em en mn , where
em and en denote the basis vectors
of direct and reciprocal lattice respectively and
mn is the Kronecker delta function.
The subscripts m and n take the integer values 1 and 2 for the two-dimensional lattice. Therefore, we can obtain the first Brillouin zone and irretrievable Brillouin zone of the lattice as shown in Fig. 2 (black). The details of the vectors are also listed in Table 1. 7 / 34
e*2
e*1
Fig. 2 The first Brillouin zone and irretrievable Brillouin zone of the lattice
Table 1 the direct and reciprocal vectors of the square lattice
Square lattice
Cartesian basis
Reciprocal basis
e1 a (1, 0)T
e1* 2a (1,0)T
e 2 a (0,1)T
e*2 2a (0,1)T
Based on the concept of Bloch’s theorem, the displacement of the point in the reference unit cell corresponding to a wave propagation can be described as q rj q j e
where
q
j
is wave amplitude,
( it k r j )
(1)
is circular frequency, and k is wave vector of
plane wave. The point corresponding to the j th point in the reference unit cell, is denoted by the vector r rj ni ei ,(i 1,2) , and
r j denotes the position of the point
corresponding to the reference cell (0,0). So the displacement at the j th point in any cell identified by the integer
ni in the direct lattice is given by
q ( r ) q ( rj )e where
k ( r rj )
q (rj )e ki ni
(2)
ki i i represents the wave vector k along the reciprocal basis vector 8 / 34
e i
, and and represent the attenuation and phase constants respectively [43].
2.3 The dynamic modeling of unit cell It is important to establish the dynamic model of the considered structure for wave propagation in the periodic lattice. Normally, the most commonly used methods for establishing the dynamic equations of periodic structures are plane wave expansion method [46], transfer matrix method [47] and finite element method[11]. Due the auxetic chiral structure is assembled with a number of slender straight beams, which has the organizational regularity, therefore the finite element method is appropriately selected to establish the dynamic model of the structure. By considering the deformation modes of the auxetic chiral structures [48], we select the Timoshenko beam theory to describe the mechanical characteristics of the beams of auxetic chiral structure for more exact results. In this study, each unit cell is considered be assembled with a rigid-jointed network of beams and each beam is discretized into a four elements as depicted in Fig. 3.
ui , fi
u3 , f3
u4 , f 4 u2 , f 2
u1 , f1 Fig. 3 Degrees of freedom of the unit cell and interaction with neighboring cells
By applying the standard finite element method [34], we can assemble the global stiffness and mass matrix matrices. When we obtain the matrices, the equation of motion of unit cell can be expressed as
9 / 34
(K 2 M )u f ,
(3)
where Κ and Μ are the global mass and stiffness matrices of the unit cell, and
u
and f are the generalized nodal displacement vector and force vector of the unit cell respectively (see Fig. 4): u u1 u2 f = f1 f 2
u3 f3
ui
T
u4 f4
fi
T
.
(4)
In Eq. (4), the subscripts of 1, 2, 3, and 4 follows the notations as described in Fig. 4, while subscript i denotes the internal degrees of freedom of nodes of the unit cell. According to the Bloch’s theorem, periodic boundary conditions including the relating unit cell’s generalized displacements and the equilibrium conditions of the generalized forces can be expressed as :
u3 ek2 u1 f3 ek2 f1 , u4 ek1 u2 f 4 ek1 f 2
(5)
Eq. (5) can be rewritten in matrix form as follows:
u Aur , f Bfr where u r u1 u 2 ui . Substituting Eq. (6) into Eq. (4), and assuming T
(6)
fi 0 can
give K r (k1 , k2 ) 2 M r (k1 , k2 ) u r 0
(7)
where K r (k1 , k2 ) , M r (k1 , k2 ) are the reduced stiffness and mass matrices. Eq. (7) is an eigenvalue problem whose solution defines the dispersion characteristics of the lattice. The solution yields the frequency of wave propagation corresponding to the assigned pair k1 and k2 , and the complete solution obtained with the varying k1 and k2 can form a surface (k1 , k2 ) denoted as the dispersion surface. By take
advantage of the symmetry of the first Brillouin zone, the solution of Eq. (7) can be obtained by varying the wave vector along the contour of the first irreducible Brillouin zone. 10 / 34
The directionality of the wave is another important property of the wave propagation of the structure, which indicates whether the structure is isotropic or anisotropic. The indication of wave directions characteristics of the lattice is provided by the group velocity which can be expressed as: T
cg , . k1 k 2
(8)
Group velocity defines the directions of energy flows in the structure, so the preferential or forbidden directions of wave propagation can be determined. The determinations of the dispersion behaviors of the lattice are highlighted by the frequency-dependent group velocity, which describes the anisotropy of the domain in the plane wave propagation. Additionally, the topologies have important effects on the wave propagation behavior. In what follows, band structures and group velocities will be calculated and discussed for the auxetic chiral lattices with different geometrical parameters.
3 Results and discussions To investigate the wave propagation behavior of the auxetic chiral lattices, we calculate and analyze the band structures and the group velocities of two different types of structures. Especially, we also investigate the relations between the wave propagation properties and the geometrical parameters. To directly compare the wave propagation behaviors of two different lattices, the frequency values are presented in terms of the normalized frequency which is defined as
a , where a is the spatial 2 c
period of the structure and c E / is the velocity of longitudinal wave for the material. At the same time, to verify the existence of band gap and explore the relationship between the formation of band gap and vibrational mode, we also carry out the numerical simulation of vibrational characteristics of typical finite structure by using the commercial finite element software COMSOL to obtain the transmission 11 / 34
coefficients and vibrational modes. The auxetic chiral structure with a finite 21 21 unit cells for the finite element simulation is shown in Fig. 4. The harmonic excitations with different frequencies are applied on the central point of the left boundary of the lattice along the
x
direction and the responses are collected on the other side of the
lattice. Except for the excited points, all the points of the boundaries are free. The frequency response function is obtained by calculating the steady-state dynamic response of auxetic chiral structure. We also define the transmission coefficient T as a function of the non-dimensional frequency, which can be expressed as:
T 20 log10 (
U res ), U exc
(9)
where Uexc and Ures are the displacements of the excitation point and the response collection point respectively. The transmission coefficient
T
provides useful
information on the frequencies of the waves propagate throughout the structure. For example, the small value of T means that the wave propagation of the corresponding frequency is prohibited. The local applied fluctuations are still local and the response rapidly decay near the excitation points.
U1 (t ) Response
Excitation
y
o
x
Fig. 4 Schematics of auxetic chiral lattice subjected to imposed displacement U1 t
12 / 34
along the
x
direction. The excitation is imposed at a node on the left side of the
lattice’s boundary, while the response is collected at a point located on the other side of the lattice.
3.1 Band structures of the auxetic chiral structures A convenient dispersion characteristic representation is provided in the form of band structures, in which the propagation frequency of the wave is plotted against the magnitude of wave vector along the edge of irreducible Brillouin zone as shown in Fig. 2. Because the wave propagation are highly affect by the geometrical parameters, we also analyzes the band gap distributions associated with chiral angles. The band diagram of Type I lattice with the chiral angle =15 is shown in Fig. 5(a). Two complete band gaps are located at [0.0624, 0.1275] and [0.1441, 0.1830], respectively. The first band gap is located between the fourth and fifth branches, and the other one appears between the fifth and sixth branches. With the increase of the chiral angles, the fifth frequency curve of the band structure of Type I lattice moves down to the fourth frequency curve resulting the first band gap becomes small, and the sixth and seventh frequency curves move down to the fifth frequency curve which makes the second band gap appears at higher-frequency ranges as shown in Fig. 5(c). Only one band gap appears between the sixth and seventh branches as shown in Figs. 5(e) and (g). Fig. 5(b) shows the transmission coefficient of the Type I lattice obtained by the numerical simulation for verifying the band structure obtained by the theoretical calculation. A significantly low transmission rate of the wave can be found in the frequency range corresponding to the band gap (shaded area in the band structure) which demonstrates that the transmission coefficient can well express the location of the band gaps. These phenomena infer that the frequency range of the bandgap in the infinite structures can be investigated based on the vibration attenuation characteristics of the finite structure. 13 / 34
0.20
0.20
0.20
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
Γ
Χ
Μ
Γ
0.00
Ω
Ω
0.20
(b)
0.15
0.15
0.10
0.10
0.05
Μ
Γ
0.00
Γ
0.00
-300 -150
-300 -150
0
(d) 0.20 0.15
0.15 0.10
0.10
0.05
0.05
0.00
Γ
Χ
T
Μ
Γ
0.00
-300 -150
(g)
0
T
k
(f)
0
T
0.20
k
(e)
Μ
(c)
0.05
Χ
Χ
k
Ω
Ω
(a)
Γ
Γ
T
k
0.00
0.00
-300 -150 0
(h)
Fig. 5 Band structures of the Type I lattice with =1/10 where (a) = 1 5 , (c) =30°, (e) =60° and (g) =75°, and Frequency response diagrams of the Type I lattices with =1/10 where (b) = 1 5 , (d) =30°, (f) =60° and (h) =75°
From the above analysis, we can see that the chiral angles have significant effects on the wave propagation of the structure, especially in the low frequency range. Therefore, we investigate the band-gap width with the variation of chiral angles to further analyze the effects of chiral angles on the band structures of Type I lattice. The distributions of band gap at different chiral angles are shown in Fig. 6. When the chiral angle is 5.7°, two band gaps start to appear, whose widths gradually increase with the increase of chiral angles. The band gap between 4th-5th order frequencies reaches its maximum width when the chiral angle is 21.5° and its width of the band gap between 4th-5th order frequencies gradually decreases when the chiral angle is greater than 21.5°. At last, the band gap disappears completely when the chiral angle is 55°. The band gap between 5th6th order frequencies disappears when the chiral angle is 26.4°, and the third band gap appears in the higher-frequency range. 14 / 34
When the chiral angle is greater than 29.5°, the band gap between the 6th-7th order frequencies appears, whose upper boundary moves up and the lower boundary moves down as the increase of chiral angles. Therefore, the width of the band gap gradually increases. When the chiral angle is 46.3°, the width of the band gap reaches its maximum value, then the band gap disappears when the chiral is 66.4°. The band gap reappears when the chiral angle is in the range of 68.8°-80.6°. Based on the above analysis, we find that the chiral angle plays significant effects on the opening, closing, and position of band gaps of the Type I lattice. By adjusting the chiral angles of the structure, we can obtain band gaps in the desired frequency range.
0.25
0.20
0.15
0.10
0.05
0.00
0
10
20
30
40
50
60
70
80
Fig. 6 Band gap distribution of Type I lattice with different between the 4th-5th (green), 5th-6th (purple), 7th-8th (cyan-blue) and 6th-7th (blue) order frequencies
We also analyze the wave propagation properties of the Type II lattice. Fig. 7(a) plots the band structure of the Type II lattice with the chiral angle 15 . We can see that one complete band gap locates at [0.1021, 0.142] between the fourth and fifth branches. The band gap of Type II lattice with chiral angle is 30 (Fig. 7(c)) shows that the first four frequency curves move down resulting the first band gap moves down. The fifth and sixth frequency curves move down to the fifth frequency curve makes the appearance of the second band gap as shown in Fig. 7(c). As shown in Figs. 7(e) and 15 / 34
(g), there are two complete band gaps in the band structure when the chiral angles are 4 5 and 6 0 . The first band gap is located between the fourth and fifth branches and
the other one appears between the sixth and seventh branches. We also can see that the shape of the frequency curves in the band structure is identical, but the frequency of the structure with the chiral angle 60 is lower than that of the structure with chiral angle 45 . Meanwhile, the vibration transmission properties of Type II lattice are analyzed. Fig. 7(b), (d), (f) and (e) shows that the transmission coefficient of Type II lattice for verifying and comparing the results in the band structures. A significantly low transmission coefficient can be found in the frequency range corresponding to the band gap in Fig. 7(a), (c), (e) and (g). Obviously, the transmission coefficient can
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10 0.05 0.00
Γ
Μ
Χ
Γ
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
Ω
Ω
accurately demonstrate the position of the band gap.
-300 -150
(a)
Μ
Χ
Γ
0.00
-300 -150
(c)
0.20
0
T
k
(b)
0.20
(d)
0.10
0.10
0.05
0.05
0.15
0.15 0.10
0.10
0.05
0.05
Γ
Χ
Μ
Γ
0.00
Ω
Ω
Γ
T
k
0.00
0
-300 -150
0.00
Γ
Χ
T
k
(e)
0
Μ
Γ
0.00
-300 -150
T
k
(f)
(g)
0
(h)
Fig. 7 Band structures of the Type II lattice with =1/10 where (a) = 1 5 , (c) =30°, (e) =45° and (f) =60°, and frequency response diagrams of the Type II lattices with at =1/10 where (b) = 1 5 , (d) =30°, (f) =45° and (h) =60°
16 / 34
Fig. 8 shows the distribution of the band gap as the function of chiral angle. We can find that the band gap between 4th-5th order frequency curves appears at =7.4° and always exists when the chiral angle is greater than 7.4°. The upper boundary of the band gap shows upward trend while the lower boundary shows downward trend when the chiral angle is between 7.4°-10.8°, so that the width of the band gap increase firstly. The width of the band gap between 4th-5th order frequencies reaches the maximum value with the chiral angle is 10.8°. Then the upper and lower boundaries of band gaps show downward trends, and the upper boundary decrease faster than the lower boundary, therefore the width of the band gap decreases gradually. The similar trend also can be found in the distribution of the band gap between 6th-7th order frequencies. The band gap between 6th-7th order frequencies appears when the chiral angle is 16.1°, and decreases as the chiral angle is greater than 28.4°. The width of the band gap between 6th-7th order frequencies is significantly larger than that of the band gap between 4th-5th order frequency curves. Based on the above study, the desired band gap characteristics of Type II lattice can be obtained by selecting an appropriate chiral angle. Therefore, the optimal chiral angle can be obtained to achieve effective vibration and wave propagation control. 0.30 0.25 0.20
0.15 0.10 0.05 0.00
0
10
20
30
40
50
60
Fig. 8 Band gap distribution of Type II lattice with different between the 4th-5th (green), 5th-6th (purple) and 6th-7th (blue) order frequencies
17 / 34
The slenderness ratio is an important geometrical parameter and has significant effects on the band gap characteristics of chiral auxetic structures. Therefore, we study and compare the band structures of the auxetic chiral structures with different slenderness ratios which are 1/8 and 1/15, where the chiral angle is 45°. Fig. 9(a) shows the band structure of the auxetic chiral structure with =1/8 and there are two complete band gaps locate at [0.061, 0.086] and [0.1235, 0.213], respectively. The first band gap locates between the fourth and fifth branches, and the second band gap is between the fifth and sixth branches. We also see that the width of the second band gap is greater than that of the first band gap. The patterns of the frequency curves of the band structures with different slenderness ratios are the same, but the frequency with =1/8 is bigger than the frequency with =1/15 . Furthermore, the width of the band gap has deep relation with the slenderness ratio as shown in Fig.10. As the slenderness ratio decreases, the width of the two band gaps gradually decreases. The reason is a smaller slenderness ratio results a softer structure. Therefore the Eigen-frequencies decrease and the corresponding band gap is also gradually reduced. The band gap can be adjusted by changing the slenderness ratio of the structure without altering the overall topology of the structure, which gives us an alternative way to tune the band gap characteristics of the auxetic chiral structure.
0.15
0.25 0.20
0.10
Ω
Ω
0.15 0.10
0.05
0.05 0.00
Γ
Χ
Μ
Γ
k
0.00
Γ
Χ
Μ
k
18 / 34
Γ
(a)
(b)
Fig. 9 The band structures of the auxetic chiral structures with different slenderness ratio: (a) =1/8 and (b) =1/15, where the chiral angle is = 4 5 0.30 0.25
0.20 0.15 0.10 0.05 0.00 1/5
1/10
1/15
1/20
Fig. 10 The distribution of the band gap of the auxetic chiral structures with different , where the chiral angle is = 45
3.2 Mechanism of the generation of the band gap Base on the analysis, we find that the geometrical parameters can significantly affect the number and location of the band gap. In order to understand the fundamental mechanism of generation of band gap, the vibration modes of the auxetic chiral structure with different geometrical parameters at the boundaries of band gap are investigated. We only take the Type I lattice as the example to analyze the mechanism of generation of band gap. Additionally, as shown in Fig. 5, the lowest band gap which is between the fourth and fifth order frequencies is important for engineering applications. Therefore, we will analyze occurrence and disappearance of the band gap between the fourth and fifth order frequencies. As shown in Figs. 11(a) and (b), the lowest band gap begins to appear at =5.7 due to the increase of the fifth order frequency, and disappear at =55. The upper edge mode from Point A to Point B when the chiral angle is from 5.7 to 55 and the modes of Point A and Point B all contain the rotational vibration. However, there are four complete 19 / 34
standing points at the ends of Beam II at Point A, and four incomplete standing points in the ends of Beam I at Point B. Thus, these two modes will play different roles. The former one will result the generation of band gap, while the latter one can narrow and close the band gap. When the chiral angle is 25, the seventh and eighth order frequencies cross with each other, forming a mode crossing point, which prevents the generation of band gap. The points C1 and C2 stand for the crossing points of the seventh and eighth-order frequencies, respectively. To further analyze the relation between the crossing points and the vibration modes of the points, we plot the vibration modes of points C1 and C2. We can find that the two modes are similar and the mode of point C1 can be obtained by rotating the vibration mode of point C2 by 90 as shown in Fig. 11(c). When the chiral angle is 68, the fourth and fifth order frequencies cross with each other, forming a mode crossing point, which are labeled as points D1 and D2 respectively. We also find that the mode of point D1 can be obtained by rotating the vibration mode of point D2 by 90. Furthermore, we have analyzed the vibration modes of the mode crossing points in the band structures with different chiral angles. It can be concluded that the vibration modes of the bigger frequencies can be obtained by rotating the vibration modes of the smaller frequencies by 90 simply. Based on the above analysis, we find that the vibration mode plays a key role in the generation and disappearance of the band gap. The generation of the lowest band gap is due to the increase of the frequency of rotational mode of the unit cell. The appearance of the mode crossing point prevents the generation of band gap.
20 / 34
0.20
0.25 0.20
0.15
0.15 0.10
0.10
A B
0.05 0.05 0.00
Γ
Χ
Μ
0.00
Γ
Γ
Χ
Μ
Band structure for θ=5.7°
Band structure for θ=55°
Vibration mode of Point A
Vibration mode of Point B
(a)
(b) 0.20
C2 C1
0.20
Γ
0.15
0.15
0.10
D2 D1 0.05
0.05
0.00
0.10
Γ
Χ
Μ
0.00
Γ
Band structure for θ=25°
Γ
Χ
Μ
Band structure for θ=68°
21 / 34
Γ
Vibration mode of
Vibration mode of
Vibration mode of
Vibration mode of
Point C1
Point C2
Point D1
Point D2
(c)
(d)
Fig. 11 The band gap of the Type I lattice for =0.1 with various and vibration modes at the marked points
3.3 Directionality of wave propagation The group velocity is one of the important properties which have deep relationship with the directions of energy flows. The wave propagation along the specific directions can be obtained according to the group velocity. The propagation mode of the elastic wave in the heterogeneous material is more complicated, and the group velocity comparison is needed in the analysis of the dispersion characteristics. Additionally, elastic energy propagates with different speeds throughout the structure usually depend on the directional wave propagation. In a general way, the investigation of the directional wave propagation in the anisotropic structure are carried out on the basis of the group velocity associated with a particular frequency, which also contains the information of the propagation speed in different directions. For example, if a wave propagates in an isotropic structure, the group velocity corresponding to a certain mode is a circular graph and there will no preferential propagating directions. In contrast, the wave exhibits non-isotropic behavior if the group velocity shape is not a circle. For the Type I lattice with different varying angles, the group velocities as a function of frequency are plotted in Fig. 12. When the chiral angle is 15°, the group velocity of the first mode at the low-frequency limit is almost the same, indicating that the group velocity is insensitive to the low frequency. The group velocity with the chiral angle is 60° at the low-frequency limit is almost the same. However, as the frequency increases, the group velocity exhibits a very complex behavior characterized by the caustics (spikes in the group velocity distribution) which usually existed in the anisotropic medium [49]. This caustic is related to the energy focusing of the wave propagation due 22 / 34
to the interference between various wave components propagating in the lattice. Additionally, we also find that the direction of the group velocity will rotate counterclockwise at a certain angle as the chiral angle increases. We further analyze the effects of the chiral angles on the wave propagation in the low-frequency range. As shown in Fig. 13, the group velocities at the frequency 0.001 are plotted where =15˚, 30˚, 45˚, 60˚ and 75˚ respectively. We can see that the group velocities are concentrated in the directions of 0°, 180°, and
cg,2
0
-50
0
-25 -50
0
cg,1
50
-25
(a)
25
20
cg,2
cg,2
0
cg,1 (b)
20
0
-20 -20
all cases.
25
50
cg,2
° in
90
0 cg,1
20
0
-20 -20
(c)
0
cg,1
20
(d)
Fig. 12 The group velocity of the first mode of the Type I lattice: (a) =15˚, (b) =30˚, (c) =60˚ and (d) =75˚ where 0.001 (red), 0.003 (green),
23 / 34
0.005 (blue) and 0.007 (cyan-blue)
cg,2
50
0
-50 -50
0
cg,1
50
Fig. 13 The group velocity of the first mode of the Type I at 0.001 with different : =15˚ (red), =30˚(green), =45˚(blue), =60˚ (cyan-blue) and =75˚(purple)
To verify the above results and further analyze the preferential energy flow within the auxetic chiral lattice, we establish the finite element model to carry out the numerical simulations via the commercial finite element software COMSOL. The finite lattice model containing 21 unit cells is already shown in Fig. 4. Each beam of the lattice is considered as the Timoshenko beam model with a square cross section. The dimensions and material’s properties are also given in Section 2.1. The local rotational harmonic excitation with the corresponding normalized frequency are applied at the center of the structure. Fig. 14 shows the transient displacement response of structures at the considered time instant with different chiral angles. We can see that the vibrational energy propagates along the specific directions in auxetic chiral structures which is consistent with calculation results of the group velocities, and the auxetic chiral structure allows energy to preferentially propagate in both vertical and horizontal directions. As the chiral angle increases, the direction of the group velocity rotates counterclockwise. Therefore, the 24 / 34
auxetic chiral lattice provides the flexibility to make the elastic waves propagate along the desired directions, which can be adjusted by changing the topology of the auxetic chiral structure.
(a)
(b)
(c)
(d)
Fig. 14 Transient response of the Type I lattice under the harmonic excitations at
0.001 with different : (a) =15˚, (b) =30˚, (c) =60˚ and (d) =75˚
The group velocity of Type II lattice with varying chiral angles as a function of frequency and direction is shown in Fig. 15. When the chiral angle of Type II lattice is 25 / 34
15˚, the group velocity of the first-order mode at the low-frequency range is almost the same, which indicates elastic waves in the low frequency range travel in roughly the same directions in the auxetic chiral structure. The group velocity of Type II lattice with different chiral angles also shows the same characteristics. As the chiral angle increases, the group velocity rotates counterclockwise at a certain angle as shown in Fig. 15. To further investigate the effect of the chiral angle on the wave propagation, we plot the group velocities of Type II lattice at frequency 0.001 where =15˚, 30˚, 45˚, and 60˚ as shown in Fig. 16. Meanwhile, the transient displacement field of Type II lattice with different chiral angles is shown in Fig. 17. It is confirmed that the direction of energy propagation in the finite lattice is consistent with the group velocity prediction. The chiral angle of Type II lattice also has a great effect on the direction of the elastic wave.
140
cg,2
cg,2
50
0
0
-50 -140 -140
0
cg,1
140
(a)
-50
0
cg,1 (b)
26 / 34
50
12
cg,2
cg,2
30
0
-30 -30
0
-12 -12
30
cg,1
0
0
cg,1
(c)
12
(d)
Fig. 15 The group velocity of the first mode of the Type II lattice: (a) =15˚, (b) =30˚, (c) =45˚ and (d) =60˚ where 0.001 (red), 0.003 (green),
0.005 (blue) and 0.007 (cyan-blue)
150 100
cg,2
50 0 -50 -100 -150 -150
-100
-50
0
cg,1
50
100
150
Fig. 16 The group velocity of the first mode of the Type II lattice at 0.001 with different : =15˚ (red), =30˚(green), =45˚(blue) and =60˚ (cyan-blue)
27 / 34
(a)
(b)
(c)
(d)
Fig. 17 Transient response of the Type II lattice under the harmonic excitations at
0.001 with different : (a) =15˚, (b) =30˚, (c) =45˚ and (d) =60˚
4 Conclusions This study investigates the wave propagations in auxetic chiral structure. Based on the Bloch’s theorem and finite element method, the dynamic equation of auxetic chiral structure are established and eigenvalue are solved to analyze the elastic wave propagation in chiral lattices. The main findings are as follows:
28 / 34
(1) The auxetic chiral structures have special wave propagation properties. By investigating the band structure and vibrational mode, we can find that the vibration mode plays a key role in the generation and disappearance of the band gap. The band gap at different frequency ranges can be obtained by adjusting topologies of the auxetic chiral structure to meet different engineering needs. (2) Chiral angle of auxetic chiral structure has significant effect on the direction of wave propagation. With the increase of chiral angle, the direction of the wave propagation rotates counterclockwise. (3) The transient response of the finite structure under harmonic excitation is simulated by the commercial finite element software. The frequencies where the transmission efficiency reduces is the same as the frequencies of the band gap. The directions of the wave propagation results of the numerical simulation and the theoretical calculation are identical, which provides another way to predict the band gap and directional waveguide.
Acknowledgements Funding for this work has been provided by the National Key R&D Program of China (2017YFB1102801), National Natural Science Foundation of China (Nos. 11872313 and 11502202), Fundamental Research Funds for the Central Universities.
References [1] Fozdar DY, Soman P, Lee JW, Han LH, Chen S. Three-Dimensional Polymer Constructs Exhibiting a Tunable Negative Poisson's Ratio. Advanced Functional Materials. 2011;21:2712-20. [2] Lee J-H, Singer J, Thomas E. Micro-/Nanostructured Mechanical Metamaterials. Advanced materials. 2012;24:4782-810. [3] Chen Y, Jia Z, Wang L. Hierarchical honeycomb lattice metamaterials with improved thermal resistance and mechanical properties. Composite Structures. 2016;152:395-402. [4] Zheng X, Lee H, Weisgraber TH, Shusteff M, DeOtte J, Duoss EB, et al. Ultralight, ultrastiff mechanical metamaterials. Science. 2014;344:1373-7. 29 / 34
[5] Wu W, Hu W, Qian G, Liao H, Xu X, Berto F. Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review. Materials and Design. 2019;180:107950. [6] Zhang K, Ge M, Zhao C, Deng Z, Xu X. Free vibration of nonlocal Timoshenko beams made of functionally graded materials by Symplectic method. Composites Part B: Engineering. 2019;156:174-84. [7] Wang J, Wang G, Feng X, Kitamura T, Kang Y, Yu S, et al. Hierarchical chirality transfer in the growth of Towel Gourd tendrils. Scientific Reports. 2013;3:1-7. [8] Ai L, Gao X-L. Metamaterials with negative Poisson’s ratio and non-positive thermal expansion. Composite Structures. 2017;162:70-84. [9] Buckmann T, Stenger N, Kadic M, Kaschke J, Frolich A, Kennerknecht T, et al. Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Advanced materials. 2012;24:2710-4. [10] Prawoto Y. Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio. Computational Materials Science. 2012;58:140-53. [11] Tee KF, Spadoni A, Scarpa F, Ruzzene M. Wave Propagation in Auxetic Tetrachiral Honeycombs. Journal of Vibration and Acoustics. 2010;132:031007-1-8. [12] Scarpa F, Tomlin PJ. On the transverse shear modulus of negative Poisson's ratio honeycomb structures. Fatigue and Fracture of Engineering Materials and Structures. 2000;23:717-20. [13] Tang Y, Lin G, Han L, Qiu S, Yang S, Yin J. Design of Hierarchically Cut Hinges for Highly Stretchable and Reconfigurable Metamaterials with Enhanced Strength. Advanced materials. 2015;27:7181-90. [14] Grima JN, Evans KE. Auxetic behavior from rotating triangles. Journal of Materials Science. 2006;41:3193-6. [15] Shan S, Kang SH, Zhao Z, Fang L, Bertoldi K. Design of planar isotropic negative Poisson’s ratio structures. Extreme Mechanics Letters. 2015;4:96-102. [16] Lakes R. Foam structures with a negative poisson's ratio. Science. 1987;235:1038-40. [17] Cho Y, Shin JH, Costa A, Kim TA, Kunin V, Li J, et al. Engineering the shape and structure of materials by fractal cut. Proceedings of the National Academy of Sciences of the United States of America. 2014;111:17390-5. [18] Wagner M, Chen T, Shea K. Large shape transforming 4D auxetic structures. 3D Printing and Additive Manufacturing. 2017;4:133-42. [19] Ebrahimi H, Mousanezhad D, Haghpanah B, Ghosh R, Vaziri A. Lattice Materials with Reversible Foldability Advanced Engineering Materials. 2017;19:1600646-1-6. [20] Gatt R, Caruana G, Roberto, Attard D, Casha A, Wolak W, Dudek K, et al. On the properties of real finite-sized planar and tubular stent-like auxetic structures. Physica Status Solidi (B). 2014;251:321-7. [21] Mizzi L, Attard D, Casha A, Grima JN, Gatt R. On the suitability of hexagonal honeycombs as stent geometries. Physica Status Solidi B. 2014;251:328-37. [22] Tan TW, Douglas GR, Bond T, Phani AS. Compliance and Longitudinal Strain of Cardiovascular Stents: Influence of Cell Geometry. Journal of Medical Devices. 2011;5:041002-1-6. 30 / 34
[23] Scarpa F, Smith FC, Chambers B, Burriesci G. Mechanical and electromagnetic behaviour of auxetic honeycomb structures. Aeronautical Journal. 2003;107:175-83. [24] Scarpa F, Ciffo LG, Yates JR. Dynamic properties of high structural integrity auxetic open cell foam. Smart Materials and Structures. 2004;13:49-56. [25] Babaee S, Shim J, Weaver JC, Chen ER, Patel N, Bertoldi K. 3D soft metamaterials with negative Poisson's ratio. Advanced materials. 2013;25:5044-9. [26] Mousanezhad D, Babaee S, Ebrahimi H, Ghosh R, Hamouda AS, Bertoldi K, et al. Hierarchical honeycomb auxetic metamaterials. Scientific Reports. 2015;5:1-8. [27] Bertoldi K, Reis PM, Willshaw S, Mullin T. Negative Poisson's ratio behavior induced by an elastic instability. Advanced materials. 2010;22:361-6. [28] Jiang Y, Li Y. 3D Printed Auxetic Mechanical Metamaterial with Chiral Cells and Re-entrant Cores. Scientific Reports. 2018;8:1-11. [29] Gaspar N, Ren X, Smith C, Grima J, Evans K. Novel honeycombs with auxetic behaviour. Acta Materialia. 2005;53:2439-45. [30] Smith CW, Grima JN, Evans KE. A novel mechanism for generating auxetic behaviour in reticulated foams: missing rib foam model. Acta Materialia. 2000;48:4349-56. [31] Nemat-Nasser S. Anti-plane shear waves in periodic elastic composites: band structure and anomalous wave refraction. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2015;471. [32] Mokhtari AA, Lu Y, Srivastava A. On the emergence of negative effective density and modulus in 2-phase phononic crystals. Journal of the Mechanics and Physics of Solids. 2019;126:25671. [33] Shmuel G, Band R. Universality of the frequency spectrum of laminates. Journal of the Mechanics and Physics of Solids. 2016;92:127-36. [34] Phani AS, Woodhouse J, Fleck NA. Wave propagation in two-dimensional periodic lattices. The Journal of the Acoustical Society of America. 2006;119:1995-2005. [35] Trainiti G, Rimoli JJ, Ruzzene M. Wave propagation in undulated structural lattices. International Journal of Solids and Structures. 2016;97-98:431-44. [36] Zhang K, Zhao P, Hong F, Yu Y, Deng Z. On the directional wave propagation in the tetrachiral and hexachiral lattices with local resonators. Smart Materials and Structures. 2020;29:015017. [37] Zhang K, Zhao C, Luo J, Ma Y, Deng Z. Analysis of temperature-dependent wave propagation for programmable lattices. International Journal of Mechanical Sciences. 2020;171:105372. [38] Schaeffer M, Ruzzene M. Wave propagation in multistable magneto-elastic lattices. International Journal of Solids and Structures. 2015;56-57:78-95. [39] Warmuth F, Korner C. Phononic Band Gaps in 2D Quadratic and 3D Cubic Cellular Structures. Materials. 2015;8:8327-37. [40] Bertoldi K, Boyce MC. Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations. Physical Review B. 2008;78. [41] Trainiti G, Rimoli JJ, Ruzzene M. Wave propagation in periodically undulated beams and plates. International Journal of Solids and Structures. 2015;75–76:260-76. [42] Martinsson PG, Movchan AB. Vibrations of lattice structures and phononic band gaps. Quarterly Journal Of Mechanics And Applied Mathematics. 2003;569(1):45-64. 31 / 34
[43] Spadoni A, Ruzzene M, Gonella S, Scarpa F. Phononic properties of hexagonal chiral lattices. Wave Motion. 2009;46:435-50. [44] Hussein MI, Leamy MJ, Ruzzene M. Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook. Applied Mechanics Reviews. 2014;66:040802-1-38. [45] Wang Y, Wang Y, Zhang C. Bandgaps and directional propagation of elastic waves in 2D square zigzag lattice structures. Journal of Physics D: Applied Physics. 2014;47:1-14. [46] Su Xl, Gao Yw, Zhou Yh. The influence of material properties on the elastic band structures of one-dimensional functionally graded phononic crystals. Journal of Applied Physics. 2012;112. [47] Li F, Zhang C, Liu C. Active tuning of vibration and wave propagation in elastic beams with periodically placed piezoelectric actuator/sensor pairs. Journal of Sound and Vibration. 2017;393:14-29. [48] Jin S, Korkolis YP, Li Y. Shear resistance of an auxetic chiral mechanical metamaterial. International Journal of Solids and Structures. 2019;174-175:28-37. [49] Taylor B, Maris HJ, Elbaum C. Focusing of Phonons in Crystalline Solids due to Elastic Anisotropy. Physical Review B. 1971;3:1462-72.
32 / 34
Author statement
Kai Zhang: Conceptualization, Methodology, Writing - Original Draft, Writing Review & Editing. Pengcheng Zhao: Conceptualization, Formal analysis, Validation, Writing - Original Draft. Cheng Zhao: Software, Investigation. Fang Hong: Investigation, Resources. Zichen Deng: Supervision.
33 / 34
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
34 / 34
S0263-8223(19)34229-1 https://doi.org/10.1016/j.compstruct.2020.111952 COST 111952
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
7 November 2019 8 January 2020 17 January 2020
Please cite this article as: Zhang, K., Zhao, P., Zhao, C., Hong, F., Deng, Z., Study on the mechanism of band gap and directional wave propagation of the auxetic chiral lattices, Composite Structures (2020), doi: https://doi.org/ 10.1016/j.compstruct.2020.111952
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Ltd.
Study on the mechanism of band gap and directional wave propagation of the auxetic chiral lattices Kai Zhang1,2*, Pengcheng Zhao1,2, Cheng Zhao1,2, Fang Hong1,2, Zichen Deng1,2 1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, P. R. China 2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072, P. R. China
Abstract In this study, the wave propagation properties in terms of the band gap and directions of wave propagation of the auxetic chiral structure are analyzed. The mechanism of generation of the band gap are carefully investigated. The auxetic chiral structure are assembled with the repeat unit cells and the unit cell contains a number of rigidly connected beams. The dynamic model of the unit cell are established by the principle of finite element method. The wave behaviors of the lattices are calculated by solving the dynamic model with the help of the Bloch’s theorem. The band structure are obtained and the effects of the chiral angles on the width and position of the band gap distributions are carefully studied. Especially, the mechanism of formation of the band gap are also analyzed by investigating the vibrational mode calculated by the commercial finite element software. The group velocities are calculated to analyze the effects of the geometrical parameters on the directional frequency-dependent energy flows in the structures. We also use the commercial finite element software to simulate the directional wave behaviors in the structure. We find that the first mode of the elastic wave spread only along certain specific directions in the auxetic chiral structure. The *
Corresponding author, Email: [email protected] (Kai Zhang) 1 / 34
speed of wave propagation will be reduced, and the direction of wave propagation rotates counterclockwise when the chiral angle increases. Keywords: Wave propagation, Auxetic chiral structure, Band gap, Group velocity
2 / 34
1 Introduction Mechanical metamaterials are a kind of artificial archiechtured materials have broad engineering applications like the energy absorption foams, sensors and actuators, smart composites [1] in aerospace, automobile, civil industries, etc. Their unique mechanical properties can be achieved via innovative geometric design [2-4]. In the past few years, scholars had used rigid and flexible components to carefully design microstructures with special configurations. A number of unique mechanical properties was found, such as negative Poisson's ratio, vibration attenuation, impact resistance, etc [5, 6]. Additionally, the fast growing additive manufacturing technology enables rapid production of the complex geometry, which provides a new way to design and manufacture the innovative metamaterials [7-9]. The auxetic metamaterial is one important category of the metamaterials, and has the light weight while retaining high rigidity and strength [10]. It also has excellent energy absorption capability, and special acoustic properties [11-13]. Furthermore, auxeticity can increase fracture toughness, and shear stiffness [14-16] of the structure and let the structure to undergo dramatic shape changes [17-19]. The above properties highlight the potential applications of auxetic metamaterials in developing novel lattice metal [20-22], corrugated panels [23], and foam metal [24]. Recently, chiral structures with auxetic properties have attracted great interests around the academic world as its advantages for potential applications in flexible and smart structures, and components. Normally, the existing auxetic mechanical metamaterials have two basic deformation mechanisms. One is the deformations of the symmetric units with re-entrant angles (also be called as instability-induced auxetic effects) [2527], and the other one is the rotations of the chiral units when the structure deformed (chirality-induced auxetic effects) [28-30]. Although the deformations of the auxetic chiral lattice have been extensively studied, the study of the dynamic properties of the auxetic chiral lattice from the viewpoint of wave propagation has not been fully studied. In recent years, the study of wave propagation in periodic metamaterial lattice structures 3 / 34
with tunable wave properties and directional waveguides has aroused considerable interests, and the progress largely depends on determining the wave propagation characteristics in the periodic systems [31-33]. In these wave propagation properties, the band structure is one the important properties and it has deep relation with topologies and geometries of structures [34]. The group velocity is also another important properties to analyze the anisotropic characteristics and the directional waveguide of the structure, which is the elastic wave propagates along a specific direction in the structure [35, 36]. There are also many efforts were carried out to tune the wave propagation in the lattices by applying the external effects like the temperature fields [37] and magneto-elastic effects [38]. However, the wave behavior is still highly determined by the topology and geometry of the structures [39-41]. Therefore, the relations between the wave behaviors and geometrical parameters become significantly important topics for the analysis of wave propagation and design of the innovational structure. Because of the special mechanical properties, the chiral lattices have the unique vibrational filtering [42] and wave steering capabilities [43] which make the lattice to be one of the best choice for the sound isolation [44]. We also found that the chirality-induced rotational efficiency can deduce the effective auxetic effect, which can significantly amplify the effects of chiral metamaterial [28]. Additionally, by utilizing the two basic deformation mechanisms of re-entrant angle and chirality [28], new auxetic mechanical metamaterials are developed. Furthermore, the study on the wave propagation behaviors of lattice structure with zigzag arms [29, 30, 45] shows that lower-frequencies complete band gaps can be generated and good directional properties are produced in such structure compared with the traditional lattices with straight arms. These results infer that the rotating auxetic chiral structures can exhibit unique wave propagation behaviors. Therefore, we have great interests to carry out the study on the wave propagation analysis to find the relationship between the structural deformation and wave behaviors. To study the wave propagation characteristics of the auxetic chiral lattices, we establish 4 / 34
the dynamic model of the structure and analyze the effects of configurations and geometrical parameters of the structure on the band structures. The relationship between the band gap and the vibration mode is carefully investigated. We compare the analytical results with the results obtained by using the commercial finite element software COMSOL for verifying the band structures and transmission spectra of the structures. The vibration modes at the band gap edges are calculated to analyze the mechanism of the generation of band gap. The directional characteristics of the wave propagation in terms of the group velocities in the auxetic chiral lattice are analyzed. The transient propagation of elastic waves in finite structures, are calculated to show the directional properties of the considered systems. The paper is organized in four sections, including the Introduction section above. In Section 2, brief descriptions of the two kinds of auxetic chiral lattices are given, and analytical models for the dynamical behaviors of the structures are established. Section 3 investigates the performance of various types of structures in terms of band gaps and directionality of the wave propagation. Additionally, the finite element (FE) simulations are carried out to verify the theoretical calculations and to discuss the mechanism of formation of band gaps. Finally, Section 4 summarizes the important conclusions drawn by this study.
2 The dynamical model of the auxetic chiral lattices 2.1 Geometries of the auxetic chiral lattices In this study, the characteristics of wave propagation in two types of auxetic chiral structures are investigated. The periodic auxetic chiral lattices which are Type I lattice and Type II lattice are shown in Figs. 1(a) and (b). The unit cell of chiral auxetic lattices are four-fold symmetry, and two different types of unit cells are designed respectively as shown in Figs. 1(c) and (d). The dimensions of the unit cells are a a , and the unit cells are assembled with a series of rigid connections of beams. For simplicity, we name
5 / 34
the two beams as Beam I and Beam II and the lengths of Beam I and II are
l1 and l2,
respectively. Furthermore, the slenderness ratio of the beam is defined as . The crosssection of the beam is supposed as the square and the width of the beam is d a / (2 ) . The chiral angle between the Beam I and the X-axis is . Type I lattice is established by the orthogonally connected Beams I and II and their lengths have the relationship: l12 l22
a2 . Furthermore, the relationships between the 4
lengths of the Beam I and II and the chiral angle are l1 a cos and l 2 a sin . 2
2
For the Type II lattice, it is assembled with Beam I and Beam II which have the same lengths. The relationship between the length and chiral angle is l1 l 2
a . 4 cos
Therefore, the independent geometrical parameters of two types lattices are chiral angle
and slenderness ratio . The maximum chiral angle of Type II lattice is
max arctan2=63 . At the same time, material parameters of the structures are assumed as: Young’s modulus E=200 GPa, Poisson’s ratio
=0.3 and mass density
=7850 kg·m-3.
(a)
(b)
6 / 34
d
d l1
l1
l2 BeamІ
l2
BeamІ BeamІІ
BeamІІ
(c)
(d)
Fig. 1 Two types of auxetic chiral lattices: (a) Type I lattice, (b) Type II lattice, (c) Unit cell of Type I lattice, and (d) Unit cell of Type II lattice
2.2 The Bloch’s theorem Wave propagation in periodic systems is traditionally studied through the application of Bloch’s theorem, which is a relevant concept that originated in solid state physics. By applying the Bloch's theorem, we can study the wave characteristics in the whole lattice by analyzing the wave motion within a single unit cell, which greatly improves the calculation efficiency and saves the calculation cost. The joints of any lattice structure can be regarded as a collection of points, called lattice points, and these could be represented by a set of basis vectors. By selecting a suitable unit cell, the entire lattice can be obtained by tessellating the unit cell along the basis vectors. We also should define a reciprocal lattice in the wave vector space and the basis vectors of the direct and reciprocal lattice satisfy em en mn , where
em and en denote the basis vectors
of direct and reciprocal lattice respectively and
mn is the Kronecker delta function.
The subscripts m and n take the integer values 1 and 2 for the two-dimensional lattice. Therefore, we can obtain the first Brillouin zone and irretrievable Brillouin zone of the lattice as shown in Fig. 2 (black). The details of the vectors are also listed in Table 1. 7 / 34
e*2
e*1
Fig. 2 The first Brillouin zone and irretrievable Brillouin zone of the lattice
Table 1 the direct and reciprocal vectors of the square lattice
Square lattice
Cartesian basis
Reciprocal basis
e1 a (1, 0)T
e1* 2a (1,0)T
e 2 a (0,1)T
e*2 2a (0,1)T
Based on the concept of Bloch’s theorem, the displacement of the point in the reference unit cell corresponding to a wave propagation can be described as q rj q j e
where
q
j
is wave amplitude,
( it k r j )
(1)
is circular frequency, and k is wave vector of
plane wave. The point corresponding to the j th point in the reference unit cell, is denoted by the vector r rj ni ei ,(i 1,2) , and
r j denotes the position of the point
corresponding to the reference cell (0,0). So the displacement at the j th point in any cell identified by the integer
ni in the direct lattice is given by
q ( r ) q ( rj )e where
k ( r rj )
q (rj )e ki ni
(2)
ki i i represents the wave vector k along the reciprocal basis vector 8 / 34
e i
, and and represent the attenuation and phase constants respectively [43].
2.3 The dynamic modeling of unit cell It is important to establish the dynamic model of the considered structure for wave propagation in the periodic lattice. Normally, the most commonly used methods for establishing the dynamic equations of periodic structures are plane wave expansion method [46], transfer matrix method [47] and finite element method[11]. Due the auxetic chiral structure is assembled with a number of slender straight beams, which has the organizational regularity, therefore the finite element method is appropriately selected to establish the dynamic model of the structure. By considering the deformation modes of the auxetic chiral structures [48], we select the Timoshenko beam theory to describe the mechanical characteristics of the beams of auxetic chiral structure for more exact results. In this study, each unit cell is considered be assembled with a rigid-jointed network of beams and each beam is discretized into a four elements as depicted in Fig. 3.
ui , fi
u3 , f3
u4 , f 4 u2 , f 2
u1 , f1 Fig. 3 Degrees of freedom of the unit cell and interaction with neighboring cells
By applying the standard finite element method [34], we can assemble the global stiffness and mass matrix matrices. When we obtain the matrices, the equation of motion of unit cell can be expressed as
9 / 34
(K 2 M )u f ,
(3)
where Κ and Μ are the global mass and stiffness matrices of the unit cell, and
u
and f are the generalized nodal displacement vector and force vector of the unit cell respectively (see Fig. 4): u u1 u2 f = f1 f 2
u3 f3
ui
T
u4 f4
fi
T
.
(4)
In Eq. (4), the subscripts of 1, 2, 3, and 4 follows the notations as described in Fig. 4, while subscript i denotes the internal degrees of freedom of nodes of the unit cell. According to the Bloch’s theorem, periodic boundary conditions including the relating unit cell’s generalized displacements and the equilibrium conditions of the generalized forces can be expressed as :
u3 ek2 u1 f3 ek2 f1 , u4 ek1 u2 f 4 ek1 f 2
(5)
Eq. (5) can be rewritten in matrix form as follows:
u Aur , f Bfr where u r u1 u 2 ui . Substituting Eq. (6) into Eq. (4), and assuming T
(6)
fi 0 can
give K r (k1 , k2 ) 2 M r (k1 , k2 ) u r 0
(7)
where K r (k1 , k2 ) , M r (k1 , k2 ) are the reduced stiffness and mass matrices. Eq. (7) is an eigenvalue problem whose solution defines the dispersion characteristics of the lattice. The solution yields the frequency of wave propagation corresponding to the assigned pair k1 and k2 , and the complete solution obtained with the varying k1 and k2 can form a surface (k1 , k2 ) denoted as the dispersion surface. By take
advantage of the symmetry of the first Brillouin zone, the solution of Eq. (7) can be obtained by varying the wave vector along the contour of the first irreducible Brillouin zone. 10 / 34
The directionality of the wave is another important property of the wave propagation of the structure, which indicates whether the structure is isotropic or anisotropic. The indication of wave directions characteristics of the lattice is provided by the group velocity which can be expressed as: T
cg , . k1 k 2
(8)
Group velocity defines the directions of energy flows in the structure, so the preferential or forbidden directions of wave propagation can be determined. The determinations of the dispersion behaviors of the lattice are highlighted by the frequency-dependent group velocity, which describes the anisotropy of the domain in the plane wave propagation. Additionally, the topologies have important effects on the wave propagation behavior. In what follows, band structures and group velocities will be calculated and discussed for the auxetic chiral lattices with different geometrical parameters.
3 Results and discussions To investigate the wave propagation behavior of the auxetic chiral lattices, we calculate and analyze the band structures and the group velocities of two different types of structures. Especially, we also investigate the relations between the wave propagation properties and the geometrical parameters. To directly compare the wave propagation behaviors of two different lattices, the frequency values are presented in terms of the normalized frequency which is defined as
a , where a is the spatial 2 c
period of the structure and c E / is the velocity of longitudinal wave for the material. At the same time, to verify the existence of band gap and explore the relationship between the formation of band gap and vibrational mode, we also carry out the numerical simulation of vibrational characteristics of typical finite structure by using the commercial finite element software COMSOL to obtain the transmission 11 / 34
coefficients and vibrational modes. The auxetic chiral structure with a finite 21 21 unit cells for the finite element simulation is shown in Fig. 4. The harmonic excitations with different frequencies are applied on the central point of the left boundary of the lattice along the
x
direction and the responses are collected on the other side of the
lattice. Except for the excited points, all the points of the boundaries are free. The frequency response function is obtained by calculating the steady-state dynamic response of auxetic chiral structure. We also define the transmission coefficient T as a function of the non-dimensional frequency, which can be expressed as:
T 20 log10 (
U res ), U exc
(9)
where Uexc and Ures are the displacements of the excitation point and the response collection point respectively. The transmission coefficient
T
provides useful
information on the frequencies of the waves propagate throughout the structure. For example, the small value of T means that the wave propagation of the corresponding frequency is prohibited. The local applied fluctuations are still local and the response rapidly decay near the excitation points.
U1 (t ) Response
Excitation
y
o
x
Fig. 4 Schematics of auxetic chiral lattice subjected to imposed displacement U1 t
12 / 34
along the
x
direction. The excitation is imposed at a node on the left side of the
lattice’s boundary, while the response is collected at a point located on the other side of the lattice.
3.1 Band structures of the auxetic chiral structures A convenient dispersion characteristic representation is provided in the form of band structures, in which the propagation frequency of the wave is plotted against the magnitude of wave vector along the edge of irreducible Brillouin zone as shown in Fig. 2. Because the wave propagation are highly affect by the geometrical parameters, we also analyzes the band gap distributions associated with chiral angles. The band diagram of Type I lattice with the chiral angle =15 is shown in Fig. 5(a). Two complete band gaps are located at [0.0624, 0.1275] and [0.1441, 0.1830], respectively. The first band gap is located between the fourth and fifth branches, and the other one appears between the fifth and sixth branches. With the increase of the chiral angles, the fifth frequency curve of the band structure of Type I lattice moves down to the fourth frequency curve resulting the first band gap becomes small, and the sixth and seventh frequency curves move down to the fifth frequency curve which makes the second band gap appears at higher-frequency ranges as shown in Fig. 5(c). Only one band gap appears between the sixth and seventh branches as shown in Figs. 5(e) and (g). Fig. 5(b) shows the transmission coefficient of the Type I lattice obtained by the numerical simulation for verifying the band structure obtained by the theoretical calculation. A significantly low transmission rate of the wave can be found in the frequency range corresponding to the band gap (shaded area in the band structure) which demonstrates that the transmission coefficient can well express the location of the band gaps. These phenomena infer that the frequency range of the bandgap in the infinite structures can be investigated based on the vibration attenuation characteristics of the finite structure. 13 / 34
0.20
0.20
0.20
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
Γ
Χ
Μ
Γ
0.00
Ω
Ω
0.20
(b)
0.15
0.15
0.10
0.10
0.05
Μ
Γ
0.00
Γ
0.00
-300 -150
-300 -150
0
(d) 0.20 0.15
0.15 0.10
0.10
0.05
0.05
0.00
Γ
Χ
T
Μ
Γ
0.00
-300 -150
(g)
0
T
k
(f)
0
T
0.20
k
(e)
Μ
(c)
0.05
Χ
Χ
k
Ω
Ω
(a)
Γ
Γ
T
k
0.00
0.00
-300 -150 0
(h)
Fig. 5 Band structures of the Type I lattice with =1/10 where (a) = 1 5 , (c) =30°, (e) =60° and (g) =75°, and Frequency response diagrams of the Type I lattices with =1/10 where (b) = 1 5 , (d) =30°, (f) =60° and (h) =75°
From the above analysis, we can see that the chiral angles have significant effects on the wave propagation of the structure, especially in the low frequency range. Therefore, we investigate the band-gap width with the variation of chiral angles to further analyze the effects of chiral angles on the band structures of Type I lattice. The distributions of band gap at different chiral angles are shown in Fig. 6. When the chiral angle is 5.7°, two band gaps start to appear, whose widths gradually increase with the increase of chiral angles. The band gap between 4th-5th order frequencies reaches its maximum width when the chiral angle is 21.5° and its width of the band gap between 4th-5th order frequencies gradually decreases when the chiral angle is greater than 21.5°. At last, the band gap disappears completely when the chiral angle is 55°. The band gap between 5th6th order frequencies disappears when the chiral angle is 26.4°, and the third band gap appears in the higher-frequency range. 14 / 34
When the chiral angle is greater than 29.5°, the band gap between the 6th-7th order frequencies appears, whose upper boundary moves up and the lower boundary moves down as the increase of chiral angles. Therefore, the width of the band gap gradually increases. When the chiral angle is 46.3°, the width of the band gap reaches its maximum value, then the band gap disappears when the chiral is 66.4°. The band gap reappears when the chiral angle is in the range of 68.8°-80.6°. Based on the above analysis, we find that the chiral angle plays significant effects on the opening, closing, and position of band gaps of the Type I lattice. By adjusting the chiral angles of the structure, we can obtain band gaps in the desired frequency range.
0.25
0.20
0.15
0.10
0.05
0.00
0
10
20
30
40
50
60
70
80
Fig. 6 Band gap distribution of Type I lattice with different between the 4th-5th (green), 5th-6th (purple), 7th-8th (cyan-blue) and 6th-7th (blue) order frequencies
We also analyze the wave propagation properties of the Type II lattice. Fig. 7(a) plots the band structure of the Type II lattice with the chiral angle 15 . We can see that one complete band gap locates at [0.1021, 0.142] between the fourth and fifth branches. The band gap of Type II lattice with chiral angle is 30 (Fig. 7(c)) shows that the first four frequency curves move down resulting the first band gap moves down. The fifth and sixth frequency curves move down to the fifth frequency curve makes the appearance of the second band gap as shown in Fig. 7(c). As shown in Figs. 7(e) and 15 / 34
(g), there are two complete band gaps in the band structure when the chiral angles are 4 5 and 6 0 . The first band gap is located between the fourth and fifth branches and
the other one appears between the sixth and seventh branches. We also can see that the shape of the frequency curves in the band structure is identical, but the frequency of the structure with the chiral angle 60 is lower than that of the structure with chiral angle 45 . Meanwhile, the vibration transmission properties of Type II lattice are analyzed. Fig. 7(b), (d), (f) and (e) shows that the transmission coefficient of Type II lattice for verifying and comparing the results in the band structures. A significantly low transmission coefficient can be found in the frequency range corresponding to the band gap in Fig. 7(a), (c), (e) and (g). Obviously, the transmission coefficient can
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10 0.05 0.00
Γ
Μ
Χ
Γ
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.00
0.00
Ω
Ω
accurately demonstrate the position of the band gap.
-300 -150
(a)
Μ
Χ
Γ
0.00
-300 -150
(c)
0.20
0
T
k
(b)
0.20
(d)
0.10
0.10
0.05
0.05
0.15
0.15 0.10
0.10
0.05
0.05
Γ
Χ
Μ
Γ
0.00
Ω
Ω
Γ
T
k
0.00
0
-300 -150
0.00
Γ
Χ
T
k
(e)
0
Μ
Γ
0.00
-300 -150
T
k
(f)
(g)
0
(h)
Fig. 7 Band structures of the Type II lattice with =1/10 where (a) = 1 5 , (c) =30°, (e) =45° and (f) =60°, and frequency response diagrams of the Type II lattices with at =1/10 where (b) = 1 5 , (d) =30°, (f) =45° and (h) =60°
16 / 34
Fig. 8 shows the distribution of the band gap as the function of chiral angle. We can find that the band gap between 4th-5th order frequency curves appears at =7.4° and always exists when the chiral angle is greater than 7.4°. The upper boundary of the band gap shows upward trend while the lower boundary shows downward trend when the chiral angle is between 7.4°-10.8°, so that the width of the band gap increase firstly. The width of the band gap between 4th-5th order frequencies reaches the maximum value with the chiral angle is 10.8°. Then the upper and lower boundaries of band gaps show downward trends, and the upper boundary decrease faster than the lower boundary, therefore the width of the band gap decreases gradually. The similar trend also can be found in the distribution of the band gap between 6th-7th order frequencies. The band gap between 6th-7th order frequencies appears when the chiral angle is 16.1°, and decreases as the chiral angle is greater than 28.4°. The width of the band gap between 6th-7th order frequencies is significantly larger than that of the band gap between 4th-5th order frequency curves. Based on the above study, the desired band gap characteristics of Type II lattice can be obtained by selecting an appropriate chiral angle. Therefore, the optimal chiral angle can be obtained to achieve effective vibration and wave propagation control. 0.30 0.25 0.20
0.15 0.10 0.05 0.00
0
10
20
30
40
50
60
Fig. 8 Band gap distribution of Type II lattice with different between the 4th-5th (green), 5th-6th (purple) and 6th-7th (blue) order frequencies
17 / 34
The slenderness ratio is an important geometrical parameter and has significant effects on the band gap characteristics of chiral auxetic structures. Therefore, we study and compare the band structures of the auxetic chiral structures with different slenderness ratios which are 1/8 and 1/15, where the chiral angle is 45°. Fig. 9(a) shows the band structure of the auxetic chiral structure with =1/8 and there are two complete band gaps locate at [0.061, 0.086] and [0.1235, 0.213], respectively. The first band gap locates between the fourth and fifth branches, and the second band gap is between the fifth and sixth branches. We also see that the width of the second band gap is greater than that of the first band gap. The patterns of the frequency curves of the band structures with different slenderness ratios are the same, but the frequency with =1/8 is bigger than the frequency with =1/15 . Furthermore, the width of the band gap has deep relation with the slenderness ratio as shown in Fig.10. As the slenderness ratio decreases, the width of the two band gaps gradually decreases. The reason is a smaller slenderness ratio results a softer structure. Therefore the Eigen-frequencies decrease and the corresponding band gap is also gradually reduced. The band gap can be adjusted by changing the slenderness ratio of the structure without altering the overall topology of the structure, which gives us an alternative way to tune the band gap characteristics of the auxetic chiral structure.
0.15
0.25 0.20
0.10
Ω
Ω
0.15 0.10
0.05
0.05 0.00
Γ
Χ
Μ
Γ
k
0.00
Γ
Χ
Μ
k
18 / 34
Γ
(a)
(b)
Fig. 9 The band structures of the auxetic chiral structures with different slenderness ratio: (a) =1/8 and (b) =1/15, where the chiral angle is = 4 5 0.30 0.25
0.20 0.15 0.10 0.05 0.00 1/5
1/10
1/15
1/20
Fig. 10 The distribution of the band gap of the auxetic chiral structures with different , where the chiral angle is = 45
3.2 Mechanism of the generation of the band gap Base on the analysis, we find that the geometrical parameters can significantly affect the number and location of the band gap. In order to understand the fundamental mechanism of generation of band gap, the vibration modes of the auxetic chiral structure with different geometrical parameters at the boundaries of band gap are investigated. We only take the Type I lattice as the example to analyze the mechanism of generation of band gap. Additionally, as shown in Fig. 5, the lowest band gap which is between the fourth and fifth order frequencies is important for engineering applications. Therefore, we will analyze occurrence and disappearance of the band gap between the fourth and fifth order frequencies. As shown in Figs. 11(a) and (b), the lowest band gap begins to appear at =5.7 due to the increase of the fifth order frequency, and disappear at =55. The upper edge mode from Point A to Point B when the chiral angle is from 5.7 to 55 and the modes of Point A and Point B all contain the rotational vibration. However, there are four complete 19 / 34
standing points at the ends of Beam II at Point A, and four incomplete standing points in the ends of Beam I at Point B. Thus, these two modes will play different roles. The former one will result the generation of band gap, while the latter one can narrow and close the band gap. When the chiral angle is 25, the seventh and eighth order frequencies cross with each other, forming a mode crossing point, which prevents the generation of band gap. The points C1 and C2 stand for the crossing points of the seventh and eighth-order frequencies, respectively. To further analyze the relation between the crossing points and the vibration modes of the points, we plot the vibration modes of points C1 and C2. We can find that the two modes are similar and the mode of point C1 can be obtained by rotating the vibration mode of point C2 by 90 as shown in Fig. 11(c). When the chiral angle is 68, the fourth and fifth order frequencies cross with each other, forming a mode crossing point, which are labeled as points D1 and D2 respectively. We also find that the mode of point D1 can be obtained by rotating the vibration mode of point D2 by 90. Furthermore, we have analyzed the vibration modes of the mode crossing points in the band structures with different chiral angles. It can be concluded that the vibration modes of the bigger frequencies can be obtained by rotating the vibration modes of the smaller frequencies by 90 simply. Based on the above analysis, we find that the vibration mode plays a key role in the generation and disappearance of the band gap. The generation of the lowest band gap is due to the increase of the frequency of rotational mode of the unit cell. The appearance of the mode crossing point prevents the generation of band gap.
20 / 34
0.20
0.25 0.20
0.15
0.15 0.10
0.10
A B
0.05 0.05 0.00
Γ
Χ
Μ
0.00
Γ
Γ
Χ
Μ
Band structure for θ=5.7°
Band structure for θ=55°
Vibration mode of Point A
Vibration mode of Point B
(a)
(b) 0.20
C2 C1
0.20
Γ
0.15
0.15
0.10
D2 D1 0.05
0.05
0.00
0.10
Γ
Χ
Μ
0.00
Γ
Band structure for θ=25°
Γ
Χ
Μ
Band structure for θ=68°
21 / 34
Γ
Vibration mode of
Vibration mode of
Vibration mode of
Vibration mode of
Point C1
Point C2
Point D1
Point D2
(c)
(d)
Fig. 11 The band gap of the Type I lattice for =0.1 with various and vibration modes at the marked points
3.3 Directionality of wave propagation The group velocity is one of the important properties which have deep relationship with the directions of energy flows. The wave propagation along the specific directions can be obtained according to the group velocity. The propagation mode of the elastic wave in the heterogeneous material is more complicated, and the group velocity comparison is needed in the analysis of the dispersion characteristics. Additionally, elastic energy propagates with different speeds throughout the structure usually depend on the directional wave propagation. In a general way, the investigation of the directional wave propagation in the anisotropic structure are carried out on the basis of the group velocity associated with a particular frequency, which also contains the information of the propagation speed in different directions. For example, if a wave propagates in an isotropic structure, the group velocity corresponding to a certain mode is a circular graph and there will no preferential propagating directions. In contrast, the wave exhibits non-isotropic behavior if the group velocity shape is not a circle. For the Type I lattice with different varying angles, the group velocities as a function of frequency are plotted in Fig. 12. When the chiral angle is 15°, the group velocity of the first mode at the low-frequency limit is almost the same, indicating that the group velocity is insensitive to the low frequency. The group velocity with the chiral angle is 60° at the low-frequency limit is almost the same. However, as the frequency increases, the group velocity exhibits a very complex behavior characterized by the caustics (spikes in the group velocity distribution) which usually existed in the anisotropic medium [49]. This caustic is related to the energy focusing of the wave propagation due 22 / 34
to the interference between various wave components propagating in the lattice. Additionally, we also find that the direction of the group velocity will rotate counterclockwise at a certain angle as the chiral angle increases. We further analyze the effects of the chiral angles on the wave propagation in the low-frequency range. As shown in Fig. 13, the group velocities at the frequency 0.001 are plotted where =15˚, 30˚, 45˚, 60˚ and 75˚ respectively. We can see that the group velocities are concentrated in the directions of 0°, 180°, and
cg,2
0
-50
0
-25 -50
0
cg,1
50
-25
(a)
25
20
cg,2
cg,2
0
cg,1 (b)
20
0
-20 -20
all cases.
25
50
cg,2
° in
90
0 cg,1
20
0
-20 -20
(c)
0
cg,1
20
(d)
Fig. 12 The group velocity of the first mode of the Type I lattice: (a) =15˚, (b) =30˚, (c) =60˚ and (d) =75˚ where 0.001 (red), 0.003 (green),
23 / 34
0.005 (blue) and 0.007 (cyan-blue)
cg,2
50
0
-50 -50
0
cg,1
50
Fig. 13 The group velocity of the first mode of the Type I at 0.001 with different : =15˚ (red), =30˚(green), =45˚(blue), =60˚ (cyan-blue) and =75˚(purple)
To verify the above results and further analyze the preferential energy flow within the auxetic chiral lattice, we establish the finite element model to carry out the numerical simulations via the commercial finite element software COMSOL. The finite lattice model containing 21 unit cells is already shown in Fig. 4. Each beam of the lattice is considered as the Timoshenko beam model with a square cross section. The dimensions and material’s properties are also given in Section 2.1. The local rotational harmonic excitation with the corresponding normalized frequency are applied at the center of the structure. Fig. 14 shows the transient displacement response of structures at the considered time instant with different chiral angles. We can see that the vibrational energy propagates along the specific directions in auxetic chiral structures which is consistent with calculation results of the group velocities, and the auxetic chiral structure allows energy to preferentially propagate in both vertical and horizontal directions. As the chiral angle increases, the direction of the group velocity rotates counterclockwise. Therefore, the 24 / 34
auxetic chiral lattice provides the flexibility to make the elastic waves propagate along the desired directions, which can be adjusted by changing the topology of the auxetic chiral structure.
(a)
(b)
(c)
(d)
Fig. 14 Transient response of the Type I lattice under the harmonic excitations at
0.001 with different : (a) =15˚, (b) =30˚, (c) =60˚ and (d) =75˚
The group velocity of Type II lattice with varying chiral angles as a function of frequency and direction is shown in Fig. 15. When the chiral angle of Type II lattice is 25 / 34
15˚, the group velocity of the first-order mode at the low-frequency range is almost the same, which indicates elastic waves in the low frequency range travel in roughly the same directions in the auxetic chiral structure. The group velocity of Type II lattice with different chiral angles also shows the same characteristics. As the chiral angle increases, the group velocity rotates counterclockwise at a certain angle as shown in Fig. 15. To further investigate the effect of the chiral angle on the wave propagation, we plot the group velocities of Type II lattice at frequency 0.001 where =15˚, 30˚, 45˚, and 60˚ as shown in Fig. 16. Meanwhile, the transient displacement field of Type II lattice with different chiral angles is shown in Fig. 17. It is confirmed that the direction of energy propagation in the finite lattice is consistent with the group velocity prediction. The chiral angle of Type II lattice also has a great effect on the direction of the elastic wave.
140
cg,2
cg,2
50
0
0
-50 -140 -140
0
cg,1
140
(a)
-50
0
cg,1 (b)
26 / 34
50
12
cg,2
cg,2
30
0
-30 -30
0
-12 -12
30
cg,1
0
0
cg,1
(c)
12
(d)
Fig. 15 The group velocity of the first mode of the Type II lattice: (a) =15˚, (b) =30˚, (c) =45˚ and (d) =60˚ where 0.001 (red), 0.003 (green),
0.005 (blue) and 0.007 (cyan-blue)
150 100
cg,2
50 0 -50 -100 -150 -150
-100
-50
0
cg,1
50
100
150
Fig. 16 The group velocity of the first mode of the Type II lattice at 0.001 with different : =15˚ (red), =30˚(green), =45˚(blue) and =60˚ (cyan-blue)
27 / 34
(a)
(b)
(c)
(d)
Fig. 17 Transient response of the Type II lattice under the harmonic excitations at
0.001 with different : (a) =15˚, (b) =30˚, (c) =45˚ and (d) =60˚
4 Conclusions This study investigates the wave propagations in auxetic chiral structure. Based on the Bloch’s theorem and finite element method, the dynamic equation of auxetic chiral structure are established and eigenvalue are solved to analyze the elastic wave propagation in chiral lattices. The main findings are as follows:
28 / 34
(1) The auxetic chiral structures have special wave propagation properties. By investigating the band structure and vibrational mode, we can find that the vibration mode plays a key role in the generation and disappearance of the band gap. The band gap at different frequency ranges can be obtained by adjusting topologies of the auxetic chiral structure to meet different engineering needs. (2) Chiral angle of auxetic chiral structure has significant effect on the direction of wave propagation. With the increase of chiral angle, the direction of the wave propagation rotates counterclockwise. (3) The transient response of the finite structure under harmonic excitation is simulated by the commercial finite element software. The frequencies where the transmission efficiency reduces is the same as the frequencies of the band gap. The directions of the wave propagation results of the numerical simulation and the theoretical calculation are identical, which provides another way to predict the band gap and directional waveguide.
Acknowledgements Funding for this work has been provided by the National Key R&D Program of China (2017YFB1102801), National Natural Science Foundation of China (Nos. 11872313 and 11502202), Fundamental Research Funds for the Central Universities.
References [1] Fozdar DY, Soman P, Lee JW, Han LH, Chen S. Three-Dimensional Polymer Constructs Exhibiting a Tunable Negative Poisson's Ratio. Advanced Functional Materials. 2011;21:2712-20. [2] Lee J-H, Singer J, Thomas E. Micro-/Nanostructured Mechanical Metamaterials. Advanced materials. 2012;24:4782-810. [3] Chen Y, Jia Z, Wang L. Hierarchical honeycomb lattice metamaterials with improved thermal resistance and mechanical properties. Composite Structures. 2016;152:395-402. [4] Zheng X, Lee H, Weisgraber TH, Shusteff M, DeOtte J, Duoss EB, et al. Ultralight, ultrastiff mechanical metamaterials. Science. 2014;344:1373-7. 29 / 34
[5] Wu W, Hu W, Qian G, Liao H, Xu X, Berto F. Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review. Materials and Design. 2019;180:107950. [6] Zhang K, Ge M, Zhao C, Deng Z, Xu X. Free vibration of nonlocal Timoshenko beams made of functionally graded materials by Symplectic method. Composites Part B: Engineering. 2019;156:174-84. [7] Wang J, Wang G, Feng X, Kitamura T, Kang Y, Yu S, et al. Hierarchical chirality transfer in the growth of Towel Gourd tendrils. Scientific Reports. 2013;3:1-7. [8] Ai L, Gao X-L. Metamaterials with negative Poisson’s ratio and non-positive thermal expansion. Composite Structures. 2017;162:70-84. [9] Buckmann T, Stenger N, Kadic M, Kaschke J, Frolich A, Kennerknecht T, et al. Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Advanced materials. 2012;24:2710-4. [10] Prawoto Y. Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio. Computational Materials Science. 2012;58:140-53. [11] Tee KF, Spadoni A, Scarpa F, Ruzzene M. Wave Propagation in Auxetic Tetrachiral Honeycombs. Journal of Vibration and Acoustics. 2010;132:031007-1-8. [12] Scarpa F, Tomlin PJ. On the transverse shear modulus of negative Poisson's ratio honeycomb structures. Fatigue and Fracture of Engineering Materials and Structures. 2000;23:717-20. [13] Tang Y, Lin G, Han L, Qiu S, Yang S, Yin J. Design of Hierarchically Cut Hinges for Highly Stretchable and Reconfigurable Metamaterials with Enhanced Strength. Advanced materials. 2015;27:7181-90. [14] Grima JN, Evans KE. Auxetic behavior from rotating triangles. Journal of Materials Science. 2006;41:3193-6. [15] Shan S, Kang SH, Zhao Z, Fang L, Bertoldi K. Design of planar isotropic negative Poisson’s ratio structures. Extreme Mechanics Letters. 2015;4:96-102. [16] Lakes R. Foam structures with a negative poisson's ratio. Science. 1987;235:1038-40. [17] Cho Y, Shin JH, Costa A, Kim TA, Kunin V, Li J, et al. Engineering the shape and structure of materials by fractal cut. Proceedings of the National Academy of Sciences of the United States of America. 2014;111:17390-5. [18] Wagner M, Chen T, Shea K. Large shape transforming 4D auxetic structures. 3D Printing and Additive Manufacturing. 2017;4:133-42. [19] Ebrahimi H, Mousanezhad D, Haghpanah B, Ghosh R, Vaziri A. Lattice Materials with Reversible Foldability Advanced Engineering Materials. 2017;19:1600646-1-6. [20] Gatt R, Caruana G, Roberto, Attard D, Casha A, Wolak W, Dudek K, et al. On the properties of real finite-sized planar and tubular stent-like auxetic structures. Physica Status Solidi (B). 2014;251:321-7. [21] Mizzi L, Attard D, Casha A, Grima JN, Gatt R. On the suitability of hexagonal honeycombs as stent geometries. Physica Status Solidi B. 2014;251:328-37. [22] Tan TW, Douglas GR, Bond T, Phani AS. Compliance and Longitudinal Strain of Cardiovascular Stents: Influence of Cell Geometry. Journal of Medical Devices. 2011;5:041002-1-6. 30 / 34
[23] Scarpa F, Smith FC, Chambers B, Burriesci G. Mechanical and electromagnetic behaviour of auxetic honeycomb structures. Aeronautical Journal. 2003;107:175-83. [24] Scarpa F, Ciffo LG, Yates JR. Dynamic properties of high structural integrity auxetic open cell foam. Smart Materials and Structures. 2004;13:49-56. [25] Babaee S, Shim J, Weaver JC, Chen ER, Patel N, Bertoldi K. 3D soft metamaterials with negative Poisson's ratio. Advanced materials. 2013;25:5044-9. [26] Mousanezhad D, Babaee S, Ebrahimi H, Ghosh R, Hamouda AS, Bertoldi K, et al. Hierarchical honeycomb auxetic metamaterials. Scientific Reports. 2015;5:1-8. [27] Bertoldi K, Reis PM, Willshaw S, Mullin T. Negative Poisson's ratio behavior induced by an elastic instability. Advanced materials. 2010;22:361-6. [28] Jiang Y, Li Y. 3D Printed Auxetic Mechanical Metamaterial with Chiral Cells and Re-entrant Cores. Scientific Reports. 2018;8:1-11. [29] Gaspar N, Ren X, Smith C, Grima J, Evans K. Novel honeycombs with auxetic behaviour. Acta Materialia. 2005;53:2439-45. [30] Smith CW, Grima JN, Evans KE. A novel mechanism for generating auxetic behaviour in reticulated foams: missing rib foam model. Acta Materialia. 2000;48:4349-56. [31] Nemat-Nasser S. Anti-plane shear waves in periodic elastic composites: band structure and anomalous wave refraction. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2015;471. [32] Mokhtari AA, Lu Y, Srivastava A. On the emergence of negative effective density and modulus in 2-phase phononic crystals. Journal of the Mechanics and Physics of Solids. 2019;126:25671. [33] Shmuel G, Band R. Universality of the frequency spectrum of laminates. Journal of the Mechanics and Physics of Solids. 2016;92:127-36. [34] Phani AS, Woodhouse J, Fleck NA. Wave propagation in two-dimensional periodic lattices. The Journal of the Acoustical Society of America. 2006;119:1995-2005. [35] Trainiti G, Rimoli JJ, Ruzzene M. Wave propagation in undulated structural lattices. International Journal of Solids and Structures. 2016;97-98:431-44. [36] Zhang K, Zhao P, Hong F, Yu Y, Deng Z. On the directional wave propagation in the tetrachiral and hexachiral lattices with local resonators. Smart Materials and Structures. 2020;29:015017. [37] Zhang K, Zhao C, Luo J, Ma Y, Deng Z. Analysis of temperature-dependent wave propagation for programmable lattices. International Journal of Mechanical Sciences. 2020;171:105372. [38] Schaeffer M, Ruzzene M. Wave propagation in multistable magneto-elastic lattices. International Journal of Solids and Structures. 2015;56-57:78-95. [39] Warmuth F, Korner C. Phononic Band Gaps in 2D Quadratic and 3D Cubic Cellular Structures. Materials. 2015;8:8327-37. [40] Bertoldi K, Boyce MC. Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations. Physical Review B. 2008;78. [41] Trainiti G, Rimoli JJ, Ruzzene M. Wave propagation in periodically undulated beams and plates. International Journal of Solids and Structures. 2015;75–76:260-76. [42] Martinsson PG, Movchan AB. Vibrations of lattice structures and phononic band gaps. Quarterly Journal Of Mechanics And Applied Mathematics. 2003;569(1):45-64. 31 / 34
[43] Spadoni A, Ruzzene M, Gonella S, Scarpa F. Phononic properties of hexagonal chiral lattices. Wave Motion. 2009;46:435-50. [44] Hussein MI, Leamy MJ, Ruzzene M. Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook. Applied Mechanics Reviews. 2014;66:040802-1-38. [45] Wang Y, Wang Y, Zhang C. Bandgaps and directional propagation of elastic waves in 2D square zigzag lattice structures. Journal of Physics D: Applied Physics. 2014;47:1-14. [46] Su Xl, Gao Yw, Zhou Yh. The influence of material properties on the elastic band structures of one-dimensional functionally graded phononic crystals. Journal of Applied Physics. 2012;112. [47] Li F, Zhang C, Liu C. Active tuning of vibration and wave propagation in elastic beams with periodically placed piezoelectric actuator/sensor pairs. Journal of Sound and Vibration. 2017;393:14-29. [48] Jin S, Korkolis YP, Li Y. Shear resistance of an auxetic chiral mechanical metamaterial. International Journal of Solids and Structures. 2019;174-175:28-37. [49] Taylor B, Maris HJ, Elbaum C. Focusing of Phonons in Crystalline Solids due to Elastic Anisotropy. Physical Review B. 1971;3:1462-72.
32 / 34
Author statement
Kai Zhang: Conceptualization, Methodology, Writing - Original Draft, Writing Review & Editing. Pengcheng Zhao: Conceptualization, Formal analysis, Validation, Writing - Original Draft. Cheng Zhao: Software, Investigation. Fang Hong: Investigation, Resources. Zichen Deng: Supervision.
33 / 34
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
34 / 34