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Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures
Accepted Manuscript Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures Bin Wang, Qian Ding, Yongtao Sun, Shihui Yu, Fuguang Ren, Xinyu Cao, Yinghong Du PII: DOI: Reference:
S2211-3797(18)32931-0 https://doi.org/10.1016/j.rinp.2018.12.068 RINP 1938
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
9 November 2018 13 December 2018 17 December 2018
Please cite this article as: Wang, B., Ding, Q., Sun, Y., Yu, S., Ren, F., Cao, X., Du, Y., Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures, Results in Physics (2018), doi: https://doi.org/10.1016/j.rinp.2018.12.068
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Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures Bin Wanga,b, Qian Dinga,b, Yongtao Suna,b,c*, Shihui Yub,d**, Fuguang Renb, Xinyu Caoa,b, Yinghong Due a
b
Department of Mechanics, Tianjin University, Road Yaguan 135, 300350, Tianjin, China Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin University, Road Yaguan 135, 300350, Tianjin, China c State
d e
Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China
School of Microelectronics, Tianjin University, Road Weijin 92, 300072, Tianjin, China
Department of Materials Science and Engineering, Tianjin University, Road Yaguan 135, 300350, Tianjin, China
*Corresponding authors: [email protected]; [email protected]
Abstract For multifunctional optimization design of honeycomb structures, the high stiffness hierarchical honeycombs with stochastic Voronoi substructures (HHSVS) are proposed by substituting cell walls of the regular hexagonal honeycombs (ORHH) with Voronoi honeycomb lattices of equal mass. In this study, the tunable linear elastic fracture properties of the HHSVS are investigated by finite element analysis. Results demonstrate that size effect on fracture toughness of the HHSVS is noticeable and a brittleness number is suggested to determine it. At the same time, compared with the ORHH and conventional Voronoi honeycombs of equal mass/density, the in-plane fracture toughness of the HHSVS could be more than 2 times larger and regardless of the cell regularity and hierarchical parameters, fracture toughness of the HHSVS exhibits a weaker quadratic dependence on the relative density and is the highest. As a whole, the HHSVS exhibit the combined properties of tunable Poisson’s ratio, higher stiffness, enhanced tunable fracture toughness, lower imperfection sensitivity and better structural-acoustic performance etc. The research provides a novel strategy for the multifunctional optimization design of the honeycombs structures widely used in the engineering fields. 1
Keywords: Hierarchical honeycombs with stochastic Voronoi substructures (HHSVS); Tunable fracture toughness; Size effect; T-stress.
1. Introduction Owing to their superior stiffness-to-weight and strength-to-weight ratio, honeycomb structures have been widely used in the fields of mechanical, aerospace and civil engineering. To optimize their topologies and thus promote their multifunctional structural integrity applications, many types of hierarchical honeycombs have been proposed, such as hierarchical honeycombs with hexagonal [1,2], triangular [2,3], Kagome [3] and chiral [4] honeycomb substructures, self-similar isotropic [5] and anisotropic [6] hierarchical honeycombs etc. With regard to mechanical properties, analytical or numerical analyses show that compared to conventional honeycombs of equal mass/density, hierarchical honeycombs could exhibit tunable Poisson’s ratio and higher in-plane stiffness or strength [1-11]. With respect to multifunctional properties, hierarchical honeycombs structures could have better wave filtering properties [12-16], enhanced in-plane [17,18] or out-of-plane [19-23] energy absorption properties, excellent crashworthiness under uniaxial dynamic impact [24-31], improved thermal conductivity properties [2] and so on. The substructures of the aforementioned hierarchical honeycombs are largely regular honeycomb lattices. However, studies on hierarchical honeycombs with stochastic Voronoi honeycomb substructures are lacking [32]. In fact, cellular materials with random irregular cells and structural hierarchy can be found in many living natural materials such as bones or plant stems [33-38]. The degree of cell irregularity may have great influence on mechanical properties of the hierarchical cellular materials [39-44]. Hence, hierarchical honeycombs with stochastic Voronoi substructures (HHSVS) [45] were proposed by substituting cell walls of the regular hexagonal honeycombs (ORHH) with stochastic Voronoi substructures of equal mass. The stiffness of these types of irregular hierarchical honeycombs could be as three times higher as that of the
2
ORHH of equal mass as predicted by numerical analyses. Except the improvement in stiffness, HHSVS also exhibit excellent structural-acoustic properties [46] than the ORHH. In addition to exhibiting excellent stiffness, strength, wave filtering, and imperfection sensitivity, the damage resistance of the honeycomb structures plays a crucial role in engineering applications. To explore damage resistance of the honeycomb structures and promote their engineering applications, extensive research has been conducted. The group of Fleck [47,48] numerically studied fracture toughness of the elastic-brittle regular hexagonal, Kagome, triangular, and diamond honeycombs and analytically determined the effect of relative density on the fracture toughness. Moreover, they explored the defect sensitivity of modulus and fracture toughness for five morphologies of the 2D lattices [49]. Wang and McDowell [50] analyzed the effects of defects on the in-plane properties of periodic metal honeycombs including the regular hexagonal, square, and triangular honeycombs. Ajdari et al. [51] reported the effects of missing walls and filled cells on the elastic–plastic behavior of both regular hexagonal and non-periodic Voronoi structures using finite element (FE) analysis. Christodoulou and Tan investigated the effects of cell-regularity, relative density [52], and size [53] on the toughness of conventional Voronoi honeycombs using FE analysis. With regard to the fracture properties of the hierarchical honeycombs, Fan et al. [54] analytically analyzed hierarchical cellular materials made of sandwiched struts and indicated that the hierarchical honeycombs are much more damage tolerant and insensitive to wavy imperfections in the cell wall. Chen et al. [55] constructed a type of hierarchical honeycombs by substituting the three-edge joints of the regular hexagonal honeycomb with hollow-cylindrical joints and reported that the fracture strength could be improved by 303% with respect to the conventional ones. Wang et al. [56] proved that the hierarchical honeycombs exhibit insensitivity to missing bars defect compared to conventional honeycombs. Except their tunable Poisson’s ratio [45], higher stiffness[45], better structural-acoustic properties [46]
3
and lower imperfection sensitivity [56] compared to the ORHH and conventional Voronoi honeycombs of equal mass/density [52,53], fracture properties of the HHSVS, which play very important role for their multifunctional applications, are still not clear. To comprehensively understand tunable fracture properties of the HHSVS and thus promote their multifunctional structural integrity design for engineering applications, in this paper, effects of size, cell regularity, relative density, thickness-to-length ratio, hierarchical length ratio, mixed-mode loading and T-stress on fracture toughness of the HHSVS are systematically analyzed through the numerical simulations. The study of this paper provides a new strategy for the multifunctional structural integrity design of the lightweight components, such as sandwich panels, widely used in the engineering fields.
2. Geometrical models, elastic properties and FE models of the HHSVS for fracture analysis 2.1 Geometrical models of the HHSVS Fig. 1 shows schematic of the unit cell of the HHSVS [45]. Various methods are used to generate foam-like representative volume elements (RVEs) based on Voronoi tessellation. The stochastic Voronoi substructures were developed using a method proposed in Ref. [57,58]. First, a uniform distribution of the seed positions is generated, corresponding to a perfectly regular honeycomb pattern. Next, the positions of the seeds are perturbed via pseudo-random motion within a range of limits. The unit cell of the HHSVS is finally obtained by removing the conventional stochastic hexagonal honeycomb structures with regular ones to ensure that the pairs of counterpart nodes are located on opposing boundaries. The unit cell of the HHSVS is then copied along the two directions to obtain a larger model. The cell regularity is defined as d is the minimum distance between the centers of any two neighboring cells in a randomly irregular honeycomb, and d 0 4
2 A0 is the distance between any two N 3
neighboring cells in a regular hexagonal honeycomb with the same number of cells. Ao is the controlling area, and N is the number of cells. →1 indicates that the hexagonal honeycomb is perfectly regular and →0 represents a completely random Voronoi honeycomb. The relative density is defined as the ratio between the macroscopic density of the cellular structure and that of the cell wall material. It is assumed that all the cell walls have the same thickness t, and the relative non-dimensional density is determined as:
t
N
l
i 1 i
(1)
A0
where li and N are the length and number of the cell walls. The cell-wall thickness-to-length ratio of the superstructure is defined as the ratio of its thickness tsup to its length lsup, i.e., tsup
lsup
. The ratio
between the lengths in the super and sub-structures is termed as the hierarchical length ratio, i.e.
lsup l sub , where l sub is the average value of the sub-structure cell wall length .
(b) →0
(a)
(c) = 0.4
(d) = 0.7 5
Fig. 1. Schematic of unit cells of (a) the original regular hexagonal honeycomb (ORHH) and the HHSVS of equal mass/density: (b)→0, (c)4 and (d) .
2.2 Elastic properties of the HHSVS Elastic properties of the HHSVS, which will be used for analysis of the linear elastic fracture properties of the HHSVS, are discussed in this section. The FE models are developed to explore the effects of the stochastic Voronoi sub-structures on the in-plane elastic properties using the software ABAQUS/standard. Each cell wall is simulated using 2D Timoshenko beam elements (B21). The periodic boundary conditions (PBC) were applied along the boundaries of RVEs [59], and the details are shown in Ref. [45]. Under plane-stress conditions, the in-plane elastic modulus is represented using the power-law expression
E Es B b , where Es is the Young’s modulus of the solid of which the HHSVS are made. Under the plane-strain condition, the elastic modulus in the prismatic x3 direction is defined as E33 Es , and the longitudinal Poisson’s ratios are
31 32 s , where νs is the Poisson’s ratio of the solid of which the
HHSVS are made. Consequently, the plane-strain elastic moduli are represented as [49]: 1 E, 1 B b 1
(2a)
s2 B b 1 . 1 s2 B b 1
(2b)
E ps
ps
2 s
2.3 FE models of the HHSVS for fracture analysis To analyze fracture properties of the HHSVS, the linear elastic fracture mechanics method is applied. Figure 2 shows the schematic illustrations of the cracked FE models. The displacements associated to the crack-tip K-field for homogeneous and isotropic solid are applied along the model boundary as follows [60,61]:
u1
1 KⅠ 1/ 2 1 KⅡ 1/ 2 r ( cos )cos r ( 2 cos )sin , 2 2 2 G ps 2 2 2 G ps
6
(3a)
u2
1 KⅠ 1/ 2 1 KⅡ 1/ 2 r ( cos )sin r ( 2 cos )cos . 2 2 2 G ps 2 2 2 G ps
(3b)
In the above expressions, u1 and u2 are the displacements aligned along the Cartesian axes x1 and x2 , respectively, and r-θ is the polar coordinate system centered at the stationary crack tip. KI and KII are the mode I and mode II stress intensity factors, respectively, and = (3 −ps)/(1 +ps). In addition, the in-plane rotation ω is expressed as:
1 2
(u2,1 u1,2 )
1 1 1/ 2 r ( KⅠsin KⅡ cos ) . 2 2 4 2 G ps
(3c)
1 Under mixed-mode failure, the mixture of KI and KII is determined by a mode-mixity M 2 tan ( KⅡ KⅠ)
with 0≤M <1 (M=0 and M=1 correspond to pure mode I and mode II loading, respectively). The lattice walls are assumed to be linear-elastic, and fail at the ultimate stress f . The macroscopic toughness KC can be obtained if the maximum local stress reaches the strength f [47,49,52]. Considering the stochastic properties of the HHSVS, at least seven samples are analyzed for several degrees of regularity, and the average value of them is chosen for the results. u2 u1
x2 θ
x1
0.0
0.4
0.7
Fig. 2. Schematic of the FE model of the HHSVS for fracture analyses.
3. Fracture properties of the HHSVS In this part, effects of size, cell regularity, relative density, thickness-to-length ratio, hierarchical length 7
ratio, mixed-mode loading and T-stress on fracture toughness of the HHSVS are studied, respectively.
3.1 Size effect on fracture toughness of the HHSVS First, size effect on fracture toughness of the HHSVS is investigated. Because of the strain gradient effects, surface elasticity and initial stress [62], mechanical properties of the hierarchical honeycombs are found to be size-dependent. In fact, fracture toughness of the cellular solid has been found to be dependent on the specimen scales [47,52]. Christodoulou and Tan [53] discussed the physical origin of the size effects. It is found that the effect of size in lattice materials depends on the specimen boundary constraint on fracture process zone, thereby disrupting the K-field in the vicinity of the crack-tip. With the increase in the specimen size, the interaction becomes weaker. The fracture toughness is measured by summing up the contributions from the two physically distinct processes, i.e. surface energy and energy dissipation in the process zone. Based on dimensional analysis and the process zone, Carpinteri [63] proposed an energy brittleness number to calibrate the specimen size effects of the specimens, which is the ratio of the process zone size to the characteristic structural dimension. Therefore, the process zone size d around the crack tip in the plane-strain state is calculated as [64]:
d a[sec(
( ) ) 1] 2 f
(4)
Here, a is the crack length; σ(∞) and σf denote the uniform tension at infinity and uniaxial tensile strength of the material, respectively; β is a parameter describing the behavior of the cellular material and the stress state, which is defined as 1 1 2 0.02 (1 ) according to Triantafillou and Gibson [65]. To show the size effect on fracture toughness of the HHSVS, the parameters = 0.01, and are chosen as the examples for illustration. The corresponding results, for the ratio of the calculated process zone size d to the overall specimen length L and the normalized fracture toughness
KC / ( 2 f
lˆ )
predicted from several scale models, are plotted in Fig. 3 as a function of the number of cells. 8
From Fig. 3 we can see that as the size increases, both the normalized toughness and the normalized process zone size converge to constant values, thus indicating a strong size dependence at small scale sizes. Accordingly, to eliminate the scaling phenomena, the simulations were performed with lattices of 10000 cells considering the computational cost and accuracy.
Fig. 3. Normalized fracture toughness and d L 1011 vs the number of cells in the model.
3.2 Effect of cell regularity In this section, the effect of cell regularity on fracture toughness of the HHSVS is investigated. Fig. 4 shows the predicted toughness values KIC and KIIC plotted as a function of the cell regularity . The results are obtained for = 0.01, and Moreover, the fracture toughness is normalized by 2
f
lˆ
. From Fig. 4 we can see that the relationship between the cell regularity and the loading modes are
unclear. However, the hierarchical lattices exhibit larger crack initiation resistance to mode I cracking than that of mode II, regardless of cell regularity. For comparison, both the mode I and mode II fracture toughness of the regular hexagonal [47] and conventional Voronoi honeycomb [52] of equal masses and under the same conditions are shown in Fig. 4. It is apparent that fracture toughness of the HHSVS is at least twice that of the Voronoi honeycombs and ORHH, regardless of the loading mode. Hence, the introduction of hierarchy can effectively improve fracture initiation resistance of the honeycombs. 9
Fig. 4. Normalized mode I and mode II fracture toughness vs cell regularity of the HHSVS, the conventional Voronoi honeycombs and the regular hexagonal honeycombs (“Hierarchy”, “Voronoi” and “Hexagonal” stand for the HHSVS, the conventional Voronoi honeycombs and the ORHH, respectively).
3.3 Effect of the relative density In this section, effect of the relative density on fracture toughness of the HHSVS is studied. It is well known that the fracture toughness KC of the cellular material has a power-law dependence on the relative density
KC
f l
as follows:
D d
(5)
As demonstrated by Fleck and Qiu [47], the exponent d is approximately 2 for the hexagonal honeycomb, unity for the triangular honeycomb, and 0.5 for the Kagome lattice, respectively. Moreover, the exponent d ranges from 1.994 to 1.988 for mode I toughness for the Voronoi honeycombs [52]. Fig. 5 shows the variations in the mode I and mode II fracture toughness of the HHSVS with the fixed relative density for
and . It is also shown in Fig. 5 that the fracture toughness can be best-fitted by the power-law in Eq. 4. For mode I loading, Fig. 5 shows that the exponent d equals to 1.910, 1.954, and 2.060 for , and , respectively. For mode II loading, the d values are 1.944, 1.82, and 2.094 for 10
, andrespectively. A relatively weak quadratic dependence of the fracture toughness of the
K C / ( f
lˆ )
HHSVS on the relative density is found compared to that of the ORHH and Voronoi honeycombs [47,52].
Fig. 5. Mode I and mode II KC vs the relative density (“Hexagonal” stands for the ORHH of equal mass/density with the HHSVS).
3.4 Effect of the thickness-to-length ratio In this section, the effect of the thickness-to-length ratio on fracture toughness of the HHSVS is investigated. Theoretically, the ratio of thickness to length α ranges from 0 to →0 and α→
, and the two limit cases α
correspond to the regular hexagonal and Voronoi honeycombs, respectively. In this study, the
models with different values of ranging from 0.05 to 1.2 with fixed = 0.01 and are employed in the FE analyses under mode I loading. From Fig. 6 we can see that fracture toughness of the HHSVS is very sensitive to the ratio of thickness to length of the superstructure. It increases to a maximum value of 2.16, and subsequently, rapidly decreases to 0.70 with the increase of α. In addition, Fig.6 shows the fracture toughness values of the ORHH with =0.01 [47] and Voronoi honeycomb with and =0.01 [52] for comparison. Apparently, the introduction of hierarchy could significantly increase fracture toughness of the HHSVS at small thickness-to-length ratios of the cell wall of the superstructure. Moreover, tunable elastic properties can be achieved for a wide range.
11
Fig. 6. Predicted fracture toughness for various thickness-to-length ratio α.
3.5 Effect of the hierarchical length ratio In this section, effect of the hierarchical length ratio λ on fracture toughness of the HHSVS is analyzed. For illustration, Fig. 7 shows the variation of the normalized mode I fracture toughness with the increase of λ for = 0.01, and . Apparently, the variation is unclear. It is worth noting that the true fracture toughness decreases with the increase of λ. The smaller the sub-cell, the smaller the value of the fracture toughness.
Fig. 7. Predicted fracture toughness with respect to various hierarchical length ratio λ.
3.6 Effect of the mixed-mode loading 12
In this section, the effect of the mixed-mode loading on the tunable fracture toughness of the HHSVS is analyzed. The KI/KII ratio is calculated for different values of mode mixity. Fig. 8 shows the fracture locus in the mode space for = 0.01, and, wherein both KI and KII are normalized by KIC. It is obvious that the locus of the values takes the shape of an inner convex envelope. With the variation in the mode-mixity, the fracture initiation site of the cell wall changes.
Fig. 8. Failure envelope under mixed-mode loading.
3.7 Effect of T-stress The T-stress is the second term in the William’s series expansion and parallel to the crack plane. The T-stress can be neglected under mode II loading with the assumption of symmetry and plays a negligible role at the crack tip for solids. However, Fleck and Qiu [47] demonstrated the need to include the effect of the T-stress to describe the fracture behaviors of a regular hexagonal honeycomb. Moreover, Christodoulou and Tan [52] demonstrated the need to include the T-stress in predicting the fracture toughness of a Voronoi lattice. To apply the T-stress around the periphery of the model, the displacement components in Eq. (2) are modified for T-stress terms:
u1 T
2 1- ps
E ps
r cos ,
(6a) 13
u2 T
ps (1+ ps ) E ps
r sin .
(6b)
The stress biaxiality ratio can be obtained as follows [66]: B
T a
(6c)
K
where a is the crack length. For illustration, the effect of the T-stress on fracture toughness of the HHSVS is shown in Fig. 9 for = 0.01, and. K C0 is the critical fracture toughness at T-stress equal to zero. Fig. 9 shows that the normalized fracture toughness increases rapidly with the increase in the T-stress under either mode I or mode II loading conditions. However, the mode I value is more sensitive to the negative T-stresses. Moreover, at the same T-stress level, the fracture toughness decreases and becomes a constant as the irregularity increases to 0.7. Therefore, compared to the regular hexagonal and Voronoi honeycombs of equal mass/density, the defect sensitivity of the HHSVS can be tuned more flexibly by
KC / KC0
changing the crack length ratio, loading configuration and out-of-plane geometries.
Fig. 9. T-stress vs fracture toughness.
4. Conclusions In the work, the FE-based method is used to calculate the tunable fracture toughness of the high stiffness HHSVS. Effects of size, cell regularity, relative density, thickness-to-length ratio, hierarchical length ratio, 14
mixed-mode loading and T-stress on fracture toughness of the HHSVS are systematically investigated. The following conclusions are drawn from this study: 1. Although the effects of the cell regularity on the mode I and mode II toughness values are unclear, the crack initiation resistance of the HHSVS is found to be better than that of the Voronoi and regular hexagonal honeycombs of equal masses. And fracture toughness of the HHSVS is very sensitive to the thickness-to-length ratio of the cell walls of the superstructure. 2. The effect of size on fracture toughness of the HHSVS is noticeable. A brittleness number is suggested to determine the effect of the specimen size, which is explicitly a non-dimensional quantity used to describe the ratio of the size of the process zone to the characteristic structural dimension. 3. Compared to the ORHH and Voronoi honeycombs of equal mass/density, in-plane fracture toughness of the HHSVS could be more than 2 times larger and the HHSVS exhibit a relatively weak quadratic dependence of the nominal fracture toughness on the relative density. 4. Fracture toughness of the HHSVS increases with the increase of the T-stress under both mode I and mode II loading conditions. At a fixed T-stress level, fracture toughness of the HHSVS becomes a constant with the increase in the cell irregularity. And compared to the regular hexagonal and Voronoi structures, the defect sensitivity of the HHSVS can be flexibly tuned in a wider range by changing the crack length ratio, loading configuration or out-of-plane geometries. As a whole, compared with the ORHH and conventional Voronoi honeycombs of equal mass/density, the HHSVS exhibit the combined advantages of tunable Poisson’s ratio, higher stiffness, enhanced tunable fracture toughness, lower imperfection sensitivity and better structural-acoustic performance and so on.
Acknowledgements The research is supported by the National Natural Science Foundation of China under Grant 15
No. 11502162 and the Natural Science Foundation of Tianjin under Grant No. 15JCQNJC05100. YS is also supported by the fund (No. SV2015-KF-02) from State Key Laboratory for Strength and Vibration of Mechanical Structures (Xi’an Jiaotong University), the Research Project Key
Laboratory
of
Mechanical
System
and
of State
Vibration MSV201611(Shanghai Jiaotong
University), the Seed Foundation of Tianjin University (No. 2017XRG-0035 and 2017XYF-0009).
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[40] Cui S, Gong B, Ding Q, Sun Y, Ren F, Liu X, Yan Q, Yang H, Wang X, Song B, Mechanical metamaterials foams with tunable negative Poisson’s ratio for enhanced energy absorption and damage resistance. Materials 2018;11,1869: doi:10.3390/ma11101869. [41] Zhu HX, Thorpe SM, Windle AH, The effect of cell irregularity on the high strain compression of 2D Voronoi honeycombs, Int. J. Solids Struct. 2006; 43(5): 1061–78. [42] Zhu HX, Zhang P, Balintb D, Thorpe SM, Elliott JA, Windle AH, Lin J, The effects of regularity on the geometrical properties of Voronoi tessellations, Physica A 2014; 406(15): 42–58. [43] Mukhopadhyay T, Adhikari S, Equivalent in-plane elastic properties of irregular honeycombs: An analytical approach, Int. J. Solids Struct. 2016; 91: 169–84. [44] Mukhopadhyay T, Adhikari S. Stochastic mechanics of metamaterials, Compos. Struct. 2017; 162: 85–97. [45] Du YH, Pugno N, Gong BM, Wang DP, Sun YT, Ding Q, Mechanical properties of the hierarchical honeycombs with stochastic Voronoi sub-structures, Europhys. Lett. 2015;111(5):56007. [46] Denli H, Sun J, Structural-acoustic optimization of sandwich structures with cellular cores for minimum sound radiation, J. Sound Vib. 2007;301(1):93-105. [47] Fleck NA, Qiu X, The damage tolerance of elastic-brittle, two-dimensional isotropic lattices, J. Mech. Phys. Solids 2007; 55(3): 562-88. [48] Alonso IQ, Fleck NA, Damage tolerance of an elastic-brittle diamond-celled honeycomb, Scr. Mater. 2007; 56(8): 693-6. [49] Romijn NE, Fleck NA, The fracture toughness of planar lattices: imperfection sensitivity, J. Mech. Phys. Solids 2007; 55(12): 2538-64. [50] Wang AJ, Mcdowell DL, Effects of defects on in-plane properties of periodic metal honeycombs, Int. J. Mech. Sci. 2003; 45(11): 1799-1813. [51] Ajdari A, Nayeb-Hashemi H, Canavan P, Warner G, Effect of defects on elastic–plastic behavior of cellular materials, Mater. Sci. Eng. A 2008; 487: 558–67. [52] Christodoulou I, Tan PJ, Crack initiation and fracture toughness of random Voronoi honeycombs, Eng. Fract. Mech. 2013; 104: 140-61. [53] Christodoulou I, Tan PJ, Role of specimen size upon the measured toughness of cellular solids, J. Phys. Conf. Ser. 2013; 451(1): 012004. [54] Fan HL, Jin FN, Fang DN, Mechanical properties of hierarchical cellular materials, Part I: Analysis. Compos. Sci. Technol. 2008; 68(15): 3380-87. [55] Chen Q, Pugno N, Zhao K, Li ZY, Mechanical properties of a hollow-cylindrical-joint honeycomb, Compos. Struct. 2014; 109: 68-74. [56] Wang B, Shi Y, Chen Y and Mo Y, 2D hierarchical lattices’ imperfection sensitivity to missing bars defect, J. Theor. App. Mech-Pol. 2015; 5: 141-5. [57] Grenestedt JL, Tanaka K, Influence of cell shape variations on elastic stiffness of closed cell cellular solids, Scripta. Mater. 1998; 40: 71-7. [58] Zhu HX, Hobdell JR, Windle AH, Effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs, J. Mech. Phys. Solids 2001; 49(4): 857-70. [59] Chen C, Lu TJ, Fleck NA, Effect of imperfections on the yielding of two-dimensional foams, J. Mech. Phys. Solids 1999; 47(11): 2235-72. [60] Sih GC, Paris PC, Irwin GR, On cracks in rectilinearly anisotropic bodies, Int. J. Fract. Mech. 1965; 1(3): 189-203. [61] Williams ML, On the stress distribution at the base of a stationary crack, ASME J. Appl. Mech. 1957; 24: 109-14. 18
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S2211-3797(18)32931-0 https://doi.org/10.1016/j.rinp.2018.12.068 RINP 1938
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
9 November 2018 13 December 2018 17 December 2018
Please cite this article as: Wang, B., Ding, Q., Sun, Y., Yu, S., Ren, F., Cao, X., Du, Y., Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures, Results in Physics (2018), doi: https://doi.org/10.1016/j.rinp.2018.12.068
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Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures Bin Wanga,b, Qian Dinga,b, Yongtao Suna,b,c*, Shihui Yub,d**, Fuguang Renb, Xinyu Caoa,b, Yinghong Due a
b
Department of Mechanics, Tianjin University, Road Yaguan 135, 300350, Tianjin, China Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin University, Road Yaguan 135, 300350, Tianjin, China c State
d e
Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China
School of Microelectronics, Tianjin University, Road Weijin 92, 300072, Tianjin, China
Department of Materials Science and Engineering, Tianjin University, Road Yaguan 135, 300350, Tianjin, China
*Corresponding authors: [email protected]; [email protected]
Abstract For multifunctional optimization design of honeycomb structures, the high stiffness hierarchical honeycombs with stochastic Voronoi substructures (HHSVS) are proposed by substituting cell walls of the regular hexagonal honeycombs (ORHH) with Voronoi honeycomb lattices of equal mass. In this study, the tunable linear elastic fracture properties of the HHSVS are investigated by finite element analysis. Results demonstrate that size effect on fracture toughness of the HHSVS is noticeable and a brittleness number is suggested to determine it. At the same time, compared with the ORHH and conventional Voronoi honeycombs of equal mass/density, the in-plane fracture toughness of the HHSVS could be more than 2 times larger and regardless of the cell regularity and hierarchical parameters, fracture toughness of the HHSVS exhibits a weaker quadratic dependence on the relative density and is the highest. As a whole, the HHSVS exhibit the combined properties of tunable Poisson’s ratio, higher stiffness, enhanced tunable fracture toughness, lower imperfection sensitivity and better structural-acoustic performance etc. The research provides a novel strategy for the multifunctional optimization design of the honeycombs structures widely used in the engineering fields. 1
Keywords: Hierarchical honeycombs with stochastic Voronoi substructures (HHSVS); Tunable fracture toughness; Size effect; T-stress.
1. Introduction Owing to their superior stiffness-to-weight and strength-to-weight ratio, honeycomb structures have been widely used in the fields of mechanical, aerospace and civil engineering. To optimize their topologies and thus promote their multifunctional structural integrity applications, many types of hierarchical honeycombs have been proposed, such as hierarchical honeycombs with hexagonal [1,2], triangular [2,3], Kagome [3] and chiral [4] honeycomb substructures, self-similar isotropic [5] and anisotropic [6] hierarchical honeycombs etc. With regard to mechanical properties, analytical or numerical analyses show that compared to conventional honeycombs of equal mass/density, hierarchical honeycombs could exhibit tunable Poisson’s ratio and higher in-plane stiffness or strength [1-11]. With respect to multifunctional properties, hierarchical honeycombs structures could have better wave filtering properties [12-16], enhanced in-plane [17,18] or out-of-plane [19-23] energy absorption properties, excellent crashworthiness under uniaxial dynamic impact [24-31], improved thermal conductivity properties [2] and so on. The substructures of the aforementioned hierarchical honeycombs are largely regular honeycomb lattices. However, studies on hierarchical honeycombs with stochastic Voronoi honeycomb substructures are lacking [32]. In fact, cellular materials with random irregular cells and structural hierarchy can be found in many living natural materials such as bones or plant stems [33-38]. The degree of cell irregularity may have great influence on mechanical properties of the hierarchical cellular materials [39-44]. Hence, hierarchical honeycombs with stochastic Voronoi substructures (HHSVS) [45] were proposed by substituting cell walls of the regular hexagonal honeycombs (ORHH) with stochastic Voronoi substructures of equal mass. The stiffness of these types of irregular hierarchical honeycombs could be as three times higher as that of the
2
ORHH of equal mass as predicted by numerical analyses. Except the improvement in stiffness, HHSVS also exhibit excellent structural-acoustic properties [46] than the ORHH. In addition to exhibiting excellent stiffness, strength, wave filtering, and imperfection sensitivity, the damage resistance of the honeycomb structures plays a crucial role in engineering applications. To explore damage resistance of the honeycomb structures and promote their engineering applications, extensive research has been conducted. The group of Fleck [47,48] numerically studied fracture toughness of the elastic-brittle regular hexagonal, Kagome, triangular, and diamond honeycombs and analytically determined the effect of relative density on the fracture toughness. Moreover, they explored the defect sensitivity of modulus and fracture toughness for five morphologies of the 2D lattices [49]. Wang and McDowell [50] analyzed the effects of defects on the in-plane properties of periodic metal honeycombs including the regular hexagonal, square, and triangular honeycombs. Ajdari et al. [51] reported the effects of missing walls and filled cells on the elastic–plastic behavior of both regular hexagonal and non-periodic Voronoi structures using finite element (FE) analysis. Christodoulou and Tan investigated the effects of cell-regularity, relative density [52], and size [53] on the toughness of conventional Voronoi honeycombs using FE analysis. With regard to the fracture properties of the hierarchical honeycombs, Fan et al. [54] analytically analyzed hierarchical cellular materials made of sandwiched struts and indicated that the hierarchical honeycombs are much more damage tolerant and insensitive to wavy imperfections in the cell wall. Chen et al. [55] constructed a type of hierarchical honeycombs by substituting the three-edge joints of the regular hexagonal honeycomb with hollow-cylindrical joints and reported that the fracture strength could be improved by 303% with respect to the conventional ones. Wang et al. [56] proved that the hierarchical honeycombs exhibit insensitivity to missing bars defect compared to conventional honeycombs. Except their tunable Poisson’s ratio [45], higher stiffness[45], better structural-acoustic properties [46]
3
and lower imperfection sensitivity [56] compared to the ORHH and conventional Voronoi honeycombs of equal mass/density [52,53], fracture properties of the HHSVS, which play very important role for their multifunctional applications, are still not clear. To comprehensively understand tunable fracture properties of the HHSVS and thus promote their multifunctional structural integrity design for engineering applications, in this paper, effects of size, cell regularity, relative density, thickness-to-length ratio, hierarchical length ratio, mixed-mode loading and T-stress on fracture toughness of the HHSVS are systematically analyzed through the numerical simulations. The study of this paper provides a new strategy for the multifunctional structural integrity design of the lightweight components, such as sandwich panels, widely used in the engineering fields.
2. Geometrical models, elastic properties and FE models of the HHSVS for fracture analysis 2.1 Geometrical models of the HHSVS Fig. 1 shows schematic of the unit cell of the HHSVS [45]. Various methods are used to generate foam-like representative volume elements (RVEs) based on Voronoi tessellation. The stochastic Voronoi substructures were developed using a method proposed in Ref. [57,58]. First, a uniform distribution of the seed positions is generated, corresponding to a perfectly regular honeycomb pattern. Next, the positions of the seeds are perturbed via pseudo-random motion within a range of limits. The unit cell of the HHSVS is finally obtained by removing the conventional stochastic hexagonal honeycomb structures with regular ones to ensure that the pairs of counterpart nodes are located on opposing boundaries. The unit cell of the HHSVS is then copied along the two directions to obtain a larger model. The cell regularity is defined as d is the minimum distance between the centers of any two neighboring cells in a randomly irregular honeycomb, and d 0 4
2 A0 is the distance between any two N 3
neighboring cells in a regular hexagonal honeycomb with the same number of cells. Ao is the controlling area, and N is the number of cells. →1 indicates that the hexagonal honeycomb is perfectly regular and →0 represents a completely random Voronoi honeycomb. The relative density is defined as the ratio between the macroscopic density of the cellular structure and that of the cell wall material. It is assumed that all the cell walls have the same thickness t, and the relative non-dimensional density is determined as:
t
N
l
i 1 i
(1)
A0
where li and N are the length and number of the cell walls. The cell-wall thickness-to-length ratio of the superstructure is defined as the ratio of its thickness tsup to its length lsup, i.e., tsup
lsup
. The ratio
between the lengths in the super and sub-structures is termed as the hierarchical length ratio, i.e.
lsup l sub , where l sub is the average value of the sub-structure cell wall length .
(b) →0
(a)
(c) = 0.4
(d) = 0.7 5
Fig. 1. Schematic of unit cells of (a) the original regular hexagonal honeycomb (ORHH) and the HHSVS of equal mass/density: (b)→0, (c)4 and (d) .
2.2 Elastic properties of the HHSVS Elastic properties of the HHSVS, which will be used for analysis of the linear elastic fracture properties of the HHSVS, are discussed in this section. The FE models are developed to explore the effects of the stochastic Voronoi sub-structures on the in-plane elastic properties using the software ABAQUS/standard. Each cell wall is simulated using 2D Timoshenko beam elements (B21). The periodic boundary conditions (PBC) were applied along the boundaries of RVEs [59], and the details are shown in Ref. [45]. Under plane-stress conditions, the in-plane elastic modulus is represented using the power-law expression
E Es B b , where Es is the Young’s modulus of the solid of which the HHSVS are made. Under the plane-strain condition, the elastic modulus in the prismatic x3 direction is defined as E33 Es , and the longitudinal Poisson’s ratios are
31 32 s , where νs is the Poisson’s ratio of the solid of which the
HHSVS are made. Consequently, the plane-strain elastic moduli are represented as [49]: 1 E, 1 B b 1
(2a)
s2 B b 1 . 1 s2 B b 1
(2b)
E ps
ps
2 s
2.3 FE models of the HHSVS for fracture analysis To analyze fracture properties of the HHSVS, the linear elastic fracture mechanics method is applied. Figure 2 shows the schematic illustrations of the cracked FE models. The displacements associated to the crack-tip K-field for homogeneous and isotropic solid are applied along the model boundary as follows [60,61]:
u1
1 KⅠ 1/ 2 1 KⅡ 1/ 2 r ( cos )cos r ( 2 cos )sin , 2 2 2 G ps 2 2 2 G ps
6
(3a)
u2
1 KⅠ 1/ 2 1 KⅡ 1/ 2 r ( cos )sin r ( 2 cos )cos . 2 2 2 G ps 2 2 2 G ps
(3b)
In the above expressions, u1 and u2 are the displacements aligned along the Cartesian axes x1 and x2 , respectively, and r-θ is the polar coordinate system centered at the stationary crack tip. KI and KII are the mode I and mode II stress intensity factors, respectively, and = (3 −ps)/(1 +ps). In addition, the in-plane rotation ω is expressed as:
1 2
(u2,1 u1,2 )
1 1 1/ 2 r ( KⅠsin KⅡ cos ) . 2 2 4 2 G ps
(3c)
1 Under mixed-mode failure, the mixture of KI and KII is determined by a mode-mixity M 2 tan ( KⅡ KⅠ)
with 0≤M <1 (M=0 and M=1 correspond to pure mode I and mode II loading, respectively). The lattice walls are assumed to be linear-elastic, and fail at the ultimate stress f . The macroscopic toughness KC can be obtained if the maximum local stress reaches the strength f [47,49,52]. Considering the stochastic properties of the HHSVS, at least seven samples are analyzed for several degrees of regularity, and the average value of them is chosen for the results. u2 u1
x2 θ
x1
0.0
0.4
0.7
Fig. 2. Schematic of the FE model of the HHSVS for fracture analyses.
3. Fracture properties of the HHSVS In this part, effects of size, cell regularity, relative density, thickness-to-length ratio, hierarchical length 7
ratio, mixed-mode loading and T-stress on fracture toughness of the HHSVS are studied, respectively.
3.1 Size effect on fracture toughness of the HHSVS First, size effect on fracture toughness of the HHSVS is investigated. Because of the strain gradient effects, surface elasticity and initial stress [62], mechanical properties of the hierarchical honeycombs are found to be size-dependent. In fact, fracture toughness of the cellular solid has been found to be dependent on the specimen scales [47,52]. Christodoulou and Tan [53] discussed the physical origin of the size effects. It is found that the effect of size in lattice materials depends on the specimen boundary constraint on fracture process zone, thereby disrupting the K-field in the vicinity of the crack-tip. With the increase in the specimen size, the interaction becomes weaker. The fracture toughness is measured by summing up the contributions from the two physically distinct processes, i.e. surface energy and energy dissipation in the process zone. Based on dimensional analysis and the process zone, Carpinteri [63] proposed an energy brittleness number to calibrate the specimen size effects of the specimens, which is the ratio of the process zone size to the characteristic structural dimension. Therefore, the process zone size d around the crack tip in the plane-strain state is calculated as [64]:
d a[sec(
( ) ) 1] 2 f
(4)
Here, a is the crack length; σ(∞) and σf denote the uniform tension at infinity and uniaxial tensile strength of the material, respectively; β is a parameter describing the behavior of the cellular material and the stress state, which is defined as 1 1 2 0.02 (1 ) according to Triantafillou and Gibson [65]. To show the size effect on fracture toughness of the HHSVS, the parameters = 0.01, and are chosen as the examples for illustration. The corresponding results, for the ratio of the calculated process zone size d to the overall specimen length L and the normalized fracture toughness
KC / ( 2 f
lˆ )
predicted from several scale models, are plotted in Fig. 3 as a function of the number of cells. 8
From Fig. 3 we can see that as the size increases, both the normalized toughness and the normalized process zone size converge to constant values, thus indicating a strong size dependence at small scale sizes. Accordingly, to eliminate the scaling phenomena, the simulations were performed with lattices of 10000 cells considering the computational cost and accuracy.
Fig. 3. Normalized fracture toughness and d L 1011 vs the number of cells in the model.
3.2 Effect of cell regularity In this section, the effect of cell regularity on fracture toughness of the HHSVS is investigated. Fig. 4 shows the predicted toughness values KIC and KIIC plotted as a function of the cell regularity . The results are obtained for = 0.01, and Moreover, the fracture toughness is normalized by 2
f
lˆ
. From Fig. 4 we can see that the relationship between the cell regularity and the loading modes are
unclear. However, the hierarchical lattices exhibit larger crack initiation resistance to mode I cracking than that of mode II, regardless of cell regularity. For comparison, both the mode I and mode II fracture toughness of the regular hexagonal [47] and conventional Voronoi honeycomb [52] of equal masses and under the same conditions are shown in Fig. 4. It is apparent that fracture toughness of the HHSVS is at least twice that of the Voronoi honeycombs and ORHH, regardless of the loading mode. Hence, the introduction of hierarchy can effectively improve fracture initiation resistance of the honeycombs. 9
Fig. 4. Normalized mode I and mode II fracture toughness vs cell regularity of the HHSVS, the conventional Voronoi honeycombs and the regular hexagonal honeycombs (“Hierarchy”, “Voronoi” and “Hexagonal” stand for the HHSVS, the conventional Voronoi honeycombs and the ORHH, respectively).
3.3 Effect of the relative density In this section, effect of the relative density on fracture toughness of the HHSVS is studied. It is well known that the fracture toughness KC of the cellular material has a power-law dependence on the relative density
KC
f l
as follows:
D d
(5)
As demonstrated by Fleck and Qiu [47], the exponent d is approximately 2 for the hexagonal honeycomb, unity for the triangular honeycomb, and 0.5 for the Kagome lattice, respectively. Moreover, the exponent d ranges from 1.994 to 1.988 for mode I toughness for the Voronoi honeycombs [52]. Fig. 5 shows the variations in the mode I and mode II fracture toughness of the HHSVS with the fixed relative density for
and . It is also shown in Fig. 5 that the fracture toughness can be best-fitted by the power-law in Eq. 4. For mode I loading, Fig. 5 shows that the exponent d equals to 1.910, 1.954, and 2.060 for , and , respectively. For mode II loading, the d values are 1.944, 1.82, and 2.094 for 10
, andrespectively. A relatively weak quadratic dependence of the fracture toughness of the
K C / ( f
lˆ )
HHSVS on the relative density is found compared to that of the ORHH and Voronoi honeycombs [47,52].
Fig. 5. Mode I and mode II KC vs the relative density (“Hexagonal” stands for the ORHH of equal mass/density with the HHSVS).
3.4 Effect of the thickness-to-length ratio In this section, the effect of the thickness-to-length ratio on fracture toughness of the HHSVS is investigated. Theoretically, the ratio of thickness to length α ranges from 0 to →0 and α→
, and the two limit cases α
correspond to the regular hexagonal and Voronoi honeycombs, respectively. In this study, the
models with different values of ranging from 0.05 to 1.2 with fixed = 0.01 and are employed in the FE analyses under mode I loading. From Fig. 6 we can see that fracture toughness of the HHSVS is very sensitive to the ratio of thickness to length of the superstructure. It increases to a maximum value of 2.16, and subsequently, rapidly decreases to 0.70 with the increase of α. In addition, Fig.6 shows the fracture toughness values of the ORHH with =0.01 [47] and Voronoi honeycomb with and =0.01 [52] for comparison. Apparently, the introduction of hierarchy could significantly increase fracture toughness of the HHSVS at small thickness-to-length ratios of the cell wall of the superstructure. Moreover, tunable elastic properties can be achieved for a wide range.
11
Fig. 6. Predicted fracture toughness for various thickness-to-length ratio α.
3.5 Effect of the hierarchical length ratio In this section, effect of the hierarchical length ratio λ on fracture toughness of the HHSVS is analyzed. For illustration, Fig. 7 shows the variation of the normalized mode I fracture toughness with the increase of λ for = 0.01, and . Apparently, the variation is unclear. It is worth noting that the true fracture toughness decreases with the increase of λ. The smaller the sub-cell, the smaller the value of the fracture toughness.
Fig. 7. Predicted fracture toughness with respect to various hierarchical length ratio λ.
3.6 Effect of the mixed-mode loading 12
In this section, the effect of the mixed-mode loading on the tunable fracture toughness of the HHSVS is analyzed. The KI/KII ratio is calculated for different values of mode mixity. Fig. 8 shows the fracture locus in the mode space for = 0.01, and, wherein both KI and KII are normalized by KIC. It is obvious that the locus of the values takes the shape of an inner convex envelope. With the variation in the mode-mixity, the fracture initiation site of the cell wall changes.
Fig. 8. Failure envelope under mixed-mode loading.
3.7 Effect of T-stress The T-stress is the second term in the William’s series expansion and parallel to the crack plane. The T-stress can be neglected under mode II loading with the assumption of symmetry and plays a negligible role at the crack tip for solids. However, Fleck and Qiu [47] demonstrated the need to include the effect of the T-stress to describe the fracture behaviors of a regular hexagonal honeycomb. Moreover, Christodoulou and Tan [52] demonstrated the need to include the T-stress in predicting the fracture toughness of a Voronoi lattice. To apply the T-stress around the periphery of the model, the displacement components in Eq. (2) are modified for T-stress terms:
u1 T
2 1- ps
E ps
r cos ,
(6a) 13
u2 T
ps (1+ ps ) E ps
r sin .
(6b)
The stress biaxiality ratio can be obtained as follows [66]: B
T a
(6c)
K
where a is the crack length. For illustration, the effect of the T-stress on fracture toughness of the HHSVS is shown in Fig. 9 for = 0.01, and. K C0 is the critical fracture toughness at T-stress equal to zero. Fig. 9 shows that the normalized fracture toughness increases rapidly with the increase in the T-stress under either mode I or mode II loading conditions. However, the mode I value is more sensitive to the negative T-stresses. Moreover, at the same T-stress level, the fracture toughness decreases and becomes a constant as the irregularity increases to 0.7. Therefore, compared to the regular hexagonal and Voronoi honeycombs of equal mass/density, the defect sensitivity of the HHSVS can be tuned more flexibly by
KC / KC0
changing the crack length ratio, loading configuration and out-of-plane geometries.
Fig. 9. T-stress vs fracture toughness.
4. Conclusions In the work, the FE-based method is used to calculate the tunable fracture toughness of the high stiffness HHSVS. Effects of size, cell regularity, relative density, thickness-to-length ratio, hierarchical length ratio, 14
mixed-mode loading and T-stress on fracture toughness of the HHSVS are systematically investigated. The following conclusions are drawn from this study: 1. Although the effects of the cell regularity on the mode I and mode II toughness values are unclear, the crack initiation resistance of the HHSVS is found to be better than that of the Voronoi and regular hexagonal honeycombs of equal masses. And fracture toughness of the HHSVS is very sensitive to the thickness-to-length ratio of the cell walls of the superstructure. 2. The effect of size on fracture toughness of the HHSVS is noticeable. A brittleness number is suggested to determine the effect of the specimen size, which is explicitly a non-dimensional quantity used to describe the ratio of the size of the process zone to the characteristic structural dimension. 3. Compared to the ORHH and Voronoi honeycombs of equal mass/density, in-plane fracture toughness of the HHSVS could be more than 2 times larger and the HHSVS exhibit a relatively weak quadratic dependence of the nominal fracture toughness on the relative density. 4. Fracture toughness of the HHSVS increases with the increase of the T-stress under both mode I and mode II loading conditions. At a fixed T-stress level, fracture toughness of the HHSVS becomes a constant with the increase in the cell irregularity. And compared to the regular hexagonal and Voronoi structures, the defect sensitivity of the HHSVS can be flexibly tuned in a wider range by changing the crack length ratio, loading configuration or out-of-plane geometries. As a whole, compared with the ORHH and conventional Voronoi honeycombs of equal mass/density, the HHSVS exhibit the combined advantages of tunable Poisson’s ratio, higher stiffness, enhanced tunable fracture toughness, lower imperfection sensitivity and better structural-acoustic performance and so on.
Acknowledgements The research is supported by the National Natural Science Foundation of China under Grant 15
No. 11502162 and the Natural Science Foundation of Tianjin under Grant No. 15JCQNJC05100. YS is also supported by the fund (No. SV2015-KF-02) from State Key Laboratory for Strength and Vibration of Mechanical Structures (Xi’an Jiaotong University), the Research Project Key
Laboratory
of
Mechanical
System
and
of State
Vibration MSV201611(Shanghai Jiaotong
University), the Seed Foundation of Tianjin University (No. 2017XRG-0035 and 2017XYF-0009).
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