Elastic SH wave propagation in periodic layered composites with a periodic array of interface cracks

Acta Mechanica Solida Sinica, Vol. 28, No. 5, October, 2015 Published by AMSS Press, Wuhan, China

ISSN 0894-9166

ELASTIC SH WAVE PROPAGATION IN PERIODIC LAYERED COMPOSITES WITH A PERIODIC ARRAY OF INTERFACE CRACKS⋆⋆ Zhizhong Yan1⋆

Chunqiu Wei1

Chuanzeng Zhang2

1

( School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China) (2 Department of Civil Engineering, University of Siegen, D-57078 Siegen, Germany) Received 30 June 2014, revision received 15 May 2015

ABSTRACT The interaction of anti-plane elastic SH waves with a periodic array of interface cracks in a multi-layered periodic medium is analyzed in this paper. A perfect periodic structure without interface cracks is first studied and the transmission displacement coefficient is obtained based on the transfer matrix method in conjunction with the Bloch-Floquet theorem. This is then generalized to a single and periodic distribution of cracks at the center interface and the result is compared with that of perfect periodic cases without interface cracks. The dependence of the transmission displacement coefficient on the frequency of the incident wave, the influences of material combination, crack configuration and incident angle are discussed in detail. Compared with the corresponding perfect periodic structure without interface cracks, a new phenomenon is found in the periodic layered system with a single and periodic array of interface cracks.

KEY WORDS layered composite, transmission coefficient, interface crack, transfer matrix method

I. INTRODUCTION Recently, the propagation of elastic waves in periodic structures is a topic which has received considerable attention due to its anomalous physics, such as band gaps and negative refraction[1, 2]. These unusual effects promise a wide range of potential important applications such as acoustic filters, control of vibration isolation, noise suppression and design of new transducers; as well as for pure physics concerned with the Anderson localization. So far, tremendous results were reported about the perfect periodic structures and those with determinant defects[3, 4] . Compared to the above systems, the periodic systems containing cracks have received less attention. As is well known, interfaces play an important role in composites as well as in bonded materials. Distributed defects such as micro-cracks and debonded zones often occur on the interface as a result of material processing, manufacturing, bonding methods and in-service conditions. These defects may cause stiffness degradations, and thus influence the integrative properties of the composites. Such damage should be possibly detected by ultrasonic nondestructive testing. A good source of knowledge in the area of wave propagation and scattering in the presence of cracks is the monographs by Zhang and Gross[5] and Wang and Gross[6] , which also contain numerous references to earlier work. More recently, Bostr¨om and Golub[7] studied Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11002026 and 11372039), Beijing Natural Science Foundation (No. 3133039) and the Scientific Research Foundation for the Returned (No. 20121832001). ⋆

⋆⋆

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elastic SH wave propagation in a layered anisotropic plate with interface damage modeled by spring boundary conditions. To our knowledge, few works have been devoted to the periodic structures containing interface cracks[8, 9]. In these papers, in-plane wave motion[8] and SH wave propagation[9] in periodic layered composites with a crack were studied by a Galerkin method. In these studies, only a single crack was considered by assuming the wave propagation direction normal to the layers. With the above mentioned motivation in mind, a natural question to ask is, how would the wave propagation properties be altered if considering wave propagating obliquely in periodic layered composites with a periodic array of interface cracks? Wang and Gross[6] developed a universal method to solve SH-wave propagation in a multi-layered medium with an arbitrary number of interface cracks. The method makes use of the transfer matrix, Fourier integral transform and singular integral equation techniques. In the present paper, we extend the method to the interaction of SH waves with a periodic array of interface cracks in a multi-layered periodic medium. In this paper, a study is performed for elastic SH wave propagation in 1D periodic layered composites with a periodic array of interface cracks. The transfer matrix method (TM) is adopted for this purpose. The displacement transmission coefficient is calculated. Numerical results are presented and discussed to show the transmission displacement coefficients in the layered composites with and without interface cracks. In the case of a single or a periodic array of interface cracks, the system obeys some unique features which are in sharp contrast to the case of perfect periodic layered structures without interface cracks.

II. PROBLEM FORMULATION We consider the propagation of anti-plane time-harmonic SH elastic waves in a one-dimensional (1D) layered periodic composite structure. A schematic sketch of a periodic composite structure with a periodic array of interface cracks studied in the paper is shown in Fig.1. It is constructed by inserting two layers A and B, alternately, which have different material properties, and the cracks are periodically distributed on the center interface. The local coordinates of the monolayers are also given in the figure. The thicknesses or lengths of the monolayers are ai (i = 1, 2) respectively. So the lattice constant a = a1 + a2 . Assume the layered composite consists of n unit cells. Periodic Griffith cracks of length 2c and midpoint distance 2L between two adjacent cracks are distributed on the kth interface. Let us consider anti-plane time-harmonic SH elastic waves propagating in an arbitrary direction, which is described by the inclination angle θ. We assume that the wave number ky is one component of the wave vector k that is parallel to the interfaces between different elastic layers and kx is the other component of k that is perpendicular to the interfaces. For time-harmonic steady state and in the absence of body forces, the layered composite structure satisfies the following equations of motion. µj (wj,11 + wj,22 ) + ρj ω 2 wj = 0

(j = 1, 2)

Fig. 1. An array of periodic interface cracks in a periodic layered medium subjected to SH waves.

(1)

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where µ are Lame’s elastic constants, ρ is the mass density, w are the elastic displacements, and ω is the angular frequency. By introducing the dimensionless local coordinates, ξj =

xj , a1

ηj =

yj a1

(2)

combined with Eq.(2), the general solution of Eqs.(1) for the jth layer can be written as wj =

∞ h i X Aj (n)e(iα sin θ−ina1 π/L)ηj −iαqj ξj + Bj (n)e(iα sin θ−ina1 π/L)ηj +iαqj ξj

(3)

n=−∞

where ωa1 α= , c

qj =

s

c2 − sin2 θ c2j

(4)

Then the interface conditions for the case without interface cracks, the stress and displacement are continuous, (i)

w1R = w1 (ς1 , η1 ),

(i)

w2R = w2 (ς2 , η2 ),

w1L = w1i (0, η1 ),

w2L = w2i (0, η1 ),

(i)

(i)

(i)

τxz1L = µ1

(i)

(i)

(i)

τxz2L = µ2

∂w1 (0, η1 ), ∂ξ1 (i)

(i)

∂w2 (0, η2 ), ∂ξ2

(i)

(i)

τxz1R = µ1

∂w1 (ς1 , η1 ) ∂ξ1

(5)

(i)

(i)

τxz2R = µ2

∂w2 (ς2 , η2 ) ∂ξ2

(6)

Equations (5) and (6) are substituted into Eq.(3), then (

w1R (i) τxz1R

(

w2R (i) τxz2R

(i)

(i)

) )

= T ′1

(

w1L (i) τxz1L

T ′2

(

w2L (i) τxz2L

=

(i)

(i)

)

(7)

)

(8)

In the ith unit cell, according to the continuity conditions, we have (

(i)

w1R (i) τxz1R

)

=

(

(i)

w2L (i) τxz2L

)

(9)

Considering Eqs.(7), (8) and (9), we have (

(i)

w2R (i) τxz2R

)

= T ′2 T ′1

(

(i)

w1L (i) τxz1L

)

= T ′2 T ′1

(

(i−1)

w2R (i−1) τxz2R

)

(10)

Suppose that the periodic distribution of interface cracks is located in the middle of the kth unit cell, as illustrated in Fig.1, we can obtain from Eqs.(7)-(10): (

(k)

w1R (k) τxz1R

)

= T ′1

(

(k)

w1L (k) τxz1L

)

= T ′1

(

(k−1)

w2R (k−1) τxz2R

)

= T ′1 (T ′2 T ′1 )k−1

(

(0)

w2R (0) τxz2R

)

(11)

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where the elements of the transfer matrix T ′j for anti-plane waves are given below: [exp (−iαq1 ξ1 ) + exp (iαq1 ξ1 )] 2 [exp (iαq1 ξ1 ) − exp (−iαq1 ξ1 )] ′ T1 (1, 2) = 2iαq1 µ1 iαq µ [exp (iαq1 ξ1 ) − exp (−iαq1 ξ1 )] 1 1 T1′ (2, 1) = 2 [exp (−iαq ξ ) + exp (iαq1 ξ1 )] 1 1 T1′ (2, 2) = 2 [exp (−iαq2 ξ2 ) + exp (iαq2 ξ2 )] ′ T2 (1, 1) = 2 [exp (iαq ξ ) − exp (−iαq2 ξ2 )] 2 2 T2′ (1, 2) = 2iαq2 µ2 iαq2 µ2 [exp (iαq2 ξ2 ) − exp (−iαq2 ξ2 )] T2′ (2, 1) = 2 [exp (−iαq ξ ) + exp (iαq2 ξ2 )] 2 2 T2′ (2, 2) = 2 According to the relations between the unit cells, T1′ (1, 1) =

(i)

(i−1)

w2R = T11 w1L (i)

(i−1)

τxz2R = T21 w1L where T11 , T12 , T21 , T22 are the elements of matrix Substitute Eq.(3) into (1), then (i)

(i−1)

+ T12 τxz1L

(12)

(i−1)

+ T22 τxz1L

(13)

(T ′2 T ′1 )2 .

(i)

(i−1)

e−iαq2 ς2 A2 + eiαq2 ς2 B2 = [T11 − T12 µ1 (iαq1 )] A1 (i)

(i−1)

+ [T11 + T12 µ1 (iαq1 )] B1

(i)

(i−1)

−µ2 (αq2 ς2 )e−iαq2 ς2 A2 + µ2 (αq2 ς2 )eiαq2 ς2 B2 = [T21 − T22 µ1 (iαq1 )] A1 (i−1)

+ [T21 + T22 µ1 (iαq1 )] B1

(14)

And we obtain   T21 − T22 µ1 (iαq1 ) + T12 µ1 (iαq1 )µ2 (iαq2 ) − T11 µ2 (iαq2 )         −2µ2 (iαq2 )e−iαq2 ς2           T + T µ (iαq ) − T µ (iαq ) − T µ (iαq )µ (iαq ) 21 22 1 1 11 2 2 12 1 1 2 2  ) ( )  (     (i) (i) −2µ2 (iαq2 )e−iαq2 ς2 A1 A2 = (i)  B1(i)  T21 − T22 µ1 (iαq1 ) − T12 µ1 (iαq1 )µ2 (iαq2 ) + T11 µ2 (iαq2 )  B2      iαq ς   2µ2 (iαq2 )e 2 2          T + T µ (iαq ) + T µ (iαq ) + T µ (iαq )µ (iαq ) 21 22 1 1 11 2 2 12 1 2 2 2      iαq ς 2 2 2µ2 (iαq2 )e

(15)

In addition,

(i)

(i−1)

w1L = w2R We obtain  (i)   A1 

(i)

,

−µ1 (iαq1 ξ1 ) − µ2 (iαq2 ξ2 )  −2µ1 (iαq1 ξ1 ) =   (i)  µ1 (iαq1 ξ1 ) − µ2 (iαq2 ξ2 )eiαq2 ξ2 B1 2µ1 (iαq1 ξ1 ) 

µ1

(i−1)

∂w1L ∂w = µ2 2R ∂ξ1 ∂ξ2

 µ2 (iαq2 ξ2 ) − µ1 (iαq1 ξ1 )eiαq2 ξ2  (i−1)     A2 −2µ1 (iαq1 ξ1 )  µ2 (iαq2 ξ2 ) + µ1 (iαq1 ξ1 )eiαq2 ξ2   B (i−1)  2 2µ1 (iαq1 ξ1 )

In the kth unit cell including the periodically distributed interface cracks, we have ∞ h i X (k) (k) (k) w5 = A5 e(iα sin θ−ina1 π/L)η2 + B5 e(iα sin θ−ina1 π/L)η2 n=−∞

(16)

(17)

(18)

Vol. 28, No. 5 Zhizhong Yan et al.: Elastic SH Wave Propagation in Periodic Layered Composites

(k)

w4

=

· 457 ·

∞ h i X (k) (k) A4 e(iα sin θ−ina1 π/L)η2 + B4 e(iα sin θ−ina1 π/L)η2

(19)

n=−∞

The displacement is not continuous because of the periodically distributed cracks, △w(η2 ) =

∞ X

(iα sin θ−ina1 π/L)η2 △w(n)e ¯

(20)

n=−∞

From Eqs.(19) and (20), we can obtain ∞ X

(iα sin θ−ina1 π/L)η2 △w(n)e ¯ =

n=−∞

∞ h X

(k)

(k)

A2 e(iα sin θ−ina1 π/L)η2 + B2 e(iα sin θ−ina1 π/L)η2

n=−∞ (k) −A1 e(iα sin θ−ina1 π/L)η1 −iαq1 ξ1

(k)

− B1 e(iα sin θ−ina1 π/L)η1 +iαq1 ξ1

i

As a special case, if only a single interface crack is located on the interface, that is c c − < η2 < , x2 = 0 a1 a1 ∞ h i X (k) (k) (k) (k) A5 + B5 − A4 − B4 eiα sin(θη2 ) dη2 △w(η2 ) =

(21)

(22) (23)

n=−∞

then, we have (k)

(k)

(k)

(k)

A5 + B5 − A4 − B4

=

1 2π

At x2 = 0, the stress is continuous

Z

(k)

µ2

c/a1

e−iα sin(θη2 ) △w(η2 )dη2

(24)

−c/a1

(k)

∂w5 ∂w = µ1 4 ∂ξ2 ∂ξ1

(25)

We have

 µ q   µ1 (iαq1 )  1 1 (k) (k) (k) (k) −A4 + B4 = −A4 + B4 µ2 (iαq2 ) µ2 q2 Setting κ = µ1 q1 /(µ2 q2 ), we have Z c/a1 1 (k) (k) (k) (k) e−iα sin(θη2 ) △w(η2 )dη2 A5 + B5 = A4 + B4 + 2π −c/a1 (k)

(k)

−A5 + B5

(k)

=

(k)

−A5 + B5 Then

(k)

(k)

= −κA4 + κB4

 1 + κ 1 + κ  (k)    Z c/a1 A  5 4 1 1  2  = 2 + e−iα sin(θη2 ) △w(η2 )dη2  1 − κ 1 + κ  (k)  1  B (k)  2π −c/a 1 B4 5 2 2 At the interface of cracks, we have    A(k) 



(26)

(27) (28)

(29)

(k)

µ2

∂w5 (0) = −τyz ∂ξ2

(30)

III. TRANSMITTED WAVE FIELDS As seen in Fig.1, considering the finite thickness and connecting the two semi-infinite solids at each ends, the transmission coefficients based on the transfer matrix method are given in the following. The subscripts ‘0’ and ‘e’ will be used to denote the left and right semi-infinite solids, respectively. When the plane wave propagates normal to the layers, the transmission coefficients for the displacement fields are written as Ue 2E0 (T11 T22 − T12 T21 ) = (31) U0 E0 (T11 − Ee T21 ) − (T12 − Ee T22 )

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where Ue is the transmitted displacement amplitude of the right semi-infinite solid; and Tij = T (i, j) are the elements of the total transfer matrix; T = T n T n−1 · · · T m · · · T 1 . When the plane wave propagates obliquely to the layers, the transmission coefficients for the displacement fields in the x-direction are given by     Uxt sin θ0 Be qLe Ae + (32) = U0 qL0 A0 qL0 A0 where the transmission coefficients for the displacement potentials can be obtained as 

Ae Be A′1 B0 , , , A0 A0 A0 A0

T

= M −1 b

(33)

where the elements of M and b can be referred to Ref.[10].

IV. NUMERICAL RESULTS AND DISCUSSIONS In this section, numerical results are presented and discussed. Special attention is devoted to comparison with interface cracks and without interface cracks. 4.1. Transmission in the Composites without and with a Single Crack 4.1.1. The influence of material combination on the band structure First, to check the validity of the method, we consider a periodic structure consisting of Pb and Epoxy layers. The used material constants are given in Table 1 and the thicknesses of Pb and Epoxy layers are taken as a1 = 2a2 . Figure 2 shows a comparison of the band structure calculated by the plane wave expansion method and the transmission coefficient calculated by the present method. It can be seen that they are in very good agreement and the accuracy of the results with transfer matrix method is therefore validated. Table 1. Material constants

Materials Pb Epoxy Al Zn

Mass density ρ (kg/m3 ) 11400 1200 2700 6919

Transverse wave veloctiy cT (m/s) 860 1160 3110 2409

Fig. 2. The band structure of SH waves propagating normally as a function of the dimensionless frequency (a) and the transmission coefficient for the periodic case (b).

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As comparison examples with interface cracks and without cracks, we consider two cases: (1) higher contrast material combination, Pb and Epoxy layers, (2) lower contrast material combination, Al and Zn layers. In this section, the emphasis is placed on the transmission phenomenon. First, we consider the higher contrast material combination case-SH waves propagating normally to the layers. The transmission coefficient in the x-direction through a 36-layer (n = 18) phononic crystal bonded to two semi-infinite solids (the same material as material A) at the two ends is presented in Fig.3(a) for a structure without interface cracks. In addition, Fig.3(b) also gives the transmission coefficient for the case of SH waves propagating with a single crack. From Fig.3, we can see that for the very small single crack, in most frequencies there are band-gaps in the system without cracks, and there are still no band-gaps in the presence of a very small single crack. From mathematical point of view, in the case of the system without cracks, the transmission tends to be zero, then stresses and displacement fields are much less than the input. So on the right side of the integral equation as shown in Eq.(20), we have also very small terms compared with the input, and thus the addition of a very small crack leads to small enough extra wavefield except in special resonance situations. In the following, we consider the lower contrast case, that is to say, the Al and Zn combination. From Fig.4, we can see the same phenomenon as in Fig.3. A very small single crack has nearly no influence on the band structures. However, because of the different material combinations, the corresponding frequency band gaps have sharp differences between Fig.3 and Fig.4. So, we can conclude that the material combination has evident influence on the band structures whether the system has cracks or not. 4.1.2. The influence of material length/thickness ratio on the band structure As an example, we choose the Al and Zn material combination, and Fig.5 and Fig.6 show the comparison between the system with and without cracks. From Figs.5 and 6, we can see that the length or thickness ratio of the components has prominent influence on the band structures. And with the increase of the length ratio, the number of band gaps decreases, gradually. 4.1.3. The influence of crack configuration on the band structure Here, considering the crack size influence on the band structures, we choose the Pb and Epoxy combination as an example. From Fig.7, we can see that the crack size has obvious influence on the band structures. The low frequency band gaps are easier to occur with the increase of the crack size.

Fig. 3. Transmission coefficients in the Γx direction when the waves propagate normally to the perfect periodic structures: higher contrast material combination.

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Fig. 4. Transmission coefficients in the Γx direction when the waves propagate normally to the periodic structures: lower contrast material combination.

Fig. 5. The band structure varies with the increase of length/thickness ratio for the system without cracks.

Fig. 6. The band structure varies with the increase of length/thickness ratio for the system with a single crack.

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Fig. 7. The band structure varies with the increase of crack size for the system with a single crack.

4.2. Transmission in the Composites with a Periodic Array of Cracks As comparison examples without and with a single and periodically distributed interface cracks, we still choose Pb and Epoxy layers. The results are shown in Fig.8. From Fig.8, we can see clearly that the low frequency band gaps are enlarged in the presence of periodic cracks.

V. CONCLUSIONS In this paper, wave propagation in periodic layered composite structures without and with interface cracks are investigated and compared. The transmission coefficients are calculated. The transfer matrix method in conjunction with the Bloch-Floquet theorem is applied for this purpose. Special attention of the analysis is paid to the wave transmission phenomenon. Different from the composite structure without cracks, the periodic composite structure with interface cracks shows a new phenomenon. The main conclusions from this study can be summarized as follows: (1) For the composite structure with a single crack, a very small crack has nearly no influence on the band structures. However, the material combination has evident influence on the band structures whether the system has cracks or not. (2) The length or thickness ratio of the components has prominent influence on the band structures. And with the increase of the length ratio, the number of band gaps decreases, gradually.

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Fig. 8. The comparison among without, single crack and periodic cracks.

(3) The crack size has influence on the band structures. The low frequency band gaps are easier to occur with the increase of the crack size. (4) For the composite structure with periodic distribution of cracks, the low frequency band gaps are enlarged by the periodic cracks.

References [1] Hussein,M.I., Hamza,K., Hulbert,G.M. and Saitou,K., Optimal synthesis of 2D phononic crystals for broadband frequency isolation. Waves in Random and Complex Media, 2007, 17: 491-510. [2] Cheng,Y., Xu,J.Y. and Liu,X.J., One-dimensional structured ultrasonic metamaterials with simultaneously negative dynamic density and modulus. Physical Review B, 2008, 77: 045134. [3] Torres,M., Montero de Espinosa,F.R. and Arag´ on,J.L., Ultrasonic wedges for elastic wave bending and splitting without requiring a full band gap. Phyical Review Letters, 2001, 86: 4282-4285.

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[4] Kafesaki,M., Sigalas,M.M. and Garc´ıa,N., Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials. Physical Review Letters, 2000, 85: 4044-4047. [5] Zhang,C.H. and Gross,D., On Wave Propagation in Elastic Solids with Cracks. Southampton, UK: Computational Mechanics Publications, 1998. [6] Wang,Y.S., Gross,D., Interaction of harmonic waves with a periodic array of interface cracks in a multilayered medium: anti-plane case. International Journal of Solids and Structures, 2001, 38: 4631-4655. [7] Bostrom,A. and Golub,M., Elastic SH wave propagation in a layered anisotropic plate with interface damage modeled by spring boundary conditions. The Quarterly Journal of Mechanics & Applied Mathematics, 2009, 62(1): 39-52. [8] Golub,Mikhail.V. and Zhang,C.Z., In-plane wave motion and resonance phenomena in periodically layered composites with a crack. Wave Motion, 2014, 51: 308-322. [9] Golub,Mikhail.V. and Zhang,C.Z., Wang,Y.S., SH-wave propagation and resonance phenomena in a periodically layered composite with a crack. Journal of Sound and Vibration, 2011, 330: 3141-3154. [10] Chen,A.L. and Wang,Y.S., Study on band gaps of elastic waves propagating in one-dimensional disordered phononic crystals. Physica B, 2007, 392: 369-378.