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Scattering of a Rayleigh wave by a near surface crack which is normal to the free surface
International Journal of Engineering Science 145 (2019) 103162
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Scattering of a Rayleigh wave by a near surface crack which is normal to the free surface Chuanyong Wang a, Oluwaseyi Balogun b, Jan D. Achenbach b,∗ a b
State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, People’s Republic of China Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA
a r t i c l e
i n f o
Article history: Received 15 July 2019 Revised 28 August 2019 Accepted 31 August 2019
Keywords: Near surface crack Reciprocity theorem Rayleigh wave Equivalent body force
a b s t r a c t In this paper, scattering of a Rayleigh surface wave by a 2D near-surface crack which is normal to the free surface of a homogenous, isotropic and linearly elastic half-space has been investigated. The scattered field is represented by the radiation from equivalent body forces, which are expressed in terms of the crack-opening volumes generated by the incident Rayleigh wave. The reciprocity theorem together with a virtual wave has been used to determine the amplitude of the far field of the scattered Rayleigh wave. A special case of low frequency, and small ratio of the crack length to the wavelength of the incident Rayleigh wave, has been considered to illustrate the method for different values of the ratio of crack length to the wavelength of the incident surface wave. The displacements of the analytical solution on the free surface at some distance from the crack are compared with the displacements obtained numerically. Analytical and numerical results show excellent agreement when the Rayleigh wavelength is sufficiently larger than the crack length. The results of this paper should be useful for the quantitative non-destructive evaluation of a near-surface crack. © 2019 Published by Elsevier Ltd.
1. Introduction Near surface cracks are common defects in hard and brittle materials. They are usually induced during precision machining or even ultra-precision machining, and may lead to failures of materials and structures (Li, Zhang & Luo, 2018; Shaw, 1995). Thus, near surface cracks must be detected and characterized, not only for location but also for size, to guide the subsequent machining processes to remove the crack, so as to improve machining efficiency and quality, and reduce manufacturing costs. It is easy to locate the crack, but, to measure its size is still a challenge. One way to characterize a near surface crack is to assess a wave scattered from the crack. Since Rayleigh waves only propagate near the free surface and since they are very sensitive to cracks, they have been employed to locate and characterize surface and near surface defects. Rayleigh waves were first investigated by Rayleigh (1885), and their propagation in an isotropic and homogenous elastic half-space is well understood (Achenbach, 2012; Graff, 2012). Scattering of Rayleigh waves by vertical and horizontal near-surface cracks in an elastic half-space have been investigated by Achenbach and Brind (1981) and Achenbach, Lin and Keer (1983). Domany and Entin-Wohlman (1984) applied a multiple scattering formalism to treat scattering of elastic waves by near surface defects. Other problems of longitudinal and transverse waves scattering from defects or structures can be found in the papers by Jain and Kanwal (1980) and Norris (1986). ∗
Corresponding author. E-mail address: [email protected] (J.D. Achenbach).
https://doi.org/10.1016/j.ijengsci.2019.103162 0020-7225/© 2019 Published by Elsevier Ltd.
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The methods used in these studies require complicated integral transform calculations. It has, however, been shown that by representing scattering from the crack by radiation from equivalent body forces (Yang & Achenbach, 2017, 2018; Zhang & Achenbach, 2015) and using the reciprocity theorem in elastodynamics (Achenbach, 20 0 0, 20 03; 2014; Phan, Cho & Achenbach, 2013; 2014; Rose, Chiu, Nadarajah & Vien, 2017), the scattering problem can be solved while avoiding the integral transform formulations. The reciprocity theorem (Betti, 1872), was discussed in detail in the book by Achenbach (2003). Recently, Yang and Achenbach (2017, 2018) investigated the use of equivalent body forces for a near surface cavity in an elastic body, and obtained the scattered Rayleigh wave in the frequency domain radiating from these equivalent body forces by using the reciprocity approach. Together with the virtual wave technique, the scattering of different elastic wave modes can be solved by selecting suitable virtual wave solutions (Achenbach, 2014). Verification of the Rayleigh wave solutions obtained by the reciprocity theorem was presented by Phan, Cho and Achenbach (2014). They also applied the reciprocity method to determine the scattering field of Rayleigh waves from a surface cavity (Phan et al., 2013). In this paper, the scattering of a Rayleigh wave from a 2D near surface crack is represented by the radiation from equivalent body forces, which take the form of double forces. The amplitude of the scattered field is obtained by using the reciprocity theorem together with a virtual wave technique, by selecting a Rayleigh wave as a virtual wave. The displacements in the far field on the free surface of the analytical solution are compared with the results of the scattering problem obtained from numerical finite element calculations for different ratios of crack length to Rayleigh wavelength. The analytical and numerical results are in excellent agreement, particularly for cases where the Rayleigh wavelength is sufficiently longer than the crack length. In conjunction with experimental results, the analytical calculations presented in this paper provide information on the crack length and the distance of the crack center from the free surface for the low frequency case. The paper is organized as follows: In Section 2, the theoretical framework for the Rayleigh scattering problem in 2D plane strain is presented. The approach for deriving the equivalent body forces for the scattering problem is presented in Section 3, Section 4 discusses the application of the reciprocity problem to obtain the amplitudes of the scattered waves, and Sections 5 and 6 provides analytical and numerical results for the scattering problem for the special case of low frequency and small crack length-to-Rayleigh wavelength ratios. 2. Formulation of the scattering problem Fig. 1 shows a 2D near surface crack which is normal to the free surface of a homogeneous, isotropic, linearly elastic half-space, which is defined by a Cartesian coordinate system (x, z) such that the free surface of the half-space is the plane at z = 0. In Fig. 1, the crack depth is expressed as d, h denotes the distance from the crack center to the free surface, and l is the crack length, where z1 ≤ l ≤ z2 . As shown in Fig. 2(a), a plane Rayleigh wave, uin , propagates along the free surface and interacts with the crack, and generates forward-scattered and backward-scattered Rayleigh waves, usc , which are investigated in this paper. It should
Fig. 1. Schematic of the near surface crack.
Fig. 2. Schematic of the superposition principle for scattering from a near surface crack.
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be noted that the scattered waves include both body waves and Rayleigh waves, but the Rayleigh waves predominate at some distance from the scatterer, since the body waves suffer geometrical attenuation. The total field can be written as utotal (x ) = uin (x ) + usc (x ). For a plane Rayleigh wave the following displacements and corresponding stresses can be found in many textbooks, for example, Achenbach (2003). The displacements of Rayleigh waves may be written as
ux = ±iAinU R (z )e±ikx ,
(1a)
uz = AinW R (z )e±ikx ,
(1b)
where Ain is the amplitude, k is the wave number, k = ω/cR , where ω is the circular frequency and cR is the phase velocity of surface waves. The plus and the minus signs are for propagation in the positive and negative x-direction, respectively. The term e−iωt has been omitted for the convenience of calculation. The functions UR (z) and WR (z) are defined by
U R (z ) = d1 e(−pz ) + d2 e(−qz ) ,
(2a)
W R (z ) = d3 e(−pz ) − e(−qz ) ,
(2b)
where
d1 = − k2 + q2 /(2kp),
(3a)
d3 = ( k2 + q2 )/ ( 2k2 ),
d2 = q/k,
(3b)
and the quantities p and q are given by
2
2
p2 = k2 1 − cR2 /cL2 ,
(4a)
q2 = k 1 − cR2 /cT ,
(4b)
where cT and cL are the phase velocity of transverse and longitudinal waves, respectively. The stress components corresponding to Eq. (1a, b) can be represented by
τxx = Ain Txx (z )e±ikx ,
(5a)
τzz = Ain Tzz (z )e±ikx ,
(5b)
τzx = τxz = ±iAin Txz (z )e±ikx ,
(5c)
where
−qz
−qz
Txx (z ) = μ d4 e−pz + d5 e−qz ,
(6a)
Txz (z ) = μ d6 e−pz + d7 e
(6b)
Tzz (z ) = μ d8 e−pz + d9 e
d4 = k2 + q
2
d5 = −2q,
,
,
(6c)
2
2
2 p2 + k2 − q / 2 pk ,
(7a) (7b)
d6 = k2 + q2 /k,
(8a)
d7 = − k2 + q2 /k,
(8b)
2
d8 = − k2 + q2
/ 2 pk2 ,
(9a)
d9 = 2q,
(9b)
where μ is the shear modulus. In addition, the boundary conditions at z = 0, which are
τzz (x, 0 ) = 0,
(10a)
τzx (x, 0 ) = 0,
(10b)
yield the following well-known equation for the phase velocity of Rayleigh waves
2 − cR2 /cT2
2
− 4 1 − cR2 /cL2
1 / 2
1 − cR2 /cT2
1 / 2
= 0.
(11)
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Fig. 2 shows that the scattered surface waves are equivalent to the waves radiated by applying tractions on the crack surface. These tractions are equal in magnitude but opposite in direction to the corresponding tractions generated by the incident surface wave on the boundary of a virtual crack in the undamaged body. However, it is difficult to rigorously solve for the field scattered by a crack in a half space. Following the papers by Yang and Achenbach (2017, 2018), the crack is represented by a system of equivalent body forces in the undamaged half-space that produces elastodynamic radiation to represent the scattered field. The complicated scattering problem from a crack then changes to a relatively simple radiation problem. The equivalent body forces are calculated in Section 3 in terms of the crack opening volumes due to the incident Rayleigh waves. The reciprocity theorem is then applied to the equivalent system of body forces to obtain the displacement amplitudes of the scattered Rayleigh waves in Section 4. 3. Equivalent body forces As shown in Fig. 2(c), an incident Rayleigh wave loads the crack. This loading can be divided into a normal loading (P0 , given by Eq. 5(a)) and a shear loading (T0 , given by Eq. 5(c)), as shown in Fig. 2(c). To determine the Rayleigh wave radiated from the loaded crack, the tractions on the crack surface are represented by equivalent body forces in the uncracked body. The radiation from the equivalent body forces represents the scattering of the incident wave by the crack. As discussed by Zhang and Achenbach (2015), the equivalent body forces, and hence their radiated (scattered) wave, may be expressed in terms of strain discontinuities due to the incident wave, which, over the length l of the crack may be written in terms of the crack opening displacements ux and uz , respectively, as D εxx (0, z ) = ux (x, z )δ (x ),
(12a)
D εxz (0, z ) = uz (x, z )δ (x ).
(12b)
According to the stress-strain relation: τi j = λεkk δi j + 2μεi j , the corresponding stresses are D τxxD = (λ + 2μ)εxx (0, z ) = (λ + 2μ)ux (x, z )δ (x ),
(13)
D τzzD = λεxx (0, z ) = λux (x, z )δ (x ),
(14)
D τxzD = τzxD = 2μεxz (0, z ) = 2μuz (x, z )δ (x ),
(15)
where λ and μ are the Lamé elastic constants, and δ (x) is the Dirac delta function, the superscript D indicates stresses due to the strain discontinuity. Further integration over z yields the tractions, which are applied at (0, h). We have
τxxD = (λ + 2μ)Vx δ (x )δ (z − h ),
(16)
τzzD = λVx δ (x )δ (z − h ),
(17)
τzxD = τxzD = 2μVz δ (x )δ (z − h ).
(18)
The equivalent body forces are applied at x = 0, z = h, where z = h is the center of the crack, see Fig. 1, h = d + l/2, and
Vx = Vz =
z2
z1
z2 z1
ux dz,
(19a)
uz dz.
(19b)
It should be noted that Vx and Vz are crack opening volumes. In the usual manner, the double forces may be written as
fx = −
∂τxxD = −(λ + 2μ )Vx δ (x )δ (z − h ), ∂x
(20)
fz = −
∂τzzD = −λVx δ (x )δ (z − h ), ∂z
(21)
fzx = −
∂τzxD = −2μVz δ (x )δ (z − h ), ∂z
(22)
fxz = −
∂τxzD = −2μVz δ (x )δ (z − h ), ∂x
(23)
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Fig. 3. Equivalent double forces generated by the incident surface wave at the center of the crack.
where the δ (x) and δ (z − h ) are derivatives of the Dirac delta function, which are physically equivalent to dipoles or double forces. Note that the signs of these double forces have been changed to make the fields generated by Eqs. (20)-(23) applicable to the half-space with the actual crack. As shown in Fig. 3, there are four pairs of equivalent double forces, with fx , fz , fxz and fzx , where fx and fz , which are due to the normal loading, are symmetric with respect to x = 0, and fxz and fzx , which are due to the shear loading, are antisymmetric with respect to x = 0. In the next section, the scattered field are determined by using these equivalent double forces in conjunction with the reciprocity theorem. 4. Application of the reciprocity theorem to determine the scattered field 4.1. Scattered far-field The equivalent double forces with fx and fz are symmetric with respect to x = 0. Thus, the corresponding radiated surface waves are also symmetric with respect to x = 0. The general forms of the corresponding surface wave displacements and stresses components due to the forces fx and fz are
uxfx = ±iA±fx U R (z )e±ikx ,
(24a)
uzfx = A±fx W R (z )e±ikx ,
(24b)
τxxfx = A±fx Txx (z )e±ikx ,
(25a)
τxzfx = ±iA±fx Txz (z )e±ikx ,
(25b)
uxfz = ±iA±fz U R (z )e±ikx ,
(26a)
uzfz = A±fz W R (z )e±ikx ,
(26b)
τxxfz = A±fz Txx (z )e±ikx ,
(27a)
τxzfz = ±iA±fz Txz (z )e±ikx .
(27b)
Due to the antisymmetry of the equivalent forces fxz and fzx with respect to x = 0, the corresponding radiated surface waves are also antisymmetric. Thus, the corresponding displacement and stress components may be written as
uxfzx = iA±fzx U R (z )e±ikx ,
(28a)
uzfzx = ±A±fzx W R (z )e±ikx ,
(28b)
τxxfzx = ±A±fzx Txx (z )e±ikx ,
(29a)
τxzfzx = iA±fzx Txz (z )e±ikx .
(29b)
uxfxz = iA±fxz U R (z )e±ikx ,
(30a)
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uzfz = ±A±fxz W R (z )e±ikx ,
(30b)
τxxfxz = ±A±fxz Txx (z )e±ikx ,
(31a)
τxzfz = iA±fxz Txz (z )e±ikx .
(31b)
where A±f , A±f , A±f and A±f are the amplitudes of the radiated Rayleigh waves (interpreted as scattered Rayleigh waves) due x
z
zx
xz
to the equivalent double forces with fx , fz , fzx and fxz , respectively. The plus and the minus signs in the amplitude denote the amplitudes of forward and back scattered Rayleigh wave. It can be easily verified that A−f = A+f , A−f = A+f for the symmetric x
x
z
z
situation, A−f = −A+f , A−f = −A+f , for the antisymmetric case. The functions UR (z), WR (z), Txx (z) and Txz (z) are defined by xz
xz
zx
zx
Eqs. (2a, b) and (6a, b). The plus and the minus signs apply to scattered Rayleigh waves traveling in the positive and negative x-direction, respectively. In these expressions, the term e−iωt has again been omitted for the convenience of calculation. 4.2. Elastodynamic reciprocity theorem and virtual wave To determine the amplitude of the scattered surface waves, the elastodynamic reciprocity theorem for time-harmonic fields has been employed. This technique has been employed elsewhere (Achenbach, 20 0 0; J. D. 2014; Phan et al., 2013; H. 2014; Rose et al., 2017). For a body of region V and surface S, the reciprocity theorem may be written as
V
fiA uBi − fiB uAi dV =
S
uAi τiBj − uBi τiAj n j dS.
(32)
Here nj are the components of the outward normal to S, and fi , ui and τ ij are the components of body forces, displacements and stresses, respectively. In Eq. (32), the superscripts denote two elastodynamic solutions for the same body, State A and State B. In the present situation, State A is generated by the double forces defined by Eqs. (20)–(23) and the corresponding scattered surface wave fields are defined by Eqs. (24)–(31). State B defines a virtual wave, which is chosen to be a surface wave propagating in the positive x-direction with amplitude B:
V
uBx = iBU R (z )eikx ,
(33a)
uBz = BW R (z )eikx ,
(33b)
τxxB = BTxx (z )eikx ,
(34a)
τxzB = iBTxz (z )eikx .
(34b)
For this application, since there are no body forces for State B, and the left-hand side (LHS) of Eq. (32) reduces to fiA uBi dV . Thus,
for fx : LHS =
fx uBx dV = −(λ + 2μ )Vx kBU R (h ),
V
for fz : LHS =
V
for fzx : LHS =
V
for fxz : LHS =
fz uBz dV = λVx BW R (h )
V
(35)
(36)
fzx uBx dV = 2iμVz BU R (h )
(37)
fxz uBz dV = 2iμVz BkW R (h )
(38)
As for the right-hand side (RHS) of Eq. (32), the contour S is shown in Fig. 3. Thus, the integration has four contributions:
S
uAi τiBj − uBi τiAj n j dS = J1 + J2 + J3 + J4 .
(39)
Since the surface displacements are exponentially small far away from the free surface, we have J2 = 0. The integration vanishes on the free surface due to the traction-free boundary conditions, and thus J4 = 0. In earlier calculations (Achenbach, 2003), it has been shown that the contour integral only yields contributions from counter-propagating waves, so the integration along x = c, 0 ≤ z < ∞, does not produce a contribution, an thus J3 = 0. Hence, the contour integral reduces to RHS = J1 .
C. Wang, O. Balogun and J.D. Achenbach / International Journal of Engineering Science 145 (2019) 103162
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For the symmetric scattered field generated by fx ,
RHS = J1 =
∞ 0
= 2iBA−fx where
I=
+∞ 0
B B uxfx τxx + uzfx τxz − uBx τxxfx − uBz τxzfx
+∞
0
|x=a ∗(−1 )dz
U R (z )Txx (z ) − W R (z )Txz (z ) dz = 2iBA−fx I,
(U R (z )Txx (z ) − W R (z )Txz (z ) )dz.
(40)
(41)
Note that I can be evaluated by Eqs. (2a, b) and (6a, b) and may be presented as
I = μJ,
(42)
where
J=
d1 d4 − d3 d6 d1 d5 + d2 d4 − d3 d7 + d6 d2 d5 + d7 + + , 2p p+q 2q
(43)
and J is dimensionless. Similarly, for the double force fz ,
RHS = 2iBA−fz I.
(44)
For the antisymmetric scattered field generated by fzx ,
RHS = J1 =
∞ 0
= −2iBA−fzx
B B uxfzx τxx + uzfzx τxz − uBx τxxfzx − uBz τxzfzx
+∞
0
|x=a ∗ (−1 )dz
U R (z )Txx (z ) − W R (z )Txz (z ) dz = −2iBA−fzx I
(45)
Similarly, for the double force fxz ,
RHS = −2iBA−fxz I.
(46)
The amplitudes of the scattered surface waves due to these double forces are then obtained as
A−fx =
i(λ + 2μ )Vx kU R (h ) . 2I
A−fz =
−iλVxW R (h ) . 2I
(47)
(48)
A−fzx = − A−fxz = −
μVzU R (h ) I
.
μVz kW R (h ) I
(49)
.
(50)
It can be seen that the scattered amplitude expressions are in terms of the incident wave functions UR (h) and WR (h), the crack opening volumes and the elastic constants. They do not provide information on the shape and orientation of the crack, but for a specific value of h they provide information on the crack opening volumes. 5. Special case of small ratio of crack length to wavelength of the incident Rayleigh wave For the special case of low frequency of the incident Rayleigh wave, and small ratio of crack length to incident Rayleigh wavelength, the tractions generated by the incident Rayleigh wave on the crack surface can be taken as uniform. For this case the crack opening displacements are the static crack opening displacements due to a uniform normal loading P0 and a uniform shear loading T0 which can be obtained from the Handbook of Elasticity Solutions of Kachanov, Shafiro and Tsukrov (2013) as
ux = 2(1 − ν )P0 (l/2 )2 − (z − h )2 /μ,
(51)
uz = 2(1 − ν )T0 (l/2 )2 − (z − h )2 /μ,
(52)
where ν is Poisson’s ratio of the material, and P0 and T0 are defined by Eqs. 5(a) and (c), respectively.
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Thus, by using Eqs. 19(a) and (b), the crack opening volume can be written as
Vx = π (1 − ν )P0 l 2 /4μ,
(53)
Vz = π (1 − ν )T0 l 2 /4μ.
(54)
Submitting Vx and Vz into Eqs. (47)–(50), the amplitude of the scattered Rayleigh wave due to the double forces can be represented by
A−fx =
iπ (1 − ν )2 P0 l 2 kU R (h ) . 4I ( 1 − 2ν )
A−fz =
−iπ ν (1 − ν )P0 l 2W R (h ) . 4I ( 1 − 2ν )
(55)
(56)
A−fzx = − A−fxz = −
π (1 − ν )T0 l 2U R (h ) 4I
.
π (1 − ν )T0 kl 2W R (h ) 4I
(57)
.
(58)
These amplitudes which approximate the amplitudes of the vertical and horizontal displacements of the scattered Rayleigh wave are in terms of the crack length l and the depth of the center of the crack h. Hence one can obtain the crack length and h from a scattered Rayleigh wave obtained in experiments by using these expressions, together with the quantitative measurement of the size of the scattered field obtained by non-destructive evaluation. 6. Comparison of analytical and numerical results For the special case of Section 5, numerical results obtained by a commercial finite element method (FEM) using the ABAQUS package are presented in this section for comparison with the analytical solutions. The material considered is Aluminum, and the corresponding longitudinal wave velocity vL is 6400 m/s, the shear wave velocity vS is 3150 m/s and the Rayleigh wave velocity vR is 2900 m/s. The frequency of the incident Rayleigh wave is set to be 1 MHz, the corresponding wavelength ( ) is 2.9 mm. The amplitude of the incident Rayleigh wave is taken as 10 nm. The incident Rayleigh wave applies shear and normal tractions simultaneously on the crack surface, these tractions are equal in magnitude but opposite in direction to the corresponding stresses generated by the incident Rayleigh wave at the position of the near surface crack in the intact body. For Aluminum and an incident stress field given by Eq. (5), the normal traction is P0 = 75, 997 N/m2 , and the shear traction is T0 = 152, 790 N/m2 . The center point of the crack is fixed at h = 1.45 mm. The crack lengths (l) are taken from 50 μm to 300 μm with an interval of 50 μm. The corresponding ratio of crack length to wavelength varies from
Fig. 4. The amplitude of the scattered waves with different distance from the crack.
C. Wang, O. Balogun and J.D. Achenbach / International Journal of Engineering Science 145 (2019) 103162
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Fig. 5. Analytical and numerical amplitude results versus crack length (l) and crack depth (d): (a) horizontal displacements; (b) vertical displacements.
0.017 to 0.103, which indicates the special case of low frequency. The corresponding crack depth d, shown in Fig. 1, varies from 1.425 mm to 1.3 mm. In the numerical analysis, not only a Rayleigh wave but also bulk waves scatter from the crack. As shown in Fig. 4, the amplitude of the scattered wave from the crack (l = 200 μm) decays with increasing distance from the crack, since the scattered bulk waves suffer geometrical attenuation. As shown in Fig. 4, after 90 mm (about 30 times the wavelength), the amplitude of the scattered Rayleigh wave predominates. For x = 110 mm, where the scattered Rayleigh wave predominates. The results of the numerical analysis together with the analytical solutions, based on the use of equivalent body forces in the uncracked body, are shown in Fig. 5. The amplitudes increase as the ratio of crack length to wavelength increases. The analytical and FEM results show excellent agreement when the crack length is much smaller than the wavelength. This indicates that the method proposed in this paper is valid for low frequency testing. The use of the combination of equivalent body forces with the reciprocity and virtual wave approach provides a simple and convenient method to calculate the amplitude of the scattered Rayleigh wave from a near surface crack, for example, from a tiny subsurface crack produced during the machining processes. When the crack length increases, the two curves slightly diverge. It should be noted that the analytical solution in this paper is equally suitable for a surface-breaking crack. The example would, however, have to be replaced by using the crack opening displacement of a surface-breaking crack, which is different from that of a subsurface crack. 7. Concluding comments In this paper, the amplitude of the far-field Rayleigh wave scattered by an incident Rayleigh wave from a 2D near-surface crack which is normal to the free surface, has been determined. The analytical calculations of the scattering problem show that the Rayleigh wave scattered from the crack can be represented by radiation from equivalent body forces in the undamaged body. These equivalent body forces were calculated in terms of the crack opening volumes which are generated by the incident Rayleigh wave and the depth (h) of the center of the crack. By using the reciprocity theorem together with a virtual surface wave, the amplitudes of the Rayleigh waves radiating from the equivalent body forces were determined. The special case of a small ratio of crack length to Rayleigh wavelength and low frequency was considered, and the displacements at some distance from the crack on the surface of the half-space were obtained by numerical computations and the analytical approach based on equivalent body forces, for different ratios of crack length to Rayleigh wavelength. Analytical and numerical results agree well with each other when the wavelength of the incident Rayleigh wave is much larger than the crack length. The scattered field is expressed in terms of the crack length l and the depth parameter h. The results provide useful information for the quantitative measurement of the length and depth of a near-surface crack. Acknowledgements This work was supported by the National Natural Science Foundation of China (grant numbers 51425504, 51575488, 51605429), Science Fund for Creative Research Groups of National Natural Science Foundation of China (grant number 51521064), Zhejiang Provincial Natural Science Foundation of China LZ13E050 0 01. We appreciate the assistance of Dr. HongCin Liou in numerical calculation. We thank the China Scholarship Council for the funding to support Chuanyong wang’s study at Northwestern University.
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References Achenbach, J. D. (20 0 0). Calculation of wave fields using elastodynamic reciprocity. International Journal of Solids and Structures, 37(46–47), 7043–7053. Achenbach, J. D. (2003). Reciprocity in elastodynamics. Cambridge University Press. Achenbach, J. D. (2014). A new use of the elastodynamic reciprocity theorem. Mathematics and Mechanics of Solids, 19(1), 5–18. Achenbach, J. D., & Brind, R. J. (1981). Scattering of surface waves by a sub-surface crack. Journal of Sound and Vibration, 76(1), 43–56. Achenbach, J. D., Lin, W., & Keer, L. M. (1983). Surface waves due to scattering by a near-surface parallel crack. IEEE Transactions on Sonics and Ultrasonics, 30(4), 270–276. Achenbach, J. (2012). Wave propagation in elastic solids: 16. Elsevier. Betti, E. (1872). Teoria della elasticita. Il Nuovo Cimento (1869-1876), 7(1), 158–180. Domany, E., & Entin-Wohlman, O. (1984). Application of multiple scattering theory to subsurface defects. Journal of Applied Physics, 56(1), 137–142. Graff, K. F. (2012). Wave motion in elastic solids. Courier Corporation. Jain, D. , L., & Kanwal, R. P. (1980). Scattering of elastic waves by an elastic sphere. International Journal of Engineering Science, 18(6), 829–839. Kachanov, M. L., Shafiro, B., & Tsukrov, I. (2013). Handbook of elasticity solutions. Springer Science & Business Media. Li, Z., Zhang, F., & Luo, X. (2018). Subsurface damages beneath fracture pits of reaction-bonded silicon carbide after ultra-precision grinding. Applied Surface Science, 448, 341–350. Norris, A. N. (1986). Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites. International Journal of Engineering Science, 24(8), 1271–1282. Phan, H., Cho, Y., & Achenbach, J. D. (2013). Application of the reciprocity theorem to scattering of surface waves by a cavity. International Journal of Solids and Structures, 50(24), 4080–4088. Phan, H., Cho, Y., & Achenbach, J. D. (2014). Verification of surface wave solutions obtained by the reciprocity theorem. Ultrasonics, 54(7), 1891–1894. Rayleigh, L. (1885). On waves propagated along the plane surface of an elastic solid. Proceedings of the London Mathematical Society, 1(1), 4–11. Rose, L. F., Chiu, W. K., Nadarajah, N., & Vien, B. S. (2017). Using reciprocity to derive the far field displacements due to buried sources and scatterers. The Journal of the Acoustical Society of America, 142(5), 2979–2987. Shaw, M. C. (1995). Precision finishing. CIRP Annals, 44(1), 343–348. Yang, C., & Achenbach, J. D. (2017). Time domain scattering of elastic waves by a cavity, represented by radiation from equivalent body forces. International Journal of Engineering Science, 115, 43–50. Yang, C., & Achenbach, J. D. (2018). Radiation from equivalent body forces for scattering of surface waves by a near-surface cylindrical cavity. In Generalized models and non-classical approaches in complex materials: 2 (pp. 307–328). Cham: Springer. Zhang, H., & Achenbach, J. D. (2015). Use of equivalent body forces for acoustic emission from a crack in a plate. Mechanics Research Communications, 68, 105–108.
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Scattering of a Rayleigh wave by a near surface crack which is normal to the free surface Chuanyong Wang a, Oluwaseyi Balogun b, Jan D. Achenbach b,∗ a b
State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, People’s Republic of China Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA
a r t i c l e
i n f o
Article history: Received 15 July 2019 Revised 28 August 2019 Accepted 31 August 2019
Keywords: Near surface crack Reciprocity theorem Rayleigh wave Equivalent body force
a b s t r a c t In this paper, scattering of a Rayleigh surface wave by a 2D near-surface crack which is normal to the free surface of a homogenous, isotropic and linearly elastic half-space has been investigated. The scattered field is represented by the radiation from equivalent body forces, which are expressed in terms of the crack-opening volumes generated by the incident Rayleigh wave. The reciprocity theorem together with a virtual wave has been used to determine the amplitude of the far field of the scattered Rayleigh wave. A special case of low frequency, and small ratio of the crack length to the wavelength of the incident Rayleigh wave, has been considered to illustrate the method for different values of the ratio of crack length to the wavelength of the incident surface wave. The displacements of the analytical solution on the free surface at some distance from the crack are compared with the displacements obtained numerically. Analytical and numerical results show excellent agreement when the Rayleigh wavelength is sufficiently larger than the crack length. The results of this paper should be useful for the quantitative non-destructive evaluation of a near-surface crack. © 2019 Published by Elsevier Ltd.
1. Introduction Near surface cracks are common defects in hard and brittle materials. They are usually induced during precision machining or even ultra-precision machining, and may lead to failures of materials and structures (Li, Zhang & Luo, 2018; Shaw, 1995). Thus, near surface cracks must be detected and characterized, not only for location but also for size, to guide the subsequent machining processes to remove the crack, so as to improve machining efficiency and quality, and reduce manufacturing costs. It is easy to locate the crack, but, to measure its size is still a challenge. One way to characterize a near surface crack is to assess a wave scattered from the crack. Since Rayleigh waves only propagate near the free surface and since they are very sensitive to cracks, they have been employed to locate and characterize surface and near surface defects. Rayleigh waves were first investigated by Rayleigh (1885), and their propagation in an isotropic and homogenous elastic half-space is well understood (Achenbach, 2012; Graff, 2012). Scattering of Rayleigh waves by vertical and horizontal near-surface cracks in an elastic half-space have been investigated by Achenbach and Brind (1981) and Achenbach, Lin and Keer (1983). Domany and Entin-Wohlman (1984) applied a multiple scattering formalism to treat scattering of elastic waves by near surface defects. Other problems of longitudinal and transverse waves scattering from defects or structures can be found in the papers by Jain and Kanwal (1980) and Norris (1986). ∗
Corresponding author. E-mail address: [email protected] (J.D. Achenbach).
https://doi.org/10.1016/j.ijengsci.2019.103162 0020-7225/© 2019 Published by Elsevier Ltd.
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The methods used in these studies require complicated integral transform calculations. It has, however, been shown that by representing scattering from the crack by radiation from equivalent body forces (Yang & Achenbach, 2017, 2018; Zhang & Achenbach, 2015) and using the reciprocity theorem in elastodynamics (Achenbach, 20 0 0, 20 03; 2014; Phan, Cho & Achenbach, 2013; 2014; Rose, Chiu, Nadarajah & Vien, 2017), the scattering problem can be solved while avoiding the integral transform formulations. The reciprocity theorem (Betti, 1872), was discussed in detail in the book by Achenbach (2003). Recently, Yang and Achenbach (2017, 2018) investigated the use of equivalent body forces for a near surface cavity in an elastic body, and obtained the scattered Rayleigh wave in the frequency domain radiating from these equivalent body forces by using the reciprocity approach. Together with the virtual wave technique, the scattering of different elastic wave modes can be solved by selecting suitable virtual wave solutions (Achenbach, 2014). Verification of the Rayleigh wave solutions obtained by the reciprocity theorem was presented by Phan, Cho and Achenbach (2014). They also applied the reciprocity method to determine the scattering field of Rayleigh waves from a surface cavity (Phan et al., 2013). In this paper, the scattering of a Rayleigh wave from a 2D near surface crack is represented by the radiation from equivalent body forces, which take the form of double forces. The amplitude of the scattered field is obtained by using the reciprocity theorem together with a virtual wave technique, by selecting a Rayleigh wave as a virtual wave. The displacements in the far field on the free surface of the analytical solution are compared with the results of the scattering problem obtained from numerical finite element calculations for different ratios of crack length to Rayleigh wavelength. The analytical and numerical results are in excellent agreement, particularly for cases where the Rayleigh wavelength is sufficiently longer than the crack length. In conjunction with experimental results, the analytical calculations presented in this paper provide information on the crack length and the distance of the crack center from the free surface for the low frequency case. The paper is organized as follows: In Section 2, the theoretical framework for the Rayleigh scattering problem in 2D plane strain is presented. The approach for deriving the equivalent body forces for the scattering problem is presented in Section 3, Section 4 discusses the application of the reciprocity problem to obtain the amplitudes of the scattered waves, and Sections 5 and 6 provides analytical and numerical results for the scattering problem for the special case of low frequency and small crack length-to-Rayleigh wavelength ratios. 2. Formulation of the scattering problem Fig. 1 shows a 2D near surface crack which is normal to the free surface of a homogeneous, isotropic, linearly elastic half-space, which is defined by a Cartesian coordinate system (x, z) such that the free surface of the half-space is the plane at z = 0. In Fig. 1, the crack depth is expressed as d, h denotes the distance from the crack center to the free surface, and l is the crack length, where z1 ≤ l ≤ z2 . As shown in Fig. 2(a), a plane Rayleigh wave, uin , propagates along the free surface and interacts with the crack, and generates forward-scattered and backward-scattered Rayleigh waves, usc , which are investigated in this paper. It should
Fig. 1. Schematic of the near surface crack.
Fig. 2. Schematic of the superposition principle for scattering from a near surface crack.
C. Wang, O. Balogun and J.D. Achenbach / International Journal of Engineering Science 145 (2019) 103162
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be noted that the scattered waves include both body waves and Rayleigh waves, but the Rayleigh waves predominate at some distance from the scatterer, since the body waves suffer geometrical attenuation. The total field can be written as utotal (x ) = uin (x ) + usc (x ). For a plane Rayleigh wave the following displacements and corresponding stresses can be found in many textbooks, for example, Achenbach (2003). The displacements of Rayleigh waves may be written as
ux = ±iAinU R (z )e±ikx ,
(1a)
uz = AinW R (z )e±ikx ,
(1b)
where Ain is the amplitude, k is the wave number, k = ω/cR , where ω is the circular frequency and cR is the phase velocity of surface waves. The plus and the minus signs are for propagation in the positive and negative x-direction, respectively. The term e−iωt has been omitted for the convenience of calculation. The functions UR (z) and WR (z) are defined by
U R (z ) = d1 e(−pz ) + d2 e(−qz ) ,
(2a)
W R (z ) = d3 e(−pz ) − e(−qz ) ,
(2b)
where
d1 = − k2 + q2 /(2kp),
(3a)
d3 = ( k2 + q2 )/ ( 2k2 ),
d2 = q/k,
(3b)
and the quantities p and q are given by
2
2
p2 = k2 1 − cR2 /cL2 ,
(4a)
q2 = k 1 − cR2 /cT ,
(4b)
where cT and cL are the phase velocity of transverse and longitudinal waves, respectively. The stress components corresponding to Eq. (1a, b) can be represented by
τxx = Ain Txx (z )e±ikx ,
(5a)
τzz = Ain Tzz (z )e±ikx ,
(5b)
τzx = τxz = ±iAin Txz (z )e±ikx ,
(5c)
where
−qz
−qz
Txx (z ) = μ d4 e−pz + d5 e−qz ,
(6a)
Txz (z ) = μ d6 e−pz + d7 e
(6b)
Tzz (z ) = μ d8 e−pz + d9 e
d4 = k2 + q
2
d5 = −2q,
,
,
(6c)
2
2
2 p2 + k2 − q / 2 pk ,
(7a) (7b)
d6 = k2 + q2 /k,
(8a)
d7 = − k2 + q2 /k,
(8b)
2
d8 = − k2 + q2
/ 2 pk2 ,
(9a)
d9 = 2q,
(9b)
where μ is the shear modulus. In addition, the boundary conditions at z = 0, which are
τzz (x, 0 ) = 0,
(10a)
τzx (x, 0 ) = 0,
(10b)
yield the following well-known equation for the phase velocity of Rayleigh waves
2 − cR2 /cT2
2
− 4 1 − cR2 /cL2
1 / 2
1 − cR2 /cT2
1 / 2
= 0.
(11)
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Fig. 2 shows that the scattered surface waves are equivalent to the waves radiated by applying tractions on the crack surface. These tractions are equal in magnitude but opposite in direction to the corresponding tractions generated by the incident surface wave on the boundary of a virtual crack in the undamaged body. However, it is difficult to rigorously solve for the field scattered by a crack in a half space. Following the papers by Yang and Achenbach (2017, 2018), the crack is represented by a system of equivalent body forces in the undamaged half-space that produces elastodynamic radiation to represent the scattered field. The complicated scattering problem from a crack then changes to a relatively simple radiation problem. The equivalent body forces are calculated in Section 3 in terms of the crack opening volumes due to the incident Rayleigh waves. The reciprocity theorem is then applied to the equivalent system of body forces to obtain the displacement amplitudes of the scattered Rayleigh waves in Section 4. 3. Equivalent body forces As shown in Fig. 2(c), an incident Rayleigh wave loads the crack. This loading can be divided into a normal loading (P0 , given by Eq. 5(a)) and a shear loading (T0 , given by Eq. 5(c)), as shown in Fig. 2(c). To determine the Rayleigh wave radiated from the loaded crack, the tractions on the crack surface are represented by equivalent body forces in the uncracked body. The radiation from the equivalent body forces represents the scattering of the incident wave by the crack. As discussed by Zhang and Achenbach (2015), the equivalent body forces, and hence their radiated (scattered) wave, may be expressed in terms of strain discontinuities due to the incident wave, which, over the length l of the crack may be written in terms of the crack opening displacements ux and uz , respectively, as D εxx (0, z ) = ux (x, z )δ (x ),
(12a)
D εxz (0, z ) = uz (x, z )δ (x ).
(12b)
According to the stress-strain relation: τi j = λεkk δi j + 2μεi j , the corresponding stresses are D τxxD = (λ + 2μ)εxx (0, z ) = (λ + 2μ)ux (x, z )δ (x ),
(13)
D τzzD = λεxx (0, z ) = λux (x, z )δ (x ),
(14)
D τxzD = τzxD = 2μεxz (0, z ) = 2μuz (x, z )δ (x ),
(15)
where λ and μ are the Lamé elastic constants, and δ (x) is the Dirac delta function, the superscript D indicates stresses due to the strain discontinuity. Further integration over z yields the tractions, which are applied at (0, h). We have
τxxD = (λ + 2μ)Vx δ (x )δ (z − h ),
(16)
τzzD = λVx δ (x )δ (z − h ),
(17)
τzxD = τxzD = 2μVz δ (x )δ (z − h ).
(18)
The equivalent body forces are applied at x = 0, z = h, where z = h is the center of the crack, see Fig. 1, h = d + l/2, and
Vx = Vz =
z2
z1
z2 z1
ux dz,
(19a)
uz dz.
(19b)
It should be noted that Vx and Vz are crack opening volumes. In the usual manner, the double forces may be written as
fx = −
∂τxxD = −(λ + 2μ )Vx δ (x )δ (z − h ), ∂x
(20)
fz = −
∂τzzD = −λVx δ (x )δ (z − h ), ∂z
(21)
fzx = −
∂τzxD = −2μVz δ (x )δ (z − h ), ∂z
(22)
fxz = −
∂τxzD = −2μVz δ (x )δ (z − h ), ∂x
(23)
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Fig. 3. Equivalent double forces generated by the incident surface wave at the center of the crack.
where the δ (x) and δ (z − h ) are derivatives of the Dirac delta function, which are physically equivalent to dipoles or double forces. Note that the signs of these double forces have been changed to make the fields generated by Eqs. (20)-(23) applicable to the half-space with the actual crack. As shown in Fig. 3, there are four pairs of equivalent double forces, with fx , fz , fxz and fzx , where fx and fz , which are due to the normal loading, are symmetric with respect to x = 0, and fxz and fzx , which are due to the shear loading, are antisymmetric with respect to x = 0. In the next section, the scattered field are determined by using these equivalent double forces in conjunction with the reciprocity theorem. 4. Application of the reciprocity theorem to determine the scattered field 4.1. Scattered far-field The equivalent double forces with fx and fz are symmetric with respect to x = 0. Thus, the corresponding radiated surface waves are also symmetric with respect to x = 0. The general forms of the corresponding surface wave displacements and stresses components due to the forces fx and fz are
uxfx = ±iA±fx U R (z )e±ikx ,
(24a)
uzfx = A±fx W R (z )e±ikx ,
(24b)
τxxfx = A±fx Txx (z )e±ikx ,
(25a)
τxzfx = ±iA±fx Txz (z )e±ikx ,
(25b)
uxfz = ±iA±fz U R (z )e±ikx ,
(26a)
uzfz = A±fz W R (z )e±ikx ,
(26b)
τxxfz = A±fz Txx (z )e±ikx ,
(27a)
τxzfz = ±iA±fz Txz (z )e±ikx .
(27b)
Due to the antisymmetry of the equivalent forces fxz and fzx with respect to x = 0, the corresponding radiated surface waves are also antisymmetric. Thus, the corresponding displacement and stress components may be written as
uxfzx = iA±fzx U R (z )e±ikx ,
(28a)
uzfzx = ±A±fzx W R (z )e±ikx ,
(28b)
τxxfzx = ±A±fzx Txx (z )e±ikx ,
(29a)
τxzfzx = iA±fzx Txz (z )e±ikx .
(29b)
uxfxz = iA±fxz U R (z )e±ikx ,
(30a)
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uzfz = ±A±fxz W R (z )e±ikx ,
(30b)
τxxfxz = ±A±fxz Txx (z )e±ikx ,
(31a)
τxzfz = iA±fxz Txz (z )e±ikx .
(31b)
where A±f , A±f , A±f and A±f are the amplitudes of the radiated Rayleigh waves (interpreted as scattered Rayleigh waves) due x
z
zx
xz
to the equivalent double forces with fx , fz , fzx and fxz , respectively. The plus and the minus signs in the amplitude denote the amplitudes of forward and back scattered Rayleigh wave. It can be easily verified that A−f = A+f , A−f = A+f for the symmetric x
x
z
z
situation, A−f = −A+f , A−f = −A+f , for the antisymmetric case. The functions UR (z), WR (z), Txx (z) and Txz (z) are defined by xz
xz
zx
zx
Eqs. (2a, b) and (6a, b). The plus and the minus signs apply to scattered Rayleigh waves traveling in the positive and negative x-direction, respectively. In these expressions, the term e−iωt has again been omitted for the convenience of calculation. 4.2. Elastodynamic reciprocity theorem and virtual wave To determine the amplitude of the scattered surface waves, the elastodynamic reciprocity theorem for time-harmonic fields has been employed. This technique has been employed elsewhere (Achenbach, 20 0 0; J. D. 2014; Phan et al., 2013; H. 2014; Rose et al., 2017). For a body of region V and surface S, the reciprocity theorem may be written as
V
fiA uBi − fiB uAi dV =
S
uAi τiBj − uBi τiAj n j dS.
(32)
Here nj are the components of the outward normal to S, and fi , ui and τ ij are the components of body forces, displacements and stresses, respectively. In Eq. (32), the superscripts denote two elastodynamic solutions for the same body, State A and State B. In the present situation, State A is generated by the double forces defined by Eqs. (20)–(23) and the corresponding scattered surface wave fields are defined by Eqs. (24)–(31). State B defines a virtual wave, which is chosen to be a surface wave propagating in the positive x-direction with amplitude B:
V
uBx = iBU R (z )eikx ,
(33a)
uBz = BW R (z )eikx ,
(33b)
τxxB = BTxx (z )eikx ,
(34a)
τxzB = iBTxz (z )eikx .
(34b)
For this application, since there are no body forces for State B, and the left-hand side (LHS) of Eq. (32) reduces to fiA uBi dV . Thus,
for fx : LHS =
fx uBx dV = −(λ + 2μ )Vx kBU R (h ),
V
for fz : LHS =
V
for fzx : LHS =
V
for fxz : LHS =
fz uBz dV = λVx BW R (h )
V
(35)
(36)
fzx uBx dV = 2iμVz BU R (h )
(37)
fxz uBz dV = 2iμVz BkW R (h )
(38)
As for the right-hand side (RHS) of Eq. (32), the contour S is shown in Fig. 3. Thus, the integration has four contributions:
S
uAi τiBj − uBi τiAj n j dS = J1 + J2 + J3 + J4 .
(39)
Since the surface displacements are exponentially small far away from the free surface, we have J2 = 0. The integration vanishes on the free surface due to the traction-free boundary conditions, and thus J4 = 0. In earlier calculations (Achenbach, 2003), it has been shown that the contour integral only yields contributions from counter-propagating waves, so the integration along x = c, 0 ≤ z < ∞, does not produce a contribution, an thus J3 = 0. Hence, the contour integral reduces to RHS = J1 .
C. Wang, O. Balogun and J.D. Achenbach / International Journal of Engineering Science 145 (2019) 103162
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For the symmetric scattered field generated by fx ,
RHS = J1 =
∞ 0
= 2iBA−fx where
I=
+∞ 0
B B uxfx τxx + uzfx τxz − uBx τxxfx − uBz τxzfx
+∞
0
|x=a ∗(−1 )dz
U R (z )Txx (z ) − W R (z )Txz (z ) dz = 2iBA−fx I,
(U R (z )Txx (z ) − W R (z )Txz (z ) )dz.
(40)
(41)
Note that I can be evaluated by Eqs. (2a, b) and (6a, b) and may be presented as
I = μJ,
(42)
where
J=
d1 d4 − d3 d6 d1 d5 + d2 d4 − d3 d7 + d6 d2 d5 + d7 + + , 2p p+q 2q
(43)
and J is dimensionless. Similarly, for the double force fz ,
RHS = 2iBA−fz I.
(44)
For the antisymmetric scattered field generated by fzx ,
RHS = J1 =
∞ 0
= −2iBA−fzx
B B uxfzx τxx + uzfzx τxz − uBx τxxfzx − uBz τxzfzx
+∞
0
|x=a ∗ (−1 )dz
U R (z )Txx (z ) − W R (z )Txz (z ) dz = −2iBA−fzx I
(45)
Similarly, for the double force fxz ,
RHS = −2iBA−fxz I.
(46)
The amplitudes of the scattered surface waves due to these double forces are then obtained as
A−fx =
i(λ + 2μ )Vx kU R (h ) . 2I
A−fz =
−iλVxW R (h ) . 2I
(47)
(48)
A−fzx = − A−fxz = −
μVzU R (h ) I
.
μVz kW R (h ) I
(49)
.
(50)
It can be seen that the scattered amplitude expressions are in terms of the incident wave functions UR (h) and WR (h), the crack opening volumes and the elastic constants. They do not provide information on the shape and orientation of the crack, but for a specific value of h they provide information on the crack opening volumes. 5. Special case of small ratio of crack length to wavelength of the incident Rayleigh wave For the special case of low frequency of the incident Rayleigh wave, and small ratio of crack length to incident Rayleigh wavelength, the tractions generated by the incident Rayleigh wave on the crack surface can be taken as uniform. For this case the crack opening displacements are the static crack opening displacements due to a uniform normal loading P0 and a uniform shear loading T0 which can be obtained from the Handbook of Elasticity Solutions of Kachanov, Shafiro and Tsukrov (2013) as
ux = 2(1 − ν )P0 (l/2 )2 − (z − h )2 /μ,
(51)
uz = 2(1 − ν )T0 (l/2 )2 − (z − h )2 /μ,
(52)
where ν is Poisson’s ratio of the material, and P0 and T0 are defined by Eqs. 5(a) and (c), respectively.
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C. Wang, O. Balogun and J.D. Achenbach / International Journal of Engineering Science 145 (2019) 103162
Thus, by using Eqs. 19(a) and (b), the crack opening volume can be written as
Vx = π (1 − ν )P0 l 2 /4μ,
(53)
Vz = π (1 − ν )T0 l 2 /4μ.
(54)
Submitting Vx and Vz into Eqs. (47)–(50), the amplitude of the scattered Rayleigh wave due to the double forces can be represented by
A−fx =
iπ (1 − ν )2 P0 l 2 kU R (h ) . 4I ( 1 − 2ν )
A−fz =
−iπ ν (1 − ν )P0 l 2W R (h ) . 4I ( 1 − 2ν )
(55)
(56)
A−fzx = − A−fxz = −
π (1 − ν )T0 l 2U R (h ) 4I
.
π (1 − ν )T0 kl 2W R (h ) 4I
(57)
.
(58)
These amplitudes which approximate the amplitudes of the vertical and horizontal displacements of the scattered Rayleigh wave are in terms of the crack length l and the depth of the center of the crack h. Hence one can obtain the crack length and h from a scattered Rayleigh wave obtained in experiments by using these expressions, together with the quantitative measurement of the size of the scattered field obtained by non-destructive evaluation. 6. Comparison of analytical and numerical results For the special case of Section 5, numerical results obtained by a commercial finite element method (FEM) using the ABAQUS package are presented in this section for comparison with the analytical solutions. The material considered is Aluminum, and the corresponding longitudinal wave velocity vL is 6400 m/s, the shear wave velocity vS is 3150 m/s and the Rayleigh wave velocity vR is 2900 m/s. The frequency of the incident Rayleigh wave is set to be 1 MHz, the corresponding wavelength ( ) is 2.9 mm. The amplitude of the incident Rayleigh wave is taken as 10 nm. The incident Rayleigh wave applies shear and normal tractions simultaneously on the crack surface, these tractions are equal in magnitude but opposite in direction to the corresponding stresses generated by the incident Rayleigh wave at the position of the near surface crack in the intact body. For Aluminum and an incident stress field given by Eq. (5), the normal traction is P0 = 75, 997 N/m2 , and the shear traction is T0 = 152, 790 N/m2 . The center point of the crack is fixed at h = 1.45 mm. The crack lengths (l) are taken from 50 μm to 300 μm with an interval of 50 μm. The corresponding ratio of crack length to wavelength varies from
Fig. 4. The amplitude of the scattered waves with different distance from the crack.
C. Wang, O. Balogun and J.D. Achenbach / International Journal of Engineering Science 145 (2019) 103162
9
Fig. 5. Analytical and numerical amplitude results versus crack length (l) and crack depth (d): (a) horizontal displacements; (b) vertical displacements.
0.017 to 0.103, which indicates the special case of low frequency. The corresponding crack depth d, shown in Fig. 1, varies from 1.425 mm to 1.3 mm. In the numerical analysis, not only a Rayleigh wave but also bulk waves scatter from the crack. As shown in Fig. 4, the amplitude of the scattered wave from the crack (l = 200 μm) decays with increasing distance from the crack, since the scattered bulk waves suffer geometrical attenuation. As shown in Fig. 4, after 90 mm (about 30 times the wavelength), the amplitude of the scattered Rayleigh wave predominates. For x = 110 mm, where the scattered Rayleigh wave predominates. The results of the numerical analysis together with the analytical solutions, based on the use of equivalent body forces in the uncracked body, are shown in Fig. 5. The amplitudes increase as the ratio of crack length to wavelength increases. The analytical and FEM results show excellent agreement when the crack length is much smaller than the wavelength. This indicates that the method proposed in this paper is valid for low frequency testing. The use of the combination of equivalent body forces with the reciprocity and virtual wave approach provides a simple and convenient method to calculate the amplitude of the scattered Rayleigh wave from a near surface crack, for example, from a tiny subsurface crack produced during the machining processes. When the crack length increases, the two curves slightly diverge. It should be noted that the analytical solution in this paper is equally suitable for a surface-breaking crack. The example would, however, have to be replaced by using the crack opening displacement of a surface-breaking crack, which is different from that of a subsurface crack. 7. Concluding comments In this paper, the amplitude of the far-field Rayleigh wave scattered by an incident Rayleigh wave from a 2D near-surface crack which is normal to the free surface, has been determined. The analytical calculations of the scattering problem show that the Rayleigh wave scattered from the crack can be represented by radiation from equivalent body forces in the undamaged body. These equivalent body forces were calculated in terms of the crack opening volumes which are generated by the incident Rayleigh wave and the depth (h) of the center of the crack. By using the reciprocity theorem together with a virtual surface wave, the amplitudes of the Rayleigh waves radiating from the equivalent body forces were determined. The special case of a small ratio of crack length to Rayleigh wavelength and low frequency was considered, and the displacements at some distance from the crack on the surface of the half-space were obtained by numerical computations and the analytical approach based on equivalent body forces, for different ratios of crack length to Rayleigh wavelength. Analytical and numerical results agree well with each other when the wavelength of the incident Rayleigh wave is much larger than the crack length. The scattered field is expressed in terms of the crack length l and the depth parameter h. The results provide useful information for the quantitative measurement of the length and depth of a near-surface crack. Acknowledgements This work was supported by the National Natural Science Foundation of China (grant numbers 51425504, 51575488, 51605429), Science Fund for Creative Research Groups of National Natural Science Foundation of China (grant number 51521064), Zhejiang Provincial Natural Science Foundation of China LZ13E050 0 01. We appreciate the assistance of Dr. HongCin Liou in numerical calculation. We thank the China Scholarship Council for the funding to support Chuanyong wang’s study at Northwestern University.
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