scattering of sh waves by an arbitrarily orientated closed crack

Acta MechDOI 10.1007/s00707-013-0887-1

B. J. Chen · C. L. Du · J. L. Zhang · Z. M. Xiao

Scattering of SH waves by an arbitrarily orientated closedcrack

Received: 6 September 2012 / Revised: 19 February 2013© Springer-Verlag Wien 2013

Abstract The 2D problem of a time-harmonic plane shear horizontal (SH) wave scattered by a finite closedcrack in an isotropic material is presented in the paper. The crack is arbitrarily orientated with regard to theincident wave. A spring model based on the assumption that the traction components on the crack surfaces arelinearly related to the crack opening displacement (COD) is used to model the closed crack. The problem isformulated in a set of boundary integral equations which contains the CODs as unknowns. Numerical examplesare presented for the CODs, elastodynamic stress intensity factors, and the scattered displacement field forvarious parameters, such as spring stiffness, crack sizes and crack orientations. The results show that both thecrack closure and orientation have significant effects on the scattered displacement field for the closed crack.

1 Introduction

Engineering materials often contain various types of micro-defects such as cracks, voids and inhomogenei-ties [1–5]. These flaws may be caused by materials manufacturing, processing and in-service conditions. Theinvestigation of the wave scattering by obstacles of various shapes is the key problem of the theory of non-destructive testing. When an elastic wave propagates through a solid with an obstacle of finite dimensions, ascattered wave is generated. The scattered field includes wave motion reflected by the illuminated side of theobstacle, wave motion which cancels the incident field in the shadow zone, as well as wave motion diffractedinto the shadow zone [6]. The scattered field is the difference between the total field and the incident field. Itdescribes the overall effect generated by the presence of the obstacle in an incident wave field and thus carriesall the information related to the obstacle. The investigation of the scattering field is of particular interest inthe ultrasonic quantitative nondestructive evaluation for detecting and characterizing the location, shape, sizeand orientation of the obstacle, and thus has attracted many researchers [7–13].

Among those defects contained in engineering materials, cracks are the most dangerous ones since theyinduce stress concentrations around the crack tips which can result in the catastrophic failure of key compo-nents. In addition to the crack size, the crack orientation is also a critical parameter which has a significantinfluence on the wave propagation [14–17]. The problem of diffraction of waves by finite cracks is studiedin numerous works by using various analytic and numerical methods. Most of these works are based on theapplication of the method of integral equations. Loeber and Sih [18] considered the scattering of polarizedharmonic shear waves by a sharp crack of finite length under antiplane strain in a homogenous material. Luong

B. J. Chen (B) · C. L. Du · J. L. ZhangData Storage Institute (DSI), A*STAR, 5 Engineering Drive 1, Singapore 117608, SingaporeE-mail: [email protected].: +65-68747825

Z. M. XiaoSchool of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, SingaporeE-mail: [email protected]

B. J. Chen et al.

et al. [19] studied the diffraction of plane waves by a crack perpendicular to the plane interface of two media.The case of a crack located on the plane interface of two anisotropic media was studied by Kuo [20,21]. Thesolutions of the problem of diffraction of shear SH waves by a crack located on the plane interface of twoisotropic homogeneous materials were analyzed by Loeber and Sih [22] and Boström [23]. The problems ofdiffraction of SH waves by systems of cracks located on the plane interfaces of layered materials were solvedby Kundu [24,25]. Kurylyak et al. [26] addressed the problem of diffraction of plane elastic SH waves on acrack of finite width located in the plane interface of two different perfectly joined isotropic elastic materials.

All the above-mentioned studies assumed that the cracks are open. However, the probability of detectingand sizing the crack in solids using ultrasonic techniques is severely reduced in the case that the crack isclosed [27]. The conventional linear ultrasonic techniques overlook the crack closure and thus underestimatethe crack size. This is due to the fact that the scattered wave is weakened by sound transmission through theclosed crack surfaces. Crack closure is a result of individual contact of asperities caused by a mismatch ofthe fracture surfaces. The difficulties encountered in nondestructive evaluation of a closed crack by means ofusual ultrasonic techniques were summarized by Saka and Abe [28].

A thorough understanding of the direct problem that how the crack closure affects the scattered field is anecessary prerequisite to solving the inverse problem of obtaining the crack geometry from the scatter fielddue to crack closure. For a theoretical model of the closed crack, the spring model is often used [27,29–32].A spring model assumes that the traction components on the crack surfaces are linearly related to the crackopening displacement (COD). Maugin et al. [33] considered the interfacial waves in the presence of a slipbetween two dissimilar materials and gave the diffraction patterns for the far-field amplitude for a glass–steelinterface.

Here, we visit the problem of an SH wave scattered by an arbitrarily orientated finite closed crack in anisotropic material by using a spring model. The influences of the crack closure and orientation on the CODs,the dynamical stress intensity factors (SIFs), and the scattering field are investigated and discussed in detail.

2 Problem formulations

The physical problem to be studied is shown in Fig. 1. A crack with a length of 2a and tilted at an angle ofθ relative to the x-axis is located in an infinite homogenous linearly elastic solid. The crack is assumed to bepartially closed. The solid is subjected to an incident plane time-harmonic SH wave coming from the negativey-axis with the displacement ui

z(t), which can be written as

uiz(t) = uz0eiky−iωt . (1)

Fig. 1 An arbitrarily orientated crack of length 2a subjected to an incident SH wave

Scattering of SH waves by an arbitrarily orientated closed crack

In the above equation, uz0 is the wave amplitude, ω is the angular frequency, and k is the wave number definedby

k = ω/ct , ct = √μ/ρ, (2)

where μ is the shear rigidity, and ρ is the mass density of the solid.We are interested in studying how this wave is scattered by the closed crack. The closed crack is modeled

with a spring model [27,31]. It is convenient to solve the current problem in the coordinate system x ′oy′,as shown in Fig. 1, where the nonzero quantities are the displacement component uz and the shear stresscomponents τx ′z, τy′z .

In the coordinate system x ′oy′, Eq. (1) can be written as

uiz(t) = uz0eik(y′cos θ + x ′sin θ)− iωt . (3)

The boundary conditions along the crack plane are:

τy′z(x′, 0+) = τy′z(x

′, 0−),∣∣x ′∣∣ < a, (4)

uz(x′, 0+) = uz(x

′, 0−),∣∣x ′∣∣ > a, (5)

τy′z(x′, 0) = C[uz(x

′, 0+)− uz(x′, 0−)], ∣

∣x ′∣∣ < a (6)

where C is the spring stiffness. It indicates the degree of the crack closure. Zero spring stiffness correspondsto an open crack and infinite spring stiffness to a fully closed crack.

The total wave field can be written as a sum of the incident wave field and the scattered wave field:

uz = uiz + us

z, τy′z = τ iy′z + τ s

y′z, τx ′z = τ ix ′z + τ s

x ′z, (7)

where the superscripts “i” and “s” denote “incident” and “scattered”, respectively. The incident wave field canbe written as in Eqs. (3), (8) and (9),

, τ iy′z(x

′, y′, t) = ikμ cos θuz0eik(y′cos θ + x ′sin θ)− iωt , (8)

τ ix ′z(x

′, y′, t) = ikμ sin θuz0eik(y′cos θ + x ′sin θ)− iωt . (9)

Hereafter, the common term e−iωt that all field quantities contain is dropped for the sake of brevity.The scattered wave field us

z can be expressed as a Fourier integral of the form:

usz(x

′, y′) =+∞∫

−∞f1(s)e

−γ (s)y′eisx ′

ds, y′ > 0, (10)

and

usz(x

′, y′) =+∞∫

−∞f2(s)e

γ (s)y′eisx ′

ds, y′ < 0, (11)

where

γ (s) ={√

s2 − k2, s2 ≥ k2 = (ω/ct )2

−i√

k2 − s2, s2 < k2 = (ω/ct )2. (12)

The traction continuity condition along the crack plane (4) yields

f2 = − f1. (13)

The displacement continuity condition along the crack plane (5) can be rewritten as

uz(x′, 0+) = uz(x

′, 0−),∣∣x ′∣∣ > a, (14.1)

uz(x′, 0+)− uz(x

′, 0−) = D(x ′),∣∣x ′∣∣ < a, (14.2)

where D(x ′) is the COD. By using (23), Eq. (14) arrives at

B. J. Chen et al.

2

+∞∫

−∞f1(s)e

isx ′ds =

{0, |x ′| > a,D(x ′), |x ′| < a (15)

or

f1(s) = 1

4π

+a∫

−a

D(ξ)e−isξ dξ . (16)

The scattered wave field is thus rewritten in terms of COD as

usz(x

′,±y′) = ± 1

4π

+a∫

−a

D(ξ) dξ

+∞∫

−∞e∓γ (s)y′

eis(x ′−ξ) ds, (17)

τ syz(x

′,±y′) = − μ

4π

+a∫

−a

D(ξ) dξ

+∞∫

−∞γ (s)e∓γ (s)y′

eis(x ′−ξ)ds. (18)

The unknown COD function D(ξ) is determined by the spring condition (6), which leads to

μ

4π

+a∫

−a

[G1(x

′; ξ)+ 2C

μG2(x

′; ξ)]

D(ξ) dξ = τ iy′z(x

′, 0), (19)

where

G1(x′; ξ) =

+∞∫

−∞γ (s)eis(x ′−ξ) ds, (20)

G2(x′; ξ) =

+∞∫

−∞eis(x ′−ξ) ds, (21)

τ iy′z(x

′, 0) = ikμ cos θuz0eikx ′sin θ . (22)

Equation (19) is the boundary integral equation (BIE) for the unknown COD function D(ξ). It can besolved numerically.

3 Solution of the problem

The BIE (19) can be numerically solved by using the Galerkin method [12]. Since the stress component at thecrack tip has the same square root singularity as that for the open crack [31], the unknown COD D(ξ) can beexpanded into an infinite series of the form:

D(ξ) = (a2 − ξ2)1/2∞∑

m=1

XmUm−1(ξ/a), (23)

where Xm are the unknown complex numbers to be determined, and Um−1(ξ/a) are the Chebyshev polynomialsof the second kind.


Substituting Eq. (23) into Eq. (19) and performing some manipulations, we obtained the following set oflinear algebraic equations for the unknown Xm :

∞∑

m=1

(A1mn + C∗ A2mn)Xm = Bn, (24)

A1mn =[1+(−1)m + n](−1)m + n

2 mn

⎧⎨

⎩

+∞∫

ka

√s2−k2a2

s2 Jm(s)Jn(s) ds−i

ka∫

0

√k2a2−s2

s2 Jm(s)Jn(s) ds

⎫⎬

⎭,

(25)

A2mn = [1 + (−1)m + n](−1)m + n

2 mn

+∞∫

0

1

s2 Jm(s)Jn(s) ds, (26)

Bn = 4(−1)nkuz0n cos θ[i cos(n − 1)

π

2− sin(n − 1)

π

2

] Jn(ka sin θ)

ka sin θ, (27)

where C∗ = 2aC/μ is a non-dimensional constant describing the contact degree of the closed crack. OnceXm are determined by solving Eq. (24), the COD can be immediately calculated by using Eq. (23).

The dynamic SIFs are then evaluated by

K ±III = lim

x ′→±a

√2π(a ∓ x ′)τ s

y′z(x′, 0), (28)

where “±” denotes the SIFs at the crack tip x ′ = a and x ′ = −a. Substituting Eq. (18) into Eq. (28) and usingthe relationship Um−1(±1) = (±1)m−1m, we obtain

K ±III =

√πaμ

2

∞∑

m=1

(±1)m − 1m Xm . (29)

4 Scattered displacement fields

The scattered far-field represents a plane wave field. It is related to the shape, size, and the orientation of thecrack. The calculation of the scattered far-field is of particular interest to the ultrasonic quantitative nonde-structive evaluation. Once the COD function D(ξ) is solved, the scattered displacement field can be calculatedby Eq. (17). By substituting Eq. (23) into Eq. (17), the scattered displacement field can be expressed as

usz(x

′,±y′) = ± ia

4

∞∑

m=1

m Xmei(3m) π2

+∞∫

−∞

Jm(sa)

sei[±γ̄ (s)y′+sx ′) ds, (30)

where

γ̄ (s) = iγ (s). (31)

For an arbitrary observation point S(x, y) or S(x ′, y′) shown in Fig. 1, the integrations in Eq. (30) can onlybe evaluated numerically. We use commercial software Mathematica to do the numerical integration.

However, if the observation point is far enough from the crack, an asymptotic expression for the scattereddisplacement field can be obtained. Let x ′ = r cosψ, y′ = r sinψ , then in case of r >> a, the integrationsin Eq. (30) can be evaluated analytically by using the method of the stationary phase [34]. The scattered fardisplacement fields are thus written as

usz(r, ψ) = a

4

∞∑

m=1

m sinψJm(ka cosψ)

ka cosψ

(2πka

r/a

)1/2

(ER + i EI ), 0 ≤ ψ ≤ π, (32)

usz(r, ψ) = −a

4

∞∑

m=1

m sinψJm(ka cosψ)

ka cosψ

(2πka

r/a

)1/2

(ER + i EI ), π < ψ < 2π (33)

where

B. J. Chen et al.

ER = − sin

(kr − π

4+ 3mπ

2

)X R

m − cos

(kr − π

4+ 3mπ

2

)X I

m, (34)

EI = cos

(kr − π

4+ 3mπ

2

)X R

m − sin

(kr − π

4+ 3mπ

2

)X I

m, (35)

Xm = X Rm + i X I

m . (36)

The scattered far field can also be directly evaluated numerically for Eq. (30) which is much more time-consuming.

5 Numerical results and discussions

5.1 Verification of results

Before performing the detailed numerical examples on the COD, the dynamic SIFs, and the scattered far-field,we verify our solution by the comparison of our results with the published ones. Figure 2 depicts the amplitudesof the dynamic SIFs at both crack tips varied with the crack orientation angle with ka = 1 and C∗ = 0. TheSIFs are normalized with K 0

III = τ0√πa, where τ0 = kμuz0. In the numerical calculation, the infinite sum

is truncated at m = 20, so that the error due to the truncation is less than 1 %. The calculated SIFs are alittle higher than those published by Huang [35] who investigated the problem of SH waves interacting with afinite crack in a half-space by using the dislocation density function as unknowns and solved the equations byGalerkin’s method with m = 5. However, if we also choose m = 5, the results perfectly agree with each other(Fig. 3).

Ueda et al. [31] used the spring model to study the stress distributions due to a finite closed crack undera normally incident SH wave by using the dislocation density method. Figure 4 shows the amplitude of CODfor low frequency ka = 1 along the crack plane at three different spring constants by letting θ = 0◦ in thecurrent solution. The results agree very well with Fig.10a in reference [31].

5.2 Effect of crack orientations on CODs

Figures 5 and 6 illustrate the effect of crack orientations on CODs for ka = 1 and ka = 5, respectively. Asan example, the spring stiffness is chosen as C∗ = 2. At low ka value, the crack opens nearly as an ellipse ofrevolution, and the asymmetry is not obvious. In addition, the larger the crack orientation angle, the lower theCOD profile; the COD is the largest when the incident wave is perpendicular to the crack surface. However,

Fig. 2 SIFs at both crack tips versus crack orientations with ka = 1 and C∗ = 0: The infinite sum is truncated at m = 20


Fig. 3 SIFs at both crack tips versus crack orientations with ka = 1 and C∗ = 0: The infinite sum is truncated at m = 5

Fig. 4 The amplitude of COD for low frequency ka = 1 along the crack plane (C∗ = 2aC/μ)

Fig. 5 CODs versus crack orientations for ka = 1,C∗ = 2

B. J. Chen et al.

Fig. 6 CODs versus crack orientations for ka = 5,C∗ = 2

Fig. 7 SIFs versus crack orientations for ka = 1,C∗ = 2

Fig. 8 SIFs versus crack orientations for ka = 5,C∗ = 2


Fig. 9 Comparison of the evaluated scattered displacements numerically or analytically for ka = 1,C∗ = 2 at x = 0 for normallyincident case

Fig. 10 Scattering displacement at the observation point (x, y) = (0, 15a) versus crack orientations for ka = 1

at high ka value, waveforms develop on the crack faces; the asymmetry of the COD profile becomes obvious.In both cases, the corresponding CODs vanish for the case of grazing incidence (θ = 90◦). It is noted that thesolution reduces to that of Ueda et al. [31] when θ = 0◦. In the figures, the CODs are normalized by kauz0,where uz0 is the amplitude of the incident wave.

5.3 Effect of crack orientations on SIFs

Figures 7 and 8 depict the effect of crack orientations on the amplitudes of SIFs for ka = 1 and ka = 5,respectively. The spring stiffness is again chosen as C∗ = 2. The SIFs at the crack tip A are higher than thoseat the crack tip B. The SIFs decrease with the increasing of θ at low ka value; however, they oscillate with θ athigh ka value. For grazing incidence case (θ = 90◦), the corresponding SIFs vanish since the incident waveproduces zero-tractions on the crack faces. The SIFs in the figures are normalized with K 0

III = τ0√πa, where

τ0 = kμuz0.

B. J. Chen et al.



5.4 Effect of crack orientations and crack closures on the scattered displacement fields

Once the CODs are obtained by solving the BIE, the scattered displacement fields can be either evaluatedby Eq. (30) numerically or by (32) and (33) analytically if the observation point is at large distance from thecrack. We found that the analytical formula is valid as long as r/a < 1, where r2 = x ′2 + y′2 (see Fig. 9 for anillustration for the normally incident case). Figures 10, 11, 12, 13, 14, 15, 16 and 17 illustrate the calculatedforward and backward scattered displacements based on the analytical expression for the observation pointslocated along the y-axis S(x, y) = S(0, 15a) and S(x, y) = S(0,−15a), respectively, as shown in Fig. 1.

The forward scattering displacements as the functions of crack orientations θ are depicted in Figs. 10, 11, 12and 13 for ka = 1, ka = 3, ka = 5, and ka = 8, respectively. The backward scattering displacements asfunctions of crack orientations θ are depicted in Figs. 14, 15, 16 and 17 for ka = 1, ka = 3, ka = 5,and ka = 8, respectively. For low ka values, the difference between the forward and backward scatteringdisplacements is small in all ranges of the crack orientations, as shown in Figs. 10 and 14. For high ka values,the difference becomes significant. The scattering displacements oscillate with crack orientations. The forwardscattering displacement is generally larger than the corresponding backward scattering displacement. However,the forward and backward scattering displacements are always the same for normal incidence (θ = 0◦) andvanish for grazing incidence (θ = 90◦). Here, the scattering displacements are normalized by uz0.



Fig. 14 Scattering displacement at the observation point (x, y) = (0,−15a) versus crack orientations for ka = 1


B. J. Chen et al.



Fig. 18 Angular dependence of the near-tip scattered displacement field when r/a = 0.1 and ka = 1


It is also shown from Figs. 10, 11, 12, 13, 14, 15, 16 and 17 that the scattered displacement decreaseswith the increasing spring stiffness. This indicates that crack closure decreases the amplitudes of the scattereddisplacement, thus lowers the probability of detecting and sizing the crack in the solid. The measured cracksize is underestimated if the crack closure exists.

5.5 Angular dependence of near-crack-tip field

Once the COD function D(ξ) is solved, the scattered field can be calculated by Eq. (17) for the displacementfield and Eq. (18) for the stress field. As an example, Fig. 18 depicts the angular dependence of the scattereddisplacement field when r/a = 0.1, where r is the radical distance from the right crack tip and α is the anglebetween the observation point and the x ′-axis for the case of normal incidence with ka = 1.

6 Concluding remarks

The scattering problem of a time-harmonic plane SH wave by an arbitrarily orientated finite closed crack isstudied. The closed crack is simulated with a spring model based on the assumption that the traction com-ponents on the crack surfaces are linearly related to the COD. The problem is formulated into a set of BIEswith the CODs as unknowns and solved numerically. The solution is verified through the comparison withthe literature. Numerical examples are illustrated for the CODs, the elastodynamic SIFs, and the scatteredfar-field for various parameters, such as spring stiffness, crack sizes, wave numbers and crack orientations.The results show that the crack orientation has significant effects on crack profile, the dynamical SIFs, as wellas the scattered displacement field for the closed crack. The results also show that crack closure decreases theamplitudes of the scattered displacement, thus lowers the probability of detecting and sizing the crack in thesolid.

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scattering of sh waves by an arbitrarily orientated closed crack

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