Depth evaluation of surface-breaking cracks using laser-generated

JOURNAL OF APPLIED PHYSICS 103, 084911 共2008兲

Depth evaluation of surface-breaking cracks using laser-generated transmitted Rayleigh waves André Moura, Alexey M. Lomonosov, and Peter Hessa兲 Institute of Physical Chemistry, University of Heidelberg, Im Neuenheimer Feld 253, D-69120 Heidelberg, Germany

共Received 19 November 2007; accepted 12 March 2008; published online 30 April 2008兲 A theoretical and experimental study of nondestructive evaluation of surface-breaking cracks with linear surface acoustic wave 共SAW兲 pulses is presented. Schwarz–Christoffel conformal mapping was used to introduce a special orthogonal coordinate system that conserves the profile of the cracked surface. The inverse problem for two dimensions has been solved by means of conformal mapping. Thermoelastically generated broadband SAW pulses were employed to study the scattering of linear SAW pulses by a single crack. The surface wave component transmitted through the isolated microcrack was recorded as a function of distance by the cw laser probe-beam-deflection method. The Fourier transform of the transmitted SAW waveforms provides a stationary solution for any frequency. With this procedure the depth of the crack, produced in a separate experiment with a strongly nonlinear SAW pulse in a silica sample, was evaluated. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2910897兴 I. INTRODUCTION

Although the ultrasonic characterization of surfacebreaking cracks is a classical subject of experimental investigation, there is still no robust, simple, and accurate scheme to solve the inverse problem related to crack depth evaluation. This is true even in the simplified two-dimensional case. Especially the scattering of surface acoustic waves 共SAWs兲 has been extensively studied experimentally and by theoretical simulations.1–3 In these publications real cracks were represented by rectangular slots with separated fracture faces. In many practical cases of fatigue or thermally induced fracture, the cracks have interacting or touching faces, which makes the theoretical treatment more complicated. The theoretical method that is proposed here deals only with the SAW transmitted through the crack, regardless of the incident one. In this case the evolution of the transmitted wave depends on the geometry of the crack and the difficulties encountered in describing the real crack are by-passed. A detailed theoretical and experimental study of the twodimensional problem is the aim of the work presented here. Our main purpose is to describe a methodology that allows the inverse problem related to the characterization of the depth of a surface-breaking crack to be solved. A theory is presented that models the corresponding scattering processes with a novel efficient approach. The direct problem is solved by using an appropriate conformal mapping, the so-called Schwarz-Christoffel mapping. As will be demonstrated, this procedure enables one to introduce a natural and efficient methodology. In contrast to classical approximations in surface wave scattering, there is no limitation involved in the wavelength or in the distance between the crack and the point at which the scattered wave is measured. The theoretical approach is compared with experimental data on the basis of measured SAW profiles transmitted a兲

Electronic mail: [email protected].

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through a finite crack. The laser-induced SAW pulses were launched with a pulsed NIR laser and detected by the cw laser probe-beam-deflection technique. By varying the position of the crack relative to the probe and the source of the Rayleigh wave, the scattered acoustic field was monitored. The part of this field behind the crack, corresponding to the transmitted SAW, was used to estimate the depth of an isolated laser-induced crack. II. THEORY A. Statement of problem

It is well known that Lamb’s problem is suitable for modeling laser-generated ultrasonic waves in materials,4 especially SAWs. As will be discussed below, it is possible to classify a half-plane with a surface crack as a Lamb-type problem. The geometrical details of the problem are depicted in Fig. 1. For the sake of simplicity the optical absorption of laser energy is assumed to be large, and thermal diffusion from the heat source is neglected in our treatment,5 to avoid

FIG. 1. Top and side views of the crack-evaluation problem investigated. 103, 084911-1

© 2008 American Institute of Physics

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unnecessary details in the resulting equations. These conditions normally hold in practice, so the simplifications have essentially no influence on the general validity of the final solution obtained. Let us consider a transient line force at time t, located on the free surface of an isotropic homogeneous elastic halfspace V 共see Fig. 1兲. A straight half-plane crack opens at the free surface with both faces parallel to the line force and orthogonal to the free surface. The crack depth is denoted h and located at a distance d from the laser source at the origin O of the Cartesian coordinate system 共Ox1x2x3兲. The problem is invariant to translation along the line source, which is located along the x2-axis. The linear elastodynamic problem then takes a two-dimensional form. In the Cartesian orthogonal coordinate system and in tensor notation the following initial boundary-value problem models the following elastic wave equation: ui,tt = ␴ij,j,

共1兲

in V,

␴ij = ␭␦ij␧kk + 2␮␧ij,

␧ij = 共ui,j + u j,i兲/2,

␴11 = − F0 f共x1兲g共t兲, or ␴13 = G0 f ,1共x1兲

冕

共2兲

␴13 = 0, t

g共␶兲d␶,

the origin O, where the source acts. In the Ox1x3-plane, S consists of four parts, namely, S1, S−, S+, and S2, which are associated, respectively, with the segment x1 ⬍ d, x3 = 0 on the left-hand side of the crack, the two segments x1 = d⫾, 0 ⬍ x3 ⬍ h, which characterize the lips of the crack, and the segment x1 ⬎ d, x3 = 0 on the right-hand side. The vector n is not defined at the tip and in the corners of the crack, so Eq. 共4兲 is not valid at these points. In the coordinate system introduced here the boundary conditions for the cracked surface can be stated with a single equation. The stress is singular at the crack, and therefore the use of numerical methods to solve this problem by applying the finite difference or finite element method is delicate.6 A very small error at this singular point results in a substantial error in the calculated scattered field. This loss of accuracy also occurs at the two right-angle corners of the crack opening on the free surface. This geometrical complexity renders any boundary-integral method inefficient.7 The purpose of the next section is to rewrite the problem in an appropriate orthogonal coordinate system via Schwarz-Christoffel conformal mapping, which conserves the profile of the cracked surface S, in order to take advantage of some simple mathematical properties emerging from this change in the representation.

␴33 = 0关f共x1兲 = 0,x1

0

艌 d兴

␴i1n1 = 0

on S1 艛 S2 ,

共3兲

on S− 艛 S+ ,

共4兲

where the components of the displacement field ui共x , t兲 are the unknowns and ␴ij, ␧ij, and ␦ij are the components of the stress, strain, and Kronecker tensors, respectively. Einstein’s summation convention of the indices is used, and no difference is made between the covariant and contravariant indices, as we are concerned with orthogonal coordinate systems throughout the whole paper. The indices i and j belong to the set 兵1,3其 corresponding to the Ox1x3-plane. The notations ,i and ,t mean the spatial derivative ⳵ / ⳵xi and time derivative ⳵ / ⳵t, respectively. The scalars ␭ and ␮ are the Lamé constants. The ni are the components of the unit vector normal to S = S− 艛 S+ 艛 S1 艛 S2. The constant modulus F0 of the normal monopolar force has the dimension of a force per unit length, and the modulus G0 of the dipolar force also has the dimension of force per unit length. The functions f and g are distributions, i.e.,

冕

⬁

0

f共x1兲dx1 = 1

and

冕

⬁

g共t兲dt = 1.

共5兲

−⬁

The function f, which represents the normalized spatial energy distribution across the laser line source, is almost a Gaussian centered at the origin O. The quantity F0 共G0兲 is zero in the laser-controlled thermoelastic 共ablation兲 regime. Thus, in the thermoelastic 共ablation兲 regime ␴13 ⫽ 0 and ␴33 = 0 共␴13 = 0 and ␴33 ⫽ 0兲. Moreover, the components ui, ui,j, and ui,t must tend to zero as we let t or the modulus of x, i.e., 兩x兩, approach infinity, and the medium is at rest for negative times. Thus, the elastodynamic field is stressfree everywhere on S except at

B. Conformal mapping between uncracked and cracked half-space

It is possible to find a special orthogonal coordinate system that perfectly matches the presence of an orthogonal single crack to the x1-axis thanks to Schwarz-Christoffel conformal mapping.8 For instance, the following analytic transformation w = x1 + x3i = f共z = x + yi兲 = h共z2 − 1兲1/2 + d maps the upper half-plane Im共z兲 ⬎ 0 onto the upper half-plane Im共w兲 ⬎ 0 slit along the line segment from w = d to w = d + hi, where i is the imaginary unit. The procedure of such a mapping is standard. We intend to rewrite the wave propagation problem in the orthogonal coordinate system defined by the coordinates 共x , y兲, instead of 共x1 , x3兲, with the transformation x1 = ⫾ 冑共− ⍀ + 冑⌬兲/2 + d,

x3 = 冑共⍀ + 冑⌬兲/2,

共6兲

where ⍀ = y 2 − x 2 + h 2,

⌬ = ⍀2 + 4xy,

共7兲

as visualized in Fig. 2. The coordinates of the crack tip E共x1 = d , x3 = h兲 become x = 0 and y = 0, and the whole surface S = S− 艛 S+ 艛 S1 艛 S2 is now described by the unique equation y = 0. Note that the square of the scale factor k of the transform has the following form: k2 =

共x2 + y 2兲共x2 + y 2 + h2兲 + 4x2y 2 . 共x2 + y 2 + h2兲2 + 4x2y 2

共8兲

The elastodynamic in-plane problem described in the previous section can be rewritten by using the usual scalar and vector potentials ␸ and ␺, respectively, via the decomposition of the displacement field

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needed later. It is worth noting that the problem defined by Eqs. 共10兲–共12兲 is similar to the conventional Lamb problem.

C. Characterization of the surface-breaking crack depth from the stationary solution

FIG. 2. 共Color online兲 In the example, the Cartesian coordinates of the crack tip E are 共2, 3兲 in arbitrary units, and the coordinates 共x , y兲 are dimensionless with respect to the crack depth h = 3. Two units separate the laser source O from the crack. The horizontal and vertical straight continuous bold lines represent the free surface and the crack, respectively, and correspond together to y = 0. The dashed bold line corresponds to x = 0. The curves x = const and y = const are represented by dashed and dotted lines, respectively.

u = grad ␸ + rot ␺ 共rot ␾ = 0,

div ␺ = 0兲,

共9兲

where the bold quantities represent vectors, with u ⬅ 共u1 , 0 , u3兲 and ␺ ⬅ 共0 , ␺ , 0兲. Equations 共1兲–共4兲 are replaced by the following ones:

␸,tt = 共cd/kh兲2⌬␸,

␺,tt = 共cs/kh兲2⌬␺,

␴ij = ␭␦ij␧kk + 2␮␧ij,

in V,

␧ij = k 共ui,j + u j,i兲/2, 2

共10兲 共11兲

␭共␸,11 + ␸,22兲 + 2␮共␸,22 + ␺,12兲 = − kF0¯f 共x兲g共t兲,共f¯共x兲 = 0,x 艌 − h兲,

冋

2␮ k2共2␸,12 + ␸,22 − ␺,11兲 − = kG0¯f ,1共x兲

冕

⳵k2 共␸,2 − ␺,1兲 ⳵x

册

t

g共␶兲d␶,

on S,

共12兲

0

where the notation, i means the spatial derivatives ⳵ / ⳵x and ⳵ / ⳵y for i = 1 and 2, respectively, and the scale factor k共x , y兲 on S can be simply written as k共x,0兲 = 兩x兩/冑x2 + h2 .

共13兲

The displacement components are obtained from Eq. 共9兲, ux = 共⳵␸/⳵x + ⳵␺/⳵y兲/k,

uy = 共⳵␸/⳵y − ⳵␺/⳵x兲/k.

共14兲

Note that the covariant or the contravariant derivatives via the Christoffel symbols were used to write Eqs. 共10兲–共14兲.9 From the Cauchy relations for analytic functions it follows that the metric tensor is diagonal. We will not look for the solution of Eqs. 共10兲–共12兲, since only Eq. 共14兲 will be

It is possible to state a solution for the cracked halfspace problem from conformal mapping if we consider the associated stationary problem.10 This corresponds to a source depending on time such that g共t兲 ⬀ e−i␻t, where ␻ is the pulsation. In this case Eqs. 共10兲 and 共14兲 remain unchanged except for the operator ,tt ⬅ ⳵ / ⳵t2 in the two wave equations, which transforms into −␻2. The conformal mapping maps the upper uncracked half-plane Im共z兲 ⬎ 0 onto the upper cracked half-plane Im共w兲 ⬎ 0 slit along the line segment from w = d to w = d + hi. Hence, the solution of the uncracked half-plane problem is also the solution of the cracked halfplane problem but in the new coordinate system, and therefore we can write ¯␸共x1,x3 ; ␻兲 ⬅ ␸共x,y; ␻兲,

¯␺共x ,x ; ␻兲 ⬅ ␺共x,y; ␻兲, 共15兲 1 3

where all the quantities are complex-valued, because they represent a stationary solution proportional to e−i␻t. If in the expressions for ␸ and ␺, which are solutions of Eqs. 共10兲–共12兲, the change of variables 共x , y兲 into 共x1 , x3兲 is performed, we obtain the expressions for ¯␸ and ¯␺, which in turn should satisfy Eqs. 共1兲–共4兲. In the particular case of points at the surface S1 艛 S2, the displacements 共ux , uy兲 are related to 共u1 , u3兲 as follows: ux共x,0; ␻兲 ⬅ u1共x1,0; ␻兲/k共x,0兲, ⬅ u3共x1,0; ␻兲/k共x,0兲.

uy共x,0; ␻兲 共16兲

The surface displacements u1共x1 , 0 ; ␻兲 and u3共x1 , 0 ; ␻兲 on S1 艛 S2 can be directly obtained from experimental measurements in the original coordinate system. A simple Fourier transform in time of the recorded transient waveform that passed through the crack leads to a stationary solution for any arbitrary frequency. For the specific thermoelastic two-dimensional problem, the normal displacement of the Rayleigh wave is monopolar and undergoes no dispersion, whereas for a point-source configuration the displacement decreases proportionally to 1 / 冑x1 in the far field, if the condition ␻x1 / cR ⬎ 1 is fulfilled, where cR is the Rayleigh velocity. We model the thermoelastic source as a transient stress dipole applied to the surface and acting parallel to the surface along the x1-axis. In this case thermodiffusion and finite light absorption are not taken into account. Note that this simplification has no influence on the solution of the inverse problem considered below. Moreover, the surface displacements are largely dominated by the displacements associated with the Rayleigh wave. Now, let us introduce experimental data for the set of frequencies ␻共p兲共p = 1 · · m兲 and detection positions x共q兲 1 共q = 1 · · n兲 at the free surface that are calculated from the transmitted waveform. For every frequency ␻共p兲, the amplitude of the normal displacement U共x共q兲 , ␻共p兲兲, for instance, can be written as

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J. Appl. Phys. 103, 084911 共2008兲

Moura, Lomonosov, and Hess

共p兲 共q兲 共p兲 U共x共q兲 1 , ␻ 兲 = U共x , ␻ 兲

x共q兲

冑关x共q兲兴2 + h2 ,

共x共q兲 1 ⬎ d兲, 共17兲

F共h;q1,2 苸 兵1 ¯ n其兲 =

where Eq. 共16兲 is used with the correspondence between the two systems of coordinates for the points at the free surface, x = + 冑共x1 − d兲2 + h2,

x1 ⬎ d,

x = − 冑共x1 − d兲2 + h2,

x1 ⬍ d.

− 1+ 共18兲

It is worth noting that the distance between the two edges of the crack at the free surface corresponding to x1 → d− and x1 → d+ is essentially zero in the original coordinate system, whereas it is 2h in the space of the transformed coordinate system. We recall here that Eqs. 共10兲–共12兲 model a Lamb-type 1兲 problem. Thus, given the waveforms of the two points x共q 1 共q2兲 and x1 共q = q1 , q2兲, and one frequency, one has U共x共q1兲, ␻共p兲兲 = U共x共q2兲, ␻共p兲兲,

共19兲

in the transformed coordinate system, because the Rayleigh wave is not dispersive in such a formulation. Thus, the transmitted Rayleigh wave inherits the characteristics of the incident one. This enables us to write the following ratio: 共p兲 2兲 Uexp共x共q 1 ,␻ 兲 共p兲 1兲 Uexp共x共q 1 ,␻ 兲

=

冑

1 + 关h/x共q1兲兴2 , 1 + 关h/x共q2兲兴2

共20兲

where the left-hand side is known from experimental data. As long as only the plane Rayleigh waves are involved, the crack depth h can be estimated by minimizing the following functional with respect to h:

F共h,d;q1,2 苸 兵1 · · n其兲 =

1+ − 1+

冨

共p兲 2 2兲 兺 p=1..m关Uexp共x共q 1 , ␻ 兲兴 共p兲 2 1兲 兺 p=1..m关Uexp共x共q 1 , ␻ 兲兴

1 1兲 关共x共q 1

冨

− d兲/h兴2 + 1 , 1

2 2兲 关共x共q 1 − d兲/h兴 + 1

1+

共21兲

where we used Eq. 共18兲 for x1 ⬎ d, and the vertical bars mean the modulus. F does not depend on d, the position of the source with respect to the crack, but only on x1 − d, which is the distance of the observation point from the crack. The evaluation of the crack depth on the basis of Eq. 共21兲 is straightforward. Summation over all frequency components ␻共p兲 reduces the effects of experimental inaccuracy. Finally, let us note a transient approach to the inverse problem by using the well-known Parseval theorem in Fourier analysis, based on the minimization of the following functional:

冨

2 2兲 兰关Sexp共x共q 1 ,t兲兴 dt 2 1兲 兰关Sexp共x共q 1 ,t兲兴 dt

1 1兲 关共x共q 1

冨

− d兲/h兴2 + 1 , 1

2 2兲 关共x共q 1 − d兲/h兴 + 1

共22兲

共q2兲 共q1兲 1兲 where Sexp共x共q 1 , t兲 and Sexp共x1 , t兲 are the waveforms at x1 共q2兲 and x1 , respectively. Note that although the formulation of the direct problem considered above does not take into account interactions of the crack faces, the inverse problem should work even in the presence of interactions. From an elastodynamic point of view, this can be reduced to a linear contact interface problem.11 The linear interaction between the fracture surfaces appears in the transformed coordinate system in the form of spatial dispersion. One can then imagine a flat free surface in the transformed domain, where a small portion of length 2h embodies some dispersion effects that model the elastodynamic wave propagation at a linear-contact interface. Hence, as we are interested in the transmitted Rayleigh wave only, our methodology to solve the inverse problem remains unmodified even in the presence of interactions.

III. EXPERIMENT A. Laser-generated Rayleigh waves and their detection

In the experiments small-amplitude linear SAW pulses propagated through a surface-breaking intrinsic microcrack in isotropic fused silica. The microcrack was generated separately by means of strongly nonlinear SAW pulses developing a steep shock front during propagation. The tensile stress associated with this front gradually grows as the front steepens due to elastic nonlinearity of the material. Dynamic fracture occurs when the critical strength of the material is reached. The characteristic crack size was 艋100 ␮m at the surface. Such cracks have an approximately half-circle shape with a radius of about 50 ␮m, and under certain cracking conditions they are oriented perpendicularly to the sample surface. Linear SAWs were launched by the thermoelastic effect generated by the absorption of 1 ns pulses of a Nd: yttrium aluminum garnet 共YAG兲 laser, operating at the fundamental wavelength of 1.06 ␮m. The source had the shape of a narrow strip, generating a straight-crested SAW pulse with its wave front parallel to the crack. A cw Nd:YAG probe-laser beam 共532 nm兲 was used to detect the acoustic field at the probe spots on the sample surface. The arrangement of source, crack, and probe, employed in the present transmission experiments, is illustrated in Fig. 3. In order to enhance the reflectivity of the cw probe beam and absorption of the pulsed excitation radiation, the sample was coated with a 300 nm aluminum film. Its influence on the SAW pulse waveform was negligible, at least at the in-

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J. Appl. Phys. 103, 084911 共2008兲

Moura, Lomonosov, and Hess

FIG. 3. Scheme of the special configuration of source, crack, and probe investigated in the present experiments.

vestigated source-to-probe distance of 400 ␮m. This distance was fixed throughout the experiments, whereas the position of the crack was scanned in 5 ␮m steps, as indicated in Fig. 3. In principle, a complete picture of the acoustic field scattered at the crack, and also the original undisturbed SAW pulse, can be measured with this setup. Then the resulting scattered field consists of SAWs transmitted through the crack and reflected back from the crack, and bulk modes resulting from mode conversion at the crack. Several of these wave components can be used to extract the crack size, but in this work we focus only on the SAW transmitted through the crack. B. Crack depth evaluation using transmitted SAWs

In the numerical study reported here, we were concerned with the transmitted Rayleigh wave only, i.e., the SAW component that was detected behind the crack, after it was irradiated by the launched Rayleigh pulse. Actually, the thermoelastic source generates transversal and longitudinal waves as well. All these waves are scattered at the crack too, contributing to the surface perturbation, but in this work we assume their contributions to be negligibly small, and therefore they were not taken into account. As the Rayleigh wave passes through the crack, it is significantly attenuated directly behind the crack, but recovers gradually at longer propagation distances, as presented in Fig. 4. Here the measured SAW field is represented in terms of the surface slope.

FIG. 5. Fitting procedure for estimation of the crack depth. The squares and the continuous curve represent the dimensionless experimental data Rexp共␹共k兲兲 and the function f共␹兲 = 关1 + 1 / 共10/ 67兲2 + 1兴 / 关1 + 1 / 共␹ / 67兲2 + 1兴, respectively, with the estimate h = 67 ␮m.

When the SAW hits the crack, the surface motion is significantly enhanced at both edges, exhibiting narrow peaks in Fig. 4. The growth rate of the transmitted SAW amplitude behind the crack depends on its size, and has been used for its evaluation. The experimental frequency range was 5 – 200 MHz at the −20 dB level, which corresponds to SAW wavelengths from about 17 to 700 ␮m. This range approximately defines the range of crack sizes that can be studied. The transmitted acoustic field was measured at 60 locations behind the crack, spaced by 5 ␮m. The first one was at a distance of 10 ␮m from the crack. The waveform measured at this point was used as a reference, so that all other 59 waveforms were related to the reference using Eq. 共22兲. The distances were taken with respect to the crack, ␹ = 共x1 − d兲. Minimization of Eq. 共21兲 or 共22兲 was obtained from the following ratios: Rexp共␹共q2兲兲 =

兰关Sexp共␹共q2兲,t兲兴2dt , 兰关Sexp共␹共q1兲,t兲兴2dt

共23兲

for different positions ␹共q2兲, q2 = 2 · · 60, but with respect to the same waveform at q1 = 2, with the correspondence

␹共k兲 1 = k ⫻ 5 ␮m + 5 ␮m,

共24兲

k = 2 ¯ 60.

Finally the following functional was minimized with respect to h:

60

F共h兲 = 兺

k=2

FIG. 4. Measured SAW field, scattered at the crack, providing the dependence of the waveform on the location relative to the crack.

冨

冨

1 关␹ /h兴2 + 1 Rexp共␹共k兲兲 − . 1 1 + 共k兲 2 关␹ /h兴 + 1 1+

共2兲

共25兲

The use of the golden-section method results in a crack depth of about 67 ␮m with a residual ⬍0.05 per point. The corresponding fit is shown in Fig. 5, where the data with square markers and the continuous curve correspond to Rexp共␹共k兲兲 and

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084911-6

J. Appl. Phys. 103, 084911 共2008兲

Moura, Lomonosov, and Hess

冋

f共␹兲 = 1 +

1 共10/67兲2 + 1

册冒冋

1+

册

1 , 共␹/67兲2 + 1

共26兲

respectively. The physical meaning of these quantities corresponds to the dimensionless energy of the wave. Although the fit is quite good, its accuracy can be further increased with the choice of a more suitable metric that better matches the real crack shape. The simplified model applied here to describe the crack is very rough and is certainly still far from reality but represents a good starting point for further generalizations. In the vicinity of the crack the metric takes on a more complex form than the one considered above. This is why detection points less than 10 ␮m from the crack were avoided. Furthermore, for detection points far from the crack line, the finite aspect of the crack must be taken into account, and thus the metric should be rewritten in three-dimensional space. That is why distances larger than 300 ␮m were not considered. These aspects, however, were secondary in the present study, where we were primarily interested in a rough estimate of the crack depth. The two-dimensional geometry considered in this paper describes the interaction of a straight-crested surface wave with an infinite crack of constant depth, which is parallel to the wave front. The finite size of the crack affects the transmitted wave even in the near field, where diffraction effects are negligible. Let us estimate, as an example, the change caused by the finite size of a half-circular 共or half-pennyshaped兲 crack similar to that depicted in Fig. 3, with the radius h and its center on the upper surface. We compare the surface wave transmitted through the center of such a crack with the two-dimensional solution, i.e., the SAW transmitted through the infinite crack with the same depth h. For this particular symmetric geometry the transmitted wave V3D can be built on the basis of the two-dimensional solution V2D. Both waves depend on the crack size normalized by the wavelength V = V共h / ␭兲. Making use of the symmetry one can obtain the following expression for the wave behind the halfpenny crack: V3D

冉冊 冕 h = ␭

␲/2

−␲/2

V2D

冋

册

h cos共␸兲 cos共␸兲d␸ . ␭

共27兲

The function V2D共x兲 has been calculated analytically and numerically in several publications, for example.3,12 At the low-frequency limit 共x → 0兲 it tends to unity and for higher frequencies it asymptotically decays as a + 1 / x,13 with a small constant a ⬇ 0.08.14 For the sake of simplicity we calculate V3D just for these two limiting cases V3D

V3D

冉冊 冉冊

冉冊 冉冊

h h = 2V2D , ␭ ␭

h Ⰶ 1, ␭

h h = ␲V2D , ␭ ␭

h Ⰷ 1. ␭

共28兲

Thus, the half-circular crack enhances the transmitted field in comparison to an infinitely long crack of the same depth by a factor of about 2 for small cracks or 3.14 for relatively large cracks. This focusing effect occurs because of the symmetry of the problem; all the partial waves scattered at the edges of the circular crack arrive at the observation point with the same phase, in contrast to the case of an infinite crack. Note that this “focusing process” takes place at every point along the central line of the circle, in other words the wave is enhanced everywhere on this line. Departure from circularity reduces focusing, so Eqs. 共28兲 provide an upper estimate for this effect. The SAW pulse behind the crack measured in the present experiments was amplified by the focusing effect, and the transmission was apparently greater than the theoretical prediction. Moreover, the frequency dependence of the transmission is somewhat weaker for the half-penny crack, as follows from Eq. 共28兲. IV. CONCLUSIONS

A methodology has been developed to solve the inverse problem related to the characterization of the depth of surface-breaking cracks using laser-generated Rayleigh waves. First, the theory that models the inherent scattering processes was presented. The specific method of solution uses an appropriate conformal mapping, so-called Schwarz– Christoffel mapping, which enabled us to introduce a natural and efficient methodology to solve the inverse problem. However, the two-dimensional problem considered theoretically corresponds to an infinite laser-generated line source and a half-plane breaking crack. Hence edge effects that are due to the bounded nature of the crack are not taken into consideration. Work is currently in progress to envisage the corresponding three-dimensional problem. ACKNOWLEDGMENTS

Financial support of this work by the Deutsche Forschungsgemeninschaft 共DFG兲 is gratefully acknowledged. 1

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