Laser excitation of surface wave motion

Journal of the Mechanics and Physics of Solids 51 (2003) 1885 – 1902

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Laser excitation of surface wave motion Jan D. Achenbach∗ McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208, USA

Abstract A simpli*ed analysis of surface wave generation by laser irradiation of a homogeneous, isotropic, linearly elastic body is presented. The thermoelastic process of expansion of a surface element is examined, and a direct derivation of equivalent mechanical surface loading is presented. A novel representation of surface wave motion is given in terms of a single-wave potential for a carrier wave propagating on the free surface. Finally, for time-harmonic laser irradiation the elastodynamic reciprocity theorem is used to relate the generated surface wave motion to a virtual surface wave, which leads to a straightforward analytical determination of the generated surface waves for the cases of laser line- and point-focused illumination. The surface-wave pulses for pulsed irradiation are obtained by Fourier superposition. For the line-focus case the surface-wave pulse is proportional to the laser pulse, while for point-focus illumination the surface-wave pulse consists of a principal pulse followed by a smaller pulse of opposite sign. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Ultrasonics; Laser excitation; Surface wave; Elastic

1. Introduction Surface-breaking or near-surface cracks can be detected by observing their interaction with ultrasonic surface waves. In a conventional approach either re6ected echoes or transmitted signals are monitored in the pulse-echo or pitch-catch modes of operation by using one or more piezoelectric transducers coupled to the surface. An attractive alternative to piezoelectric transducers is to use laser-based ultrasonics (LBU), whereby a high-power pulsed laser is used to generate ultrasound thermoelastically (see, Scruby

∗

Corresponding author. Tel.: +1-847-491-5527; fax: +1-847-491-5227. E-mail address: [email protected] (J.D. Achenbach).

0022-5096/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2003.09.021

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and Drain (1990)) and a laser interferometer is used for detection. Laser-based ultrasonic (LBU) techniques provide a number of advantages over conventional ultrasonic methods, such as higher spatial resolution, non-contact generation and detection of ultrasonic waves, use of *ber optics, narrow- and broad-band generation, absolute measurements, and ability to operate on curved and rough surfaces and in hard-to-access locations. The *rst realistic modeling of laser generation of ultrasound is probably due to Scruby et al. (1980). In their work, a point representation for the laser source was proposed. The authors recognized the fact that in the thermoelastic regime the laser-heated region acts as a surface center of expansion. This model was partly based on some theoretical derivations which Sinclair (1979) had presented for an epicentral waveform due to a surface traction dipole source. Thus, Scruby et al. (1980) related the laser source with a purely mechanical surface source. Rose (1984) gave a rigorous mathematical basis for the point source representation on an elastic half-space by presenting an analysis based on the use of integral transform techniques. He demonstrated rigorously what had earlier been deduced intuitively, namely that by neglecting the eCects of thermal diCusion the laser source could be approximated by a surface center of expansion. The analogous two-dimensional case of line-source excitation has been analyzed by Bernstein and Spicer (2000). The surface-center of expansion model predicts the major features of the waveform and agrees with experiments particularly well for highly focused (small spot size) and short (nanosecond duration) laser pulses, see Aussel et al. (1988) and Hutchins (1988). In the context of thermoelasticity various theories of thermoelasticity oCer a basis for laser ultrasonic modeling in cases where the complete thermoelastic nature of the source may be important for the correct interpretation of experimental results such as near-*eld data. Based, for most part, on the work by McDonald (1990), Spicer (1991) derived a complete model for the laser source. In this paper a self-contained simpli*ed analysis is presented of surface wave generation by laser irradiation, both for line- and point-focused laser beams. The analysis includes the direct determination of the mechanical excitation that is equivalent to laser irradiation and heating of the surface in the thermoelastic range. A novel analysis of surface waves in a homogeneous, isotropic, linearly elastic solid, based on the use of a single potential for a carrier wave on the free surface, is employed. Finally, the actual surface wave generated by laser irradiation is determined for time-harmonic excitation by use of the reciprocity theorem, whereby the two reciprocal solutions are the actually generated surface waves and a “virtual” surface wave. Once surface wave excitation has been obtained for time-harmonic laser irradiation, the surface waves for pulsed radiation are obtained by Fourier superposition. The analysis avoids the more complicated use of integral transform techniques. 2. Governing equations The thermoelastic *elds are governed by the coupled equations of thermoelasticity. Relative to the (x1 ; x2 ; z) coordinate system, shown in Fig. 1, and for a homogeneous,

J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

1887

laser beam

x1

x2 z Fig. 1. Geometry of irradiated half-space.

isotropic, linearly elastic solid, the governing equations are K∇2 T = cV T˙ + T0 ∇ · u˙ − q;

(1)

∇2 u + ( + )∇(∇ · u) = uJ + ∇T;

(2)

where T and T0 are the actual and ambient temperatures, respectively, u(x; t) the displacement vector, K the coeKcient of heat conductivity, cV the speci*c heat at constant deformation, the thermoelastic coupling constant, = (3 + 2), and  the coeKcient of linear thermal expansion. For the generation of ultrasound by laser irradiation, the heat produced by mechanical deformation, given by the term T0 ∇ · u, ˙ can be neglected. With this approximation, Eq. (1) reduces to q ;  = K=cV ; (3) ∇2 T − T˙ = − cV where q represents the heat deposition per unit volume per unit time. Also,  is the thermal diCusivity,  = K=cV ;  being the mass density. For applications in non-destructive evaluation, generation of elastic waves is required in the ultrasonic frequency range. This can be achieved without damage to the specimen’s surface with short-pulsed lasers. The majority of experimental work has employed Q-switched laser pulses of duration of 10 –40 ns. A suitable expression for the heat deposition over a circular area is q = E(1 − Ri )e−z f(r)g(t)

(4)

1 2 −2r 2 =R2G e f(r) = √ 2 R2G

(5)

with

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J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

and g(t) =

8t 3 −2t 2 =2 e : 4

(6)

In these equations E is the energy of the laser pulse, Ri the surface re6ectivity, RG the Gaussian beam radius,  a laser pulse rise-time parameter and  the extinction coeKcient. Eq. (5) represents a disc of illumination since it is de*ned by a Gaussian in r. The Gaussian does not vanish completely with distance, but its value becomes negligible outside a disc. The laser pulse depends on time according to the function proposed by Schleichert et al. (1989). For both temporal and spatial pro*le, the functional dependence has been constructed so that in the limit  → 0 and RG → 0 an equivalent concentrated point source is obtained q = E(1 − Ri )(z)

(r) (t): 2r

(7)

This expression for q also implies the assumption that the energy is absorbed at the surface ( → ∞).

3. Equivalent mechanical loading It is well established that a thermoelastic source at a point in an unbounded medium can be modeled as three mutually orthogonal force dipoles. The magnitude of the dipoles, D, depends on the temperature change and certain mechanical and thermal constants of the material. On the basis of intuitive arguments, Scruby and Drain (1990) assumed that when the source is acting at a point on the surface, the dipole directed along the normal to the surface vanishes and only the dipoles on the surface remain, their strength left unaltered. This assumption does not yield the correct result as discussed by Arias and Achenbach (2003). Here, we employ a simple approach to de*ne the equivalent mechanical loading of the irradiated surface. First, we consider the two-dimensional case. To simplify the heat deposition process, we consider a Gaussian distribution as given by Eq. (5), but with respect to the x1 coordinate, which implies that in the denominator R2G is replaced by RG . We consider the limit RG → 0, and we also assume that all the energy is absorbed at the surface ( → ∞). The expression for q then assumes the following form equivalent to Eq. (4) O − Ri )(x1 )(z)g(t); q = E(1

(8)

where EO is the energy of the laser pulse per unit length in the x2 direction. This expression can be interpreted as the energy deposited in an in*nitesimal element. For an element of length l, width Px2 and depth l3 , the de*nition of q in combination with Eq. (8) implies O − Ri )Px2 g(t) qlPx2 l3 = E(1

or

O − Ri ) q = E(1

1 1 g(t): l l3

(9)

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In addition, heat conduction will be neglected, so that the equation for the temperature reduces to EO 11 T˙ = g(t): (1 − Ri ) cV l l3

(10)

In the next step, we represent g(t) by an integral superposition of harmonic components as:
∞ −i!t g(t) = d!: (11) g(!)e ˆ −∞

In conjunction with Eq. (10) the harmonic components of temperature may then be written as PTˆ = PT

g(!) ˆ e−i!t ; (−i!)

(12)

where PT =

EO 1 1 : (1 − Ri ) l l3 cV

(13)

When the laser illuminates the surface of the half-space, the very thin element undergoes thermal expansion due to a temperature increment de*ned by PT . The harmonic components of PT are given by Eq. (12). Here, we leave out the term [g(!)=(−i!)] ˆ exp(−i!t) not to reintroduce it until towards the end of the paper. Thus, in the next considerations PT is de*ned by Eq. (13). The element is maintained in plane strain in the x2 direction, and since it is located at the free surface the normal stress vanishes. The element is shown in Fig. 2. If the element is removed from the half-space, it can deform freely in the x1 and z directions. We then have 11 = (

11

+

zz )

+ 2

11

zz = (

11

+

zz )

+ 2

zz

− PT = 0;

(14)

− PT = 0:

(15)

Subtraction of one of these equations from the other one, yields 11

=

zz :

(16)

By substituting this result in Eq. (14) we obtain 11

=

zz

=

PT 3 + 2 = PT: 2( + ) 2( + )

(17)

To place the element back into the half-space, we subject the surface element to a compression in the x1 direction with a strain 11 = − PT=2( + ). Since the normal stress still vanishes, we obtain   3 + 2 zz = − PT + zz + 2 zz = 0: (18) 2( + )

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Fig. 2. Heated surface element for line-focus laser-irradiation, with equivalent mechanical loading of a half-plane.

It follows from this equation that zz

=

(3 + 2) PT: 2( + )( + 2)

(19)

The stress 11 is then obtained as 11 = −

2 (3 + 2)PT: + 2

(20)

The reactions to this stress produce forces in the opposite direction on the half-space, that generate the ultrasonic surface waves that will be investigated in this paper. The force on the half-space per unit length in the x2 direction becomes F1+ = 11 l3 =

DO ; l

(21)

where DO =

EO 2 (1 − Ri ): (3 + 2) + 2 cV

(22)

Similarly, a force F1− of the same magnitude, but in the negative x1 direction, works at x1 = −l=2. The reacting forces F1+ and F1− on the surface of the half-space clearly de*ne a double force, or dipole, of magnitude DO per unit length. Next, we consider the axially symmetric case. When the laser irradiates the surface of the half-space the temperature increase of a small disk can be written equivalently

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τzz = 0

dθ r0 l3 F

Fig. 3. Heated surface element for point-focus laser-irradiation of a half-space.

to Eq. (13) as PT =

1 1 E (1 − Ri ) 2 : cV r0 l3

(23)

In this expression E is the energy of the laser pulse, as opposed to EO for the twodimensional case which is the energy per unit length of line irradiation. The thin circular surface element undergoes thermal expansion due to a temperature increment given by Eq. (23). The element is located at the surface and therefore the normal stress in the z direction is zero. The elementary disc is shown in Fig. 3. If the element is removed from the half-plane, it can deform freely in its plane, so that the stresses in the radial and circumferential directions are also zero. The thermal strains in the radial, z and the circumferential directions can then be written as r

=

zz

=

"

= PT:

(24)

To place the disc back into the half-space, we impose a radial pressure which will yield radial and circumferential strains of magnitude −PT , still keeping the normal stress equal to zero. The isothermal Hooke’s law then yields the condition zz = ( + 2) which yields zz

=

zz .

zz

− 2 PT = 0

(25)

Substitution of

2 PT ; + 2

r

= −PT;

"

= −PT

(26)

into Hooke’s law produces the required radial stress as rr = −

2 (3 + 2)PT: + 2

(27)

It should be noted that this expression is just the same as the equivalent stress 11 for the plane strain case, given by Eq. (20), except that here PT is de*ned by Eq. (23).

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J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

Fig. 4. Application of the reciprocity theorem to the two-dimensional case.

The tractions reacting to the compressive stress rr on the disk are interpreted as radial surface tractions with harmonic components D (28) Fr = 2 ; r0 acting along the circle of radius r0 → 0, on the surface of the half-space. Here, D has the same form as given by Eq. (22) for the plane strain case, except that EO is replaced by E. It has been shown by Arias and Achenbach (2003) that, in the limit r0 → 0 the circumferential distribution of radial surface tractions is equivalent to a set of orthogonal dipoles of magnitude D acting on the surface of the half-space. 4. Surface waves generated by laser irradiation As discussed in the previous section, the wave motion generated by laser illumination of a surface can be approximated by the waves generated by an equivalent mechanical loading applied at the surface area of illumination. The analysis of the generated wave motion is a purely isothermal problem of elastodynamics. The application of a surface disturbance to a homogeneous linearly elastic half-space generates a set of body waves and a surface wave. Due to the diCerent rates of geometrical decay, only the surface wave needs to be considered suKciently far from the area of application of a surface disturbance. For ultrasound generation by laser irradiation, the prominence of surface waves in the complete solution can be seen from results presented by Arias and Achenbach (2003, Fig. 5). We will consider both the two-dimensional problem of line illumination and the axially symmetric problem of point illumination. First, the two-dimensional problem which is one of deformation in plane strain. The geometry is shown in Fig. 4. As discussed in the previous section, line source illumination is equivalent to the application of line forces. These forces are taken as time-harmonic, with a view towards later superposition in the frequency domain by using Fourier integrals. Taking the displacements also as time-harmonic with time factor [g(!)=(−i!)] ˆ exp(−i!t) the boundary conditions at z = 0 may be written as     l l z1 = −F1+  x1 − + F1−  x1 + ; (29) 2 2

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Fig. 5. Equivalent axially symmetric surface tractions for point-focus laser-irradiation of a half-space.

zz = 0;

(30)

9T = 0: 9z

(31)

In Eqs. (29)–(31), the displacement and stress components depend only on the spatial coordinates x1 and z. The time-factor is omitted, not to be reintroduced until the time harmonic displacement solutions are used by Fourier superposition to construct the response to a laser pulse of prescribed time dependence. In a conventional approach, an exponential Fourier transform is usually applied with respect to the x1 coordinate, and the resulting system of equations is solved, to yield an inverse transform in the form of an in*nite integral, see e.g. Achenbach (1973) for the usual procedure. The inverse integral cannot be evaluated in exact form, but expressions for the surface waves can be obtained as contributions from poles due to zero’s in the integrand’s denominator. Clearly, this is a long and arduous procedure with some treacherous possibilities for errors. In this paper a very simple technique, earlier proposed by the author (Achenbach, 2000) will be used, based on an application of the reciprocity theorem for elastodynamics in conjunction with the use of a virtual wave. For a region V with boundary S, the elastodynamic reciprocity theorem for two time-harmonic states de*ned by superscripts A and B may be written as:

A B B A B (fj uj − fj uj ) dV = (Bij ujA − A (32) ij uj )ni dS; V

S

where n is the outward normal. For the two-dimensional application, V is the half-plane z ¿ 0, while S is de*ned by z = 0, vertical lines at x1 = a and b, and a horizontal line as z → ∞, whose contribution vanishes. For state A, we select the surface waves that are generated by the surface loads and propagate away from their area of application in the +x1 and −x1 directions. From Eqs. (A.1)–(A.3) these surface waves may be written in the general form u1 (x1 ; z) = ±iAV R (z)e±ikR x1 ;

(33)

uz (x1 ; z) = AW R (z)e±ikR x1 ;

(34)

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J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

where V R (z) and W R (z) are de*ned by Eqs. (A.4) and (A.5). The relevant related stresses follow from Eqs. (A.12) and (A.13) as R 11 (x1 ; z) = AT11 (z)e±ikR x1 ;

(35)

1z (x1 ; z) = ±iAT1zR (z)e±ikR x1 :

(36)

In these expressions exp(−i!t) has been omitted, and kR =!=cR , where cR is the phase velocity of surface waves. The reciprocity relation given by Eq. (32), will now be applied to determine the amplitude of surface waves generated by the time-harmonic surface tractions F1+ and F1− , by using reciprocity with a virtual surface wave. For V we take the domain de*ned by a 6 x1 6 b, z ¿ 0. For the surface waves of state A, the displacements, which are symmetric with respect to x1 = 0, and the stresses are given by Eqs. (33) and (34), where ’ = exp(ikR x1 ) applies for x1 ¿ 0 and ’ = exp(−ikR x1 ) for x1 ¡ 0 For state B, the virtual wave, we select a surface wave propagating in the positive x1 direction u1B (x1 ; z) = iBV R (z)eikR x1 ;

(37)

uzB (x1 ; z) = BW R (z)eikR x1 ;

(38)

R B11 (x1 ; z) = BT11 (z)eikR x1 ;

(39)

B1z (x1 ; z) = iBT1zR (z)eikR x1 :

(40)

For the problem at hand, and considering the contour shown in Fig. 4, the reciprocity relations, Eq. (32), becomes 
0 
∞ 1 ikR l=2 1 −ikR l=2 O = iBD e − e FAB |x1 =b d z − FAB |x1 =a d z: (41) l l ∞ 0 The contribution from the integration along the line at constant z is not included since it vanishes as z → ∞. In Eq. (41), B A FAB (x1 ; z) = u1A B11 + uzA B1z − u1B A 11 − uz 1z :

(42)

Substitution of the relevant displacements and stresses of states A and B in FAB (x1 ; z) yields
0 FAB (b; z) d z = 0; (43) ∞




0

∞

where

FAB (a; z) d z = 2iABI;


I=

0

∞

R [T11 (z)V R (z) − T1zR (z)W R (z)] d z:

(44)

(45)

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By substitution of the expressions for the stresses and the displacements it is found that I = J;

(46)

where J is a generally negative constant, which has been calculated by Achenbach (2000). As has been noted in other calculations, see Achenbach (2000), the integration along x1 = a; 0 6 z ¡ ∞ and x1 = b; 0 6 z ¡ ∞ only yield contributions from counterpropagating waves. By the present choice of the virtual wave, states A and B are counterpropagating at x1 = a; 0 6 z ¡ ∞. Substitution of Eqs. (43) and (44) into Eq. (41) then yields O R (0) = 2iABI; kR BDV

(47)

where the left-hand side has been obtained in the limit of l → 0. The amplitude of the surface waves generated by the dipole follows from Eq. (47) as A(!) =

O R (0) kR DV i! V R (0) DO =− : 2iI cR J 2

(48)

Substitution of A(!) into Eq. (34), and reintroduction of the time-factor yields for the harmonic components of the normal displacement on the surface uˆ z (x1 ; !) = −

1 V R (0)W R (0) DO ±ikR x1 −i!t g(!)e ˆ : cR J 2

(49)

By the de*nition of g(!) ˆ as given by Eq. (18), the result implies that the Rayleigh pulse has the same time dependence as the laser pulse. In fact, by virtue of Eq. (18), we conclude uz (0; t) = −

1 V R (0)W R (0) DO g(t ∓ x1 =cR ): cR J 2

(50)

Next, we consider the axially symmetric problem of ultrasound generation by surface tractions that are equivalent to thermal expansion of a circular surface element. From Eq. (27) we *nd that the relevant boundary condition may be written as zr = −

D (r − r0 ): r02

(51)

The surface wave that is generated by this condition can again be obtained by an application of the reciprocity theorem, now over a cylindrical body, z ¿ 0; 0 6 r 6 b, 0 6 " 6 2 and bounded by z = 0 and r = b. For axial symmetry the carrier wave solutions are given by Eq. (A.21). For state A we select surface waves that propagate away from the circular area of application of the axially symmetric surface tractions, as shown in Fig. 5. These surface waves may be written in the general form urA = −AV R (z)H1(1) (kR r);

(52)

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J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

uzA = AW R (z)H0(1) (kR r);

(53)

where V R (z) and W R (z) are given by Eqs. (A.4) and (A.5), and A(!) is an unknown amplitude factor. The relevant corresponding stresses may be written as (1) A rz = −ATrz (z)H1 (kR r);

(54)

1 (1) (1) O A H (kR r)]: rr = A[Trr (z)H0 (kR r) + T rr (z) kR r 1

(55)

In Eqs. (54) and (55), Trz (z) = T1z (z)

and

Trr (z) = T11 (z);

TO rr (z) = 2kR V R (z);

(56) (57)

where T11 (z) and T1z (z) are de*ned by Eqs. (A.12) and (A.13). For state B, the virtual wave, we select a combination of outgoing and incoming surface waves so that the displacements are bounded at r = 0 1 urB = − BV R (z)[H1(1) (kR r) + H1(2) (kR r)]; 2 1 uzB = BW R (z)[H0(1) (kR r) + H0(2) (kR r)]; 2  1 Brr = B Trr (z)[H0(1) (kR r) + H0(2) (kR r)] 2  1 (1) (2) O +T rr (z) [H (kR r) + H1 (kR r)] ; kR r 1 1 Brz = − BTrz (z)[H1(1) (kR r) + H1(2) (kR r)]: 2

(58) (59)

(60) (61)

Since there are no body forces, the left-hand side of Eq. (32) vanishes. By the use of B A zr , given by Eq. (51), and ur , given by Eq. (58), we then obtain −2

D BV R (0)J1 (kR r0 ) r0
2
∞ A B B A {[urA Brr − urB A =b rr ] + [uz rz − uz rz ]} d" d z: 0

0

(62)

It turns out that the right-hand side of Eq. (62) reduces to a simple expression. Upon substitution of Eqs. (58)–(61) into Eq. (62) we *nd that several terms cancel each other. Eq. (62) then reduces to −2

D R V (0)J1 (kR r0 ) = A(!)IH; r0

(63)

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where by virtue of Eq. (56), the term I is just the same as given by Eq. (45). Also     d (1) d (2) H0 (3) H0(2) (3) − H0(1) (3) H0 (3) H = b : (64) d3 d3 3=kR b The expression for H can be further simpli*ed by using an identity for Hankel functions, see McLachlan (1961, p. 198, 129). The result is H=

4i kR

(65)

and Eq. (63) yields A(!) = −

kR D V R (0) J1 (kR r0 ); 2i r0 J

(66)

where the relation I = J , Eq. (46), has also been used. In the next step, Eq. (66) is further simpli*ed by considering the Bessel function J1 (kR r0 ) at very small values of kR r0 . Then we have 1 1 J1 (kR r0 ) ∼ kR r0 2

(67)

and Eq. (66) reduces to i A(!) = − C(−i!)2 ; 2

(68)

where kR = !=cR has been used, and C=

D 1 V R (0) : 2 cR2 J

(69)

In the frequency domain, the vertical displacement at z = 0 follows from Eq. (53), after restoring the time factor, as i (1) −i!t uˆ z (r; !) = − CW R (0)(−i!)g(!)H ˆ : 0 (kR r)e 2

(70)

This expression is the exponential Fourier transform of uz (r; t). Compatible with the de*nition of g(!) ˆ given by Eq. (18), we write
∞ uz (r; t) = uˆ z (r; !)e−i!t d!: (71) −∞

Substitution of uˆ z (r; !) from Eq. (69) yields
∞ i (1) −i!t uz (r; t) = − CW R (0) (−i!)g(!)H ˆ d!: 0 (!r=cR )e 2 −∞

(72)

To evaluate this integral we use the following representation for the Hankel function (Magnus and Oberhettinger, 1954, p. 28)
2i ∞ ei!(r=cR )s ds: (73) H0(1) (!r=cR ) = −  1 (s2 − 1)1=2

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Fig. 6. Surface-wave pulse for line- and point-focus laser-irradiation of a half-space where g(t) is the laser pulse de*ned by Eq. (6) and  = 10 ns.

Substitution into Eq. (72) yields after a change of the order of integration
∞
∞ 1 1 −i![t−(r=cR )s] uz (r; t) = − CW R (0) (−i!)g(!)e ˆ d! ds:  (s2 − 1)1=2 −∞ 1

(74)

The second integral can be recognized as g˙ at argument t −(r=cR )s. By using the initial condition that g˙ vanishes for negative values of its argument, and after introduction of the new variable tO = t − r=cR , we *nd
g( ˙ tO) d tO 1 D 1 V R (0)W R (0) t−r=cR uz (r; t) = − : (75) 2  2 cR J [(t − tO)2 − (r=cR )2 ]1=2 0 Based on the work of Rose (1984), Berthelot (1994) presented a further simpli*cation of the point-focus pulse referred to as the half-order derivative of g(t). This simpli*cation can be obtained from Eq. (75) by noting that the term [t − tO + rcR ]1=2 gets most of its contribution from the vicinity of the singularity of the integrand at tO = t − r=cR . Hence, we substitute this value for tO in [t − tO + r=cR ]1=2 . Eq. (75) then becomes
g( ˙ tO) d tO 1 D 1 V R (0)W R (0) cR 1=2 t−r=cR uz (r; t) = − : (76) 1=2  2 cR2 J 2r (t − r=c R − tO) 0 In Eqs. (75) and (76), the constant J is generally negative. A comparison of Eqs. (50) and (76) shows that the surface wave pulses for lineand point-focus laser-irradiation are quite diCerent in shape. The two pulses are shown in Fig. 6. For line-focus irradiation the pulse shape is proportional to the laser pulse,

J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

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while for the point-focus case an initial pulse is followed by a smaller pulse with a somewhat extended tail. These pulse shapes agree with observations from experiments. Doyle and Scala (1996, Fig. 2) show surface waveforms of surface waves generated by a laser line source. They note that the pulses are monopolar “with a positive displacement indicating a depression of the surface.” For point source illumination Hutchins (1988, Fig. 26) shows a bipolar waveform similar to the one in Fig. 6. 5. Concluding comments In this paper we have given a simple recipe to determine the mechanical loading of a surface that is equivalent to thermoelastic loading due to surface heating by laser-pulse irradiation. In addition we have presented a simple way to determine the induced surface waves, by an application of elastodynamic reciprocity in conjunction with a suitable virtual wave. Both the cases of surface wave generation by line- and point-source laser irradiation have been considered. The approach presented here can be extended to a class of anisotropic materials, certainly a transversely isotropic solid with the axis of symmetry normal to the free surfaces, to thin *lms with or without a substrate, and to layered solids. Acknowledgements This paper is based on the work partially supported by the OKce of Naval Research under Contract N00014-89-J-1362. It is a pleasure to acknowledge the assistance of Irene Arias. Appendix The author has shown elsewhere (Achenbach, 2000) that surface waves propagating along the free surface of a homogeneous, isotropic, linearly elastic half-space can be expressed in a general form of a carrier wave propagating on the free surface which carries along motions that decay exponentially with distance from the free surface. In terms of the carrier wave, ’(x1 ; x2 ), the displacements may be expressed as u = A

1 R 9’ V (z) (x1 ; x2 ); kR 9xa

uz = AW R (z)’(x1 ; x2 );

(A.1) (A.2)

where  = 1; 2, and 92 ’ 9 2 ’ + 2 + kR2 ’ = 0; 9x2 9x12

kR = !=cR :

(A.3)

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J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

Here, cR is the phase velocity of surface waves. Also V R (z) = d1 e−pz + d2 e−qz ;

(A.4)

W R (z) = d3 e−pz − e−qz :

(A.5)

In Eqs. (A.4) and (A.5), d1 = − d3 =

1 kR2 + q2 ; 2 kR p

d2 =

q ; kR

(A.6)

1 kR2 + q2 : 2 kR2

(A.7)

The quantities p and q are de*ned as p2 = kR2 −

!2 cL2

and

q2 = kR2 −

!2 : cT2

(A.8)

The single constant A in Eqs. (A.1) and (A.2) de*nes the amplitude of the surface wave. The relevant stresses are R 11 = AT11 (z)’(x1 ; x2 ) − A R z = ATz (z)

 R 92 ’ V (z) 2 ; kR 9x2

1 9’ ; kR 9x

(A.9) (A.10)

zz = ATzzR (z)’;

(A.11)

R (z) = [d4 e−pz + d5 e−qz ]; T11

(A.12)

R Tz (z) = [d6 e−pz + d7 e−qz ];

(A.13)

TzzR (z) = [d8 e−pz + d9 e−qz ]:

(A.14)

where

In Eqs. (A.12) and (A.13), 1 2p2 + kR2 − q2 d4 = (kR2 + q2 ) ; 2 pkR2 d6 =

kR2 + q2 ; kR

d8 = −

d7 = −

1 (kR2 + q2 )2 ; 2 pk 2

kR2 + q2 ; kR

d9 = 2q:

d5 = −2q;

(A.15) (A.16) (A.17)

J.D. Achenbach / J. Mech. Phys. Solids 51 (2003) 1885 – 1902

1901

It may be checked that u and uz as given by Eqs. (A.1) and (A.2) satisfy the homogeneous displacement equations of motion. In addition, the surface z = 0 should be free of surface tractions. It is immediately seen that Tz (0) = 0, while Tzz (0) = 0 requires that F(k) = (kR2 + q2 )2 − 4kR2 pq = 0:

(A.18)

By substituting p and q from Eq. (A.8) using ! = kR cR , Eq. (A.18) assumes the better-known form  2 1=2  1=2  c2 c2 c2 2 − R2 − 4 1 − R2 1 − R2 = 0: (A.19) cL cT cT Equation (A.19) is the equation for the phase velocity, cR , of Rayleigh surface waves. The interesting result is that the speci*c dependence on z displayed by Eqs. (A.1) and (A.2), which is well known for the plane-strain case, applies to a general class of surface wave motions in the half-space, where ’(x1 ; x2 ) is the solution of the reduced wave equation (A.3) on the free surface of the half-space. For the cases of plane strain and axial symmetry the carrier waves follow from Eq. (A.3) as ’(x1 ) = e±ikR x1

(A.20)

and ’(r) = H0(1) (kR r)

or

’(r) = H0(2) (kR r);

(A.21)

respectively, where r =(x12 +x22 )1=2 . In conjunction with the term exp(−i!t), Eq. (A.20) represents waves propagating in the positive and negative x1 directions, while Eq. (A.21) represents diverging and converging axially symmetric surface waves. References Achenbach, J.D., 1973. Wave Propagation in Elastic Solids. North-Holland/Elsevier Science Ltd., Amsterdam. Achenbach, J.D., 2000. Calculation of surface wave motions due to a sub-surface point force: an application of elastodynamic reciprocity. J. Acoust. Soc. Am. 107, 1892–1897. Arias, I., Achenbach, J.D., 2003. Thermoelastic generation of ultrasound by laser line-source irradiation. Int. J. Solids Struct., in press. Aussel, J., Brun, A.L., Badoux, J., 1988. Generating acoustic waves by laser: theoretical and experimental study of the emission source. Ultrasonics 25, 245–255. Bernstein, J., Spicer, J., 2000. Line source representation for laser-generated ultrasound in aluminum. J. Acoust. Soc. Am. 107 (3), 1352–1357. Berthelot, Y.H., 1994. Half-order derivative formulation for the analysis of laser-generated Rayleigh waves. Ultrasonics 32 (2), 153–154. Doyle, P., Scala, C., 1996. Near-*eld ultrasonic Rayleigh waves from a laser line source. Ultrasonics 34, 1–8. Hutchins, D., 1988. Ultrasonic generation by pulsed lasers. Physical Acoustics, Vol. XVIII. Academic Press, San Diego. Magnus, W., Oberhettinger, F., 1954. Formulas and Theorems for the Functions of Mathematical Physics. Chelsea Publishing Company, New York.

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McDonald, F.A., 1990. On the precursor in laser-generated ultrasound wave-forms in metals. Appl. Phys. Lett. 56 (3), 230–232. McLachlan, N.W., 1961. Bessel Functions for Engineers. Clarendon Press, Oxford. Rose, L., 1984. Point-source representation for laser-generated ultrasound. J. Acoust. Soc. Am. 75 (3), 723–732. Schleichert, U., Langenberg, K., Arnold, W., Fassbender, S., 1989. A quantitative theory of laser-generated ultrasound. Review of Progress in Quantitative Nondestructive Evaluation, Vol. 8A. Plenum Press, New York, pp. 489 – 496. Scruby, C., Drain, L., 1990. Laser Ultrasonics: Techniques and Applications. Adam Hilger, New York. Scruby, C., Dewhurst, R., Hutchins, D., Palmer, S., 1980. Quantitative studies of thermally-generated elastic waves in laser irradiated metals. J. Appl. Phys. 51, 6210–6216. Sinclair, J., 1979. Epicentre solutions for point multipole sources in an elastic half-space J. Phys. D: Appl. Phys. 12, 1309–1315. Spicer, J., 1991. Laser ultrasonics in *nite structures: comprehensive modelling with supporting experiment. Ph.D. Thesis, The Johns Hopkins University.