Use of equivalent body forces for acoustic emission from a crack in a plate

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Use of equivalent body forces for acoustic emission from a crack in a plate Han Zhang a , Jan D. Achenbach b,∗ a b

Acoustics Institute, Chinese Academy of Sciences, Beijing, China McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208, United States

a r t i c l e

i n f o

Article history: Received 30 January 2015 Received in revised form 6 April 2015 Accepted 7 April 2015 Available online xxx Keywords: Crack Acoustic emission Surface waves Equivalent body forces

a b s t r a c t A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x0 of the free edge comes out as depending on df/dt evaluated at x0 for t > x0 /cR , where cR is the velocity of surface waves on the free edge. © 2015 Published by Elsevier Ltd.

1. Introduction Acoustic emission (AE) is the wave motion produced in a solid body by damage processes, particularly by the opening of a crack in a stressed body. For the opening of a crack the emanating acoustic emission can provide information on the location of the crack and possibly also on its size. For that reason there is a continuing interest in the measurement and interpretation of acoustic emission. For a few recent contributions we refer to Okafor and Natarajan [1] and Zhao et al. [2]. The interpretation of AE requires a measurement model based on a theoretical or numerical analysis of the ultrasonic effects from generation to measurement. A complete analysis of that kind is very complicated due to the variety of effects that may play a role. Clearly it is important that the generation by the opening crack is described in as simple of a manner as possible. In this paper we consider the two-dimensional plane stress problem of acoustic emission of surface waves from a through crack near the free edge of a thin plate. The crack, of length l, is located in a plane normal to the free edge. The acoustic emission actually includes body waves and surface waves, but at some distance the body waves have attenuated and only the surface waves are detectable.

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

To simplify the analysis, it is shown that for wave lengths that are larger than the largest characteristic dimension of the crack we can define equivalent concentrated body forces, which when applied in the absence of the crack, produce the same radiation as is produced by the actual opening of the crack. The system of equivalent body forces consist of two double forces in the plane of the plate. The magnitudes of these double forces are in terms of elastic constants and the crack-opening volume of the crack per unit thickness of the plate, produced in the process of acoustic emission. Analytical expressions for the acoustic emission are obtained by a novel method based on the elastodynamic reciprocity theorem. This theorem is formulated for two states, A and B, where state A is the actual radiation from the concentrated double forces and state B is a suitable virtual wave. This method has been discussed in some detail by Achenbach [3].

2. Acoustic emission The two-dimensional configuration of the crack near a free surface is shown in Fig. 1. The free surface is defined by z = 0 in a Cartesian (x,z) coordinate system. The plate of thickness d is thin, and a state of plane stress defined by ∂/∂y ≡ 0 and  yy = 0 is considered. This implies that the usual elastic constant  must be replace ¯ where by , ¯ = 

2 1 − 2  = 1−  + 2

(1)

http://dx.doi.org/10.1016/j.mechrescom.2015.04.002 0093-6413/© 2015 Published by Elsevier Ltd.

Please cite this article in press as: H. Zhang, J.D. Achenbach, Use of equivalent body forces for acoustic emission from a crack in a plate, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.04.002

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Fig. 1. Acoustic emission from a sub-surface crack.

where v is Poisson’s ratio. The acoustic emission of surface waves is indicated by AE. Prior to crack formation the surface of the impending crack can be thought of as being loaded by tensile stresses xx (0, z) generated by the loading F, which keep the crack closed as if there were no crack. At time t = 0 these tensile stresses are decreasing with time dependence f(t). The crack which is located in the plane x = 0 opens up in Mode I, over a time defined by the function f(t). The crack opening displacements over the length of the crack, l, may be defined by









ux x− , z f (t) = −ux x+ , z f (t)

(2)

It is convenient to convert f(t) to frequency dependence by the exponential Fourier transform

∞ f (t) e+iωt dt

fˆ (ω) =

(3)

0

1 Re 

∞ fˆ (ω) e−iωt dω

(4)

0

The corresponding time-harmonic displacements on the crack faces may then be written as









ux x− , z fˆ (ω) e−iωt = −ux x+ , z fˆ (ω) e−iωt

(5)

We temporarily omit fˆ (ω) and employ displacements as









ux x− , z e−iωt = −u x+ , z e−iωt

(6)

 





COD = ux x+ , z − ux x− , z



= ux (0, z) e−iωt





Vx e

εD (0, z) = ux (x, z) ı (x) e−iωt

(9)

(7)

(10)

D ∂ D ∂zx + zz + P = 0 ∂x ∂z

(11)

where the superscript indicates stresses due to the strain discontinuity, and



ux (0, z) dze

−iωt



D ¯ + 2 ux (x, z) ı (x) xx = 

(12)

D ¯ zz =  u x (x, z) ı (x)

(13)

(8)

l

here l is the length of the crack. For further convenience the term exp (−iωt) is also temporarily omitted in the sequel.



D ¯ + 2 Vx ı (x) ı (z − h) =  ¯ xx

(14)

D ¯ =  V ¯ zz x ı (x) ı (z − h)

(15)

These integrated stresses are now integrated quantities centered at z = h. From Eqs. (10) and (11) we then obtain, as shown in Fig. 2:



 =

D ∂ D ∂xx + xz + Q = 0 ∂x ∂z



e−iωt = 2ux x+ , z e−iωt

The crack opening volume (COV) per unit plate thickness follows

−iωt

In the acoustic emission process the crack can be represented by a system of body forces which, when applied in the absence of the crack, will produce the same radiation as the acoustic emission from the crack. Such body forces are called equivalent body forces. For more general configurations such equivalent body forces have been derived in earthquake seismology, by Burridge and Knopoff [4] and for cracks in structures by Rice [5]. They come out in the form of double forces or dipoles. For the present geometry they have been derived in the usual forms in a somewhat simplified manner. The acoustic emission is considered a field that has been generated by a strain discontinuity which over the length l may be written as

D is not discontinuous. Next we integrate along the The stress zx length of the crack to obtain

The crack opening displacement (COD) is

as

3. Equivalent body forces

where ux (0, z) is defined by Eq. (7). The quasi-static equations of motion for plane stress are used to determine the corresponding body forces P and Q:

and f (t) =

Fig. 2. V with boundary S for application of the elastodynamic reciprocity theorem.



¯ + 2 Vx ı (x) ı (z − h) Q¯ = − 

(16)

 ¯ P¯ = −V x ı (x) ı (z − h)

(17)

where P¯ and Q¯ are body forces per unit plate thickness, integrated over the length of the crack. In Eqs. (16) and (17) the derivative of the delta function indicates a double force.

Please cite this article in press as: H. Zhang, J.D. Achenbach, Use of equivalent body forces for acoustic emission from a crack in a plate, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.04.002

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4. Radiated surface waves

elastodynamic reciprocity theorem for time-harmonic fields. For a region V with boundary S it may be written as,

For the acoustic emission, the displacements and stresses for x > 0 are in the usual surface wave forms uAx (x, z) = iA (ω) U R (z) eikR x

(18)

uAz (x, z) = A (ω) W R (z) eikR x

(19)

A R xx (x, z) = A (ω) Txx (z) eikR x

(20)

A xz

(x, z) =

R iA (ω) Txz

(z) e

ikR x

,

(21)

while for x < 0 we have uAx (x, z) = −iA (ω) U R (z) e−ikR x

(22)

uAz (x, z) = A (ω) W R (z) e−ikR x

(23)

A R xx (x, z) = A (ω) Txx (z) e−ikR x

(24)

A R xz (x, z) = −iA (ω) Txz (z) e−ikR x

(25)

here kR = ω/cR , where cR is the velocity of surface waves, but in the sequel we omit the subscript R and use k for simplicity. The term e−iωt has been omitted in Eqs. (18)–(25). The functions U R (z) and W R (z) are [Achenbach, p. 121], U R (z) = d1 e−pz + d2 e−qz , R

W (z) = d3 e

−pz

−e

−qz

(26)

,

(27)

where p2 = k2 −

ω2

and

cL2

q2 = k2 −

ω2

(28)

cT2

here cL and cT are the velocities of longitudinal and transverse waves, respectively. Also d1 = −

1 2

k2

+ q2 kp

,

q k

d2 =

(29)





V

(z) =  d4 e



+ d5 e

R Txz (z) =  d6 e−pz + d7 e

−qz



dV = S





uAi ijB − uBi ijA nj dS

(35)

(36)

uBz (x, z) = BW R (z) eikx

(37)

B xx

(38)

R BTxx

(x, z) =

(z) e

ikx

B R xz (x, z) = iBTxz (z) eikx .

(39)

The reciprocity theorem given by Eq. (35) is used to calculate the amplitude, A(␻), of the emitted surface waves. The contour of integration is shown in Fig. 2. The contour integral vanishes on the surface z = 0 due to the traction-free boundary conditions, and on z = ∝ because far from the free surface the stresses and displacements from the Rayleigh waves vanish. Note that along x = 0 the terms of the virtual wave contain exp(ika) while terms for the acoustic emission contain exp(−ika) As a consequence, products of these terms are of the form B A R xx ux = −iATxx (z) × BU R (z) .

(40)

Evaluation of the integral along the first contour, x = a, 0≤ z < ∞, then leads to

∞



B A B A A B A B xx ux + xz uz − xx ux − xz uz

=

J3

∞



 

x=a

(−1) dz (41)



R R Txx W R (z) dz (z) U R (z) − Txz

= 2iAB = 2iABI

0

where (32)

∞

d5 = −2q

(34)

For a linearly elastic isotropic solid, the reciprocity theorem has been derived elsewhere (1, p. 41). The reciprocity theorem is an

∞ J1



0

∞

(43)

 

x=b

(−1) dz



(44)

R R R R Txx W R (z) e2ikR b − Txx W R (z) e2ikR b dz (z) U R (z) e2ikR b + Txz (z) U R (z) e2ikR b − Txz

= ABi =0



d1 d4 − d3 d6 d2 d4 + d1 d5 − d3 d7 + d6 d2 d5 + d7 + + , 2p p+q 2q

On the other hand, at x = b both the virtual wave and the acoustic emission contain exp (ikR b) . Now the integration along x = b, 0≤ z ≤ ∞ reduces to

B A B A A B A B xx ux + xz uz − xx ux − xz uz

=

(42)

In integrating Eq. (42) frequency dependence disappears and I may be integrated to yield I = E, where E is dimensionless E=

5. Elastodynamic reciprocity theorem



0

(33)

k2 + q2 d7 = − k



R R R Txx U − Txz W R dz

I=

where

k2 + q2 , d6 = k



(31)

 −qz

 2p2 + k2 − q2 1 2 d4 = k + q2 , 2 pk2



uBx (x, z) = iBU R (z) eikx

(30)

−pz

− fiB uAi

For the present problem the region V is defined by a ≤ x ≤ b, and 0≤ z ≤ ∞. For state A we have body forces defined by Eqs. (16) and (17) and a radiated surface wave fields defined by Eqs. (18)–(25). For State B, the virtual wave, is chosen to be a surface wave propagating in the x-direction with amplitude B:

R R The functions Txx (z) and Txz (z) are [Achenbach, p. 122]



fiA uBi

0

1 k2 + q2 d3 = . 2 k2 R Txx

3

0

integral relation over the interior V of a body, and its boundary S, of the displacements, the surface tractions, and the body forces of two elastodynamic states, State A and State B. We will use the

Thus the integrations along z only provide a contribution when the actual wave and the virtual wave are counter propagating, as they are at x = a.

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6. Surface waves radiated by equivalent forces

Similarly the normal displacement at x0 follows from Eqs. (23) and (52) as

First we consider the case of the double force, Q¯ , defined by Eq. (16). This double force is part of system A, the acoustic emission. System B, the virtual wave, does not have a double force. Thus in the elastodynamic reciprocity theorem given by Eq. (35) the only term which appears in the integration over V is









¯ + 2) Vx ı (z − h) ı,x (x) iBU R (z) eikx dxdz (

LQ = V





¯ + 2 Vx BU R (h) =k 

u¯ Az (x, 0)

 P¯

=−



dW  i ω ¯  Vx W R (0)  fˆ (ω) e−iω(t−(x/cR )) 2I cR dz z=0

Eq. (54) is rewritten as



u¯ Az (x, 0)

 P¯

=−

i ω ¯ Vx D2 D3 fˆ (ω) e−iω 2I cR



t− cx

(54)

 (55)

R

where (45)

D3 = W R (0) = −

1 cR2 2 c2

(56)

T

The integration over S has been considered in the previous section. The result is given as J3 by Eq. (41). The equality of LQ and J3 yields AQ¯ = −

 iω  ¯  + 2 Vx U R (h) 2IcR

(46)

At a position on the surface defined by x0 , the vertical displacement of the acoustic emission in the frequency domain follows from Eqs. (23) and (46) as



uAz (x0 , 0)



=−

Q¯

 iω  ¯  + 2 Vx U R (h) W R (0) e−iω(t−(x0 /cR )) 2IcR (47)

At this point we reintroduce the function fˆ (ω) which was earlier omitted for convenience. Since the crack is close to the free surface, i.e. h  x0 , we simplify the remaining analysis by replacing U R (h) by U R (0). Eq. (47) then simplifies to



uAz (x0 , 0)



 iω  ¯  + 2 Vx D1 fˆ (ω) e−iω(t−(x0 /cR )) =−

Q¯

2IcR

(48)

where

df ↔ −iωfˆ dt

(57)

Here the notation ↔ indicates that the functions df/dt and −iωfˆ are related by the integrals (3) and (4). The second theorem is concerned with time shifting and may be expressed as f (t − t0 ) ↔ fˆ (ω) eiωt0

(58)

By employing Eqs. (57) and (58) we can conclude from Eq. (48) that [uz (x0 , 0, t)]Q¯ =



and from Eq. (55) that [uz (x0 , 0, t)]P¯ =



 df  D1  ¯ x0  + 2 Vx H t−  2IcR cR ds s=t−(x0 /cR ) 



D2 D3 ¯ x0 df   Vx H t−  2IcR cR ds s=t−(x0 /cR )

(59)

(60)

7. Concluding comments

D1 = U R (0) W R (0) = (d1 + d2 ) (d3 − 1) 1 cR2 = 2 c2 T

 1 2

2−

cR2

1−

cT2

cR2

−1/2

−

cL2

1−

cR2

1/2 

cT2

(49)

In a similar manner we can calculate the term in the volume ¯ see Fig. 2. In this case the integral for the case of the double-force P, acoustic emission is again symmetric with respect to x = 0, and the displacements and stresses for case A are defined by Eqs. (18)–(25). We find







dW R ikx ¯ ¯  V dxdz = − V x ı (x) ı,z (z − h) BW (z) e xB dz V

LP =

To complete the conversion of the results from the frequency domain to the time domain we use two well-known theorems from the theory of Fourier transforms. The first one concerns time differentiation. Taking the derivative with respect to time of both sides of Eq. (4) we conclude

R

(50)

In this paper we have presented a method to determine acoustic emission of surface waves from a crack near the free edge of a plate, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x0 of the free edge comes out as depending on df/dt evaluated at x0 for t > x0 /cR , where cR is the velocity of surface waves on the free edge. References

The right hand side of Eq. (35) is defined by J3 . Equality of LP and J3 yields AP¯ =

i ¯ dW R  Vx 2I dz

(51)

As before dW/dz at z = h is replaced by dW/dz at z = 0 and AP becomes AP¯ = − where 1 D2 = 2

iω ¯  Vx D2 2IcR

2−

cR2 cL2

(52)

1−

cR2 cL2

1/2

−

1−

cR2 cT2

1/2 (53)

[1] C. Okafor, S. Natarajan, Acoustic emission monitoring of tensile testing of corroded and un-corroded clad aluminum 2024-T3 and characterization of effects of corrosion on AE source events and material tensile properties, in: D.E. Chimenti, L.J. Bond, D.O. Thompson (Eds.), 40th Annual Review of Progress in Quantitative Nondestructive Evaluation, AIP Proceedings, Melville, N.Y., 2014, pp. 492–500, vol. 58. [2] X. Zhao, K. Qi, Q. Tao, G. Mei, Feasibility of fatigue crack detection and tracking with a multi-sensor in-situ monitoring system, in: D.E. Chimenti, L.J. Bond, D.O. Thompson (Eds.), 40th Annual Review of Progress in Quantitative Nondestructive Evaluation, AIP Proceedings, Melville, N.Y., 2014, pp. 880–887, vol. 58. [3] J.D. Achenbach, Reciprocity in Elastodynamics, Cambridge University Press, Cambridge, UK, 2003. [4] R. Burridge, L. Knopoff, Body-force equivalents for seismic dislocations, Bull. Seism. Soc. Am. 54 (1964) 1901–1914. [5] J.R. Rice, Elastic wave emission from damage processes, J. Nondestr. Eval. 1 (1980) 215–224.

Please cite this article in press as: H. Zhang, J.D. Achenbach, Use of equivalent body forces for acoustic emission from a crack in a plate, Mech. Re. Commun. (2015), http://dx.doi.org/10.1016/j.mechrescom.2015.04.002