Analysis of acoustic emission caused by internal cracks

Engineering Fracture Mechanics 68 (2001) 1317±1333

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Analysis of acoustic emission caused by internal cracks Olexandr Ye. Andreykiv *,1, Mykola V. Lysak 2, Oleh M. Serhiyenko 3, Valentyn R. Skalsky 4 Department of Structural Strength of Materials in Operating Environments, Karpenko Physico-mechanical Institute of the National Academy of Sciences of Ukraine, 5, Naukova Str., Lviv 79601, Ukraine Received 9 May 2000; accepted 29 December 2000

Abstract A calculation model for determination of acoustic emission (AE) caused by formation and subcritical growth of an internal crack in local areas near its contour is proposed. The problem is divided into the following ones: (1) instant formation in the previously loaded elastic body of a penny-shaped crack; a microcrack formation at the macrocrack front; (2) an interaction of elastic waves caused by the crack with the elastic half-space boundary and (3) the e€ect of a cylindrical waveguide on AE signal parameters. Using the obtained solutions the dependencies between amplitudes of AE signal and the area of subcritical crack growth are established. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Acoustic emission; Transducer; Crack; Displacement vector; Waveguide

1. Introduction Important scienti®c and technical problem that could be solved using acoustic emission (AE) is the possibility to control nucleation and subcritical growth of internal crack-like defects in structure elements caused, for example, by hydrogen embrittlement of metals [1±4]. These defects, owing to the loading applied to a structure can subcritically grow up, in particular, in local areas near a contour of a previously existed crack. Such character of destruction is caused by the nature of stress±strain state near the crack front, the e€ect of stress±corrosion as well as microstructure of the material. In many cases the fracture occurs inside the material and is not observed visually, even in the case of local subcritical growth of a mode I crack. However, elastic waves caused by these processes could be received by AE transducers (AET) placed on the surface of inspected object.

*

Corresponding author. Tel.: +7-380-322-632044; fax: +7-380-322-649427. E-mail address: [email protected] (O.Ye. Andreykiv). 1 Professor (Sc.D.), Principal Research Fellow. 2 Ph.D., Senior Research Fellow. 3 Also corresponding author, Ph.D., Senior Research Fellow. 4 Ph.D., Senior Research Fellow. 0013-7944/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 0 2 6 - 1

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Fig. 1. Crack-like defect radiating AE and problems that should be solved for the estimation of parameters of such defect by AE signals: (1) nucleation and subcritical crack growth; (2) in¯uences of free surface on AE signal parameters; (3) waveguide in¯uence on AE signal parameters.

For a quantitative estimation of defects by means of AE method, it is necessary to establish relationships between AE signals and crack parameters. Let us assume that a newly formed crack is considerably less than the geometrical size of a body. Then to investigate the given problem it would be necessary to consider a non-stationary dynamic problem of crack formation and its growth in a half-space (Fig. 1). Besides, AE inspection is carried out often in the conditions considerably distinguished from the ambient medium (corrosive environment, high or low temperatures or when the access to the inspected surface is complicated). In such cases waveguides are used in AE measurements. So, one should take into account the e€ect of a waveguide on AE parameters. Thus, the problem to be solved is the establishment of relationship between AE parameters at the waveguide face and characteristics of a nucleating or growing crack. In such a general statement, the solution of this problem is rather dicult. Therefore, to simplify this problem we divide it into separate problems (see Fig. 1): (1) determining AE parameters caused by formation and subcritical growth of an internal crack-type defect at local areas near its contour in the in®nite elastic body; (2) the e€ect of the body surfaces on AE signals; (3) the e€ect of a waveguide on AE signals. 2. Nucleation and subcritical crack growth 2.1. Nucleation of a penny-shaped crack According to approach proposed in Ref. [5] we replace an arbitrary-shaped crack with a penny-shaped crack of the same area. Suppose that a penny-shaped crack nucleates when the tensile stresses in certain region of elastic body achieve the certain critical value r0 (the integral characteristics of material breaking strength). The crack formation is accompanied by the instant drop of normal stresses on its surfaces from an initial level r0 to zero. Let us consider a system of cylindrical coordinates Orhz. The origin O coincides with the center of the crack of radius r0 and the axis Oz is normal to the crack plane (see Fig. 2). At in®nity tensile stresses r are applied along Oz axes. At the time t ˆ 0 they achieve certain critical value r0 , resulting in a penny-shaped crack nucleation. Using known approaches [3,6,7] this problem can be reduced to the wave equations Du

1 o2 u ˆ 0; c21 ot2

Dw

w r

1 o2 w ˆ0 c22 ot2

with respect to unknown scalar potentials u…r; z; t† and w…r; z; t†.

…1†

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Fig. 2. A penny-shaped crack in an elastic body.

Eq. (1) should satisfy the boundary conditions for half-space z > 0 rzz …r; 0; t† ˆ

r0 H …t†;

r 6 r0 ; …2†

uz …r; 0; t† ˆ 0;

r > r0 ;

srz …r; 0; t† ˆ 0;

0 < r < 1;

and zero initial conditions

ou ow u…r; z; 0† ˆ w…r; z; 0† ˆ ˆ ˆ 0: ot tˆ0 ot tˆ0

…3†

1=2

1=2

Here c1 ˆ ……k ‡ 2l†=q† is the velocity of longitudinal wave, c2 ˆ …l=q† is the velocity of shear wave, k, l are LameÕs constants, q is the material density, H …t† is the Heaviside function. The dependence between potentials u and w and components of a displacement vector ur and uz has the form: ur ˆ

ou or

ow ; oz

uz ˆ

ou ow w ‡ ‡ : oz or r

…4†

The boundary problem (1)±(3) can be solved by the method of integral transforms. Similar to Ref. [7], using Hankel transform over the spatial coordinate r and Laplace transform over time t we reduce the problem to the Fredholm integral equation of the second kind: Z 1 K…q; s† ˆ K…u; s†K…u; q; s† du ˆ q; 0 6 q 6 1; …5† 0

having the kernel K…u; q; s† symmetric with respect to u and q in the form Z 1 2 K…u; q; s† ˆ g…g=r0 ; s† sin …gq† sin …gu† dg; 0 6 q 6 1; 0 6 g 6 1; p…1 e2 † 0 where g…g=r0 ; s† ˆ 1

e

2




"

1 1‡ 2 2 2g n

2 

1 1‡ 2 2 gn

1=2   e2 1‡ 2 2 gn

1=2

…6†

# 2g2 n2 ;

nˆ

c2 ; e ˆ c2 =c1 : sr0

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The solution of integrated equation (5) was found numerically and approximated by dependence proposed in [8]: b…q†m…q† K…q; s† ˆ p ; m2 ‡ p 2

…7†

where b(q) and m(q) are the parameters of approximation obtained by the least squares method, p ˆ sr0 =c2 . Taking into account the dependence in Eq. (7), after inversion of Laplace and Hankel transforms using the Parseval and convolution theorems, for the components of displacement vector we obtain the following equations: # "Z Z 1 2 q  r‡n X ui … r; z; T † ˆ ABi b…q†m…q† Sij …a; z; sj ; q†H T 2 =e2j z2 a Gi …a; r; q† da dq; 0

r n

jˆ1

r > 1 …i ˆ z; r†;

…8†

where r0 r0 2 2 qc1 e p…1

r ˆ r=r0 ; T ˆ c2 t=r0 ; e1 ˆ e; e2 ˆ 1; ; z ˆ z=r0 ;  e2 † p p p 1 p sj ˆ T ej z2 ‡ a2 ; Bz ˆ z=…p r†; Br ˆ …p r† ; Gz ˆ aK…k†; q p 2 Gr ˆ a a‰2E…k† K…k†Š; k ˆ q2 …a r† =…4ar†; Aˆ

E…k† and K…k† are complete elliptic integrals of the ®rst and second kind respectively, Sij …
† are given in Ref. [9]. Asymptotic dependence of the displacement vector components one can obtain by leaving in Eq. (8) terms decaying weakly than 1=R …R2 ˆ r2 ‡ z2 †. In spherical coordinate system originated in the center of the penny-shaped crack the components of displacement vector are as follows uR ˆ ur cos h ‡ uz sin h;

uh ˆ uz cos h

ur sin h:

…9†

Replacing internal integrals over a in Eq. (8) by their average values as a tends to r, for large R after simpli®cation we shall obtain the approximation: Z 1   h; T † ˆ ABi …1=R†  ui …R; 2qb…q†m…q†J0 …msi †Mi …k† dq H …si † ‡ O…R 2 †; …10† 0

 R ˆ R=r0 , M1 …k† ˆ K…k†, where i ˆ R; h, B1 …h† ˆ e=p…1 2e2 cos2 h†, B2 …h† ˆ 1=p sin 2h, si ˆ T ei R, M2 …k† ˆ 2E…k† K…k†, k ˆ q=2R cosh, J0 …
† is the zero order Bessel function of the ®rst kind, B1 and B2 determine the angular dependence of radiation when a crack is modeled by a system of three mutually perpendicular dipoles [10]. As a result of numerical calculations according to Eq. (8), it was found that for distances from a crack greater than 28r0 the amplitude values both for longitudinal and shear waves decay as 1=R that is predicted in Eq. (10). When performing the AE inspection the distances between AE sources and AET are signi®cantly larger than the dimensions of AE sources. Therefore, more attention will be given to consideration AE caused by a penny-shaped  1=2 crack for a large R. The dependencies of a dimensionless value U … r; z; T † ˆ u2z … r; z; T † ‡ u2r … r; z; T † =A vs dimensionless time T =e calculated according to Eq. (8) for R ˆ 1000, e ˆ 0:535 and PoissonÕs ratio m ˆ 0:32 are given in Fig. 3a (longitudinal wave) and in Fig. 3b (shear wave). The curves 1 and 2 correspond to angles h ˆ 15° and h ˆ 75°, respectively. Similar calculations were carried out for other angles of crack orientation in the range from 0 up to p=2. As a result of the analysis of the maximal values of oscillations for longitudinal and shear waves the approximation

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(a)

(b)

Fig. 3. Dependence of the dimensionless module of a displacement vector on dimensionless time T =e at R ˆ 1000 for (a) longitudinal and (b) transversal waves: curve 1 corresponds to angle of observation h ˆ 15°, curve 2 ± h ˆ 75°.

expressions are proposed. They characterize angular dependencies of peak values of the displacement vector module for R  1. For longitudinal waves 1 2e2 cos2 h …d† U1 …h† ˆ p ; 1 ‡ v1 cos2 h

…11†

and for shear waves j sin 2hj …d† U2 …h† ˆ p ; 1 ‡ v2 cos2 h

…12†

where v1 and v2 are the parameters of approximation. Their numerical values v1 ˆ 0:68 and v2 ˆ 2:69 at e ˆ 0:535 and m ˆ 0:32 were obtained by the least squares method. The angular dependencies of the peak values of U … r; z; T † calculated according to Eq. (8) were compared with Eqs. (11) and (12). These expressions describe the angular distribution of amplitudes with an error not exceeding 4% and correlate with

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Fig. 4. Angular dependencies of the maximum values of the module of a displacement vector for (a) longitudinal and (b) shear waves.

Eq. (10). They are presented in Fig. 4a and b for longitudinal and shear waves, respectively. Taking into account Eqs. (11) and (12) and the character of spatial decay (see Eq. (10)) for estimation of the maximum values of the displacement vector module the following approximation expressions are given: …d†

umax jci ˆ di

r0 Ui …h†r02 ; qc21 R

…13†

where i ˆ 1 corresponds to longitudinal wave and i ˆ 2 to shear wave, d1 ˆ 0:452, d2 ˆ 0:832. In these expressions R is dimensional. The analysis of dependence (13) and the calculations carried out according to Eq. (8) show that for the angles within the range 15° 6 h 6 70° peak values of the displacement vector for the longitudinal wave are somewhat lower than for the shear wave (see Fig. 3). It is necessary to account this fact when receiving AE signals and ®nding the location of AE source. It was found also, that the time interval to achieve the ®rst maximum essentially decreases with increase of the orientation angle h, and the duration of its rear front changes insigni®cantly (see Fig. 3). During experimental researches of AE signals generated by formation and propagation of cracks in materials, a question on determination of frequency characteristics of these signals is important. It is caused by the necessity of choosing such AET whose bandwidth corresponds to the spectrum of AE signals radiated by these sources, that will promote increase of sensitivity of AE storage hardware. Therefore, we carried out the investigation of frequency dependencies for the obtained module of displacement vector. For this purpose the method of fast Fourier transform (FFT) [11] was used. The results of calculations of frequency dependencies of the displacement vector module are given in Fig. 5a and b …R ˆ 1000† for longitudinal and transversal waves, respectively. A step digitizing over the time was chosen equal to 0.5, and the number of digitizing points N ˆ 1024. In these ®gures ordinate axis corresponds to the normalized value of UX ˆ u…X†=u…X†jmax and abscissa axis to dimensionless frequency X ˆ xr0 =c1 (x is angular frequency). It is seen from these ®gures that the width of the spectrum is greater for h ˆ 75° than for h ˆ 15° correlating with the width of the ®rst maximum of the module of a displacement vector for these angles and R (see Fig. 3). The width of the spectrum in Fig. 5 DX  2:5. Assuming

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(a)

(b)

Fig. 5. Dependence of dimensionless value of the module of a displacement vector UX on dimensionless frequency X at R ˆ 1000 for (a) longitudinal wave and (b) shear wave: curve 1 corresponds to angle of observation h ˆ 15°, curve 2 ± h ˆ 75°.

that the radius of a penny-shaped microcrack is r0 ˆ 5  10 MHz …x ˆ 2pf †:

6

m and c1 ˆ 5  103 m/s, we obtain Df  40

2.2. Modeling of subcritical crack growth at local areas of its contour as a source of acoustic emission signals For developing the appropriate model of AE signals radiation caused by subcritical crack growth at local areas of its front, consider an elastic half-space with a ¯at mode I macrocrack bounded by a smooth contour L. Let at the time t ˆ 0 in local area, where stresses (or deformations) achieve certain critical value, due to application of external tensile forces to a body, a microcrack nucleates closely a contour of this macrocrack (see Fig. 6). As a result of unloading of the surfaces of this newly formed microcrack from an initial level down to zero, elastic waves are radiated. They reach the inspected object surfaces and can be received by AET (see Fig. 1). For simpli®cation of the problem we replace this microcrack with a penny-shaped mode I crack equal to it by the area. Let us suppose also, that radius of this penny-shaped microcrack is much less than the radius of macrocrack contour curvature. In this case instead of the above mentioned problem we may have to

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Fig. 6. The local growth of internal crack.

consider the problem of a sudden nucleation of a penny-shaped microcrack near the front of a through semi-in®nite macrocrack. Let us estimate the resulting components of the dynamic displacement ®eld. Consider now dynamic problem of growth of a semi-in®nite through crack in homogeneous isotropic elastic body. For this problem the following angular dependencies of Uhi …b† were obtained in Ref. [12]. They correspond to longitudinal and transverse waves that are valid for the wavefront region. Uhi …b† ˆ Vi …b†Ci …b†Pi …b†;

i ˆ 1; 2:

…14†

Here the functions Vi …b† ˆ

1

1 cc =ci cos b

…15†

describe the change in angular dependence of radiation caused by the crack edge propagation with velocity cc for a component of longitudinal …i ˆ 1† and shear …i ˆ 2† waves. Functions p 1 ‡ c1 =ci cosb ; …16† Ci …b† ˆ …1 ‡ cR =ci cosb†K … si cos b† where



K …g† ˆ exp ( g…z† ˆ tan

1

1 p 4z2

Z

s2 s1

 g…z†dz ; zg

si ˆ 1=ci ;

pp ) z2 s21 s22 z2 …s2

2z2 †

2

…17†

…18†

describe the in¯uence of the free surface of a semi-in®nite crack on the angular distribution of radiation. Functions P1 …b† ˆ 1

2

2…c2 =c1 † cos2 b;

P2 …b† ˆ sin 2b

…19†

determine the angular dependencies of maximum values of the displacement vector module at the fronts of longitudinal and shear waves radiated by the edge of this crack; cR is the Rayleigh wave velocity. The comparison of expression (19) with the dependencies (11) and (12) for case of a penny-shaped crack nucleation shows their similarity. So, if the cause of elastic waves radiation is the nucleation of a pennyshaped microcrack closely to the contour of a macrocrack, then the in¯uence of the free surfaces of the

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macrocrack on a displacement ®eld caused by formation of this penny-shaped crack can be estimated using dependence (16). Therefore, we express a radiation ®eld at a distance signi®cantly larger than the microcrack radius as a product of displacement components for instant nucleation of a penny-shaped crack in elastic body (Eq. (9)) by functions Ci …b† (Eq. (16)) accounting for the e€ect of a free surface. In spherical coordinate system ORhu whose origin coincides with a center of a penny-shaped crack, the components of a displacement vector at distances R  r0 can be written as follows: …d†

uR …R; t† ˆ C1 …b†uR …R; t†;

…20†

…d†

uh …R; t† ˆ C2 …b†uh …R; t†; …d†

…d†

where functions uR …R; t† and uh …R; t† are determined by Eqs. (8) and (9), cos h sin u ‡ d cosb ˆ q ; … cos h sin u ‡ d†2 ‡ sin2 h

…21†

d ˆ D=R, D is the distance between the center of a penny-shaped crack and the edge of a semi-in®nite crack. For large R, d ! 0. The components uR and uh obtained by Eqs. (20) have the same time dependence as …d† …d† appropriate components uR and uh . However, the directional characteristic of radiation in this case will di€er from those obtained for a penny-shaped crack. The numerical results calculated according to dependencies (20) allow to get the directional characteristic of radiation for the case of a penny-shaped crack nucleation in the vicinity of the macrocrack front. In Fig. 7 this directional characteristic of the maximum values of components uR …R; t† is presented for longitudinal wave and angle u ˆ p=2: The points mark experimental data obtained in Ref. [13]. We can see in this ®gure that the obtained angular dependence di€ers most of all from the similar one for an isolated penny-shaped crack (see Fig. 4a) for h close to p. In the case jhj 6 p=2 the characteristics practically coincide. Therefore, if an AET is placed in this region of angles h the in¯uence of the free surface on AE signals is insigni®cant.

Fig. 7. Angular distribution of a maximum of the displacement vector module for longitudinal wave during nucleation of a pennyshaped crack near an internal crack front.

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It follows from dependencies (20) that the magnitude of displacements at the front of a radiated longitudinal wave is proportional to r02 : umax jc1 ˆ

d1 r0 r02 …d† U …h†C1 …b†; qc21 R 1

…22†

i.e. to the area of a nucleated defect. Let us go back to consideration of the problem on a microdefect of area S nucleation in the vicinity of the internal macrocrack contour. Assuming that S is equal to the area of a microcrack and that amplitude A of AE signal is proportional to magnitude of displacement at the front of longitudinal wave, we ®nd Aˆ

kS …d† U …h†C1 …b†; R 1

…23†

where k ˆ k0 k1 and k0 is a proportionality factor between electrical signals in the output of AET and maximum values of displacements at the front of longitudinal wave, k1 ˆ d1 r0 =…pqc21 †. If during local growth of an internal crack in the vicinity of its contour N of such microdefects were formed then the total area of internal macrocrack growth DS ˆ

N X

bAk ;

…24†

kˆ1

where b is a proportionality factor. Thus, as follows from dependencies (24), the amplitudes of AE signals caused by the internal ¯at crack growth in local areas of its front are proportional to the total growth area. Such a form of relationship between A and S was con®rmed experimentally [14,15].

3. In¯uence of a body boundaries on acoustic emission signals, caused by internal defect growth The registration of AE signals is carried by an AET, located on the surface of the inspected object. For this reason the important question of AE diagnostics is the study of the in¯uence of the free surface on AE parameters. For investigation of the change in the displacement vector component caused by the boundaries of the body we shall consider a problem on determining the surface motion due to sudden nucleation of the internal penny-shaped crack in a body. Such a problem can be solved, in particular, by the method of Helmholtz potentials [16,17]. In this approach the solution of the given problem could be reduced to the numerical±analytical solving of the system of singular integral equations with respect to unknown Fourier transforms over the time of displacement vector jumps on the crack surfaces. The necessity of employing the numerical methods, both for solving the singular integrated equations, and inverting Fourier transforms, complicates the obtaining of the results and their analysis, in particular, ®nding the dependencies between the maxima of surface motion and orientation of a penny-shaped crack. Therefore, in order to estimate surface displacement caused by sudden nucleation in a half-space of such crack we shall use the approximation approaches employed in Ref. [18]. The geometry of the problem is shown in Fig. 8. The plane of a crack forms the angle U with normal to the boundary of a half-space. In toroidal coordinate system ORaw the angle w de®nes a running point on the crack contour. For each plane w ˆ constant coordinates …r0 ; a† form a polar coordinate system, whose origin coincides with this point. The plane w ˆ 0 is a plane of symmetry, d is the distance between the nearest point of the contour of a penny-shaped crack and the boundary of a half-space. Cartesian coor-

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Fig. 8. A penny-shaped crack in elastic half-space.

dinate system Oxyz will be also used. Its origin is the point of intersection of the crack plane, the plane of symmetry and the boundary of a half-space. For this model using results obtained in Ref. [18] in the near zone the overall (incident plus re¯ected) displacement on the surface caused by an incident longitudinal wave emitted by the crack in the plane of symmetry w ˆ 0 is given by h i …d† …1† U1 …R; a† ˆ R…1† …25† x …a†i ‡ Ry …a†j uR …R; h†; where R…1† x …a† ˆ R…1† y …a† …d†

2e 1 sin 2…a ‡ U† sin d ; e2 cos2 2d sin 2d sin 2…a ‡ U†

2e 2 cos …a ‡ U† cos2d ; ˆ 2 e cos2 2d sin 2d sin 2…a ‡ U† …d†

…26†

…d†

uR …R; h† ˆ uR …R; h; t† or uR …R; h; x† are components of a displacement vector for the problem of nucleation of a penny-shaped crack in the in®nite body both for the time and frequency domains, respectively, determined by Eqs. (8) and (9), i and j are the unit vectors, R is the distance from the crack center to the observation point located on the boundary of the half-space, R ˆ …r0 ‡ d cosa= cos…a ‡ U††= cosh, h is the angle between the plane of the crack and the direction vector originated in the center of the defect to the observation point, the angle d satis®es equation cosd ˆ

e sin …a ‡ U†:

…27†

The motion on the free surface caused by an incident transverse wave for angles a ‡ U such that j sin …a ‡ U†j < e, is given by: h i …d† …2† U2 …R; a† ˆ R…2† …28† x …a†i ‡ Ry …a†j uh …R; h†; where R…2† x …a† ˆ R…2† y …a†

2e 2 cos…a ‡ U† cos2…a ‡ U† ; e 2 cos2 2…a ‡ U† sin 2c sin 2…a ‡ U†

2 sin 2c cos …a ‡ U† ; ˆ 2 2 e cos 2…a ‡ U† sin 2c sin 2…a ‡ U†

…29†

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…d†

…d†

…d†

uh …R; h† is equal to uh …R; h; t† or uh …R; h; x† that are the components of a displacement vector for the problem of a penny-shaped crack nucleation. They are determined by Eqs. (8) and (9) both for the time and frequency domains, respectively, the angle c should satisfy equation e cosc ˆ

sin …a ‡ U†:

…30†

Eqs. (25) and (28) are correct for times corresponding to the near-front wave region and condition d  r0 . Taking into account Eqs. (13), (14) and (26), (29) for the maximum value of the module of the displacement vector caused by incident longitudinal and shear waves, we obtain the following equation u…k† max jci ˆ

…d†

di r0 Ui …h†r02 …i† R…k† …a† qc21 R …d†


i; k ˆ 1; 2 ;

…31†

…d†

where functions U1 …h† and U2 …h† are de®ned by Eqs. (11) and (12) for longitudinal and shear waves, respectively. Note that for d  r0 the di€erence between angles h and a is insigni®cant. In Figs. 9 and 10 dimensionless maximum values of the displacement vector module on the surface of the 1=2 …y†2 half-space Umax jci ˆ ‰u…x†2 =A for longitudinal and shear waves, respectively, are given with max jci ‡ umax jci Š respect to the dimensionless distance l=d between the epicenter (point E in Fig. 8) and the observation point located on the boundary of the half-space, l ˆ dtgU ‡ x. The calculations were carried out for crack orientation U ˆ 0° (Figs. 9a and 10a) and U ˆ 75° (Figs. 9b and 10b), d=r0 ˆ 200: In these ®gures 1 marks the maximum values of the displacement vector module for an incident wave and 2 marks the maximum values of the surface displacement, both calculated by Eq. (31). As is shown in Fig. 9 for the case of longitudinal wave the maximum values of surface motion exceed the appropriate ones for an incident wave by nearly a factor of 2. For distances l=d greater than about seven depths of the defect location the di€erence between the maxima of motion caused by incident wave and overall motion on the surface is insigni®cant. In the case of the shear wave this di€erence becomes small at the distance from epicenter equal approximately to four depths of the defect location.

4. Waveguide in¯uence on change of acoustic emission signal parameters While using waveguides in AE measurements it is necessary to take into account the following. A waveguide has eigenfrequencies that depend on its geometrical sizes. AET has also a resonance frequency band. Therefore with the purpose of the least losses during AE signals transfer it is important to match these frequencies to the components of the measuring system. A cylindrical waveguide is used most frequently in AE measurements since it is most simple in manufacturing and could be easily attached to the specimen or structure element. The ®nding of frequency eigenvalues for a cylindrical waveguide of the arbitrary sizes is a rather dicult problem [19,20]. Besides, the usage of the solution, obtained in these papers, for engineering calculations is rather problematic. Therefore in this study the approximation approach for selection of the sizes of cylindrical waveguide is proposed. This approach is based on a method of boundary interpolation [21], whose eciency is shown for many problems of the fracture mechanics and AE. As a result the following approximation expression for calculation of frequency eigenvalues for cylindrical waveguide of both arbitrary both length and diameter [22] is obtained: nh  im h im o 1=m …d † …s† …d † …s† …d † ap;q ˆ a…d† ap;q ‡ a…d† ; …32† a…d† p;q ap;q ap;q p;q ap;q p;q ‡ ap;q …s† where a…d† p;q , and ap;q are the functions appropriate to frequency eigenvalues for a disk and a rod respectively, † a…d is the value of a…d† p;q p;q at l=a  1, l is the length of the cylinder, a is its radius (Fig. 11). Eq. (32) precisely

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1329

(a)

(b)

Fig. 9. Dependence of the dimensionless value of Umax jc1 on the half-space surface on dimensionless distance l=d at d=r0 ˆ 200 for longitudinal wave for the angles of a crack orientation (a) U ˆ 0° and (b) U ˆ 75°; curve 1 corresponds to the surface motion caused only by incident wave, curve 2 corresponds to the surface motion caused by overall wave.

takes into account limiting cases of interpolation (disk and long rod). Parameter m can be found from the exact solution of the problem in some intermediate points. Let us consider now limiting cases of interpolation. For a thin disk the stresses will be absent in a plane, perpendicular to Oz axis (see Fig. 11). The equation for ®nding frequency eigenvalues for this case can be written as follows [23]: thbl 4abg2 ˆ
2 ; thal b2 ‡ g2

…33†

where a and b are phase constants for tension and shear along the cylinder axis, g is a phase constant for a radial direction. These constants are related by the dependence g2 ˆ a2 ‡ h2 ˆ b2 ‡ k 2 ;

…34†

1330

O.Ye. Andreykiv et al. / Engineering Fracture Mechanics 68 (2001) 1317±1333

(a)

(b)

Fig. 10. Dependence of the dimensionless value Umax jc2 on the half-space surface on dimensionless distance l=d at d=r0 ˆ 200 for shear wave for the angles of a crack orientation (a) U ˆ 0° and (b) U ˆ 75°; curve 1 corresponds to the surface motion caused only by incident wave, curve 2 corresponds to the surface motion caused by overall wave.

where h ˆ xc1 , k ˆ xc2 and g is de®ned by the following equation [23]: d2 J0 …ga† d…ga†

2

ˆ

g 2 ‡ b2 J0 …ga†: 2g2

…35†

Frequency eigenvalues of oscillation for a thin disk fp;q can be found by solving the system of equations (33) and (35):  fp;q ˆ ap;q …2plc2 †; …36†

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1331

Fig. 11. The cylindrical waveguide.

where ap;q ˆ …kl†p;q , p and q are integers, …al†q and …ga†p are qth and pth roots of Eqs. (34) and (36), respectively. The equation for determining frequency eigenvalues of a rod has the form

 …37† … x 1†2 u h0 a ‡ …1 ex† x u k 0 a ˆ 0; p   where u…y† ˆ yJ0 …y†=J1 …y†, x ˆ c2 =c20 …1 ‡ m†, e ˆ …1 2m†=…1 m†, c0 ˆ E=q is the velocity of sound in a rod, E is YoungÕs modulus, c is a phase velocity. Phase constant g could be divided for a long rod into two terms h0 and k 0 corresponding to tension and rotation components. A phase constant c of wave propagation in axial direction is unique. The relationship between these constants is the following: h0 ˆ h2

c2 ;

k0 ˆ k2

c2 ;

…38†

where h and k are given by Eq. (34). If the rod has a ®nite length, the following condition should be held cq ˆ

pq ; 2l

…39†

where q is a positive odd number. The eigenvalues of ap;q could be found from the solution of the system of equations (37) and (39): q …40† ap;q ˆ pq xp =2: Thus, Eqs. (32), (36) and (40) enable us to estimate frequency eigenvalues of cylindrical waveguide of arbitrary geometrical sizes. As the example the value of a1;1 at m  4, 6 was calculated (see Fig. 12). Obtained theoretical results were used to choose the optimum sizes of the waveguide used for estimation of the threshold value of KIscc . According to the design requirements to the test equipment it was important, that the waveguide had the length of 132 mm. The frequency band of the resonance type AET was 200±350 kHz. The waveguide diameter should be chosen so that frequency eigenvalues lay within the bandwidth of AET. The calculations by approximation equation (32) have shown, that the given condition is satis®ed if p ˆ 3, q ˆ 3 and p ˆ 1, q ˆ 13. According to dependence (36) fp;q will be >200 kHz if ap;q P 27:2. The condition l=a P 10 follows from here. As the length of a rod is equal to 132 mm the waveguide diameter of 13 mm was chosen. Note that while l=a P 10 the value of frequency eigenvalues calculated by approximation expression (32) and by dependence (40) are identical. The normal displacement W at the end face of the waveguide for frequencies close to eigenvalues can be estimated by the expression given in Ref. [23]. Then, using dependence (31) we obtain the following

1332

O.Ye. Andreykiv et al. / Engineering Fracture Mechanics 68 (2001) 1317±1333

Fig. 12. Dependencies of value of a1;1 on l=a calculated for a disk (curve 1), rod (curve 2) and by approximation expression (32) (curve 3).

estimation for the maximum normal displacement at the end face of the waveguide caused by a crack nucleation in elastic half-space: …d†

Umax jci /

di Ui …h†r02 …i† R…y† …a†W ; qc21 R

i ˆ 1; 2:

…41†

Moreover, using dependencies (23) and (41) amplitudes of AE signals can be expressed as follows: …d†

A/

kSUi …h† …i† R…y† …a†W ; R

i ˆ 1; 2:

…42†

The investigation of AE changes caused by the waveguide of chosen dimension was carried out in experimental determination of the threshold value of KIscc [24]. The research has shown the insigni®cant loss in amplitude due to the waveguide. That con®rms the validity of the method of boundary interpolation for the solution of such type of problems.

5. Conclusions Using both proposed models of nucleation and growth of a crack and the research of the body boundary and waveguide in¯uence on change of AE signals parameters, the analytical dependencies between parameters of a crack and parameters of AE signals are obtained. These dependencies can be used for development of appropriate techniques of AE testing and equipment as applied to crack detection.

References [1] Hartbower CE, Gerberich WW, Liebowitz H. Investigation of crack-growth stress wave relationships. Engng Fract Mech 1968;1(2):291±308. [2] Gerberich WW, Hartbower CE. Monitoring crack growth of hydrogen embrittlement and stress corrosion cracking by acoustic emission. Fundamental aspects of stress corrosion cracking. Houston: NACE; 1969. p. 420±38.

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[3] Andreykiv AYe, Lysak NV. A method of acoustic emission in investigation of fracture processes. Kiev: Naukova Dumka; 1989 (in Russian). [4] Dunegan HL, Tetelman AS. Nondestructive characterization of hydrogen embrittlement cracking by acoustic emission. Engng Fract Mech 1971;2(4):387±402. [5] Lysak MV. Acoustic emission during jumps in subcritical growth of cracks in three-dimensional bodies. Engng Fract Mech 1994;47:873±9. [6] Lysak MV. Development of the theory of acoustic emission by propagating cracks in terms of fracture mechanics. Engng Fract Mech 1996;55:443±52. [7] Chen EP, Sih GC. Transient response of cracks to impact loads. Elastodynamic crack problems in series on mechanics of fracture, vol. 4. 1977. p. 1±58. [8] Sih GC, Embley GT, Ravera RS. Impact response of a ®nite crack in plane extension. Int J Solids Struct 1972;8(7):977±93. [9] Andreykiv AYe, Lysak NV, Serhiyenko ON, Skalsy VR. Theoretical concepts of a method of acoustic emission in investigation of fracture processes. Lviv, PhMI AN USSR, reprint no. 137, 1987 (in Russian). [10] Wadley HWG, Scruby CB. Elastic wave radiation from cleavage extension. Int J Fract 1983;23(2):111±28. [11] Ahmed N, Rao KR. Orthogonal transformations for digital signal processing. New York: Springer; 1975. [12] Achenbach JD, Harris JG. Acoustic emission from a brief crack propagation event. J Appl Mech 1979;46(1):107±12. [13] Scruby CB, Wadley HNG, Rusbridge KL. Origin in acoustic emission in Al±Zn±Mg alloys. Mater Sci Eng 1983;59(2):169±83. [14] Nesmashnyi YeV, Kuznetsov BA, Maslov BYa, et al. On relation of the amplitude of AE signals with increment of the crack area. Theses for the 1st All Union Conf ``Acoustic Emission in Materials and Structures'', Part 1, 11±13 September, 1984. Rostovon-Don, 1984. p. 224±5 (in Russian). [15] Gerberich WW, Alteridge DG, Lessar JF. Acoustic emission investigation of microscopic ductile fracture. Met Trans A 1975;6A(2):797±801. [16] Mykhaskiv VV, Stankevych VZ, Khai MV. The boundary integral equations for three-dimensional problems on the steady-state vibration of a half-space with ¯at cracks. Notes Acad Sci Russia, Solid mechanics 1993;(6):44±53 (in Russian). [17] Khai MV. Two-dimensional integral equations of a Newton type potential and their application. Kiev: Naukova Dumka; 1993 (in Russian). [18] Harris JG, Pott J. Surface motion excited by acoustic emission from a buried crack. Trans ASME: J Appl Mech 1984;51(1):77±83. [19] Hrinchenko YeV. Balance and steady-state vibration of elastic bodies of the ®nite dimension. Kiev: Naukova Dumka; 1978 (in Russian). [20] Imenitova YeV, Chernyshov VV, Shegai VV. On calculation of free vibration for elastic cylinders of ®nite length. Rep Acad Sci USSR 1976;226(2):315±7 (in Russian). [21] Andreykiv AYe. Spatial problems in the crack theory. Kiev: Naukova Dumka; 1982 (in Russian). [22] Lysak MV, Skalsky VR, Serhiyenko OM. Investigation of waveguide in¯uence on change of parameters of acoustic emission signals. Physico-chemical mechanics of materials, vol. 3. 1994. p. 64±71 (in Ukrainian). [23] Kikuchi Y. Ultrasonic transducers. Tokyo: Corona Publishing; 1969. [24] Andreykiv AYe, Lysak, MV, Skalsky VR, Serhiyenko OM. A method of determining KIscc values of steel in hydrogen using acoustic emission. Technical diagnostics and nondestructive testing, vol. 4(1). 1992. p. 15±21.