- Email: [email protected]
Acoustic emission estimation of crack formation in aluminium alloys
Engineering Fracture Mechanics 77 (2010) 759–767
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Acoustic emission estimation of crack formation in aluminium alloys Olexander Andreykiv, Valentyn Skalsky, Oleh Serhiyenko, Denys Rudavskyy * Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, 5, Naukova Str., Lviv 79601, Ukraine
a r t i c l e
i n f o
Article history: Received 16 December 2008 Received in revised form 23 November 2009 Accepted 12 January 2010 Available online 28 January 2010 Keywords: Penny-shaped crack Acoustic emission Aluminium alloy
a b s t r a c t In the paper a problem on the radiation of waves of acoustic emission (AE) during formation of penny-shaped crack in aluminium alloy elastic body under tensile and twisting loading is considered. By the method of integral transforms, the problem is reduced to the solving of Fredholm integral equation of the second kind. The dependences of component of displacement vector on time and distance to the viewpoint as well as amplitudes of the AE signals on the area of the crack and its spatial orientation are obtained. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Aluminium and its alloys are widely used in modern engineering mainly in aviation and space industry. Various units, structural elements, etc. are made of them. During their operation under action of physical and chemical factors crack-like defects can form. They are especially dangerous for the structural integrity of these elements. For non-destructive testing of crack nucleation and propagation the phenomenon of acoustic emission (AE) is used. Quantitative application of AE as a method of non-destructive testing needs to find the relationships between AE signals and crack parameters (its size, spatial position, etc.). For this purpose we consider a problem of sudden formation of a penny-shaped crack in homogeneous elastic body under tensile and twisting loading (mode I and mode III, respectively). We will try to find the displacement vector components caused by this crack formation directly from equations of motion with appropriate boundary conditions corresponding to abrupt formation of the rupture without any additional conditions. We restricted our analysis to consideration of penny-shaped cracks of mode I and mode III because solving of the problem for mode II crack is more complicated and not completed yet. 2. Problem formulation and solving 2.1. Nucleation of a penny-shaped crack of mode I According to approach proposed in [1] 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 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. 1). At infinity tensile stresses r are applied along Oz axes. At the time * Corresponding author. E-mail address: [email protected] (D. Rudavskyy). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.01.009
760
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Nomenclature b, m parameters of approximation velocity of longitudinal and shear wave, respectively c1, c2 r, h, z cylindrical coordinates radius of crack r0 t time s parameter of Laplace integral transform ur, uh, uz components of a displacement vector V amplitude of the electric signal Lame’s constants k, l m Poisson’s ratio q material density r0, s0 the integral characteristics of material breaking strength rzz, szh, srz components of a stress tensor scalar potentials u, w v1, v2 parameters of approximation E(k), K(k) complete elliptic integrals of the first and second kind, respectively J0() and J1() is the zero and first order Bessel functions of the first kind F(), E() elliptic integrals of the first and second kind, respectively H(t) the Heaviside function
Fig. 1. A penny-shaped crack in an elastic body (mode I).
t = 0 they achieve certain critical value r0, resulting in a penny-shaped crack nucleation. Using known approaches [2–4] this problem can be reduced to the wave equations
Du
1 @2u ¼ 0; c21 @t2
Dw
w 1 @2w ¼ 0; r c22 @t2
ð1Þ
with respect to unknown scalar potentials /(r, z, t) and w(r, z, t). Eq. (1) should satisfy the boundary conditions for half-space z > 0
rzz ðr; 0; tÞ ¼ r0 HðtÞ; r 6 r0 ; uz ðr; 0; tÞ ¼ 0;
r > r0 ;
srz ðr; 0; tÞ ¼ 0; 0 < r < 1;
ð2Þ
761
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
and zero initial conditions
@ u @w ¼ ¼ 0: @t t¼0 @t t¼0
uðr; z; 0Þ ¼ wðr; z; 0Þ ¼ Here c1 ¼
ð3Þ
qffiffiffiffiffiffiffiffi qffiffiffi kþ2l l q , c2 ¼ q. The dependence between potentials / and w and components of a displacement vector ur and uz
has the form:
ur ¼
@ u @w ; @z @r
uz ¼
@ u @w w þ : þ @z @r r
ð4Þ
The boundary problem (1)–(3) can be solved by the method of integral transforms. Similarly to [4], 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:
Kðq; sÞ ¼
Z
1
Kðu; sÞKðu; q; sÞdu ¼ q;
0 6 q 6 1;
ð5Þ
0
having the kernel K(u, q, s) symmetric with respect to u and q in the form
Kðu; q; sÞ ¼
2 pð1 e2 Þ
Z
1
gðg=r 0 ; sÞ sinðgqÞ sinðguÞdg;
ð6Þ
0
where 2
gðg=r 0 ; sÞ ¼ ð1 e Þ
"
# 2 1=2 1 e2 1=2 2 2 2g n ; 1þ 2 2 1þ 2 2 1þ 2 2 2g n gn gn 1
n¼
c2 ; sr0
e ¼ c2 =c1 :
The solution of integrated Eq. (5) was found numerically and approximated by dependence proposed in [5]:
bðqÞmðqÞ Kðq; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; m2 þ p2
ð7Þ
where b(q) and m(q) are the parameters of approximation obtained by the least squares method, p = sr0/c2. Taking into account the dependence in (7), after inversion of Laplace and Hankel transforms using the Parseval and convolution theorems, for the components of displacement vector we obtain the following equation:
ui ðr ; z; TÞ ¼ ABi
Z 0
1
bðqÞmðqÞ
"Z
rþq
rq
2 X
# pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Sij ða; z; sj ; qÞH T ej z þ a Gi ða; r ; qÞda dq;
r > 1;
ði ¼ z; rÞ;
ð8Þ
j¼1
r0 where A ¼ qc2 e2rp0 ð1 , z ¼ z=r 0 , r ¼ r=r0 , T ¼ c2 t=r 0 is the dimensionless time, e1 = e, e2 = 1, e2 Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi Br ¼ ðp r Þ1 , Gz ¼ aKðkÞ, Gr ¼ a a½2EðkÞ KðkÞ k ¼ q2 a rÞ2 =ð4ar Þ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
sj ¼ T ej z2 þ a2 , Bz ¼ z=ðp rÞ,
Sij ða; z; sj ; qÞ ¼ Dij ða; zÞJ 0 ðmðqÞsj Þ þ Eða; zÞJ 1 ðmðqÞsj Þ þ F ij ða; zÞIðmðqÞsj Þ; D11 ða; z; qÞ ¼ eðz2 þ a2 Þ1 ½1 2a2 e2 ðz2 þ a2 Þ1 þ 3m2 ðqÞs1 ðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; E11 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 e2 ðz2 þ a2 Þðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; n o F 11 ða; z; qÞ ¼ mðqÞ1 ðz2 þ a2 Þ3=2 ½1 2e2 þ 3e2 a2 ðz2 þ a2 Þ1 þ 3ðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ðT 2 m2 Þ ; D12 ða; z; qÞ ¼ 2a2 ðz2 þ a2 Þ2 3m2 ðqÞs2 ðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; E12 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 ðz2 þ a2 Þðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; n o F 12 ða; z; qÞ ¼ mðqÞ1 3ðz2 þ a2 Þ5=2 ½5a2 ðz2 þ a2 Þ1 2ðT 2 mðqÞ2 Þ þ ðz2 þ a2 Þ3=2 ½2 3a2 ðz2 þ a2 Þ1 ; D21 ða; z; qÞ ¼ eðz2 þ a2 Þ1 ½2e2 z2 ðz2 þ a2 Þ1 þ 1 2e þ 3m2 ðqÞs1 ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1; E21 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 e2 ðz2 þ a2 Þðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1; n o F 21 ða; z; qÞ ¼ mðqÞ1 3ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1ðT 2 mðqÞ2 Þ þ ðz2 þ a2 Þ3=2 ½1 e2 3z2 e2 ðz2 þ a2 Þ1 ;
762
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Fig. 2. A penny-shaped crack in an elastic body (mode III).
D22 ða; z; qÞ ¼ 3m2 ðqÞs2 ðz2 þ a2 Þ5=2 ½1 5z2 ðz2 þ a2 Þ1 2z2 ðz2 þ a2 Þ2 ; E22 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 ðz2 þ a2 Þ þ ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1; n o F 22 ða; z; qÞ ¼ mðqÞ1 3ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1ðT 2 mðqÞ2 Þ þ ðz2 þ a2 Þ3=2 ½1 3z2 ðz2 þ a2 Þ1 ; Z x J 0 ðtÞdt: IðxÞ ¼ 0
Asymptotic dependence of the displacement vector components one can obtain by leaving in Eq. (8) terms decaying weakly than 1/R (R2 = r2 + z2). In spherical coordinate system originated in the center of the penny-shaped crack the components of displacement vector are as follows
uR ¼ ur cos h þ uz sin h;
uh ¼ uz cos h ur sin h:
ð9Þ
Replacing internal integrals over a in Eq. (8) by their average values as a tends to r, for large R after simplification we shall obtain the asymptotic expression for Eq. (8):
ui ðR; h; TÞ ¼ ABi ð1=RÞ
Z
1
2qbðqÞmðqÞJ 0 ðmsi ÞMi ðkÞdq Hðsi Þ þ OðR2 Þ;
ð10Þ
0
where i = R, h, B1(h) = e/p(1 2e2 cos2 h), B2(h) = 1/p sin 2h, si ¼ T ei R, R ¼ R=r 0 , M 1 ðkÞ ¼ KðkÞ, M 2 ðkÞ ¼ 2EðkÞ KðkÞ; k ¼ q=2R cos h, B1(h) and B2(h) determine the angular dependence of radiation when a crack is modeled as a point source by a system of three mutually perpendicular dipoles [6,7]. 2.2. Nucleation of a penny-shaped crack of mode III This type of loading may be simulated by subjecting equal and opposite tractions ±szh, which vary linearly with distance from the center of the crack. 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 the time t = 0 ± szh attain to a certain critical value s0, resulting in a penny-shaped crack nucleation. Taking account the symmetry conditions with respect to the plane z = 0 the problem can be formulated for upper half-space z > 0 with zero initial conditions and boundary conditions [4]
szh ðr; 0; tÞ ¼ s0 ðr=r0 ÞHðtÞ; r < r0 ; uh ðr; 0; tÞ ¼ 0;
r P r0 ;
ð11Þ
where uh(r, z, t) is the only non-zero component of displacement vector, governed by the dynamic equation of torsion:
@ 2 uh 1 @uh uh @ 2 uh 1 @ 2 uh þ þ ¼ : r @r @r2 r2 @z2 c22 @t2
ð12Þ
Away from the crack uh (r, z, t) tends to zero as (r2 + z2)1/2 ? 1. In the Laplace transform domain the boundary condition and equation of torsion become
szh ðr; 0; sÞ ¼ s0 ðr=r0 Þð1=sÞ; r < r0 ; uh ðr; 0; sÞ ¼ 0; r P r 0 ;
ð13Þ
@ 2 uh 1 @uh uh @ 2 uh s þ 2 þ 2 ¼ 2 uh : r @r @r 2 r @z c2
ð14Þ
763
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Applying Hankel transform over spatial coordinate r to Eq. (14) we obtain the ordinary differential equation
@ 2 uh ðn2 þ s2 =c22 Þuh : @z2
ð15Þ
The solution of this problem we may find in a similarly to the considered above problem As a result, we obtain the following equation:
uh ðr; z; TÞ ¼ A
Z
1
bðqÞmðqÞ½V 1 ðr; z; T; qÞ þ V 2 ðr; z; T; qÞ þ V 3 ðr; z; T; qÞdq;
ð16Þ
0
s0 r 0 where A ¼ p4ffiffiffiffi , 2p3l
V 1 ðr; z; T; qÞ ¼ z
Z
rþq
½½z2 þ a2 1=2
V 2 ðr; z; T; qÞ ¼ z
T
J 0 ðmðqÞðT sÞÞdsFða; r; qÞdaH
s
rq
Z
Z
Z ½z2 þ a2 1=2
rþq
s
rq
V 3 ðr; z; T; qÞ ¼ z
Z
rþq
0
H Rða; r; qÞ ¼
Z
T2
J 0 ðmðqÞðT sÞÞdsFða; r; qÞH
a½z2 þ a2 3=2 z2 þ a2
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i T 2 z2 a da;
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i T 2 z2 a ;
s¼
1=2
J 0 ðmðqÞðT sÞÞ þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 þ a2 ;
Z s
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i T 2 z2 a da; T
J 0 ðm1 ðqÞðT sÞÞds Rða; r; qÞ
1
cosðuqÞJ 1 ðruÞJ 0 ðauÞdu;
ð17Þ
0
8 0; q < a r >
> 1=2 > > > 1 1 2a > ffi f1 K sin ; k þ papqffiffiffi > 0 qþrþa
Rða; r; qÞ ¼ 1=2 pffiffiffiffiffiffiffiffi 1 > qþar 1 1 > > 1r 1 K0 sin1 qþrþ ; a Þ ; q>aþr r= Kð > k k a pk > > > :1 ; jq aj < q þ a þ r r 2
K0 ð/; kÞ ¼
p
Fða; r; qÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 EðkÞF /; 1 k þ KðkÞE /; 1 k KðkÞF /; 1 k ;k ¼
Z
1
sinðuqÞJ 0 ðruÞJ 0 ðauÞdu;
0
Fða; r; qÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 ða rÞ2 pffiffiffiffiffi ; 2 ar
8 0; > < > :
ð18Þ
q < ja rj;
1ffiffiffiffi p p ar KðkÞ;
1 ffiffiffiffi 1‘ p pk ar K k
ja rj < q < a þ r; q; a; r > 0 ; ;
q>aþr
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where F /; 1 k and E /; 1 k are the elliptic integrals of the first and second kind, respectively. The integrals in Eqs. (17) and (18) are found using [8]. 3. Numerical results 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 AE sensor are significantly larger than the dimensions of AE sources. Therefore, more attention will be given to consideration AE caused by a penny-shaped crack for a large R. The dependencies of a dimensionless value Uðr; z; TÞ ¼ ½u2z ðr; z; TÞ þ u2r ðr; z; TÞ1=2 =A vs. dimensionless time T/e calculated according to Eq. (8) for R ¼ 1000, e = 0.535 and the Poisson’s ratio m = 0.32 are given in Fig. 3a (longitudinal wave) and in Fig. 3b (shear wave). The curves 1 and 2 correspond to angles h = 15° and h = 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 expressions are proposed. They characterise angular dependencies of peak values of the displacement vector module for R 1. For longitudinal waves
1 2e2 cos2 h
UðdÞ ; 1 ðhÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ v10 cos h
ð19Þ
764
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
¼ 1000 for; (a) longitudinal and (b) transversal Fig. 3. Dependence of the dimensionless module of a displacement vector on dimensionless time T/e at R waves: curve 1 corresponds to angle of observation h = 15°, curve 2 – h = 75°.
and for shear waves
j sin 2hj
UðdÞ ; 2 ðhÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ v20 cos h
ð20Þ
where v1 and v2 are the parameters of approximation. Their numerical values v1 = 0.68 and v2 = 2.69 at e = 0.535 and m = 0.32 were obtained by the least squares method. The angular dependencies of the peak values of Uðr; z; TÞ calculated according to Eq. (8) were compared with Eqs. (19) and (20). These expressions describe the angular distribution of amplitudes with an error not exceeding 4% and correlate with Eq. (10). They are presented in Fig. 4a and b for longitudinal and shear waves, respectively. Taking into account Eqs. (11), (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 obtained:
umax jci ¼ di
2 r0 UðdÞ i ðhÞr 0 ; 2 qc1 R
ð21Þ
where i = 1 corresponds to longitudinal wave and i = 2 to shear wave, d1 = 0.452, d2 = 0.832. In these expressions R is dimensional. The analysis of dependence (21) 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 finding the location of AE source. It was found also, that the time interval to achieve the first maximum essentially decreases with increase of the orientation angle h, and the duration of its rear front changes insignificantly (see Fig. 3). The numerical calculation of dimensionless value of U h ðr; z; TÞ ¼ uh ðr; z; TÞ=A for mode III crack depending on dimensionless time T was carried out according to Eq. (16). The results for R = 100 (R ¼ ðr 2 þ z2 Þ1=2 ) are shown in Fig. 5. The magnitude of the first maximum that corresponds to arriving of the shear wave front to the observation point is highest. The distances between peaks are approximately the same. The calculation of this normalized displacement U X ¼ uðXÞ=uðXÞjmax depending
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
765
Fig. 4. Angular dependencies of the maximum values of the module of a displacement vector for (a) longitudinal and (b) shear waves.
Fig. 5. Dependence of Uh(r, z, T) on T: 1 – u = 15°, 2 – u = 45°, R = 100.
on dimensionless frequency X ¼ xr 0 =c2 was made using fast Fourier transform method [9] and is shown in Fig. 6 for the same values of R and u. Similar calculations of U h ðr; z; TÞ depending on dimensionless time T were carried out for various values of R and u. After the analysis of maximal values of U h ðr; z; TÞ we obtain the following approximation dependence (for R > 30r0): ðdÞ
s0 r20 U2 ðuÞ ffi ; uhj max ¼ 1:138 pffiffiffiffiffiffi 2p3lR ðdÞ
ð22Þ
where U2 ðuÞ is determined by Eq. (20). The approximation error has not exceed 7.5%.The angular distribution of uhjmax value is the same as for mode I crack shear wave (see Fig. 4b).
766
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Fig. 6. Dependence of UX on X: 1 – u = 15°, 2 – u = 45°, R = 100.
Fig. 7. Maximal displacements calculated according to Eqs. (21) (a) and (22) (b): curve 1 – for D18 Al alloy and 2 – for B95 Al alloy.
The results of calculation according to Eqs. (21) and (22) for two aluminium alloys D18 (chemical composition – Si 0.5%, Fe 0.5%, Cu 2.2–3.0%, Mn 0.2%, Mg 0.5%, Cr 0.1%, Zn 0.1%, other 0.15%) and B95 (chemical composition – Zn 5.0–6.5%, Mg 1.8–2.8%, Cu 1.4–2.0%, Mn 0.2–0.6%, Cr 0.1–0.25%, Fe 0.25–0.5%, Si 0.1%, Ni 0.1%, Ti 0.05%, others <0.1%) used in airplane production are shown in Fig. 7. The following characteristics were used for D18 Al alloy: r0 = 300 MPa, s0 = 200 MPa, E = 71,000 MPa, m = 0.32; for B95 Al alloy: r0 = 480 MPa, s0 = 32 MPa, E = 70,000 MPa, m = 0.33 and q = 2700 kg m3 R = 0.5 m pr 20 = 106 m2 for both materials [10].
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
767
Supposing that umax jci and uh jmax are directly proportional to amplitude V of the electric signal at the sensor output we can rewrite Eqs. (21) and (22) as follows.
V ¼ bi
2 r0 UðdÞ s0 r20 UðdÞ ðuÞ i ðhÞr 0 ffi2 : ; V ¼ bh pffiffiffiffiffiffi 2 qc1 R 2p3lR
ð23Þ
for mode I and mode III cracks, respectively, bi and bh are the proportionality factors for these cracks. Most AE results are obtained by using piezoelectric sensors. Their sensitivity depends mainly on the piezoelectric ceramic sensitivity, of which they are made. But other parameters like surface roughness of the inspected element, the lubricant characteristics used for sensor coupling also influence the piezoelectric sensor sensitivity. Therefore, before AE recording the calibration of the sensitivity of the entire measuring system should be performed using for instance capacity sensors. The parameters bi and bh should be determined from such procedure considering also the AE device characteristics (the amplification factor, frequency operating range, etc.). 4. Conclusion Formation of modes I and III crack we simulated by a sudden fall of the stress on the surface of the rupture from the initial level r0 in elastic continuum to zero. Solving the equations of motion with appropriate boundary conditions we have obtained the components of displacement vector as a function of time and coordinates for arbitrary distances from the center of the crack to the observation point. Applying the method of fast Fourier transform we calculated the frequency dependence of the displacement vector components and found that the width of the spectrum maximum is inversely proportional to the crack size. Analyzing the maximum values of displacement vector components, depending on the angle between the plane of the crack location and direction of observation point, we have obtained the approximation formulas that determine the radiation patterns for longitudinal and shear waves, the parameters of approximation we calculated by the least squares method, and found that the maximum values of the vector displacement components at a distance greater than 28r0 are proportional to the crack area. Asymptotic dependence (10) of the displacement vector components for large R, when the crack is considered as a point source, determines angular dependence of radiation. The functions B1(h) and B2(h) determine the radiation patterns for longitudinal and shear waves, respectively, are the same as obtained in MT analysis [6] applied for estimation of AE caused by the formation and growth of cracks. Thus, Eq. (20) establishes a relationship between our analysis and MT analysis. However, in our analysis we found the time dependence of the displacement vector components. Both asymptotic dependence (20) and general expression (7) demonstrate that the maximum values of displacement vector components are proportional to the crack area (r 20 ) and decay as 1/R. Approximation expression (23) for penny-shaped cracks of modes I and III allows to assess the crack area by the AE signal amplitude. Measuring the angular distribution of AE amplitudes by a set of AE sensors one can find from approximation equations, which describe the radiation pattern, the spatial location of the crack. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Lysak MV. Acoustic emission during jumps in subcritical growth of cracks in three-dimensional bodies. Engng Fract Mech 1994;47:873–9. Andreykiv AYe, Lysak NV. A method of acoustic emission in investigation of fracture processes. Kiev: Naukova Dumka; 1989. 176p. [in Russian]. Lysak MV. Development of the theory of acoustic emission by propagating cracks in terms of fracture mechanics. Engng Fract Mech 1996;55:443–52. Chen EP, Sih GC. Transient response of cracks to impact loads. Elastodynamic Crack Probl Series Mech Fract 1977;4:1–58. Sih GC, Embley GT, Ravera RS. Impact response of a finite crack in plane extension. Int J Solids Struct 1972;8(7):977–93. Ohtsu M. Source inversion of acoustic emission waveform. Struct Engng/Earthquake Engng 1988;5(2):275–83. Wadley HWG, Scruby CB. Elastic wave radiation from cleavage extension. Int J Fract 1983;23(2):111–28. Prudnikov AP, Brychkov Yu A, Marichev OI. Integrals and Series. Special Functions. Moscow: ‘‘Nauka”; 1983. 751p. Ahmed N, Rao KR. Orthogonal transformations for digital signal processing. New-York: Springer-Verlag, Berlin-Heidelberg; 1975. 248p. Ostash OP, Fedirko VM, Uchanin VM, et al. In: Ostash OP, Fedirko VM, editors. Fracture Mechanics and Strength of Materials. Panasyuk (Editor-in-chief) VV, editors. Strength and Durability of Airplane Materials and Structural Elements, vol. 9. Lviv; 2007. p. 1068.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Acoustic emission estimation of crack formation in aluminium alloys Olexander Andreykiv, Valentyn Skalsky, Oleh Serhiyenko, Denys Rudavskyy * Karpenko Physico-Mechanical Institute of National Academy of Sciences of Ukraine, 5, Naukova Str., Lviv 79601, Ukraine
a r t i c l e
i n f o
Article history: Received 16 December 2008 Received in revised form 23 November 2009 Accepted 12 January 2010 Available online 28 January 2010 Keywords: Penny-shaped crack Acoustic emission Aluminium alloy
a b s t r a c t In the paper a problem on the radiation of waves of acoustic emission (AE) during formation of penny-shaped crack in aluminium alloy elastic body under tensile and twisting loading is considered. By the method of integral transforms, the problem is reduced to the solving of Fredholm integral equation of the second kind. The dependences of component of displacement vector on time and distance to the viewpoint as well as amplitudes of the AE signals on the area of the crack and its spatial orientation are obtained. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Aluminium and its alloys are widely used in modern engineering mainly in aviation and space industry. Various units, structural elements, etc. are made of them. During their operation under action of physical and chemical factors crack-like defects can form. They are especially dangerous for the structural integrity of these elements. For non-destructive testing of crack nucleation and propagation the phenomenon of acoustic emission (AE) is used. Quantitative application of AE as a method of non-destructive testing needs to find the relationships between AE signals and crack parameters (its size, spatial position, etc.). For this purpose we consider a problem of sudden formation of a penny-shaped crack in homogeneous elastic body under tensile and twisting loading (mode I and mode III, respectively). We will try to find the displacement vector components caused by this crack formation directly from equations of motion with appropriate boundary conditions corresponding to abrupt formation of the rupture without any additional conditions. We restricted our analysis to consideration of penny-shaped cracks of mode I and mode III because solving of the problem for mode II crack is more complicated and not completed yet. 2. Problem formulation and solving 2.1. Nucleation of a penny-shaped crack of mode I According to approach proposed in [1] 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 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. 1). At infinity tensile stresses r are applied along Oz axes. At the time * Corresponding author. E-mail address: [email protected] (D. Rudavskyy). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.01.009
760
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Nomenclature b, m parameters of approximation velocity of longitudinal and shear wave, respectively c1, c2 r, h, z cylindrical coordinates radius of crack r0 t time s parameter of Laplace integral transform ur, uh, uz components of a displacement vector V amplitude of the electric signal Lame’s constants k, l m Poisson’s ratio q material density r0, s0 the integral characteristics of material breaking strength rzz, szh, srz components of a stress tensor scalar potentials u, w v1, v2 parameters of approximation E(k), K(k) complete elliptic integrals of the first and second kind, respectively J0() and J1() is the zero and first order Bessel functions of the first kind F(), E() elliptic integrals of the first and second kind, respectively H(t) the Heaviside function
Fig. 1. A penny-shaped crack in an elastic body (mode I).
t = 0 they achieve certain critical value r0, resulting in a penny-shaped crack nucleation. Using known approaches [2–4] this problem can be reduced to the wave equations
Du
1 @2u ¼ 0; c21 @t2
Dw
w 1 @2w ¼ 0; r c22 @t2
ð1Þ
with respect to unknown scalar potentials /(r, z, t) and w(r, z, t). Eq. (1) should satisfy the boundary conditions for half-space z > 0
rzz ðr; 0; tÞ ¼ r0 HðtÞ; r 6 r0 ; uz ðr; 0; tÞ ¼ 0;
r > r0 ;
srz ðr; 0; tÞ ¼ 0; 0 < r < 1;
ð2Þ
761
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
and zero initial conditions
@ u @w ¼ ¼ 0: @t t¼0 @t t¼0
uðr; z; 0Þ ¼ wðr; z; 0Þ ¼ Here c1 ¼
ð3Þ
qffiffiffiffiffiffiffiffi qffiffiffi kþ2l l q , c2 ¼ q. The dependence between potentials / and w and components of a displacement vector ur and uz
has the form:
ur ¼
@ u @w ; @z @r
uz ¼
@ u @w w þ : þ @z @r r
ð4Þ
The boundary problem (1)–(3) can be solved by the method of integral transforms. Similarly to [4], 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:
Kðq; sÞ ¼
Z
1
Kðu; sÞKðu; q; sÞdu ¼ q;
0 6 q 6 1;
ð5Þ
0
having the kernel K(u, q, s) symmetric with respect to u and q in the form
Kðu; q; sÞ ¼
2 pð1 e2 Þ
Z
1
gðg=r 0 ; sÞ sinðgqÞ sinðguÞdg;
ð6Þ
0
where 2
gðg=r 0 ; sÞ ¼ ð1 e Þ
"
# 2 1=2 1 e2 1=2 2 2 2g n ; 1þ 2 2 1þ 2 2 1þ 2 2 2g n gn gn 1
n¼
c2 ; sr0
e ¼ c2 =c1 :
The solution of integrated Eq. (5) was found numerically and approximated by dependence proposed in [5]:
bðqÞmðqÞ Kðq; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; m2 þ p2
ð7Þ
where b(q) and m(q) are the parameters of approximation obtained by the least squares method, p = sr0/c2. Taking into account the dependence in (7), after inversion of Laplace and Hankel transforms using the Parseval and convolution theorems, for the components of displacement vector we obtain the following equation:
ui ðr ; z; TÞ ¼ ABi
Z 0
1
bðqÞmðqÞ
"Z
rþq
rq
2 X
# pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Sij ða; z; sj ; qÞH T ej z þ a Gi ða; r ; qÞda dq;
r > 1;
ði ¼ z; rÞ;
ð8Þ
j¼1
r0 where A ¼ qc2 e2rp0 ð1 , z ¼ z=r 0 , r ¼ r=r0 , T ¼ c2 t=r 0 is the dimensionless time, e1 = e, e2 = 1, e2 Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi Br ¼ ðp r Þ1 , Gz ¼ aKðkÞ, Gr ¼ a a½2EðkÞ KðkÞ k ¼ q2 a rÞ2 =ð4ar Þ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
sj ¼ T ej z2 þ a2 , Bz ¼ z=ðp rÞ,
Sij ða; z; sj ; qÞ ¼ Dij ða; zÞJ 0 ðmðqÞsj Þ þ Eða; zÞJ 1 ðmðqÞsj Þ þ F ij ða; zÞIðmðqÞsj Þ; D11 ða; z; qÞ ¼ eðz2 þ a2 Þ1 ½1 2a2 e2 ðz2 þ a2 Þ1 þ 3m2 ðqÞs1 ðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; E11 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 e2 ðz2 þ a2 Þðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; n o F 11 ða; z; qÞ ¼ mðqÞ1 ðz2 þ a2 Þ3=2 ½1 2e2 þ 3e2 a2 ðz2 þ a2 Þ1 þ 3ðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ðT 2 m2 Þ ; D12 ða; z; qÞ ¼ 2a2 ðz2 þ a2 Þ2 3m2 ðqÞs2 ðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; E12 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 ðz2 þ a2 Þðz2 þ a2 Þ5=2 ½2 5a2 ðz2 þ a2 Þ1 ; n o F 12 ða; z; qÞ ¼ mðqÞ1 3ðz2 þ a2 Þ5=2 ½5a2 ðz2 þ a2 Þ1 2ðT 2 mðqÞ2 Þ þ ðz2 þ a2 Þ3=2 ½2 3a2 ðz2 þ a2 Þ1 ; D21 ða; z; qÞ ¼ eðz2 þ a2 Þ1 ½2e2 z2 ðz2 þ a2 Þ1 þ 1 2e þ 3m2 ðqÞs1 ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1; E21 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 e2 ðz2 þ a2 Þðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1; n o F 21 ða; z; qÞ ¼ mðqÞ1 3ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1ðT 2 mðqÞ2 Þ þ ðz2 þ a2 Þ3=2 ½1 e2 3z2 e2 ðz2 þ a2 Þ1 ;
762
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Fig. 2. A penny-shaped crack in an elastic body (mode III).
D22 ða; z; qÞ ¼ 3m2 ðqÞs2 ðz2 þ a2 Þ5=2 ½1 5z2 ðz2 þ a2 Þ1 2z2 ðz2 þ a2 Þ2 ; E22 ða; z; qÞ ¼ 3m1 ðqÞ½T 2 ðz2 þ a2 Þ þ ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1; n o F 22 ða; z; qÞ ¼ mðqÞ1 3ðz2 þ a2 Þ5=2 ½5z2 ðz2 þ a2 Þ1 1ðT 2 mðqÞ2 Þ þ ðz2 þ a2 Þ3=2 ½1 3z2 ðz2 þ a2 Þ1 ; Z x J 0 ðtÞdt: IðxÞ ¼ 0
Asymptotic dependence of the displacement vector components one can obtain by leaving in Eq. (8) terms decaying weakly than 1/R (R2 = r2 + z2). In spherical coordinate system originated in the center of the penny-shaped crack the components of displacement vector are as follows
uR ¼ ur cos h þ uz sin h;
uh ¼ uz cos h ur sin h:
ð9Þ
Replacing internal integrals over a in Eq. (8) by their average values as a tends to r, for large R after simplification we shall obtain the asymptotic expression for Eq. (8):
ui ðR; h; TÞ ¼ ABi ð1=RÞ
Z
1
2qbðqÞmðqÞJ 0 ðmsi ÞMi ðkÞdq Hðsi Þ þ OðR2 Þ;
ð10Þ
0
where i = R, h, B1(h) = e/p(1 2e2 cos2 h), B2(h) = 1/p sin 2h, si ¼ T ei R, R ¼ R=r 0 , M 1 ðkÞ ¼ KðkÞ, M 2 ðkÞ ¼ 2EðkÞ KðkÞ; k ¼ q=2R cos h, B1(h) and B2(h) determine the angular dependence of radiation when a crack is modeled as a point source by a system of three mutually perpendicular dipoles [6,7]. 2.2. Nucleation of a penny-shaped crack of mode III This type of loading may be simulated by subjecting equal and opposite tractions ±szh, which vary linearly with distance from the center of the crack. 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 the time t = 0 ± szh attain to a certain critical value s0, resulting in a penny-shaped crack nucleation. Taking account the symmetry conditions with respect to the plane z = 0 the problem can be formulated for upper half-space z > 0 with zero initial conditions and boundary conditions [4]
szh ðr; 0; tÞ ¼ s0 ðr=r0 ÞHðtÞ; r < r0 ; uh ðr; 0; tÞ ¼ 0;
r P r0 ;
ð11Þ
where uh(r, z, t) is the only non-zero component of displacement vector, governed by the dynamic equation of torsion:
@ 2 uh 1 @uh uh @ 2 uh 1 @ 2 uh þ þ ¼ : r @r @r2 r2 @z2 c22 @t2
ð12Þ
Away from the crack uh (r, z, t) tends to zero as (r2 + z2)1/2 ? 1. In the Laplace transform domain the boundary condition and equation of torsion become
szh ðr; 0; sÞ ¼ s0 ðr=r0 Þð1=sÞ; r < r0 ; uh ðr; 0; sÞ ¼ 0; r P r 0 ;
ð13Þ
@ 2 uh 1 @uh uh @ 2 uh s þ 2 þ 2 ¼ 2 uh : r @r @r 2 r @z c2
ð14Þ
763
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Applying Hankel transform over spatial coordinate r to Eq. (14) we obtain the ordinary differential equation
@ 2 uh ðn2 þ s2 =c22 Þuh : @z2
ð15Þ
The solution of this problem we may find in a similarly to the considered above problem As a result, we obtain the following equation:
uh ðr; z; TÞ ¼ A
Z
1
bðqÞmðqÞ½V 1 ðr; z; T; qÞ þ V 2 ðr; z; T; qÞ þ V 3 ðr; z; T; qÞdq;
ð16Þ
0
s0 r 0 where A ¼ p4ffiffiffiffi , 2p3l
V 1 ðr; z; T; qÞ ¼ z
Z
rþq
½½z2 þ a2 1=2
V 2 ðr; z; T; qÞ ¼ z
T
J 0 ðmðqÞðT sÞÞdsFða; r; qÞdaH
s
rq
Z
Z
Z ½z2 þ a2 1=2
rþq
s
rq
V 3 ðr; z; T; qÞ ¼ z
Z
rþq
0
H Rða; r; qÞ ¼
Z
T2
J 0 ðmðqÞðT sÞÞdsFða; r; qÞH
a½z2 þ a2 3=2 z2 þ a2
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i T 2 z2 a da;
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i T 2 z2 a ;
s¼
1=2
J 0 ðmðqÞðT sÞÞ þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 þ a2 ;
Z s
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i T 2 z2 a da; T
J 0 ðm1 ðqÞðT sÞÞds Rða; r; qÞ
1
cosðuqÞJ 1 ðruÞJ 0 ðauÞdu;
ð17Þ
0
8 0; q < a r >
> 1=2 > > > 1 1 2a > ffi f1 K sin ; k þ papqffiffiffi > 0 qþrþa
Rða; r; qÞ ¼ 1=2 pffiffiffiffiffiffiffiffi 1 > qþar 1 1 > > 1r 1 K0 sin1 qþrþ ; a Þ ; q>aþr r= Kð > k k a pk > > > :1 ; jq aj < q þ a þ r r 2
K0 ð/; kÞ ¼
p
Fða; r; qÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 EðkÞF /; 1 k þ KðkÞE /; 1 k KðkÞF /; 1 k ;k ¼
Z
1
sinðuqÞJ 0 ðruÞJ 0 ðauÞdu;
0
Fða; r; qÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 ða rÞ2 pffiffiffiffiffi ; 2 ar
8 0; > < > :
ð18Þ
q < ja rj;
1ffiffiffiffi p p ar KðkÞ;
1 ffiffiffiffi 1‘ p pk ar K k
ja rj < q < a þ r; q; a; r > 0 ; ;
q>aþr
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where F /; 1 k and E /; 1 k are the elliptic integrals of the first and second kind, respectively. The integrals in Eqs. (17) and (18) are found using [8]. 3. Numerical results 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 AE sensor are significantly larger than the dimensions of AE sources. Therefore, more attention will be given to consideration AE caused by a penny-shaped crack for a large R. The dependencies of a dimensionless value Uðr; z; TÞ ¼ ½u2z ðr; z; TÞ þ u2r ðr; z; TÞ1=2 =A vs. dimensionless time T/e calculated according to Eq. (8) for R ¼ 1000, e = 0.535 and the Poisson’s ratio m = 0.32 are given in Fig. 3a (longitudinal wave) and in Fig. 3b (shear wave). The curves 1 and 2 correspond to angles h = 15° and h = 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 expressions are proposed. They characterise angular dependencies of peak values of the displacement vector module for R 1. For longitudinal waves
1 2e2 cos2 h
UðdÞ ; 1 ðhÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ v10 cos h
ð19Þ
764
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
¼ 1000 for; (a) longitudinal and (b) transversal Fig. 3. Dependence of the dimensionless module of a displacement vector on dimensionless time T/e at R waves: curve 1 corresponds to angle of observation h = 15°, curve 2 – h = 75°.
and for shear waves
j sin 2hj
UðdÞ ; 2 ðhÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ v20 cos h
ð20Þ
where v1 and v2 are the parameters of approximation. Their numerical values v1 = 0.68 and v2 = 2.69 at e = 0.535 and m = 0.32 were obtained by the least squares method. The angular dependencies of the peak values of Uðr; z; TÞ calculated according to Eq. (8) were compared with Eqs. (19) and (20). These expressions describe the angular distribution of amplitudes with an error not exceeding 4% and correlate with Eq. (10). They are presented in Fig. 4a and b for longitudinal and shear waves, respectively. Taking into account Eqs. (11), (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 obtained:
umax jci ¼ di
2 r0 UðdÞ i ðhÞr 0 ; 2 qc1 R
ð21Þ
where i = 1 corresponds to longitudinal wave and i = 2 to shear wave, d1 = 0.452, d2 = 0.832. In these expressions R is dimensional. The analysis of dependence (21) 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 finding the location of AE source. It was found also, that the time interval to achieve the first maximum essentially decreases with increase of the orientation angle h, and the duration of its rear front changes insignificantly (see Fig. 3). The numerical calculation of dimensionless value of U h ðr; z; TÞ ¼ uh ðr; z; TÞ=A for mode III crack depending on dimensionless time T was carried out according to Eq. (16). The results for R = 100 (R ¼ ðr 2 þ z2 Þ1=2 ) are shown in Fig. 5. The magnitude of the first maximum that corresponds to arriving of the shear wave front to the observation point is highest. The distances between peaks are approximately the same. The calculation of this normalized displacement U X ¼ uðXÞ=uðXÞjmax depending
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
765
Fig. 4. Angular dependencies of the maximum values of the module of a displacement vector for (a) longitudinal and (b) shear waves.
Fig. 5. Dependence of Uh(r, z, T) on T: 1 – u = 15°, 2 – u = 45°, R = 100.
on dimensionless frequency X ¼ xr 0 =c2 was made using fast Fourier transform method [9] and is shown in Fig. 6 for the same values of R and u. Similar calculations of U h ðr; z; TÞ depending on dimensionless time T were carried out for various values of R and u. After the analysis of maximal values of U h ðr; z; TÞ we obtain the following approximation dependence (for R > 30r0): ðdÞ
s0 r20 U2 ðuÞ ffi ; uhj max ¼ 1:138 pffiffiffiffiffiffi 2p3lR ðdÞ
ð22Þ
where U2 ðuÞ is determined by Eq. (20). The approximation error has not exceed 7.5%.The angular distribution of uhjmax value is the same as for mode I crack shear wave (see Fig. 4b).
766
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
Fig. 6. Dependence of UX on X: 1 – u = 15°, 2 – u = 45°, R = 100.
Fig. 7. Maximal displacements calculated according to Eqs. (21) (a) and (22) (b): curve 1 – for D18 Al alloy and 2 – for B95 Al alloy.
The results of calculation according to Eqs. (21) and (22) for two aluminium alloys D18 (chemical composition – Si 0.5%, Fe 0.5%, Cu 2.2–3.0%, Mn 0.2%, Mg 0.5%, Cr 0.1%, Zn 0.1%, other 0.15%) and B95 (chemical composition – Zn 5.0–6.5%, Mg 1.8–2.8%, Cu 1.4–2.0%, Mn 0.2–0.6%, Cr 0.1–0.25%, Fe 0.25–0.5%, Si 0.1%, Ni 0.1%, Ti 0.05%, others <0.1%) used in airplane production are shown in Fig. 7. The following characteristics were used for D18 Al alloy: r0 = 300 MPa, s0 = 200 MPa, E = 71,000 MPa, m = 0.32; for B95 Al alloy: r0 = 480 MPa, s0 = 32 MPa, E = 70,000 MPa, m = 0.33 and q = 2700 kg m3 R = 0.5 m pr 20 = 106 m2 for both materials [10].
O. Andreykiv et al. / Engineering Fracture Mechanics 77 (2010) 759–767
767
Supposing that umax jci and uh jmax are directly proportional to amplitude V of the electric signal at the sensor output we can rewrite Eqs. (21) and (22) as follows.
V ¼ bi
2 r0 UðdÞ s0 r20 UðdÞ ðuÞ i ðhÞr 0 ffi2 : ; V ¼ bh pffiffiffiffiffiffi 2 qc1 R 2p3lR
ð23Þ
for mode I and mode III cracks, respectively, bi and bh are the proportionality factors for these cracks. Most AE results are obtained by using piezoelectric sensors. Their sensitivity depends mainly on the piezoelectric ceramic sensitivity, of which they are made. But other parameters like surface roughness of the inspected element, the lubricant characteristics used for sensor coupling also influence the piezoelectric sensor sensitivity. Therefore, before AE recording the calibration of the sensitivity of the entire measuring system should be performed using for instance capacity sensors. The parameters bi and bh should be determined from such procedure considering also the AE device characteristics (the amplification factor, frequency operating range, etc.). 4. Conclusion Formation of modes I and III crack we simulated by a sudden fall of the stress on the surface of the rupture from the initial level r0 in elastic continuum to zero. Solving the equations of motion with appropriate boundary conditions we have obtained the components of displacement vector as a function of time and coordinates for arbitrary distances from the center of the crack to the observation point. Applying the method of fast Fourier transform we calculated the frequency dependence of the displacement vector components and found that the width of the spectrum maximum is inversely proportional to the crack size. Analyzing the maximum values of displacement vector components, depending on the angle between the plane of the crack location and direction of observation point, we have obtained the approximation formulas that determine the radiation patterns for longitudinal and shear waves, the parameters of approximation we calculated by the least squares method, and found that the maximum values of the vector displacement components at a distance greater than 28r0 are proportional to the crack area. Asymptotic dependence (10) of the displacement vector components for large R, when the crack is considered as a point source, determines angular dependence of radiation. The functions B1(h) and B2(h) determine the radiation patterns for longitudinal and shear waves, respectively, are the same as obtained in MT analysis [6] applied for estimation of AE caused by the formation and growth of cracks. Thus, Eq. (20) establishes a relationship between our analysis and MT analysis. However, in our analysis we found the time dependence of the displacement vector components. Both asymptotic dependence (20) and general expression (7) demonstrate that the maximum values of displacement vector components are proportional to the crack area (r 20 ) and decay as 1/R. Approximation expression (23) for penny-shaped cracks of modes I and III allows to assess the crack area by the AE signal amplitude. Measuring the angular distribution of AE amplitudes by a set of AE sensors one can find from approximation equations, which describe the radiation pattern, the spatial location of the crack. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Lysak MV. Acoustic emission during jumps in subcritical growth of cracks in three-dimensional bodies. Engng Fract Mech 1994;47:873–9. Andreykiv AYe, Lysak NV. A method of acoustic emission in investigation of fracture processes. Kiev: Naukova Dumka; 1989. 176p. [in Russian]. Lysak MV. Development of the theory of acoustic emission by propagating cracks in terms of fracture mechanics. Engng Fract Mech 1996;55:443–52. Chen EP, Sih GC. Transient response of cracks to impact loads. Elastodynamic Crack Probl Series Mech Fract 1977;4:1–58. Sih GC, Embley GT, Ravera RS. Impact response of a finite crack in plane extension. Int J Solids Struct 1972;8(7):977–93. Ohtsu M. Source inversion of acoustic emission waveform. Struct Engng/Earthquake Engng 1988;5(2):275–83. Wadley HWG, Scruby CB. Elastic wave radiation from cleavage extension. Int J Fract 1983;23(2):111–28. Prudnikov AP, Brychkov Yu A, Marichev OI. Integrals and Series. Special Functions. Moscow: ‘‘Nauka”; 1983. 751p. Ahmed N, Rao KR. Orthogonal transformations for digital signal processing. New-York: Springer-Verlag, Berlin-Heidelberg; 1975. 248p. Ostash OP, Fedirko VM, Uchanin VM, et al. In: Ostash OP, Fedirko VM, editors. Fracture Mechanics and Strength of Materials. Panasyuk (Editor-in-chief) VV, editors. Strength and Durability of Airplane Materials and Structural Elements, vol. 9. Lviv; 2007. p. 1068.