- Email: [email protected]
Data interpretation in ACPD crack inspection
Data interpretation in ACPD crack inspection M. C. Lugg The voltages measured in the ACPD crack-sizing technique are sharply peaked at the crack edge for a short crack in a material of small skin depth, or for any crack in a material of large skin depth. The conventional sizing method uses these voltages as a first estimate of crack depth. A number of situations arise where these voltages may not be measured accurately, however, including rapid automated scanning and hand-held readings at weld toes. In such cases, more accurate sizing may be obtained using methods based on the saddle voltage with the probe straddling the crack. Multipliers have been calculated which allow crack depths to be estimated from the saddle voltage for circular arc or semi-elliptical cracks in the extremes of either small or large skin depth. These multipliers are presented in graphical form together with guidelines for their use.
Keywords: AC potential drop (ACPD) technique, crack sizing, saddle voltage, circular arc and semi-elliptical cracks The AC potential drop (ACPD) technique is used to size surface-breaking defects in two main areas. In crack growth monitoring, a fatigue crack is sized at regular intervals throughout a fatigue test in order to produce crack growth rate data. In this situation it is usual to have a fixed number of voltage probe sites and so the distribution of data points along a crack can be quite sparse, particularly when the crack is small. An earlier paper [~] described a method for optimizing the accuracy of crack sizing with sparse data. The other main use of the ACPD technique is for crack inspection. In this case potential difference readings are taken around a crack using either a hand-held probe or, if the specimen geometry permits, a probe driven by a stepper motor under computer control. In this way, the electric field around a crack can be mapped in as much detail as the time available allows. A review of the use of the ACPD technique is found in Collins et alt2J. Crack inspection conventionally involves obtaining pairs of potential differences at a series of points along the crack length: V1 when the probe is just off the crack, and V2 when the probe just straddles the crack (see Figure la). This information is sufficient to determine the crack shape in most situations.
The c o n v e n t i o n a l i n t e r p r e t a t i o n f o r a thi n skin A thin skin material is one in which the skin depth 6 is much less than the probe spacing A and the depth d of the crack being measured. In practice, as long as 6/A and f/d are less than about 0.1, the thin skin approximation will be valid. At frequencies of a few kilohertz, this applies to cracks of depth 1 mm or more in magnetic steels or 10 mm in aluminium. If the crack depth is in turn small
compared with the crack length a, the crack can be cotlsidered as essentially of uniform depth, and the voltage trace from a probe moving perpendicularly across the crack far from the ends follows the simple 'top-hat' shape shown in Figure lb. This arises because the current flow is undisturbed by the presence of the crack right up to the crack edge. The voltages V1 and V2 a r e then constant, and accurate probe positioning is not required. The crack depth is found from the formula
d=d~=~- ~ - 1
(1)
v~
g Y,
o~
4--
b
I1
Probe positioo
C
Probe position
Fig. 1 Voltages measured around a surface-breaking crack: (a) probe positions; (b) voltage trace for a probe traversing a uniform depth crack with a small skin depth &; (c) voltage trace for a finite length crack or for a large skin depth
0308-9126/89/030149-06/$03.00 © 1989 Butterworth & Co (Publishers) Ltd NDT International Volume 22 Number 3 June 1989
149
If the crack aspect ratio is not large, however, the voltage trace from a traversing probe has a shape similar to that shown in Figure lc. The current flow is now perturbed by the presence of the crack, and the voltage trace peaks sharply at V~ and V2. For cases where the crack depth is not uniform, a series of solutions have been obtained for various crack shapes, the most realistic one being that for a semi-elliptical crack. In this case no analytic equation for crack depth can be written down, but a series of multipliers M have been calculated TM from which the crack depth can be found:
d=diM
(2)
where M is a function of aspect ratio, probe gap and distance from the crack centre-line.
The c o n v e n t i o n a l i n t e r p r e t a t i o n f o r a t h i c k skin A thick skin material is one in which the skin depth is much larger than the crack depth, although the thick skin approximation is valid in practice as long as 6/d is greater than about 2. At frequencies of a few kilohertz, this applies to cracks of depth 3 mm or less in titanium and stainless steel or 0.5 mm in aluminium. As with a low aspect ratio crack in a thin skin material, the voltage trace from a probe traversing any crack in a thick skin material has a shape like that in Figure lc. In the thick skin extreme the surface electric potential ~b is given by t4] = E o ( y 2 +d2) 1/2
(3)
where E o is the uniform upstream electric field strength and y is the perpendicular distance from the crack edge. The potential is of opposite sign either side of the crack, so there is a jump in ~b of 2Eod across the crack. From Equation (3) it can be seen that
d2)x/2 - Eod V2=c~(A)-c~(-O)=Eo(a2 +d2)~/2+ Eod V1 = ~b(A) - ~ ( + 0 ) = Eo(A 2 +
A (k V l~l/2(~-l) V:/
From Equation (2) the measured off-crack voltage would now be V~ = 4~(1.1A) - ~b(0.1A) = EoA(~/2.21 - x/1.01) = 0.4816EoA which is 16% higher than the true V1. At the same time, the measured cross-crack voltage would be V~ = ~b(0.9A) - ~b(-0.1A) = EoA(x/1.81 + x/1.01) = 2.3504EoA which is 2.6 % lower than the true V2. The combination of these results substituted into Equation (5) gives a measured crack depth of d' =0.878A, 12% lower than the true depth. The same considerations apply to measurements on cracks of finite aspect ratio. The use of V1 and V2 to determine crack depth can thus lead to results that are highly sensitive to probe position. In cases where this would cause a problem, such as when rapid automated scanning is required, an alternative approach should be considered. The two other voltages uniquely determined in the trace of Figure lc are the saddle point voltage V~, when the probe is bisected by the crack, and the undisturbed upstream voltage Vo. Both of these voltages are relatively insensitive to probe position and may be of more use for crack sizing in certain situations. This alternative approach to crack sizing is developed in .the rest of the paper.
The a l t e r n a t i v e i n t e r p r e t a t i o n f o r a t h i c k skin and a u n i f o r m d e p t h crack
(6)
In order to use the stable voltages V~ and Vo to size a large aspect ratio crack in a thick skin material, an alternative formula for crack depth is needed to replace Equation (5). From Equation (2) it can be seen that the upstream voltage is simply
The case f o r a l t e r n a t i v e m u l t i p l i e r s Accurate crack sizes can be obtained using the conventional multipliers in a wide variety of situations, as long as V~ and V2 can be measured accurately. However, since I,'1 and V2 are sharply peaked, slight changes in probe position will give rise to large changes in the measured voltages. In practice, care must be taken to place the leg of the voltage probe as close as possible to the crack edge. With a hand-held probe this can be very time consuming, while with a stepper motor drive the step size has to be made very small, giving rise to a long scanning time. The effect of poor probe positioning can be illustrated by the case of the uniform crack in a thick skin material. Consider the case where A = d. From Equation (3) the true off-crack voltage is
150
Substitution into Equation (5) gives the true crack depth d = A. Now consider the result when the probe is positioned a distance A/10 away from the crack edge for each reading. This would follow from a misplacement of only 1 mm for a standard 10 mm probe.
(5)
For semi-elliptical cracks, thick skin multipliers have recently been calculated TM.
I/1 = EoA(x/2 - 1) = 0.4142EoA
V2 = EoA(x/2 + 1) = 2.4142EoA
(4)
from which the crack depth is given by d=
and from Equation (4) the true cross-crack voltage is
Vo = Lim [tp(y + A) - q~(y)] = EoA
(7)
while the saddle voltage is given by V~= q ~ ( A ) - q~(_ A ) = Eo(A2 + 4d2)1/2
(8)
Rearranging Equations (7) and (8), the alternative formula for crack depth is found to be
d=~kVg
1
(9)
The desensitivity to probe position obtained with this equation can be demonstrated by applying the same test used in the previous section. If the probe is displaced by a distance A/10, the measured upstream voltage will be unchanged since V~ = Eo A
NDT International June 1989
The measured saddle voltage will be
107 .
v; = 4 , ( 0 . 6 A ) - 4'( - 0.4A) = EoA(x/1.36 + ,,/1.16)
I0 s .
= 2.2432EoA
iOs
which is only 0.3 % higher than the true saddle voltage given by
tO4 tO~
V~= 25(0.5A) = 2EoA~/1.25 = 2.2361EoA Substituting the measured voltages into Equation (9) gives the measured depth as
i0 z o
z
vz i01
d ' = 1.004A
I
which is only 0.4% higher than the true depth. The sensitivity to probe positioning is thus seen to be very much reduced. There are disadvantages to this alternative interpretation of ACPD readings, however. The first disadvantage is a practical one: both the conventional and alternative interpretations require that the incident electric field be of uniform strength, at least over a region covering the crack and the probe positions. The alternative approach relies on this to ensure that the stable voltage Vo can be measured far from the crack, and thus means that the field must be uniform over a wider area. In some situations involving non-uniform specimen geometries the incident field may not be sufficiently uniform to reach a steady value for Vo. The second disadvantage is more fundamental. Since the ratio V~/Vo is always less than V2/V1, the alternative interpretation is less sensitive to crack depth. Equations (7) and (8) give the voltage ratio _~_V~=(1
Vo
4d2"~ z/2_
\ +A2/
(10)
The sensitivity S~ of this ratio to crack depth is then found to be S a(K/Vo) 4d ( 4d2'x -1/2 (11) a-= - X 1+ In contrast, Equations (3) and (4) give V2
(A2 + d2) 1/2 + d d
= ( A 2 + d 2 ) 1/2 - -
(12)
from which the sensitivity to crack depth is found to be O(V2/V1) 2(1 +d2/a2) 1/2 S== O(d/A) = [ ( 1 +d2/A2)l/2--d/A]2 (13) Figure 2 shows how these two sensitivities vary as a function of d/A. For deep cracks where d/A --, oo, Sa --*2 while Sc~4d/A. At the other extreme, where cracks are shallow and d/A-,O, S,--.4d/A and S¢--.2. The conventional signal interpretation is therefore much more sensitive to changes in crack depth at both extremes, and the sensitivity of the alternative interpretation drops to zero for very shallow cracks. For shallow cracks the improved accuracy of the alternative method of crack sizing due to its insensitivity to probe position will thus be negated by this poor sensitivity to crack depth. In the intermediate region, putting A = d gives S~ = 4/5 ~/2 and S== 23/2/(2~/2-1) 2, so that here the conventional interpretation is more sensitive by a factor of 9.2. The use of the alternative interpretation for measuring long cracks in a thick skin material should thus be confined to cases in which d > A.
NDT International June 1989
~ / ~
/ - ~ - ' - V I
"
iO -I
10.2
........
10-3
I
........
I0 "z
1
I0 -I
........
f
I d/A
........
1
I01
........
I
.......
102
103
Fig. 2 Sensitivity o f voltage ratio to crack depth for a thick skin material
The alternative interpretation for a thin skin and a semi-elliptical crack
The alternative method of data interpretation can also be applied to the measurement of two-dimensional cracks, in particular to those of circular arc or semi-elliptical shape. In a thin skin material the distribution of current around a crack is determined by the value of the parameters I6] m = l/lt,6 (14) where l is the length of the crack and/*r is the relative magnetic permeability of the material. When m is large compared with unity, the currents on the surface are undisturbed by the presence of the crack, and the voltage trace of a traversing probe retains the 'top-hat' shape. There is then no problem associated with poor probe positioning. Since, with a thin skin, 6 is necessarily much less than l, m is large when/.t r ,-, 1. For frequencies of a few kilohertz, this combination occurs for high conductivity nonmagnetic metals such as aluminium. The multipliers to be used in Equation (2) in such a situation are currently under investigation. For ferromagnetic materials with large /~r, m will be smaller than unity for short cracks. In this case the electric field around a crack can be written as the gradient of a potential ~b which satisfies the plane Laplacian equation v2~ = 0 (15) on both the metal surface and the crack face, This leads
to the 'unfolding theorem' of Dover et a~7] which enables the crack face to be folded up to form a single twodimensional domain (see Figure 3a). The Laplace equation is then solved in this domain, subject to the boundary condition that the bottom of the crack is an equipotential. As shown in Figure 3a, the surface electric field in this case is disturbed by the presence of the crack, so an alternative interpretation of the data using the saddle voltage will again be useful in avoiding possible errors in the measurement of sharply peaked voltages. For example, the potential differences measured on the centre-line of a semi-circular crack of depth a by a probe of length A = a are (see Appendix, Equations (25)-(27)) I/2 = ~b, + ~bo = 1.539a I'1 = qga- ~bo = 0.77a
151
a
b
. - plane
i0 z
~- plane
"t
,
- -
Conventional multipliers
....
Alternative multipliers
11111111111111111 IIIIIIIIIII
,,,
IIIIII
III Ill
III I I
III
Ill
Ill
III
III
Ill
2o/d
~" i0 I 3
5 /';- I0
i....- " O
D
/"
and 1/s = 249a/2
5 - - I 0
I
. . . . . . .
I0 -i
Fig. 4 =
----'_C--
_ _ _ _ _ _ _
Fig. 3 Conformal transformation used for a circular arc crack: (a) equipotentials and streamlines in the true z plane; (b) equipotentials and streamlines in the transformed ~ plane
//
I
.
.
.
.
.
.
.
I A/O
.
I0 i
C o m p a r i s o n of thin skin m u l t i p l i e r s for s e m i - e l l i p t i c a l c r a c k s
1.465a
The far-field voltage is Vo = a. The conventional onedimensional estimate for depth (Equation (1)) then gives
Since the true crack depth is a, the multiplier needed to correct the one-dimensional estimate is M = 2. Using a similar method with the alternative voltage,
:=
4"
4-
5-
d~,=~
Vo-1
=0.233a
(16)
and hence M. = d/d~, = 4.29. Multipliers for other probe spacings and for positions off the crack centre-line may be found for other circular arc cracks using Equation (21). As for the uniform crack in a thick skin of the previous section, the fact that V~< V2 and Vo > V~ means that the alternative multiplier is higher than the conventional one. The improvement obtained by avoiding the peaked voltages is again negated somewhat by the reduced sensitivity to crack depth. The potential distribution around a semi-elliptical crack in a thin skin field has been found by Collins et al p] and the results can be used to obtain a series of alternative multipliers for finding true crack depths from measurements of the saddle and far-field voltages. Figure 4 shows a comparison between the conventional and alternative multipliers for readings taken on the centre-lines of semi-elliptical cracks of aspect ratio 2, 5 and 10. The alternative multiplier is seen to diverge as the ratio of probe gap to crack length becomes large, indicating that it should be used with caution with large probes or short cracks. A multiplier of 10, for instance, means that for a semi-circular crack of depth 1 ram, measured with a probe of length 5 mm, the saddle voltage V~ will be only 4 % higher than the far-field Vo. The alternative multipliers for centre-line readings on semi-elliptical cracks are plotted in Figure 5 as a function of d~,/A for a range of crack lengths. Using this plot, the true crack depth can be found from readings of V~ and Vo under the assumption that the crack is of semi-elliptical shape. It is recommended that the alternative multipliers are not used when A/a > 1 because they are then very sensitive to the measured value of daJA. The region where this is so is shown shaded in Figure 5.
152
2
2-
00
0.5
1.0
1.5
2.0
2.5
dla/A Fig. 5 Alternative centre-line multipliers for semi-elliptical cracks in a thin skin material using the ratio Vs/Vo
The a l t e r n a t i v e i n t e r p r e t a t i o n for a t h i c k skin and a semi-elliptical crack The solution for the surface potential around a semielliptical crack when the skin depth is large compared with the crack size was given by Michael et al tS]. This solution has been used to calculate alternative multipliers using the ratio ~ / V o. These multipliers are again found to be larger than the conventional ones obtained from the ratio V2/I"1, and diverge rapidly for small cracks or large probes. The large divergence found makes it more convenient to plot the reciprocal multiplier 1/Ma, shown in Figure 6 for a range of crack sizes. The value of 1/M~ is seen to reach a maximum and M= is always greater than unity. The asymptote for the crack of uniform depth shown in Figure 6 is found from Equations (9) and (16) to be M,~
1 +2--~1~}
as a ~
(17)
In cases where the multiplier is larger than about 5, it is likely that errors will arise in crack sizing that make the alternative approach unsuitable for use. In such cases the conventional multipliers should be used instead. The regions where this is so are shown shaded in Figure 6.
NDT International June 1989
_______-------
Appendix. The thin skin potential distribution around a circular arc crack
20/A+-
The potential distribution around a circular arc crack can be found using a transformation to coaxial coordinatest’l. However, an alternative method involving conformal mapping will be used here which has the advantage of enabling the potential to be determined explicitly as a function of position. The Karman-Trefftz transformationt81 z-a -= z+a
0.0
0.5
a
1.0
1.5
2.0
2.5
%'A
(-a/n
n (18)
( 5 + a/n >
maps a circular arc of half-length a and depth b in the complex plane z = x + iy onto a straight line of half-length a/n in the complex plane [ = $ + i4. The parameter n is given by n=2-2RJz
(19)
where Q = tan- ’ a/b, so that n = 312 corresponds to the transformation of a semi-circle and increasingly shallow circular arcs are mapped as n + 1. The transformation is shown in Figure 3. Equation (18) is used to model fluid flow past a circular arc dip in which the streamlines are parallel to the real axis in the far-field. In the 5 plane the problem is transformed to flow parallel to this plate, which gives rise to streamlines everywhere parallel to the real axis. In order to model current flow past a circular arc crack, the equipotentials and streamlines in the z plane need to be interchanged to match Figure 3a, so that after mapping to the 5 plane the potential is given by 4=Im(5)
(20)
Rearrangement of (18) gives the explicit determination of potential as
b
*,,‘A
Fig. 6 Alternative centre-line multipliers for semi-elliptical cracks in a thick skin material using the ratio VJ V,,: (a) large a/A; (b) small a/A
As an example of the use of the saddle voltage, consider the case of a probe of length A = a, measuring on the centre-line of a semi-circular crack. In this case n = 3/2 and z = iy, so Equation (21) becomes
Conclusions The conventional method of sizing surface-breaking cracks in the ACPD technique uses voltages measured just over and just off a crack. These voltages are then used to produce a simple first estimate of depth which is modified by the use of theoretically calculated multipliers to take account of crack shape, skin depth, etc. The voltage readings are often very sensitive to probe position, however, and in such cases are likely to produce an underestimate of depth. An alternative approach has been proposed which makes use of the relatively stable saddle voltage obtained when the probe straddles the crack. Solutions of the potential problems involved have enabled alternative multipliers to be found for this approach for situations involving circular arc or semi-elliptical cracks in the extremes of either small or large skin depth. The main disadvantage of the alternative approach is that the voltages involved are less sensitive to crack depth, which means that the multipliers produced are larger than the conventional ones, particularly for short or shallow cracks. For this reason it would be wise to avoid the use of the alternative multipliers when they are found to be larger than about 5.
NDT International
June 1989
(22) where A=y2-a2
.
7--T+’ Y +a
2ya
(23)
Y2
Writing A = r(cos 8 + i sin 0) and using the fact that A2/3 = r213[ cos f( 8 + 2kn) + i sin $( 8 + 2k7c)]
k=O, 1, 2
(24)
it can be shown that &=
*2a/3$
at y=O
(25)
at y=a
(26)
at y = a/2
(27)
References 1
2
Lugg, M.C. ‘The analysis of sparse data in APCD crack growth monitoring’ NDT International 21 (1988) pp 153-158 Collins, R., Dover, W.D. and Michael, D.H. ‘The use of AC field
153
3
4 5
6
measurements for non-destructive testing' in Research Techniques in Non-Destructive Testing vol 8, ed R.S. Sharpe, Academic Press, London (1985) pp 211-267 Collins, R., Dover, W.D. and Ranger, K.B. 'The AC field around a plane semi-elliptical crack in a metal surface' in Proc 13th Syrup on Non-Destructive Evaluation ed B.E. Leonard, NTIAC, San Antonio, TX (1981) pp 470-479 Mirshekar-Synhknl, D., Collins, R. and Michael, D.H. 'The influence of skin depth on crack measurement by the AC field technique' J Nondestructive Eval 3 (1982) pp 65-76 Michael, D.H., Collins, R., Parramore, D.R., Aldoujaily, M. and Travis, R.P. 'Thick-ski0 modelling for surface-breaking cracks' in Review of Progress in Quantitative Non-Destructive Evaluation vol 7A, eds D.O. Thompson and D.E. Chimenti, Plenum Press, New York (1988) pp 191-197 Lewis,A.M., Michael, D.H., Lugg, M.C. and Collins, R. 'Thin-skin electromagnetic fields around surface-breaking cracks in metals', J
Appl Phys 64 (1988) pp 3777-3784 7 Dover, W.D., Charlesworth, F.D.W., Taylor, K.A., Collins, R. and Michael, D.H. 'AC field measurement - theory and practice' in The Measurement of Crack Length and Shape during Fracture and Fatigue
ed C.J. Beevers, EMAS, Warley, West Midlands, UK (1980) pp 222-260 8 Durand, W.F. Aerodynamic Theory vol 2, Dover, New York (1963)
Author The a u t h o r was formerly in the D e p a r t m e n t of Mechanical Engineering, University College of L o n d o n , T o r r i n g t o n Place, L o n d o n W C 1 E 7JE, U K . His present address is T S C Ltd, 34 L i n f o r d F o r u m , R o c k i n g h a m Drive, L i n f o r d W o o d , M i l t o n K e y n e s M K 1 4 6LY, U K .
Paper received 28 July 1988. Revised 6 October 1988
154
NDT International June 1989
Keywords: AC potential drop (ACPD) technique, crack sizing, saddle voltage, circular arc and semi-elliptical cracks The AC potential drop (ACPD) technique is used to size surface-breaking defects in two main areas. In crack growth monitoring, a fatigue crack is sized at regular intervals throughout a fatigue test in order to produce crack growth rate data. In this situation it is usual to have a fixed number of voltage probe sites and so the distribution of data points along a crack can be quite sparse, particularly when the crack is small. An earlier paper [~] described a method for optimizing the accuracy of crack sizing with sparse data. The other main use of the ACPD technique is for crack inspection. In this case potential difference readings are taken around a crack using either a hand-held probe or, if the specimen geometry permits, a probe driven by a stepper motor under computer control. In this way, the electric field around a crack can be mapped in as much detail as the time available allows. A review of the use of the ACPD technique is found in Collins et alt2J. Crack inspection conventionally involves obtaining pairs of potential differences at a series of points along the crack length: V1 when the probe is just off the crack, and V2 when the probe just straddles the crack (see Figure la). This information is sufficient to determine the crack shape in most situations.
The c o n v e n t i o n a l i n t e r p r e t a t i o n f o r a thi n skin A thin skin material is one in which the skin depth 6 is much less than the probe spacing A and the depth d of the crack being measured. In practice, as long as 6/A and f/d are less than about 0.1, the thin skin approximation will be valid. At frequencies of a few kilohertz, this applies to cracks of depth 1 mm or more in magnetic steels or 10 mm in aluminium. If the crack depth is in turn small
compared with the crack length a, the crack can be cotlsidered as essentially of uniform depth, and the voltage trace from a probe moving perpendicularly across the crack far from the ends follows the simple 'top-hat' shape shown in Figure lb. This arises because the current flow is undisturbed by the presence of the crack right up to the crack edge. The voltages V1 and V2 a r e then constant, and accurate probe positioning is not required. The crack depth is found from the formula
d=d~=~- ~ - 1
(1)
v~
g Y,
o~
4--
b
I1
Probe positioo
C
Probe position
Fig. 1 Voltages measured around a surface-breaking crack: (a) probe positions; (b) voltage trace for a probe traversing a uniform depth crack with a small skin depth &; (c) voltage trace for a finite length crack or for a large skin depth
0308-9126/89/030149-06/$03.00 © 1989 Butterworth & Co (Publishers) Ltd NDT International Volume 22 Number 3 June 1989
149
If the crack aspect ratio is not large, however, the voltage trace from a traversing probe has a shape similar to that shown in Figure lc. The current flow is now perturbed by the presence of the crack, and the voltage trace peaks sharply at V~ and V2. For cases where the crack depth is not uniform, a series of solutions have been obtained for various crack shapes, the most realistic one being that for a semi-elliptical crack. In this case no analytic equation for crack depth can be written down, but a series of multipliers M have been calculated TM from which the crack depth can be found:
d=diM
(2)
where M is a function of aspect ratio, probe gap and distance from the crack centre-line.
The c o n v e n t i o n a l i n t e r p r e t a t i o n f o r a t h i c k skin A thick skin material is one in which the skin depth is much larger than the crack depth, although the thick skin approximation is valid in practice as long as 6/d is greater than about 2. At frequencies of a few kilohertz, this applies to cracks of depth 3 mm or less in titanium and stainless steel or 0.5 mm in aluminium. As with a low aspect ratio crack in a thin skin material, the voltage trace from a probe traversing any crack in a thick skin material has a shape like that in Figure lc. In the thick skin extreme the surface electric potential ~b is given by t4] = E o ( y 2 +d2) 1/2
(3)
where E o is the uniform upstream electric field strength and y is the perpendicular distance from the crack edge. The potential is of opposite sign either side of the crack, so there is a jump in ~b of 2Eod across the crack. From Equation (3) it can be seen that
d2)x/2 - Eod V2=c~(A)-c~(-O)=Eo(a2 +d2)~/2+ Eod V1 = ~b(A) - ~ ( + 0 ) = Eo(A 2 +
A (k V l~l/2(~-l) V:/
From Equation (2) the measured off-crack voltage would now be V~ = 4~(1.1A) - ~b(0.1A) = EoA(~/2.21 - x/1.01) = 0.4816EoA which is 16% higher than the true V1. At the same time, the measured cross-crack voltage would be V~ = ~b(0.9A) - ~b(-0.1A) = EoA(x/1.81 + x/1.01) = 2.3504EoA which is 2.6 % lower than the true V2. The combination of these results substituted into Equation (5) gives a measured crack depth of d' =0.878A, 12% lower than the true depth. The same considerations apply to measurements on cracks of finite aspect ratio. The use of V1 and V2 to determine crack depth can thus lead to results that are highly sensitive to probe position. In cases where this would cause a problem, such as when rapid automated scanning is required, an alternative approach should be considered. The two other voltages uniquely determined in the trace of Figure lc are the saddle point voltage V~, when the probe is bisected by the crack, and the undisturbed upstream voltage Vo. Both of these voltages are relatively insensitive to probe position and may be of more use for crack sizing in certain situations. This alternative approach to crack sizing is developed in .the rest of the paper.
The a l t e r n a t i v e i n t e r p r e t a t i o n f o r a t h i c k skin and a u n i f o r m d e p t h crack
(6)
In order to use the stable voltages V~ and Vo to size a large aspect ratio crack in a thick skin material, an alternative formula for crack depth is needed to replace Equation (5). From Equation (2) it can be seen that the upstream voltage is simply
The case f o r a l t e r n a t i v e m u l t i p l i e r s Accurate crack sizes can be obtained using the conventional multipliers in a wide variety of situations, as long as V~ and V2 can be measured accurately. However, since I,'1 and V2 are sharply peaked, slight changes in probe position will give rise to large changes in the measured voltages. In practice, care must be taken to place the leg of the voltage probe as close as possible to the crack edge. With a hand-held probe this can be very time consuming, while with a stepper motor drive the step size has to be made very small, giving rise to a long scanning time. The effect of poor probe positioning can be illustrated by the case of the uniform crack in a thick skin material. Consider the case where A = d. From Equation (3) the true off-crack voltage is
150
Substitution into Equation (5) gives the true crack depth d = A. Now consider the result when the probe is positioned a distance A/10 away from the crack edge for each reading. This would follow from a misplacement of only 1 mm for a standard 10 mm probe.
(5)
For semi-elliptical cracks, thick skin multipliers have recently been calculated TM.
I/1 = EoA(x/2 - 1) = 0.4142EoA
V2 = EoA(x/2 + 1) = 2.4142EoA
(4)
from which the crack depth is given by d=
and from Equation (4) the true cross-crack voltage is
Vo = Lim [tp(y + A) - q~(y)] = EoA
(7)
while the saddle voltage is given by V~= q ~ ( A ) - q~(_ A ) = Eo(A2 + 4d2)1/2
(8)
Rearranging Equations (7) and (8), the alternative formula for crack depth is found to be
d=~kVg
1
(9)
The desensitivity to probe position obtained with this equation can be demonstrated by applying the same test used in the previous section. If the probe is displaced by a distance A/10, the measured upstream voltage will be unchanged since V~ = Eo A
NDT International June 1989
The measured saddle voltage will be
107 .
v; = 4 , ( 0 . 6 A ) - 4'( - 0.4A) = EoA(x/1.36 + ,,/1.16)
I0 s .
= 2.2432EoA
iOs
which is only 0.3 % higher than the true saddle voltage given by
tO4 tO~
V~= 25(0.5A) = 2EoA~/1.25 = 2.2361EoA Substituting the measured voltages into Equation (9) gives the measured depth as
i0 z o
z
vz i01
d ' = 1.004A
I
which is only 0.4% higher than the true depth. The sensitivity to probe positioning is thus seen to be very much reduced. There are disadvantages to this alternative interpretation of ACPD readings, however. The first disadvantage is a practical one: both the conventional and alternative interpretations require that the incident electric field be of uniform strength, at least over a region covering the crack and the probe positions. The alternative approach relies on this to ensure that the stable voltage Vo can be measured far from the crack, and thus means that the field must be uniform over a wider area. In some situations involving non-uniform specimen geometries the incident field may not be sufficiently uniform to reach a steady value for Vo. The second disadvantage is more fundamental. Since the ratio V~/Vo is always less than V2/V1, the alternative interpretation is less sensitive to crack depth. Equations (7) and (8) give the voltage ratio _~_V~=(1
Vo
4d2"~ z/2_
\ +A2/
(10)
The sensitivity S~ of this ratio to crack depth is then found to be S a(K/Vo) 4d ( 4d2'x -1/2 (11) a-= - X 1+ In contrast, Equations (3) and (4) give V2
(A2 + d2) 1/2 + d d
= ( A 2 + d 2 ) 1/2 - -
(12)
from which the sensitivity to crack depth is found to be O(V2/V1) 2(1 +d2/a2) 1/2 S== O(d/A) = [ ( 1 +d2/A2)l/2--d/A]2 (13) Figure 2 shows how these two sensitivities vary as a function of d/A. For deep cracks where d/A --, oo, Sa --*2 while Sc~4d/A. At the other extreme, where cracks are shallow and d/A-,O, S,--.4d/A and S¢--.2. The conventional signal interpretation is therefore much more sensitive to changes in crack depth at both extremes, and the sensitivity of the alternative interpretation drops to zero for very shallow cracks. For shallow cracks the improved accuracy of the alternative method of crack sizing due to its insensitivity to probe position will thus be negated by this poor sensitivity to crack depth. In the intermediate region, putting A = d gives S~ = 4/5 ~/2 and S== 23/2/(2~/2-1) 2, so that here the conventional interpretation is more sensitive by a factor of 9.2. The use of the alternative interpretation for measuring long cracks in a thick skin material should thus be confined to cases in which d > A.
NDT International June 1989
~ / ~
/ - ~ - ' - V I
"
iO -I
10.2
........
10-3
I
........
I0 "z
1
I0 -I
........
f
I d/A
........
1
I01
........
I
.......
102
103
Fig. 2 Sensitivity o f voltage ratio to crack depth for a thick skin material
The alternative interpretation for a thin skin and a semi-elliptical crack
The alternative method of data interpretation can also be applied to the measurement of two-dimensional cracks, in particular to those of circular arc or semi-elliptical shape. In a thin skin material the distribution of current around a crack is determined by the value of the parameters I6] m = l/lt,6 (14) where l is the length of the crack and/*r is the relative magnetic permeability of the material. When m is large compared with unity, the currents on the surface are undisturbed by the presence of the crack, and the voltage trace of a traversing probe retains the 'top-hat' shape. There is then no problem associated with poor probe positioning. Since, with a thin skin, 6 is necessarily much less than l, m is large when/.t r ,-, 1. For frequencies of a few kilohertz, this combination occurs for high conductivity nonmagnetic metals such as aluminium. The multipliers to be used in Equation (2) in such a situation are currently under investigation. For ferromagnetic materials with large /~r, m will be smaller than unity for short cracks. In this case the electric field around a crack can be written as the gradient of a potential ~b which satisfies the plane Laplacian equation v2~ = 0 (15) on both the metal surface and the crack face, This leads
to the 'unfolding theorem' of Dover et a~7] which enables the crack face to be folded up to form a single twodimensional domain (see Figure 3a). The Laplace equation is then solved in this domain, subject to the boundary condition that the bottom of the crack is an equipotential. As shown in Figure 3a, the surface electric field in this case is disturbed by the presence of the crack, so an alternative interpretation of the data using the saddle voltage will again be useful in avoiding possible errors in the measurement of sharply peaked voltages. For example, the potential differences measured on the centre-line of a semi-circular crack of depth a by a probe of length A = a are (see Appendix, Equations (25)-(27)) I/2 = ~b, + ~bo = 1.539a I'1 = qga- ~bo = 0.77a
151
a
b
. - plane
i0 z
~- plane
"t
,
- -
Conventional multipliers
....
Alternative multipliers
11111111111111111 IIIIIIIIIII
,,,
IIIIII
III Ill
III I I
III
Ill
Ill
III
III
Ill
2o/d
~" i0 I 3
5 /';- I0
i....- " O
D
/"
and 1/s = 249a/2
5 - - I 0
I
. . . . . . .
I0 -i
Fig. 4 =
----'_C--
_ _ _ _ _ _ _
Fig. 3 Conformal transformation used for a circular arc crack: (a) equipotentials and streamlines in the true z plane; (b) equipotentials and streamlines in the transformed ~ plane
//
I
.
.
.
.
.
.
.
I A/O
.
I0 i
C o m p a r i s o n of thin skin m u l t i p l i e r s for s e m i - e l l i p t i c a l c r a c k s
1.465a
The far-field voltage is Vo = a. The conventional onedimensional estimate for depth (Equation (1)) then gives
Since the true crack depth is a, the multiplier needed to correct the one-dimensional estimate is M = 2. Using a similar method with the alternative voltage,
:=
4"
4-
5-
d~,=~
Vo-1
=0.233a
(16)
and hence M. = d/d~, = 4.29. Multipliers for other probe spacings and for positions off the crack centre-line may be found for other circular arc cracks using Equation (21). As for the uniform crack in a thick skin of the previous section, the fact that V~< V2 and Vo > V~ means that the alternative multiplier is higher than the conventional one. The improvement obtained by avoiding the peaked voltages is again negated somewhat by the reduced sensitivity to crack depth. The potential distribution around a semi-elliptical crack in a thin skin field has been found by Collins et al p] and the results can be used to obtain a series of alternative multipliers for finding true crack depths from measurements of the saddle and far-field voltages. Figure 4 shows a comparison between the conventional and alternative multipliers for readings taken on the centre-lines of semi-elliptical cracks of aspect ratio 2, 5 and 10. The alternative multiplier is seen to diverge as the ratio of probe gap to crack length becomes large, indicating that it should be used with caution with large probes or short cracks. A multiplier of 10, for instance, means that for a semi-circular crack of depth 1 ram, measured with a probe of length 5 mm, the saddle voltage V~ will be only 4 % higher than the far-field Vo. The alternative multipliers for centre-line readings on semi-elliptical cracks are plotted in Figure 5 as a function of d~,/A for a range of crack lengths. Using this plot, the true crack depth can be found from readings of V~ and Vo under the assumption that the crack is of semi-elliptical shape. It is recommended that the alternative multipliers are not used when A/a > 1 because they are then very sensitive to the measured value of daJA. The region where this is so is shown shaded in Figure 5.
152
2
2-
00
0.5
1.0
1.5
2.0
2.5
dla/A Fig. 5 Alternative centre-line multipliers for semi-elliptical cracks in a thin skin material using the ratio Vs/Vo
The a l t e r n a t i v e i n t e r p r e t a t i o n for a t h i c k skin and a semi-elliptical crack The solution for the surface potential around a semielliptical crack when the skin depth is large compared with the crack size was given by Michael et al tS]. This solution has been used to calculate alternative multipliers using the ratio ~ / V o. These multipliers are again found to be larger than the conventional ones obtained from the ratio V2/I"1, and diverge rapidly for small cracks or large probes. The large divergence found makes it more convenient to plot the reciprocal multiplier 1/Ma, shown in Figure 6 for a range of crack sizes. The value of 1/M~ is seen to reach a maximum and M= is always greater than unity. The asymptote for the crack of uniform depth shown in Figure 6 is found from Equations (9) and (16) to be M,~
1 +2--~1~}
as a ~
(17)
In cases where the multiplier is larger than about 5, it is likely that errors will arise in crack sizing that make the alternative approach unsuitable for use. In such cases the conventional multipliers should be used instead. The regions where this is so are shown shaded in Figure 6.
NDT International June 1989
_______-------
Appendix. The thin skin potential distribution around a circular arc crack
20/A+-
The potential distribution around a circular arc crack can be found using a transformation to coaxial coordinatest’l. However, an alternative method involving conformal mapping will be used here which has the advantage of enabling the potential to be determined explicitly as a function of position. The Karman-Trefftz transformationt81 z-a -= z+a
0.0
0.5
a
1.0
1.5
2.0
2.5
%'A
(-a/n
n (18)
( 5 + a/n >
maps a circular arc of half-length a and depth b in the complex plane z = x + iy onto a straight line of half-length a/n in the complex plane [ = $ + i4. The parameter n is given by n=2-2RJz
(19)
where Q = tan- ’ a/b, so that n = 312 corresponds to the transformation of a semi-circle and increasingly shallow circular arcs are mapped as n + 1. The transformation is shown in Figure 3. Equation (18) is used to model fluid flow past a circular arc dip in which the streamlines are parallel to the real axis in the far-field. In the 5 plane the problem is transformed to flow parallel to this plate, which gives rise to streamlines everywhere parallel to the real axis. In order to model current flow past a circular arc crack, the equipotentials and streamlines in the z plane need to be interchanged to match Figure 3a, so that after mapping to the 5 plane the potential is given by 4=Im(5)
(20)
Rearrangement of (18) gives the explicit determination of potential as
b
*,,‘A
Fig. 6 Alternative centre-line multipliers for semi-elliptical cracks in a thick skin material using the ratio VJ V,,: (a) large a/A; (b) small a/A
As an example of the use of the saddle voltage, consider the case of a probe of length A = a, measuring on the centre-line of a semi-circular crack. In this case n = 3/2 and z = iy, so Equation (21) becomes
Conclusions The conventional method of sizing surface-breaking cracks in the ACPD technique uses voltages measured just over and just off a crack. These voltages are then used to produce a simple first estimate of depth which is modified by the use of theoretically calculated multipliers to take account of crack shape, skin depth, etc. The voltage readings are often very sensitive to probe position, however, and in such cases are likely to produce an underestimate of depth. An alternative approach has been proposed which makes use of the relatively stable saddle voltage obtained when the probe straddles the crack. Solutions of the potential problems involved have enabled alternative multipliers to be found for this approach for situations involving circular arc or semi-elliptical cracks in the extremes of either small or large skin depth. The main disadvantage of the alternative approach is that the voltages involved are less sensitive to crack depth, which means that the multipliers produced are larger than the conventional ones, particularly for short or shallow cracks. For this reason it would be wise to avoid the use of the alternative multipliers when they are found to be larger than about 5.
NDT International
June 1989
(22) where A=y2-a2
.
7--T+’ Y +a
2ya
(23)
Y2
Writing A = r(cos 8 + i sin 0) and using the fact that A2/3 = r213[ cos f( 8 + 2kn) + i sin $( 8 + 2k7c)]
k=O, 1, 2
(24)
it can be shown that &=
*2a/3$
at y=O
(25)
at y=a
(26)
at y = a/2
(27)
References 1
2
Lugg, M.C. ‘The analysis of sparse data in APCD crack growth monitoring’ NDT International 21 (1988) pp 153-158 Collins, R., Dover, W.D. and Michael, D.H. ‘The use of AC field
153
3
4 5
6
measurements for non-destructive testing' in Research Techniques in Non-Destructive Testing vol 8, ed R.S. Sharpe, Academic Press, London (1985) pp 211-267 Collins, R., Dover, W.D. and Ranger, K.B. 'The AC field around a plane semi-elliptical crack in a metal surface' in Proc 13th Syrup on Non-Destructive Evaluation ed B.E. Leonard, NTIAC, San Antonio, TX (1981) pp 470-479 Mirshekar-Synhknl, D., Collins, R. and Michael, D.H. 'The influence of skin depth on crack measurement by the AC field technique' J Nondestructive Eval 3 (1982) pp 65-76 Michael, D.H., Collins, R., Parramore, D.R., Aldoujaily, M. and Travis, R.P. 'Thick-ski0 modelling for surface-breaking cracks' in Review of Progress in Quantitative Non-Destructive Evaluation vol 7A, eds D.O. Thompson and D.E. Chimenti, Plenum Press, New York (1988) pp 191-197 Lewis,A.M., Michael, D.H., Lugg, M.C. and Collins, R. 'Thin-skin electromagnetic fields around surface-breaking cracks in metals', J
Appl Phys 64 (1988) pp 3777-3784 7 Dover, W.D., Charlesworth, F.D.W., Taylor, K.A., Collins, R. and Michael, D.H. 'AC field measurement - theory and practice' in The Measurement of Crack Length and Shape during Fracture and Fatigue
ed C.J. Beevers, EMAS, Warley, West Midlands, UK (1980) pp 222-260 8 Durand, W.F. Aerodynamic Theory vol 2, Dover, New York (1963)
Author The a u t h o r was formerly in the D e p a r t m e n t of Mechanical Engineering, University College of L o n d o n , T o r r i n g t o n Place, L o n d o n W C 1 E 7JE, U K . His present address is T S C Ltd, 34 L i n f o r d F o r u m , R o c k i n g h a m Drive, L i n f o r d W o o d , M i l t o n K e y n e s M K 1 4 6LY, U K .
Paper received 28 July 1988. Revised 6 October 1988
154
NDT International June 1989