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Magnetic leakage fields calculated by the method of finite differences
Magnetic leakage fields calculated by the method of finite differences B. Brudar Is the signal indicating magnetic leakage fields proportional to the depth of the crack? This question needs to be answered to show the effectiveness of the testing method. However, contradictory reports have appeared in the literature in the past few years. The results given in this paper obtained by the method of finite differences illustrate a new aspect of the problem throwing a new light on previous results obtained by the method of finite elements.
Keywords: non-destructive testing, magnetic leakage fields, relative permeability, crack depth estimation
NDT methods whereby means of magnetic leakage fields, various surface and subsurface defects can be detected are very well known. When magnetic flux flows perpendicularly into a gap in a ferromagnetic material some of the magnetic flux lines leave the ferromagnetic material to avoid the gap. One can therefore observe an increase in magnetic flux density outside the material in the close neighbourhood of the defect Magnetic leakage flux can be registered visually by means of a suspension of very fine ferromagnetic particles in oil.
The problem itself is very interesting and so we calculated the distribution of the DC magnetic field in the neighbourhood of a surface gap in a ferromagnetic plate using the method of finite differences. It is interesting to note that our results agree with the results of FbrsterPl, but for some extreme cases our results are similar to those published by Dobmannl21. There are quite obvious limitations of the leakage field method for NDT.
Various kinds of fluorescent magnetic powders can be used and ultraviolet light gives in some cases better information about the defect However, it is very tiring for the operator to watch visually a certain part of a ferromagnetic product over a period of hours, especially in mass production. Several attempts have been made to exclude the human factor. The magnetic leakage fields can be registered by magnetic tape or measured by Hall probes. Any measurable deformation of the magnetic field can also be stored and processed by computer.
Let us assume the two-dimensional case of a ferromagnetic plate placed in a homogeneous DC magnetic field. The crack is assumed to be perpendicular to the magnetic flux, which is tangential to the surface of the plate The magnetic properties (the relative permeability bt) of the plate are given in Figure 1, and the data correspond to carbon steel (ST 37). It is further assumed
In the NDT literature some articles appeared to explain the whole phenomenon by the results of experimental measurements and by mathematical calculationI~-31. However the results of the calculations in Refs 2 and 3 show a certain discrepancy, and it is still not clear whether the signal measured by the leakage flux method is a good indication of the depth of the crack or whether the width of the crack also influences the measured signal significantly. This question is very important because in practice in nearly all cases the depth of a crack is much more significant than its width. In the literaturel~.21 the experimental results are interpreted by mathematical calculations using the method of finite elements,
Mathematical formulation
=L ¢ =3
IOO0
.g ~
-
I O F=g. 1 ST 37
I
I
I I
I
I
I
I I I I I I I I I I I I000 ZOOO MognetN: field strength, H [Am "t)
Relative permeab=hty as a function of magnetic field strength for
0308-9126/85/060353-05 $3.00 © 1985 Butterworth 8 Co (Publishers) Ltd NDT INTERNATIONAL. VOL 18. NO 6. DECEMBER1985
353
4 2
2
2
2
2
2
2
9
0
0
0
0
0
0
0
9
0
Z)
0
0
0
0
0
6
4
4
4
4
4
4
5
B
8
8
B
8
8
7
B
8
8
8
8
8
B
8
8
B
8
8
8
3
3
3
3
3
3
3
0
0
0
0
0
0
0
9
0
0
O
0
0
0
Z)
9
2
2
2
2
2
2
2
9
,¥
L
Fig 2
Lattice of points
that a D( field with magnetic strength H0 = 1000 A m -~ is applied, and the magnetic field a r o u n d the crack is calculated by the method of finite differences,
the relative permeability/.t should be chosen so that the relation between B and H according to Figure 1 is fulfilled`
The principal arrangement is given in Figure 2 A rectangular mesh is chosen with unequal mesh distances in t h e X and Y directions so that various widths o f cracks could be assumed` The depth of the gap is represented by D, the halt'width by W, and the thickness of the plate(L) is chosen as 20 mesh distances.
For points (6) and (7) a special procedurel4l is used. The difference equation written for these two points is in principle the arithmetm mean of two equations: one that holds for the h o m o g e n e o u s regions (outside or Inside the ferromagnetm material) and one that holds for the points if they were lying on the diagonal b o u n d a r y (le the line passing through (6) and the point (0. - 1 )
To calculate the magnetic flux density B and the magnetic field strength H m the vicinity of the defecL one starts with the differential equation for the scalar potential U V2U =
0
(1)
This equation holds for both inside and outside the plate Due to the symmetry only one half of the cross-section has to be taken into a c c o u n t The different n u m b e r s at the mesh points indicate the type of difference equation being used at that p o i n t The points (1) he on the line of symmetry and so m the difference equation for these points this has to be taken Into a c c o u n t The points (3) and (4) as well as (5). (6) and (7) he on the b o u n d a r y between the ferromagnetic material and air At any b o u n d a r y the normal c o m p o n e n t of the magnetic flux density remains u n c h a n g e d when crossing the b o u n d a r y From this fact It is possible to formulate the corresponding difference equation. In the relationship B = - - /*0/.t grad U
354
(2)
The points (2) and (9) are assumed to lie far enough from the defect so that it can be assumed t h a t H v = 0 a n d H , = / / 0 At the points (0) and (8) the difference equations corresponding to Equation (1) are the same. written in the usual form. In the computer program it is foreseen that the mesh distances in the X and Y directions may be different For the solving procedure the so-called L l e b m a n n extrapolanon relaxation methodt 4t was used. The property of the ferromagnetic material was taken into account so that we found lteratively the value o f p until the difference between the calculated and real values ofp, was smaller than or equal to 10. Alter some thousands of iterations we obtained the results for the scalar potentml. The magnetic field strength and the magnetic flux density outside and inside the material were then calculated. All the data represented m the followmg figures are drawn for the case when the distance above the surface of the plate is equal to 1 (y= 1). Various widths were simulated by varying the mesh distance in the X direction
NDT INTERNATIONAL. DECEMBER 1985
I
-0
5
io x
F~g. 3
-5
Normal component of the magnet,c held strength at Y = 1
t
o
5
x F=g. 5
Tangentml component
Hx(X) at vanous widths
and depths
conclude that the proportionality as given in Ref. 3 is preserved. Also Figure 5 shows that this proportionality is preserved when the width is varied. The difference (Hx - Ho) can be supposed to be proportional to the depth of the crack at any constant width. So far everything seems to be in order and the statements by Forster that the leakage field signal H v is proportional to the depth of the crack and that it ~s practically independent of the width could be verified. Also his statements about the influence of the measuring distance y on the distribution H x (x) could be verified.
W=I
O--
One can imagine that the magnetic flux lines are prevented from penetrating the walls of the gap and that the magnetic flux has to flow partially out of the ferromagnetic material. In such a simplified picture the width of the crack can be neglected.
2v.
--
1
I I I I II -5 F=g. 4
0 x
~
i i i I I 5
Tangentml component of the magnet=c field strength at Y = 1
Results
From Figure 3 it could be further concluded that the H.y compoffent has its maximal value above point (6) m Figure2 and across the width of the gap it changes its sign. Let us suppose that the gap is extremely narrow and deep. According to the results mentioned above Hy should be proportional to the depth and across some extremely short distance it should change its sign! We also changed the mesh distance in theX direction and calculated the field near defects with the same depth (D = 9) and different widths(W= 0.5, W = 0.2, W = 0.1). The results are given in Figure 6. It is quite obvious that the curvesH~(x) aty = 1 depend on the width of the crack. The signal Hy mdx is much smaller if, for example, W/D = 1/90, instead of W/D = !/9.
Figure 3 shows the Hy component outside the plate one mesh distance above the plate for different widths of the crack. It is obvious that the maximal value is proportional to the depth of the crack One can see that there are only very small variations due to the variations in width of the gap. This result corresponds to the results of F6rsterl31.
Figure 7 shows the function Hx(x ) aty = 1 for the cases of D=9andW=3, W = 2 , W = l , W = 0 . 5 , W = 0 . 2 and W = 0.1. Also this component of the magnetic field strongly depends on the crack width.
Figure 4 shows the H, component as a function of different depth s at a fixed width (w = 1). Here one can also
We also chose at W = 0.2 and W = 0.1 three cases: D = 9, D = 6 and D = 3 to see if(Hx - H0) is proportional to D
NDT INTERNATIONAL.
DECEMBER 1985
355
m e n t i o n e d above show some interesting facts 1
If the ratio D / W ~ I O it p, possible to ',a) that the magnetic field strengthH~ m,,x a t y = 1 Is p r o p o r t i o n a l to the depth o f the crack a n d does not d e p e n d on its width Also, at a defimte w~dth the m a x i m u m (H, - H0) ~s p r o p o m o n a l to the depth of the crack.
2
II the ratio D/W ~ 1 0 the H, and H, c o m p o n e n t s at i = 1 b e c o m e m u c h s m a l l e r a n d are no longer simply related to the depth o f the crack. In this case the width of the crack has to be taken into account.
150C
IOOC
)E
T 500
3000- t
O 0
5
I0
F~g 6
Normal component
Hv(x ) at D =
~
2000.
x 9 for different w~dths
O=9
W=02
0--6
W=OI
0=3
W=02
... ~¢~1000.
I I I I I
-5
I l l l l
0
5
x Fig 8
Tangential component
Hx(X ) for
the extreme cases
1500
I
I
I
I
I 7
-5
I 0
I
I
I
I ,
~
I000
5
x
Fig 7
Tangential component
Hx(x )
as a funct)on of different wtdths
( F i g u r e 8) O b v i o u s l y it is n o t Also the m a x i m a l values o f H~ c a l c u l a t e d a t y = 1 for different d e p t h s (D = 3, 6, 9) are strongly d e p e n d e n t on the width o f the gap ( F i g u r e 9)
500
T h e results o f these extreme cases ( F i g u r e s 8 a n d 9) can be c o m p a r e d with the results o f Lordlll a n d Dobmannf21
Conclusion T h e d e t a d e d c a l c u l a t i o n of the DC" m a g n e t i c leakage field in f e r r o m a g n e t i c material will be published. T h e results
356
0
3
6
9
D F)g 9
Maximal value of Hv as a function of different depths
NDT INTERNATIONAL.
DECEMBER 1985
.
The NDT method using leakage fields has some limitations. The measuring device should also provide some information about the width of the 'pulse" Hx(x). This width depends only on the width of the gap and not on its depth. For wider gaps it is directly proportional to the width of the gap. With the thinner gaps it seems to approach a certain limiting value defined by the magnetic field strengthH0 and by the properties of the ferromagnetic material (/t). However, within this limiting value, knowing the width of the 'pulse' one could calculate (perhaps in advance) the corresponding possible depth of the crack
There is still much practical work needed to verify this mathematical result and also to define the practical limits of the method.
Author The author is with the Iron and Steel Works, 64270 Jesenice, SRS, Yugoslavia.
References 1 Lord,W. and Hwan&J.H. "Defectcharacterizatmn from magnetic leakage fields"Bnt J NDT (January 1977) pp 14-18 2 Dobmann, G. and Walle, G. 'New set-ups for mathematlcalnumeriai solutions of magnetic leakage-flux testing with d.c and a.c mode in the FRG" lOth WCNDZ Moscow (1982) paper IC-5 3 Forster, F. "Neue Erkenntmsse auf dem Gebiet der zerstorungsfreten Prufung mit magneuschem Streufluss" 3 rd European Conference on ND T. Florence ( 15-18 October 1984) pp 287-303 4 Binns, K.J. and Lawrenson, P.J. "Analysis and computauon of electric and magnettc field problems, Pergamon Press Oxford
Paper rece,ved 26 April 1985. Rev,sed 12 July 1985
NDT INTERNATIONAL. DECEMBER 1985
357
Keywords: non-destructive testing, magnetic leakage fields, relative permeability, crack depth estimation
NDT methods whereby means of magnetic leakage fields, various surface and subsurface defects can be detected are very well known. When magnetic flux flows perpendicularly into a gap in a ferromagnetic material some of the magnetic flux lines leave the ferromagnetic material to avoid the gap. One can therefore observe an increase in magnetic flux density outside the material in the close neighbourhood of the defect Magnetic leakage flux can be registered visually by means of a suspension of very fine ferromagnetic particles in oil.
The problem itself is very interesting and so we calculated the distribution of the DC magnetic field in the neighbourhood of a surface gap in a ferromagnetic plate using the method of finite differences. It is interesting to note that our results agree with the results of FbrsterPl, but for some extreme cases our results are similar to those published by Dobmannl21. There are quite obvious limitations of the leakage field method for NDT.
Various kinds of fluorescent magnetic powders can be used and ultraviolet light gives in some cases better information about the defect However, it is very tiring for the operator to watch visually a certain part of a ferromagnetic product over a period of hours, especially in mass production. Several attempts have been made to exclude the human factor. The magnetic leakage fields can be registered by magnetic tape or measured by Hall probes. Any measurable deformation of the magnetic field can also be stored and processed by computer.
Let us assume the two-dimensional case of a ferromagnetic plate placed in a homogeneous DC magnetic field. The crack is assumed to be perpendicular to the magnetic flux, which is tangential to the surface of the plate The magnetic properties (the relative permeability bt) of the plate are given in Figure 1, and the data correspond to carbon steel (ST 37). It is further assumed
In the NDT literature some articles appeared to explain the whole phenomenon by the results of experimental measurements and by mathematical calculationI~-31. However the results of the calculations in Refs 2 and 3 show a certain discrepancy, and it is still not clear whether the signal measured by the leakage flux method is a good indication of the depth of the crack or whether the width of the crack also influences the measured signal significantly. This question is very important because in practice in nearly all cases the depth of a crack is much more significant than its width. In the literaturel~.21 the experimental results are interpreted by mathematical calculations using the method of finite elements,
Mathematical formulation
=L ¢ =3
IOO0
.g ~
-
I O F=g. 1 ST 37
I
I
I I
I
I
I
I I I I I I I I I I I I000 ZOOO MognetN: field strength, H [Am "t)
Relative permeab=hty as a function of magnetic field strength for
0308-9126/85/060353-05 $3.00 © 1985 Butterworth 8 Co (Publishers) Ltd NDT INTERNATIONAL. VOL 18. NO 6. DECEMBER1985
353
4 2
2
2
2
2
2
2
9
0
0
0
0
0
0
0
9
0
Z)
0
0
0
0
0
6
4
4
4
4
4
4
5
B
8
8
B
8
8
7
B
8
8
8
8
8
B
8
8
B
8
8
8
3
3
3
3
3
3
3
0
0
0
0
0
0
0
9
0
0
O
0
0
0
Z)
9
2
2
2
2
2
2
2
9
,¥
L
Fig 2
Lattice of points
that a D( field with magnetic strength H0 = 1000 A m -~ is applied, and the magnetic field a r o u n d the crack is calculated by the method of finite differences,
the relative permeability/.t should be chosen so that the relation between B and H according to Figure 1 is fulfilled`
The principal arrangement is given in Figure 2 A rectangular mesh is chosen with unequal mesh distances in t h e X and Y directions so that various widths o f cracks could be assumed` The depth of the gap is represented by D, the halt'width by W, and the thickness of the plate(L) is chosen as 20 mesh distances.
For points (6) and (7) a special procedurel4l is used. The difference equation written for these two points is in principle the arithmetm mean of two equations: one that holds for the h o m o g e n e o u s regions (outside or Inside the ferromagnetm material) and one that holds for the points if they were lying on the diagonal b o u n d a r y (le the line passing through (6) and the point (0. - 1 )
To calculate the magnetic flux density B and the magnetic field strength H m the vicinity of the defecL one starts with the differential equation for the scalar potential U V2U =
0
(1)
This equation holds for both inside and outside the plate Due to the symmetry only one half of the cross-section has to be taken into a c c o u n t The different n u m b e r s at the mesh points indicate the type of difference equation being used at that p o i n t The points (1) he on the line of symmetry and so m the difference equation for these points this has to be taken Into a c c o u n t The points (3) and (4) as well as (5). (6) and (7) he on the b o u n d a r y between the ferromagnetic material and air At any b o u n d a r y the normal c o m p o n e n t of the magnetic flux density remains u n c h a n g e d when crossing the b o u n d a r y From this fact It is possible to formulate the corresponding difference equation. In the relationship B = - - /*0/.t grad U
354
(2)
The points (2) and (9) are assumed to lie far enough from the defect so that it can be assumed t h a t H v = 0 a n d H , = / / 0 At the points (0) and (8) the difference equations corresponding to Equation (1) are the same. written in the usual form. In the computer program it is foreseen that the mesh distances in the X and Y directions may be different For the solving procedure the so-called L l e b m a n n extrapolanon relaxation methodt 4t was used. The property of the ferromagnetic material was taken into account so that we found lteratively the value o f p until the difference between the calculated and real values ofp, was smaller than or equal to 10. Alter some thousands of iterations we obtained the results for the scalar potentml. The magnetic field strength and the magnetic flux density outside and inside the material were then calculated. All the data represented m the followmg figures are drawn for the case when the distance above the surface of the plate is equal to 1 (y= 1). Various widths were simulated by varying the mesh distance in the X direction
NDT INTERNATIONAL. DECEMBER 1985
I
-0
5
io x
F~g. 3
-5
Normal component of the magnet,c held strength at Y = 1
t
o
5
x F=g. 5
Tangentml component
Hx(X) at vanous widths
and depths
conclude that the proportionality as given in Ref. 3 is preserved. Also Figure 5 shows that this proportionality is preserved when the width is varied. The difference (Hx - Ho) can be supposed to be proportional to the depth of the crack at any constant width. So far everything seems to be in order and the statements by Forster that the leakage field signal H v is proportional to the depth of the crack and that it ~s practically independent of the width could be verified. Also his statements about the influence of the measuring distance y on the distribution H x (x) could be verified.
W=I
O--
One can imagine that the magnetic flux lines are prevented from penetrating the walls of the gap and that the magnetic flux has to flow partially out of the ferromagnetic material. In such a simplified picture the width of the crack can be neglected.
2v.
--
1
I I I I II -5 F=g. 4
0 x
~
i i i I I 5
Tangentml component of the magnet=c field strength at Y = 1
Results
From Figure 3 it could be further concluded that the H.y compoffent has its maximal value above point (6) m Figure2 and across the width of the gap it changes its sign. Let us suppose that the gap is extremely narrow and deep. According to the results mentioned above Hy should be proportional to the depth and across some extremely short distance it should change its sign! We also changed the mesh distance in theX direction and calculated the field near defects with the same depth (D = 9) and different widths(W= 0.5, W = 0.2, W = 0.1). The results are given in Figure 6. It is quite obvious that the curvesH~(x) aty = 1 depend on the width of the crack. The signal Hy mdx is much smaller if, for example, W/D = 1/90, instead of W/D = !/9.
Figure 3 shows the Hy component outside the plate one mesh distance above the plate for different widths of the crack. It is obvious that the maximal value is proportional to the depth of the crack One can see that there are only very small variations due to the variations in width of the gap. This result corresponds to the results of F6rsterl31.
Figure 7 shows the function Hx(x ) aty = 1 for the cases of D=9andW=3, W = 2 , W = l , W = 0 . 5 , W = 0 . 2 and W = 0.1. Also this component of the magnetic field strongly depends on the crack width.
Figure 4 shows the H, component as a function of different depth s at a fixed width (w = 1). Here one can also
We also chose at W = 0.2 and W = 0.1 three cases: D = 9, D = 6 and D = 3 to see if(Hx - H0) is proportional to D
NDT INTERNATIONAL.
DECEMBER 1985
355
m e n t i o n e d above show some interesting facts 1
If the ratio D / W ~ I O it p, possible to ',a) that the magnetic field strengthH~ m,,x a t y = 1 Is p r o p o r t i o n a l to the depth o f the crack a n d does not d e p e n d on its width Also, at a defimte w~dth the m a x i m u m (H, - H0) ~s p r o p o m o n a l to the depth of the crack.
2
II the ratio D/W ~ 1 0 the H, and H, c o m p o n e n t s at i = 1 b e c o m e m u c h s m a l l e r a n d are no longer simply related to the depth o f the crack. In this case the width of the crack has to be taken into account.
150C
IOOC
)E
T 500
3000- t
O 0
5
I0
F~g 6
Normal component
Hv(x ) at D =
~
2000.
x 9 for different w~dths
O=9
W=02
0--6
W=OI
0=3
W=02
... ~¢~1000.
I I I I I
-5
I l l l l
0
5
x Fig 8
Tangential component
Hx(X ) for
the extreme cases
1500
I
I
I
I
I 7
-5
I 0
I
I
I
I ,
~
I000
5
x
Fig 7
Tangential component
Hx(x )
as a funct)on of different wtdths
( F i g u r e 8) O b v i o u s l y it is n o t Also the m a x i m a l values o f H~ c a l c u l a t e d a t y = 1 for different d e p t h s (D = 3, 6, 9) are strongly d e p e n d e n t on the width o f the gap ( F i g u r e 9)
500
T h e results o f these extreme cases ( F i g u r e s 8 a n d 9) can be c o m p a r e d with the results o f Lordlll a n d Dobmannf21
Conclusion T h e d e t a d e d c a l c u l a t i o n of the DC" m a g n e t i c leakage field in f e r r o m a g n e t i c material will be published. T h e results
356
0
3
6
9
D F)g 9
Maximal value of Hv as a function of different depths
NDT INTERNATIONAL.
DECEMBER 1985
.
The NDT method using leakage fields has some limitations. The measuring device should also provide some information about the width of the 'pulse" Hx(x). This width depends only on the width of the gap and not on its depth. For wider gaps it is directly proportional to the width of the gap. With the thinner gaps it seems to approach a certain limiting value defined by the magnetic field strengthH0 and by the properties of the ferromagnetic material (/t). However, within this limiting value, knowing the width of the 'pulse' one could calculate (perhaps in advance) the corresponding possible depth of the crack
There is still much practical work needed to verify this mathematical result and also to define the practical limits of the method.
Author The author is with the Iron and Steel Works, 64270 Jesenice, SRS, Yugoslavia.
References 1 Lord,W. and Hwan&J.H. "Defectcharacterizatmn from magnetic leakage fields"Bnt J NDT (January 1977) pp 14-18 2 Dobmann, G. and Walle, G. 'New set-ups for mathematlcalnumeriai solutions of magnetic leakage-flux testing with d.c and a.c mode in the FRG" lOth WCNDZ Moscow (1982) paper IC-5 3 Forster, F. "Neue Erkenntmsse auf dem Gebiet der zerstorungsfreten Prufung mit magneuschem Streufluss" 3 rd European Conference on ND T. Florence ( 15-18 October 1984) pp 287-303 4 Binns, K.J. and Lawrenson, P.J. "Analysis and computauon of electric and magnettc field problems, Pergamon Press Oxford
Paper rece,ved 26 April 1985. Rev,sed 12 July 1985
NDT INTERNATIONAL. DECEMBER 1985
357