MATHEMATICAL MODELING AT ACCOUNT of PARAMETERS of
INCOMPLETE FUSION of a WELD
E. Krouglova, L. Knyzuk, V. Kortov
Ural State Technical University, Ekaterinburg, Russia
ABSTRACT. In the report the questions of mathematical modeling at the radiographic control
of welded seams are considered in particular, at definition of depth of defect.
Introduction
The radiographic examination is one of the widespread kinds of the control of products, in
particular welding. It allows to reveal various defects of welds, which essentially influence
serviceability of metal construction: cracks, incomplete fusion, porosity, slag, tungsten and other
inclusions. The minimal size of defects revealed by a radiographic method is 1 - 2 % from thickness
of a controlled detail (for thickness from 5 up to 500 mm) [1].
Incomplete fusion are the most widespread and dangerous defects of welds. Under the standards
regulating realization of the radiographic control, the size of incomplete fusion is length of its image
on a radiographs. The size of defect in a direction of radiation (depth) is not determined. Now the
weld with long incomplete fusion is rejected irrespective of its depth. To avoid it, the physical
model and the method of account of depth of incomplete fusion are offered.
Physical model incomplete fusion
The most simple example is incomplete fusion in a root of a weld, since it can be presented as
extended defect with rectangular section. More complex variant is incomplete fusion on section of a
weld, i.e. incomplete fusion, inclined to a surface of a weld on a angle (fig. 1).
For improvement and the checks of a method of account of depth of incomplete fusion
simulators of defect with the known sizes of defects are created.
The simulator of incomplete fusion in a root of a weld represents two plates of the given sizes
fixed on a substrate. This design allows to change depth and width of defect, thickness of a product.
The simulator of incomplete fusion on section of a weld represents a core made from two parts, so,
that between them there is a backlash of given width. The core is inserted into an aperture of the
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basis. This design allows to change width and angle of an inclination of defect, thickness of a
product.
a
b
Fig. 1. The circuit of an arrangement of incomplete fusion in a root (a)
and on section of a weld (b)
Mathematical model of incomplete fusion
Incomplete fusion in a root of a weld on a radiographs looks as a dark direct line of appropriate
width and density, taking place along a weld on his(its) middle. Together with weld the flute type
penetrameter was transparent. It contains simulators of defects as flutes of rectangular section.
Width of all flutes of the penetrameter is identical, and the depth varies.
Using a radiographs of weld and penetrameter, it is necessary to determine the characteristics
incomplete fusion, to decide for acceptance of a detail. As such characteristics we shall consider
depth, width incomplete fusion and unsharpness of the image.
For account of the required characteristics the distribution of optical density on film is analyzed
[2]. As on a radiographs there is an image of the penetrameter, width and depth of which flutes are
known, it is possible to determine the characteristics of researched defect, comparing the image of
defect with the images of flutes of the penetrameter. Account of the characteristics of flutes of the
penetrameter or defect needs quantitative measurement of optical density of a film. For definition of
depth of defect it is necessary to measure contrast of the image of defect, i.e. difference of optical
density of the images of the defect and background.
We shall consider defect of rectangular section. On a film the step form of curve optical density
is not kept (fig.2). The change of density is distributed to some distance, which refers to as
unsharpness.
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Fig. 2. A curve of density in area of flutes of the penetrameter
Let’s approximate a curve of density by four straight line and calculate the characteristics of a
flute of the penetrameter:
Unsharpness: H  x 2  x1
(1)
Contrast: K  y 2  y1
(2)
Width: S  x 4  x 2  x 4  x1  H
(3)
Depth: h  k 2  K  b2
(4)
When unsharpness exceeds width of defect, the defect loses the true sizes, the contrast decreases
[3]. Hence, for narrow defects the «direct» definition of width and depth will result in a mistake.
For account of narrow defect (S  H) it is necessary to restore true meaning of contrast (fig. 3),
determining unsharpness of its image depending on tg  [4].
We consider tg  and unsharpness of the image:
tg 
y 2  y1
x 2  x1
H  k1  tg  b1  k1 
(5)
y2  y1
 b1
x2  x1
(6)
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Fig. 3. A curve of density in the field of defect
Postponing on an axis Х meanings of unsharpness, we restore contrast of the image:
x 4  x1  H
(7)
For definition of ordinate of a point of crossing again expression for tg  is written down:
tg 
y 4  y1
x 4  x1
(8)
From (6) follows:
y 4   x 4  x1   tg  y1
(9)
The coordinates of nominal contrast of narrow defect are received. After that we consider all
characteristics of narrow defect:
S  x 4  x1  H  x 4  x1  k 1 
y 2  y1
 b1
x 2  x1
K  y 4  y1  ( x 4  x1 )  tg  ( k1 
y 2  y1
y  y1
 b1 )  2
x 2  x1
x 2  x1
h  k 2  K  b2
(10)
(11)
(12)
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Experimental check of modeling
For experimental confirmation of the offered technique of account of depth of incomplete
fusion of a weld the radiographic films of simulators of defects with the various known
characteristics are executed, the meanings of depth, width and unsharpness of the image of defects
are designed.
On an example of one radiographs of the simulator incomplete fusion in a root of a seam in
detail we shall consider the basic laws of realization of account of depth of defect on the offered
technique. For this purpose we shall choose a sample № 1n.
First of all it is necessary to consider the penetrameter, which image necessarily is present on a
film. For each concrete film the peaks of the image of the flute of the penetrameter are analyzed, the
characteristics of each peak are measured and the dependence of unsharpness of the image of a flute
from its depth is constructed(Fig.4). The depth of flutes is known.
unsharpness, mm
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,5
1
1,5
2
2,5
depth, mm
Fig. 4. Dependence of unsharpness of the image of flutes of the penetrameter from their
depth for a sample № 1n
Till a fig. 4 it is visible, that the dependence of unsharpness of the images of flutes from their
depth is possible to approximate by a direct line and to prolong it for limits of depths of flutes.
As the information about unsharpness of the image of defect can be received only on tg we
shall construct dependence tg from unsharpness (Fig. 5).
Similarly dependence in a fig. 5 is approximated by direct line.
After that it is necessary to consider a curve of density of the image of defect. As any of the
characteristics of defect is unknown, it is impossible on curve density to determine unsharpness of
the image of defect. tg is one characteristic, which can be determined directly on curve density. By
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measuring it, from fig. 5 it is determined unsharpness of the image of defect. By finding out
unsharpness, we consider other characteristics of defect on the offered technique.
8,00
tg a
6,00
4,00
2,00
0,00
-2,00
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
unsharpness, mm
Fig. 5. Dependence of tg from unsharpness of the image of flutes
the penetrameter for a sample № 1н
For defect and flutes of the penetrameter on a sample № 1н, the data of unsharpness and depth
of defect are received. Let's put the received points on dependence of unsharpness from depth (Fig.
6). It is visible, that points of the designed characteristics practically lay on dependence of
unsharpness of the image from depth.
It is necessary to note, incomplete fusion on section looks as a wide strip, and, accordingly, the
curve of density has other kind.
In this case with the account unsharpness of the image it is considered, that width of peak on
curve density characterizes depth of incomplete fusion on section of a weld, and its height - width.
The further account is conducted similarly to account of the characteristics incomplete fusion in a
root of a weld.
Conclusion
The technique of definition of depth incomplete fusion in a root and on section of a weld is
developed. The technique is tested on welds by thickness up to 10 mm.
The offered mathematical model of incomplete fusion in a root and on section of a weld has
received experimental confirmation.
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unsharpness, mm
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,5
1
1,5
2
2,5
depth, mm
Fig.6. Dependence of unsharpness of the image from depth for a sample 1н
 - penetrameter
 - defect
REFERENCES
1.
Sosnin F.R.: Nondestructive testing. Moscow, Mechanical engineering, 2003. (In
Russian)
2.
E. Krouglova, L. Knyzuk: ‘Definition of the sizes of defects of weld on x-ray film’.
Nondestructive testing 2004 1 71-75. (In Russian)
3.
Dmhovskiy V.V.: Bases of X-ray engineering, Moscow, Medical publishing house,
1960. (In Russian)
4.
L. Knyzuk, E. Krouglova: ‘Way of restoration of contrast of the image narrow
incomplete fusion for account of its depth’, 22nd Conference on Welding, Kirov, 2003. (In
Russian)
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