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УДК 539.3 + 621.791
Анализ поля напряжений вблизи сварного шва на основе измерения деформаций в рамках физической мезомеханики
S. Yoshida1, T. Sasaki2, M. Usui2, and Ik-K. Park3
1 Юго-восточный университет штата Луизиана, Хаммонд, Луизиана, 70402, США
2 Ниигагский университет, Ниигата, 950-2181, Япония 3 Сеульский национальный университет науки и технологий, Сеул, 139-743, Корея
Напряжения, возникающие при сварке, анализируются с учетом характера деформации материала. Для измерения остаточных напряжений используются тензодатчики, образца со сварным швом на внешнее воздействие анализируется с помощью электронной спекл-интерферометрии. К стальному образцу в форме тонкой пластинки со стыковым швом прикладывали растягивающую нагрузку и с помощью электронной спекл-интерферометрии изучали полученное поле деформаций. Проведено сравнение с образцом из того же материала тех же размеров без сварного соединения. Показано, что возникающее при сварке остаточное напряжение вызывает асимметрию нормальной деформации под воздействием внешней растягивающей нагрузки. Асимметрия усиливает сдвиговую и ротационную моды деформации, что приводит к концентрации напряжений на удалении от сварного шва, где остаточные напряжения пренебрежимо малы. Выявленные особенности обсуждаются с точки зрения физической мезомеханики. Показано, что поведение сварного образца под воздействием внешней нагрузки подобно пластической деформации.
Ключевые слова: остаточные напряжения при сварке, физическая мезомеханика, динамика пластической деформации, электронная спекл-интерферометрия
Analysis of near weld stress field based on strain measurement and physical mesomechanics
S. Yoshida1, T. Sasaki2, M. Usui2, and Ik-K. Park3
1 Southeastern Louisiana University, Hammond, 70402, USA 2 Niigata University, Niigata, 950-2181, Japan 3 Seoul National University of Science and Technology, Seoul, 139-743, Korea
Stresses induced by welding are analyzed from the viewpoint of material deformation behavior. Strain gauges are used to measure the residual stresses, and electronic speckle-pattern interferometry is used to analyze the response of the welded work to external force. A tensile load is applied to a butt-welded, thin-plate steel specimen, and the resultant strain field is analyzed with the electronic speckle-pattern interferometry. Comparison is made with the case of a non-welded specimen of the same material and dimension. The analyses indicate that the residual stress due to welding makes the normal strain due to the external tensile load asymmetric. The asymmetry enhances shear and rotational modes of deformation, generating stress concentration at a point away from the weld where the residual stress is substantially negligible. The observed features are discussed based on physical mesomechanics. Analysis reveals plastic deformation like behavior in the response of the welded specimen to the external force.
Keywords: residual stress due to welding, physical mesomechanics, plastic deformation dynamics, electronic speckle-pattern interfer-ometry
1. Introduction
The stress field near a welded joint is complicated. The thermal load imposed by the welding torch and the subse-
quent cooling process form a sharply varying spatial temperature gradient, and consequently, generate stresses locked in the material [1]. This type of hidden stress is known as the residual stress [2], and is harmful as it can lead to distortion, fatigue cracking, premature failures of the welded
work. A number of destructive [2-5] and nondestructive [6-10] techniques have been developed to assess residual stress. The destructive techniques, in general, remove a small portion of the material to relax the residual stress, and evaluate it by comparing the deformation status before and after the removal of the material. For the analysis of residual stress in welded works, destructive methods are not suitable for at least two reasons. First, since the thermal load is
© Yoshida S., Sasaki T., Usui M., Park Ik-K., 2015
the main cause of it, residual stress in a welded work is mostly formed at the region known as the heat affected zone. In most cases, as generally known well, the heat affected zone is weaker than the welded zone. Removal of material in the heat affected zone enhances the weakness of the welded work, and should be avoided as much as possible. Second, residual stress is not uniform near the weld. Oftentimes, compressive and tensile residual stress alternates, which is understandable if we note that the stress is caused by the thermal load from the welding torch that repeats the heating and cooling processes as it moves along the weld line. To identify such a spatial variation of the residual stress, it is necessary to assess it at a number of points with a substantially small increment. This further weakens the welded work if the technique is destructive. Thus, the application of a nondestructive technique is essential. This situation causes a self-conflicting situation. Being hidden in the material, residual stress cannot be revealed unless the locking mechanism is removed, but we do not want to remove the material.
While the evaluation of residual stress is important, similarly important and often overlooked is the analysis of the response of residually stressed materials to external loads. When a stress is locked in a material, it is likely that the local elastic modulus is altered from the original value. A tensile residual stress is locked by the neighboring materials that are either plastically and hence permanently elongated, or that are elastically compressed as they equilibrate the tensile residual stress1. In either case, the local elastic modulus is altered from the unstressed situation. The change in the local elastic modulus of the region with a compres-sive residual stress can be argued in the same fashion. Furthermore, it is expected that the elastic modulus varies in a complex fashion near the weld as a function of the residual stress. Thus, when a welded work is subject to an external load, its response can be substantially different from the case when a specimen of the same material and size with no residual stress is subject to the same external load. It is widely known that a compressive residual stress raises the elastic modulus because the decrease in the inter-atomic distance resulting from the compression pushes the equilibrium point on the strain-energy potential curve towards the steeper side of the potential well. Here the potential well corresponds to the equilibrium point when the material is unstressed. Similarly, a tensile residual stress pushes the equilibrium point towards the longer atomic-distance side of the potential well where the potential curve is less steeper than near the well. Consequently, on the application of an external load, the material stretches less near a
1 Imagine a column of elastic material is sandwiched by a pair of slightly longer columns of the same material from the sides. As the central, shorter column is stretched the side columns are compressed.
compressive residual stress and stretches more near a tensile residual stress, as compared with the region of no residual stress. In the context of weld-induced residual stresses, since compressive and tensile residual stress can alternates near the weld, it is possible that a welded work behaves in a completely unexpected fashion when an external load is applied.
The aim of this study was to explore this exact question of "how do the residual stresses alter the response of welded works to external force?" To this end, a low-level tensile load was applied to a butt-welded thin-plate specimen and its response was compared with the response of a non-welded, annealed specimen of the same dimension and material (called the annealed specimen). Electronic speckle pattern interferometry (ESPI) was used to analyze the specimen behavior as a two dimensional, full field strain map. After the strain measurement, a strain gauge was used to evaluate the residual stress at several points where the specimen was cut into a small piece around the strain gauge so that the residual stress was relaxed. The results indicated that the change in the elastic modulus due to the residual stress near the weld altered the pattern of overall deformation, making it less symmetric around the central line of the specimen (both parallel and perpendicular to the tensile axis). This generated strain concentration at a point away from the weld line, causing more bodily rotation in response to the applied tensile load. The observed behavior of the annealed and welded specimens were analyzed based on physical mesomechanics [11, 12]. The use of physical meso-mechanics was important because being able to describe elastic and plastic deformation on the same theoretical foundation [13], it allowed us to evaluate the effect of the residual stress in conjunction with plastic deformation.
2. Experimental
2.1. Specimens
The material of the specimen tested was a cold-rolled carbon steel. Two plates of 90 mm (wide) x 75 mm (long) x 3 mm (thick) were butt-welded into a plate of 90 mm x 150 mm x 3 mm, and cut into 50 mm (wide) x 150 mm (long) specimens. The weld overlay was polished off so that the entire specimen had the same thickness. A non-welded specimen of the same dimension was prepared as the control. Since the plates were initially annealed at 1173K for 30 s, the non-welded specimen is called the annealed specimen. As indicated by Fig. 1, the specimen had four holes at the two ends that were used to attach to the grips of the tensile machine with bolts and nuts. The arrays of six circles shown in Fig. 1 indicate where the residual strain was measured with the strain gauges. Although this figure shows the circles on the right hand side of the weld line only for clarity, the residual strain was measured on the left hand side of the specimen as well at the six points symmetric to the above-mentioned six points about the weld-
Tensile load
I ......... ''I" | ! ' i'' I j. ■ I ■ |! III j I II !|:lj I .......|ll l| I | . I .1 | |
an чо Ю0 1,0 iir '"о mm'70 160 190 200 210 2
40 mm
40 mm
150 mm
Illuminated Mirror
Specimen ^ area
Fig. 1. Specimen used in the present study. The six circles along y = 5 and 20 mm indicate the location where the residual stress was measured with strain gauges. The residual stress measurement was made on the left side of the specimen at six points symmetric about the weld line
line; the residual strain was measured at twelve points in total.
2.2. Experimental procedures
Since the evaluation of residual strain with the strain gauge was destructive, strain measurement with an external load and ESPI was made first. A care was taken to assure that the applied load did not relax the residual stress or cause plastic deformation. The tensile load was increased from0.311to 3.68k№. The corresponding stress was 3.68 kN/(3 mmx50 mm) = 24.5 MPa, which was 10 % of the yield stress 245 MPa.
Figure 2 illustrates the optical arrangement of the ESPI setup used in this study. The ESPI setup consisted of two interferometers, one sensitive to the x component of the inplane displacement vector (called the x interferometer) and the other sensitive to the y component (the y interferometer). Here the coordinate axes x and y were set on the plane of the specimen surface where the x axis is parallel to the weld line and the y axis parallel to the tensile axis. For each interferometer, a semiconductor laser (660 nm in wavelength) beam was split into two interferometric paths with a beam splitter. The two steering mirrors placed in the respective optical paths steered the beams so that they would overlap on top of each other on the specimen surface. Beam expanders (not shown in Fig. 2 to avoid complexity) were used so that the two beams would cover the area of interest on the specimen. The in-plane displacement measurement was conducted through the following steps.
Step 1: An interferometric image of the specimen was taken with the two interferometric beams of the x interferometer on top of each other on the specimen surface. The image was taken by a digital camera and stored in computer memory (called the base image). This procedure was repeated for the y interferometer.
1 The initial load was set at 0.311 kN rather than 0N to avoid the slippage of the specimen in the grip of the tensile machine.
splitter Laser
(horizontal)
/' \ ' ^
Stationary Beam
< end ^ T i<> expander - Laser
(vertical)
Fig. 2. Optical arrangement for the ESPI setup
Step 2: The tensile machine was engaged for the application of a tensile load to the specimen. When the load reached a preset value, the cross head of the tensile machine was stopped and another interferometric image was taken. The resultant image was stored in computer memory (called the deformed image). The procedure was taken for the x interferometer and repeated for the y interferometer.
Steps 1 and 2 were to acquire the interferometric image data. These steps yielded two sets of the base image and deformed image, a set for the x component and the other for the y component of the in-plane displacement. The following steps, the data analysis procedures, were taken after the data acquisition procedures were completed. The data analysis procedures were common to both x and y components of the in-plane displacement. Below the displacement component is denoted by i.
Step 3: The base image was subtracted from the deformed image on a pixel-by-pixel basis. The resultant subtracted image was stored in computer memory. This subtracted image contained the phase information associated with the phase change due to the deformation caused by the tensile load The image was called the subtraction fringe pattern. Figure 3 shows sample subtraction-fringe patterns.
Step 4: An integer representing the fringe order was assigned to each of the subtraction fringes. Here the fringe order is defined as follows. In accordance with electronic speckle-pattern interferometry, each dark fringe represents the contour of a constant displacement that corresponds to the phase difference of 2nn where n is an integer. Depending on the angle of incidence of the interferometric beams and other geometric factors, 2n corresponds to the unit displacement ti0 where t denotes the displacement vector and i is the coordinate variable xt that the pair of the interferometric beams is sensitive too. Accordingly, n = 0 was assigned to the dark fringe corresponding to zero displacement. Subsequently, n = 1 or n = -1 was assigned to the fringe next to the fringe corresponding to n = 0. Here, if the next fringe corresponded to displacement of tx0, n = 1 was assigned and if it corresponded to -t,x 0, n = -1 was assigned. Similarly, a signed integer was assigned to the next dark fringe one by one. The lower two images of Fig. 3 illustrate how the fringe-orders were assigned. Once an or-
^-interferometer x-interferometer
Fig. 3. Sample subtraction-fringe patterns and assigned fringe orders
der was assigned to all the fringes respectively, grid lines parallel to the x axis were drawn at y = yj through y = yk. Here, yj denotes the y-coordinate of the first grid line and yk denotes the last grid line. For each of the k grid lines, the coordinate points where each fringe crossed the grid line were found. This yielded a vector (xn, n) where xn was the x coordinate at which the grid line crossed the nth fringe. After all the grid lines were processed, a table containing the coordinate points where grid lines crossed all the fringes was formed. Here the (xn, n) vector for the grid line at y = yk constituted the kth row of the table. The resultant table was called the fringe order table. A fringe order table was generated for the respective components of the in-plane displacement vector; tx and ty.
Step 5: Once a fringe order table was generated, the fringe order was replaced with corresponding displacement vector component according to the relation ti (n) = n£i0. Here ti (n) denotes the displacement vector component that corresponds to the fringe order n. Subsequently, for each row the vector (xn, n£0) was interpolated with respect to the discrete values of xn from the lowest to highest values of n for the row. Iteration of this procedure for all the rows yielded a map of ti that provided the displacement component ti for all the grid points.
Completion of Steps 4 and 5 for tx and ty yielded a table of displacement vector t evaluated at all the grid points. Once the displacement vector table was obtained, strain and rotation tables were obtained through numerical differentiation as follow.
_ 1 2
d^
dx dx,
(
(1)
% _ 2
dx dx,
(2)
After the ESPI measurement was over, the residual strain was measured at the reference points shown in Fig. 1. Each strain gauge comprised a stack of two layers where the first layer contained a conductor sensitive to strain along the x axis and the other contained a conductor sensitive to strain along the y-axis. The conductor was approximately 1 mm long whose initial (undeformed) resistance was 120 ± 0.8 W. Thus, the strain gauge was sensitive to both e xx and e yy. The residual strain measurement was conducted in the following way. After the specimen was dismounted from the tensile machine, the conductivity of the conductors was recorded for each strain gauge. Then, a square segment of the specimen approximately 8 mmx8 mm was cut off around each strain gauge. The conductivity was remeasured for the conductors of each strain gauge and recorded. The difference in the conductivity between before and after the square segment of the material was cut off was interpreted as the difference in the strain, and hence representing the residual strain. The normal strain in the x and y directions were evaluated in this way for all the twelve reference points shown in Fig. 1.
3. Results and discussions
3.1. Strain data from strain gauge and ESPI
Figure 4 summarizes the results of strain measurement made with the strain gauge and ESPI. The top two rows are the strain data from the strain gauge measurement and the other rows are from the ESPI. Each column presents data
obtained along the same line on the specimen parallel to the tensile axis. The left column (a) labeled x = 143 presents data along 143 th pixel, which corresponds to the line labeled x = -16 mm in Fig. 1. Similarly, the middle (b) and right (c) columns present data along 255th pixel or x = 0 mm, and 367th pixel or x = 16 mm, respectively. The strain (rotation) data from ESPI are calculated from the interpolated in-plane displacement tx (x, y) and ty (x, y) data as explained above.
As mentioned above, the strain data measured with the strain gauge presents residual strains and those with the ESPI displacement measurement present strains due to the external load applied by the tensile machine. Therefore, when the ESPI measurement is applied to the welded specimen, the resultant displacement presents the response of the specimen to the applied tensile load with the elastic modulus altered by the residual stress. Each of the strain data from ESPI compares the cases of the annealed specimen (the solid line) and the welded specimen (the dot-dashed line). The x axis, whose positive direction is downward in Fig. 1, is parallel to the weld line, and the y axis is parallel to the tensile axis. The positive direction of the y axis is right-ward in Fig. 1 and the tensile load is applied in the negative y direction. The normal strain 8xx is perpendicular to the tensile load and called the lateral normal strain. The nor-
mal strain 8yy is called the longitudinal normal strain, hereafter. The following observations can be made.
3.1.1. Residual strain
1. Residual lateral normal strain is compressive near the weld and slightly tensile at the two ends where the specimen is gripped by the tensile machine. The peak compressive strain near the weld is greater along the middle line (x = 0 mm in Fig. 1) than the upper (x = -16 mm) or lower (x = 16 mm) lines. The peak compressive residual strain of the upper and lower lines are similar to each other.
2. Residual longitudinal normal strain is tensile for the entire specimen, where the tensile strain is much lower along the middle line (x = 0 mm) than the upper (x = -16 mm) or lower (x = 16 mm) line. Along the upper and lower lines, the tensile strain is greater near the weld than the two ends.
3. The above observations indicate the following tendency regarding the elastic modulus in the welded specimen. In the direction parallel to the weld line, the elastic modulus is greater near the weld than the other part. In the direction parallel to the tensile axis, the elastic modulus near the weld is lower toward the upper and lower sides of the specimen than the middle. In other words, referring to Fig. 1, the welded specimen is less rigid horizontally toward the upper and lower sides of the specimen than
Ь 1
1 0 w -1
Strain gauge x, x = 143
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0
1 5
J* 0
0 200 400 600 Strain gauges, x = 143
.xr \
0
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о
to _
200 400 ESPI, x= 143
600
Ь 1
1°
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о
Strain gauge x, x = 255 b
\
/
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200 400 600 Strain gauges, x = 255
2 1 co*_l
Strain gauge x, x = 367 0
к.
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0
°0
200 400 ESPI. x= 143
600
? 4
2 о
Й-!
200 400 ESPI. x = 255
600
T 10
0
1 5
о
0 200 400 600 Strain gauges, x = 367
0
°0
200 400 ESPI, x = 255
600
'П
i 4 о H
^ 0
И
to —
200 400 ESPI, x = 367
600
'0
200 400 ESPI, x = 367
600
200 400 ESPI, x= 143
200 400 ESPI. x = 255
200 400 ESPI, x = 367
200 400 v, pixel
200 400 v, pixel
400 v, pixel
Fig. 4. Longitudinal and lateral strain measured with strain gauge (top two rows) and ESPI (bottom four rows). Annealed (solid lines), welded specimens (dot-dashed lines). Column (a)-(c) show data along the top, middle and bottom horizontal lines (labeled x = -16, 0 and 16 mm in Fig. 1), respectively
the middle, and vertically more rigid near the weld than the two ends towards the grip of the tensile machine.
3.1.2. Strain due to external load
1. Both the annealed and welded specimens show compressive lateral normal strain exx. The compressive strain perpendicular to the tensile axis can be understood as reflecting the Poisson's effect. However, the welded specimen shows greater compressive strain than the annealed specimen. This is counter-intuitive because the residual strain measurement indicates that the welded specimen has a higher elastic modulus than the annealed specimen in the direction perpendicular to the tensile axis as we discussed under "Residual stress observation 1" above. A possible explanation is that the high rigidity parallel to the weld line hinders the specimen from being compressed near the horizontal center according to the Poisson's effect when pulled horizontally. This possibility will be discussed later in this paper.
2. Neither the longitudinal normal strain or shear strain of the welded specimen shows substantial difference from the annealed case, whereas the welded specimen shows much higher rotation than the annealed specimen.
Figure 5 displays the same external load induced deformation as Fig. 4 in the form of contour plots. The left column is for the annealed specimen and the right column is
g 2ОО pi
,Г
4ОО
for the welded specimen. At a glance, it appears that the deformation of the annealed specimen is symmetric about the horizontal and vertical centers of the specimen as expected. However, the deformation of the welded specimen appears totally asymmetric. The lateral normal strain exx appears more compressive near the top side than the bottom. The longitudinal normal strain e^ is concentrated near the bottom middle of the right hand side of the weld (centered around (x, y) ~ (350, 480)). On the other hand, the shear strain and rotation in the welded specimen are much greater than the annealed specimen.
It is possible that the hindrance of the Poisson-effect-like natural lateral compression due to the elevated elastic modulus near the weld is the main cause of this asymmetric behavior. To look into this more precisely, refer to Fig. 6 where the lateral normal strain e xx near the weld is compared with the case of the annealed specimen. It is seen that the annealed specimen shows the natural compression along the x-axis (parallel to the weld) where the strain is null near the center (along the line parallel to the y-axis going through the vertical center of the specimen where the lateral displacement changes the sign), and compressive at both sides of the weld line. This behavior clearly reflects the Poisson's effect. On the other hand, the welded specimen shows almost monotonic decrease in the compressive lateral nor-
^ Normal strain exx, welded 10-4 b
О 2ОО 4ОО 6ОО Normal strain annealed
2ОО 4ОО 6ОО Shear strain exy, annealed
2ОО 4ОО 6ОО Rotation annealed
ix2 pi
,Г
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-
■D. - V
■ ¿r
w-4
О -l
-2
a
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2ОО 4ОО 6ОО y, pixel
ю-4 l
О -l
2ОО 4ОО 6ОО Normal strain e^, welded
2ОО 4ОО 6ОО Rotation ol, welded
0 200
Fig. 5. Contour plots of strain and rotation due to external load for annealed (a) and for welded specimens (b)
4ОО 6ОО y, pixel
Normal strain exx, annealed
Ю-4
I
О
Annealed 0 Welded b
Fig. 6. Lateral strain near weld. The annealed specimen shows the natural compression due to the Poisson's effect. The welded specimen does not. Note that the x axis is reversed for better view
mal strain from the bottom side of the specimen (x = 400 pixel) to the upper side (x = 100 pixel.) Note that for the better visibility of the asymmetric behavior of the lateral normal strain the x axis of Fig. 6 is reversed as compared with Fig. 1, and the front side of the graph is the bottom side of the specimen and the back side is the top side of the specimen.
The natural symmetric compression due to the Poisson's effect can be viewed as the conjugate rotations by the four blocks around the center of the specimen, as illustrated schematically by the upper illustration of Fig. 7, so that the total angular momentum is null. The total angular momentum must be zero because the tensile machine does not exert torque on the specimen as it pulls the specimen in a purely translational manner. The monotonic variation in the lateral compressive normal strain along with the above-argued asymmetric nature of the deformation indicate that the welded specimen undergoes rotation in a certain direction near the weld. In order for the entire system to conserve the angular momentum, some other part of the specimen must undergo rotation of the opposite sign. The rotation №z of
Fig. 7. Rotational situation when the specimen undergoes natural compression due to the Poisson's effect (upper), and when the specimen exhibits the asymmetric behavior due to the residual stress (lower). As the x and y axes are defined as shown in this figure, the clockwise and counterclockwise rotations are negative and positive rotation around the z axis, respectively
the welded specimen seen in Fig. 5 indicates the situation where the entire specimen undergoes clockwise rotation in the range y = 100to 500 pixel and counterclockwise rotation in the range y > 500 pixel with the bottom right near x = 400 pixel and y = 500 pixel as the hinge. The lower illustration of Fig. 7 illustrates the situation schematically with some exaggeration. This observation is consistent with the feature observed in the longitudinal normal strain of the welded specimen (the second row in the right column of Fig. 5) that the region near this hinge point undergoes much greater stretch. It is also consistent with the quiver plot of the displacement field shown in Fig. 8.
The contour plots, Fig. 5, illustrate the two-dimensional symmetry/asymmetry clearly. However, the magnitude of each mode of strain is not easy to read from the contour plot. To discuss the magnitude of each strain and its spatial variation, refer to Fig. 9 where the same strain data as Fig. 5 are displayed in the form of three-dimensional surface plots. The arrangement is the same as Fig. 5; the left column is for the annealed specimen and the right column is for the welded specimen. The dashed lines near y = 300 pixel indicates the location of the weld. The following features are seen.
Fig. 8. Contour plots of displacement. Upper for annealed and lower for welded specimens. Tensile load to the left
Annealed, normal strain perpendicular
tn t^ncil^ íivic P
Annealed, normal strain perpendicular to tensile axis svv
Annealed, shear strain perpendicular to tensile axis sYV
Welded, normal strain perpendicular
to tensile axis s
Welded, normal strain perpendicular
to fp>nci" qyi'c с
Welded, normal strain perpendicular
tn t n С 1 1 QVIC С
Fig. 9. Normal strain in annealed and welded specimens. Dashed lines indicate location of weld
1. The lateral normal strain 8xx of the welded specimen
shows more oscillatory behavior in the y dependence than the annealed specimen. The oscillation along the y axis even appears somewhat periodical. This naively indicates that the welded specimen undergoes plastic deformation. According to the plastic deformation criterion [13, 14] based on physical mesomechanics [11, 12], as will be discussed later in this paper in more detail, curly feature of the rota-
tion field (Vxw ^ 0) is an indicator of plastic deformation. In the two-dimensional dynamics in the xy plane, they dePendence of 8 xx, 9e XJ dy = dfâ J dx )/dy = d\l dxdy constitutes part of the y component of the Vxw vector as
(Vx©) y = -dcoz/ dx = -dfây/dx-dljdy )/dx = = d2 %Jdxdy-a2 %y/ dx2.
Thus, the y dependence of exx indirectly indicates plastic deformation of the specimen.
2. The y dependence of the longitudinal normal strain e^ for the welded case appears similar to the annealed case. As for the x dependence, it appears that the welded case shows slightly higher tensile strain near the bottom side of the specimen than the annealed case. This tendency reflects the above-argued rotational behavior of the welded specimen with the hinge near (x,y) = (400, 500) pixel.
3. The shear strain exy of the welded case shows a greater oscillatory feature both in the x and y directions than the annealed case. This can be explained as the development of plastic deformation for the same reason as the y dependence of exx discussed above.
4. Rotation roz plots show the above-argued effect that for most part of the specimen the welded case exhibits negatively greater rotation than the annealed case. Also note that the annealed and welded cases respectively show the sign of mz consistent with Fig. 7; the clockwise/ counterclockwise are negative/positive rotation.
3.2. Physical mesomechanical interpretation 3.2.1. Physical mesomechanics
Details of the physical mesomechanical view of deformation and fracture can be found elsewhere [11-13]. In short, deformation of all stages, from elastic deformation through fracture, can be concisely expressed by the following field equations that govern the displacement field of solids under deformation.
V-v = -jo, (3)
Vx v =
dt '
1 dv
c dt
(4)
(5)
V-ra = 0. (6)
Here v and ra are the temporal and spatial derivatives of the vector potential A, as defined below, j0 and j are the temporal and spatial components of the quantity known as the charge of symmetry. In the present context, since they behave as source terms that determine the dynamics of deformation, j0 is referred to as the deformation charge1 and j as its flow. The charges of symmetry are involved in the field equations because physical mesomechanics is a gauge theory based on the symmetry of local elasticity [13]. The vector potential A is the gauge potential that logically connects local entity referred to as the deformation structural element within which the local deformation can be described by the law of linear elasticity, v and ra are the two field
1 In some contexts, p(V - v) is called the deformation charge. Here p is the density of the material and assumed to be constant. So, the quantities p(V-v) and (V-v) are different by a factor of a constant, and are called "charge" or "deformation charge" interchangeably.
variables that represent the gauge field. Field equations (3)-(6) are derived via application of the least action principle to the gauge field. The phase velocity c in Eq. (5) represents the spatiotemporal behavior of the gauge field
3A (7)
v=~dt' (7) ra =VxA. (8)
Mechanical consideration allows us to put the phase velocity in the following form, which is normally interpreted as the phase velocity of a shear wave,
c =f '
(9)
where G is the shear modulus and p is the density of the solid. Substitution of Eq. (9) into Eq. (5) yields the following equation that can be interpreted as the equation of motion governing the dynamics of a unit volume in a solid under deformation
dv
p— = -GVx ro - Gj. dt
(10)
The first term on the right-hand side of Eq. (10) represents the elastic shear force due to differential rotation and the second term the longitudinal force.
Application of divergence to Eq. (10) and the use of the mathematical identity V-(Vxm) = 0 lead to the following equation
3(V- v)
dt
- = -V- (Gj).
(11)
Equation (11) can be viewed as an equation of continuity where the quantity Gj is the current of p(V - v). Accordingly, it follows that p(V - v) is a conserved quantity and its temporal change is compensated by the current Gj. p(V - v) = V - (pv) is differential of the momentum density. In this view, the left-hand side of Eq. (11) represents the temporal change of the momentum density, i.e., the momentum of a unit volume. Thus, Eq. (11) can be viewed as representing Newton's second law stating that the temporal change of the momentum is equal to the differential force acting on the unit volume. As will be explained shortly, whether the deformation is elastic or plastic is determined by whether the longitudinal force Gj represents elastic force or energy dissipative (velocity damping) force.
The interpretation that Gj is the flow of p(V - v) allows us to relate Gj with p(V - v) via drift velocity Wd
Gj = Wd p(V- v). (12)
Here the drift velocity Wd can be interpreted as the velocity of the spatial pattern of the displacement field expressed by (V-v). If the deformation is elastic, the longitudinal force is proportional to the volume expansion V - £ where X is the displacement vector, and the drift velocity is the phase velocity of the elastic compression wave. If the deformation is plastic, the longitudinal force is energy dissi-pative force proportional to the local velocity v, and the drift velocity represents how much faster the pattern (V - v)
Fig. 10. Shear force and deformation charge in annealed (left) and welded (right) specimens
flows as compared with the particle velocity v. As described in Appendix, the longitudinal force Gj in the elastic and plastic cases can be expressed in the following forms, respectively
G jel = -(X + 2G )VV ), (13)
G jpi = p(V • v ) Wd = GoP(V.v ) v. (14)
Here l is the first Lamé parameter (G is the second Lamé parameter equal to the shear modulus) and s is the ratio of drift velocity Wd to the particle velocity v whose meaning will be discussed in Appendix. Importantly, even in the pure plastic regime, the first term on the right-hand side of Eq. (10) represents elastic shear force. It follows that in the pure plastic regime, the elastic recovery dynamics is rotational whereas in the pure elastic regime it is translational in proportion to the volume expansion as indicated by Eq. (13). In the pure plastic regime, the translational dynamics is represented by the energy dissipative longitudinal force (14). From this standpoint, the following condition can be viewed as an indicator of plastic deformation occurring in an initially linear elastic material.
Vx© * 0. (15)
Having the above argument in mind, compare the curl of
rotation Vx© and the divergence of velocity V • v between the annealed and welded cases. Here the curl of rotation is proportional to the shear force as -GVx © and the divergence of the velocity is to the deformation charge as pV • v. Figure 10 compares these quantities for the annealed specimen (left column) and the welded specimen (right column) as three-dimensional plots. Figure 11 compares the same quantities as contour plots where the top row is for the annealed and the bottom row is for the welded case. The following observations can be made.
3.2.2. Three dimensional plots
1. The x component of Vx©, which represents the shear force component perpendicular to the tensile axis, appears substantially different between the annealed and welded specimens. Here the difference can be characterized in two ways; (i) overall, the welded specimen shows higher value in the entire area and (ii) the welded specimens show more oscillatory feature in the spatial profile of the Vx© field.
2. The y component of Vx © appears to be lower in the welded specimen than the annealed specimen, contrary to the case of the x component. The more oscillatory feature in the spatial profile of the welded case is the same as the x component.
Fig. 11. Shear force and deformation charge in annealed (a) and welded specimens (b)
3. (V • v) appears substantially different between the annealed and welded cases as well. Unlike the case of (Vxw)j, the welded specimen shows lower values than the annealed case. The spatial distributions are also different; the annealed specimen shows more uniform distribution than the welded specimen.
3.2.3. Contour plots
1. The distribution of (Vxra)x in the annealed specimen appears more symmetric about the vertical and horizontal centers (the lines going through the center of the specimen as seen in Fig. 1 parallel to the x and y axes, respectively) than the welded case. In fact, in the welded specimen, (Vxra)x is distributed in a slant fashion from the top left to the bottom right. As G(Vxro)x represents the x component of the shear elastic force (Eq. (13)), this observation is consistent with the symmetry/asymmetry argument made in conjunction with Fig. 7. The welded specimen exhibits less Poisson-effect-like compression in the x direction.
2. The y component (Vxra)y shows similar distribution between the annealed and welded cases. However, the welded case shows substantially lower values over the entire specimens than the annealed case. As G(Vxw) represents the y component of the shear elastic force, this indicates that the welded specimen possesses substantially lower elasticity in the direction of the tensile axis.
3. The charge (V • v) distributions are completely different between the annealed and welded cases. The annealed case shows uniform distribution whereas the welded case shows a concentrated charge at the bottom right.
It is interesting to discuss the above observations in conjunction with plastic deformation. The uniform distribution of the charge in the annealed specimen indicates that the deformation charges represent the elastic force proportional to the volume expansion represented by Eq. (13). The symmetric charge distribution is consistent with the natural Poisson-effect like behavior. The concentrated charge observed in the welded specimen indicates that the deformation is advanced to plastic deformation near the hinge of the bodily rotation, and the plastic longitudinal force
a0p(V- v)v (Eq. (14)) is more effective near the hinge than the other part of the specimen. This further indicates that the specimen exerts less elastic force near the hinge, which is consistent with the asymmetric nature of the vertical elastic force reflecting the less Poisson's effect exhibited by the welded specimen.
4. Conclusions
The effect of residual stresses induced by welding has been discussed. It has been found that the residual stress alters the material elastic modulus, and consequently, it hinders the specimen from being deformed in accordance with the Poisson's effect when it is subject to an external tensile load. This in turn causes a stress concentration in a certain point of the specimen, causing a bodily-rotation like behavior when a tensile load is applied. The resultant deformation has been analyzed based on physical mesomechanics. The analyses indicate that the asymmetric behavior of the deformation is closely related to local plastic deformation of the specimen. The present study indicates the importance of assessing the effect of welding induced stress not only in the context of residual stresses but also of the change in the material response to an external load.
Acknowledgment
The present study was supported by the National Research Foundation of Korea grants funded by the Korean government MEST, NRF-2013R1A2A2A05005713, NRF-2013M2A2A9043274.
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Appendix
Plastic longitudinal force
Figure A1 illustrates a system of particles that flow in the positive s direction. The particle velocity is a function of xs as vs (xs). Consider that one particular particle undergoes an infinitesimal displacement from xs = s0 to xs = s0 + ds in an infinitesimal duration dt. The change in the velocity of this particle during this time can be expressed as follows
dV 5
dt
_d dt
dxs
ds
dxs
ds dt
dv*
dxs
(A1)
From Eq. (A1), it follows that the change in the momentum of a unit volume per unit time resulting from the same displacement ds can be given as follows
Fig. A1. The particle displacement in a flow from x0 to x0 + ds changing its velocity by dv s
dp s dv s dvs —- = p—- = p—- v s. dt dt dxs s
(A2)
Rewriting the spatial dependence of the velocity in terms of the coordinate variables x, y and z, we obtain the following expression
dvs _ dvs dx + dvs dy + dvs dz _dvs ^ +
dxs dx dxs dy 9 xs d z d xs 9 x
dvs n dvs dvx dvy dvz ^ + —-P +—-y _—^ + —y- + —^_ V- v, (A3) 9y dz dx dy 9 z
where a, P and y are the directional cosines, and vx _ vsa, vy _ vs P, and vz _ vs y are the x, y and z component of the velocity vector. Since the change in momentum is due to an external force, substituting Eq. (A3) into Eq. (A2), we obtain the following expression for the external force
1 long
:p(V' V )V.
(A4)
Here the subscript "long" indicates that the external force is longitudinal in the sense that it changes the momentum of the unit volume in the direction of the local particle velocity v s.
Compare Eq. (A4) with Eq. (14), and notice that the only difference is that vs is replaced with Wd. Assume Wd is in the same direction as vs and greater by a factor of constant g0 as follows
Wd = ao v* (A5)
Now consider Fig. A2 that schematically illustrates the situation where dvs/dxs moves faster than the particles. Figure A2, a presents the situation where the local velocity field has a positive charge (dvs/dxs > 0) at t = t0 between xs = a1 and xs = b1. Here, the velocity at these two points are vs (a1) = vl and vs (b1) = vh where vl < vh and it is assumed that the dependence of vs on xs is linear and the particles behind xs = a1 and those in front of xs = b1 are constant at vl and vh, respectively. Velocity v represents the average velocity between lines a1 and b1.1 Figure A2, b illustrates the situation where after the passage of an infinitesimal duration dt the particles are displaced by ds according to Eq. (A4) with the xs dependence of the velocity field unchanged. The definitions of a2 and b2 are the same as t = t0; the particles behind a2 and those in front of b2 flow respectively with velocities vl and vh. The longitudinal force (A4) causes this displacement and the associated momentum change. Figure A2, c illustrates the situation where the pattern of dvs/dxs moves faster than the particles during the same duration dt where line b2 defines the position beyond which the particle velocity becomes constant at the higher value of vh. Comparison of Figs. A2, b and c indicates that the regions between lines a2 and b2, and b2 and b2, the faster movement of dvs/dxs causes the particles to lose their momentum. If dvs /dxs moved at the
1 This assumption does not cause the argument to lose its generality.
t = t0
0
vh
ах
t = +
V х vh
a2 I
H
I
0
t = t0 +dt
.-J
vh
Vi
t»
Wa
Fig. A2. Momentum loss associated with drift velocity of V • v higher than particle velocity: a1 — xs position, where v = vl at t = t1, b — xs position, where v = vh at t = t1, a2 — xs position, where v = vl at t = t2, b2 — xs position, where v = vh at t = t2, b2 — xs position, where v = vh at t = t2, when V • v moves faster than particles
particle velocity in accordance with Eq. (A4), the particle velocity would be v in the region between a2 and b2, and vh between b2 and b/2 at t = t0 + dt. However, because the pattern dvjdx^ moves faster than the particles, the
particle velocity in these regions are vl and v, respectively. The faster movement of the pattern dv5 /dx^ causes the particle velocity to decrease from v to vl, and vh to v. This means that if the velocity of V • v is higher than the particle velocity there is momentum loss, and the higher the value of a0 defined in Eq. (20) the greater the momentum loss. Here, the longitudinal force acting on this group of particles causes these momentum decreases. Thus, we can argue that if the drift velocity of V • v is higher than the particle velocity, there is additional longitudinal force exerting in the opposite direction to the particle movement. Since this force causes the momentum loss and is proportional to the particle velocity as indicated by Eq. (A5), it can be identified as velocity damping force. This force causes the irreversibility of plastic deformation. Replacing v in Eq. (A4) with Wd defined in Eq. (A5), we can express this force as follows, and call it the plastic longitudinal force G jpl as introduced in Eq. (14)
Gjpi = p(V • v ) Wd = a0P(V • v ) v. (A6)
Elastic longitudinal force
The elastic version of longitudinal force can be derived from Eq. (14). In wave dynamics where the general wave solution has the form of y(t, r) = t - k • r), the temporal and spatial derivatives are connected as dt = =- (Vy) • c, and therefore the right hand side of Eq. (14) can be put as follows
Wd P(V • v ) = Wd pdt (V • Ç) = -pWd V (V • £) • c. (A7)
Here ^ is the angular frequency, k is the wave number in rad (the magnitude of the propagation vector), and c = Q/ k is the phase velocity. Putting Wd = c and using the well-known phase velocity of the elastic compression wave •n/(à + 2G)/p for c, we can rewrite Eq. (A7) in the following form
Wd p(V • v ) = -pWd V (V • £) • c = -pV (V • £)c • c =
= -p V(V • £)•c2 = -p V(V • =
p
= -V(À + 2G )(V^ £). (A8)
The right hand side of Eq. (A8) is the differential of the elastic force proportional to the volume expansion V • £ with the Lamé parameters À and G as the constant of the proportionality. Thus, this force can be interpreted as the longitudinal elastic force introduced in Eq. (13)
Gjei =-(À + 2G )V(V • £). (A9)
Поступила в редакцию
--22.07.2015 г.
Сведения об авторах
Sanichiro Yoshida, Prof., Southeastern Louisiana University, USA, syoshida@selu.edu Tomohiro Sasaki, Dr., Assoc. Prof., Niigata University, Japan, tomodx@eng.niigata-u.ac.jp Masaru Usui, Graduate Student, Niigata University, Japan, masa_baske_usubou07@yahoo.co.jp Ik-Keun Park, Prof., Seoul National University of Science and Technology, Korea, ikpark@seoultech.ac.kr
