Applied Mathematics and Mechanics (English Edition), 2006, 27(3):371–376 c Editorial Committee of Appl. Math. Mech., ISSN 0253-4827
THERMAL STRESS FIELD WHEN CRACK ARREST IN AN AXIAL SYMMETRY METAL DIE USING ELECTROMAGNETIC HEATING ∗ FU Yu-ming (
),
TIAN Zhen-guo (
¨),
ÜÛ
ZHENG Li-juan (
)
(College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, Hebei Province, P. R. China) (Communicated by CHEN Zheng-han)
Abstract: In order to solve the thermal stress field around crack tip in metal die when crack prevention using electromagnetic heating, a metal die with a half-embedded round crack was selected as the study object. The complex function method was used as a basis for the theoretical model of the space crack prevention in metal dies using electromagnetic heating. The crack arrest was accomplished by a pulse current discharge through the inner and outer. The theoretical analysis results show that the temperature around the crack tip rises instantly above the melting point of the metal. Small welded joints are formed at a small sphere near the crack tip inside the metal die by metal melting as a result of the heat concentration effect when the current pulse discharged. The thermal compressive stress field appears around the crack tip at the moment. The research results show that the crack prevention using electromagnetic heating can decrease the stress concentration and forms a compressive stress area around the crack tip, and also prevents the main crack from propagating further, and the goal of crack preventing can be reached. Key words: crack; crack arrest; pulse current discharge; complex function; thermal stress field Chinese Library Classification: O482.6 2000 Mathematics Subject Classification: 74F05; 74F15; 74S30 Digital Object Identifier(DOI): 10.1007/s 10483-006-0313-z
Introduction Electromagnetic heating is an effective method for preventing crack propagation in a metal die[1,2,3] . Currently, the integral transform is used to solve the problem in theory analysis on crack arrest using electromagnetic heating. However, the methods have some limitation on solving the crack problem. Particularly, the shortcoming is presented when it is used to assay the space crack problem[4] . The complex function method is used to study the theoretical problem of crack arrest using electromagnetic heating, its benefit is that the complex functions can be used to determine the crack problem on infinite half-plane by the conformal transforming, thus the complex question is converted to personification question. The boundary condition is difficultly educed in theory analysis on thermal-elastic problem with crack. In complex functions, the current flow through the conductor with crack is simulated to the fluid flowing through the barricade, their potential function is similarity. ∗ Received May 26, 2004; Revised Nov.15, 2005 Project supported by the National Natural Science Foundation of China (No.50275128) and the Natural Science Foundation of Hebei Province of China (No.599255) Corresponding author FU Yu-ming, Associate Professor, Doctor, E-mail:
[email protected]
372
1
FU Yu-ming, TIAN Zhen-guo and ZHENG Li-juan
Foundational of Theory for Analytical Model
1.1 Problem statement Figure 1 shows a half-infinite length metal die, where the outer radius is 2b + R0 , the inner radius is R0 . There exists a circle crack on the top surface of the die, and the radius is b + R0 , the depth is a. The depth of the crack compared to the length of the die is very short, so the effect of the crack on the metal die can be ignored in the analysis. The super density pulse current is discharged through the metal die using inside and outside electrodes. The temperature goes up sharply near the annular crack tip caused by electromagnetic heating and the thermal compressive stresses appear at the tips of the crack. So the formation of the main crack source is accomplished. The problem is equal to finding the thermal stress distribution when the circular heat source exists inside the metal die.
Fig.1
The metal die with halfembedding round crack
Fig.2
Cut of the die
1.2 Conversion of problem Because the metal die with the half-embedding round crack is symmetrical to the axis, the exerting of electrodes is symmetrical, the heat source and the temperature field are too independent of the coordinate. Therefore, the problem shown in Fig.1 may be transformed to the model shown in Fig.2. This new model is used to find the stress field that is determined completely by the imposed heat source. 1.3 Conformal transformation Take B1 (B2 ) as the origin to rebuild a Cartesian system with coordinates xBy, shown in Fig.3(a), and the conformal transformations are shown in Figs.3(b), (c), (d). B1 , B2 are located left and right beside the crack. The width of the crack is disregarded in the study. So the final problem is a thermal stress problem acting on a heat source at point D on the z3 -half-plane[5] . Because the transformation is conformal, the density of the heat source power is unchanged. The conformal transformation equation from Fig.3(a) to Fig.3(d) is π πa . z3 = − sin2 z + sh2 2b 2b
(1)
The inverse function of Eq.(1) is z=
2b arcsin π
πa . z32 − sh2 2b
(2)
Thermal Stress Field in Metal Die
(a) z-plane
(b) z1 -plane
(c) z2 -plane
(d) z3 -plane
Fig.3
2
373
Transform of surfaces
Calculation for Heat Source Power
The required power from the heat source is determined by first knowing how the current flow is distributed through current loading on the half-infinite plane containing the edge crack that is shown in Fig.3(c). The heat source power is then found by using the transformation from Fig.3(c) to Fig.3(d). If the origin is located at B1 (B2 ), Ref.[6] gives the current density J = |J0 | √
z , + z2
a2
(3)
√ where J0 is the current density, a is the depth of the crack, z = x + iy, J = Jx + iJy , i = −1, the sign | · | designates the absolute value inside the sign. According to the electromagnetic heating effect, and the shifting relationship of the coordinate: r = x − R0 − b, z = −y, the heat source power expression in the polar coordinate system becomes |J|2 (R0 + b + r)2 + z 2 Q= = |J0 | × , (4) σ σ [a2 + (R0 + b + r)2 − z 2 ]2 + 4(R0 + b + r)2 z 2 where σ is the electrical conductance.
3
Calculation of Temperature Field
Owing to the short time, the influence of the boundary can be relaxed. So the model is considered to be an infinite system. Taking account of the existence of the crack, the temperature field is (5) T = T0 + T∗ , where T0 is the temperature when there is a point heat source at D without a crack, T∗ is the temperature disturbing caused by the crack.
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FU Yu-ming, TIAN Zhen-guo and ZHENG Li-juan
According to Ref.[7], considering the width of the straight crack is zero, the heat source produced by the current detouring is in the same line with the crack, so there is not temperature disturbing caused by the crack, T∗ (x, y) = 0. For this condition, the temperature is not influenced by the crack. Using the coordinate transformation, the temperature field in polar coordinates becomes: 2 +z 2 |J0 |2 [(R0 + b + r)2 + z 2 ] exp − (R0 +b+r) 4βτ . (6) T (r, z) = √ (2 πβτ )2 σ [a2 + (R0 + b + r)2 − z 2 ]2 + 4(R0 + b + r)2 z 2
4
Calculation of Stress Field
The temperature field comes into being in an axial symmetry metal die when the current discharging. Because the temperature distributing disequilibrium, the thermal stress can be generated in all metal die. The distribution of thermal stress is an importance factor of effecting crack prevention using the electromagnetic heating. The thermal stress problem in axial symmetry metal die is calculated in converted model using conformal transformation, so the radial stress and axial stress can be solved, and the circumferential stress can be solved by using stress-strain relationship. 4.1 Base stress equation indicated by complex function Through transformation, initial question can be simplified as the thermal stress at D point on the underside z3 -half-plane with existing the electricity heart source. Taking into account of the question, there existing the point heat source with power of Q at D point the expression of function Φ(z3 ) can be obtained around D and D points
Φ(z3 ) = A(z3 ) + Φ− ∗ (z3 ),
z3 → D ;
(7)
Φ+ ∗ (z3 ),
D ,
(8)
Φ(z3 ) = B(z3 ) +
(D −D )A0 −A0 z3 −D βQ 2Eα0 4πλ(1+κ) , β = 1+ν .
where A(z3 ) = A0 ln(z3 −D ), B(z3 ) = 3−ν 1+ν , A0
z3 →
ln(z3 −D ), Φ(z3 ) is blending function
in area of y < 0, κ = = 4.2 Stress boundary condition If there exists mechanical load around inner of the metal die justly, and the other boundaries are free, so the expression on z-plane can be obtained: ⎧ σy = τxy = 0, ⎨ y = 0, x = 0, y < a or x = −b, σx = τxy = 0, (9) ⎩ x = b, σx − iτxy = g(t),
where g(τ ) is assigned function on boundary of x = b, t is the point on boundary of x = b . After conformal transformation, the boundary condition of the expression 9 can be conformed, and the expression on z3 -half-plane is y3 = 0,
σy3 − iτx3 y3 = f (τ ),
where τ is the point on boundary of y3 = 0. Based on the basic stress equation of complex function the boundary (10 plified to the following expression: Φ− (τ ) − Φ+ (τ ) = f (τ ),
y3 = 0,
(10)
can be sim(11)
where Φ− (τ ) and Φ+ (τ ) are the limiting of function Φ(z3 ) on upper boundary along y3 < 0 and y3 > 0 direction.
Thermal Stress Field in Metal Die
375
4.3 Calculation of stress field without mechanical load Solving the expression (11), and contented the conditions (7) and (8) furthermore it tends to zero at infinite point, so the expression can be determined using[8] , f (τ )dτ 1 + A(z3 ) + B(z3 ), (12) Φ(z3 ) = − 2πi τ − z3 where A(z3 ) and B(z3 ) are the functions defined by the expressions (7) and (8). Without regarding the mechanical load, and only taking into account the current, so the problem is converted to the thermal stress question with the point heat source at D point on z-plane. g(t) = 0 is established in Eq.(9) on z-plane. The expression can be converted to f (τ ) = 0 on z3 -plane so the expression 12 can be simplified with taking into account the expressions (7) and (8):
Φ(z3 ) = A0 ln
z3 − D z3 − D
+ A0
D − D . z3 − D
(13)
The stress expression can be obtained by generalizing ⎞ ⎛ 2 πa 2 π 2 πa sin2 πz + sh − i sh (a + d) − sh 2b 2b 2b 2b ⎠ σx3 + σy3 =4A0 Re ⎝ln 2 πa 2 π 2 πa 2 πz sin 2b + sh 2b + i sh 2b (a + d) − sh 2b ⎞ ⎛ π (a + d) − sh2 πa 2i sh2 2b 2b ⎠, + 4A0 Re ⎝ 2 πa 2 π 2 πz sin 2b + sh 2b − i sh 2b (a + d) + sh2 πa 2b
σy3 − iτx3 y3
sin2 = A0 ln sin2
+A0 sin2
πz 2b
+ sh2
πa 2b
πz 2b
+ sh2
πa 2b
2i sh2 πz 2b
sin2 −A0 ln sin2
−A0 sin2
+ sh2
+ d) − sh2
πa 2b
π 2b (a
+ d) − sh2
πa 2b
+ b) − sh2 πa 2b 2 π − i sh 2b (a + d) + sh2
πz 2b
+ sh2
πa 2b
πz 2b
+ sh2
πa 2b
+ sh2
π 2b (a
π 2b (a
πa 2b
2i sh2 πz 2b
− i sh2 + i sh2
− i sh2 + i sh2
πa 2b
π 2b (a
+ d) − sh2
πa 2b
π 2b (a
+ d) − sh2
πa 2b
+ d) − sh2 πa 2b π − i sh2 2b (a + d) + sh2
π 2b (a
πa 2b
2 πa 2 πa 2 πz 2 πz + sh − sin + sh + sin 2b 2b 2b 2b
πa 2b
⎛ ×⎝− sin2
πz 2b
+ sh2
πa 2b
1 − i sh2
π 2b (a
+ d) − sh2
πa 2b
(14)
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FU Yu-ming, TIAN Zhen-guo and ZHENG Li-juan
+ sin2
+
πz 2b
sin2
+ sh2
πa 2b
1 + i sh2
2i sin2 πz 2b
+ sh2
πa 2b
π 2b (a
+ d) − sh2
πa 2b
⎞ + d) − sh2 πa 2b π + i sh2 2b (a + d) − sh2
π 2b (a
⎟ ⎟ 2 ⎟. ⎠ πa
(15)
2b
Separating the real part and virtual part from the expressions (15) and (14), the stresses σx3 , σy3 , τx3 y3 can be obtained. Using the shifting relationship of the coordinate, the stresses σr , σz , τrx can be obtained in the polar coordinate system. Because the question is axial symmetry, so the stress σθ can be obtained.
5
Conclusion
By using the complex function methods, the thermal stress field can be obtained when the space crack arresting in metal die with engineering significance using electromagnetic heating, it can be used to extend the practical application, and put the train of the thought in the theoretical analysis on crack arrest using electromagnetic heating from plane problem to space problem.
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