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Stresses in rapidly heated wires
Volume 115, number 5
PHYSICS LETTERS A
14 April 1986
S T R E S S E S IN R A P I D L Y H E A T E D W I R E S
J.G. T E R N A N Materials Research Laboratories, Defence Science and Technology Organisation, Melbourne 3032, Australia Received 19 November 1985; revised manuscript received 3 February 1986; accepted for publication 5 February 1986
Wires that carry high currents may suffer tensile fracture. Theories for this behaviour in terms of longitudinal magnetic or electric forces are refuted. The stress waves developed by rapid thermal expansion are shown to be large enough to break the wire.
Two explanations have been offered for the tensile fracture of wires that carry high currents. Graneau [ 1, 2] argues from Ampere's law [ 1 - 4 ] that the breaks are caused by magnetic forces acting in the direction of the current. However, Ampere's law has been proved equivalent to the magnetic part of the Lorentz law when the divergence of the current is zero [3,4] and since, according to the Lorentz law, the force is normal to the current, Ampere's law does not predict a longitudinal force. Graneau obtains incorrect results by partial instead of complete integration over the current distribution. Pappas and Moyssides [5] have questioned the equivalence proofs because both proofs use Gauss' theorem in the form
fr-la.nds= f r-lV.adv+ f a.vr-ldv, S
v
(1)
v
where r is the position vector, a is a vector with a continuous derivative, V is a region containing the origin, and S is the bounding surface of V. They object that the integrands in eq. (1) are inf'mite at the origin. Now, strictly, Gauss' theorem would only directly apply if (1/r)a were continuously differentiable over V, and this is not the case. However, if the origin is surrounded by a small sphere of radius e and surface Z, then the theorem applies over the r e # o n between S and ~. The surface integral over Y. approaches zero as e, and the volume integrals converge as e -~ 0, the first as e 2 and the second as e. Eq. (1) is therefore valid 230
and the objection of Pappas and Moyssides is groundless. There is no longitudinal magnetic force and Graneau's explanation for the wire fracture is refuted. Aspden has proposed an alternative theory [6] in which he asserts there is an excess longitudinal electric field E = (la/8n)dI/dtwhich acts on the bound charge of the conductor. Here, #/8n is the internal self-inductance per unit length of the wire [7]. The longitudinal electric force per unit volume is then qE, where q is the charge density. The accepted view is that the current obeys Ohm's law g = oE and there is no excess field within the conductor. Apart from physical objections, the values predicted by Aspden's theory are too large at moderate frequencies and far too large at high frequencies. For example, if his argument is followed at f r e q u e n c y f w h e r e the current is confined to the skin depth/5 = 1/(nlaof)1/2 for a wire of radius a, then the reactive field is E = I/2rroa6.The charge per unit length in the area of the current is 2nagq,which gives a force per unit length oflq/o. For aluminium, q = 3.4 X 1010 C m -3 [6] and o = 3.5 X 107 S m - 1 . I f / = 1 A, the force is 970 N m -1 which is sufficient to break a 1 mm wire of 1 m length. Some other reason must be sought for the wire fracture. When the wire is heated, electrically or otherwise, a longitudinal standing wave of stress will be generated by the thermal expansion if the ends of the wire are free, as they are in Graneau's experiments [2]. This stress wave may lead to tensile fracture if the heating rate is high enough. 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 115, number 5
PHYSICS LETTERS A
Consider a thin unrestrained wire of length l directed along the x-axis, [xI <~l/2. Let k be the (constant) heating rate and a the coefficient of linear expansion. Then, neglecting damping and lateral effects, the stress X, particle displacement u and velocity v satisfy
~v/~x = a k + ( 1 / E ) ~ X / a t ,
v = ~u/~t,
OX/Ox = p a y / a t ,
(2) (3)
where E is Young's modulus and p is the density. These equations together form the wave equation, with wave velocity c = (E/p) 1/2 , and must be solved with the initial conditions v(x, O) = O, X(x, O) = O. The boundary conditions are X ( - U 2 , t) = XQ/2, t) = 0 if the ends are free. The solution is best expressed as a pair of travelling triangular waves:
X = (akEl/4c) × [g(n(x - cO~l) - g(n(x + cO~l)] ,
(4)
where g is a unit triangular function with period 2n and the properties g ( n / 2 ) = 1, g(O) = - g ( - O ) = -g(O + n). Eq. (4) is a standing wave. The wire vibrates with a frequency f = c/2l, and initially goes into compression as it expands. The stress at the centre, x = 0, is a triangular function of time with maximum modulus Xm(0 ) = akEl/2e. The stress at other points is a trapezoidal time function with maximum X m ( x ) = Xm(0)(1 - 21x[/l). The strain rate s = Ov/Ox, and eqs. (2), (4) give the maximum rate s m = 2ak. The heating rate for a direct current I is k = 12/ ppA2o, where A is the cross-sectional area. For aluminium the specific heat u = 865 J kg -1 K -1 , the electrical conductivity o = 3.5 × 107 S m - 1 and the density p = 2700 kg m - 3 . Based on Graneau's experiments, I is taken as 5000 A, and the wire radius a = 0.6 mm. These values yield k = 2.4 × 105 K s -1 . The coefficient of expansion of aluminium is a = 2.4 X 10 - 5 K -1 which gives s m = 11.5 s -1 . This is not excessive for ahiminium and static mechanical properties may be assumed to apply. Thus, with E = 70 GPa, c = 5.1 k m s -1 and i f / = 1 re, then f = 2.55 kHz and Xm(0 ) = 39.5 MPa. The ultimate tensile strength U of the aluminium used by Graneau is not known at room temperature but it may be in the range 1 0 0 - 2 0 0 MPa. The strength decreases with increasing temperature and Judge [8] gives U = 180
14 April 1986
MPa at room temperature falling roughly linearly to 37 MPa at T-- 670 K. With the given value of k, a wire of this material is expected to break in tension on the fifth cycle when t ~- 1.8 ms, T "~ 700 K. The first break should occur within the central 0.5 m of the wire. A strength of U = 23 MPa at T = 570 K is given by Graneau for his material and the first break would then occur on the third or fourth cycle with T in the range 5 5 0 - 6 5 0 K. This analysis will apply if damping attenuation of the first few vibrations is small. Damping introduced by the wire supports will be negligible if the wire is lightly suspended. Internal friction gives a damping factor ~ < 0.1 for aluminium at T < 700 K [9] ; the factor could possibly be less than 0.01. If the attenuation per cycle is taken as e x p ( - n ~ ) , ~ < 0.01 can be neglected at the assumed heating rate, but if ~ = 0.1, k must be increased by a factor of about 2.5 to obtain fracture which would then occur on the second or third cycle. Graneau used the sinusoidal current of an LC-circuit to heat the wire. This circuit gives reasonably long discharge times because the inductance is able to strike an arc between the separating pieces of wire and so maintain the current. However, with a sinusoidal current, the term ak in eq. (2) must be replaced by ak(1 - cos Coht), where the heating frequencYfh is twice the driving frequency. The rms value o f / i s used in the expression for the mean heating rate k. The stress now becomes
cos(~hX/C) X = Xm(O)[w@/(1
8 n2
~
COS(COhl/2c))Sin COht
a n cos[(2n + 1)nx/l]
0
× sin [(2n + 1)trot~l]]
,
(5)
where Xm(0 ) = akEl/2e and
an =
(-1)" (2n + 1)2{1 - [(2n + 1)Trc/whl] 2}
(6)
The series in eq. (5) is a free vibration which would eventually decay under damping if there were no fracture. I f f h ] f > 0.8, the driven vibration of frequency f h has an amplitude greater than (4f[zrfh)Xm(O) at some x. A stress of at least Xm(0 ) would therefore be 231
Volume 115, number 5
PHYSICS LETTERS A
expected at x = 0, when 0.8 < f h / f < 1.3, even if the wire were heavily damped. When fh = 3.66 kHz and l = 1 m [2], the stress at x = 0 is approximately X(0, t ) = Xm(0 ) X [1.14 sin c o h t - 1.57 sin(rrct/l)] ,
(7)
which has a maximum value of about 2.5 Xm(0), so the sinusoidal current produces a greater tension than a direct current of the same mean heating value. I f f h ~ f , eq. (5) gives, approximately, X = X m (O)(2/rr)
14 April 1986
If, for a constant heating rate, the wire happened to break at x = 0 when X(0, t ) = Xm(0), the near-faces separate with an initial relative velocity o f akl = 5.8 m s -1 when l = 1 m. If there were no further breaks, they would move apart to a distance ctkl2/2c = 0.56 mm before coming together again. The two pieces vibrate with a frequency 2f. The initial stress X(x, O) = Xm(0)(1 - 2[~cl/l)is maintained for a time ~ = [xl/c where it undergoes a jump o f - X m ( 0 ) before returning to its initial value at time (l/c - "c). In summary, the search for longitudinal applied forces to explain the tensile fracture of wires appears to be unnecessary. These forces can be generated by thermal expansion.
× {[1 - (2x/l) sin@x/l) - (511r) cos(Trx/l)] sin(Trot~l)
+ (2ct/t) cosOrx/l) cosOrct/l)),
(8)
where all terms of the series except the first have been neglected, The stress at x = 0 has an amplitude of about (8ft/Tr)Xm(O) which equals 2.6 Xm(0 ) after only one cycle. The wire is evidently easier to break if it is driven at a resonant frequency. The motion of the wire after the first break has not been explored in any detail, but some interior points o f the two pieces appear to develop stresses comparable with the initial breaking stress. This will lead to many further breaks if the current is maintained by arcs and the temperature continues to rise.
232
References [1] [2] [3] [4] [5]
P. Graneau, Phys. Lett. A 97 (1983) 253. P. Graneau, IEEE Trans. Magn. MAG-20 (1984) 444. D.C. Jolly, Phys. Lett. A 107 (1985) 231. J.G. Ternan, J. Appl. Phys. 57 (1985) 1743. P.T. Pappas and P.G. Moyssides, Phys. Lett. A 111 (1985) 193. [6] H. Aspden, Phys. Lett A 107 (1985) 238. [7] W.R. Smythe, Static and dynamic electricity, 3rd Ed. (McGraw-Hill, New York, 1968)p. 338. [8] A.W. Judge, Engineering materials, 2nd Ed., Vol. 2 (Pitman, London, 1943) p. 30. [9] W.P. Mason, Physical acoustics and the properties of solids (Van Nostrand, New York, 1958) ch. 9.
PHYSICS LETTERS A
14 April 1986
S T R E S S E S IN R A P I D L Y H E A T E D W I R E S
J.G. T E R N A N Materials Research Laboratories, Defence Science and Technology Organisation, Melbourne 3032, Australia Received 19 November 1985; revised manuscript received 3 February 1986; accepted for publication 5 February 1986
Wires that carry high currents may suffer tensile fracture. Theories for this behaviour in terms of longitudinal magnetic or electric forces are refuted. The stress waves developed by rapid thermal expansion are shown to be large enough to break the wire.
Two explanations have been offered for the tensile fracture of wires that carry high currents. Graneau [ 1, 2] argues from Ampere's law [ 1 - 4 ] that the breaks are caused by magnetic forces acting in the direction of the current. However, Ampere's law has been proved equivalent to the magnetic part of the Lorentz law when the divergence of the current is zero [3,4] and since, according to the Lorentz law, the force is normal to the current, Ampere's law does not predict a longitudinal force. Graneau obtains incorrect results by partial instead of complete integration over the current distribution. Pappas and Moyssides [5] have questioned the equivalence proofs because both proofs use Gauss' theorem in the form
fr-la.nds= f r-lV.adv+ f a.vr-ldv, S
v
(1)
v
where r is the position vector, a is a vector with a continuous derivative, V is a region containing the origin, and S is the bounding surface of V. They object that the integrands in eq. (1) are inf'mite at the origin. Now, strictly, Gauss' theorem would only directly apply if (1/r)a were continuously differentiable over V, and this is not the case. However, if the origin is surrounded by a small sphere of radius e and surface Z, then the theorem applies over the r e # o n between S and ~. The surface integral over Y. approaches zero as e, and the volume integrals converge as e -~ 0, the first as e 2 and the second as e. Eq. (1) is therefore valid 230
and the objection of Pappas and Moyssides is groundless. There is no longitudinal magnetic force and Graneau's explanation for the wire fracture is refuted. Aspden has proposed an alternative theory [6] in which he asserts there is an excess longitudinal electric field E = (la/8n)dI/dtwhich acts on the bound charge of the conductor. Here, #/8n is the internal self-inductance per unit length of the wire [7]. The longitudinal electric force per unit volume is then qE, where q is the charge density. The accepted view is that the current obeys Ohm's law g = oE and there is no excess field within the conductor. Apart from physical objections, the values predicted by Aspden's theory are too large at moderate frequencies and far too large at high frequencies. For example, if his argument is followed at f r e q u e n c y f w h e r e the current is confined to the skin depth/5 = 1/(nlaof)1/2 for a wire of radius a, then the reactive field is E = I/2rroa6.The charge per unit length in the area of the current is 2nagq,which gives a force per unit length oflq/o. For aluminium, q = 3.4 X 1010 C m -3 [6] and o = 3.5 X 107 S m - 1 . I f / = 1 A, the force is 970 N m -1 which is sufficient to break a 1 mm wire of 1 m length. Some other reason must be sought for the wire fracture. When the wire is heated, electrically or otherwise, a longitudinal standing wave of stress will be generated by the thermal expansion if the ends of the wire are free, as they are in Graneau's experiments [2]. This stress wave may lead to tensile fracture if the heating rate is high enough. 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 115, number 5
PHYSICS LETTERS A
Consider a thin unrestrained wire of length l directed along the x-axis, [xI <~l/2. Let k be the (constant) heating rate and a the coefficient of linear expansion. Then, neglecting damping and lateral effects, the stress X, particle displacement u and velocity v satisfy
~v/~x = a k + ( 1 / E ) ~ X / a t ,
v = ~u/~t,
OX/Ox = p a y / a t ,
(2) (3)
where E is Young's modulus and p is the density. These equations together form the wave equation, with wave velocity c = (E/p) 1/2 , and must be solved with the initial conditions v(x, O) = O, X(x, O) = O. The boundary conditions are X ( - U 2 , t) = XQ/2, t) = 0 if the ends are free. The solution is best expressed as a pair of travelling triangular waves:
X = (akEl/4c) × [g(n(x - cO~l) - g(n(x + cO~l)] ,
(4)
where g is a unit triangular function with period 2n and the properties g ( n / 2 ) = 1, g(O) = - g ( - O ) = -g(O + n). Eq. (4) is a standing wave. The wire vibrates with a frequency f = c/2l, and initially goes into compression as it expands. The stress at the centre, x = 0, is a triangular function of time with maximum modulus Xm(0 ) = akEl/2e. The stress at other points is a trapezoidal time function with maximum X m ( x ) = Xm(0)(1 - 21x[/l). The strain rate s = Ov/Ox, and eqs. (2), (4) give the maximum rate s m = 2ak. The heating rate for a direct current I is k = 12/ ppA2o, where A is the cross-sectional area. For aluminium the specific heat u = 865 J kg -1 K -1 , the electrical conductivity o = 3.5 × 107 S m - 1 and the density p = 2700 kg m - 3 . Based on Graneau's experiments, I is taken as 5000 A, and the wire radius a = 0.6 mm. These values yield k = 2.4 × 105 K s -1 . The coefficient of expansion of aluminium is a = 2.4 X 10 - 5 K -1 which gives s m = 11.5 s -1 . This is not excessive for ahiminium and static mechanical properties may be assumed to apply. Thus, with E = 70 GPa, c = 5.1 k m s -1 and i f / = 1 re, then f = 2.55 kHz and Xm(0 ) = 39.5 MPa. The ultimate tensile strength U of the aluminium used by Graneau is not known at room temperature but it may be in the range 1 0 0 - 2 0 0 MPa. The strength decreases with increasing temperature and Judge [8] gives U = 180
14 April 1986
MPa at room temperature falling roughly linearly to 37 MPa at T-- 670 K. With the given value of k, a wire of this material is expected to break in tension on the fifth cycle when t ~- 1.8 ms, T "~ 700 K. The first break should occur within the central 0.5 m of the wire. A strength of U = 23 MPa at T = 570 K is given by Graneau for his material and the first break would then occur on the third or fourth cycle with T in the range 5 5 0 - 6 5 0 K. This analysis will apply if damping attenuation of the first few vibrations is small. Damping introduced by the wire supports will be negligible if the wire is lightly suspended. Internal friction gives a damping factor ~ < 0.1 for aluminium at T < 700 K [9] ; the factor could possibly be less than 0.01. If the attenuation per cycle is taken as e x p ( - n ~ ) , ~ < 0.01 can be neglected at the assumed heating rate, but if ~ = 0.1, k must be increased by a factor of about 2.5 to obtain fracture which would then occur on the second or third cycle. Graneau used the sinusoidal current of an LC-circuit to heat the wire. This circuit gives reasonably long discharge times because the inductance is able to strike an arc between the separating pieces of wire and so maintain the current. However, with a sinusoidal current, the term ak in eq. (2) must be replaced by ak(1 - cos Coht), where the heating frequencYfh is twice the driving frequency. The rms value o f / i s used in the expression for the mean heating rate k. The stress now becomes
cos(~hX/C) X = Xm(O)[w@/(1
8 n2
~
COS(COhl/2c))Sin COht
a n cos[(2n + 1)nx/l]
0
× sin [(2n + 1)trot~l]]
,
(5)
where Xm(0 ) = akEl/2e and
an =
(-1)" (2n + 1)2{1 - [(2n + 1)Trc/whl] 2}
(6)
The series in eq. (5) is a free vibration which would eventually decay under damping if there were no fracture. I f f h ] f > 0.8, the driven vibration of frequency f h has an amplitude greater than (4f[zrfh)Xm(O) at some x. A stress of at least Xm(0 ) would therefore be 231
Volume 115, number 5
PHYSICS LETTERS A
expected at x = 0, when 0.8 < f h / f < 1.3, even if the wire were heavily damped. When fh = 3.66 kHz and l = 1 m [2], the stress at x = 0 is approximately X(0, t ) = Xm(0 ) X [1.14 sin c o h t - 1.57 sin(rrct/l)] ,
(7)
which has a maximum value of about 2.5 Xm(0), so the sinusoidal current produces a greater tension than a direct current of the same mean heating value. I f f h ~ f , eq. (5) gives, approximately, X = X m (O)(2/rr)
14 April 1986
If, for a constant heating rate, the wire happened to break at x = 0 when X(0, t ) = Xm(0), the near-faces separate with an initial relative velocity o f akl = 5.8 m s -1 when l = 1 m. If there were no further breaks, they would move apart to a distance ctkl2/2c = 0.56 mm before coming together again. The two pieces vibrate with a frequency 2f. The initial stress X(x, O) = Xm(0)(1 - 2[~cl/l)is maintained for a time ~ = [xl/c where it undergoes a jump o f - X m ( 0 ) before returning to its initial value at time (l/c - "c). In summary, the search for longitudinal applied forces to explain the tensile fracture of wires appears to be unnecessary. These forces can be generated by thermal expansion.
× {[1 - (2x/l) sin@x/l) - (511r) cos(Trx/l)] sin(Trot~l)
+ (2ct/t) cosOrx/l) cosOrct/l)),
(8)
where all terms of the series except the first have been neglected, The stress at x = 0 has an amplitude of about (8ft/Tr)Xm(O) which equals 2.6 Xm(0 ) after only one cycle. The wire is evidently easier to break if it is driven at a resonant frequency. The motion of the wire after the first break has not been explored in any detail, but some interior points o f the two pieces appear to develop stresses comparable with the initial breaking stress. This will lead to many further breaks if the current is maintained by arcs and the temperature continues to rise.
232
References [1] [2] [3] [4] [5]
P. Graneau, Phys. Lett. A 97 (1983) 253. P. Graneau, IEEE Trans. Magn. MAG-20 (1984) 444. D.C. Jolly, Phys. Lett. A 107 (1985) 231. J.G. Ternan, J. Appl. Phys. 57 (1985) 1743. P.T. Pappas and P.G. Moyssides, Phys. Lett. A 111 (1985) 193. [6] H. Aspden, Phys. Lett A 107 (1985) 238. [7] W.R. Smythe, Static and dynamic electricity, 3rd Ed. (McGraw-Hill, New York, 1968)p. 338. [8] A.W. Judge, Engineering materials, 2nd Ed., Vol. 2 (Pitman, London, 1943) p. 30. [9] W.P. Mason, Physical acoustics and the properties of solids (Van Nostrand, New York, 1958) ch. 9.