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Forces on bubbles and voids in a stress gradient
Journal of Nuclear Materials 58 (1975) 55-58 0 North-Holland Publishing Company
FORCES ON BUBBLES AND VOIDS IN A STRESS GRADIENT M.V. SPEIGHT Central Electricity
Generating Board, Berkeley Nuclear Laboratories, Berkeley,
Gloucestershire,
UK
Received 7 April 1975
Forces on voids and gas bubbles in a solid under a stress gradient are evaluated. The bubbles are allowed to change volume to reach equilibrium with the local stress, and it is shown that the force originates solely with the discrepancy between surface energy and surface tension. The force on non-equilibrium voids is derived on the assumption that the temperature is sufficiently low to permit them to migrate at constant volume. Les forces agissant sur les vides et sur les bulles de gaz dans un solide sous l’effet d’un gradient de contraintes sont dvaludes Les bulles sont amen&es a changer de volume pour atteindre 1’6quilibre avec la contrainte locale et l’on montre que la force prend uniquement son origine dans l’tcart entre l’energie superficielle et la tension superficielle. La force agissant sur les vides hors d’equilibre est deduite en supposant que la temperature est suffisamment basse pour permettre ?I ces vides de migrer a volume constant. Die Krlfte, die an Poren und Gasblasen in einem FestkBrper unter einem Spannungsgradienten angreifen, werden berechnet. Die Blasen kiinnen ihr Volumen zur Einstellung des Gleichgewichts mit lokalen Spannungen Bndern. Es wird gezeigt, dass die Kraft allein von der Diskrepanz zwischen der Oberfllchenenergie und der Oberflachenspannung ausgeht. Die Kraft an nicht im Gleichgewicht befindlichen Poren wird unter der Annahme abgeleitet, dass die Temperatur fiir einen Transport bei konstantem Volumen hinreiched niedrig ist.
1. Introduction Many tissile and non-tissile materials develop bubbles or voids under thermal and fast neutron irradiation. Bubbles are usually defined as cavities containing insoluble gas, often produced during the irradiation by transmutation, at a pressure which exactly balances the total collapse force due to surface energy and applied mechanical compression. That is, a bubble tends neither to shrink nor grow and, except for the second order effect due to the inequality of surface tension and surface free energy [ 11, does not disturb the stresses existing in the surrounding solid in the bubble’s absence. On the contrary, a void has an internal gas pressure below that necessary for equilibrium, and is prevented from shrinking only by the excessive vacancy concentration maintained in the solid by the net transfer of atoms to dislocations induced by the irradiation damage process (see ref. [2] for a review of this subject). In general, voids form during low-temperature (less than half the absolute melting temperature) ir-
radiation when the thermal equilibrium concentration of vacancies is significantly enhanced. At higher temperatures, in the absence of assistance from irradiation damage, cavities must be stabilised wholly by internal gas pressure. Both bubbles and voids are produced in reactor corn ponents, many of which are differentially stressed during operation. For example, at the high heat fluxes in the fast reactor, a temperature gradient exists across the wall of fuel-element cladding. On first reaching power, the thermal stresses in the cladding are tensile towards the outside surface and compressive in the inner regions. Although these stresses subsequently relax by irradiation-creep [3], further internal stresses are generated by the tendency of the clad to swell, by void growth, by different amounts corresponding to the temperature variation through the wall. Thus a balance between the continuous development of stress by differential swelling and relaxation by creep leads to steady state non-uniform stress levels in the clad. The possibility arises that the stress gradients impose forces
56
M. V. Speight /Forces on bubbles and voids
on the voids and cause them to migrate, providing a suitable mechanism exists for void mobility. Previous analyses [4,5,6] of th.: effects of stress gradients are largely inaccurate and incomplete. Here a comprehensive approach is developed which emphasises the different response of bubbles, filled with an equilibrium pressure of gas, and gas-deficient voids to a particular stress gradient. The effects are demonstrated, making the compatible assumptions of spherical cavities and a purely hydrostatic stress system. Similar conclusions, but with greatly increased mathematical complexity, could be obtained with deviatoric stress components and a corresponding non-spherical equilibrium cavity shape.
chanism the stress in the solid is restored to its hydrostatic state existing before the introduction of the bubble. The combined work done on the solid by the applied stress and the internal gas pressure is (utp)dV, whilst there is a concurrent increase of sdA in the surface energy. The presence of the bubble in the second region of crystal reduces the strain energy there by an amount (u + du)2 V/2K, where K is the bulk modulus of elasticity. The total free energy change dG2 associated with this second region of crystal is then dG, = -(u t p)dV + sdA - (a + du)2 V/2K.
Consider an equilibrium spherical gas bubble of radius r, volume V, surface area A, containing gas at pressure p and placed in a region of crystal under hydrostatic tension u. The condition for bubble equilibrium, representing a state of minimum total free energy is [ I]
In the first region of crystal, externally stressed at u, the bubble volume was replaced by material stressed at (a + do). In contrast to the equilibrium bubble this new material causes a distortion in the stress field which can be completely relieved by a plastic deformation process permitting a total volume relaxation Vdu/K, so that the stresses throughout the volume of the new material match those in the surrounding solid. The work done by the solid against the applied stress is then Vudu/K. Allowing for the increase in strain energy of u2 V/2K caused by the removal of the bubble, the free energy change dG, associated with the first region of crystal is
p+u=2s/r
dG, = u2 V/2K + Vudu/K
2. Derivation of forces 2.1. Equilibrium bubble in ideal solid
(1)
where s is the surface energy. However, the radial stress which the bubble surface exerts on the bulk solid is (27/r - p) which deviates from the externally applied tension, u, when the surface tension, y, differs from the surface energy. Whilst y is the work required to elastically stretch a surface, s is the energy required for its creation [7]. Where they differ the presence of even equilibrium gas bubbles perturbs the stress field in the material. In the following argument it is assumed initially that s and y are equivalent so that the stress at every point in the solid is purely hydrostatic and equal to the applied tension, u. Suppose the bubble is removed to a region of crystal where the hydrostatic stress is (u + da) and the initial bubble site filled with the displaced material stressed at (a t do). The bubble is not in mechanical equilibrium in its new surroundings. The perturbation it induces in the elastic stresses in the solid can be fully relaxed by a mass transfer process which allows the bubble to attain equilibrium through expanding by an incremental volume d V. By this me-
The total free energy change on transferring bubble to a region of higher stress is then
the
dG = dG, + dG, =-(u+p)dV+sdA
+ [u2+2udu-(u+du)2]V/2K
For a spherical bubble, dA/dV = 2/r, so that from eq.(l) the first two terms in the above expression sum identically to zero. From the final term the derivative dG/du is seen to vanish as do tends to zero. Thus, in a solid where surface energy and surface tension are identical, a stress gradient exerts no force on an equilibrium gas bubble. This conclusion is not at all surprising; the bubble induces no distortion in the stress field so that the solid does not detect its presence. In the case where surface energy and surface tension are not equivalent this is no longer true and a finite fBrce does act upon the bubble. This situation will now be considered. 2.2. Inequality of surface tension and surface energy With the bubble in a region stressed at u a plastic
51
M. V. Speight / Froces on bubbles and voids
flow process allows its volume to change to reach an equilibrium value V. At equilibrium the tensile stress, t(= 2-y/r - p), on the solid at the bubble surface does not, in general, match the applied stress, and elastic relaxation occurs resulting in a further increase in bubble volume Ak’,. During relaxation the external stress does an amount of work oAv, on the solid and part of this appears as an increase (u - t)Av,/2 in elastic stored energy of the bulk solid. Additionally an amount of energy (27/r) A P’, is stored as elastic strain in the surface layer of material bounding the bubble. The total change in free energy AC, associated with attaining elastic equilibrium is then AC, = -oAv,
2.3. Voids ofconstant volume
+ (a - t)AP’,/2 + (2y/r)AV,
= [(27/r) - (u + t)/2]AV,
.
(2)
The elastic relaxation Av, is proportional to the differential stress (u - t), and in an elastically isotropic medium its magnitude is readily shown to be [8] AL’, = 3 f’(u - t)/4/~
(3)
where p is the shear modulus. For a gas bubble in equilibrium according to eq. (1) u - t = (2/r) (s - y)
(4)
so that, in this case, elastic relaxation is directly determined by the difference between surface energy and surface tension. Substituting for A& from eq. (3) and for t from eq. (4) the expression for AC, becomes AC, = -3 V(s - r)[u -(s + -r)/r]/2v = 3 Us - 7) ]P - (s -
7)lrlDw.
Voids, containing a negligible pressure of enclosed gas, cannot exist in a material in a state of thermal and mechanical equilibrium. They persist at low temperatures, particularly under irradiation, because plastic flow processes relying on volume diffusion are too slow to effect significant shrinkage in a realistic time. Although the temperature is too low to permit volume changes, the more rapid rate of surface diffusion may enable voids to move in an imposed stress gradient. Like the equilibrium gas bubble the force on a void originates solely with the degree of elastic relaxation. With no internal gas pressure and applied tension ut the free energy change associated with relaxation is, from eq. (2)
ql AV$
AC, = [(2-r/r) and from eq. (3)
(5)
If the applied stress is compressive it is easily shown that eq. (5) still describes the free energy change occurring during the elastic relaxation of an equilibrium gas bubble. When (s - y) is non-zero the derivative -(dAG,/dx) defines the total force, F, acting on the bubble. Assuming that the enclosed gas obeys the perfect gas law (i.e. the product pV is constant) the force on the bubble in a given stress gradient du/dx becomes F = ]27@ -
as to move the bubble to a region of greater tension or lower compression. Conversely, when Y > s and p > (y - s)/r the tendency is for bubble movement into areas of the solid under lower tension or greater compression. If a bubble is, at all times, to remain in equilibrium with the applied stress, dr/du must be finite and, from eq. (l), this requires that p > 2s/3r. Hence, except when y % s, the restriction that p must exceed (y - s)/r is unlikely to be violated for equilibrium bubbles. This being the case, in a given stress gradient, the algebraic sign of the quantity (s - y) alone determines the direction of the force.
YYPIb + 6 - r)l (WW (d&.x) .
Since the equilibrium bubble size increases as the applied hydrostatic stress becomes more tensile the derivative dr/du obtained from eq. (1) is always positive. Hence, where s > y the force is always such
A& = 3V]ot -
CW)lI+ .
In a gradient of hydrostatic tension, do,/&, the force on a void which migrates without change in volume is then F = (3 V/41-1)]ot -
(Wr)l do,/& .
The direction of the force depends exclusively on the differential stress ut - (2-y/r). When this is positive the void moves to a region of increased applied tension, and in the opposite direction when the stress due to surface tension exceeds the applied tensile stress. By completely analogous arguments the force on a void in a material under compression, uc, is readily shown to be F = (3 V/~P) [oe + GW)l
do&
58
M, V. Speight /Forces
so that under all circumstances the void tends to move to a region of increased compression.
on bubbles and voids
which at temperatures sufficiently high for volume diffusion would promote unlimited void growth, the direction of motion is towards increased tension.
3. Conclusions Acknowledgement A stress gradient in a solid where surface tension y and surface energy s are equivalent exerts no force on an equilibrium gas bubble. Where these surface properties diverge the force on a gas bubble, which migrates and changes volume to establish equilibrium with the local stress, is proportional to the difference between surface energy and surface tension. In a given stress gradient, unless y + s, the direction of the force is determined exclusively by the algebraic sign of this difference. This applies whether the applied stress is tensile or compressive. Non-equilibrium voids, which because they exist at low temperatures are assumed to migrate without change in volume, generally gravitate towards regions of greater compression or lower tension. The exception is when the applied stress is tensile and exceeds the surface tension stress (27/r). Under these conditions,
This paper is published by permission of the Central Electricity Generating Board.
References
[l] A.B. Lidiard and R.S. Nelson, Phil. Mag. i7 (1968) 425. [2] DIR. Norris, Radiation Effects 14 (1972) 1. [3] E.P. Hicks and If. Hughes, Proc. B.N.E.S. Conf. on lrradiation Embrittlcment and Creep in Fuel Cladding and Core Components (1972) 13. [4] B.L. Eyre and R. Bullough, J. Nucl. Mater 26 (1968) 249. (51 F.A. Nichols, J. Nucl. Mater 30 (1969) 143. [6] D.G. Martin, J. Nucl. Mater 33 (1969) 23. [7] P.R. Couchman and W.A. Jesser, Surface Science 34 (1973) [8] F.C. Frank,
J. Nucl. Mater 48 (1973)
199
FORCES ON BUBBLES AND VOIDS IN A STRESS GRADIENT M.V. SPEIGHT Central Electricity
Generating Board, Berkeley Nuclear Laboratories, Berkeley,
Gloucestershire,
UK
Received 7 April 1975
Forces on voids and gas bubbles in a solid under a stress gradient are evaluated. The bubbles are allowed to change volume to reach equilibrium with the local stress, and it is shown that the force originates solely with the discrepancy between surface energy and surface tension. The force on non-equilibrium voids is derived on the assumption that the temperature is sufficiently low to permit them to migrate at constant volume. Les forces agissant sur les vides et sur les bulles de gaz dans un solide sous l’effet d’un gradient de contraintes sont dvaludes Les bulles sont amen&es a changer de volume pour atteindre 1’6quilibre avec la contrainte locale et l’on montre que la force prend uniquement son origine dans l’tcart entre l’energie superficielle et la tension superficielle. La force agissant sur les vides hors d’equilibre est deduite en supposant que la temperature est suffisamment basse pour permettre ?I ces vides de migrer a volume constant. Die Krlfte, die an Poren und Gasblasen in einem FestkBrper unter einem Spannungsgradienten angreifen, werden berechnet. Die Blasen kiinnen ihr Volumen zur Einstellung des Gleichgewichts mit lokalen Spannungen Bndern. Es wird gezeigt, dass die Kraft allein von der Diskrepanz zwischen der Oberfllchenenergie und der Oberflachenspannung ausgeht. Die Kraft an nicht im Gleichgewicht befindlichen Poren wird unter der Annahme abgeleitet, dass die Temperatur fiir einen Transport bei konstantem Volumen hinreiched niedrig ist.
1. Introduction Many tissile and non-tissile materials develop bubbles or voids under thermal and fast neutron irradiation. Bubbles are usually defined as cavities containing insoluble gas, often produced during the irradiation by transmutation, at a pressure which exactly balances the total collapse force due to surface energy and applied mechanical compression. That is, a bubble tends neither to shrink nor grow and, except for the second order effect due to the inequality of surface tension and surface free energy [ 11, does not disturb the stresses existing in the surrounding solid in the bubble’s absence. On the contrary, a void has an internal gas pressure below that necessary for equilibrium, and is prevented from shrinking only by the excessive vacancy concentration maintained in the solid by the net transfer of atoms to dislocations induced by the irradiation damage process (see ref. [2] for a review of this subject). In general, voids form during low-temperature (less than half the absolute melting temperature) ir-
radiation when the thermal equilibrium concentration of vacancies is significantly enhanced. At higher temperatures, in the absence of assistance from irradiation damage, cavities must be stabilised wholly by internal gas pressure. Both bubbles and voids are produced in reactor corn ponents, many of which are differentially stressed during operation. For example, at the high heat fluxes in the fast reactor, a temperature gradient exists across the wall of fuel-element cladding. On first reaching power, the thermal stresses in the cladding are tensile towards the outside surface and compressive in the inner regions. Although these stresses subsequently relax by irradiation-creep [3], further internal stresses are generated by the tendency of the clad to swell, by void growth, by different amounts corresponding to the temperature variation through the wall. Thus a balance between the continuous development of stress by differential swelling and relaxation by creep leads to steady state non-uniform stress levels in the clad. The possibility arises that the stress gradients impose forces
56
M. V. Speight /Forces on bubbles and voids
on the voids and cause them to migrate, providing a suitable mechanism exists for void mobility. Previous analyses [4,5,6] of th.: effects of stress gradients are largely inaccurate and incomplete. Here a comprehensive approach is developed which emphasises the different response of bubbles, filled with an equilibrium pressure of gas, and gas-deficient voids to a particular stress gradient. The effects are demonstrated, making the compatible assumptions of spherical cavities and a purely hydrostatic stress system. Similar conclusions, but with greatly increased mathematical complexity, could be obtained with deviatoric stress components and a corresponding non-spherical equilibrium cavity shape.
chanism the stress in the solid is restored to its hydrostatic state existing before the introduction of the bubble. The combined work done on the solid by the applied stress and the internal gas pressure is (utp)dV, whilst there is a concurrent increase of sdA in the surface energy. The presence of the bubble in the second region of crystal reduces the strain energy there by an amount (u + du)2 V/2K, where K is the bulk modulus of elasticity. The total free energy change dG2 associated with this second region of crystal is then dG, = -(u t p)dV + sdA - (a + du)2 V/2K.
Consider an equilibrium spherical gas bubble of radius r, volume V, surface area A, containing gas at pressure p and placed in a region of crystal under hydrostatic tension u. The condition for bubble equilibrium, representing a state of minimum total free energy is [ I]
In the first region of crystal, externally stressed at u, the bubble volume was replaced by material stressed at (a + do). In contrast to the equilibrium bubble this new material causes a distortion in the stress field which can be completely relieved by a plastic deformation process permitting a total volume relaxation Vdu/K, so that the stresses throughout the volume of the new material match those in the surrounding solid. The work done by the solid against the applied stress is then Vudu/K. Allowing for the increase in strain energy of u2 V/2K caused by the removal of the bubble, the free energy change dG, associated with the first region of crystal is
p+u=2s/r
dG, = u2 V/2K + Vudu/K
2. Derivation of forces 2.1. Equilibrium bubble in ideal solid
(1)
where s is the surface energy. However, the radial stress which the bubble surface exerts on the bulk solid is (27/r - p) which deviates from the externally applied tension, u, when the surface tension, y, differs from the surface energy. Whilst y is the work required to elastically stretch a surface, s is the energy required for its creation [7]. Where they differ the presence of even equilibrium gas bubbles perturbs the stress field in the material. In the following argument it is assumed initially that s and y are equivalent so that the stress at every point in the solid is purely hydrostatic and equal to the applied tension, u. Suppose the bubble is removed to a region of crystal where the hydrostatic stress is (u + da) and the initial bubble site filled with the displaced material stressed at (a t do). The bubble is not in mechanical equilibrium in its new surroundings. The perturbation it induces in the elastic stresses in the solid can be fully relaxed by a mass transfer process which allows the bubble to attain equilibrium through expanding by an incremental volume d V. By this me-
The total free energy change on transferring bubble to a region of higher stress is then
the
dG = dG, + dG, =-(u+p)dV+sdA
+ [u2+2udu-(u+du)2]V/2K
For a spherical bubble, dA/dV = 2/r, so that from eq.(l) the first two terms in the above expression sum identically to zero. From the final term the derivative dG/du is seen to vanish as do tends to zero. Thus, in a solid where surface energy and surface tension are identical, a stress gradient exerts no force on an equilibrium gas bubble. This conclusion is not at all surprising; the bubble induces no distortion in the stress field so that the solid does not detect its presence. In the case where surface energy and surface tension are not equivalent this is no longer true and a finite fBrce does act upon the bubble. This situation will now be considered. 2.2. Inequality of surface tension and surface energy With the bubble in a region stressed at u a plastic
51
M. V. Speight / Froces on bubbles and voids
flow process allows its volume to change to reach an equilibrium value V. At equilibrium the tensile stress, t(= 2-y/r - p), on the solid at the bubble surface does not, in general, match the applied stress, and elastic relaxation occurs resulting in a further increase in bubble volume Ak’,. During relaxation the external stress does an amount of work oAv, on the solid and part of this appears as an increase (u - t)Av,/2 in elastic stored energy of the bulk solid. Additionally an amount of energy (27/r) A P’, is stored as elastic strain in the surface layer of material bounding the bubble. The total change in free energy AC, associated with attaining elastic equilibrium is then AC, = -oAv,
2.3. Voids ofconstant volume
+ (a - t)AP’,/2 + (2y/r)AV,
= [(27/r) - (u + t)/2]AV,
.
(2)
The elastic relaxation Av, is proportional to the differential stress (u - t), and in an elastically isotropic medium its magnitude is readily shown to be [8] AL’, = 3 f’(u - t)/4/~
(3)
where p is the shear modulus. For a gas bubble in equilibrium according to eq. (1) u - t = (2/r) (s - y)
(4)
so that, in this case, elastic relaxation is directly determined by the difference between surface energy and surface tension. Substituting for A& from eq. (3) and for t from eq. (4) the expression for AC, becomes AC, = -3 V(s - r)[u -(s + -r)/r]/2v = 3 Us - 7) ]P - (s -
7)lrlDw.
Voids, containing a negligible pressure of enclosed gas, cannot exist in a material in a state of thermal and mechanical equilibrium. They persist at low temperatures, particularly under irradiation, because plastic flow processes relying on volume diffusion are too slow to effect significant shrinkage in a realistic time. Although the temperature is too low to permit volume changes, the more rapid rate of surface diffusion may enable voids to move in an imposed stress gradient. Like the equilibrium gas bubble the force on a void originates solely with the degree of elastic relaxation. With no internal gas pressure and applied tension ut the free energy change associated with relaxation is, from eq. (2)
ql AV$
AC, = [(2-r/r) and from eq. (3)
(5)
If the applied stress is compressive it is easily shown that eq. (5) still describes the free energy change occurring during the elastic relaxation of an equilibrium gas bubble. When (s - y) is non-zero the derivative -(dAG,/dx) defines the total force, F, acting on the bubble. Assuming that the enclosed gas obeys the perfect gas law (i.e. the product pV is constant) the force on the bubble in a given stress gradient du/dx becomes F = ]27@ -
as to move the bubble to a region of greater tension or lower compression. Conversely, when Y > s and p > (y - s)/r the tendency is for bubble movement into areas of the solid under lower tension or greater compression. If a bubble is, at all times, to remain in equilibrium with the applied stress, dr/du must be finite and, from eq. (l), this requires that p > 2s/3r. Hence, except when y % s, the restriction that p must exceed (y - s)/r is unlikely to be violated for equilibrium bubbles. This being the case, in a given stress gradient, the algebraic sign of the quantity (s - y) alone determines the direction of the force.
YYPIb + 6 - r)l (WW (d&.x) .
Since the equilibrium bubble size increases as the applied hydrostatic stress becomes more tensile the derivative dr/du obtained from eq. (1) is always positive. Hence, where s > y the force is always such
A& = 3V]ot -
CW)lI+ .
In a gradient of hydrostatic tension, do,/&, the force on a void which migrates without change in volume is then F = (3 V/41-1)]ot -
(Wr)l do,/& .
The direction of the force depends exclusively on the differential stress ut - (2-y/r). When this is positive the void moves to a region of increased applied tension, and in the opposite direction when the stress due to surface tension exceeds the applied tensile stress. By completely analogous arguments the force on a void in a material under compression, uc, is readily shown to be F = (3 V/~P) [oe + GW)l
do&
58
M, V. Speight /Forces
so that under all circumstances the void tends to move to a region of increased compression.
on bubbles and voids
which at temperatures sufficiently high for volume diffusion would promote unlimited void growth, the direction of motion is towards increased tension.
3. Conclusions Acknowledgement A stress gradient in a solid where surface tension y and surface energy s are equivalent exerts no force on an equilibrium gas bubble. Where these surface properties diverge the force on a gas bubble, which migrates and changes volume to establish equilibrium with the local stress, is proportional to the difference between surface energy and surface tension. In a given stress gradient, unless y + s, the direction of the force is determined exclusively by the algebraic sign of this difference. This applies whether the applied stress is tensile or compressive. Non-equilibrium voids, which because they exist at low temperatures are assumed to migrate without change in volume, generally gravitate towards regions of greater compression or lower tension. The exception is when the applied stress is tensile and exceeds the surface tension stress (27/r). Under these conditions,
This paper is published by permission of the Central Electricity Generating Board.
References
[l] A.B. Lidiard and R.S. Nelson, Phil. Mag. i7 (1968) 425. [2] DIR. Norris, Radiation Effects 14 (1972) 1. [3] E.P. Hicks and If. Hughes, Proc. B.N.E.S. Conf. on lrradiation Embrittlcment and Creep in Fuel Cladding and Core Components (1972) 13. [4] B.L. Eyre and R. Bullough, J. Nucl. Mater 26 (1968) 249. (51 F.A. Nichols, J. Nucl. Mater 30 (1969) 143. [6] D.G. Martin, J. Nucl. Mater 33 (1969) 23. [7] P.R. Couchman and W.A. Jesser, Surface Science 34 (1973) [8] F.C. Frank,
J. Nucl. Mater 48 (1973)
199