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A cavity growth diagram for high temperature creep
Scripta
METALLURGICA
Vol. 14, pp. 179-182, 1980 P r i n t e d in the U.S.A.
P e r g a m o n Press Ltd. All rights r e s e r v e d .
A CAVITY GROWTH DIAGRAM FOR HIGH TEMPERATURE CREEP
David A. Miller t and Terence G. Langdon Department of Materials Science, University of Southern California Los Angeles, California 90007 tNow at Department of Mechanical Engineering, University of Bristol, Bristol BS8 ITR, England. (Received October (Revised D e c e m b e r I.
i, 1979) 26, 1979)
Introduction
Under conditions of high temperature creep, polycrystalline materials often fail by the nucleation, growth, and interlinkage of grain boundary cavities. Considerable attention has been devoted to the growth of cavities, and it is now clear that there are two major growth processes. First, cavities may grow by accepting vacancies from a sphere immediately around the cavity. Second, growth may be controlled by creep deformation in the surrounding material. These two processes are termed diffusion growth and power-law growth, respectively. In addition, diffusion growth may be either unconstrained or constrained, with these two types of behavior operating sequentially (i). Most theories of cavity growth analyze only one of the possibilities of diffusion and power-law growth, but a unified approach for the combined processes was developed recently by Beer& and Speight (2) and Edward and Ashby (3). These two growth processes were first examined in detail by Dyson and Taplin (4), and a diagram was constructed showing the transition from diffusion to power-law growth for a cavity spacing of i0 vm and a range of strain rates from 10 -4 to i0 -I0 s-I. The purpose of this paper is to present an alternative, simplified cavity growth diagram which directly indicates the growth process under any selected experimental conditions. 2.
The Processes of Cavity Growth
The process of diffusion growth is based on an increase in cavity size when the applied stress, a, is sufficiently large to overcome the tendency for shrinkage due to surface tension. This requires that ~ > 2y/r, where y is the surface energy of the cavity and r is the cavity radius. There are several theories of diffusion growth (5-8) which lead to essentially similar relationships for the rate of change of cavity radius with time, t. Following Speight and Harris (6), the rate of diffusion growth is given by
dt
d
2kTr 2 [in (a/2r) - ½ ]
where ~ is the atomic volume, d is the width of the grain boundary, Dg b is the coefficient for grain boundary diffusion, k is Boltzmann's constant, T is the absolute temperature, and a is the cavity spacing. For power-law growth, Hancock (9) showed that the cavity growth rate is given by
[ ]__dt dr
=
r~ Y ---2~
[2]
P 179 0036-9748/80/020179-04502.00/0 C o p y r i g h t (c) 1980 P e r g a m o n Press
Ltd.
180
CAVITY
GROWTH
IN C R E E P
Vol.
14,
No.
2
where ~ is the strain rate and ~ is the coefficient of viscosity in shear. Equation [2] is derived from an extension of low temperature ductile fracture to high temperatures, but several similar relationships are also available (2,10,11). Inspection of equations [i] and [2] shows that diffusion growth is favored at small cavity radii and low strain rates, but there is a transition to power-law growth at large cavity radii and fast strain rates. For each process, the rate of change of cavity radius with strain, e, may be expressed as
d
~
2kTr 2 [in (a/2r) - ½1
[3]
and
Edrlp = rE3J,o Following the procedure of Hancock (9), equations [3] and [4] are logarithmically plotted in schematic form in Fig. i as dr/dE vs r. This diagram shows the transition from diffusion growth at small cavity radii to power-law growth at large radii, with the transition occurring at a critical radius, r c. It should be noted that the two growth processes approximate to straight lines on a plot of dr/de vs r at large cavity radii. In order to illustrate the transition between these two processes, Dyson and Taplin (4) plotted 2rc/a vs T/Tm for different values of {, where Tm is the absolute melting point of the material. However, an alternative and more simple procedure is to note that, at large values of r when surface tension effects are negligible, the linear portions of Fig. i may be expressed as
Edrl de
d
[
2kTr 2 [in (a/2r) - ½ ]
t
[5]
and
E1
=
r
[6]
de
P
An exact solution of equations [5] and [6] is complicated because of the presence of the term (a/2r) in the denominator of equation [5]. However, as noted by Dyson and Taplln (4), an approximate solution may be obtained by setting [in (a/2r) - ½] = i. It follows that, to a first approximation, the critical radius is independent of the cavity spacing, and r c is given by
[7]
The first term on the right of the equality in equation [7] is a constant for any material at the testing temperature, so that the value of r c is determined exclusively by the magnitude of ~/~. Figure 2 shows a cavity growth diagram in the logarithmic form of r c vs (~6D~b/2kT) (o/~). The line having a slope of 1/3 marks the theoretical boundary between the two growth processes, and the broken extension for r c < 1.0 Bm indicates that the critical radius is underestimated at low values of u/~ due to the increasing importance of the surface tension term in equation [4]. This diagram provides a convenient method of estimating the approximate critical transition radius for any experimentally determined value of a/~.
Vol.
14,
No.
2
CAVITY
GROWTH
I
IN C R E E P
181
I
// I,LI I-<1 n"
/ #,
/ / ~ " - P O W ER - LAW
"lF-
DIFFUSION~ ~ GROWTH ~'
on. >i-
(at~aCid
r
>
q
// X
Nt
~'/
(dr/dE)p
/\ ,\ ; I
I ! I
r¢
I
LOG. CAVITY RADIUS (r) FIG. i Cavity growth rate vs cavity radius for the diffusion and power-law growth processes, showing the critical transition radius, r c.
I,.,;n
I0 _-:
I
I
' ' ~'l'"l
POWER-LAW GROWTH "
!. 1.0--
Volue
I iO-II 10-19
of rc
un~restimoted
ot ~ow (o'It,) 10-18
I ........ 10-17 ( ~ S Dgb~ ~'--~-~'~'--j ("~) ( m31
I
10-16
10-15
FIG. 2 Cavity growth diagram showing the predicted transition from diffusion to power-law growth.
182
CAVITY
GROWTH
3.
IN C R E E P
Vol.
14,
No.
2
Discussion
There is an important difference in the appearance of grain boundary cavities after diffusion and power-law growth. Whereas diffusion growth leads to cavities which are reasonably spherical in cross-section and lying preferentially on grain boundaries oriented almost perpendicular to the tensile axis, power-law growth gives elliptical cavities which are elongated in the direction of the tensile stress. These two types of cavities have been observed in a 1% Cr-½% Mo ferritic steel under creep conditions (9). A similar transition in cavity appearance has been reported also in a fine-grained superplastic material (12). However, it is important to exercise care in using Fig. 2 to interpret cavitation in superplastic metals because the grain sizes are usually very small. Equation [I] applies to an isolated cavity on a single grain boundary facet, but at small grain sizes the situation is rapidly reached where a/2 = d/2, where d is the average spatial grain diameter. The rate of diffusion growth is then enhanced because the cavity intersects a number of grain boundaries (13), and thus an additional restriction on Fig. 2 is that r c ~ d/2. In summary, Fig. 2 is a cavity growth diagram which may be used to estimate the approximate critical radius for a transition from diffusion to power-law growth. The procedure breaks down when the cavity radius is very small (
D.A. Miller and T.G. Langdon, "Independent and Sequential Cavity Growth Mechanisms," Scripta Met. (January 1980). W. Beer~ and M.V. Speight, Metal Sci. 12, 172 (1978). G.H. Edward and M.F. Ashby, Acta Met. 27, 1505 (1979). B.F. Dyson and D.M.R. Taplln, Grain Boundaries, p. E23. The Institution of Metallurgists, London (1976). D. Hull and D.E. Rimmer, Phil. Mag. 4, 673 (1959). M.V. Speight and J.E. Harris, Metal Sci. J. _i, 83 (1967). M.V. Speight and W. Beer~, Metal Sci. 9, 190 (1975). R. Raj and M.F. Ashby, Acta Met. 23, 653 (1975). J.W. Hancock, Metal Sci. IO, 319 (1976). R.J. DJ~Melfi and W.D. Nix, Intl. J. Fracture 13, 341 (1977). W.D. Nix, D.K. Matlock and R.J. DiMelfi, Acta Met. 25, 495 (1977). S.-A. Shei and T.G. Langdon, J. Mater. Sci. 13, 1084 (1978). D.A. Miller and T.G. Langdon, "An Analysis of Cavity Growth During Superplasticity," Met. Trans. A (December 1979).
METALLURGICA
Vol. 14, pp. 179-182, 1980 P r i n t e d in the U.S.A.
P e r g a m o n Press Ltd. All rights r e s e r v e d .
A CAVITY GROWTH DIAGRAM FOR HIGH TEMPERATURE CREEP
David A. Miller t and Terence G. Langdon Department of Materials Science, University of Southern California Los Angeles, California 90007 tNow at Department of Mechanical Engineering, University of Bristol, Bristol BS8 ITR, England. (Received October (Revised D e c e m b e r I.
i, 1979) 26, 1979)
Introduction
Under conditions of high temperature creep, polycrystalline materials often fail by the nucleation, growth, and interlinkage of grain boundary cavities. Considerable attention has been devoted to the growth of cavities, and it is now clear that there are two major growth processes. First, cavities may grow by accepting vacancies from a sphere immediately around the cavity. Second, growth may be controlled by creep deformation in the surrounding material. These two processes are termed diffusion growth and power-law growth, respectively. In addition, diffusion growth may be either unconstrained or constrained, with these two types of behavior operating sequentially (i). Most theories of cavity growth analyze only one of the possibilities of diffusion and power-law growth, but a unified approach for the combined processes was developed recently by Beer& and Speight (2) and Edward and Ashby (3). These two growth processes were first examined in detail by Dyson and Taplin (4), and a diagram was constructed showing the transition from diffusion to power-law growth for a cavity spacing of i0 vm and a range of strain rates from 10 -4 to i0 -I0 s-I. The purpose of this paper is to present an alternative, simplified cavity growth diagram which directly indicates the growth process under any selected experimental conditions. 2.
The Processes of Cavity Growth
The process of diffusion growth is based on an increase in cavity size when the applied stress, a, is sufficiently large to overcome the tendency for shrinkage due to surface tension. This requires that ~ > 2y/r, where y is the surface energy of the cavity and r is the cavity radius. There are several theories of diffusion growth (5-8) which lead to essentially similar relationships for the rate of change of cavity radius with time, t. Following Speight and Harris (6), the rate of diffusion growth is given by
dt
d
2kTr 2 [in (a/2r) - ½ ]
where ~ is the atomic volume, d is the width of the grain boundary, Dg b is the coefficient for grain boundary diffusion, k is Boltzmann's constant, T is the absolute temperature, and a is the cavity spacing. For power-law growth, Hancock (9) showed that the cavity growth rate is given by
[ ]__dt dr
=
r~ Y ---2~
[2]
P 179 0036-9748/80/020179-04502.00/0 C o p y r i g h t (c) 1980 P e r g a m o n Press
Ltd.
180
CAVITY
GROWTH
IN C R E E P
Vol.
14,
No.
2
where ~ is the strain rate and ~ is the coefficient of viscosity in shear. Equation [2] is derived from an extension of low temperature ductile fracture to high temperatures, but several similar relationships are also available (2,10,11). Inspection of equations [i] and [2] shows that diffusion growth is favored at small cavity radii and low strain rates, but there is a transition to power-law growth at large cavity radii and fast strain rates. For each process, the rate of change of cavity radius with strain, e, may be expressed as
d
~
2kTr 2 [in (a/2r) - ½1
[3]
and
Edrlp = rE3J,o Following the procedure of Hancock (9), equations [3] and [4] are logarithmically plotted in schematic form in Fig. i as dr/dE vs r. This diagram shows the transition from diffusion growth at small cavity radii to power-law growth at large radii, with the transition occurring at a critical radius, r c. It should be noted that the two growth processes approximate to straight lines on a plot of dr/de vs r at large cavity radii. In order to illustrate the transition between these two processes, Dyson and Taplin (4) plotted 2rc/a vs T/Tm for different values of {, where Tm is the absolute melting point of the material. However, an alternative and more simple procedure is to note that, at large values of r when surface tension effects are negligible, the linear portions of Fig. i may be expressed as
Edrl de
d
[
2kTr 2 [in (a/2r) - ½ ]
t
[5]
and
E1
=
r
[6]
de
P
An exact solution of equations [5] and [6] is complicated because of the presence of the term (a/2r) in the denominator of equation [5]. However, as noted by Dyson and Taplln (4), an approximate solution may be obtained by setting [in (a/2r) - ½] = i. It follows that, to a first approximation, the critical radius is independent of the cavity spacing, and r c is given by
[7]
The first term on the right of the equality in equation [7] is a constant for any material at the testing temperature, so that the value of r c is determined exclusively by the magnitude of ~/~. Figure 2 shows a cavity growth diagram in the logarithmic form of r c vs (~6D~b/2kT) (o/~). The line having a slope of 1/3 marks the theoretical boundary between the two growth processes, and the broken extension for r c < 1.0 Bm indicates that the critical radius is underestimated at low values of u/~ due to the increasing importance of the surface tension term in equation [4]. This diagram provides a convenient method of estimating the approximate critical transition radius for any experimentally determined value of a/~.
Vol.
14,
No.
2
CAVITY
GROWTH
I
IN C R E E P
181
I
// I,LI I-<1 n"
/ #,
/ / ~ " - P O W ER - LAW
"lF-
DIFFUSION~ ~ GROWTH ~'
on. >i-
(at~aCid
r
>
q
// X
Nt
~'/
(dr/dE)p
/\ ,\ ; I
I ! I
r¢
I
LOG. CAVITY RADIUS (r) FIG. i Cavity growth rate vs cavity radius for the diffusion and power-law growth processes, showing the critical transition radius, r c.
I,.,;n
I0 _-:
I
I
' ' ~'l'"l
POWER-LAW GROWTH "
!. 1.0--
Volue
I iO-II 10-19
of rc
un~restimoted
ot ~ow (o'It,) 10-18
I ........ 10-17 ( ~ S Dgb~ ~'--~-~'~'--j ("~) ( m31
I
10-16
10-15
FIG. 2 Cavity growth diagram showing the predicted transition from diffusion to power-law growth.
182
CAVITY
GROWTH
3.
IN C R E E P
Vol.
14,
No.
2
Discussion
There is an important difference in the appearance of grain boundary cavities after diffusion and power-law growth. Whereas diffusion growth leads to cavities which are reasonably spherical in cross-section and lying preferentially on grain boundaries oriented almost perpendicular to the tensile axis, power-law growth gives elliptical cavities which are elongated in the direction of the tensile stress. These two types of cavities have been observed in a 1% Cr-½% Mo ferritic steel under creep conditions (9). A similar transition in cavity appearance has been reported also in a fine-grained superplastic material (12). However, it is important to exercise care in using Fig. 2 to interpret cavitation in superplastic metals because the grain sizes are usually very small. Equation [I] applies to an isolated cavity on a single grain boundary facet, but at small grain sizes the situation is rapidly reached where a/2 = d/2, where d is the average spatial grain diameter. The rate of diffusion growth is then enhanced because the cavity intersects a number of grain boundaries (13), and thus an additional restriction on Fig. 2 is that r c ~ d/2. In summary, Fig. 2 is a cavity growth diagram which may be used to estimate the approximate critical radius for a transition from diffusion to power-law growth. The procedure breaks down when the cavity radius is very small (
D.A. Miller and T.G. Langdon, "Independent and Sequential Cavity Growth Mechanisms," Scripta Met. (January 1980). W. Beer~ and M.V. Speight, Metal Sci. 12, 172 (1978). G.H. Edward and M.F. Ashby, Acta Met. 27, 1505 (1979). B.F. Dyson and D.M.R. Taplln, Grain Boundaries, p. E23. The Institution of Metallurgists, London (1976). D. Hull and D.E. Rimmer, Phil. Mag. 4, 673 (1959). M.V. Speight and J.E. Harris, Metal Sci. J. _i, 83 (1967). M.V. Speight and W. Beer~, Metal Sci. 9, 190 (1975). R. Raj and M.F. Ashby, Acta Met. 23, 653 (1975). J.W. Hancock, Metal Sci. IO, 319 (1976). R.J. DJ~Melfi and W.D. Nix, Intl. J. Fracture 13, 341 (1977). W.D. Nix, D.K. Matlock and R.J. DiMelfi, Acta Met. 25, 495 (1977). S.-A. Shei and T.G. Langdon, J. Mater. Sci. 13, 1084 (1978). D.A. Miller and T.G. Langdon, "An Analysis of Cavity Growth During Superplasticity," Met. Trans. A (December 1979).