A Formulation for Anisotropy in Diffusional Creep

A Formulation for Anisotropy in Diffusional CreepAuthor(s): G. W. GreenwoodSource: Proceedings: Mathematical and Physical Sciences, Vol. 436, No. 1896 (Jan. 8, 1992), pp.187-196Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52028 .

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A formulation for anisotropy in diffusional creep

BY G. W. GREENWOOD

School of Materials, University of Sheffield, Mappin Street, Sheffield SI 3JD, U.K.

Grain shape can introduce anisotropy in creep which depends on the diffusion of vacancies between grain boundary sources and sinks. Such anisotropy is examined to determine the rate of creep under multiaxial stresses both for lattice and grain boundary diffusion. Noting the role of grain boundary sliding in this form of creep it is shown that, with some approximations that only become significant in an identified case, complete and fully self-consistent formulae can be derived for the rate of creep in terms of grain dimensions. The results are presented in the form of compliance matrices which are analogous to those that have a well-established role in the characterization of elastic anisotropy. A comparable usefulness of these 'creep compliance coefficients' is envisaged in evaluating anisotropic diffusional creep behaviour and a similar approach can be extended to more general cases where creep rates may be interface controlled.

1. Introduction

The effect of grain size (Nabarro 1948; Herring 1950) and anisotropy of grain shape (Nix 1981) on the resistance to diffusional creep are well recognized. More recently the influence of grain shape anisotropy on the response to multiaxial stress has also been examined (Greenwood 1985) though with the restriction to the three principal stresses aligned along the axes of orthorhombic shaped grains. To overcome this restriction some assumptions must be made but it is proposed in the present paper that the grain boundary sliding, which is also governed by diffusional processes (Liftschitz 1963; Gibbs 1965; Raj & Ashby 1971; Speight 1975) and known to be a necessary accompaniment to diffusional creep, is likely to lead to a smooth response in the change of creep rate when the axes of the principal stresses are rotated. Analyses are presented for both lattice and grain boundary diffusion and the results are compiled in the form of 'creep compliance matrices'. These provide a complete description of the effect of grain shape on creep anisotropy in response to any stress system that may be superimposed.

2. Creep through lattice diffusion

2.1. The case of principal stresses aligned with grain axes

This situation has been rigorously analysed (Greenwood (1985) and, with the application of three principal tensile stresses ox, ry, and oz acting along the axes of orthorhombic grains of respective dimensions X, Y and Z, the corresponding creep rates ex, dy and dz are given by

x = 12(DQ/kT) [o'x(y2 + Z2)-ry Z2 - y2]/[X2Y2+ Y2Z2 +Z2X2], (2.1) Proc. R. Soc. Lond. A (1992) 436, 187-196

Printed in Great Britain 187



where D is the self-diffusion coefficient, 2 is atomic volume, k is Boltzmann's constant and T is absolute temperature.

It is convenient to write a = 12(DQ1/kT) and / = [X2Y2 + Y2Z2 + Z2X2] so that the

corresponding equations for Gy and ez now become

,y -= a Z0Xy(Z2 +X2) _ - OX2 _ Z2]/] (2.2)

z = O-Z(X2 + y2)- -

X2]/. (2.3)

These three equations fully define the creep response for the geometrical conditions prescribed and have been obtained through a full treatment of the diffusional creep problem when the process is not inhibited by limitation of the grain boundaries to act as vacancy sources and sinks (Harris et al. 1969).

The equations, however, do not describe the creep response when the stress axes are rotated and some further propositions are required before such a calculation may be attempted. Such considerations are presented in ?2.2. The importance in practical situations may also be noted, for example in materials strengthened in a longitudinal direction by a parallel elongated grain structure but required also to have adequate torsional resistance about the longitudinal axis. A further example (Jones 1973) is in the coiled tungsten filaments in lamps where the elongated grain structure along the tungsten wire is a major feature in the maintenance of adequate creep strength under shear stress at a high fraction of the melting temperature. Considered alone, the diffusional creep formula suggest that adequate strength may be obtained simply by an equiaxed coarse grain size but in practice, particularly with components with a narrow dimension, this would lead to a through-section grain boundary shear not controlled by diffusion to negate the strength development. Thus there is strong interest in materials development with markedly anisotropic grain structures to match the required creep resistance under the stress application envisaged.

2.2. The effect of shear stresses on non-equiaxed grains There is a lack of experimental data in this area. Nevertheless, some analogy may

be proposed with the theory and observation of grain boundary sliding as it occurs in the diffusional creep range for equiaxed grains (Burton 1977). This form of grain boundary sliding is now recognized to be governed by the diffusional creep process and does not require any modification to the diffusional creep formula in the calculation of overall creep rate (Liftschitz 1963; Speight 1975). The situation can be regarded either as one of grain boundary sliding accommodated by diffusional creep or as diffusional creep accommodated by grain boundary sliding (Raj & Ashby 1971). This is markedly different from the form of grain boundary sliding (Rachinger 1952) that may take place without diffusional accommodation at somewhat higher stress levels and the fundamental distinctions between the two grain boundary sliding modes are now well understood.

Now considering the case of diffusional controlled grain boundary sliding in grains of anisotropic shape, it is assumed the creep strength varies smoothly as the stress axes are rotated. Here the concept of a smooth variation implies no maxima or minima in values between those of the orthogonal directions parallel to the orthorhombic grain dimensions. It is convenient to represent this situation geometrically by constructing an ellipsoid with its axes in these directions as in figure 1, where a section of the ellipse is taken in the plane with z = 0. The ellipse is orientated with regard to the grain dimensions such that a 1] X, b 11 Y and c I1 Z and the

Proc. R. Soc. Lond. A (1992)

188 G. W. Greenwood




y

x

Figure 1. Elliptical construction used for the evaluation of tensile and shear creep rates. The axes are defined by equations (2.6) and (2.7).

values of a, b and c can be related to the creep rates under given applied stresses. Consider numerically equal tensile and compressive stresses r in the a and b directions with o-r = -oy = o and with o- = 0 then equations (2.1) and (2.2) become, respectively,

= a(Y2 + 2Z2)/, (2.4)

y = -ao-(X2 + 2Z2)/l. (2.5)

Now defining the axes a = ex/or and b = y/or it follows that

a = a(Y2 + 2Z2)?, (2.6)

b = (X2 + 2Z2)/. (2.7)

This construction permits the effect of a shear stress Yxy to be evaluated by considering the resultant shear strain rate yxy. The situation is equivalent to numerically equal tensile and compressive stresses with each \ao = Y,x applied respectively at 45? to the a and b axes. From this, xy/x , = 2L, where L is the distance from the origin to the point of intersection of the lines y = _x with the ellipse (x/a)2 + (y/b)= 1. Hence L = 21ab/(a + b2)I and substituting for ft and for a and b from equations (2.6) and (2.7)

2 \/2 2.xy 2(y2 + 2Z2) (X2 + 2Z2) y 2 = [X2Y2 + Y2Z2 + Z2X2] [(Y2 + 2Z2)2 + (X2 2)2] (2.8)

This result seems initially unsatisfactory for it includes the third dimension Z of the grain whereas this situation should correspond in practice to diffusional creep with only a two-dimensional flow of vacancies independently of any third-dimensional influence. This situation, however, is resolved when equation (2.8) is examined in more detail, for it reveals that Z is only of minor significance, thus confirming that


189



the elliptical construction very closely, though not precisely, represents the

parameters indicated. Considering specific conditions, for example, evaluating equation (2.8) when X = Y = Z, then

L = a/X2. (2.9)

For the condition Z > X and Y, it is noted that

L = 2a/(X2 + 2) (2.10)

and for Z < X and Y, then L = 2x/(X4+ 4). (2. 11)

None of the above terms incorporates Z, confirming the insignificance of this

parameter in the evaluation of the ~yy against Yry relationship and further inspection shows that its numerical influence cannot exceed V2.

Thus within error which may reach a factor of \/2 when Z is relatively very small, equation (2.8) can be simplified to the form

=xy = 4x Yxy/(X2 + y2). (2.12)

By symmetry, it also follows that

yz - 4 ryz/ (Y2 + Z2) and zx = 4x Yzx/(Z2 X2).

These equations now allow a matrix formulation of these parameters analogous to

compliance matrices in the theory of anisotropic elasticity so that the complete response to any applied stress system may be determined. This aspect is considered later in ?4.

3. The anisotropy of creep controlled by grain boundary diffusion

3.1. Specific situations where rigorous analyses can be made

Because of the different path lengths for diffusion along the grain boundaries of anisotropic grains, it is clear that creep resistance will again be dependent upon the directions of the applied stresses with respect to the grain dimensions. Some specific situations have already been considered but in some cases a difficulty has been caused by the necessity to assume an unrealistically large vacancy flux occurring along the grain edges when analytical solutions have been attempted (Nix 1981). This difficulty can be overcome by assuming a cylindrical grain geometry and an axial stress to obtain a totally self-consistent solution (Burton & Greenwood 1985) leading to the definitive equation for axial creep rate,

E = 48Dg wo-Q/kTL0R(3R + 2L,), (3.1)

where Dg is the grain boundary self-diffusion coefficient w is the effective grain boundary width and Lo and R are respectively the length and radius of the cylindrical grain. Although equation (3.1) is helpful in comparing some limiting conditions which will be considered later, it does not assist at this stage in the more general case of three unequal grain dimensions where the grains fit together to occupy the entire material.

It is now suggested that a useful additional approach, avoiding the problems mentioned, but still not dealing with the most general case, lies in considering the situation where the vacancy flux lines within the grain boundaries are all parallel. Proc. R. Soc. Lond. A (1992)

190 G. W. Greenwood




This condition is achieved when principal stresses act perpendicularly to two pairs of faces of an orthorhombic grain with no stress in the third dimension which greatly exceeds the lengths of the other two directions as in figure 2.

Here, if we consider a stress ori, at a point at (y, z) on a face where x = _+ X, then

cY/2 fZ/2 fYI 7/j c-' dy dz = l

xYZ. (3.2)

With a rate of vacancy deposition jV per unit area on this face the vacancy concentration is derived from Fick's second law, in the form,

(D2c/dy2) + (d2c/z2) = - V/Dgv w2Q, (3.3) where Dgv is the grain boundary vacancy diffusion coefficient, w is the grain boundary width and Q is the atomic volume. With grain dimensions Z > X and Y, (02c/Oz2) is negligible and can be ignored. Further, on the central plane y = 0, (ac/dy) = 0 so that integration of equation (3.3) gives the vacancy concentration

C = - (Vx y2/Dgv w) + K1. (3.4)

Now C = C exp (o-r Q/kT) = Col1 + (ocx Q/kT)],

where Co is the equilibrium vacancy concentration without stress at temperature T, since usually o-xi Q < kT (Cottrell 1964). Thus

=Xi = (C- Co) kT/Co 2. (3.5)

Substituting for oxi from equation (3.5) and for C from equation (3.4) into equation (3.2) and integrating, then

(kT/Co Q) [- (V Y2/24Dgv w2Q) + (K - C0)] = -x. (3.6)

A corresponding equation can be derived for the rate of vacancy removal from the faces at x = + X, so that

(kcT/Co 2) [ (Vy X2/24Dgv w(2) + (K2 - Co)] = O. (3.7)

To evaluate these equations, it is noted that the vacancy concentrations at both grain edges where x = +_X and y = _+Y are equal so that equation (3.4) can be extended in the form

C = - (V Y2/8Dgv wQ2) + K1 =- (VyX2/SDgvw Q2) K2. (3.8)

Now K1 and K2 can be eliminated from equations (3.6), (3.7) and (3.8) to give

(Vx y2 Vy X2) = 12(x- (D,) (Dgv wQC/kT). (3.9)

To evaluate the creep rate (e it is noted that ex = 2V!/X. Similarly cy = 2Vy/Y and since volume is conserved e + (y = 0. Using these latter relationships, the creep rate can thus be written

(4 = 24Dg wQ2(0rX - y)/XY(X + Y)kT, (3.10)

where Dg =Dgv C the grain boundary diffusion coefficient. Equation (3.10) is subsequently used to compare with proposals for a more extended approach in describing the multiaxial mechanical response where deformation is controlled by the rate of grain boundary diffusion.

Moreover, despite the different geometrical conditions, equation (3.10) is seen to be


t9t



192

y

t 17

G. W. Greenwood

ya

I1 z"

Ox

x

K< x x z Figure 2. Stresses o-r and (ry are applied perpendicularly to the faces at x = +-X and y = +2Y respectively of a grain of orthorhombic shape. The third grain dimension Z > X and Y so that the lines of vacancy flux in the grain boundaries are all parallel to the xy-plane.

consistent with equation (3.1) when it is recalled that R refers to the radius in the cylindrical grain geometry whereas X and Y are the sides of a rectangle. It is also consistent with results (Burton & Greenwood 1985) for the situation where

numerically equal tensile and compressive stress act perpendicularly to respective faces of a cube. In the latter case ox =-cry, and X = Y so that equation (3.10) becomes ex = 24DgwQ2o-x/kTX3.

3.2. A generalized description of grain boundary diffusion controlled creep in materials with anisotropic grain shapes

A rigorous analytical solution to this problem analogous to that for Nabarro- Herring creep as described by equation (2.1) has not proved possible for creep controlled by grain boundary diffusion because of detailed difficulties of matching vacancy fluxes at grain edges. Nevertheless, some directions are apparent in the approach that may be taken. With three stresses o-, cry and cz applied parallel to the average grain dimensions X, Y and Z, it is noted that

(a) there is no effect of hydrostatic pressure P so that creep rates ex, ey and ez are unaffected by stress changes to (acr +P), (ry +P) and (c-Z +P);

(b) there is no volume change so that ex + Cy + ez = 0; (c) the strain rate is given by equation (3.10) for stresses arx and cy acting on

material where Z > X and Y. Inspection shows that a specific formulation, meeting all these requirements can

be expressed by

(x = (24D wQ/kT) [crx(Y2 +Z2)- _ Z2 -z y2] (3.11) XYZ(XY+YZ+ZX)

Now writing (24D)gw2/kT) = 0 and XYZ(XY+ YZ+ZX) = ?, the corresponding equations for (y and ez become

,y = O[cy(Z2 +X2)_-o-X2-_ Z2]/q, (3.12)




A formulation for anisotropy in diffusional creep 193

and ez = O[-Z(X2 + Y2) - o Y2 - oyX2]/O. (3.13)

It is readily seen that condition (a) is satisfied with the addition of hydrostatic pressure P having no effect. Further, the sum ex + &y + ,y = 0 for all values of oX, Cry and roz in accordance with (b). Finally the creep rate given by equation (3.11) becomes independent of Z when Z > X and Y as required by condition (c) and is then identical with equation (3.10).

The response to shear stress now requires examination and this can be undertaken

by a method similar to that described for Nabarro-Herring creep in ?2.2. With reference to figure 1, the major and minor axes of the ellipse are now given by

a = O(Y2 + 2Z2)/9

and b = (X2 + 2Z2)/qO.

Thus, by analogy, corresponding to equation (2.8), a shear stress Yxy produces a strain rate ixy given by

2 2rxY (Y2+ 2Z2)(X2+ 22)

XYZ(XY+ YZ + ZX) [(y2 + 2Z2)2 + (X2 + 2Z2)2]

Equation (3.14) again includes the third dimension Z and it is necessary to consider its importance. First for Z > X and Y, then

=xy = 4 YrxO[XY(X+ Y)]. (3.15)

For X =Y =Z, then

Xy = 2 YO/X3. (3.16)

Considering equations (3.15) and (3.16), the insensitivity to Z is noted. This situation, however, does not continue down to values of Z < X and Y since in this case

Yy = 2i2YxyO/Z(X4 +4)i. (3.17)

Equation (3.17) is a reflection of a difficulty previously noted (Nix 1981) in precisely matching all the boundary conditions by analytical expressions so that grain edges are not required to carry an unrealistically large vacancy flux. In fact, the present analysis indicates that vacancy fluxes can cross grain edges since, in the limit, this situation corresponds closely to the case for lattice diffusion when the Z dimension reduces to the grain boundary width w. Since the 0 parameter contains w, it is noted( that equation (3.17) is comparable with the case for lattice diffusion as in equation (2.12) when Z = w.

4. The formulation of compliance matrices for diffusional creep

The approach here follows that adopted for the listing and manipulation of values for anisotropic elastic constants (Voigt 1910). A variety of notations has been used, but the one chosen in the present paper is summarized by the equations, i = 1 to 6,

6

ei = ijj, (4.1) j=l

where Sj represents the (6 x 6) compliance coefficients. However, to clarify the stress




194 G. W. Greenwood

orientation, we take o- = Go, etc., and also distinguish the shear components by replacing c14 by ryz, etc. Thus the three equations for response to tensile strain rate have the form for ex (and similarly for ex and cy)

Cx = O 11+ Oy S12 + z 13 + yz814 + zx S15+ xy X816' (4.2)

The further three equations expressing shear strain rates y~x are written in the form

?yx = xS41+ yS42 + 43+ Y yz844 + zx 845 + xy 46 (4.3)

Now grains with orthotropic symmetry (that is, with three mutually perpendicular planes of symmetry) have been assumed in the present analysis and the compliance coefficients are consequently reduced in number to 12 but with only nine having independent values (Hearmon 1961). Thus equations (4.2) and (4.3) can be summarized in the equation,

-X -

11 S12 S13 (X

ey S21 S22 S 23 y

ez S31 S32 S33 z

yyz -84- '(4.4) Vyz S44 yz

yzx S55 Yzx

_xy_ _ 66 _xy,

in which S12 = 21, etc. It is readily seen that this provides a convenient form on which to base the

description of anisotropic diffusional creep behaviour since different values in the compliance coefficients arise through the grain dimensions.

Considering first the case of Nabarro-Herring creep occurring by lattice diffusion, equations (2.1) and (2.12) immediately become relevant. Making the appropriate substitutions and recalling that it is convenient to let (X2Y2 + Y2Z2 + Z2X2) = , the creep compliance matrix can be completed and equations summarized in the form,

-.: ~ - -(yp+ z )// - -r/t~ -/~a- ":

- Z2// (Z2 +X2)// -X2/? ,

_ _~~~~~~~~~~ Z~~~~~~~~Y

6, _ 12DO -Y? -X2// (XY2+ ?)/ TZ kY'

41( Y" + Z2) ~,,~

); z 4/ (Z2 + X2)

-_ _ - 4/(X2 + )'2 ,_

(4.5)

It is instructive to consider how analogies with compliances in the theory of elastic anisotropy can be used to check the self-consistency of these extensions to descriptions of creep. For an isotropic elastic solid, deforming without change in volume so that Poisson's ratio is 0.5, it can readily be shown that 81l = 2S12 = S44, etc. Diffusional creep becomes isotropic when X = Y= Z and it is observed from equation (4.5) that identical relationships apply. Proc. R. Soc. Lond. A (1992)



A formulation for anisotropy in diffusional creep 195

It may also be noted that the problem of possible anisotropy in interface controlled diffusional creep (Ashby 1969; Greenwood 1970) may be represented in a similar manner since flow directions are governed by linear functions of stress even in power law creep (Greenwood 1991).

A similar approach leads to the formulation of a compliance matrix for the generalization of Coble creep where vacancy flux is confined to the grain boundaries (Coble 1963). Now we make use of equations (3.11) and (3.15). Recalling that XYZ(XY+ YZ + ZX) is denoted by 0, the matrix is now written,

-~ -(Y2 + Z2)/o _ -Z 2/ - -o

y -_ Z2/0 (Z2 +X2)/ -X2/0

ez 24Dg wQ -y/ _-X2/0 (X2 + y2)/Q

yz T

4/YZ(Y+Z) ryz Yzx 4/ZX(X + ) Yzx

Jxy_ _ 4/XY(X+Y)_ _xy_

(4.6)

5. The case where lattice and grain boundary diffusion both significantly contribute to diffusional creep

Independently of stress level, but dependent on grain dimensions and temperature (because of the different activation enthalpies of lattice and grain boundary diffusion), over a certain range both lattice and grain boundary diffusion may make simultaneously significant contributions to creep. In this range it is generally agreed that the effects are simply additive. This feature can be readily incorporated into the matrix formulation because the compliance coefficients placed in corresponding rows and columns are additive. For example, the creep rate ex under uniaxial stress crx is given by

ex = (T12ox Q(Y2 + Z2)/kT) [D// + 2Dg w/09. (5.1)

Similarly the shear strain rate yxy in creep under a shear stress Yxy can be written,

x-y = (48 Yxy Q/kT) LD/(X2 + y2) + 2Dg w/XY(X + Y)]. (5.2)

6. Conclusions

Analytical expressions have been formulated that make possible the approximate calculation of diffusional creep rates, when both lattice and grain boundary diffusion make contributions in materials with anisotropic grain shape under multiaxial stresses. Errors only become large in the limiting case (to which equations of the form of (3.17) then become applicable) of a material, in which one grain dimension is substantially less than the other two, subjected to a shear stress under conditions such that creep is controlled by grain boundary diffusion and with vacancy emission and absorption on the narrow grain faces.




By the summation of terms on the right-hand side of equations (4.5) and (4.6), a full equation may be written in the form,

-(+z2)/f? -Z2/p -Y2//

Ey _ -Z2/ (Z2 +X2)/f1 -X2/fI

12Q D - y/_ -X2/P (X2 + Y2)/

7,z~ kT 4/(Y2 + 2)

~zx 4/(2 +X2)

7xy_ _ 4/(X2 + Y2)

-(Y2+Z2)/q - 2/0 -_-Y/0

- -

-z2/0q (Z2 +X2)/0 -X2/q

+2Dg Y/ - X2/- (X2+ Y2)/ .

4/YZ(Y+Z) ryz 4/X(Z + X) TZ

4/XY(X + Y) _ _xy_

where p =X2 + 2Z2 + Z2X2 and 0 = XYZ(XY +YZ+ZX).

I am grateful for valuable discussions in the School of Materials and the Structural Integrity Research Institute of the University of Sheffield and for support from the Science and Engineering Research Council and the Leverhulme Trust Foundation.

References

Ashby, M. . 1969 Scr. metall. 3, 837-842.

Burton, B. 1977 Diffusional creep in polycrystalline materials. Aedermannsdorf: Trans. Tech. Publications.

Burton, B. & Greenwood, G. W. 1985 Mat. Sci. Technol. 1, 11029-11032. Coble, R. L. 1963 J. appl. Phys. 34, 1679-1682. Cottrell, A. H. 1964 Mechanical properties of matter, pp. 203-204. New York: Wiley. Gibbs, G. B. 1965 Mem. sci. Revue Metall. 62, 781-786. Greenwood, G. W. 1970 Scr. metall. 4. 171-173. Greenwood, G. W. 1985 Phil. Mag. A51, 537-542. Greenwood, G. W. 1991 (In the press.) Harris, J. E., Jones, R. B., Greenwood, G. W. & Ward, M. J. 1969 J. Aust. Inst. Mlet. 14, 154-162. Hearnon, R. F. S. 1961 Introduction to applied elastic anisotropy. Oxford University Press.

Herring, C. 1950 J. appl. Phys. 21, 437-445. Liftschitz, I. M. 1963 Soviet Phys. JETP 17, 909-920. Nabarro, F. R. N. 1948 Report on the Conference on the Strength of Solids, pp. 75-90. The

Physical Society of London. Nix, W. -D. 1981 Metals Forum 4, 38-43.

Rachinger, W. A. 1952 J. Inst. Met. 81, 33-41.

Raj, R. & Ashby, M. F. 1971 Metall. Trans. 2, 1113-1127.

Speight, M. V. 1975 Acta metall. 23, 779-781.

Voight, W. 1910 Lehrbuch der Kristallphysik Trubner, Leipzig.

Received 26 July 199.1: accepted 2 September 1991


G. W. Greenwood 196



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A Formulation for Anisotropy in Diffusional Creep