CONTRIBUTION OF GRAIN BOUNDARY SLIDING INDIFFUSIONAL CREEP

Byung-Nam Kim and Keijiro HiragaNational Research Institute for Metals, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

(Received August 30, 1999)(Accepted in revised form October 29, 1999)

Keywords:Grain boundaries; Diffusion; Mechanical properties

Introduction

Diffusional creep is a deformation due to stress-directed mass transport along grain boundaries and/orthrough grains at low applied stresses. This type of creep is characterized by a linear constitutiveequation between strain rate and applied stress. The constitutive equation was derived by some differentmethods. Earlier studies ignoring an interaction among grains such as grain boundary sliding analyzedthe shape variation of a single spherical grain by lattice (1,2) or by grain boundary diffusion (3,4). Fora more realistic structure consisting of an aggregate of grains, Lifshitz (5) firstly analyzed diffusionalcreep and pointed out that grain boundary sliding is necessary in order for the relative motion ofindividual grains to accommodate the macroscopic deformation. From a geometrical consideration ofgrain boundary sliding, Stevens (6) quantified its contribution to the macroscopic deformation. For anaggregate of space-filling polyhedra, he predicted that grain boundary sliding contributes over 60% ofthe macroscopic strain during diffusional creep. In the present study, an alternative evaluation is givento this issue. For this purpose, first the conventional constitutive equation for a single spherical grainis extended to treat an aggregate of spherical grains. Next, considering the balance between the worksupplied by external stress and that done by diffusive matter during diffusional creep, a constitutiveequation is derived on energetic basis. From these constitutive equations, the contribution of grain-boundary sliding to the macroscopic strain is evaluated for creep deformation due either to grain-boundary diffusion or to lattice diffusion.

Grain Boundary Diffusion

For a homogeneously stressed, elastically isotropic body subjected to infinitesimal strains, Kamb (7) hasshown that the chemical potentialm at a grain boundary depends on the orientation of the boundary, andcan be written as

m 5 m02V[1 1 (s# /B)](sn2s# ) 2 V[s# 1 (s# 2/2B)2(4G)21sij9sij9], (1)

where m0 and V is the chemical potential and the molecular volume respectively, in the reference(stress-free) state,sn is the normal stress at grain boundary,s# is the mean normal stress,s9ij is thedeviatoric stress, B is the elastic bulk modulus and G is the elastic shear modulus. The last term of Eq.(1) is the elastic strain energy of distortion.

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We first consider a diffusional flow at the boundary of a single spherical grain under uniaxial tensionfor which the mass and the volume of the grain remain constant. Under an applied tensile stress ofsalong theu 5 0 (axi-symmetric) axis,sn is s cos2 u whereu is the angle from the tensile axis, as shownin Fig. 1. The initial flux of matter along the spherical grain boundary J9b (molecules per unit time perunit length) is therefore

J9b 52Dbd

VkT

­m

­s, (2)

where Db is the diffusion coefficient of grain boundary,d is the effective thickness of grain boundary,k and T have their usual meanings, and s is the distance from the tensile axis along grain boundary(5Ru). Then, the normal velocity of the grain boundary by surface diffusion can be obtained by

vn52V

x

­(J9bx)

­s5

2DbdVs

kTR2 (3cos2u 2 1), (3)

where R is the radius of the spherical grain. Diffusion on the single isolated grain can be treated as asurface diffusion. Sinces# ands9ij s9ij in Eq. (1) are independent of the location on grain boundary underuniaxial tension ands# ,, B for actual cases, their terms disappear in Eqs. (2) and (3). The maximumstrain rate«b of the spherical grain atu 5 0 is obtained geometrically by

«b 5vn(u 5 0)

R5

4DbdVs

kTR3 . (4)

This approach is almost the same with the Green’s one (4), except that surface diffusion is used forobtaining the shape variation of the spherical grain. The resultant constitutive equation, Eq. (4), isidentical to Eq. (9) of Green (4) under uniaxial tension and to Eq. (13) of Coble (3) except for a factorof 4/7.4 in the case ofs# 5 0. Although Coble (3) assumed that in areas of vacancy source the generationrate is everywhere the same and this is also the case for vacancy annihilation, the present treatment isnot restricted to any particular mechanism of diffusion and requires no assumptions beyond Fick’s firstlaw.

The conventional models for a single spherical grain assume that the maximum strain rate of thegrain (Eq. (4)) is identical to the macroscopic strain rate. This is valid, however, only when thedeformation is completely homogeneous as the case of perfect plasticity, provided that a singledeformation mechanism works. In actual, the normal velocity vn depends onu as given in Eq. (3), andthus the deformation of the grain is inherently heterogeneous. The macroscopic strain should thereforebe evaluated as an average of the individual heterogeneous strains on the whole surface of the grain

Figure 1. Schematic description of a spherical grain stressed in the uniaxial direction.

DIFFUSIONAL CREEP452 Vol. 42, No. 5

boundary. We will calculate next the macroscopic strain rate from Eq. (3) by employing the sametechnique used by Stevens (6).

Consider a flat grain boundary facet of infinitesimal area dS, whose normal vector makes an angleof u against the tensile axis, as shown in Fig. 2. Also consider a plane tangent to the boundary andextending throughout the specimen. If the whole of this plane moved in the normal direction to the planeat a velocity of vn, the strain rate of the specimen would be vn cosu/L, where L is the specimen length.If the displacement occurred on dS, the contribution to the macroscopic strain rate would be vn cos2

udS/V, where V is the volume of the specimen. The macroscopic strain rate can be obtained byintegrating the individual contribution of the boundary facet over the whole of the grain-boundarysurface.

In an aggregate of spherical grains, the normal vector is distributed uniformly in space, so that dS5S sinudu, where S is the total area of grain boundaries in the specimen. Since the grains make contactwith each other, the thickness of grain boundary becomes 2 times the thickness for a single grain (2d).We then obtain the macroscopic strain rate«t as

«t 54SDbdVs

VkTR2 E0

p/2

(3cos2u21)cos2usinudu. (5)

By using Eq. (4) and the relationship of S/V5 1.5/R in an aggregate of spherical grains, the integralof Eq. (5) gives the contribution of the diffusional deformation to the macroscopic strain rate as

«t 52

5«b. (6)

The macroscopic strain rate is smaller than the maximum strain rate of each grain by 0.4 times, whenthe deformation is caused only by grain-boundary diffusion.

Next we examine the relationship of Eq. (6) from an energetic consideration for diffusional creep.For deformation caused by grain-boundary diffusion only, let us consider the balance between theexternal work supplied to the spherical grain and the energy consumed by the flow of diffusional matter.For a grain boundary segment with an infinitesimal length ds on which the normal stress acts (see Fig.1), the work done by grain-boundary movement in the normal direction per unit time dE˙

b is given by

Figure 2. Grain boundary facet within a specimen.

DIFFUSIONAL CREEP 453Vol. 42, No. 5

dEb 5 2pRsinudssn vn 5DbdVs2

kT4p(3cos2u21)sinucos2udu. (7)

Integrating Eq. (7) in the range of 0# u # p, we obtain the external work E˙b done by the grain

boundary per unit time as

Eb532

15

pDbdVs2

kT. (8)

On the other hand, the consumed energy per unit time dE˙fb by the flux of matter at s, where a

chemical potential difference dm exists, is

dEfb 5 Jbdm 5 28pDbdVs2

kTcos2usin3udu. (9)

Integration of Eq. (9) over the whole of the grain boundary (0# u # p) yields the total consumedenergy per grain per unit time E˙

fb, and leads to E˙fb 5 2Eb, that is, the energy consumed by the

diffusional flow of matter is balanced with the external work supplied to the spherical grain. Here,variation in the surface energy due to the shape change of the spherical grain is negligible forinfinitesimal deformation. The differentiation of the grain-surface area by time or strain is zero, sincefor a constant volume of the grain the surface area takes a minimum for a spherical shape.

In this study, we assume that the above diffusional process is quasi-steady state, i.e., that shearstresses along the grain boundaries are fully relaxed by grain-boundary sliding, and that the plating andsliding rates at the grain boundaries are infinitesimal. Although the diffusional flow induces theelongation of grains, we assume that the grains keep their original equiaxed shape during deformationby certain mechanisms, such as faster migration of grain boundaries. For an aggregate of sphericalgrains subjected to uniaxial tension, the total work supplied during deformation per unit time iss«t, andthus the work done per grain per unit time E˙

t is

Et 54

3ps«tR

3. (10)

By using Eq. (4), E˙ b of Eq. (8) for a single grain can be rewritten as

Eb 58

15ps«bR

3. (11)

When deformation occurs purely by grain boundary diffusion without sliding, E˙t of Eq. (10) should be

equal to Eb of Eq. (11). This leads to the same relationship with Eq. (6). Thus, we can again concludethat the macroscopic strain rate is 40% of the maximum strain rate of each grain.

In actual, however, deformation in the aggregate of grains cannot occur by diffusion alone; that is,diffusional creep accompanies grain-boundary sliding, as pointed out by Lifshitz (5) and Stevens (6).Accordingly, the relationship given by Eq. (6) has to be modified. From a geometrical viewpoint, it isrational to assume that for the prohibition of grain-boundary separation the macroscopic strain shouldbe identical to the maximum strain of each grain («t 5 «b). By applying this boundary condition to thepresent model, the strain rate due to grain-boundary sliding«sd becomes 60% of the maximum strainrate of each grain («sd 5 0.6«b). This is well consistent with the analyses of Stevens (6) and Cannon(8).

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Lattice Diffusion

For lattice diffusion, a similar analysis is available. Green (4) has given a complete description of thechemical potentialm for a spherical grain subjected to external stress

m 5 m02V[11(s# /B)](sn2s# )(r2/R2)2V[s# 1(s# 2/2B)2(4G)21s9ijs9ij ], (12)

where r is the distance from the center of grain (x/sinu). Since the flux in the radial direction Jr at thegrain boundary determines the shape variation of the grain, the normal velocity of the grain boundaryunder uniaxial tension is obtained by

vn(r 5 R) 5dr

dt5 VJr 5 2

Dl

kT

­m

­r5

2DlVs

3kTR(3cos2u21), (13)

where Dl is the diffusion coefficient of lattice. Then the maximum strain rate« l of the spherical grainat u 5 0 is obtained geometrically by

« l 5vn(u 5 0)

R5

4DlVs

3kTR2 . (14)

This equation is identical to those of Ruoff (2) and Herring (1) for the special case ofs# 5 0. Calculatingthe macroscopic strain rate by averaging the strain rates of individual boundary facets, we obtain thecontribution of the lattice diffusion to the macroscopic strain rate as

«t 52

5«l. (15)

The ratio of the macroscopic strain rate to the strain rate of each grain for lattice diffusion is thusidentical to that for grain-boundary diffusion: the ratio is 0.4 independently of the diffusion path.

From an energetic consideration similar to that used in the previous section, we then examine therelationship represented by Eq. (15) for lattice diffusion. The external work E˙

l done by the normal stresson the grain boundary per unit time is obtained as

El 532

45

pDlVRs2

kT5

8

15ps«lR

3. (16)

On the other hand, the consumed energy per unit time dE˙fl by the flux of matter in the radial and tangent

directions within the grain can be written by

Ef l 5 216pDlVs2

kTR4 E0

R

r4drE0

p/2

[cos2usin2u1(cos2u21/3)2]sinudu. (17)

The integral of Eq. (17) yields E˙fl 5 2El, that is, the energy consumed by lattice diffusion is balanced

with the external work supplied to the spherical grain. On this basis, since Eqs. (16) and (11) have thesame form, equating Eq. (10) to Eq. (16) yields to Eq. (15). Thus the energetic consideration for latticediffusion also predicts that the macroscopic strain rate is 40% of the maximum strain rate of each grain,when deformation occurs without grain boundary sliding. Additionally, introducing the boundarycondition that the macroscopic strain is identical to the maximum strain of each grain («t 5 «l), weagain obtain the prediction that 60% of the macroscopic strain is contributed by grain boundary sliding.

DIFFUSIONAL CREEP 455Vol. 42, No. 5

Summary and Conclusions

The macroscopic strain rate of an aggregate of spherical grains was evaluated by averaging the strainrates of the individual grains deformed by grain boundary diffusion or lattice diffusion. The resultsshows that independently of the diffusion path the macroscopic strain rate is 40% of the maximum strainrate of each grain consisting the aggregate, provided that deformation occurs only by the diffusionalmechanisms without grain-boundary sliding. This result is supported by an energetic consideration,where the balance was examined between the external work supplied to the aggregate of grains and theenergy consumed by the diffusional flow of matter. When grain-boundary sliding is permitted as anaccommodation process to prohibit grain-boundary separation, the present analysis shows that the strainrate due to grain-boundary sliding becomes 60% of the maximum strain rate of each grain. Althoughspherical grains employed in the present model are not space-filling, the obtained results may representthe essential feature of the contribution of grain-boundary sliding in diffusional creep of polycrystals ofequixed grains.

Acknowledgment

The authors gratefully acknowledge helpful discussion with Dr. H. Yoshinaga.

References

1. C. Herring, J. Appl. Phys. 21, 437 (1950).2. A. L. Ruoff, J. Appl. Phys. 36, 2903 (1965).3. R. L. Coble, J. Appl. Phys. 34, 1679 (1963).4. H. W. Green, J. Appl. Phys. 41, 3899 (1970).5. I. M. Lifshitz, Soviet Phys. JETP. 17, 909 (1963).6. R. N. Stevens, Phil. Mag. 23, 265 (1971).7. W. B. Kamb, J. Geol. 67, 153 (1959).8. W. R. Cannon, Phil. Mag. 25, 1489 (1972).

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