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The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity
Tectonophysics, 28 (1975) 265-274 0 Elsevier Scientific Publishing Company,
Amsterdam - Printed in The Netherlands
THE CHANGE OF SHAPE OF A VISCOUS ELLIPSOIDAL REGION EMBEDDED IN A SLOWLY DEFORMING MATRIX HAVING A DIFFERENT VISCOSITY
B.A. BILBY, Department
(Submitted
J.D. ESHELBY of the Theory
and A.K. KUNDU* of Materials,
University
of Sheffield,
Sheffield
(Great
Britain)
December 10, 1974; revised version accepted May 30, 1975)
ABSTRACT Bilby, B.A., Eshelby, J.D. and Kundu, A.K., 1975. The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics, 28: 265-274. The general theory of the elastic fields round ellipsoidal inclusions and inhomogeneities is applied to sblve the problem of the slow deformation of a viscous material containing an ellipsoidal inhomogeneity of different viscosity. The theory is used to calculate the twodimensional finite strain of an elliptic cylinder and of prolate and oblate spheroids under applied pure-strain rates, and attention is drawn to applications in the fields of rock deformation and in the mixing and homogenization of viscous liquids.
INTRODUCTION
It has been pointed out (Eshelby, 1957, 1959) that the general theory of the elastic fields round ellipsoidal inclusions and inhomogeneities may also be used to solve the problem of the deformation of a viscous material containing an ellipsoidal inhomogeneity of different viscosity. This problem is of interest in connection with the theory of the deformation of rocks (Ramsay, 1967; Gay, 1968a, b; Dunnet, 1969; Jaeger, 1969, chapter 5; Wood, 1973; Tan, 1974) and the theory of mixing and homogenization of viscous liquids (Cable, 1968) and so in the present paper we develop the theory of the viscous inhomogeneity in more detail. We give the name ‘inclusion’ to a region bounded by a surface S in an elastic solid which has undergone a change of shape and size, which, without the constraint of the surrounding matrix would be an arbitrary homogeneous strain e; Because of the constraint the inclusion and the material around it
* A.K. Kundu is now at Berhampur College of Technology,
West Bengal, India.
266
are thrown into a state of stress. The resulting elastic field can be found as follows (Eshelby, 1957, 1959; to be referred to as I and II). Cut round the region bounded by S which is to transform and remove it from the matrix. Allow the unconstrained transformation to take place and then restore the region to its original size and shape by applying to it the surface tractions -pFnj, where: pc = heTSij + 2peT
(1)
are the stresses calculated from eT by Hooke’s law. We here assume that the material is isotropic with Lame constants X and 1-1;eT is the dilatation. Now replace the region in the matrix and reweld at the surface S; the strain in the inclusion and matrix is everywhere zero, and there is a distribution of body force -p$nj over S. Finally, to cancel this, apply a distribution of body force +pFnj over S, and let this produce the displacement UC everywhere. We can write down UC in terms of the elastic Green’s function Uq(r - r’), which is the elastic displacement in the i-direction at the point r (coordinates 3~~)produced by a unit force applied in the j-direction at the point r’ (coordinates 3~~‘). Uij(r - r’) depends on the modulus Ir - r’l of r - r’ and is given by (Love, 1927, p. 185): Uij(r -
r’)
=
1 4nplr - r’l - 16~/~(1&ij
a2 (r - r’l V) aX,aXj
Here v is Poisson’s ratio, and 6, = 1 if i = j and is zero otherwise. U,(r - r’) we have: UC =
J
Uij(r
-
(2) In terms of
r’)p$z,dS
(3)
S
It is shown in I that if the surface S is an ellipsoid the displacement UC is linear in x1, x2, x3 inside S, so that the strain and stress components in the inclusion, as constrained by the matrix, are uniform. The strain may be written in the form: eC = s..zlkl eT kl u
(4)
The constant coefficients sjj& are worked out in I; they depend only on Poisson’s ratio and the axial ratios b/u, c/a of the ellipsoidal inclusion, which we assume to have semi-axes a, b, c. If pg denotes the stress derived from ec by Hooke’s law the (non-uniform) stress in the matrix is pi, but the uniform stress in the inclusion is pg - pr, because there is no stress associated with the transformation strain e:, This useful fact enables us to determine the perturbation caused in a uniform stress field by an ellipsoidal region of differing elastic constants, an ellipsoidal inhomogeneity in the language of I. On the field of the inclusion constrained in the matrix superpose a u.niform field et. It is useful to introduce
267
the deviatoric parts ‘eij = eij - ie6ij, ‘Pij = pij - ip”ij of the strain and stress respectively, where P = - $p = - $pii is the hydrostatic pressure. The stressstrain relations then become: ‘pij = 2/.L’eij
p = 3Ke,
(5)
where K = X + :/..I is the bulk modulus. The strain within the ellipsoid is now given by the constant values ec + eA, ‘eg + ‘e$ while the stress in it is p = 3fc(eC + eA - eT), ‘pij = 2p(‘e$ + ‘et - ‘ez). Thus we could replace the material within S by an ellipsoid of elastic constants fcl, p, , while maintaining continuity of tractions and displacements across S provided that: Kl(ec+eA)=z(eC+eA-eT)
(6)
pl(‘e$ + ‘et) = p(‘e$ + ‘ef - ‘e$)
(7)
The e$ can be eliminated from eqs. 6 and 7 with the help of the S,,, eq. 4. Then, for given et, K~, pl, K, p, the quantities ez can be calculated from eqs. 6 and 7 and the field of the inhomogeneity evaluated with the help of eq. 3. In the sequel we consider only incompressible deformations and flows so that we no longer need to distinguish between eij and its deviatoric part ‘e+
THE VISCOUS INHO~O~ENEITY
We can pass to the case of an incompressible elastic medium by letting Poisson’s ratio tend to f so that K --f m and e + 0 in such a way that the hydrostatic pressure P = -Ke remains finite. The equilibrium equation, which may be written: /lV2Uf--
3 2(1+&-
aP
-0
becomes: PV 'l&i= aP/aXi
which must be supplemented aUi/aXi=
condition:
0
The Green’s function Uij(r--r')
by the incompressibility
=
(eq. 2) becomes:
6ij
4n-plr - r’l
--
I a2 -lr-r’l 8W axiaxj
To get the associated pressure Pj we may calculate associated with eq. 2 and then put v =-$ to get:
(9) the hydrostatic
pressure
Alternatively, if eq. 9 is inserted in eq. 8 it is clear that the associated pressure is equal to eq. 10 plus an arbitrary uniform pressure which may be taken to be zero. Any solution ui of eq. 8 can be interpreted as the velocity associated with a slow viscous flow in a medium of viscosity p. The stresses are:
becomes the rate of strain tensor and eq. 8 becomes the Navier-Stokes equation for slow viscous flow in which inertia terms are negligible. In the viscous context, the field defined by eqs. 9 and 10 is sometimes referred to as a Stokeslet (Batchelor, 1970, p. 240). In particular, the solution for an ellipsoidal elastic inhomogeneity in a uniform strain field et becomes the solution for an ellipsoidal viscous inhomogeneity in a flow described by the rate of strain tensor e$ . We may, if we wish, at the expense of a little artificiality, transcribe the physical argument which led to eq. 3. The interior of an ellipsoidal region in a viscous medium is removed from the surrounding material and is found to be undergoing a spontaneous change of form (due, perhaps, to the progress of some imagin~y phase change) specified by the rate of strain tensor e$‘. This deformation is cancelled by applying tractions -pznj which induce an equal and opposite viscous flow and the ellipsoid is returned to the matrix and joined to it. The layer of body force thus built in is relaxed and a uniform rate of strain e$ is superimposed overall. The (uniform) rate of strain and the (uniform) stress in the ellipsoid are then connected by eq. 7 with /J replaced by some other value E_C~ for the viscosity. Less intuitively we may simply start with the flow (eq. 3) representing a certain distribution of body force over S together with a uniform dilatationless flow et and consider the surface traction on and the deformation of the ellipsoidal surface immediately outside the layer of body force. From the detailed results given in I and II it follows that the traction and defo~ation are consistent with the presence within the surface of material with a viscosity p1 which is related to @ and the shape of the ellipsoid by eq. ‘7. To discuss the finite deformation of the viscous inhomogeneity we proceed as follows. For a given deviatoric strain rate et applied at infinity, eqs. 4 and 7 enable ec to be found in terms of e$. Then the total strain rate within the ellipsoid, which is, by eq. 7: eij
(11) can also be found in terms of e$, In eq. 8 and in eq. 12 below we have dropped the dash since for each term emm = 0 and so ‘eii = eii. Since ep is uniform
within S, the ellipsoid always remains an ellipsoid and the analysis can be applied at every instant of the deformation. Integration of the equations for efc thus enables the finite deformation of the inhomogeneity to be followed. This integration is considerably simplified if et is a pure strain rate whose principal axes coincide initially with those of the ellipsoid, for obviously they will then continue to do so. In the examples below we confine ourselves to applied strain rates of this type. Then in combining eqs. 4 and 7 to determine elT in terms of et we have only to deal with the three equations: (/Jr -
P)Sijh&
+
r.l$=
(12)
(PLkk$
where ij = 11, 22, 33. (The incompressible limit of the remaining eq. 6 adds no new information about the change of shape of the ellipsoid and we may ignore it.) The necessary coefficients are given (I 3.7 with u = +) by: S 1111=
1 -
&122=
W2/4M,,
(3a2/470(L
+ LX)
and their cyclic counterparts,
where:
lab = (Ib - 1,)/3(a2 - b2) and I,, I*, I, are the constant appear in the expression: cp= +(a” -x2)1,
coefficients
+ +(b2 -y2)Ib
for the Newtonian
potential
THE DEFORMATION
(demagnetizing
factors)
which
+ +(c” -z2)Ic
inside a solid ellipsoid of unit density.
OF AN ELLIPTIC
CYLINDER
We take first an elliptic cylinder of viscosity /..fr with axis parallel to 3ta and cross section ~r~/a~ + xz2/b2 = 1 and subject the medium (of viscosity /J) remote from it to a deviatoric pure strain rate with principal axes parallel to those of the ellipse, so that e& = -ei2 > 0 and e& = 0 (Fig. 1). Putting (a + b)-l = k we have the results (see I 3.17), I, = 4nbk, I, = 4mk, I, = 0, 3I,, = 4rk2, Ia, = 47713~~-lab, I,, = 4n/3b2 -lab, and also, as c + 00, the limiting values 3c21,, = I,, 3c21,, = Ib, c21cc = 0. From these we obtain S 1111= 1 -- a2k2, S1122 = b2k2, S,,,, = bk, Sssll = 0, Sass3 = 0. The other nonvanishing components follow from these by simultaneous interchange of (1,2) and (a, b). For these values of Sijkl, eqs. 12 give e&, = 0, e& = -eg2 and, with eq. 11: inc. =_
cl1
inc = 1 da _
p(a2 + a02)2e;41
e22
a dt
2&z&L~
- p) + p(a2 +- ao2)2
270
Fig. 1. Deformation of elliptic cylinder by a deviatoric pure strain-rate with principal axes parallel to those of the elliptical cross-section.
where we have set the constant area of the ellipse nab equal to RQ-,~. Integrating with the boundary condition that t = 0 when a = b = a0 we have:
l =efl t
In (alao) + -$a (u’uo)2 (4ao)2 + 1
where (Y= (pl - I_c)//.I.In terms of the so-called natural strain 5’ = In (alao) of the elliptical inhomogeneity and the natural strain SH = [ln(u/uo)lH = ef,t applied at infinity, and which would, of course, prevail within the ellipse in the homogeneous case (Y= 0 we have (Cable, 1968): S + +ff tanh S = S, = efr t
(13)
Figure 2 shows the general form of this equation, where the curves are labelled not by (Ybut by R = pi/p = 1 + 01.R = 1 corresponds to the homogeneous case where S = S, and R = 0 to the deformation of an elliptical hole. Increas-
Fig. 2. Total strain S = ln(a/a,-J in the elliptic cylinder versus applied total strain SH. The curves are labelled with the viscosity ratio R = pi/p= 1 + a.
271
ing values of R represent inhomogeneities of increasing viscosity; R = 00 corresponds to a perfectly rigid inclusion. For large S, the curves all become parallel to the homogeneous line R = 1, but they are translated along the S,-axis. Thus the stiff inhomogeneity ultimately reaches the same strain as the weak one but after a greater value of the applied strain SH. Ultimately the increment in S equals the increment in SH, but the stiff inhomogeneity suffers a smaller strain (and the weak one a larger) when the deformation is in its early stages. THE DEFORMATION
OF PROLATE
AND OBLATE
SPHEROIDS
We next take a prolate spheroid of viscosity pl with semi-axes a > b = c and subject the medium (of viscosity p) remote from it to an applied deviatoric pure strain-rate with principal axes parallel to those of the spheroid, chosen so that the l-axis of the strain-rate is parallel to the a-axis of the spheroid, with ef, = -2e& = -2e&. If efl is positive we expect the spheroid to become increasingly prolate; if efl is negative we expect the spheroid to tend to a spherical shape and then to become an oblate spheroid if the deformation is continued. For the prolate spheroid with a > b = c we have (I 3.16): 1, = I, = 2nu(u2 - 1)-3’2[u(u2
- 1)1’2 - cash-‘u]
where u = a/b, while the quantities
(14)
Sijhl may be expressed
in terms of:
A = (31, - 4n)/47r(a2 - b2) as follows; s, 111 = 1 - 2ha2, S2222 = S,,,, = z(l - Ab2), S1122 = S,,,, = Ab2, S 2211 = &air = Aa2, S’s233 = S’s322 = t (1 - Ab2). Eqs. 12 now show,
as we might have anticipated, that eT1 = -2eg2 = -2eT3. are solved for eT1 and e?; calculated by eq. 11 we get:
e~f=ld”= kh(a 2 a dt
b2)[a {12na2
- 31,(2a2 +
When the equations
b2.)} + 47r(a2 - b2)]-l
The volume 4rab2/3 of the spheroid is constant and we put ab2 = ao3. Introducing the variable (a/~,)~‘~ = w, writing b2-= ao3/a and integrating we get: w cash-lw
+ In w213 = eflt + const.
(w2 -1)3’s We put t = 0 when w = 1 so that at this time we have a sphere of radius ao. Taking the limit when w + 1 shows the constant to be -5ar/6 so that finally we have, with w = (~/a~)~‘~: a[(1/3)
+ (w2 - l)-l{l-
(w2 - 1)-1’2w
cash-‘w}]
= S,-S
(15)
Fig. 3. Total strain S = ln(a/a,-J in the prolate and oblate spheroids versus applied strain Sff. The curves are labelled with the viscosity ratio R = pi/p = 1 + (Y.
total
where we have again introduced the natural strain S = ln(a/ae) of the inhomogeneity and the natural strain S, = e&t applied at infinity, which is the value of S when cy = 0. For the oblate spheroid with b = c > a, the expression for 1, = Ib is again given by eq. 14, but with u < 1; it is still real because the factors inside and outside the square brackets are both imaginary. A more natural way of writing this expression is then obtained by putting cash-‘u = sinh-‘(v2 - l)r’2 and noting that when u < 1, sinh-“(u2 - 1)1’2 = sinh-‘i(l - v~)*‘~ = i sin-l(l - u2)1/2 = i cos-lu. Thus eq. 14 becomes for b = c > a (see I): 1, = 1, = 2nv(l-
u2)-a’2~cos-1u
- o(l - u2)l~2]
(16)
for 21= c > a (I 3.15). Eq. 15 therefore suffices also to describe the transition of the sphere to an ablate spheroid as w becomes less than 1 and S negative. Once again, we put cash-lw = sinhM1(w2 - 1)“’ and so obtain the more natural form : ~[(1/3)-(l-W2)-1{1-(1-Lu2)--1’2WCOS-1W~]=SH-S
(17)
as the equation describing the development of the oblate spheroid (w < 1) from the sphere (W = 1). Eq. 17 may be obtained directly, of course, by repeating the above analysis using eq. 16 instead of eq. 14. Figure 3 shows S as a function of S, for various values of cy from eq. 15 when w > 1 and S, S, > 0, and eq. 17, when w < 1 and S, SN < 0.
273 DXSCUSSION
The only previous work on the finite deformation of a viscous inhomogeneity seems to be Gay’s (1968) treatment of the problem we have dealt with in the section ‘The deformation of an elliptic cylinder’. He finds, in our notation: S = Sl$ J(1 + @)
(18)
which differs from eq. 13 in being linear in S. Even when S is small enough for eq. 13 to become:
S=S,/(l-+)
119)
eq. 18 does not agree with it. There seem to be several errors in Gay’s anaiysis. Suppose first that S is small, so that the cylindrical inhomogeneity is nearly circular. Gay uses a general three-dimensional solution given by Lamb (1932, p. 595) and used by Taylor (1932) to treat the initial deformation of a spherical viscous inhomogeneity. To find the corresponding two-Dimensions solution he simply puts z = 0 in Lamb’s formulas. Though this reproduces the genera1 shape of the formulas the numerical coefficients are incorrect. Likewise, to get the harmonic functions to insert in the formula he puts x = 0 in their three-dimensional counterp~ts, a process which does not generally generate a twodimensional harmonic function. When these shortcomings are corrected eq. 18 becomes eq. 19. To treat the general case Gay in effect maps the ellipse back on to the circle and thence deduces that eq. 18 is valid for all S. (With the correction just indicated he would have found instead that eq. 19 is valid for finite S.) It does not seem to be possible to justify this mapping process. Gay compares his formula with some experimental results but, because of their scatter and the limited range of SN and 01which they cover, it is hardly possible to decide whether they favour any one of eqs. 13, 18 or 19 rather than another. The results we have obtained have interesting implications for the mixing and homogenizing of viscous liquids, and a preliminary discussion of some of them has already been given by Cable (1968). In mixing theory the important question is how fast is an inhomogeneity stretched and thinned, and so in the present paper we have merely calculated the rate of deformation of the ellipsoid. However, when the appropriate e; have been obtained, as explained in the second section of this paper, the entire flow inside and outside the inhomogeneity can be found from the general theory presented in I and II.
274 REFERENCES Batchelor, G.K., 1970. Introduction to Fluid Dynamics. Cambridge University Press, 615 PP. Cable, M., 1968. The physical chemistry of glassmaking. Proc. Int. Congr. on Glass, London. Society of Glass Technology, pp. 163-178. Dunnet, D., 1969. A technique of finite-strain analysis using elliptical particles. Tectonophysics, 7: 117-136. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Sot. London, Ser. A, 241: 376-396. Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proc. R. Sot. London, Ser. A, ‘252: 561-569. Gay, N.C., 1968a. Pure-shear and simple-shear deformation of inhomogeneous viscous fluids. 1. Theory. Tectonophysics, 5: 211-234. Gay, N.C., 1968b. Pure-shear and simple-shear deformation of inhomogeneous viscous fluids. 2. The determination of the total finite strain in a rock from objects such as deformed pebbles. Tectonophysics, 5: 295-302. Jaeger, J.C., 1969. Elasticity, Fracture and Flow. Methuen, London, 263 pp. Lamb, H., 1932. Hydrodynamics. Cambridge University Press, 738 pp. Love, A.E.H., 1927. A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, 643 pp. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, 567 pp. Tan, B.K., 1974. Deformation of particles developed around rigid and deformable nuclei. Tectonophysics, 24: 243-257. Taylor, G.I., 1932. The viscosity of a fluid containing small drops of another fluid. Proc. R. Sot. London, Ser. A, 138: 41-48, Wood, D.S., 1973. Patterns and magnitude of natural strain in rocks. Philos. Trans. R. Sot. Lond., Ser. A, 274: 373-382.
Amsterdam - Printed in The Netherlands
THE CHANGE OF SHAPE OF A VISCOUS ELLIPSOIDAL REGION EMBEDDED IN A SLOWLY DEFORMING MATRIX HAVING A DIFFERENT VISCOSITY
B.A. BILBY, Department
(Submitted
J.D. ESHELBY of the Theory
and A.K. KUNDU* of Materials,
University
of Sheffield,
Sheffield
(Great
Britain)
December 10, 1974; revised version accepted May 30, 1975)
ABSTRACT Bilby, B.A., Eshelby, J.D. and Kundu, A.K., 1975. The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics, 28: 265-274. The general theory of the elastic fields round ellipsoidal inclusions and inhomogeneities is applied to sblve the problem of the slow deformation of a viscous material containing an ellipsoidal inhomogeneity of different viscosity. The theory is used to calculate the twodimensional finite strain of an elliptic cylinder and of prolate and oblate spheroids under applied pure-strain rates, and attention is drawn to applications in the fields of rock deformation and in the mixing and homogenization of viscous liquids.
INTRODUCTION
It has been pointed out (Eshelby, 1957, 1959) that the general theory of the elastic fields round ellipsoidal inclusions and inhomogeneities may also be used to solve the problem of the deformation of a viscous material containing an ellipsoidal inhomogeneity of different viscosity. This problem is of interest in connection with the theory of the deformation of rocks (Ramsay, 1967; Gay, 1968a, b; Dunnet, 1969; Jaeger, 1969, chapter 5; Wood, 1973; Tan, 1974) and the theory of mixing and homogenization of viscous liquids (Cable, 1968) and so in the present paper we develop the theory of the viscous inhomogeneity in more detail. We give the name ‘inclusion’ to a region bounded by a surface S in an elastic solid which has undergone a change of shape and size, which, without the constraint of the surrounding matrix would be an arbitrary homogeneous strain e; Because of the constraint the inclusion and the material around it
* A.K. Kundu is now at Berhampur College of Technology,
West Bengal, India.
266
are thrown into a state of stress. The resulting elastic field can be found as follows (Eshelby, 1957, 1959; to be referred to as I and II). Cut round the region bounded by S which is to transform and remove it from the matrix. Allow the unconstrained transformation to take place and then restore the region to its original size and shape by applying to it the surface tractions -pFnj, where: pc = heTSij + 2peT
(1)
are the stresses calculated from eT by Hooke’s law. We here assume that the material is isotropic with Lame constants X and 1-1;eT is the dilatation. Now replace the region in the matrix and reweld at the surface S; the strain in the inclusion and matrix is everywhere zero, and there is a distribution of body force -p$nj over S. Finally, to cancel this, apply a distribution of body force +pFnj over S, and let this produce the displacement UC everywhere. We can write down UC in terms of the elastic Green’s function Uq(r - r’), which is the elastic displacement in the i-direction at the point r (coordinates 3~~)produced by a unit force applied in the j-direction at the point r’ (coordinates 3~~‘). Uij(r - r’) depends on the modulus Ir - r’l of r - r’ and is given by (Love, 1927, p. 185): Uij(r -
r’)
=
1 4nplr - r’l - 16~/~(1&ij
a2 (r - r’l V) aX,aXj
Here v is Poisson’s ratio, and 6, = 1 if i = j and is zero otherwise. U,(r - r’) we have: UC =
J
Uij(r
-
(2) In terms of
r’)p$z,dS
(3)
S
It is shown in I that if the surface S is an ellipsoid the displacement UC is linear in x1, x2, x3 inside S, so that the strain and stress components in the inclusion, as constrained by the matrix, are uniform. The strain may be written in the form: eC = s..zlkl eT kl u
(4)
The constant coefficients sjj& are worked out in I; they depend only on Poisson’s ratio and the axial ratios b/u, c/a of the ellipsoidal inclusion, which we assume to have semi-axes a, b, c. If pg denotes the stress derived from ec by Hooke’s law the (non-uniform) stress in the matrix is pi, but the uniform stress in the inclusion is pg - pr, because there is no stress associated with the transformation strain e:, This useful fact enables us to determine the perturbation caused in a uniform stress field by an ellipsoidal region of differing elastic constants, an ellipsoidal inhomogeneity in the language of I. On the field of the inclusion constrained in the matrix superpose a u.niform field et. It is useful to introduce
267
the deviatoric parts ‘eij = eij - ie6ij, ‘Pij = pij - ip”ij of the strain and stress respectively, where P = - $p = - $pii is the hydrostatic pressure. The stressstrain relations then become: ‘pij = 2/.L’eij
p = 3Ke,
(5)
where K = X + :/..I is the bulk modulus. The strain within the ellipsoid is now given by the constant values ec + eA, ‘eg + ‘e$ while the stress in it is p = 3fc(eC + eA - eT), ‘pij = 2p(‘e$ + ‘et - ‘ez). Thus we could replace the material within S by an ellipsoid of elastic constants fcl, p, , while maintaining continuity of tractions and displacements across S provided that: Kl(ec+eA)=z(eC+eA-eT)
(6)
pl(‘e$ + ‘et) = p(‘e$ + ‘ef - ‘e$)
(7)
The e$ can be eliminated from eqs. 6 and 7 with the help of the S,,, eq. 4. Then, for given et, K~, pl, K, p, the quantities ez can be calculated from eqs. 6 and 7 and the field of the inhomogeneity evaluated with the help of eq. 3. In the sequel we consider only incompressible deformations and flows so that we no longer need to distinguish between eij and its deviatoric part ‘e+
THE VISCOUS INHO~O~ENEITY
We can pass to the case of an incompressible elastic medium by letting Poisson’s ratio tend to f so that K --f m and e + 0 in such a way that the hydrostatic pressure P = -Ke remains finite. The equilibrium equation, which may be written: /lV2Uf--
3 2(1+&-
aP
-0
becomes: PV 'l&i= aP/aXi
which must be supplemented aUi/aXi=
condition:
0
The Green’s function Uij(r--r')
by the incompressibility
=
(eq. 2) becomes:
6ij
4n-plr - r’l
--
I a2 -lr-r’l 8W axiaxj
To get the associated pressure Pj we may calculate associated with eq. 2 and then put v =-$ to get:
(9) the hydrostatic
pressure
Alternatively, if eq. 9 is inserted in eq. 8 it is clear that the associated pressure is equal to eq. 10 plus an arbitrary uniform pressure which may be taken to be zero. Any solution ui of eq. 8 can be interpreted as the velocity associated with a slow viscous flow in a medium of viscosity p. The stresses are:
becomes the rate of strain tensor and eq. 8 becomes the Navier-Stokes equation for slow viscous flow in which inertia terms are negligible. In the viscous context, the field defined by eqs. 9 and 10 is sometimes referred to as a Stokeslet (Batchelor, 1970, p. 240). In particular, the solution for an ellipsoidal elastic inhomogeneity in a uniform strain field et becomes the solution for an ellipsoidal viscous inhomogeneity in a flow described by the rate of strain tensor e$ . We may, if we wish, at the expense of a little artificiality, transcribe the physical argument which led to eq. 3. The interior of an ellipsoidal region in a viscous medium is removed from the surrounding material and is found to be undergoing a spontaneous change of form (due, perhaps, to the progress of some imagin~y phase change) specified by the rate of strain tensor e$‘. This deformation is cancelled by applying tractions -pznj which induce an equal and opposite viscous flow and the ellipsoid is returned to the matrix and joined to it. The layer of body force thus built in is relaxed and a uniform rate of strain e$ is superimposed overall. The (uniform) rate of strain and the (uniform) stress in the ellipsoid are then connected by eq. 7 with /J replaced by some other value E_C~ for the viscosity. Less intuitively we may simply start with the flow (eq. 3) representing a certain distribution of body force over S together with a uniform dilatationless flow et and consider the surface traction on and the deformation of the ellipsoidal surface immediately outside the layer of body force. From the detailed results given in I and II it follows that the traction and defo~ation are consistent with the presence within the surface of material with a viscosity p1 which is related to @ and the shape of the ellipsoid by eq. ‘7. To discuss the finite deformation of the viscous inhomogeneity we proceed as follows. For a given deviatoric strain rate et applied at infinity, eqs. 4 and 7 enable ec to be found in terms of e$. Then the total strain rate within the ellipsoid, which is, by eq. 7: eij
(11) can also be found in terms of e$, In eq. 8 and in eq. 12 below we have dropped the dash since for each term emm = 0 and so ‘eii = eii. Since ep is uniform
within S, the ellipsoid always remains an ellipsoid and the analysis can be applied at every instant of the deformation. Integration of the equations for efc thus enables the finite deformation of the inhomogeneity to be followed. This integration is considerably simplified if et is a pure strain rate whose principal axes coincide initially with those of the ellipsoid, for obviously they will then continue to do so. In the examples below we confine ourselves to applied strain rates of this type. Then in combining eqs. 4 and 7 to determine elT in terms of et we have only to deal with the three equations: (/Jr -
P)Sijh&
+
r.l$=
(12)
(PLkk$
where ij = 11, 22, 33. (The incompressible limit of the remaining eq. 6 adds no new information about the change of shape of the ellipsoid and we may ignore it.) The necessary coefficients are given (I 3.7 with u = +) by: S 1111=
1 -
&122=
W2/4M,,
(3a2/470(L
+ LX)
and their cyclic counterparts,
where:
lab = (Ib - 1,)/3(a2 - b2) and I,, I*, I, are the constant appear in the expression: cp= +(a” -x2)1,
coefficients
+ +(b2 -y2)Ib
for the Newtonian
potential
THE DEFORMATION
(demagnetizing
factors)
which
+ +(c” -z2)Ic
inside a solid ellipsoid of unit density.
OF AN ELLIPTIC
CYLINDER
We take first an elliptic cylinder of viscosity /..fr with axis parallel to 3ta and cross section ~r~/a~ + xz2/b2 = 1 and subject the medium (of viscosity /J) remote from it to a deviatoric pure strain rate with principal axes parallel to those of the ellipse, so that e& = -ei2 > 0 and e& = 0 (Fig. 1). Putting (a + b)-l = k we have the results (see I 3.17), I, = 4nbk, I, = 4mk, I, = 0, 3I,, = 4rk2, Ia, = 47713~~-lab, I,, = 4n/3b2 -lab, and also, as c + 00, the limiting values 3c21,, = I,, 3c21,, = Ib, c21cc = 0. From these we obtain S 1111= 1 -- a2k2, S1122 = b2k2, S,,,, = bk, Sssll = 0, Sass3 = 0. The other nonvanishing components follow from these by simultaneous interchange of (1,2) and (a, b). For these values of Sijkl, eqs. 12 give e&, = 0, e& = -eg2 and, with eq. 11: inc. =_
cl1
inc = 1 da _
p(a2 + a02)2e;41
e22
a dt
2&z&L~
- p) + p(a2 +- ao2)2
270
Fig. 1. Deformation of elliptic cylinder by a deviatoric pure strain-rate with principal axes parallel to those of the elliptical cross-section.
where we have set the constant area of the ellipse nab equal to RQ-,~. Integrating with the boundary condition that t = 0 when a = b = a0 we have:
l =efl t
In (alao) + -$a (u’uo)2 (4ao)2 + 1
where (Y= (pl - I_c)//.I.In terms of the so-called natural strain 5’ = In (alao) of the elliptical inhomogeneity and the natural strain SH = [ln(u/uo)lH = ef,t applied at infinity, and which would, of course, prevail within the ellipse in the homogeneous case (Y= 0 we have (Cable, 1968): S + +ff tanh S = S, = efr t
(13)
Figure 2 shows the general form of this equation, where the curves are labelled not by (Ybut by R = pi/p = 1 + 01.R = 1 corresponds to the homogeneous case where S = S, and R = 0 to the deformation of an elliptical hole. Increas-
Fig. 2. Total strain S = ln(a/a,-J in the elliptic cylinder versus applied total strain SH. The curves are labelled with the viscosity ratio R = pi/p= 1 + a.
271
ing values of R represent inhomogeneities of increasing viscosity; R = 00 corresponds to a perfectly rigid inclusion. For large S, the curves all become parallel to the homogeneous line R = 1, but they are translated along the S,-axis. Thus the stiff inhomogeneity ultimately reaches the same strain as the weak one but after a greater value of the applied strain SH. Ultimately the increment in S equals the increment in SH, but the stiff inhomogeneity suffers a smaller strain (and the weak one a larger) when the deformation is in its early stages. THE DEFORMATION
OF PROLATE
AND OBLATE
SPHEROIDS
We next take a prolate spheroid of viscosity pl with semi-axes a > b = c and subject the medium (of viscosity p) remote from it to an applied deviatoric pure strain-rate with principal axes parallel to those of the spheroid, chosen so that the l-axis of the strain-rate is parallel to the a-axis of the spheroid, with ef, = -2e& = -2e&. If efl is positive we expect the spheroid to become increasingly prolate; if efl is negative we expect the spheroid to tend to a spherical shape and then to become an oblate spheroid if the deformation is continued. For the prolate spheroid with a > b = c we have (I 3.16): 1, = I, = 2nu(u2 - 1)-3’2[u(u2
- 1)1’2 - cash-‘u]
where u = a/b, while the quantities
(14)
Sijhl may be expressed
in terms of:
A = (31, - 4n)/47r(a2 - b2) as follows; s, 111 = 1 - 2ha2, S2222 = S,,,, = z(l - Ab2), S1122 = S,,,, = Ab2, S 2211 = &air = Aa2, S’s233 = S’s322 = t (1 - Ab2). Eqs. 12 now show,
as we might have anticipated, that eT1 = -2eg2 = -2eT3. are solved for eT1 and e?; calculated by eq. 11 we get:
e~f=ld”= kh(a 2 a dt
b2)[a {12na2
- 31,(2a2 +
When the equations
b2.)} + 47r(a2 - b2)]-l
The volume 4rab2/3 of the spheroid is constant and we put ab2 = ao3. Introducing the variable (a/~,)~‘~ = w, writing b2-= ao3/a and integrating we get: w cash-lw
+ In w213 = eflt + const.
(w2 -1)3’s We put t = 0 when w = 1 so that at this time we have a sphere of radius ao. Taking the limit when w + 1 shows the constant to be -5ar/6 so that finally we have, with w = (~/a~)~‘~: a[(1/3)
+ (w2 - l)-l{l-
(w2 - 1)-1’2w
cash-‘w}]
= S,-S
(15)
Fig. 3. Total strain S = ln(a/a,-J in the prolate and oblate spheroids versus applied strain Sff. The curves are labelled with the viscosity ratio R = pi/p = 1 + (Y.
total
where we have again introduced the natural strain S = ln(a/ae) of the inhomogeneity and the natural strain S, = e&t applied at infinity, which is the value of S when cy = 0. For the oblate spheroid with b = c > a, the expression for 1, = Ib is again given by eq. 14, but with u < 1; it is still real because the factors inside and outside the square brackets are both imaginary. A more natural way of writing this expression is then obtained by putting cash-‘u = sinh-‘(v2 - l)r’2 and noting that when u < 1, sinh-“(u2 - 1)1’2 = sinh-‘i(l - v~)*‘~ = i sin-l(l - u2)1/2 = i cos-lu. Thus eq. 14 becomes for b = c > a (see I): 1, = 1, = 2nv(l-
u2)-a’2~cos-1u
- o(l - u2)l~2]
(16)
for 21= c > a (I 3.15). Eq. 15 therefore suffices also to describe the transition of the sphere to an ablate spheroid as w becomes less than 1 and S negative. Once again, we put cash-lw = sinhM1(w2 - 1)“’ and so obtain the more natural form : ~[(1/3)-(l-W2)-1{1-(1-Lu2)--1’2WCOS-1W~]=SH-S
(17)
as the equation describing the development of the oblate spheroid (w < 1) from the sphere (W = 1). Eq. 17 may be obtained directly, of course, by repeating the above analysis using eq. 16 instead of eq. 14. Figure 3 shows S as a function of S, for various values of cy from eq. 15 when w > 1 and S, S, > 0, and eq. 17, when w < 1 and S, SN < 0.
273 DXSCUSSION
The only previous work on the finite deformation of a viscous inhomogeneity seems to be Gay’s (1968) treatment of the problem we have dealt with in the section ‘The deformation of an elliptic cylinder’. He finds, in our notation: S = Sl$ J(1 + @)
(18)
which differs from eq. 13 in being linear in S. Even when S is small enough for eq. 13 to become:
S=S,/(l-+)
119)
eq. 18 does not agree with it. There seem to be several errors in Gay’s anaiysis. Suppose first that S is small, so that the cylindrical inhomogeneity is nearly circular. Gay uses a general three-dimensional solution given by Lamb (1932, p. 595) and used by Taylor (1932) to treat the initial deformation of a spherical viscous inhomogeneity. To find the corresponding two-Dimensions solution he simply puts z = 0 in Lamb’s formulas. Though this reproduces the genera1 shape of the formulas the numerical coefficients are incorrect. Likewise, to get the harmonic functions to insert in the formula he puts x = 0 in their three-dimensional counterp~ts, a process which does not generally generate a twodimensional harmonic function. When these shortcomings are corrected eq. 18 becomes eq. 19. To treat the general case Gay in effect maps the ellipse back on to the circle and thence deduces that eq. 18 is valid for all S. (With the correction just indicated he would have found instead that eq. 19 is valid for finite S.) It does not seem to be possible to justify this mapping process. Gay compares his formula with some experimental results but, because of their scatter and the limited range of SN and 01which they cover, it is hardly possible to decide whether they favour any one of eqs. 13, 18 or 19 rather than another. The results we have obtained have interesting implications for the mixing and homogenizing of viscous liquids, and a preliminary discussion of some of them has already been given by Cable (1968). In mixing theory the important question is how fast is an inhomogeneity stretched and thinned, and so in the present paper we have merely calculated the rate of deformation of the ellipsoid. However, when the appropriate e; have been obtained, as explained in the second section of this paper, the entire flow inside and outside the inhomogeneity can be found from the general theory presented in I and II.
274 REFERENCES Batchelor, G.K., 1970. Introduction to Fluid Dynamics. Cambridge University Press, 615 PP. Cable, M., 1968. The physical chemistry of glassmaking. Proc. Int. Congr. on Glass, London. Society of Glass Technology, pp. 163-178. Dunnet, D., 1969. A technique of finite-strain analysis using elliptical particles. Tectonophysics, 7: 117-136. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Sot. London, Ser. A, 241: 376-396. Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proc. R. Sot. London, Ser. A, ‘252: 561-569. Gay, N.C., 1968a. Pure-shear and simple-shear deformation of inhomogeneous viscous fluids. 1. Theory. Tectonophysics, 5: 211-234. Gay, N.C., 1968b. Pure-shear and simple-shear deformation of inhomogeneous viscous fluids. 2. The determination of the total finite strain in a rock from objects such as deformed pebbles. Tectonophysics, 5: 295-302. Jaeger, J.C., 1969. Elasticity, Fracture and Flow. Methuen, London, 263 pp. Lamb, H., 1932. Hydrodynamics. Cambridge University Press, 738 pp. Love, A.E.H., 1927. A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, 643 pp. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, 567 pp. Tan, B.K., 1974. Deformation of particles developed around rigid and deformable nuclei. Tectonophysics, 24: 243-257. Taylor, G.I., 1932. The viscosity of a fluid containing small drops of another fluid. Proc. R. Sot. London, Ser. A, 138: 41-48, Wood, D.S., 1973. Patterns and magnitude of natural strain in rocks. Philos. Trans. R. Sot. Lond., Ser. A, 274: 373-382.