Mechanics and Mathematics of Crystals: Selected Papers of J L Ericksen

+

P3a3) '

(3.31)

= 1 + na. ® a 2 , giving us the reduction to f3.ll). CASE 2:

One of the

BK

vanishes.

As before, renumber to get greatest common divisor of 6 1 = mp 1 , now letting

3

B and

=0.

With

6 ,

write

m

as the

B 2 = mp 2 ,

q

and

r

(3.32)

be integers such that

2

plq - p r = 1 .

(3.33)

With the allowable change of lattice vectors given by a

l

1 = p a

+p

l

2 "I 2 '

a

a 2 = ra x + qa 2

5

3 = a3 '

>

(3.34)

J

we have B K a K = mtp-'^aj^ + p 2 a 2 ) = ma 1

(3.35)

a

= ^ "K ' or

B

= I

=0.

Applying to this the analysis described in

CASE I then gives the desired reduction. CASE 3:

The

B ' are all non-zero.

Here write 2 2 3 3 B = mp , B = mp , with p integers

and p q and

(3.36)

relatively prime, now choosing r so that

p2q - p3r = 1 .

(3.37)

128 J. L. Ericksen

'^ The change of lattice vectors given by

ix = a 1 , a2

=

P2a2

a

=

ra

3

2

1

+ p3a

+

qa

3

3 '>

(3.38)

'

then gives BKaR = B 1 a 1 + m ( p 2 a 2 + p 3 a 3 ) 1= 6 a.

-K+ ma, = 6 a R ,

(3.39)

reducing this to CASE 2. To sum up, if S and u are, respectively, integers and rational numbers, (3.23) defines an element of G provided the numbers also satisfy (3.25) and the condition that the products B v>L be integers. With a K taken as a possible selection of lattice vectors, F, given by (3.8) defines one of the possible lattice-invariant shears. By using the above algorithms, we can always find lattice vectors reducing any possible F to the form (3.11). Said differently, v must be parallel to a possible reciprocal lattice vector, a to one of the corresponding perpendicular lattice vectors, its magnitude being limited by (3.12). Rather obviously, such discontinuities give rise to discontinuities in lattice vectors which agree with the Born rule, although the x-ray crystallographer would perceive lattice vectors as constant throughout. Perhaps it only reflects my lack of ingenuity, but I find it difficult to seriously consider anything more general than piecewise homogeneous deformations, as a real ambiguity involved in relating the Born hypothesis to measurements of lattice vectors. Of course, the metallurgist uses other clues, pondering how finite crystals are seen to be shaped, fit together, etc. It is not entirely easy to sort out all of the mathematical and mechanical ideas which might be involved in such considerations.

129 The Cauchy and Born Hypotheses for Crystals 4.

73

PRACTICE. It is not so hard for the uninitiated to be misled by

common descriptions of configurations.

In a certain

temperature range, a monatomic crystal might adopt a configuration sometimes described as a face-centered cubic.

At other temperatures, it might occur in the form

described as body-centered cubic. are of these forms, for example.

Different phases of iron The words suggest

picturing an homogeneous deformation of one cube to another.

Apart from a rather inconsequential translation

and rotation, this could only be a uniform dilation, all directions being stretched the same, a conformal mappina. Experience contradicts this, and it is inconsistent with the Cauchy and Born hypotheses. For the face-centered cubic, the words suggest a translation group generated by three orthogonal vectors of equal length; forming edges of a cube, say b^.

and

bj_, b 2

Application of this translation group to one atom,

located at the origin, would generate a simple lattice, a simple cubic configuration.

To get the face centered

variety, we add identical atoms at three places, say V 2 (b-L + b 2 ) ,

+ b

V 2 (b 2

3>' 1 /2(b 3 + b x )

,

(4.1)

applying the same translation group to these, to locate positions of remainder. Or, equivalently, we place atoms at positions whose components, relative to this basis, are either inteaers or half-integers. In discussing deformations likely to occur in transitions of the kind mentioned, metallurgists sometimes picture the configuration in a different way, as a bodycentered tetragonal. Here one introduces as translation vectors c K , given by cx

= l/2 (t>i - b2)

.|

c 2 =V2 (bi + b 2 ) , >

C

b

3 = 3 '

(4.2)

J

still orthogonal, with

c^

and

c2,

but not

C3

of equal

130 J. L. Ericksen

74

length, describing edges of the tetragon. With the factors of V2 occurring, the two sets are not related by G. From this alone, it follows that not both can be lattice vectors. Application of the second group to an atom at V 2 (b-^ + b2) generates most atomic positions. To get the rest, similarly translate an atom whose position vector, relative to this point, is y2 (b-L + b 2 + b 3 ) .

(4.3)

Another way of describing the same configuration is to introduce translation vectors *1

=1

/ 2 (hx - b 2 ) =

Cl

,

>|

a 2 = V2 (bx + b 2 ) = c 2 ,

\

(4.4)

a 3 = V2 (bx + b 3 ) = y2 (c1 + c 2 + c 3 ) , J Applied to one atom, this generates the whole set, describing it as a simple lattice, these a^ being one of the possible sets of lattice vectors. With the factors of V2 running in (4.4), neither b^ nor Cj^ can be lattice vectors. In the jargon used by Ericksen [12], they are sublattice vectors. Generally sub-lattice vectors b R are related to lattice vectors by equations of the form

b K = n£a L , where the

n£

(4.5)

are integers such that

detlln^ll * 0, 1, -1 ,

(4.6)

so the inverse transformation exists, with coefficients which are rational numbers, not all integers.

In at least

some cases where the Born hypothesis fails, a modification can reasonably be applied, with lattice vectors replaced by suitably selected sub-lattice vectors.

Of course, this

makes the hypothesis still more ambiguous. For the body-centered cubic, we also introduce three orthogonal vectors bK.

dK

of equal length, like the previous

From an atom at the origin, this again

generates a simple cubic lattice. V2 (<3X + d 2 + d 3 ) ,

Add an atom at (4.7)

131 The Cauchy and Born Hypotheses for Crystals

75

and similarly translate it to complete the configuration. Again, this is a simple lattice, with one set of lattice vectors a beinq qiven by a

l

= d

l '

a2 = a2 , a 3 = V 2 (^

(4.8) + d2 + d 3 ) ,

and, again, the d^ are sub—lattice vectors, but not lattice vectors. The descriptions as face- or body-centered cubes have some merit, making rather obvious the crystallographic point groups appropriate for these. By a routine calculation, one can get this from the simple lattice description. The body-centered tetragonal description suggests a different point group, and it is less routine to correctly calculate the point group, using it, but it makes it easier to picture some relevant deformations. According to either the Cauchy or the Born hypothesis, which here agree, a possible homogeneous deformation taking the face-centered cubic to the body-centered cubic configuration is defined in terms of lattice vectors described above by, as the deformation with gradient F such that a R = Fa K .

(4.9)

Of course, one can superpose lattice-invariant deformations, as described earlier, Includina use of latticeinvariant shears. On theoretical grounds, I see no easy way of deciding that one of these is more likely to be observed than another, although one expects them to be separated by energy barriers. With (4.9), elementary calculations indicate that one can picture the deformation as that taking the tetragon with edges c K to the cube with edges d K , a deformation which is clearly different from the uniform dilatation mentioned at the beginning of this section. In the metallurgical literature, this is called the Bain distortion or Bain deformation. Some discussions of this such as that of Nishiyama T5, p. 339] mention experimental confirmation that this is the deformation which occurs in at

132 J. L. Ericksen

76

least some cases. Following this is a discussion of martensitic transformations which involve much more complex patterns of deformation, with quite different crystal configurations contacting each other. Certainly, it would be nice to have better tools, to resolve such puzzles. In this discussion, I glossed a point. Conceivably, the uniform dilatation and the deformation given by (4.9) could both be consistent with the Born hypothesis. It is not very hard to show that they can't, but I won't take space to elaborate this.

1.

REFERENCES Cauchy, A.-L., Sur 1'equilibre et le mouvement d'un systdme de points materiels sollicites par forces

2.

d'attraction ou de repulsion mutuelle, Ex. de Math _3_ (1828), 227-287. Cauchy, A.-L., De la pression on tension dans un systdme de points materiels, Ex. de Math. _3_ F1828), 253-277.

3.

Cauchy, A.-L. Sur les Equations differentielles d'equilibre ou de mouvement pour un systeme de points mat£riels, sollicites par des forces d'attraction ou de repulsion mutuelle, Ex. de Math. 4 (1829), 342-369.

4.

M. Born, Dynamik der Kristallqitter, B. G. Teubner, Leipzig-Berlin (1915).

5.

Z. Nishiyama, Martensitic Transformation, Academic Press, New York-San Francisco-London

6.

(1978).

L. D. Landau, On the theory of phase transitions, in Collected Papers of L. D. Landau (ed. D. Ter Haar) Gordon and Breach and Pergamon Press, New York-LondonParis (1965).

7.

Parry, G. P., On the elasticity of monatomic crystals, Mat. Proc. Camb. Phil. Soc. S0_ (1976), 189-211.

8.

Parry, G. P., On diatomic crystals. Int. J. Solids Structures l± (1978), 283-287.

9.

Pitteri, M. Reconciliation of local and alobal symmetries of crystals, to appear in J. Elasticity.

133 The Cauchy and Born Hypotheses for Crystals 10.

Rivlin, R. S., Some thoughts on material stability, in Finite Elasticity (D. E. Carlson and R. T. Shield, eds.), Martinus Nijhoff Publishers, Hague-BostonLondon-London (1970), 105-122.

11.

James, R. D., Mechanics of coherent phase transformations in solids, Materials Research Laboratory Report MRL E-143, Brown University, Providence, October 1982.

12.

Ericksen, J. L., Crystal lattices and sub-lattices, Rend. Sem. Mat. Univ. Padova, 6Q_ (1982), 1-9.

Department of Aerospace Engineering and Mechanics and School of Mathematics University of Minnesota Minneapolis, MN 55455

77

134 Int. J. Solids Structures Vol. 22, No. 9, pp. 951-964, 1986 Printed in Great Britain.

0020-7683/86 S3.00+ .00 Pergamon Journals Lid.

CONSTITUTIVE THEORY FOR SOME CONSTRAINED ELASTIC CRYSTALS J. L. ERICKSEN Department of Aerospace Engineering and Mechanics, and School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. (Received30 January 1984; in revisedform 11 July 1985) Abstract—-Some of the observations of A-15 superconductors near cubic-tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints. Here we begin to develop a theory of this kind.

1. INTRODUCTION

In crystals, it is not unusual to encounter phase transformations involving some change in crystal symmetry. In some of these, observations indicate that some elastic modulus becomes quite small compared to the others, near the transformation. For example, Nakanishi[l] remarks that this seems to be a common feature of alloys exhibiting the shape memory effect. A similar thing also occurs in the so-called high-temperature or A-l 5 superconductors, near a transformation of the cubic-tetragonal type, as is indicated by data presented by Keller and Hanak[2], for example. Indeed, a shear modulus seems to extrapolate to zero at the transformation temperature, one of several indications that these transformations might be of second-order. Landau theory indicates that this is highly improbable, that the transformation should be of first-order, with this modulus remaining positive. As is discussed in some detail by Ericksen[3], there is room to quibble about this theory and its predictions, but it would do no harm to better understand the theories of both possibilities. Move slightly away from transformation and they share the feature that the modulus is positive, but relatively small. Such situations are not unlike the situation encountered in, say, elastomers, for which the shear modulus is small compared to the bulk modulus. There we employ an idealization, regarding the bulk modulus as becoming infinite or, more properly, treating the materials as constrained, in this case incompressible. Here we propose to play a similar game with some crystals. The decision as to what constraints are appropriate depends on which ratios of moduli are small, and for crystals, numerous possibilities exist. To be definite, we will try to model the situation occurring in the A-15 superconductors. We bias the discussion a bit, in favor of transformations which are of first-order, although much of the analysis can also be applied to those of second-order. With situations of this general kind, we have a pretty good analog or generalization of the "order parameters" which Landau[4] introduced, in his considerations of secondorder transformations. From our view, they become the deformations possible in the corresponding constrained materials. Finding the general idea useful, physicists have stretched it to cover various things encountered in first-order transformations which are in some sense weak. In this respect, we are giving one interpretation of what it means to be weak.

2. CONSTRAINTS

For the crystals of the A-15 type, unloaded crystals transform from a configuration of cubic symmetry to one of tetragonal form as the temperature is lowered through a critical value Tc. With a common labelling of elastic moduli, used by Love[5 (Chap. VI)], for SAS 22:9-A

951

135 952

J. L. ERICKSEN

example, the linear elastic strain energy of the cubic phase can be put in the form 2 W = ( C 1 1 - C , 2 ) ( ? , ? 1 + ^ i 2 + ^3) + [(Cn + 2C 12 )/3]( ei ,+8 22 + 833)2 + 4C44(e?2 + e| 3 + el1), (1) where % = %-i(«l 1+622 + «33)
(2)

e is the infinitesimal strain tensor, and Q denotes elastic moduli. I have added carets to distinguish these from components of a tensor C, to be introduced later. This describes the energy relative to an orthonormal basis, base vectors parallel to the orthogonal lattice vectors associated with the cubic phase. From this form, the usual conditions on moduli, conditions that W^O, are easily read off. Observations indicate that, as the temperature is lowered to become near Tc, the crystal softens in the manner indicated by

(Cli-Cn)l(Cn+2Cn)«l,

(3)

(Cn-C12)/C44«l. The notion that these denominators are effectively infinite then gives us an estimate of likely constraints as £11+822 + 6 3 3 = 0

(4)

«i2 = 6 2 3 = 8 3 , = 0 .

(5)

and

Given the continuous or nearly continuous nature of the transformation, it seems to me reasonable to consider the constraints as also applying to the tetragonal phases. When the transformation to tetragonal form takes place, the tetragonal phase tends to be twinned. Naturally, it can be tricky to interpret measurements made on twinned crystals. To cover the two phases and the twinning, we need nonlinear theory, with deformations which must be considered as finite. Those associated with the transformation and twinning are in fact quite small, so that we will aim at theory appropriate for relatively small deformations. Even then, we need to extrapolate (4) and (5) to apply to finite deformations, a somewhat ambiguous matter. Various extrapolations would be reasonably consistent with observations of the deformations associated with the transformation. From the viewpoint of nonlinear thermoelasticity theory, it is convenient to think of selecting as a reference configuration the unloaded cubic configuration, at the transition temperature Tc. Refer this to rectangular Cartesian coordinates x = (x,,x 2 ,x 3 ). A deformation maps x to

y=y(x),

(6)

with detF>0,

(7)

C = F TF = C T > 0

(8)

F=Vy, the usual deformation gradient. Then

is one of the commonly used measures of finite deformation. My first inclination was to interpret (4) as the condition of incompressibility, implying that det C = 1. After some

136 Constitutive theory for some constrained elastic crystals

953

exploration, I decided that a different extrapolation is more promising, viz. C,, + C 22 + C 33 = 3.

(9)

Infinitely many other possibilities exist, so that I will try to make clear in what sense this is unique. With (5), the only reasonable extrapolation seems to be the evident choice C, 2 = C, 3 = C 2 3 = 0 .

(10)

Thus we assume that deformations possible in our constrained materials are described by

1

/ 0 0 0 g 0 ,

0 with/, g and /; positive functions, satisfying f+g+h

(11)

0 h = 3,

(12)

relative to the preferred rectangular Cartesian coordinate system described above. For C near the identity, (12) does, of course, approximate the incompressibility condition, our constrained materials then becoming nearly incompressible. 3. KINEMATICAL CONSIDERATIONS

Formally, one can interpret (6) as a change of coordinates taking curvilinear coordinates xJ to Cartesian coordinates y', C being then viewed as the metric tensor in the latter coordinate system. From elementary tensor analysis, we then know that (13)

y'jk'Tty'j,

where the F's are Christoffel symbols based on C, considered as a metric tensor. Some use (13) as a definition of these symbols. Here and elsewhere, commas denote partial derivatives. Further, integrability conditions for this system are summarized by the condition that the Riemann tensor based on C must vanish, a set of differential equations which / g and h must satisfy. Without using (12), this gives the following six equations:

2{g^+h,22)-l(g,3y+g,2h,2]/g-[(h,2r+g,3hi]/h+gAhJf=o\ 2

2

2(hJI+f,i3)-[(hi) +hif.3]/h-[(f,3) +h_lhl]/f+h2f,2/g

(14)

=0)

and 2/23 -f.iM-f.a.ilg-fMh

=o

|

2g.3l-g.3g.i/g-g.3hJ>'-9.if.ilf = 0 \ = 0\ 2hA1-hih1lh-hlftllf-hrgAlg

(15)

Thus our three unknowns must satisfy seven equations. At first glance, it might seem unlikely that there are solutions other than the obvious homogeneous deformations, with /, g and h constant. That the impression is misleading can be seen by considering cases where/and g are independent of x3, with h constant, the generalized plane deformations.

137 954

J. L. ERICKSEN

With the constraint (12) applying, one can represent the possibilities in the form / = a(l+cosoc) g = a(l-msa) h=3-2a,

1 I, 0
( 16 )

with a constant, a a function of x, and x2 satisfying cos2a
(17)

One then finds that the system of equations collapses to a single equation, viz. a,n-a,22=0.

(18)

« = j»(zi) + y(z2),

(19)

As we all know, a general solution is

where ^J2z{ =x,+x2,

(20)

^/2z2=x,-x2,

a and /? being arbitrary functions, smooth enough to satisfy (18), at least in a weak sense. Of course, (17) imposes a restriction but, locally, we have a description of infinitely many possible deformations. Using (13) or the equivalent, one can get the corresponding deformations. To within a constant rotation and translation, one finds that

f

y, = c ( — sin (I dz,—sin y dz 2 )

.

J

y2 = c (cos /? dz, - c o s y dz 2 )

(21)

f

j>3 = d x 3 where

e=±V/a,

(22)

d = J^-2a,

the sign being chosen so that (23)

csin(£ + )0>0.

Obviously, the lines, or more properly, the planes z, = const, and z2 = const, are characteristics of the hyperbolic equation (18). Physically, these planes do have a particular significance. As the cubic phase transforms to the tetragonal phase, the material planes X\ ±x2 = const.,

-V2±x3 = const.,

Xi+xt=

const.

(24)

become the so-called twin planes, surfaces of discontinuity which are commonly observed. Later, we will say more about twinning. As will become clear, these planes are, in different ways, related to other possible discontinuities. In a more general way, one can explore what happens if one adopts a different extrapolation of (4), for example, det C = 1. With this choice, one gets an equation somewhat like (18), for generalized plane deformations, a nonlinear hyperbolic equation, with

138 Constitutive theory for some constrained elastic crystals

955

different characteristics, which seem not to admit any easy physical interpretation. For the general system, one can write conditions restricting jumps in, say, second derivatives, when the functions and their first derivatives are continuous, using the kinematical conditions of compatibility [/<,] = rv,vj,

[g,ij]=svivj,

[hj] = tv^

(25)

Here the square bracket denotes the jumps, and v is the unit normal to the discontinuity surface. Equations (14) and (15) give restrictions of the form rvl + sv] = 0,

rv 2 v 3 =0,

etc.

(26)

For a general constraint of the form o(f,g, h) = 0, the additional restrictions take the form da

da ,

da

Tfrv>+T9sv>

+

Thtvl

= 0

-

W

Examination of the set indicates that, for these surfaces to be the planes given by (24), one needs da

da da

Tf = Tg = Th-

(28)

Solve these partial differential equations for
Here we explore the possibility of having surfaces of discontinuity across which the deformation gradient F = Vy suffers a finite discontinuity, with y remaining continuous, a kind of situation which is encountered in twinning, in particular. Various workers use the adjective "coherent" to describe discontinuities leaving the displacement continuous, to distinguish these from defects involving slip, cracking, etc. Let overbars denote quantities evaluated on one side of the surface, the same symbols without bars indicating the corresponding quantities on the other side. The unusual kinematic conditions of compatibility then give F = F(l+A®N),

(29)

where N is the unit normal to the surface of discontinuity, in the reference configuration, and A is the so-called amplitude vector. From this we get

C = FTF = (1 + N A)C{\ + A ® N) = C+N®CA + CA®N+A-CAN®N.

(30)

Since C and C should both be compatible with (11) and (12), we must have tr(C-Q=2N-CA + A-CA=0.

(31)

139 956

J. L. ERICKSEN

We set

CA =N-CAN+IM,

M-M=\,

M-N = O,

(32)

where X is some scalar. With (31), (30) then reduces to

C-C = X(N®M+N® M) = X{E,®Ei-E1®E2),

(33)

where £, and E2 are the orthogonal unit vectors given by

(34)

sj2E1=N-M)

Clearly, this gives a spectral representation of C—C. Since C and C share the same eigenvectors, the orthonormal base vectors e, (i = 1,2,3) in which (11) holds, Et and E2 must be parallel to two of these. Analysis of any of these choices is much the same so, to be definite, we take

£, =«„

E2=e2,

(35)

JlM = ei-e2.

(36)

giving j2N = e,+e2,

Clearly, N is here normal to one of the planes given by (24), and we could arrange to get any other. With (36), it follows easily that /-/ =*

|

g-g=-X>h =h J

(37)

From this, we can read off one conclusion of interest. For a cubic configuration, f=g = h=>C=l. For one of tetragonal form, two eigenvalues of C should coincide, and be different from the third, with their sum being three. Try tofittwo such configurations to (37), and you conclude that cubic and tetragonal phases cannot coexist coherently. This is consistent with observations of A-15 superconductors. It is one of the things which has supported the notion that such transformations might be of second-order. Clearly, the present theory also excludes coexistence, if the transformation is of first-order. With the results at hand, it is a bit tedious, but not really difficult, to complete the analysis, so I will omit some of the details. To describe the results, we set / = /fc2(l+cos2,u),

g = k\\~cos2n\ / = £ = & 2 (l-cos 2/1), k = sJ(i-h)l2,

fc2(l+cos2/I),

(38)

MO Constitutive theory for some constrained elastic crystals

957

with fi and p acute angles. Introduce the rotation matrix | cos 8 R= - s i n e

sin 6 cosfl

I! 0

0

0 II 0 ,

(39)

1I

with (40)

B = H-fi. Set

I I

k cos n

R being any rotation matrix, and set

With

0

0 I

(41) 0

0

ksinn

0

k cos /Z

0

jh I

0

0

>

(41)

1

(42) 0

k sin fi 0

0

0

^

1

•

A = y/2 sin e[sin fi (cos ^)"'e, —cos /I (sin |t)"1e2],

(42) (43)

one then has the solutions of (29) conforming to (36). Special cases correspond to twinning. As this is commonly interpreted, the term refers to cases where C = HTCH ? C,

(44)

with H an element of the invariance group for relevant constitutive equations, such that H2 = \.

(45)

Note that this implies that det C = det C, which would be quite compatible with the notion that the constraint of incompressibility applies, for example. We have not yet discussed such invariance, but will do so. For the moment, we note that, with N given by (36), the matrix H= -l + 2N®N = HT

(46)

represents a 180° rotation, with N as axis, so that it satisfies (45). An elementary calculation then gives

C = HT

I/ 0

0 g

0 II II g 0 0 \H - 0 /

I0 0 h1

0| 0 .

I0 0 A I

(47)

so that (44) will hold, provided f*B,

(48)

141 958

J. L. ERICKSEN

and (37) 3 holds. Comparing (38) and (47), we now have /« = K/2-AI,

(49)

d = 2fi-n/2.

(50)

(40) then giving

It then follows from (43) that

A = -2Mcos2n,

(51)

where M is given by (36), implying that A-N

= 0,

(52)

as would be expected by those familiar with twinning analyses. Similarly familiar is the fact that (44) implies the existence of a rotation matrix R such that

F = RFH = F(\+A®N)

(53)

R2 = l.

(54)

and

For the example, a calculation gives

R=

-\+2m®m,

where m = i?(sin /xe, +cos fie2)-

(55)

Clearly, (48) excludes twinning of our cubic phases, which seems to be in accord with observations of the A-15 superconductors. Twinning is observed in the tetragonal phases. Take f=h, and you get the analysis of such cases. Rather obviously, with slight changes in the analysis, one can take N normal to any of the planes listed in (24). If one replaces (12) by another likely extrapolation of (4), for example_/#/; = 1, one gets these same planes as twin planes, etc., so one does need to consider something other than twinning to decide between the possibilities. Being purely kinematic, these analyses can be applied to crystals loaded in various ways, not necessarily statically. By the same token, the analyses are incomplete, involving no consideration of energies or forces. One thing is worth noting. The constraints (10) tend to exclude deformations of the simple shearing type. However, as is clear from (52) and (53), the relative deformation F~ 'F is of the simple shearing type, this being characteristic of twinning. Clearly, the discontinuities here considered are stronger than those considered in Section 2, so that it is not so obvious, before doing the analysis, that they should occur on the planes described by (24). 5. ENERGETICS

We aim at developing theory good enough to cover both phases involved in the cubic-tetragonal transformations, including twinning of the latter, for cases where the deformations are not very large, with absolute temperatures T near the critical value Tc. At least for static problems, it seems reasonable to try to use nonlinear thermoelasticity

142 Constitutive theory for some constrained elastic crystals

959

theory, for which we need some constitutive equation for = <j>(C, T).

(56)

In principle, <> / should be considered to be invariant under an infinite discrete group of the kind described by Ericksen[6]. However, as is discussed by Parry[7] and Pitteri[8], we need only require invariance under the point group corresponding to our cubic reference, if we are concerned with deformations meeting restrictions which they describe. It so happens that such restrictions are met by all deformations satisfying our constraints. Thus, as long as we accept the notion that the constraints apply, we need only require that 4>{HTCH, T) = 4>{C, T) = ${f,g, h, T),

(57)

with H belonging to the indicated point group. In particular, this includes orthogonal transformations interchanging pairs of our preferred base vectors, which means that $ should be a symmetric function of/, g and h, this being enough to ensure invariance under the full group. Note that, with (47), <ji is then invariant under (46). In physical terms, it is pretty obvious that twin-related configurations should have the same energy. Physically, this supports the view that $ should remain invariant under the cubic group when the crystal has transformed to configurations of the tetragonal type. Given this symmetry, we can reduce (j> to a function of elementary symmetric functions, a matter discussed carefully by Ball[9]. Bearing in mind the constraint (12), this means that 4> is expressible in the form (j> = $(J, K, T),

(58)

where 6/ = ( / - l ) 2 + ( ^ - l ) 2 + (/ ! -l) 2 = tr(C-l) 2 ,

(59)

2AT = ( / - l ) ( 0 - l ) ( * - l ) = d e t ( C - l ) .

(60)

Mathematically, potentials of essentially the same form, and similar character arise in considerations of isotropic-nematic phase transformations in liquid crystals, so that we will borrow some results occurring in Ericksen's[10] discussion of these. First, the inequality K2 sg P

(61)

always holds. When this reduces to equality, at least two of the quantities/, g and h must be equal. Refining this a bit, we have f = g = h=\oJ

= K = Q,

(62)

characterizing our cubic phases. Configurations of the tetragonal type are covered by

/ = «#* 1 f^g^h

ioK2 = J'>0.

(63)

f=h*gJ These correspond to the nematic phases in liquid crystals, the cubic phases being the analog of the isotropic phases in the latter. Also, the constraint (12), together with the condition that these functions be positive, provides another restriction. By an elementary calculation, fgh = 2K-3J+l>0,

(64)

143 960

J. L. ERICKSEN

one can show that (61) and (64), along with J > 0, cover the limitations on possible values of J and K. In general terms, we then want $ to have an absolute minimum of the kind indicated by (62) when T > Tc, switching to the kind indicated by (63) when T < Tc, it being possible that both retain some status near T = Tc, as at least relative minimizers. Expressing some of these ideas more formally, we at least want that

$(J, K,T)> $(0,0, T),

(65)

T > Tc

and, for some choice of the function / = JQ{T), and for one of the two choices of algebraic signs,

$(J, K,T)> $(J0, ±Jl'2, T) = >/,(/„, T),

T < Tc.

(66)

When they first reported these transformations in an A-15 superconductor, Batterman and Barrett[l 1] opined that they might well be of second order, this being a reasonable opinion, I think. For this, it is important that Jo -> 0 as T-> Tc, and data such as are presented by Keller and Hanak[2] indicate that this might be true. To make a long story short, experimentation still seems to leave doubt as to whether such transformations are of secondorder, or of first-order, with small discontinuities in Jo, etc. masked by experimental errors. For analyzing such small deformations, it seems natural to try to approximate 4> by a polynomial of rather low degree in C— 1, if you like by the first few terms in the Taylor expansion of a smooth function. One of the form

$ = a(T) + b(T)J+c{T)K+d(T)J2

(67)

covers the possible quartics. There should be no danger of confusing the temperaturedependent coefficients with constants similarly labelled earlier. Assume that the temperature-dependent coefficients are smooth and similarly approximated near T — Tc, and you have what is sometimes called mean field theory. As is discussed by Wilson[12], for example, such assumptions go wrong in analyses of critical points in fluids, etc., situations bearing some similarity to the kinds of transformations considered here. Still, it seems to me worthwhile to better understand what kinds of predictions are associated with such a guess, and my own understanding of this leaves much to be desired. Of course, one could try a compromise, using (67), but allowing the coefficients to have mild singularities at

T=TC.

With (61) and (67), we clearly have $>t(J,T) = a + bJ-\c\Pl2 + dJ\

(68)

from which it is clear that if has minimizers, they should be of cubic or tetragonal form, which is good, for our purposes. Were (67) exact, we would need to have d > 0,

(69)

to get the minimizers, so we will try assuming this. Were d < 0 for this term in a Taylor expansion, one might still have the minima, but one would need to consider higher order terms in the expansion to sort this out, for a smooth potential. To have even a relative minimum of cubic form {J = K = 0), for T > Tc, we must have b(T)>0,

T>Tc.

(70)

Other extremals can be located, by setting the derivative of \j/ equal to 0, giving b-3\c\JV2/2 + 2dJ = 0,

(71)

yl

a quadratic in J . It will have real roots if 9c 2 > 32bd,

(72)

144 Constitutive theory for some constrained elastic crystals

961

and we want this, at least for T < Tc. For whatever it is worth, the Landau-type argument that the transformation should not be of second-order is as follows. To have bifurcation occur at T = Tc, it is easy to see that one needs b(Tc) = 0, so J = 0 then satisfies (71). At Tc, <j> should still be a minimum, for J = K = 0. An inspection of (67) or (71) makes clear that, for this, it is necessary that c(Tc) = 0. Grant that <j> is thrice differentiable and, by essentially the same analysis, you come to this conclusion. As Landau[4] saw it, it is highly improbable that two functions of one variable should vanish simultaneously. Nowadays, experts in bifurcation theory would, I think, agree that generically, such a transformation is not of second-order. If we argue generically, b should remain positive at and near T = Tc, so that the cubic phase should retain some status, as a relative minimizer, for T < Tc. Then, as J increases, i// must take on a local maximum before it can take on another minimum. Thus the latter must correspond to the larger root of (71), when this is real. At this, we have / = JQ{T), with J]j2 = (3|c| + ^9c2 - 12bd)/$d.

(73)

By elementary calculation, the cubic phase 7 = 0 has the lowest energy when \j/{Ja,T)> iMO, T)oc2<

4bd,

(74)

and we want this for T > Tc. Similarly, the tetragonal phase does when

\l/(0,T)>\l/(J0,T)oc2>

4bd,

(75)

and we want this for T < Tc. Of course, Tc represents the temperature at which the two energies become equal, so that c2 = 4bd,

at

T=TC,

(76)

and generically, this should be an isolated temperature. Assuming this form of $ applies to the A-15 superconductors, we must have JO(TC) very small, which requires that, for T near Tc, |c|/rf«l,

b/d«l,

(77)

so that, by the indicated kind of reasoning, this gives one estimate of what $ might look like, near the transformation, certainly involving a bias in favor of the notion that the transformation is of first-order. One might introduce guesses about the temperature dependence of coefficients near Tc, based on ideas of smoothness. Otherwise, this seems to be the simplest kind of model which accommodates the two phases and twinning. It could do no harm to better understand what all it predicts, and how this compares to the behavior of real crystals, but it is a somewhat naive guess. 6. EQUILIBRIUM EQUATIONS

Here we begin by reverting to index notation. In dealing with constrained elastic materials, we follow the most common practice, which is to use the format suggested by Ericksen and Rivlin[13], to introduce kinds of Lagrange multipliers or forces of constraint. Some possible generalizations are considered by Antman[14], who argues that they might well be of import for some kinds of theories, but not elasticity theory. Here we write f = cj,-n(Cu + C22 + Cii-3) + 2(X1C2i+X2Cii+X}Cl2),

(78)

where n and the A's are arbitrary functions of position, <j> being given by a definite constitutive equation, for example, that represented by (67). Then, ignoring the constraints, treat <j> as the potential for an unconstrained material, using any of the common formulae for calculating stresses, to be used as usual in equations of equilibrium or motion. Without

145 962

J. L. ERICKSEN

really looking at such calculations, we can notice one curiosity. We have four multipliers to play with, only three equations to be satisfied, so that it seems that it should be possible to get any kinematically possible deformation to satisfy the equilibrium equations with zero body force, this still leaving only three equations to determine four unknowns. Before, we found that a naive count of equations and unknowns was misleading, so that we should look more closely at the equations. For example, the Piola-Kirchhoff stress tensor is given by 71 = dftdy'j = T'ky[k,

(79)

where fjk

=

fkj

=

af/dCjt + d$/dCkJ,

(80)

or, in matrix form,

f=b3 IU2

n2 xA,

(8i)

Xi /X3II

with H = 8<j>/dCkk — n

(no sum).

(82)

The equilibrium equations

can, with the help of (13), be reduced to the form

Ti+r}kfjk = o.

(83)

With C of the form (11) we get, after some calculation

(A,hHAih-U*h =*n (^,),2 + (^3),l-(^),2 =2 \,

(hX1X, +

(84)

{hX,\2-(hn),,=^J

where 2®, =[
2$2 = [^-2g(df/8g)h,

(85)

2$ 3 = [(0-2A(fl<£/5/!)],3. With the deformation given, (84) then reduces to three linear equations for the four unknown multipliers, seeming to reinforce our first impression. We do know that we have the generalized plane deformations given by (21). Assuming that the multipliers do not depend on x3, the above system reduces to

{fX,h = {fn + a)A (<M3),, = (
^,2 + A 2 , , = 0

J

(86)

M6 Constitutive theory for some constrained elastic crystals

963

with

2
(87)

Then (86), and (86) 2 can be viewed as integrability conditions for functions £, and rf such that

/ * , = {.„

fn + c = i,2-)

(88)

Eliminating A3 and n gives 9li-f 1.2 = 0 1 9l2-fi.i=g*-fry

(89)

One can solve for the gradient of either function, then cross-differentiate to get an equation for the other. For example, that for £ is [(g/f)i,ih -Kg/niih

= (*-g°/f),2

(90)

a linear hyperbolic equation having as characteristics our old friends, the possible twin planes. Locally, any solution of this generates a solution of the equilibrium equations. That is, one can work back from this to get A3 and n, etc. Similarly, we can satisfy (86)3 by writing A, and i 2 in terms of derivatives of an arbitrary function. Certainly, this serves to confirm the first impression. With the several constraints, we begin to approach the situation encountered in rigid body mechanics. There one might introduce stress as an essentially arbitrary tensor, restricted a bit by the condition that its divergence vanishes, for example. The idea is more cumbersome than useful, so that we use other familiar ideas to formulate and solve physical problems. It does suggest that we might need to change our thinking habits, to make effective use of these theories of highly constrained materials. Inherently, the physical situations envisaged are complex. As should be clear from our consideration of minimizers, the simplest problem, of the equilibrium of an unloaded crystal, is nontrivial, and requires stability analyses. For various static problems, one can use the notion of minimum energy to formulate problems, building stability criteria into the formulation. Certainly, this is a sensible approach, but it is hardly a panacea. I have begun to look at some of the simplest experiments from this point of view, but find it tricky, so that it seems premature to comment on this. Of course, the general format is designed to produce some agreement with a few of the observations, but not enough to firm up specific constitutive equations, or to assess the quality of predictions of such theory. Acknowledgement—This material is based on work supported by the National Science Foundation under Grant No. MEA-8304750.

REFERENCES 1. N. Nakanishi, Lattice softening and the origin of SME. In Shape Memory Effect in Alloys (Edited by J. Perkins). Plenum, New York (1975). 2. K. R. Keller and J. J. Hanak, Phys. Rev. 154, 628 (1967). 3. J. L. Ericksen, Archs Ration. Mech. Analysis 73, 99 (1980). 4. L. D. Landau, On the theory of phase transitions. In Collected Papers of L. D. Landau (Edited by D. TerHaar). Gordon and Breach, and Pergamon, New York (1965). 5. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge Univ. Press, England (1927). 6. J. L. Ericksen, Adv. Appl. Mech. 17, 189 (1977). 7. G. P. Parry, Math. Proc. Camb. Phil. Soc. 80, 189 (1976). 8. M. Pitteri, J. Elasticity 14, 175 (1984).

^ _ _ _ 964

147 J. L. ERICKSEN

9. J. M. Ball, Differentiability properties of symmetric and isotropic functions, to be published in Duke Math. 10. J. L. Ericksen, A thermodynamic view of order parameters for liquid crystals. In Orienting Polymers, SpringerVerlag Notes in Mathematics (Edited by J. L. Ericksen), Vol. 1063. Springer-Verlag, New York (1984). 11. B. W. Batterman and C. S. Barrett, Phys. Rev. Lett. 13, 290 (1964). 12. K. G. Wilson, Rev. Mod. Phys. 55, 583 (1983). 13. J. L. Ericksen and R. S. Rivlin, J. Rat. Mech. Analysis 3, 281 (1954). 14. S. S. Antman, Re. Accad. Na:. LinceilO, 256 (1982).

148 Material Instabilities in Continuum Mechanics and Related Mathematical Problems © 1988 Oxford University Press

11 SOME CONSTRAINED ELASTIC CRYSTALS J. L. ERICKSEN

1.

INTRODUCTION Often, crystals undergo phase transformations involving

some change of symmetry. When such transformations are of second-order, and often when they are of first-order but weakly so, it is observed that some linear elastic moduli become quite small compared to others, near transition. reasonable to try using simpler, idealized considering

It then seems rather theory, roughly

the larger moduli as infinite or, more properly

regarding the crystals as constrained materials.

Motivated by

observations of cubic-tetragonal transformations in A-15 superconductors, Ericksen

[1] formulated

a thermoelasticity theory

of this kind, which also shows some promise as a theory for somewhat similar transformations occuring in Indium-Thallium systems.

A variety of different constraints would be in reason-

able agreement with the observations.

Ericksen [1] gives one

theoretical argument leading to a unique choice.

Here, we give

a rather different argument leading to the same choice. In the Indium-Thallium systems, but not in the A-15 superconductors, cubic and tetragonal phases have been observed to coexist, by Burkart and Read [2], and they form rather complicated configurations.

Ball and James [3] found a way to deduce

algorithms which enable one to describe coarser features of such

149 120

J.L. ERICKSEN

configurations.

We will present and discuss the results of such

calculations, for the theory of constrained crystals. The reasoning of Ball and James does not suggest why such configurations should be stable enough to be observed.

Burkart and Read [2]

attempt to rationalize this, but a satisfactory theory for this remains to be developed. is not observed

It should explain why such coexistence

in A-15 superconductors.

It would be easy to

explain this, if such transformations are of second-order, as was suggested by the first workers to observe them, Batterman and Barrett [A]. As is discussed by Ericksen [5], reasoning accepted by many physicists leads to the conclusion that they should not be, but there is some room for argument about this. We have not seen clear experimental proof that the implied discontinuities in deformation occur. In any event, such stability questions remain open. 2.

CONSTRAINTS For the aforementioned crystals, we have cubic crystals at

higher temperatures, transforming to crystals of tetragonal form, as the temperature T is lowered through a critical value T'o and, commonly, the tetragonal phase contains twins.

The trans-

formation involves a discontinuity in deformation which is quite small.

In the A-15 superconductors, it might be zero or just

small enough to be obscured by experimental errors. In the IndiumThallium alloys, data presented by Burkart and Read [2] indicate strains of the order 10 large.

, definitely non-zero, but not terribly

Briefly, we seek a theory capable of describing both

phases, twinning in the tetragonal phase, as well as the effects of small loadings and temperature changes.

Roughly, the deforma-

tions of interest should be considered as finite, although they are not very large;

conversion of one twin to another involves

finite rotations, for example. In the cubic phase, the linear elastic strain energy W is of the form

150 121

SOME CONSTRAINED ELASTIC CRYSTALS

2(/= (c u -c 1 2 )(nf 1 + n2 2+ n|3) + [(^11+2C12)/3](eil+e22 + e33)2 + 4

^^(e12

+ £

23 + e 3 l ) .

(2.D

v-3(ell+e22+e33)6^.

<2-2>

where \Q

= e

and e is the infinitesimal strain tensor. The C.. are elastic moduli, labelled as most experimentalists do, except that we have added carets to distinguish these from components of a tensor C to be used later, Ericksen [l], noting observations indicating that, for T near To , ( C 1 1 - C 1 2 ) / ( C U + 2C 1 2 ) < 1 , 1 \

(cll-c12)/chk

< i,

(2.3)

J

thought of the denominators as effectively i n f i n i t e , to get e

ll

+ e

22+e33 = ° »

(2-4)

e

12= e 23= £ 31 = ° .

<2-5)

as a first estimate of likely constraints.

In some way, we need

and

to extrapolate these, to apply to finite measures of strain. From the viewpoint of nonlinear thermoelasticity theory, it is convenient to take as a reference configuration the (unstressed) cubic phase at the transition temperature T^. Refer this to rectangular Cartesian coordinates x=(x^ , x2, x 3 ) , with the orthonormal base vectors e. parallel to the usual cubic lattice vectors, as is presumed in (2.1).

A deformation

maps x to u = u(x) ,

(2.6)

with F =Vu , detF> 0

(2.7)

the usual deformation gradient, and the symmetric tensor C = F1F (2.8) is a commonly used measure of finite deformation. The only

151 122

J . L . ERICKSEN

reasonable e x t r a p o l a t i o n of (2.5) seems to be C12=C23=C31 = 0 .

(2.9)

As will be made clear, if it is not already so, a variety of reasonable extrapolations of (2.4) exist.

Ericksen [l] gives an

argument leading to the choice Cn+C22+C33

= 3,

(2.10)

and, shortly, we will argue this in a different way.

Given the

experimental errors involved in measuring small strains, it is hard to say which extrapolation might best fit the deformations experienced by the real crystals. From what little experience we have, if one picked the constraint to make the mathematical theory simplest, one might well pick (2.10). 3.

JUMP DISCONTINUITIES Consider a material surface, with unit normal If in the

reference configuration, across which the deformation gradient F suffers a finite discontinuity, with U remaining continuous. If the two limiting values are denoted by F and F , the usual kinematical conditions of compatibility read F = F(l + A ® N) , where A is the so-called amplitude vector.

(3.1) From t h i s , we get

C = I?! = ( 1 +N ® A) C(\+A ® N) = C + N <8 M + M ® N ,

(3.2)

where 1M =2CA + A • CAS

(3.3)

We assume that (2.9) applies, so that ? - C = aex ® ex + B e 2 ® e 2 + Y « 3 ® e3 .

(3.4)

For the present, we ignore the remaining, ambiguous constraint. From (3.2), if V is a non-zero vector satisfying V. M = V' N = 0,

(3.5)

252 SOME CONSTRAINED ELASTIC CRYSTALS then

_ (C-C) V = 0 .

123

(3.6)

Using ( 3 . 4 ) , we then get ctKj = &V2 = yV3 = 0 ,

(3.7)

with V^ = V ' e . , the components of V . If a = $ = Y = 0, C = Cy implying the existence of a rotation matrix Ft such that (3.8) F = RF . With (3.1), this gives R = F{\ + A ® N) F"1 = 1 + FA ® F~T N .

(3.9) —T

This implies that any vector perpendicular to F vector of R , with eigenvalue one.

N is an eigen-

Since a nontrivial rotation

has only one axis, we have c t = B = Y = O-»J?=l"*>l = O ,

(3.10)

the trivial case. This result is well-known to those familiar with twinning theory. If two of the three quantities vanish, say a

= g= o* y ,

(3.11)

it is easy to show that (3.2)—(3.4) imply that N , M and / must all be parallel to « 3 , so we can take N = e 3 , A = 6e 3 .

(3.12)

With the determinants of F and F positive, (3.1) implies that det (1 +A ® N) = 1 +A -N > 0 ,

(3.13)

For the particular case at hand, this gives l + 6 > 0 , a mild restriction. If just one of the three quantities vanish, there is no important loss of generality in assuming that Y = 0, a 6 # 0 , V = e 3 •

(3.14)

Then, with (3.5), we can represent the unit vector H in the form

153 124

J-L. ERICKSEN N = cos e2 .

(3.15)

Now M, also perpendicular to V, must satisfy M ® f f + W ® W =a « 1 ® e 1 + B e 2 ® e 2 .

(3.16)

An elementary analysis then shows that M has the form , P = cos c() 2j — sin < M 2

M=mP

(3.17)

where m is some non-zero scalar. This gives C 1 X — C J J = a = 2m cos2<|> , C22-C22

= S = - 2 m sin2 * ,

(3.18)

r —r L

33~°33 •

The diagonal matrices C and C must be positive definite, implying that C n + 2mcos2i))> 0 , C 2 2 - 2msin2<() > 0 .

(3.19)

Suppose that we are given (diagonal) C, m and <)) satisfying these conditions.

Then (3.15) determines N, and (3.17) determines P.

From (3.3), we must have A = C"1 (mP-aS) ,

(3.20)

with

2a =A • CA =C~l{mP-aN) • (mP-aS) ,

(3.21)

giving a quadratic equation for a , viz a 2 AT "C"1 S-2 (mP • c"1ff+ 1) a + m 2 P • C~l P = 0 .

(3.22)

It is elementary to verify that our assumptions imply that the two roots are r e a l .

Condition (3.13) implies that

1 +mP • C~* N >aN - C"1ff,

(3.23)

which selects one of the two r o o t s , that given by JV • C~l Na = 1 +mP • C~l N - /I A = (mP • C~l If + 1 ) 2 -m2{C~l

P • P){C~1 N • If) .

(3.24) (3.25)

^54 '25

SOME CONSTRAINED ELASTIC CRYSTALS With (3.20), this determines A. One can take F=/c,

the usual

positive definite square root of the matrix C , use (3.1) to determine F and verify that the corresponding C is diagonal, etc.

So, this summarizes the kind of solutions of (3.1) which

are permitted by the constraint (2.9). 4.

THE REMAINING CONSTRAINT In the A-\5 superconductors and Indium-Thallium alloys,

discontinuities of the kind considered above commonly occur in the tetragonal phase, associated with twinning.

For such twins,

observations indicate that N takes on one of the six directions given by /Iff = e. ±e. , *•

i *s .

(4.1)

3

Clearly, these do not conform to (3.12), but fit (3.15), or equivalents obtained by replacing e^ and « 2 by two other base vectors, or their negatives. Also, one has cos2c(> = sin2cj> = \ .

(4.2)

Briefly, observations indicate that, in the unstressed cubic phase C =el ,

(4.3)

a being a function of temperature, describing thermal expansion, with c(T ) = 1 a consequence of our choice of reference configuration.

In the unstressed tetragonal phase, C can take on any

one of three values, of the form C ( 1 ) = diag (v2 , \i2 , vi2) , C ( 2 ) = diag (p 2 , v 2 ,y 2 ) ,

>

(4.4)

C ( 3 ) = diag (y2 , u 2 , v 2 ) , with y and v ^ y

positive functions of temperature.

these conform to (2.9).

Clearly,

If experimentalists had doubts about

this, they would not be likely to call these tetragonal phases. As is discussed in some detail by Ericksen [l] , the

155 126

J.L. ERICKSEN

possibility of twinning is linked to the assumption that governing constitutive equations are invariant under the cubic point group.

It would seem unreasonable not to require this of the

constraints.

Assuming (2.9) holds, we thus look for a constraint

extrapolating (2.8), of the form /(Cn

,C22 , <733) = 0 ,

(A.5)

with f a smooth symmetric function of its arguments. Our assumption, that the reference configuration is attainable, gives the condition /(I,

1,1) = 0 .

(4.6)

Any such function will satisfy •JTJ—

= -M- = M-

=b

when

C= 1,

(4.7)

for some value of b. For infinitesimal deformations, we can linearize f, writing C. . = 1 + 2e .. , Iris

(no sum)

(4.8)

Iftr

to introduce infinitesimal strains, and use / « 2 6 ( e n + e 2 2 + e33) = 0 .

(4.9)

Assuming only that b=t0, we thus get (2.4) in this approximation, so the linear estimate is of little help, in selecting particular choices of

f.

With f a symmetric function, if / = 0 holds for C = C / 1 N » it will hold for C. . and C. . . Briefly, this is enough to ensure that the standard analysis of twinning in the unstressed tetragonal phase will go through, for any reasonable choice of /.

Naturally, / = 0

should hold for a value of C. > well

approximating observed values, but (4.9) comes close to guaranteeing this, for the crystals considered. Application of small loads to a twinned crystal may or may not cause the discontinuities to move through the material, perhaps to be created or destroyed.

What they seem not to want to

^56 SOME CONSTRAINED ELASTIC CRYSTALS do is rotate through the material.

127

That is, they prefer the

material planes listed in (A.I), although the loadings cause C to change. In the A-15 superconductors, I have not seen evidence of other kinds of jump discontinuities in F. In Indium-Thallium alloys, cubic and tetragonal phases coexist, in a complicated way.

Possibly, some differently oriented discontinuity is in

this picture, but I have not seen clear evidence for it, so will ignore this. So, let us find forms of / which force N to take on just the values given by (4.1).

First, suppose that, for all values

of C of interest 5/

_

061 i

3f o Cj 9

3/

(4

,0)

33

A general integral of these equations is an arbitrary function of

y = c

n

+ c2Z + c 3 3 - 3 ,

(4.H)

so, with ( 4 . 6 ) , we have / = Uy)

.

•(£>) = o .

(4.12)

Since we are only interested in the set where / = 0 with very little loss of generality, take constraint as (2.10).

f = y

we can,

giving the

Bearing in mind (3.4), we see that this

excludes possibilities like (3.11), as would various reasonable alternatives.

The rest are like (3.18).

If C and C satisfy

(2.10), we see from (3.18) that either m = 0 , the trivial case, or (4.2) holds, implying that N is included in the set given by (4.1).

So, this form of / has the desired property. If f is a symmetric function not covered by the above

argument, there will be some value of C such that f{Cu

,C22 , C 3 3 ) = 0

(4.13)

and (4.14) Now fix such a C and consider

157 128

J.L. ERICKSEN f(Cn

with

,C22 , C 3 3 ) = 0 ,

(4.15)

C,, = C,,+ 2m cos2(J) , ^-16)

2

as an equation to be solved for m, in terms of <(> . Since f is a symmetric function, (4.15) will be satisfied at

(m, <|>) = (m,<|>),

where m = C*22 — ^11 ^ ^ » <j>3cos2$ = sin2$ = \ , since this makes CJJ = C 2 2 etc.

(4.17)

Calculating the partial deriva-

tive of f with respect to m at this place, we get

3ffl

3C 2 2

"dC-y J

(4.18)

Thus, by the implicit function theorem, we can solve ( 4 . 1 5 ) — (4.16) for m , locally.

Thus, we can pick <j> near <}) , not satis^

fying (4.2), and find a corresponding m . With the analysis given in Section 3, it is then clear that we can construct jump continuities, with N not included in the set given by (4.1). Essentially, then, (2.10) is the only extrapolation of (2.8) which forces all jump discontinuities to be on the twin planes. As is discussed by Ericksen [ 1 ] , relevant equilibrium equations have a hyperbolic character and, with the constraints selected, and only with these, the characteristic surfaces also become the twin planes.

Clearly, this indicates that these become the pos-

sible bearers of various other kinds of weaker discontinuities. As he indicates, one can construct relatively simple Helmholtz free energy functions which, near T=Ta

, have absolute or rela-

tive minima at the cubic phase, described by C = 1,

(4.19)

the form of (4.3) satisfying (2.10), and at the tetragonal phase, described by (4.4), with

v = V 3 ~ l ^ , 0
(A 20)

'

L58 SOME CONSTRAINED ELASTIC CRYSTALS to satisfy (2.10).

129

At least qualitatively, such forms are capa-

ble of describing the cubic-tetragonal transformation, twinning, effects of small shear stresses, etc. There is a rather interesting interpretation of the constraints, not mentioned by Ericksen [1], In the cubic reference configuration, a diagonal of the unit cell is parallel to the unit vector E = (±el ± e2 ± e3)//3 ,

(A.21)

where the algebraic signs can be chosen arbitrarily.

If the

constraints are satisfied,

\\FE\\2 = E'CE = (Cn+C22

+ CZ3)/3 = 1

(4.22)

so the material is inextensible in all of these directions. Conversely, if it is extensible in all, the constraints (2.9) and (2.10) must hold.

After I mentioned this in a lecture, David

Parker noted that, for the plane deformations considered by Ericksen [1], the material behaves as if it were inextensible in two in-plane directions, making these deformations analogous to the plane deformations encountered in textiles.

In three

dimensions, one has something more like a block of foam rubber, with four appropriately oriented systems of inextensible fibres running through it.

Possibly, this will help some to picture

possible deformations. 5.

PHASE INTERFACES An example of twinning can be obtained as follows. F = V C ( 3 ) = p(«j ® «! +<22 ® el)

with y and V satisfying (4.20).

+ ve

3 ® e3 '

Set ^5-1)

Introduce the orthonortnal

basis

fx = « lf f2 = (02+e3)//2 , f3 = (e2-e3)//2 , /, being one of directions listed in (4.1).

A = af2, N =/ 3 . With a little calculation, one finds that

(5.2)

Set

(5.3)

_^___ 130

159

J.L. ERICKSEN F = F{\ +A ® N)

(5.4)

FTF

= C.z)

(5.5)

a = 6(l-y 2 )/(3-u 2 ) .

(5.6)

satisfies provided Often, one sees sets of these layers separated by parallel planes of discontinuity, with the deformation gradient alternating between the two values.

The thickness of these layers can

vary in a random or regular manner. In Indium-Thallium alloys, observations of coexistence of tetragonal and cubic phases, by Burkart and Read [2], involve such a system of twins in

the tetragonal phase.

Those layers with F

seem to be of about the same thickness, as do those with value F. However, the two thicknesses are not the same, one being about twice the other.

This configuration meets the cubic phase at an

interface which is roughly planar, close enough to a plane that one can estimate its crystallographic orientation.

As is dis-

cussed by Ericksen [1], one cannot fit the above F to a value consistent with another fitting (4.19).

This presumed that the

constraints are satisfied, but this is not vital.

So, indica-

tions are that the interface or transition region is more complicated, probably involving inhomogeneous deformation near the interface.

Burkart and Read [2] give a "schematic representation"

of the interface, perhaps to be regarded as a guess as to how atoms could adjust their positions and avoid great distortions of the latter.

Roughly similar configurations are observed in

various crystals which undergo martensitic transformations, and we have not yet found any theory to describe any of them, in detail. Ball and James [3] noticed that one can! construct minimizing sequences for the total energy which resemble such configurations, involving regular spaced twins, with the thickness of the layers approaching zero. energy minimizers.

Such sequences do not converge to

It is not so clear why configurations resem-

bling some of these should be stable enough to be observed.

|60 SOME CONSTRAINED ELASTIC CRYSTALS

131

However, by analyzing the limit, they derive formulae, which seem to describe coarser features of such interfaces rather well. With our constraints, there is doubt as to whether the kinds of deformations needed in the derivation can be constructed. However, it seems interesting to see what the end result predicts. Involved is a scalar A € ( 0 , l ) , interpretable as the fractional thickness of the layers in which F occurs, F occurring in the proportion

1—A.

Then, in the obvious sense, the aver-

age deformation gradient in the tetragonal phase is (F) = XF + (l-A)F = F {) + XA ® N).

(5.7)

In the cubic phase, (A.19) should hold, implying that the deformation gradient should be R, some rotation.

According to the

Ball-James analysis, we should have (F)

= i?(l + B ® D)

(5.8)

where D is the normal to the nominal plane phase interface, and B is some vector.

If \F) were the actual deformation gradient

in the tetragonal phase, this would, of course, be the usual kinematical conduction of compatibility.

Burkart and Read [2]

deduce a kind of approximation to this, using a notion of small average strain, neglecting some small rotations, etc. Here,F, A and N can be regarded as given, but X, R, B and D are not, so one uses (5.7) and (5.8), i.e. R{\ + B ® D) = F{\ + XA ® N) ,

(5.9)

to try to determine them, it being understood that F , A and If are given by (5.1) and (5.3).

We have found no way

to avoid

some tedious calculations in doing this, and will not record the grim details.

It is helpful to begin by noting that, if E is

a unit vector perpendicular to D and N , (5.9) implies that FE = RE •*• \\FE\\ = 1 .

(5.10)

It is easy to check that, as the notation suggests, E must be

161 132

J.L. ERICKSEN

one of the inextensible directions, given by (4.21), one in the subset perpendicular to II. This gives two possible directions, E = («1+e2+e3)//3 ,

(5.11)

E = («1-e2-83)//3 .

(5.12)

or

One can select either and proceed, with the knowledge that D must be perpendicular to E .

By taking determinants, one gets

1 +B 'D = detF = u2v .

(5.13)

Then, one requires other conditions guaranteeing that R , obtained by solving (5.9), is a rotation, the tedious part. the choice, one gets two solutions.

With (5.12) as

One is of the form:

\ = (3 + N / 4 V 2 - 3 ) / 6 ,

(5.14)

/ 2 u 0 = f1 + ( / 2 + V 4 y z - 3 / 3 ) / / 2 , g =

/2vV-l) [ v f l -( / 2 W4y 2 -3/ 3 )/v/2],

R being calculated from (5.9).

(5.16)

Clearly, these exist only if

Au2 ^ 3 , a condition not guaranteed by (4.20).

(5.17) To get the second solu-

tion, simply replace V 4 y — 3 by — v 4 y 2 — 3 tions.

(5.15)

in these prescrip-

As long as (5.17) holds, the two values of X lie in

(0,1), as they should, and any value in the interval can be obtained, by suitable choice of y. When equality holds in (5.17), the two solutions coalesce, with A = j . value of p, the two values of X sum to one. lar solutions using (5.11).

For a fixed

One gets two simi-

The two values of E are related by

an element of the cubic point group, reversing « 2 and e,. Clearly, this leaves F and A®S

invariant;

one can get this

by transforming the entries in (5.9) by the indicated group element.

Actually, one of the previous solutions can be converted

to the other, by a group element interchanging e2 and e 3 but this is a bit more subtle. Altogether, this gives four solutions,

162 SOME CONSTRAINED ELASTIC CRYSTALS the number we should have, according

133

to the work of Ball and

James [3], although the breakdown of solutions as X ->\ was not something we anticipated from the general considerations. In the observations of Burkart and Read [2], p is close to one.

From Eqn. (5.14), we have lim A = § , y-»i

in good agreement with their observations.

(5.18) Similarly,

lim D = [/•1 + ( f 2 + / 3 ) / / 2 ] / / 2 p-»l = (« 1 +« 2 )/^2 also in good agreement with

(5.19)

their estimate of the crystallo-

graphic orientation of the normal

to the nominal interface.

Interestingly, in this limit, D becomes normal to a twin plane, different from that associated

with the twin planes in

tetragonal phase, a possible bearer of discontinuities.

the

If the

result were precise, one might have such a plane of discontinuity as the real interface, separating the two phases, with some kind of inhomogeneous deformation near the interface.

Conceivably,

re-examination of the derivation of (5.9), with the constrained deformation, would modify the result to be in agreement with this.

In any event, this is what seems to be suggested, by the

"schematic representation" of Burkart and Read [2]. Another possibility might have the twin planes with the normals indicated by (5.19) as surfaces of discontinuity, arranged in stairstep fashion.

In places, the twin planes involved in the tetra-

gonal phase might extend, to provide risers, separating one set of twins from the cubic phase.

In this picture, the real inter-

face becomes crinkled, but need not stray far from the nominal plane.

It is not so obvious that one must have such walls

separating the phases; F could vary smoothly, at least in places, to effect the transition.

Also, some different kind of

crinkling might occur, as suggested below.

With electron

163 134

J.L. ERICKSEN

microscopy, one might decide what kind of picture of the interface fits best, but I have not seen such observations.

With our

highly constrained deformations, it is not very clear whether one can construct deformations fitting any of these pictures. If one can, it is likely that one can find Lagrange multipliers enabling one to satisfy equations of equilibrium in a weak sense, from Ericksen's [1] discussion of such matters.

One also needs

some analysis of their stability, which is likely to be tricky. So, we are far from having satisfactory analysis of such phase mixtures. It seems worthwhile to note that the twin planes in the tetragonal phase intersect those with normal indicated by (5.19) in a line parallel to E, given by (5.12), also the direction of the line of intersection of either set of twin planes with the plane with normal D.

Another set of twin planes containing

such lines exist, those with normal parallel to e , + e 3 • With three such systems of planes, one can form an interesting variety of crinkled planes,triangular cylinders, etc., geometrically. Of course, arranging patterns of discontinuities also involves considerations of compatibility conditions, complicating matters. ACKNOWLEDGEMENT Much of the research covered in Sections 3 — 5 was done while I was visiting Heriot-Watt University, where I greatly appreciated the warm hospitality, lively discussions, and having some time to think.

Especial thanks go to John Ball, for much

help in arranging this.

Also, discussions with Richard James

helped me to make a little progress in understanding the phase mixtures in Indium-Thallium alloys.

164 SOME CONSTRAINED ELASTIC CRYSTALS

135

REFERENCES 1. J.L. Ericksen, "Constitutive theory for some constrained elastic crystals", to appear in Int. J. Solids and Structures. 2. M.W. Burkart and T.A. Read, "Diffusionless phase change in Indium-Thallium systems", Trans. AIME J. Metals 197, (1953), pp.1516-1524. 3. J.M. Ball and R.D. James, "Fine phase mixtures as minimizers of energy':, to appear in Arch. Ratl. Mech. Anal. 4. B.W. Batterman and C.S. Barrett, "Crystal structure of superconducting V 3 Si", Phys. Rev. Lett. 13, (1964), pp.390-392. 5. J.L. Ericksen, "Some phase transitions in crystals", Arch. Ratl. Mech. Anal. 73, (1980), pp.99-124.

_

^

Arch. Rational Mech. Anal. 139 (1997) 181-200. © Springer-Verlag 1997

Equilibrium Theory for X-ray Observations of Crystals J. L. ERICKSEN Communicated by M. GURTIN

1. Introduction In molecular theories of crystal elasticity, one needs some way of relating changes in atomic arrangements to macroscopic deformation. The oldest idea, introduced by CAUCHY, was that the gross motion agrees with atomic motion, at the mass points representing atoms. After appreciating reasons why this is not really sound, physically, BORN1 modified this assumption in what seems to be a very reasonable way. His assumption, called the Born Rule, hereafter abbreviated to B.R., has become the standard assumption in this area. It is widely appreciated that it is not consistent with observations of phenomena involving slip, commonly associated with dislocation motion and what we call plasticity. While linear elasticity theory has been used extensively in studies of isolated dislocations, we all know that it is not a good theory to use for predicting what deformations will occur in a simple tension experiment, for example. Elasticity theory has been used successfully to deal with other kinds of deformations which can be large, associated with phase transitions involving changes of symmetry, including sophisticated descriptions2 of rather complicated arrangements of twins occurring with these. From this, it seems reasonable to include such deformations among those commonly regarded as elastic and, possibly, to use similar analyses for other kinds of twins which are rather different, physically. Those alluded to above are transformation twins, occurring naturally in unstressed samples as the temperature changes, associated with phase transitions. Then there are the growth twins, occurring naturally as a crystal solidifies from a melt or solution; some include twins occurring during annealing. Here again, elasticity theory has been useful in designing mechanical treatments to eliminate some of them. Early work of this kind, done during World War II, relating to Dauphine twins in quartz, 1 2

For a brief and critical presentation of this theory, see STAKGOLD [1]. For a recent exposition of theory of this kind, see BHATTACHARYA et al. [2].

165

166 182

J. L. ERICKSEN

was covered by the WOOSTERS [3] and THOMAS & WOOSTER [4]. So, at least some problems involving growth twins are within the province of elasticity theory. In the third category are the deformation twins, produced by applying loads involving shear, remaining when loads are removed. There is a good and recent survey of work in this area, by CHRISTIAN & MAHAJAN [5]. It is clear from this that workers have been struggling hard to find ideas which can be used to understand the phenomena observed, with limited success, including some use of elasticity theory, among other things. Studies have turned up so many complications that it is hard to be optimistic about finding any theory with a broad range of applicability. What are called mechanical twins can be either transformation or deformation twins. From a different perspective, ZANZOTTO [6] pointed out a basic difficulty which, for me at least, helps clear the air a bit. After looking at many observations, he concluded that B.R. is reliable for what might reasonably be regarded as elastic deformations for two kinds of crystals, in view of the aforementioned experience. One covers the monatomic crystals fitting the description of Bravais lattices. The other consists of the so-called shape-memory alloys. For other kinds of crystals, it sometimes applies, often does not, and there seems to be no obvious pattern in this. At least for failures relating to twinning, he gives an ingenious argument, convincing to me, that when B.R. fails to apply, so does elasticity theory. No viable alternative to B.R. suggests itself. In pondering such difficulties, I got to thinking about what kind of field theory would result, if one used the ideas of molecular theories of elasticity, but did not use B.R. Here, I present some of my thoughts on this, which lead to an equilibrium theory of lattice vector fields, etc., things more related to X-ray observations and to some made using electron microscopes. It is in a form such that, if one trusts B.R. for a limited set of deformations, one can use it to describe these. ZANZOTTO notes that, for some but not all crystals, B.R. applies pretty well to thermal expansion, although it might then fail for twinning. Also, I will set up an analog of the elementary theory of twinning used in elasticity theory, albeit by rather different reasoning.

2. On Molecular Theory Except for some earlier writers, those who work on molecular theories of crystal elasticity aim at calculating strain energy functions, using the energy arguments of thermoelasticity theory to deduce formulae for stress tensors. Unfortunately, this requires use of the B.R., to be avoided here. Various kinds of molecular theory are used, producing different strain energy functions. For continuum theory, I have looked for a format general enough to cover what is common to such theories. Here, I present some of the reasoning I used for this. Since various failings of B.R. are associated with polyatomic crystals, we will consider these. Consider a crystal filling all of space, with atoms idealized as mass points, to be definite. Commonly, it is thought of as undergoing temperature-de-

167 Equilibrium Theory for X-ray Observations of Crystals

183

pendent vibratory motion about positions of equilibrium. With some abuse of language, we interpret the latter positions as the positions of the atoms. The simplest crystal configurations are the Bravais simple lattices. For these, all atoms are identical, and the position XN of the Mh atom can be represented in the form xN = naNea + const.,

iV=I,2,...,

(1)

where ea(a = 1,2,3) are lattice vectors, required to be linearly independent. Also, the naN are integers, with every choice of integers giving a possible atomic position. Here and in the following, the summation convention is used. Actually, these simplest configurations are not of great interest here, since B.R. applies pretty well to them. Other crystal configurations can be described as a finite set of interpenetrating Bravais lattices with the same lattice vectors. Basically, the lattice vectors generate a translation group, describing the periodicity required by the classical definition of a crystal. Number the lattices by Greek indices. If there are M of these, they will have positions given by XNX = n"Nea +pa,

x= 1,...,M,

N = \,2,...,

(2)

where the pa are some vectors. These positions are to be distinct, which means that the differences pa — p^oi + p, cannot have integer components relative to the basis ea. Atoms on different lattices may or may not be identical. The relative translations like pa — p\ are called shifts, following PITTERI [7], who discusses some important features of such multi-lattices. Concerning molecular theories of elasticity, it is generally agreed that these deal only with rather short-range interactions. Since the aim is to get energy functions of the kind used in elasticity, one has to argue that, to a good approximation, they have the usual property of being additive set functions. One does not want to try to keep track of those little vibrations. For this, one either uses "zero-temperature" models, assuming they vanish or, if not, one uses statistical mechanics, typically, which introduces the temperature 6 as a control variable. One should be aware that there is some ambiguity in the multi-lattice description. For example, a body-centered cubic monatomic crystal fits the description of a Bravais lattice. However, it is more often described a 2-lattice, with lattice vectors orthogonal and of equal length, pictured as the edges of a cube, one atom in the second lattice sitting in the center of a cube. This might not be the best way to picture the maximal translation group, but it is good for picturing the point group, for example. Of course, for elasticity theory, one wants to cover some variety of configurations. Generally, if a configuration can be described as a multi-lattice with some value of M, it can also be described as one with a larger value of M, although not with the same choice of lattice vectors. Strictly speaking, lattice vectors should be associated with the maximal translation group for the configuration, which means a minimum value of M, so I am abusing language a bit. So, using individual judgement, one somehow fixes a value of M, for a calculation of a specific strain-energy function. Of course, one must make

^68 184

J. L. ERICKSEN

some assumptions about how the atoms interact with each other, and numerous possibilities for this to occur in the literature. Note also that ea and pa determine the positions of the atoms. It is then pretty clear that for say q>, the Helmholtz free-energy -function per unit mass, one gets some constitutive equation of the form cp = (p(ea,pa,6),

(3)

maybe this only for 9 = 0. With any decent molecular theory, it should be invariant under rotations and translations. For the former, it is described by ea -> Rea,

Pa

-> Rpa, R~l = RT, det R=\.

(4)

If we put this in, consider the infinitesimal approximation R = 1 + Q, QT = -Q, and differentiate (p with respect to Q at Q = 0, then we get a differential identity, asserting that a second-order tensor T is symmetric, where T = ^®ea+^®Pa = TT. (5) dpa dea From general experience with continuum mechanics, this should be associated with a balance of couples, like the condition the Cauchy stress tensor be symmetric. For the translations, we have ea -+ ea, pa -> pa + c,

(6)

where c is an arbitrary constant vector. Similarly, this gives the differential identity (7)

the experience being that this should be associated with a balance of forces. If one is familiar with B.R., one can apply it to elasticity theory to get a formula somewhat similar to (5) for the Cauchy stress, omitting the part associated with pa. If one knows the old molecular theory of Born, one might spot some equilibrium equations which eliminate those terms associated with pa, to be introduced later, I shall sketch how relevant calculations are made for a simple kind of molecular theory, a zero temperature model, with atoms subject to central forces. Essentially, these are old calculations presented by LOVE [8, note B]. Being of an older school, he calculated the Cauchy stress tensor for Bravais simple lattices and 2-lattices directly, then used elasticity theory to find the strain energy, the reverse of what is now the common practice. The generalization to Af-lattices is almost obvious, certainly easily done. I will just rearrange terms to fit the format suggested above, indicating how both are calculated directly from molecular theory of this kind. Here, the atoms are considered to be at rest. In the ath lattice, all atoms are identical, so have the same mass /iy. Pick one atom from each lattice and call this set a pseudo-molecule. It has the mass

169 Equilibrium Theory for X-ray Observations of Crystals u

0 = 5>«-

185

(8)

The central forces are described by a set of potential functions M=flk,

(9)

each depending only on the distance between one pair of atoms, consisting of one in the ath lattice, the other in the /fth. These should approach zero quite rapidly as the distances tend to infinity, so that infinite sums to be encountered converge rather rapidly. Generally, the energy of the infinite crystal is infinite. However, we can get a reasonable estimate of energy per unit mass by giving each atom a fair share of the energy. If x and y denote positions of two atoms, the pair potential for them will have a certain value, to be shared equally by them. For example, if we pick an atom in the first lattice, its share of the total energy is M

£5>«(K'e«+A.-'Pil)/2> n

a—1

(io)

where ^ n denotes the sum of over all integers n", excepting rf = 0 in f\\. It has the analogous meaning in calculations to come. Now add up the contributions of this kind for a pseudo-molecule, and divide by its mass, to get the energy per unit mass M


(11)

a special form of (3) for this theory, considered to apply at 6 = 0, where the difference between internal energy and Helmholtz free energy is no longer significant. With the rather explicit formula, one can verify aforementioned invariance properties of cp and some additional ones not yet mentioned, which also hold for other versions of molecular theory. For the positions described by (2), different choices of ea and pa give exactly the same set of positions. This gives transformations of the form ea-^rnbaeb,

pa -> pa + n % ,

(12)

where the w's andra'sare integers such that det \\mba\\ = ± 1 . Not all of these can be viewed as leaving

, one can replace this infinite group by a finite group, as is discussed by PITTERI [7]. There are the central inversions, for which the action is ea -> - e a , b

b

pa ->• - p x .

(13)

With m a = -S a in (12), we can, independently, reverse ea, so the pair could map the domain to itself. However, central inversions sometimes relate dif-

170 186

J. L. ERICKSEN

ferent constitutive equations describing enantiomorphs, so I am inclined to interpret this as indicated by <(>(ea,pa, 6) =
(14)

where q> and q>' might be the same or different functions, then describing enantiomorphs. In the latter case, one could regard the two as branches of a double-valued function which is invariant under the indicated transformation. If atoms in some lattices are identical, q> is a symmetric function of the corresponding vectors pa. I shall continue to include the improper transformations, because they can be involved in growth twins, for example, but do keep in mind the above discussion. To calculate the stress tensor, we use CAUCHY'S interpretation of it. Pick a plane with unit normal v and on it a region R of area S, bounded by a curve c. In (11), for example, the infinite sum is replaced by a finite sum, in practice, which means that interactions between atoms only count when the distance between them is less than a certain value. Roughly, a continuum point is a ball of about this size. The idea is that the region R has typical linear dimensions large compared to this, but is small enough to be regarded as infinitesimal, macroscopically. Ideas of this kind are used in justifying the notion that the Helmholtz free energy is (approximately) an additive set function, for macroscopic regions. The idea is to consider pairs of atoms whose lines of action cross R, to calculate the resultant force exerted by those on the side into which v points, on those on the opposite side. Consider one such pair, say at some positions x in the first lattice, y in the second, y being in the side into which v points, so (y-x)-v>0.

(15)

For some integers n", we have y-x

= naea+p2-p\,

and the force exerted on the one at x by that at y is

or

^+»-»i>i£S-

<16)

Various pairs of this kind have the same relative position vector and lines of action intersecting R, so multiply (16) by this number, «o- To estimate this, LOVE constructs a cylindrical region capped by R, this translated by the relative position vector, bounded on the side by the ruled surface with rulings parallel to this. Then, n0 should be the number of atoms in this region, a large number. Also, it should contain about the same number of atoms in each of the other lattices. The total mass for the region can then be estimated from microscopic and macroscopic reasoning, giving the equation

_i Equilibrium Theory for X-ray Observations of Crystals pS(y-x)

187 07)

• v = non,

where fi is the mass of a pseudo-molecule. For this set of pairs, we thus get an estimate of the resultant force as (16) multiplied by pS(naea+p2-p1)-v/fi.

(18)

This should be summed over almost all the values of n" consistent with (15). There is the point that, for some rather large relative position vectors almost perpendicular to v, their lines of action cross the plane outside R. Ignore this, for the moment. Now consider the analogous calculation for pairs in these two lattices, with that in the first lattice being in the region into which v points. If we look at the result, we see that this is equivalent to doing the first sum, for integers rf satisfying the reverse of the inequality (15). With (17), we can add in directions such that (y — x) • v — 0. So, add these two contributions, divide by S to get the force per unit area, and let R increase, tofillout the whole plane, to catch those directions temporarily ignored. For pairs in the same lattice, one uses a slightly different argument. Consult LOVE [8, note B] if this causes trouble. Adding up all the contributions of this kind gives, as a prescription for the Cauchy stress tensor t,

1

- P 1_ !_• f*A\V™P\)

2n\V «

'

(

}

where

K«p = n"ea +pa- pp.

(20)

With this and (11), it is straightforward to verify that, with T given by (5), t=pT p

= (^-®e"+w®p)-

(21)

I think that this applies quite generally, perhaps with some idea of stress which is not the same as CAUCHY'S, but do not know how to make a convincing argument for this. Among the troublesome cases is an old one, discussed in some detail by POINCARE [9], in connection with elasticity theory, covered more briefly by STAKGOLD [1], Briefly, POINCARE considered what are sometimes called pseudo-potentials, potentials depending in a rather general way on the positions of all the atoms. The negative derivative of this with respect to its position gives the force on an atom, but there is no sensible way of calculating a part of this, as exerted by subsets of other atoms. Obviously, CAUCHY'S idea of stress does not really make sense in this context. There is a counterpart of this in nonlinear field theory. Matter can produce fields, not with parts attributable to a subset, although a field can exert a force on a subset. However, in those molecular theories, with some approximations and use of energy arguments, one gets a stress tensor fitting the usual prescription used in elasticity theory, where CAUCHY'S idea is commonly used. This is not the place for a discussion of such subtleties, although the matter deserves serious thought.

]72

_^____^_^__ 188

J. L. ERICKSEN

We are not quite done, there being a matter of consistency. Certainly, any atoms subject to non-zero forces would not be at rest. For a Bravais simple lattice subject to central forces, it is easy to see that the force acting on any one vanishes but, as is familiar to workers in this area, this is not true for the more complicated configurations, in general. By calculations much like those we have done, one can calculate the resultant force fa on any atom in the ath lattice, and show that it is given by

/.-,£.

<*>

X > = 0,

(23)

From (7), what is automatic is that

01=1

i.e., that the resultant force on a pseudo-molecule vanishes, verifying the notion that (7) represents a balance of forces. However, we should also introduce, as equilibrium equations for/j a , the condition that each atom be subject to no force,

Then, when they are satisfied,

w,=°do t = p~-(g)ea.

(24)

(25)

Of course, t is here independent of position, so V • t = 0. If (24) is nicely soluble for pa, we can eliminate these variables and, with B.R., get the formula for the CAUCHY stress which is generally accepted. Note that, with (12), solutions of (24) for px are never unique. I think it likely that some of the failures of elasticity theory discussed by ZANZOTTO [6] are associated with the possibility that these equations are not so nicely soluble. For a purely mechanical theory like this, it seems to me unlikely, physically, to be able to find external forces to balance fa and not introduce other variables, perhaps electrical polarizations or magnetizations. Also, I am comfortable with assuming that (24) applies to general forms of the function q>, partly because I do not see how else to get molecular theory and elasticity theory to be consistent, when B.R. applies. So, we have scraped together a kind of format, a start toward building a continuum theory of equilibrium, albeit not of a very familiar kind. Clearly, what follows more directly from those molecular theories is not a theory of macroscopic deformation, but a theory of those vectors ea and pa, really a theory of X-ray observations.

3. Constraints and B.R. For continuum theory, consider ea and pa as vector fields, defined over some regions of space, these being smooth except on curves or surfaces or perhaps at isolated points. Singularities there occurring might well represent

^ _ _ _

173 Equilibrium Theory for X-ray Observations of Crystals

189

isolated defects, for example vacancies, dislocations or twins. Even when these fields are smooth, they can involve continuous distributions of defects. Measures of these associated with ea only are discussed in some detail by DAVINI & PARRY [10]. Probably, some failures of B.R. are associated with large numbers of defects. For example, the experiments of BAIZER & SIGVALDSON [11] on thermal expansion of zinc indicate a serious failure of B.R., which they attribute to changes in locations of vacancies, along with changes in dislocations. I do not have a very good feeling for what goes on in the motions associated with forming deformation twins. Logically, there are two possibilities which might occur when twins form without producing other defects. Either the changes in all atomic positions in a twin can be described by some linear transformation of the simple shearing kind or not. In the former case, the experience is that the macroscopic deformation gradient is likely to be such a transformation. In the latter case, workers try to fit this gradient to a lattice consisting of a subset of atoms, which seems to be possible, generally. Then, they try to estimate how the remainder have been shuffled around, to get to their final positions. Roughly, this is what "shuffling" means, in practice. There are inequivalent descriptions, more precise, in the literature. For further discussion of shuffling, including sketches of possibilities, see CHRISTIAN & MAHAJAN [5]. For multi-lattices, with pa determined by equilibrium equations depending on the material, shuffling does occur, in most cases. Here, I limit applicability of theory, by excluding continuous distributions of defects, essentially. From the discussion of DAVINI & PARRY [10], excluding continuous distributions of dislocations gives the constraint curl e" = 0,

(26)

where ea are the reciprocal lattice vectors, the basis dual to ea. At least locally, this is equivalent to having three scalar functions x" such that ea = V Z fl .

(27)

Generally, if c is a Burgers circuit, a closed oriented curve on which ea is smooth, the numbers

b" = I e" dx

(28)

c

are interpretable as certain components of the Burgers vector for the circuit, which should vanish unless the circuit encloses a line defect, usually regarded as a dislocation line, where e" can be expected to have singularities. Assuming ba = 0, for all circuits in a region where ea is smooth, is enough to make (27) apply throughout the region. Obviously, if we are to use the suggestions from molecular theory, we need some prescription for the mass density p. From the microscopic point of view, the mass per unit volume is the mass of a pseudo-molecule divided by the volume of a unit cell, suggesting what I accept, i.e.,

J74 190

J. L. ERICKSEN

p = nl\e\ • e 2 A e 3 | = p\ex •
(29)

Having numerous point defects, like vacancies or interstitials, could upset this. So, effectively, continuous distributions of point defects are excluded. From the earlier discussion of thermal expansion of zinc, one should be leery of trying to apply the present theory to it. Note that, if we use (12) to transform ea, the transformed vectors again satisfy (26) and give the same value to p. Deliberately, I have postponed describing B.R. To use it, one introduces some possible configuration as a reference and a definite choice of lattice vectors Ea for it. If F denotes the usual deformation gradient, relative to the reference, the assumption is that the vectors given by ea = FEa

(30)

are a possible set of lattice vectors in the deformed configuration. This gives ea=F~TEa.

(31)

If we regard this as a function of spatial coordinates x, we can verify that it satisfies (26). Also, (29) is consistent with the usual Jacobian rule for relating p to the reference mass density, when F is used. Commonly, Ea is taken to be constant. Then, combining (27) and (31) gives an equation which can be integrated easily, to give a

x

= X • Ea + const.

(32)

or, equivalently, X = xaEa + const.

(33)

with X representing positions of material points in the reference configuration. It is not hard to generalize this to cases where Ea is not constant, but (27) applies to it. So, if it seems safe to apply B.R. to a limited set of deformations, to link them to changes in ea and/?,*; one can use these ideas to do so. I have and will continue to assume that B.R. applies to the trivial cases of rotations and translations. With either (32) and (33) or the indicated generalization, X = const, o %a = const., so those crystallographic points are also material points, when B.R. applies. In some twinning situations, it is clear that some crystallographic or irrational planes of interest do not move as material surfaces, so B.R. cannot then apply. Here, in discussing B.R., I do not insist that ea be true lattice vectors, generating the maximal translation group: I am allowing some use of the weakened version labelled W.B.R. by ZANZOTTO [6]. I do agree with him that, for a decent theory, one must fix on some mode of description of configurations, fixing a value of M, for a given crystal. As is clear from the discussion by CHRISTIAN & MAHAJAN [5], failures of B.R. for twinning are commonly linked to shuffling which, in itself, does not produce anything recognizable as a defect. Commonly, in crystallography, workers mention (zi,z 2 ,z 3 ) crystallographic planes, where the zs are some integers, meaning a surface for which the normal has the direction

175 Equilibrium Theory for X-ray Observations of Crystals

191

zaea = V(z a X a ).

(34)

They either state or rely on common usage to identify how the ea are to be selected. Here, such a surface is described by what looks like the equation of a plane, i.e., zaIa = const.

(35)

so, I call these crystallographic planes. Workers also speak of irrational planes, described in the same way with real numbers za, not replaceable by integers. From this, and similar descriptions of crystallographic directions, it is natural to think of the x" as Cartesian coordinates of points in an affine space, which I call crystallographic space. So, instead of connecting positions in space to positions of material points in a reference, we are connecting them to these crystallographic points. Since boundaries of crystals tend to be made up of parts of such planes, this view can be of some help in thinking about possible boundary conditions. Given the various notions about atomic shuffling, it seems to me likely that making sense of macroscopic deformations might well involve averaging over a larger scale than is needed for X-ray observations, making it a bit tricky to combine these in a single theory. Clearly, adding a constant to / changes nothing of physical interest. Essentially, we have interpreted ^" as Cartesian coordinates in an affine space, which fits nicely with this translation group. Also, it has an infinite discrete group acting on it, inferred from (12). Putting these two groups together, we get transformations of the form

Xa^mtXb + C,

(36) a

where the
176 192

J. L. ERICKSEN

4. Equilibrium Equations From the previous discussion, it is pretty clear that we need to select some equilibrium equations for x", which involves some subtleties. For q>, etc., I assume that the types of constitutive equations considered in §2 apply to the fields ea and pa, although these are there assumed to be constants. However, given (27), it is better to replace ea by ea = Vx" in these, which is routine. So, we will have (37)

cp = (p(ea,px,6) = 7p(e",pa,6), with equilibrium equations given in part by

^ = ? = 0' Opa

OP*

(38)

6 being regarded as a constant, really a control parameter. With (38), the CAUCHY stress is given by

t= P

^t®ea

=

(39)

-^ ®lt» = ~9%®e">

from (5) and (21). This should satisfy the usual equilibrium equations V - f + / = 0,

(40)

/ being the body force per unit volume. Of course, in (37) and (39), e = V# a . For fitting the equations to principles of virtual work, etc., it helps to introduce the energy per unit volume a

w = plp = w(Vxa,Pa,0),

(41)

with p given by (29)2. This gives dVxa ® V * a '

~~ W

^ '

comparable to my [13, eq. (4.5)]. There, in eq. (4.3), is a prescription for a configurational stress, the obvious analog being *

= <&•

<43>

for which I use the same name. By a simple calculation, V • t = - V • caVX\

(44)

a

where det || Vx || +0, from the assumption that lattice vectors are linearly independent. Thus, the equilibrium equations have the alternative forms V.t=-f&V-ca

= -ga, f=-9aVxa-

(45)

In the latter form, they are of the usual Euler-Lagrange type, with a source term, better fitting variational formats. Given a conceptually clear statement

177 Equilibrium Theory for X-ray Observations of Crystals

193

of the physical interpretation of ca, it might be feasible to deduce the formula (43) from the molecular theory discussed earlier, or use the formula to find a good interpretation, something I have not yet tried. With (29) and the usual relations for dual bases, we have = p.e2 A e3 = pS/j1 A V* 3 ,

pe\ = /i|det Vf\e\

etc.,

(46)

For one thing, this is related to a measure of point defects discussed by & PARRY [10]. If if is a region, with e" smooth on dR, one has the measures DAVINI

H I e2 Ae 3 dS = f pex-dS, dR

etc.,

(47)

dR

where dS denotes the vector element of area. With (46), it is easy to show that V • (pea) = 0,

(48) a

so these measures vanish if R contains no holes and, in it, e is smooth. Note also that, from (29), (49) «

&

=

"

*

*

•

&

.

-

<

>

•

< « >

In language used by experts in the calculus of variations, p is a null Lagrangian. Let us turn to a principle of virtual work, much like that I [13] used for a generalization of elasticity theory. There, the discussion fails to mention that the domain variation of reference configurations can involve adding or subtracting mass. Here, I exclude all singularities and topological complications, dealing with smooth fields and smooth regions, topologically equivalent to balls. Reasonably, w can be expected to become infinite at values of its arguments for which positions of different atoms coincide. Certainly, loadings cause a body to occupy different positions in space, so we should vary domains. With no clear ideas about deformations, it is not so clear what should be meant by a material body, except that its mass should remain fixed, viz., Jt = / p dv = const.,

(50)

R

As usual, we consider one-parameter families of fields, defined over regions which similarly vary. For the domains, we have invertible mappings of the form x = x(x,t),

x(x,0)=x,

(51)

mapping some domain R(0) to R(t), with e

3x=

def VX ,

~dj(X>0)-

,

,.,,

(52)

Similarly, the functions %" n e e d to be defined on R(t), and varied independently, so we introduce functions

178 194

J. L. ERICKSEN

r(x(x,o),o) = xa(x),

r = r(x,t),

(53)-

with

«y=^M).

(54)

I assume that the fields pjj£,i) are obtained as solutions of (38) and are smooth, dpa being calculated in the same way as d%a. It turns out to be convenient to introduce the combination AXa

= Sxa + VXa • Sx,

the variational equivalent of the material derivative of the analog of a velocity field. First, a calculation gives, with

(55) a X ,

with Sx regarded as

e = / wdv

(56)

R

as energy, Ss = - I V • ca3xa dv + f {ca5Xa + wSx) • dS,

(57)

OR

R

where R = R(0). This is to be equated to virtual work, denoted by W, which can be regarded as a linear functional of dXa and Sx or, equivalently, of AXa and Sx. I use the latter, writing

W = J(f-Sx

+ gaAxa)dv + j ( h S x + kaAxa)do,

(58)

dR

R

where da denotes the scalar element of area and the coefficients of the variations are generalized forces. So we want de=W

when 5Jf = 0.

(59)

As noted earlier, I do not think it reasonable to think that one can produce external forces conjugate to pa, so I do not include a linear functional of them. Actually, one could deduce (38) from (59). It is easiest to deal with the constraint by using a Lagrange multiplier. However let us first look at balance laws obtained by using variations based on Lie groups leaving w invariant, so <5e = 0. There are the translations in physical space for which Ax" = 0,

5x = const.,

(60)

yielding

f f dv+ f hda = 0, R

(61)

dR

which I interpret as a balance of forces. For the rotations, Axfl = 0, Sx = dAx,

(62)

179 Equilibrium Theory for X-ray Observations of Crystals

195

with d any constant vector, we get the balance of moments

(63)

fxAfdv + I xAhda = 0. OR

R

Finally, there are the translations in crystallographic space indicated in (36), for which = Sxa = const.,

Af

dx = 0,

(64)

producing

f gadv+

fkada

= 0,

(65)

OR

R

which I interpret as a balance of configurational forces. We now consider general variations. As interior equations, we then get, by routine calculations V-ca + ga = 0,

/ + ^VZa = 0

in R.

(66)

With (44), these imply that (45) is satisfied, (66) emerging as the equilibrium equation favored for y_a, although (45)] is equivalent. Matching the boundary terms gives the familiar tv = h on dR,

(67)

where v is the outward directed unit normal, and the not so familiar ca • v = ka + Xpea • v o n dR,

(68)

with X as the Lagrange multiplier associated with fixing the mass. Now, at least as long as mass is conserved, we generally think of ~q> and 7p + a as physically equivalent, if a is a constant or, more generally an affine function of 9. However, they give different values to ca, the difference being the divergenceless vectors. a ^ -

= aPea.

(69)

In my estimation, we should regard such different vectors as physically equivalent. So, I suggest that boundary conditions like (68) need only to be satisfied to within this ambiguity, as is implied by (68). One can check that 7p and q> + a give the same Cauchy stress, using (41) and (42). Considering our rather meager experience with formulating boundary conditions, etc., for configurational forces, it is tempting to use the alternative formulation in terms of the stress tensor. However, considerations of configurational forces are being used increasingly, in studies of various kinds of defects, so there is reason to try to better understand them. There are recent studies emphasizing them, by GURTIN [14] and MAUGIN & TRIMARCO [15]. One should not view (51) as describing a one-parameter family of macroscopic deformations, and there is no compelling reason to do so. Even in elasticity theory, we use adjectives like "virtual", to indicate that such

^80 196

J. L. ERICKSEN

mappings might or might not be realizable as deformations, physically. Naively, one might expect that, for macroscopic deformations, the variational equivalent of the equation of continuity should hold, i.e., dp + V • (pSx) = 0,

(70) 3

and it does not for all variations considered above . On the other hand, the mass density associated with macroscopic deformation is likely to be some average of p, if this involves averaging on some larger scale. 6. On Twinning To a large extent, theories of twinning deal with twins present in crystals subject to no loads, in what are, or at least seem to be equilibrium configurations. I shall tackle these in a way which seems to me natural, for the theory here considered, although it is not completely conventional. Such twins could be of the deformation kind, for example, produced by loads involving shear. We are then doing a post-mortem of what occurs, after loads are removed, and the crystal has come to rest. We begin by considering configurations minimizing the energy £ = / pep dv,

(71)

R

subject to the constraint

f

Ji = I p dv = const., J

(72)

R

R being an unspecified region. Here, I use q> rather then 7p, to stay closer to what is conventional. Proceeding naively, we try to find constant minimizers ea and pa. For these, e = J?q>,

(73)

so it is a matter of minimizing q>. Clearly, p is constant, so all that matters about R is its volume. If ea and pa denote such a minimizer, we can apply the transformations leaving cp invariant to generate an infinite set of such minimizers. If ea and pa denotes any of these, there is a choice of symmetry transformations such that ea = Qmbaeb,

pa = Q{pa + naaea + c),

(74)

where Q is some orthogonal transformation, the rest being obtained by combining (4), (6), (12) and (13). For cases where atoms in different lattices are identical, one can generalize this a bit, allowing interchanges of corresponding/^. However, for simplicity, 3

By one way of interpreting this, our variations cover some atomic shuffling as well as gross deformations.

18J. Equilibrium Theory for X-ray Observations of Crystals

197

I ignore this. From the extremal conditions for a minimum, it follows that the stress tensor vanishes and the configurational stress is physically equivalent to zero. It might be possible to use the theory of space groups to say more about minimizers. Not having thought this through carefully, I will not say more. For elementary twinning theory, one considers a plane I call a twin4 plane, across which ea and pa undergo a jump discontinuity to a symmetryrelated configuration, some choice of ea and j?a in (74). One could get trivial examples of this kind by having one configuration throughout, with the different choices of ea and pa possible for one configuration. It is understood that these are to be excluded, that it is not possible to relate the configurations on the two sides by transformations of the form (74), with Q= \. For studies based on elasticity theory, another condition is used, kinematic conditions of compatibility coming from the assumption that the displacement is continuous on the twin plane. When B.R. fails to apply, it is at best doubtful to rely on this. However, there is a different assumption which is equivalent to this when B.R. applies, and is still usable when it does not. It is Assumption 1. For a twin, isolated from other singularities, it is possible to choose lattice vectors so that the Burgers vector vanishes for all Burgers circuits lying in a neighborhood of the plane, including those intersecting the twin plane. The statement is not conventional, so I will expend some ink, to explain how it relates to more conventional ideas used in discussions of twinning. Assuming it, one can pick some point in the neighborhood, assign a value of Xa = Xo there, then use the path-independent integrals implied by it to define Xa at any other point P in the neighborhood, as indicated by p

a

X (P) = xS + Jea-dx.

(75)

Pa

At the twin plane, #fl is then continuous, giving us an analog of the continuous displacement assumed in elasticity theory. This gives us the analogous kinematic condition of compatibility. Denoting by ea and ea the limiting values on the twin plane obtained as one-sided limits, we have ~ea = Vxa = Vxa + qav = ea + qav

(76)

for some scalars q", v being the unit normal to the plane or, equivalently, ea = He",

(77)

where H=l+v®q,

q = qaea.

(78)

Consider the equation of the twin plane in crystallographic space. With x" continuous, the two limits can be described by the same equation, so it makes sense to say that it is a certain crystallographic or irrational plane, not that it 4 In the literature, this and other adjectives are used, e.g., interface, twinning or composition.

182 198

J. L. ERICKSEN

is one thing on one side, another on the other. Certainly, this fits standard procedures in discussions of twinning. Also, for any vector u perpendicular to v, we have u • v = 0

=>

u • ea = u • ea,

(79)

which means that crystallographic directions in the plane, with the same Miller indices on both sides, coincide. An example of a growth twin in alum not satisfying this condition is mentioned by ZANZOTTO [16, note 1], so the assumption does not apply universally. As he mentions, there seems not to be a definition of growth twins which is generally accepted. However, I do not know of any observations of mechanical twins similar to those seen in alum. In using continuum theory, we do lose sight of the detailed arrangements of atoms on both sides, as is the case with X-ray observations. So, there is no way of knowing whether atoms on one side might be translated relative to the other by a fraction of lattice spacing, for example. This is something that might be seen, using electron microscopes, but one then encounters some uncertainty about how well what is observed in the small samples there used relates to what occurs in the bulk. What we have discussed relates to special choices of lattice vectors, but these must be compatible with (74), so e" must be some choice of ea. One implication is that det ea = ± det ea,

(80)

giving detH=

l+q-v

= ±\.

(81)

The relation between some growth twins does involve orientation reversing operations, so cases where the lower sign holds in (81) might be of interest for some of these. I do not think that this possibility is of interest for mechanical twins, and so accept Assumption 2. For mechanical twins, there is a way of choosing e" so that (81) applies, with the upper choice of sign. There is the logical possibility that, with different choices of e", one can get both choices of sign in (81), a possibility I have not explored, but I know that it happens, in special cases. With Assumption 2, we have det H = 1 <=> q • v = 0,

(82)

~ea=H-Tea = {\-q®v)ea.

(83)

That is, the linear transformation applying to lattice vectors has the form of that for a simple shearing deformation. For mechanical twins, it seems to be generally agreed that the relative macroscopic deformation gradient is of the simple shearing kind. To try to determine this from X-ray observations, workers try to find lattice vectors related as in (83) and, it seems, always find some. Most but not all observations indicate that twin planes are crystallographic planes. Then, theoretically, there are infinitely many, if there is one.

_ _ ^ _ Equilibrium Theory for X-ray Observations of Crystals

183 199

Many involve shears too huge to be considered seriously but, typically, one gets some different possibilities which look reasonable, on the face of it. Failures of B.R. occur when none of these agree with observations, as does happen for some crystals. Given a set of lattice vectors satisfying (82) and (83), one can work back, to show that these do conform to Assumptions 1 and 2, providing more evidence that these assumptions apply to mechanical twins. Using the fact that ea must also be one of the ea given by (74) leads to the rather standard twinning equation Qmbaeb = (\-q®v)ea,

(84)

Assumption 2 implying that d e t g d e t ||m*|| = 1.

(85)

The experience is that (84) admits various solutions not matched by observations, suggesting that some other criterion should be introduced, but I do not know what to propose for it. For Bravais simple lattices, there is no loss in generality assuming both determinants are positive, so Q is a rotation. However, one should be aware that the possibility that both are negative has a different effect on (73). Even when it fails to apply, one can use B.R. to borrow twin analysis based on the version of (84) which is used in elasticity theory, giving us the analogous elementary theory of twinning. In a similar way, one can adapt most of the theory of twin microstructures. However, I do not want to expend the ink to explain subtleties related to this. For deformation twins, it is clear from evidence discussed by CHRISTIAN & MAHAJAN [5] that interaction of twins with dislocations and other defects is important, so there is also reason to adapt theories of such defects based on elasticity theory, which seems feasible. Acknowledgement. I thank MARIO PITTERI and GIOVANNI ZANZOTTO for helpful dis-

cussion of various aspects of this theory. References [1] STAKGOLD, I., The Cauchy relations in a molecular theory of elasticity, Quarterly of Applied Mathematies, 8, 169-186 (1950). [2] BHATTACHARYA, K., Firoozye, N. B., JAMES, R. D. & KOHN, R. V., Restrictions

[3] [4] [5] [6]

on microstructure, Proceedings of the Royal Society of Edinburgh, 124A, 843-878 (1994). WOOSTER, W. A. & WOOSTER, N., Control of electrical twinning in quartz, Nature, 159, 40-406. (1946). THOMAS, L. A. & WOOSTER, W. A., Piezocrescense - the growth of Dauphine twinning in quartz under stress, Proceedings of the Royal Society of London, A208, 43-63 (1951). CHRISTIAN, J. W. & MAHAJAN, S., Deformation twinning, Progress in Materials Science, 39, 1-157 (1995). ZANZOTTO, G., On the material symmetry group of elastic crystals and the Born rule, Archive for Rational Mechanics and Analysis, 121, 1-36 (1992).

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J. L. ERICKSEN

[7] PITTERI, M., On v + 1 lattices, Journal of Elasticity, 15, 3-25 (1985). [8] LOVE, A. E. H., A treatise on the mathematical theory of elasticity, 4th ed., Cambridge University Press, Cambridge, 1927. [9] POINCARE, H., Lefons sur la theorie de I'elasticite, Georges Carre, Paris, 1892. [10] DAVINI, C. & PARRY: A complete list of invariants for defective crystals. Proceedings of the Royal Society of London, A432, 34-365 (1991). [11] BALZER, R. & SIGVALDSON, H., Equilibrium vacancy concentration measurements on Zinc single crystals. Journal of Physics F: Metal Physics, 9, 17-178 (1979). [12] ERICKSEN, J. L., On the symmetry of deformable crystals, Archive for Rational Mechanics and Analysis, 72, 1-13 (1979). [13] ERICKSEN, J. L., Remarks concerning forces on line defects. Zeitschrift fur Angewandte Mathematik und Physik, 46, Special Issue, S247-S271 (1995). [14] GURTIN, M. E., The nature of configurational forces, Archive for Rational Mechanics and Analysis, 131, 67-100 (1995). [15] MAUGIN, G. A. & TRIMARCO, C , The dynamics of configurational forces at phase-transition fronts, Meccanica, 30, 605-619 (1995). [16] ZANZOTTO, G., Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results, Notes I and II, Atti della Accademia Nazionale dei Lincei, 82, 725-741, 743-756 (1988). 5378 Buckskin Bob Road Florence, Oregon 97439 (Accepted May 2, 1996)

185 figAMtfMStl E=3ySS3

vveierstrass Institute tor Applied Analysis and Stochastics

Keport No. 18 2000

Anniversary Volume Krzysztof Wilmanski Contributions to Continuum Theories

A Minimization Problem in the X-ray Theory J. L. ERICKSEN 5378 Buckskin Bob Rd Florence, OR 97439, USA

Summary

and it is somewhat different from the analog used in elasticity theory. Here, I discuss some My purpose is to elaborate some features of ideas that I have found helpful for better una continuum theory of X-ray observations of crys- derstanding this problem. tals, relating to an energy minimization problem. The problem is similar to one commonly used in . thermoelasticity theory to analyze phenomena re- 2 Kinematics Ot 71-latticeS lating to phase transitions and twinning, but there „,. v , .. ,.a ihe X-ray theory is a continuum theory of deare some subtle differences, r i_i i • i tormable n-lattices, that is of n geometrically identical crystal lattices, translated relative to 1 Introduction each other. Atoms in any one of the lattices are required to be identical, but atoms occurThose involved in molecular theories of elas- r j n g m different lattices need not be the same, ticity theory for crystals commonly use the To within a translation, the identical lattices Cauchy-Born rule to convert functions of lat- c a n be described by three linearly independent tice vectors to functions of deformation gradi- vectors e a , the lattice vectors, determining the ents. Zanzotto (1992) pointed out that, for de- reciprocal lattice vectors (dual basis) e a , satisformations associated with twinning, for exam- tying pie, the rule seems to be reliable for Bravais lattices and shape memory alloys, but not for e" • e& = $%, e° ea = ea m & e(p ^ ea _> im-\\ael> a ray observations, what I call the X-ray theory. I ^"^ m _ | | m & | | ^ GL("i TL')' I (1997) formulated such a theory, patterned after molecular theories of elasticity, but not us- In all matrices used here, the lower index labels ing the commonly assumed Cauchy-Born rule. rows. To describe how the different lattices are There, I mentioned a relatively simple kind of energy minimization problem. More recently, I translated relative to each other, we use shift (1999) outlined a way to adapt the theory of vectors, denoted by microstructures based on elasticity theory, deP l ) ' = h---,n-l. (2.3) scribed by Ball and James (1992), for exampie. This involves analyzing finely space twins, T , . . ., ., , . J H , f ' T o determine these, pick some atom in one of T among other thmgs. In the adaptation, that t h e l a U k e s M ^ ^ ^ a n d ^ mimnuzation problem plays an .mportant role, rf t h e o t h e r ^ ^ ^ ^ ^ ^ . ^ ^ ^ 77

J_86 position vectors of the latter, relative to this origin. Again, there are infinitely many ways of choosing these for a given configuration and, as is discussed by Pitted (1985), for example, these are related by transformations of the form

where m, a and 1 are selected from the matrices occurring in (2.2) and (2.4). Here, (2.7)i describes a lattice group element for the skeletal lattice. It can happen that it -holds but (2.7)2 does not, the n-lattice having less symmetry than the skeletal. Also, there is a caveat. (2-4) g o m e descriptions of n-lattices are nonessenPi -> °qPj + ' i e a i h £%• . . . , ., .. , tial, meaning that the configuration involved ,-, 5 & ' For monatomic crystals, the a matrices are el, ,,, . ,, can also be described as a n-lattice, with ements of the group generated by ' n < n. If not, the description is called essen(a)Matrices obtained by replacing \ tial. For nonessential descriptions, one can't (2.5) rely on (2.7) to correctly describe the symmetry a column in the unit matrix by one with all entries equal to -1. . of the configuration: it can underestimate how (6)Matrices obtained by inter' much symmetry occurs in the configuration. The remaining discussion excludes nonessential changing two columns in the unit matrix. J descriptions. A lattice group element is then described by the following set of integers: This covers picking the origin and other atoms used to determine shifts to be any selection of (m, a, I) e L(e a ,p;). (2-8) atoms, taking one from each lattice and renumbering the lattices so they are, essentially, per- In particular, these satisfy mutations. For polyatomic crystals, only permP mutations permuting identical atoms are al= *> P ~ 1> 2 , 3,4, or 6, 1 ^ ^ lowed. The shifts are restricted by the condim p = 1 =^ otp = 1. J tion that two atoms must not be in the same T /n n\ ii ii .i i i In (2.9)2 the converse implication does not hold, ., , ,, , , , „c place, formalized by c J ' so it can be that a — 1 when p = 2, lor examn =£ / a P anH if 4 zt 4 "1 P le - F° r present purposes, this rather sketchy a p. _ n- J: l e f ^-"l description will suffice. A good general reference for these matters and other aspects of the theory of crystals is the book by Pitteri and for any integers If and If-. In most discussions of crystals, e a and p ; Zanzotto (1999). Results concerning various are regarded as constant vectors but, in the X- l a t t i c e groups are presented by them, Adeleke ray theory, they are vector fields, functions of ( 1 9 9 9 ). Ericksen (1999), Fadda and Zanzotto positions in space. Effectively, this is what is (1999), Parry (1998) and Pitteri and Zanzotto done in other continuum theories. With elas- (1998). Pitteri (1998) and I (1998) presented ticity theory, they are commonly regarded as different characterizations of nonessential defunctions of position in some reference config- 3 scnptions. Constitutive equations uration, which is essentially equivalent. The

concept of reference configurations is not ba- 3 sic to the X-ray theory, although one could introduce analogs of these, when it serves some useful purpose. Symmetry of particular descriptions is best described by the lattice groups L(e a , p;) introduced by Pitteri (1985). These are defined by the possible solutions of the equations Qea = mbaeb <=> Qea = (m^JJe 6 , "] > (27) Q G O(3), QPi = °^iPj + '? e a, J

Constitutive equations

The constitutive equations I (1997) proposed involve an assumption which amounts to exeluding continuous distributions of dislocations: (3.1) there are scalar functions \ a s u c n t n a t e" = Vx"(3-1) This and other entities are considered as functions of position in space, there being no material coordinates in this theory. Hereafter, (3.1)

78

187 is assumed to hold. Also, motivated by the idea phase transition problems, the configurations of that mass is a multiple of that of a unit cell, I interest can be included in a neighborhood of assumed that, for the mass density p, the kind introduced by Pitteri (1985), centered at some configuration. For such a domain, it p = ke e Ae |, (o.i) suffices to consider
ip =
(3.3) 4

Energy minimization

These are assumed to be invariant under rota- j ^ l g g 7 | tlonsi

l g g g ) b r i e f l y c o n s i d e r e d t h e p r o blem

of

minimizing the energy
R £ SO(3).

/

[

,

'

E = Jpvdx, n

In part, the equilibrium equations are z!f_ — o dpi

(4.1)

where ft is a fixed region, subject to the con' '' that the value of the mass,

stra 11

(3-5)

When these are satisfied, the other constitutive equations reduce to the form

M=

pdx

(4.2)

n

is fixed. Also, 9 is assumed to be fixed, so I will ignore it. In this, I was interested in configurations involving complicated arrays of twins, among other things, configurations not smooth enough to satisfy the equilibrium equations everywhere. Roughly, the idea is to consider the various material bodies of the same mass that could occupy the region fi, selecting those minimizin E In e thermoelasticity theory, one minimizes t h e anal ° g o f E f o r a fixed material body, instead. My general experience in such matters led me to believe that, for minimizers, the Cauchy stress should reduce to a constant hyV • t = —(V • c a )e°, (3.7) drostatic pressure, which depends on the value of M, for a given material and region Q. Correso, if the body force vanishes, as will be as- s p o n dence with knowledgeable persons has insumed here, dicated that, even for smooth minimizers, this , , is not obvious to some. I doubt that it is possi_ V- t = 0 and V • ca = 0, (3.8) , . . . r <•,,• •,. ble to give a rigorous general proof of this, with giving three independent equations for the realistic assumptions allowing for complicated three unknowns Xa• Concerning material sym- arrangements of twins, phase mixtures and the metry, the basic idea is that (2.2) and (2.4) de- l i k e a n d - certainly, I am not capable of doing so. scribe physically equivalent lattice vectors and W h a t i s feasible is to make the idea plausible, shifts. However, one needs to tailor this to u s i n g r a t h e r n a i v e f o r m a l arguments, Now what one considers to be the domain of V, and > l ( 1 9 9 7 ) s e t UP a Principle of virtual work w h i c h l u s e d t o one needs to exercise judgment in selecting this. > S e t equations satisfied by For example, for some but not all twinning and smooth extremals described previously. In this,

9

„

79

_188 I used domain perturbations, which need to be restricted to be mappings of 0 onto itself. This does affect some conclusions about boundary conditions, but not interior equations. Instead of trying to work with the extremal equations, let us approach the problem in a different way, still quite formal, possibly useful for those interested in devising a more rigorous treatment. First fix n, using your own judgment about the domain of ip. Introduce a subenergy function obtained by permitting p,to take on all values allowed, denned by cr(ea) = inLifi{ea, p{).

is the stress deviator, and p = — — — (£rt)/3

(4-9)

j S ) by a reasonable interpretation, the (possibly negative) thermodynamic pressure, also what is commonly regarded as the mechanical pressure, A formulation quite similar to this for thermoelasticity was introduced by Flory (1961), in a s t u d y of thermodynamics of high polymers, and I (1981) made use of it in another thermodynamic study. N o WWei n t r o d u c e

(4.3)

another subenergy function, an analog of one Chipot and Kinderlehrer Physically, I don't see a good reason to doubt (1998), Kinderlehrer (1988) and I (1981) used, that this can be regarded as a minimum, taken in elasticity and thermoelasticity theories, Fonon for some values of the shifts satisfying (3.5). seca and Parry (1992) using it for a more genIn the molecular theories of elasticity, workers eral theory, solve the analog of (3.5), but one also wants r ( " ) = inf. p -i =[/ cr(e a ). (4.10) solutions fitting a condition like (4.3), to get stable equilibria. With it, we have for any fields G r a n t e d t h a t i l reall y d e P e n d s o n *• 1 interpret e°(x) satisfying the mass constraint, it as what thermodynamicists commonly mean / p ( T ( e o ( x ) ) d x < E, (4.4) b y t h e H e l m h o l t z f r e e e n e r g y P e r "nit mass, ' for the kinds of physical problems being conJ with E evaluated for these fields and any ad- sidered, but they won't relate v to ea. For the missible shift fields: one needs to exercise some fields e " mentioned above, we should then have judgment about minimal smoothness require„.. r , \ , <. , E" = I pr(u)dx. = 1 ments. n I / (4 11) l Now, I digress a bit, to introduce a reJ u~ r(y)d^ < E' < E, ( ' n ' formulation of the constitutive equations which seems worth mentioning. Consider the change w h e r 6 ) ^ i n (4 5 \ of variables described by ) i'=l/p=l/(fce'-e2Ae3). (4.12) f"=ell/(e1-e2Ae3)1/3, a l i \ (4 5) e = (kv)~ / f I lr / 1 2 3M I Again, I don't think it unreasonable to regard ~ 'P ~ I \. \e e '\' J the infimum as a minimum. Then, using (4.5)u being the specific volume, the f° being homo- (4-9), it is fairly easy to show that, for the geneous of degree zero in the eb. Here, I have fields remaining after the selection implied by used (3.4). Then with (4-l°)> the Cauchy stress reduces to an hydrostatic pressure. Taking T(V) as a possible funcV = <£(f\ Pi,", 0), (4.6) t i o n ^ o n e g e t s Ei

_

obtained by substituting (4.5)2 in V, one gets, by a routine calculation, t —t

— ol

_

,

„,

'

(A 7)

a s o n e wou l d expect from this. Assuming this satisfies the equilibrium equation (3.8)i, perhaps in a weak sense, we get

(4.8)

T » = -p = const.

where, when (3.5) holds,

t D = p(V. | £ l / 3 - f ® | ^

,, ^

80

(4.14)

189 The difficulties in giving a rigorous derivation of t = - p i , p = const., for complicated miniraizers are like those discussed by Kinderlehrer (1988) for the analogous situation in elasticity theory, with additional complications associated with the mass constraint. For situations rather similar, mathematically, Chipot and Kinderlehrer (1988) and Fonseca and Parry (1992) showed that the mean stress is an hydrostatic pressure, so this might be provable for the case at hand. At least for less complicated minimizers, v should satisfy (4.14) and the mass constraint, linking the value of the pressure p to that of M. In elasticity theory, for example, workers generally assume that there are homogeneous minimizers and it is easy to see how these conditions work out for these. I believe that a thermodynamicist would be likely to simply assume (4.14), regard both p and M as control variables, and use standard techniques to analyze such problems, allowing the volume to vary. In proposing the problem described above, I was primarily concerned with adapting the theory of microstructures used in elasticity theory and, for this, it seems to me to trickier to deal with controlling p and M, since Q then needs to be variable. However, this deserves more thought: it is desirable to be able to control p.

237-277. Christian, J.W. and Mahajan, S., 1995, Deformation twinning, Progress in Materials Science, 39, 1-157. Ericksen, J.L., 1981, Some simple cases of the Gibbs phenomenon for thermoelastic solids, Journal of Thermal stresses, 4, 13-30. Ericksen, J.L., 1997, Equilibrium theory for X-ray observations, Archive for Rational Mechanics and Analysis, 139, 181-200. Ericksen, J.L., 1998, On nonessential descriptions of crystal multilattices, Journal of Mathematics and Mechanics of Solids, 3, 363-392. Ericksen, J.L., 1999, On groups occuring in the theory of crystals multilattices, to appear in Archive for Rational Mechanics and Analysis, Fadda, G. and Zanzotto, G., 1999, The arithmetic symmetry of monatomic planar S-lattices, to appear in Acta Crystallographica. Flory, P.J., 1961, Thermodynamic relations for high elastic polymers, Transactions of the Faraday Society, 57, 829-838.

Acknowledgment: I thank Marion Ericksen Fonseca, I. and Parry, G., 1992, Equilibrium for her help with the typing. configurations of defective crystals, Archive for Rational Mechanics and Analysis, 120, 245-283. References Kinderlehrer, D., 1988, Remarks about equilibAdeleke, S., 1999, On the classification of rium configurations of crystals, in: Materials monoatomic crystal multilattices, in prepara- Instabilities in Continuum Mechanics, ed. J.M. tlon Ball, Oxford University Press, 217-241. Ball, J.M. and James, R.D., 1992, Pro- P a r r V ) G ; 1 9 9 8 > Low dlmensional latUce posed experimental tests of a theory of fine groups for the continUum mechanics of phase microstructures and the two-well problem, transitions in crystals, Archive for Rational Philosophical Transactions of the Royal Soci- Mechanics and Analysis, 145, 1-22. ety of London, 333A, 389-450. Chipot, M. and Kinderlehrer, D., 1988, Equihbnum configurations of crystals, Archive for Rational Mechanics and Analysis, 103,

Pitteri, M. 1985, On (v + 1)-lattices, Journal Elasticity, 15, 3-25.

of

Pitteri)

81

M.

1998>

Geometry and

symmetry

_190 of multi-lattices, International Plasticity, 14, 139-147.

Journal of

Pitteri, M. and Zanzotto, G, 1998, Beyond space groups: the arithmetic symmetry of deformable multi-lattices, Acta Crystallographica, A54, 339-373. Pitteri, M. and Zanzotto, G, 1999, Continuum models for phase transitions and twinnings crystals, to be published by CRC/Chapman and Hall, London. Zanzotto, G, 1992, On the material symmetry groups of elastic crystals and the Born rule, Archive for Rational Mechanics and Analysis, 121, 1-36.

19J_ £ £ Journal of Elasticity 55: 201-218, 1999. ^ ^

201

© 2000 Kluwer Academic Publishers. Printed in the Netherlands.

Notes on the X-ray Theory J.L. ERICKSEN 5378 Buckskin Bob Rd., Florence, OR 97439, U.S.A. Received 24 September 1999 Abstract. My aim is to better compare thermoelasticity theory with a continuum theory of X-ray observations of crystals, to be described, to make it easier to adapt ideas and techniques from one to the other. To this end, I will present some different ways of formulating the X-ray theory. Also, I will adapt symmetry considerations from the former, partly to analyze problems of a kind not considered in thermoelasticity theory. Key words: constitutive equations for crystals, reference configurations.

1. Introduction Those involved in molecular theories of elasticity theory for crystals commonly use the Cauchy-Born rule, hereafter abbreviated to CBR, to convert functions of lattice vectors to functions of deformation gradients. Zanzotto [1] pointed out that, for deformations associated with twinning, for example, the rule seems to be reliable for Bravais lattices and shape memory alloys. However, for other kinds of crystals, it sometimes applies, but frequently does not and, in the latter cases, elasticity theory also fails to apply, in general. To provide some theory for such cases, I [2] proposed an equilibrium theory dealing only with the X-ray observations of lattice vectors, etc., hereafter called the X-ray theory. Here, my aim is to present some analogs of some useful results in thermoelasticity theory, to help in the development of the X-ray theory, and reformulations of equations emphasizing similarities between the two kinds of theories. Also, I shall discuss use of symmetry assumptions to help analyze some equilibrium equations that have no counterpart in thermoelasticity theory, as well as to analyze problems not considered in such theory. Elsewhere, I [3] outlined an adaptation of the theory of microstructures based on elasticity theory, presented by Ball and James [4], for example. This is useful in analyzing rather complicated configurations, involving many twins. Also, I [5] elaborated features of an energy minimization problem which is important for this, among other things. 2. Kinematics of n -lattices The X-ray theory is a continuum theory of deformable n -lattices, that is, of n geometrically identical crystal lattices, translated relative to each other. Atoms in

192 202

J.L. ERICKSEN

any one of the lattices are required to be identical, but atoms occurring in different lattices need not be the same. To within a translation, the identical lattices can be described by three linearly independent vectors ea, the lattice vectors, determining the reciprocal lattice vectors (dual basis) ta, satisfying ea • eb = Sab,

ea efl = efl ea = 1.

(2.1)

A specification of these vectors defines what is called a skeletal lattice, for any crystal configuration having the periodicity described by them. For any given lattice, there are infinitely many choices of lattice vectors, related by transformations of the form ea -» mbaeb <* ta -> ( m " 1 ) ^

m = \\mba \\ e GL(3, Z).

(2.2)

In all matrices used here, the lower index labels rows. To describe how the different lattices are translated relative to each other, we use shift vectors, denoted by p,.,

i = l,...,fi-l.

(2.3)

To determine these, pick some atom in one of the lattices as origin, and some atom in each of the other lattices. Then, the shifts are the position vectors of the latter, relative to this origin. Again, there are infinitely many ways of choosing these for a given configuration and, as is discussed by Pitteri [6], for example, these are related by transformations of the form p,-->a/p;+/?efl)

/?eZ.

(2.4)

For monatomic crystals, the a matrices are elements of the group generated by (a) Matrices obtained by replacing a column in the unit matrix by one with all entries equal to —I. (2 5) (b) Matrices obtained by interchanging two columns in the unit matrix. This covers picking the origin and other atoms used to determine shifts to be any selection of atoms, taking one from each lattice and renumbering the lattices so they are, essentially, permutations. For polyatomic crystals, only permutations permuting identical atoms are allowed. The shifts are restricted by the condition that two atoms must not be in the same place, formalized by p,-^/?e a

and,

ifi^j,

p, -p,-#/?•«„,

(2.6)

for any integers If and /?.. In most discussions of crystals, ea and p, are regarded as constant vectors but, in the X-ray theory, they are vector fields, functions of positions in space. Effectively, this is what is done in other continuum theories. With elasticity theory, they are commonly regarded as functions of position in some reference configuration,

193 NOTES ON THE X-RAY THEORY

203

which is essentially equivalent. The concept of reference configurations is not basic to the X-ray theory but, as will be discussed, one can introduce an analog of these, making it easier to see some similarities between the X-ray theory and thermoelasticity theory. Symmetry of particular descriptions is best described by the lattice groups L(e a , p,) introduced by Pitteri [6]. These are determined by the possible solutions of the equations Qea = mbaeb & Qea = ( m " 1 ) ^

Q e O(3),

Qp, = a/p,. +l?ea,

(2.7a) (2.7b)

where m, a and 1 are selected from the matrices occurring in (2.2) and (2.4). Here, (2.7a) describes a lattice group element for the skeletal lattice. It can happen that it holds but (2.7b) does not, a possibility to be discussed later. Also, there is a caveat. Some descriptions of n -lattices are nonessential, meaning that the configuration involved can also be described as a w'-lattice, with n' < n. If not, the description is called essential. For nonessential descriptions, one cannot rely on (2.7) to correctly describe the symmetry of the configuration: it can underestimate how much symmetry occurs in the configuration. I do not wish to deal with complications associated with this so, hereafter, any description considered is to be regarded as essential. A lattice group element is then described by the following set of integers: (m,a,I)eL(efl,p,).

(2.8)

In particular, these satisfy m* = 1, p = 1,2,3,4,6, mp = 1 =>• ap = 1.

(2.9a) (2.9b)

In (2.9b) the converse implication does not hold, and examples with a = 1 for p = 2 will be discussed later. For present purposes, this rather sketchy description will suffice. A good general reference for these matters and other aspects of the theory of crystals is the book by Pitteri and Zanzotto [7]. Results concerning various lattice groups are presented by them, Adeleke [8], Ericksen [3, 5], Fadda and Zanzotto [9], Parry [10] and Pitteri and Zanzotto [11]. Pitteri [12] and I [13] presented different characterizations of nonessential descriptions. 3. Constitutive Equations The constitutive equations I [2] proposed are of the form

t = p(ea®^+p,.®^)=tT, 11

~~~do'

(3.1)

194 204

J.L. ERICKSEN

plus a description of configurational stresses, to be mentioned later. Actually, I used position vectors relative to an arbitrary origin instead of shifts, an inconsequential difference. Here, the interpretations are that

R € SO(3).

(3.2)

T

For example, this yields t = t as an identity. Using the shifts takes care of the invariance under translations. Similarly, the molecular theory and invariance considerations led to equilibrium equations for the shifts, of the form | ^ = 0.

(3.3)

The idea that the mass is a multiple of the mass of a unit cell gave, for the mass density p, Id -e2Ae3|

(3.4)

where k is a positive constant. Also, by excluding continuous distributions of dislocations, I inferred that, at least locally, there are scalar functions xa such that efl = V x f l .

(3.5)

Henceforth, all such fields are assumed to satisfy this. Clearly,

ei • ez A e3 > 0 or ei • e2 A e3 < 0.

(3.6)

Similarly, the domain is not likely to include shifts violating (2.6). Obviously, this excludes replacing SO(3) by 0(3) in (3.2) or using (2.2) with detm = - 1 . It does not exclude combining the latter with improper orthogonal transformations, as will be considered later. I do have reservations about including such transformations in the invariance group for cp, in general, thinking it better to think hard about physical implications of this, in considering special cases, before assuming it. Later, I shall say a bit more about this. Of course, any transformations in the invariance group of

195 NOTES ON THE X-RAY THEORY

205

one of Pitteri's [6] domains, centered at an essential configuration. This picks out a finite subgroup as the group mapping the neighborhood to itself, essentially the elements occurring in the lattice group for the center, combined with (3.2). For some kinds of twinning and phase transition phenomena, elasticity theory of this kind has been quite successful. For other such phenomena, one needs domains to be larger than Pitteri neighborhoods, although it might be judged to be reasonable to use some bounded neighborhoods, in particular cases. Then, some transformations in the infinite group might map only some configurations in the domain back to the domain, and it is not so easy to properly account for this. As far as I know, no one has worked out an example of this kind. 4. Reformulations With (3.5), it is rather awkward to use the formulation (3.1), so it seems better to replace ea by e" in the list of independent variables, as I [2] did, for some analyses. Later, I will discuss a compromise, using ta, the common practice in elasticity theory, to better compare the two kinds of theories. After working with the theory a while, I found it advantageous to make an additional change. Introduce variables p" such that P,- = P>a => P? = P, • e*

(4.1)

and, in place of (3.1), use (p

= y{ea,p>[,9)

(4.2)

as the potential determining the constitutive equations. Making the change of variables, one gets t

= -"e"®|? = - ^ ® e B '

cfl = ^

+p|?p,

de°

(43a)

(4.3b)

dp? (4.3C)

in which it is not assumed that the equilibrium Equations (3.3) are satisfied: they become

•i*i| =a

(4.4,

As before, the symmetry of t indicated in (4.3a) is really an identity, derivable from (3.2). Further, the three vectors ca represent the configurational stress mentioned earlier, being analogous to the configurational stress used in elasticity theory. This

a 206

J.L. ERICKSEN

has its uses, in analyzing singular solutions representing defects, but I will not use it here. Bearing in mind (3.6), one needs to decide which orientation of lattice vectors to employ, so I choose d • e2 A e3 > 0 <s> e1 • e2 A e3 > 0.

(4.5)

Now this and the inner products Gab

def ta _eb

=

Qba

( 4 6 )

determine the ea to within a rotation, and any choice of these uniquely determines ea. Then p, is determined by (4.1), given any possible values of pf. Of course, the Gab must be coefficients of a positive definite quadratic form, to determine admissible lattice vectors. Since pf is an inner product of vectors, it is invariant under SO(3), in fact 0(3). Said differently, if ^ i s invariant under SO(3), as described by (3.2), it can also be considered to be a function of the form