Application of finite strain theory to non-cubic crystals

Application of finite strain theory to non-cubic crystals

I. Phys. Chcm.Solids, 1976,Vol. 37, pp. 711-718. Pergamon Press.Printed in Great Britain APPLICATION OF FINITE STRAIN THEORY TO NON-CUBIC CRYSTALS J...

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I. Phys. Chcm.Solids, 1976,Vol. 37, pp. 711-718.

Pergamon Press.Printed in Great Britain

APPLICATION OF FINITE STRAIN THEORY TO NON-CUBIC CRYSTALS J. Scold WEAVER Departmentof Physics,QueensCollegeof the City Universityof New York, Flushing, NY 11367,U.S.A. (Received 25 August 1975;accepted 14November 1975) Abstract-The application of finite strain theory to a crystal of orthorhombic or higher elastic symmetry, which is subjected to a purely extensional deformation parallel to the crystallographic axes, yields stress-strain relations and effective elastic constants for hydrostatic stresses. Alternative formulations of the theory are obtained when the free energy is written as Taylor series in E, the invariant analogue of the Eulerian strain tensor, or q, the Lagrangian strain tensor. In most cases, the formulation in terms of E provides a better approximation when the Taylor series are truncated at the second or third order. In the case of cubic crystals, the stress-strain relations reduce to the Birch equation. For non-cubic crystals, the P-V relations calculated using the non-cubic and Birch equations will differ due to the effects of elastic anisotropy. A comparison between the second-order approximations of the non-cubic stress-strain relations and the cubic Birch equation suggests that the difference in volume will be less than 1% for most materials. The difference in volume is reduced when the third-order approximations are used. When the third-order terms are retained, the stress-strain relations calculated using the Eulerian formulation agree with measured linear compression data for quartz to 150kbar (P/K, = 0.4). For zinc, the calculated pressure-volume relation agrees with the shock-wave Hugoniot up to 750kbar (P/K, = 1.2), although both calculated and Hugoniot P-V relations disagree with X-ray compression data. At pressures greater than 300kbar, the calculated axial ratio of zinc approaches that for other hexagonal metals (c/a = 1.63).

INTRODUCTION The pressure-volume relation known as the Birch equation or the Birch-Murnaghan equation was derived by the application of the theory of finite elastic strain to isotropic substances and to cubic crystals [l-3]. The Birch equation has been widely used to extrapolate compression data to higher pressures; to calculate P-V relations from values for the elastic constants measured at low pressure; and as a fitting equation for data smoothing or for determination of material constants from measured values of the pressure and volume. Surprisingly, the stress-strain relations analogous to the Birch equation for non-cubic crystals have not been presented in the literature. The Birch equation has sometimes been applied to materials with less than cubic elastic symmetry, although there has been no estimate of the magnitude of the error which might result. It seems likely that the Birch equation will continue to be applied to non-cubic substances when elastic constant data are not available and in situations, such as shock compression experiments, where only pressure-volume data are obtained. Hence, the non-cubic analogues of the Birch equation are needed both for direct applications to non-cubic materials and for estimation of the errors that result from use of the Birch equation for non-cubic crystals. In the following section, stress-strain relations for a crystal with orthorhombic or higher elastic symmetry, subjected to extensional deformation parallel to the crystallographic axes, are derived. Although shear stress and strain are excluded, these stress-strain relations are sufficiently general to include cases involving hydrostatic stress and/or uniaxial stress parallel to a crystal axis. When the stress is hydrostatic, these stress-strain relations constitute the analogue of the Birch equation and can be used to describe linear and volume compres-

sions or as fitting equations. A direct comparison between the pressure-volume relations calculated using the Birch equation and the non-cubic stress-strain relations can then be made and the error resulting from use of the Birch equation is estimated. THEORY The results of finite strain theory which are needed here have been presented several times [3-61, although applications have been restricted to cubic elastic symmetry. The present discussion will be based on Davies’ treatment[41 and his notation will be generally followed. The stressstrain relations are derived by considering a crystal of orthorhombic elastic symmetry which is subjected to a purely extensional elastic deformation parallel to the crystal axes. Expressions for effective elastic constants, linear and volume compressibilities and their pressure derivatives can then be obtained for the special case of hydrostatic stress. The corresponding equations for tetragonal and hexagonal crystals follow immediately from the orthorhombic results as special cases. For compactness, the results will be expressed in the Voigt notation. It will be noted that only index values I,2 and 3 are needed, since the shear components of the stress and strain tensors vanish for the deformation considered here. It will be convenient to employ a modified form of the tensor summation convention in which repeated indices (p, 4, r, s, t) are summed over the values 1,2 and 3, but the indices i, j and k are not summed. The subscript zero will be used to denote quantities evaluated in the natural or unstressed state. The zero subscript will be omitted from the elastic constants and compressibilities for the sake of brevity. The coordinates of a point in the material will be referred to Cartesian axes chosen to coincide with the

711

.I. S. WEAVER

712

crystallographic axes. The deformation carries a point with position vector (ai) in the unstressed, initial state to the position (x,) in the final, stressed state. A homogeneous extension parallel to the crystal axes can then be written as Xi= Ai&

(1)

or p/p0 = (1 c z?q,f-“*( 1+ 2rf2))1’2(1 + 27)?) ‘i2.

(7b)

The free energy associated with the deformation of a hyperelastic material can be expanded in a Taylor series in the strain measures in the form

where hi is the ‘“stretch” associated with the ith crystal axis. The stretches can be related to the initial and firmi values of the lattice parameters of the crystal by Al = alao hz = b/b

(21

A3 = c/co.

Following Davies [4], the deformation gradients G and F are introduced by @.I

and Qbf

where pO$ is the free energy per unit initial volume; the coefficients are evaluated in the unstressed state; and the Voigt notation is used. The coefficients are interpreted as elastic constants and are considered as adiabatic or isothermal when p& is taken to be the internal energy or the Helmholtz free energy respectively. When @a) is written in terms of the ni by means of the relation, Ei=ni-2$+.*s, and corresponding powers of the vi are compared with (8b), it is found that the second-order coefficients cii are the same for the two form~ations and that the third-order coetftcients satisfy ret&ions of the form

where SQis the Kronecker delta. The strain measures which will be employed are W)

and

in the full tensor notation. The strain measure E is the frame-indifferent anaiogue of the “Eulerian” strain tensor]4,7], and ‘1 is the “Lagrangian” strain tensor, Since the tensor E and the Euferian strain tensor e used by Birch/S] are identicat for the particular deformation considered here, the formulation based on the use of the strain measured E will be called the “EuXerian” formulation, although E is properly a “Lagrangian” strain measure [6,7]. Both formulations of the problem will be developed in parallel throughout this section and will be compared in the discussion. Co~espondi~~ results for the Eulerian and Lagrangian formulations will be denoted by equation numbers ending in “a” and “b” respectively and Eulerian coefficients will be distinguished, when necessary, by the superscript *. For the deformation (I), the strain measures become a = ict - A?)

@at

vi = ffh? - 1)

iTb>

and

when written in the Voigt notation. The density p, and volume V, relative to the values in the unstressed state, can be written in terms of the stretches by p/p0 = (V/V&’ = (h,Xzh$“l and in terms of the strain measures as p/p0 = (I - ZE$“fl -2&)‘“(1-

2E3)‘”

From (8) and (9) it can be seen that the elastic constants are inv~i~t under ~~terch~ge of indices so that there are 6 independent second-order and 10 independent thirdorder elastic constants for an orthorhombic crystal. The additional relations among these constants which are required by symmetry in the tetragonal, hexagonal, rhomboh~ral, and cubic systems reduce the number of independent elastic constants. as shown in Table 1 (adapted from Brugger’s tab~~a~on [8])* Table 1. Independent second and third-order elastic constants (adaptedfromBrugger[8]) Laue O?xUD

Application of finite strain theory to non-cubic crystals

The Cauchy stress tensor T is given by[4] Tii = plpoGdpodlaEmG,i

(104

?;I = p/pOE,apO+/an&S

(lob)

713

The pressure derivatives of the effective elastic constants c:~ are obtained by direct differentiation of (13). The results are unwieldy, but reduce at P = 0 to

and

where tension is taken to be positive. Substitution from (3) and (8) then yields the stress-strain relations rT;= p/pO(l -2Ei){ci,E, +fc&E,E, +. . .}

(lla)

Ti =p/po(l+217i){c,,9,+fci,~~~77~+..s}

(llb)

C&(O)=Cij(Pu+2Pi+2pj)+lt26ij-C;$~,+”’

(Ha)

~:i(O)=C~i(p,-2~~-2~,)+1_2~~~-Cii,P,t~..

(15b)

The pressure derivatives of the linear compressibilities p: = apJaP can be calculated from

and

p : = - sire:,ps

in Voigt notation. These stress-strain relations can be written in terms of the relative length of the crystal axes (alao, etc.) by means of (2), (5) and (6). When the stress is a hydrostatic compression (i.e. T, = -P), the three equations for the Eulerian formulation (lla, with i = 1, 2, 3) constitute the analogue of the Birch equation for an orthorhombic crystal. The stressstrain relations can be numerically evaluated for the strain components when the pressure and the elastic constants are given. The pressure-volume relation can then be calculated using (5a). These stress-strain relations can be fitted to linear compression data in order to determine combinations of elastic constants. To conclude this section, some useful results for the special case of hydrostatic stress will be presented. For hydrostatic stresses, the effective values of the elastic constants are [4] ~~~~~ = plpoG,iG,~(a’po~laE,,aE,~)G,,G,r

+ P(&SjI + S&k + S&!)

(124

and ciikr=

q(P)

which follows from (14) where Sijis the compliance tensor (si,cj = 6,). The pressure derivative of the volume compressibility becomes p:=-c:J3&

= p/po(l-2Ei)(l

(17)

and the derivative of the bulk modulus is given by K ’ = - ,B-$ b= /3-2c :,/3&

(18)

At P = 0, (IS) and (18) combine to yield (1%) (1%) where AEP~-‘(P~‘+P~*+~~-PIPz-PzP~-PIP~).

(20)

The parameter A, defined by (20), describes the contribution of the anisotropy of the linear compressibility to the value of Kh when only second-order terms are retained in the expansion of the free energy (8). The definition (20) can be written in the equivalent form

p/poF,,Fk,(a'p,~/a~~7pqaD,~)F,,E, - P( S&I + &l&k- &SW) A = fK’{(j?~ - Pz)‘+ (& - &)‘+ (Pj -P,)‘} (12b)

in the full tensor notation. Substitution from (3) and (8) yields

(16)

(21)

from which it is seen that A 3 0. This parameter can also be written in terms of the rates of change of the axial ratios with volume as

-2Ei){cii + c$E, +. . .}

+P(1+26ij)

(13a)

and c&P) = p/po(l + 27li)(l + 277ixcij+ ci,,n, + . .} +P(1-2&)

(13b)

in Voigt notation, where Cij(P)denotes an effective elastic constant for an infinitesimal strain superimposed on the strain resulting from the hydrostatic prestress -P. The coefficient of P in (13) is correct for i, j = 1,2 and 3. When shear constants are calculated, it must be replaced by the complete term from (12). The linear compressibilities pi = Ai-*aAJaP; the volume compressibility p. = - V-‘dV/aP = p, + pz + p,; and the bulk modulus K = /3-’ can be calculated using (13) and the relation c,,(P)/?, = 1.

(14)

In the case of tetragonal, hexagonal or rhombohedral crystals the parameter A assumes the simple forms A=(yrz(s)‘.

(23)

The numerical value of A is shown as a function of the linear compressibilities in Fig. 1, for materials with positive linear compressibilities. The minimum value A=0 occurs only when p1 = pz = /& (i.e. for cubic crystals or as a possible special case in other systems) at the corner indicated in Fig. 1. The point corresponding to a tetragonal, hexagonal or rhombohedral crystal will lie along the diagonal when p. < /?,, or along the sides when p. > &. Excluding crystals having negative linear com-

J. S.

714

WEAVER

formulation, and that K; = -48 for the Lagrangian formulation. Since 0 G As 1 (when cases with j%< 0 are excluded), 4 c K6 c 8 for the Euferian formuiatio~ and - 4 =GKAs 0 for the Lagrangian formulation. Since values for K6 for real materials tend to he in the range 3-8 approximately~9, lo], the second-order Eulerian formuiation of the stress-strain relation provides a better approximation to the third-order property K& than does the corresponding La8rangi~ equation. It is possible to estimate the magnitllde of the fourth-order terms in the case of cubic crystals. For hydrostatic compression of a cubic crystal, (11) becomes 02

P = -( V/VJ”‘(C$E where E = i{l

0 0

0.2

04

06

08

+ COE’ + CW’+ . . .)

(24a)

- ( ViVJ”“}, or

10

&‘A Fig. 1, Value of the an~s~~ropy parameterA detinedby GO).The linearcompressibilitiespi are orderedso that p,,& G &. pressibiIities, the maximum value A = 1 can occur when fi, = &=O; p3 ?r 0. For tetragonal, etc. crystals with j% > &, a maximum value A = 0.25 occurs when & = 0: &, f 0. Since values for A greater than about 0.06 imply that the ratios of the linear compressibilities exceed 2: I, the value of A will be small for most materials. However, the value of A can exceed 0.5 in some cases suc.h as zinc or cadmium.

where n = ~(V/V”)~‘3- I). The coeffcients CT and C; can be expressed in terms of the bulk modulus and its pressure derivatives, evaluated at P = 0 as follows: C4=C:=3& CT = - (912~K~K~- 4) CS=(9/2)K”(K~K~~K~2--7K~~~) C, = - (9~2~K~K~ C,-(912)K,(K,KltKd2~K~-~~.

I3ISCUSSION The Eulerian and Lagran~an formn~ations, which were The relative irnpo~~~e of the fourth-order terms in the developed in parallel in the previous section, will yield two formulations is then expressed by the ratio identical resuits when all of the terms in the Taylor series ~C~E*)/(C~~*).The author has previously pointed out11 11 expansion of the free energy (8) are retained. In practice, that both measured values of Kg for three cesium halides the series will be truncated after some order-usuahy the and values calculated from simple lattice models for ionic second or third order-and the two formulations will no solids suggest that the quantity - K&JKb tends to tie in longer yieId the same values. It is desirable to employ the the range 1-2. For 3 G Kk G 8, and 0 G - &Kl: -S2K& the formulation with more rapidly convergent series, in order ratio CHIC, < 0.36 and the effect of the fourth-order term to minimize the error due to neglected higher order terms. on the pressure will be smaller in the Eulerian formulation Since there is little information concerning the value of when VIVA> 0.41. the fourth-order elastic constants, such a choice must Finally, an important diiIerence between the two straiu primarily be based an the relative magnitude of the measures is that, as the stretch or volume approaches third-order terms in the two formulations. Althou~ there zero, E increases without bound whereas v approaches a is no general solution to this problem, several arguments finite limit TI= - f. Hence, for Iarge order terms, it seems suggest that the Eulerian formulation may be superior for likely that Cf/C, < 1 if both series in (24a) and (24b) are practical problems. In the following, the expressions to converge at large compression. If this is so, the “second-order” or “third-order” will refer to the order of contribution to the pressure from the high-order terms the elastic constants retained rather than the power of the will be smaller for the Eulerian formulation. strain measure. Since the arguments presented above all suggest that The relation (9) between the c$ and the c:~~can be the Eulerian formulation provides more rapidly conused to indicate the relative importance of the third-order vergent series expansions than does the Lagrangian terms in the two form~ations. Since the values of the CQ~ formulation, the Eulerian formulation will be used tend to be negative and the Cijare usually positive, the C& t~oughout the rest of this paper, and the Lagrangian may tend to be smaller in magnitude than the Cijkas was formulation will not be considered further. suggested for cubic crystals by BirchM, Hence, the The Eulerian formulation of the non-cubic stress-strain Tad-order terms may tend to be smaller for the Eulerian relations consists of a set of three equations (I la) which formulations. The same conclusion is reached by compar- must be soived simultaneously. Rearranging the stressing the values of K; catculated for the second-order strain relations in the form approximation to the two formu~tions. For the secondorder, {19} indicates that Kb = 4t43 for the Eulerian

715

Application of finite strain theory to non-cubic crystals

$i = -P(p/po)-‘(1 - 2Ei)-’ -ki,sE,E, -

+

(26)

allows solution for the strains by iteration when the pressure and elastic constants are given. The right-hand side of (26) is evaluated using trial values for the E, (R = 0 can be used to start the process), and the resulting values for the +, are substituted in (25) to yield improved estimates of the strains. The process is repeated using the new volume until succeeding values agree to the necessary precision. Since the sequence of trial values for the Ei oscillates about the final values, the rate of convergence is improved if the average of successive values is used as the new approximation. This procedure was successful except for a few extremely anisotropic cases found in the Monte-Carlo study to be described later, or when the third-order terms became large. In difficult cases, the instability was removed by using a weighted average of successive trial values to reduce the oscillations at the cost of slower convergence. By treating hexagonal or tetragonal crystals as orthorhombic, it is unnecessary to use a special program to handle these cases.

anticipated that the third-order contribution might be relatively large for quartz. The linear compression curves for the u-axis and c-axis of the crystal were calculated from the Eulerian stress-strain relations (lla) and are compared with experimental values determined by X-ray diffraction techniques [14] in Fig. 2. (The pressure values from [14] were recalculated using the measured NaCl lattice parameter values and the pressure scale of Weaver et al. [15]). As expected from the value for Kb, the second-order approximation is inadequate for quartz, with the larger discrepancy occurring for the c-axis. Quartz

I

099

a/a, 098

097

c/co 096

EXAMPLES

The rhombohedral material (~-quartz can be used to illustrate the use of the non-cubic stress-strain relations since measured values are available for both second-order [ 121 and third-order [ 131 elastic constants. The elastic constants and several quantities derived from them are shown in Table 2. Since the value K; = 6.28, which was calculated using the third-order elastic constants, differs significantly from the value Kh = 4 + a

quartz.

48 = 4.03 for the second-order

approximation,

it was

0

40

20

60

80

Flastic

constants Zinc II

II

ij

Cij

1

-2.10 -3.45 0.12 -2.23 -2.94 -3.12 -3.32 -8.15

113 122 123 133 222 333

Derived

hi]

Cijk

nuantities

PSI

1.6368 .3640 .5300 .6347

1.0575

!1 I I

112 111

8.32 -3.17 0.60 -1.95 -2.94 -2.64 7.10 4.54

Mb

1

-17.6 -4.4 -2.7 2.1 -2.1 -3.5 -24.1 -7.2

Yb

2.0 )"b -2.9 -0.6 3.6 -2.1 -1.4 -4.5 0.4

(at P=O) ZbC

(Mb-l)

.9375

.1479

P!b-I)

.7264

1.3286

0"

(Mh-')

2.6733

1.6243

K

mhl

.3741

.6156

1

E3

A

.0085

.5284 0.14

B;

(Mb-2)

-15.0

R;

(!4b-2)

-14.9

-18.06

R:

(m-2,

-44.9

-17.78

x'

II

Mb

CTjk

Mh

ouartz 6

=i j

Mb

:1191 0704

ijk

8)

b2J

.8680

6.28

140

constants using the Eulerian formulation for both second-order (dashed curves) and third-order (solid curves) approximations. The calculated compression is compared with measured values obtained using static X-ray techniques[l4].

ouartz

11 12 13 33

120

Fig. 2. Linear compressionof ~-quartz calculated from elastic

Table 2. Elastic properties of a-quartz and zinc r,

100

P (kbar)

6.74

716

J. W. WEAVER

However, the third-order approximation agrees with the measured values up to the greatest pressure reached in the experiment (150kbar). The agreement between calculated and measured linear compression suggests that the effect of differences between the adiabatic or mixed elastic constants and the isothe~al elastic constants, which were neglected in the c~culation, is small. It is interesting to note that the cl& are not small compared with the Cijk as was suggested earlier in the discussion. Thus, Birch’s conjecture [3] that the third-order coefficients are smaller for the Eulerian formulation may not be universally true. However, second-order Eulerian formulation should still provide a better approximation to the pressure-volume relation since the value lC6= 4.03 is in better agreement with the measured value (6.28) than the value calculated from the second-order Lagrangian formulation (-0.03). Zinc. The hexagonal metal zinc provides an example of an extremely anisotropic material for which both second and third order elastic constants are known[l6,17]. As shown in Table 2, the value K&= 6.74 is largely accounted for by the contribution 4 + 4A = 6.11 from the second-order terms. It will also be noticed that the c& tend to be smaller in magnitude than the c,,k.Hence, the difference between the second and third order approximations is expected to be smaller than in the case of quartz. The calculated values for the linear compression of zinc are compared with measured values obtained by X-ray diffraction techniques[l8] in Fig. 3. Although the calculated and measured values for the a-axis compression are

0.92 - McWhan [Isj 0 o/u0

0880

I

,

1

50

I 100

I

i 150

P (kborf Fig.

3. The linear compression of zinc calculated from elastic

constants using the Eulerian formulation. The third-order approximation (solid curves) and the second-order approximation (dashed curves) are compared with smoothed experimental data[lll].

in fair agreement up to 120kbar, the values for the c-axis disagree seriously at pressures greater than about 40 kbar. However, the pressure-volume relation for zinc shown in Fig. 4 was found to agree with the shock-wave Bugoniot [N--21] for pressures up to 750 kbar when the third-order approximation is used. The difference between the P-V relations determined by X-ray and shock-wave techniques is large compared with both the experimental uncertainties and the Hugoniot-isotherm corrections, and remains to be explained. It should be noted that the calculated P-V relation will be approximately an adiabat (since adiabatic second-order elastic

16 l-

v/v0

0

02

04

0.6

0.8

I

12

P (mbar)

Fig. 4. Axial ratio and volume compression of zinc calculated from elastic constants using the second-order (dashed curves) and third-order (solid curves) approximation to the Eulerian formulation are compared with measured values for the Hugoniot[l9-211 and values obtained by static X-ray techniques[U]. A; Walsh et al. [19], 0; McQueen and March[20], A; Altshuler et a1.[21],0; McWhan[lIl].

constants were used and the mixed third-order constants are small) and hence will lie between the Hugoniot and the room-temperature isotherm. At P = 0, the value of the axial ratio (c/a = 1.856) of zinc is unusually large for a hexagonal metall221. Since the ratio of the linear compressibilities &lfi, (= 9, P = 0) is large, the Bxial ratio decreases rapidly with pressure as shown in Fig. 4. For pressures greater than 300 kbar, the axial ratio approaches the values found for other hexagonal metals such as magnesium and titanium and for packing of rigid spheres (1.633).In addition, the difference between the linear compressibilities is reduced at high pressure so that the variation of axial ratio with pressure is reduced. Second-order calculations for cadmium metal which, like zinc, has a large initial axial ratio, also suggest that c/a approaches the ideal close-packed value at high pressure. Application of the second-order approximation to a number of tetragonal and hexagonal substances shows that the axial ratio initially increases or decreases with pressure depending on the relative values for the linear compressibilities, and that difference in linear compressibilities is reduced at high pressure. COMPARISONWITH THE BIRCH EQUATION

For some applications, the pressure-volume relation rather than the linear stress-strain relation is of primary importance. Although (lla) contains an implicit P-V relation for orthorhombic crystals, the use of this equation requires that values for 6 second-order and 10 third-order elastic constants be available. Unfortunately,

111

Application of finite strain theory to non-cubic crystals

the necessary values have not been determined for many substances, although values for K0 and K& may be available. In such cases, the Birch equation might be used if the resulting error in the P-V relation were shown to be sufficiently small. A direct comparison between the two P-V relations is difficult due to the large number of material constants involved. However, in the secondorder approximation, the possible range of difference in volume between the Birch equation and the non-cubic stress-strain relations can be estimated. The parameter KII calculated from the two secondorder P-V relations provides one means of comparing them. The second-order Birch equation yields KA =4 whereas the non-cubic equations yield KA= 4 + 48 where A is a measure of the elastic anisotropy defined by (20). Since 0 < A c 1 (if the fit are all positive) the value of Ki, can range from 4-8 for the non-cubic equations. Hence, the bulk modulus will increase more rapidly with pressure for the non-cubic equations and the compression will be less at high pressure. Thus it may be expected that: (1) at a given pressure, the Birch equation will yield a smaller volume than that calculated from the non-cubic equations; (2) the difference in volume will tend to increase with increasing pressure; and (3) the difference in volume will tend to be greater for substances with larger values of the parameter A. In order to study the effect of variations in the vahres of the 6 independent second-order elastic constants on the P-V relation, a Nlonte-Carlo procedure was employed. A random number generator was used to produce a set of 6 elastic constants (the compliance matrix s was actually generated for convenience). The values were required to satisfy the conditions sii >O and SiiSji- s$>O, which are sufficient to insure that the strain energy is a positivedefinite function of the strains. Two additional requirements: pi > 0; and sii < 0 (i # j) were arbitrarily imposed to reduce the number of extraordinary cases considered. When a set of compliances satisfying the above conditions was found, the volume was calculated for a selection of pressures using @la), and the corresponding volume was calculated using the Birch equation (24aj-both truncated after the second-order. The procedure was then repeated until a large number (usually 1000) of sets of elastic constants had been examined. The calculated difference in volume is shown in Fig. 5 as a function of the anisotropy parameter A for 1000trials at a fixed pressure P = 4Ko. The differences at a pressure P = K. show a similar dist~bution but are from 1.5 to 2 times larger. As expected, the Birch equation always yields smaller values of V/V, than those calculated from the non-cubic equations. The difference tends to increase with increasing anisotropy parameter, although there is considerable variation not accounted for by this measure of anisotropy. The difference in volume-hence the error resulting from use of the Birch equation-exceeds 0.04 in 7 cases, and is greater than 0.02 in 142 of the cases studied. In fact, since high-pressure X-ray diffraction techniques are capable of determining V/V0 with an uncertainty of about 0.005, the volume difference is sig~~cant in more than 70% of the cases shown in Fig. 5. An examination of Fig. 1 shows that values of A greater

005

004

003

002

0.01

0

n Fig. 5. Difference in volume between the Birch equation (V,) and the Eulerian formulation (V,,), both in the second-order approximation. The distribution of differences in volume at P =$X, is shown for HlOOsets of randomly generated elastic constants as a function of the anisotropy parameterA.

than about 0.1 imply that the ratios of linear compressibilities exceed 2: 1. Since most real substances are not so highly anisotropic, the large volume differences shown in Fig. 5 may be misleading. For example, in the 247 cases with A < 0.1, the volume difference exceeds 0.005 for only 1 case at P =?Ko, and does not exceed 0.01 at P = Ko. IIence, although large errors can result from use of the Birch equation to describe the P-V relation of non-cubic crystals, the error may not be significant when the anisotropy parameter A is small. Since much of the error in the volume can be attributed to the difference in I(; (= 4 for the Birch equation; = 4+4A for the non-cubic equations), the error in the P-V relation may be reduced when the third-order Birch equation is used. When the third-order approximations are compared, the difference in volume will reflect differences in the fourth-order coefficients, since both P-V relations are constrained to have the same values of K. and I(& Hence, the volume difference will be small if the fourth-order terms are small. The calculations for zinc which were presented earlier illustrate this point. The relatively large value A = 0.52 was calculated from the linear compressibilities. The difference in volume between the second-order Birch and non-cubic equations was found to be 0.016 at P = 300 kbar (i.e. P = fKo). However, when the third-order approximation is used for both equations, the volume difference is reduced to 0.002. Thus, it appears that the second-order Birch equation does indeed provide a useful approximation to the P-V relation for non-cubic materials when the anisotropy parameter A is small. However, when A ~0.1 the second-order Birch equation should be used with caution. It seems likely that the third-order Birch equation will be an adequate approximation to the P-V relation in most cases, although further examples need to be tested. SWEMARY AND CONCLUSIONS The app~cation of finite strain theory to an orthorhombit crystal subjected to an extensional elastic deformation

718

J. s.

parallel to the crystal axes, yields stress-strain relations useful for problems involving hydrostatic and/or uniaxial stresses. When the stress is hydrostatic, expressions were obtained for effective elastic constants, linear and volume compressibilities, bulk modulus, and their pressure derivatives. Alternative formulations of the theory based on either the Lagrangian strain tensor 17 or on E, the rotationally invariant analogue of the Eulerian strain tensor, were described. It appears likely that the “Eulerian” formulation based on E, will usually be more accurate when truncated at a given order in the elastic constants since the higher order terms tend to be smaller than those for the Lagrangian formulation. For cubic crystals under hydrostatic stress, the Eulerian formulation reduces to the well-known Birch equation. The results for non-cubic substances are in the form of a set of three stress-strain relations (reduced to two equations for hexagonal or tetragonal crystals) which can be solved by numerical techniques to yield linear and volume compression when the pressure and elastic constants are given. The hexagonal materials a-quartz and zinc were used as examples of the application of the non-cubic stressstrain relations. The third-order terms were found to be significant in the case of quartz but were less important for zinc. A comparison between Kb and the quantity 4+4A, which is the value of Kb in the second-order approximation, indicates whether the third-order terms can be neglected. For both second and third order approximations, the initial anisotropy in the linear compressibilities was reduced at high pressures, so that the rate of change of the axial ratio for both quartz and zinc was also reduced. Calculations for zinc suggest that the axial ratio becomes comparable to that for other hexagonal metals at pressures greater than about 300 kbar. A Monte-Carlo study suggests that the Birch equation, which was derived for cubic crystals, can be successfully used to describe the pressure-volume relation for non-cubic materials if the elastic constant values needed for the non-cubic equations are not available. For the second-order approximation, the error in volume resulting from use of the Birch equation is probably less than 1% for most real materials at pressures up to Ko, although

WEAVER

much larger errors can occur for highly anisotropic materials. The volume error is significantly reduced when the third-order approximations are used. Since the anisotropy parameter A can be calculated from linear compressibility values or from second-order elastic constants, it can be used as a guide to the range of error which might result from use of the Birch equation. Acknowledgements-The author wishes to thank T. Takahashi for many helpful discussions and suggestions. This study was supported, in part, by a grant from the City University of New

York Faculty ResearchAwardProgram.

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