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The elastic constants of a cubic crystal subjected to moderately high hydrostatic pressure
J. PhYs.Chem..%tids Pergamon Press
1968. Vol. 29, pp. 417-422.
Printed in Great Britain.
THE ELASTIC CONSTANTS OF A CUBIC CRYSTAL SUBJECTED TO MODERATELY HIGH HYDROSTATIC PRESSURE D. H. CHUNG Department of Geology and Geophysics, Massachusetts institute of Technology, Cambridge, Massachusetts O? 139, U.S.A. (Received 22 Seprember 1967: in revisedform I7 October 1967) Abstract-This paper presents the results of an analysis of the finite strain theory that goes to the next higher-order term than third-order in the strains for cubic crystals. Theoretical expressions for the effective second-order elastic constants are presented here in terms of the second-, third-, and fourth-order elastic constants and the Lagrangian strain; these have been derived from the rigorous stress-strain telationship set by the Murnaghan theory of finite deformation. Since typical ultrasonic experiments with pressure result in thermodynamicaNy mixed elastic constants, the practical ultrasonic expressions of the effective elastic constants are presented for a cubic crystal subjected to moderately high hydrostatic compression. The rigorous relationships between the pressure derivatives of the effective elastic constants and partial contractions of the higher-order elastic constants are presented also, and then the primary experimental quantities that are resulting from the usual ultrasonic-pressure experiments have been identified in terms of the r~errno~nu~fico/~y mixed second-, third-, and fourthorder elastic constants of the crystal.
1. INTRODUCTION strain theory for a medium of the cubic symmetry subjected to a finite hydrostatic compression was first developed by Birch [ 11. In this work, Birch had defined the free-energy of a crystal in terms of a power series of the energy in the strain components including terms up to the third-order in the strains and then derived expressions for the effective elastic constants of the crystal under hydrostatic pressure. These expressions contained the second- and third-order elastic constants of crystal, and they were shown to describe the effective elastic properties of cubic crystals subjected to relatively ‘low’ compression.* For a crystal subjected to moderately ‘high’ compression, however, the THE FINITE
higher order terms than the third-order in the strains would play an important role in the description of crystal elasticity under this high pressure. This paper presents the results of our analysis of the finite strain theory that goes to the next higher order term than the third-order in the strains for cubic crystals. In the following, expressions for the effective elastic constants resulting from the rigorous stress-strain relationship set by the Murnaghan theory of finite deformation[2] will be presented and compared with the earlier work. Since typical ultrasonic experiments with pressure result in thermodynamically mixed elastic constants, the derived expressions of the effective elastic constants had to be identified according to their thermodynamic boundary conditions. This is done in Section 3. In Section 4, we present the rigorous relation of pressure derivatives of the effective elastic constants to partial contractions of the higher-order elastic constants. And, in Section 5, the present work is summarized with a brief discussion.
*The terms ‘low’ and ‘high’ compressions as used here are meant to imply the following: For a given solid under hydrostatic pressure, if the change in the second-order elastic constants with pressure is linear, this pressure (and up to this pressure) is referred to as relatively ‘low’ pressure. If the change with pressure is nonhear, the pressures higher than this pressure are referred to as relatively ‘high’ pressure. Although it is difficult to make a sharp boundary between the ‘low’ and ‘high’ pressures, 2. THE EFFECTIVE ELASTIC CONSTANTS the pressure in the order of 0.05 BT (where BT is the isothermal bulk modulus of solid under the study) for Using the usual Cartesian coordinates, we take the most solids can be referred to as the relatively low presprincipal axes to coincide with the symmetry axes of an sure and the pressure in the order of 0.1 BT may be re- unstrained cubic crystal. Let us denote the coordinates ferred to as the moderately high pressure. of a point p in the crystal before strain by (a I) a%,ua) and 417
418
D. H. CHUNG
those in the crystal after a strain by (x1, x2, x3). These points determine then a displacement vector with components (x-a,), (x2- Q), (X,-U,); thus, the components of the displacements are simply y=x,-0,.
(r=
1,2,3).
@=$fJ”+l#JI+~2+&1+dh+.
.
where I#J~ contains the jth order terms of r),j and for a body with no initial stresses 4,, and 4, can be set equal to zero. Thus, &, &, and& are:
(1)
For this reason, it is convenient to discuss the strain according to the Lagrangian scheme. The Lagrangian strain components in Murnaghan’s notation are[2]
(1 I) 44
where the subscripts ranges are 1, 2, 3 and 6, are the Kronecker deltas. Using the Jacobian J of the deformation, we can write equation (2) in a compact form as
[VI= 3( J*J- 13)
( 10)
=& ukrmno~7)i~7)kl~nlnT”lJ
where the subscripts ranges are 1, 2, 3 and the Einstein summation convention for repeated subscripts is assumed. The rigorous relation of the stresses T,, to the strains n.... 1s
(?.a) or in a compact form
where./* is the transpose of./ given by (12a)
db,,%,%)
J=
(3)
Na,, a2. ~1
and /,is the unit matrix. Consider a special strain which is composed of a hydrostatic strain and a subsequent homogeneous strain. By the homogeneous strain, the point p initially at (a,, Q, uY) is brought to the point (x1”,x2”,x9”), given by x,“= (1-(*)a,,
(s=
1,2,3)
Here, pO and p are the crystal densities before and after a strain, respectively, and the compression ratio is
PO
The additional displacements geneous strain are then u:.=x
&..&“,
(13)
det J
and the gradient of @ with respect to nns is
(4)
thus, every line in the crystal is shortened by the factor (I-a),where ru-n($?-1).
1
P
Ha@ -=
(5)
(14)
aqnq t,
resulting from the homo-
J (r,s=
1,2,3)
(6)
where &, = &; thus, & represent the six homogeneous strain components and they are infinitesimal quantities. The final coordinates are then given by X, = x,O+ u:.
(7)
From this relation Birch [ 11, we can set
(12), as was done by
T, = -PS,
where-P
is hydrostatic
t T:,
(19
pressure given by
From these, we find that
(16) z=
(l-(Y)(&?+&A.
(8)
Here, since /ST8are infinitesimal quantities as noted above, only the first powers of & are retained and all others are neglected. In view of the particular strain defined by equation (7), the Lagrangian strains become nrs= &S77+tPrr(L-t2n).
(9)
The strain energy density @ per unit initial volume is usually given by a power series of the Lagrangian strains as
and T:, are expressed in terms of the p’s and ‘a new set’ of elastic constants. This new set of the elastic constants are the effective second-order elastic constants under the hydrostatic pressure and we will denote hereafter to as CI1, C,, and C,,. Since
ELASTIC
CONSTANTS
SUBJECTED
and (18)
T;z = 2C,&
after having gone through a long algebra, we find the expressions for the effective secondorder elastic constants as follows: CM =
Cl1 +
7) (2c11+
+~2(-kll
2c12 -
+kllll + 2cll12
2c12 +
+
Cl11 +
+klll cll22
PRESSURE
419
Ghate[6]. In light of the present analysis, the writer believes that the minus signs of the quantities cI1 and c12 found in the q2 term of Ghate’s equation (23) should have been plus signs. And, as for the expression for C44, the quantity (+QcI14J should be found in the q2 term of Ghate’s equation (24).
&I,)
f5cl12 +
TO HYDROSTATIC
+
cl23
(19)
%23)
3. THE ULTRASONIC EFFECTIVE ELASTIC CONSTANTS
The expressions of the effective elastic constants as given by equations (19-2 1) can +r12(cll+~c12-hill-cl12 be either the adiabatic or isothermal expressions, and the proper designation of these is (20) + clllZ+ C1122+kl123) obviously done be adding the proper superscript either‘s’ or ‘T’ to all the elastic constants. c4, = c44 -t- 7) (Cl1 + 2c12 + c44+ Cl44 + 2%x) The acoustic data resulting from the usual --c,2-k44+3clll + q2(-c11 acoustic experiments with pressure are neither + 3c,,2 + cl23 + cl44 + 2clSS thermodynamically adiabatic nor thermodynamically isothermal quantities, but they (21) +?k1144 +- Cl155 + 2Cl25, + Cl266). are ‘thermodynamically mixed’ isothermalWhere cILy, cLLvxand cfivhCare the second-, adiabatic quantities [7]. Thus, in this section, third- and fourth-order elastic constants of we seek for the expressions of the effective crystal in Voigt’s notation, respectively, and elastic constants that may be resulting from experiments at high they are expressed in accordance with the the ultrasonic-pressure pressures. thermodynamic definition [3]. Recalling the usual behaviors of ultrasonic It may be noted that, in equations (19-2 l), the coefficients of the terms in r) with the wave velocities in the medium of a cubic we note that a longitudinal second- and third-order elastic constants are crystal[8,9], stiffness cI1 and shear stiffness c44 result the conventional expressions for the effective elastic constants[l, 4-61 and they agree with directly from measurements of the longitudithose derived initially by Birch[ l] when the nal and transverse wave velocities in the [OOl] direction of the crystal, respectively. If one third-order elastic constants in Birch’s definimeasures a transverse wave velocity in [ 1lo] tion are converted into those of more general thermodynamic definition.* However, the polarized in the [ 1TO] direction, the resulting coefficients of the terms in q2 in equations (20) stiffness constant is (cl1 - c12)/2. Thus, from and (21) are at variance with ones given by this, one finds immediately the elastic constant cl2 as a typical procedure. Following exactly the same procedure as the above but subjected *The relations between the c,.,“~defined by Briigger (c”,:,) and those by Birch (CS,‘;*)are: tiLI = 6&, cflz to hydrostatic pressure, we find the ultrasonic = 2c7;:‘,, ci%= c%, &6 = Q&, c$ = k$%, and fl& effective elastic constants of cubic crystals as: = +_$&. It is noted that the relation between Birch’s clSR Cl2
=
Cl2 +
rl c-c11
-
Cl2 +
2c112+
Cl231
and Briigger’s cd56 should be as given in this paper, provided cqs6 term in the expression of the strain energy is [?~c~~~(~MMJ~~ + 7)21r)31r)1J I. However, if the term in the expression of the strain energy is [c,W(r)lz~2al)31+ 7)~1~1)13)1as in Birch’s original paper (e.g. equation 12 of [ l]), the relation should be c& = &z$&.
C ll(ultrasonic)
=
a+
7) (CL
+
Ccl”
+
3BT)
+ q2 (-&c;, + C,“” + *Cdm +C,m+tCaT+C*T-33BT)
(22)
420
D. H. CHUNG
C Wultrasonic) = cfz + ,I-/(c& + Chrt’- 3B7’)
+ r/y+& -;ca7c
44tultrasonic)
=
-I-Cb’?’-I- C,” + gcl’lz:$ C&T-t 3W)
C+j + 7) (Cd4 -t C,“”
(23)
coordinates of a material point in two reference states ai and ergaccording to (aJay) = h. The Lagrangian strain tensors corresponding to these two reference states are qij and q$, and they are related by
3B9
-t-
-+qy-+c& + C,” ++c,,n+ Cg7” -i- c,“+ c,“-
3B7‘).
(24)
Where B7‘= (CT,+ 2~:;) 13 and the Cj are c, = c,,,+2c1,, Cb = 2c112+ cl23 c,
=
cs4*
Cd
=
Cl111
c,
=
cl,,,+
c,
=
Cl144
+
(25) 06)
2e*@j +
2&1*12
+
2c,,,5
where e = 4( A”- f ) . Since from thermodynamics (alaP), = -( V/B~)(~/~V)~ and (~A/~~),~= + V”, we find by differentiating equation (32) that
(27) (28)
c,,,,+
IT,,23
(2%
(30) (31)
c, = 2Gz55+ C1266.
The superscripts ‘s’, ‘T’ and ‘MI’ designate thermodynamically adiabatic, thermodynamically isothermal, and thermodynamically mixed elastic constants, respectively. Since the Cj are related to the pressure derivatives of the linear elastic constants c,,, these relationships are to be found. 4. RELATION OF PRESSURE DERIVATIVES EFFECTIVE ELASTIC CONSTANTS
OF THE
(33) Note that the first term in equation (33) is by definition the zero-pressure second-order elastic constants. The second term is, however, thermodynamically mixed third-order elastic constants at zero pressure. Thus, from equation (3 3), it follows that [7]
TO PARTIAL
CONTRACTIONS OF THE HIGHER-ORDER ELASTIC CONSTANTS
The pressure-dependent elastic constants are [7, IO]
second-order
+
PDijki
i-g= rl
(32) where Dukl =I S&l - &$jk - &,&. V denotes the volume of crystal at reference state characterized by the hydrostatic pressure P, and +j is the strain tensor corresponding to an arbitrarily deformed state characterized by that pressure P. V” is defined by the relation (V/v”) = A”, where A is a factor given by the
+3(&&j
]I.
(35)
y(: is the Criineisen constant, j3 is the coefEcient of volume expansion, and A is the ratio of the adiabatic bulk modulus to the isothermal bulk modulus and it is given by A = 1+ T/$y(;. The quantities given by equation (35) are the primary experimentai quantities which result from the usual ultrasonic-pressure experiments at low pressures. For cubic crystals, equation (3 5) reduces to:
ELASTIC
CONSTANTS
SUBJECTED
TO HYDROSTATIC
PRESSURE
421
Therefore, solving equation (39) for C?jjklmmRn, we find explicitly the followings:
(37)
Alternately, the expressions for C,“, Cblfi and C,” can be obtained directly from the effective elastic constants given earlier by taking the pressure derivatives and evaluating them at zero-pressure. Thus, we find
2C,‘n+3C’I:23=
C~+2CBm=
(6BT-CC,T-2CbT)
-1 1
(6R’+2Cgl.)[I+($$]
+c,,-2c,“+
(3B$$‘$T=
C,“.
(43)
Or, from the effective elastic constants given by equations (22-24), we find that
c,~~=-3~~[1+{~)~=~-cf, (364
CA”’= c&- 2C,“+
c.‘fl=-3BT[1+{~}~=,]-c~~. (38a) To obtain the relation of the second pressure derivatives of the elastic constants to partial contractions of the higher-order elastic constants, we start from equation (34). By taking the pressure derivatives of (Ksj,, /#)T and arranging the result,
(6BT - CaT- 2CbT)
(4W P=O
csm = ci2 - 2cb”+
(6BT- C,T- 2&T)
x [-~+E1_,3 (42a) Cc”
=
cq*
-
2c,m + (6BT-
c,7’-
2C*T)
(39) (43a)
where C$klmmnn are certain linear combinations of the fourth-order elastic constants and
Thus, from equations
(28-31) and equations
D. H. CHUNG
422 CB”
=
2c”’ 1112 +
Cc” = cm 1144 +
w%22
‘Wm
+
+
(45)
WL123
4~7255+ 2c%,, + (46)
These are the primary experimental quantities that are resulting from the ultrasonic-pressure experiments at high pressures. 5. SUMMARY
AND CONCLUDING
NOTES
Summarizing the foregone sections, the followings may be stated: (i). Expressions for the effective elastic constants of a cubic crystal subjected to moderately high hydrostatic pressure are derived from the consideration of the rigorous stress-strain relation set by Murnaghan’s theory of finite deformations, and these have been compared with the earlier work. Discrepancies were found in the expressions of the effective second-order elastic constants C,, and C,,. The expressions as ones presented here may be derived from the strain-energy density considerations, and this method may be used to distinguish the observed discrepancies. (ii). Expressions for the effective secondorder elastic constants are C,” = CjL” + C1n+ CI,$ where C, and C,[ are certain linear combinations of the second- and the higher-order elastic constants of crystal and r] is the Lagrangian strain which depends upon pressure. The present expressions are distinguished from the expressions of Birch[ I], Seeger and Buck [4], and Thurston [S] and others mainly by the appearance of the C,& terms in the expressions of the effective elastic constants of cubic crystals. second-order (iii). Ultrasonic effective elastic constants at high hydrostatic pressures
were
derived
and
in terms of elastic constants. The rigorous relationships between the pressure derivatives of the elastic constants and partial contractions of the higher-order elastic constants were presented also. And, then, the primary experimental quantities that may be resulting from the ultrasonic-pressure experiments have been identified in a useful form in terms of the thermodynamically mixed second-, third-, and fourth-order elastic constants of the crystal under the study. thermodynamically
presented
mixed higher-order
Acknowledgements-Sincere appreciation is expressed to Professor Gene Simmons for his support and interest shown in this work. An early stage of this work was done while the author was a Post-Doctoral Research Fellow at The Pennsylvania State University and was supported by the U.S. Office of Naval Research, Contract No. Nonr-656(27). The work was completed at Massachusetts Institute of Technology and was supported by the National Aeronautics and Space Administration, Grant NGR-22-009-176, for which the author gratefully acknowledges. REFERENCES 1. BIRCH F., Phys. Rev. 71,809 (1947). F. D., Am. J. Math. 59,235 (1937); 2. MURNAGHAN and also, Finite Deformation of an Elastic Solid Wiley, New York (195 1). 3. BRUGGERK.,Phys.Rev.l33,A1611(1964). 4. SEEGER A. and BUCK O., Z. Naturf. GA, 1056 (1960). 5. THURSTON R. N.,J. acoust. Sot. Am. 37,348 (1965). 6. GHATE P. B., Phys. St&us Solidi 14,325 (1966). 7. BARSCH G. R., Phys. Status Solidi 19, 129 (1967): see also the references cited by Barsch. 8. See, for example, HUNTINGTON H. B., Solid State Physics (Edited by Seitz F. and Turnbull D.), Vol. 7. Academic Press, New York (1958). 9. THURSTON R. N., Physical Acoustics (Edited by Mason W. P.), Vol. IA. Academic Press, New York (1964). G. and LUDWIG W., Solid St&e 10. LEIBFRIED Physics (Edited by Seitz F. and Turnbull D.), Vol. 12. Academic Press, New York (196 1).
1968. Vol. 29, pp. 417-422.
Printed in Great Britain.
THE ELASTIC CONSTANTS OF A CUBIC CRYSTAL SUBJECTED TO MODERATELY HIGH HYDROSTATIC PRESSURE D. H. CHUNG Department of Geology and Geophysics, Massachusetts institute of Technology, Cambridge, Massachusetts O? 139, U.S.A. (Received 22 Seprember 1967: in revisedform I7 October 1967) Abstract-This paper presents the results of an analysis of the finite strain theory that goes to the next higher-order term than third-order in the strains for cubic crystals. Theoretical expressions for the effective second-order elastic constants are presented here in terms of the second-, third-, and fourth-order elastic constants and the Lagrangian strain; these have been derived from the rigorous stress-strain telationship set by the Murnaghan theory of finite deformation. Since typical ultrasonic experiments with pressure result in thermodynamicaNy mixed elastic constants, the practical ultrasonic expressions of the effective elastic constants are presented for a cubic crystal subjected to moderately high hydrostatic compression. The rigorous relationships between the pressure derivatives of the effective elastic constants and partial contractions of the higher-order elastic constants are presented also, and then the primary experimental quantities that are resulting from the usual ultrasonic-pressure experiments have been identified in terms of the r~errno~nu~fico/~y mixed second-, third-, and fourthorder elastic constants of the crystal.
1. INTRODUCTION strain theory for a medium of the cubic symmetry subjected to a finite hydrostatic compression was first developed by Birch [ 11. In this work, Birch had defined the free-energy of a crystal in terms of a power series of the energy in the strain components including terms up to the third-order in the strains and then derived expressions for the effective elastic constants of the crystal under hydrostatic pressure. These expressions contained the second- and third-order elastic constants of crystal, and they were shown to describe the effective elastic properties of cubic crystals subjected to relatively ‘low’ compression.* For a crystal subjected to moderately ‘high’ compression, however, the THE FINITE
higher order terms than the third-order in the strains would play an important role in the description of crystal elasticity under this high pressure. This paper presents the results of our analysis of the finite strain theory that goes to the next higher order term than the third-order in the strains for cubic crystals. In the following, expressions for the effective elastic constants resulting from the rigorous stress-strain relationship set by the Murnaghan theory of finite deformation[2] will be presented and compared with the earlier work. Since typical ultrasonic experiments with pressure result in thermodynamically mixed elastic constants, the derived expressions of the effective elastic constants had to be identified according to their thermodynamic boundary conditions. This is done in Section 3. In Section 4, we present the rigorous relation of pressure derivatives of the effective elastic constants to partial contractions of the higher-order elastic constants. And, in Section 5, the present work is summarized with a brief discussion.
*The terms ‘low’ and ‘high’ compressions as used here are meant to imply the following: For a given solid under hydrostatic pressure, if the change in the second-order elastic constants with pressure is linear, this pressure (and up to this pressure) is referred to as relatively ‘low’ pressure. If the change with pressure is nonhear, the pressures higher than this pressure are referred to as relatively ‘high’ pressure. Although it is difficult to make a sharp boundary between the ‘low’ and ‘high’ pressures, 2. THE EFFECTIVE ELASTIC CONSTANTS the pressure in the order of 0.05 BT (where BT is the isothermal bulk modulus of solid under the study) for Using the usual Cartesian coordinates, we take the most solids can be referred to as the relatively low presprincipal axes to coincide with the symmetry axes of an sure and the pressure in the order of 0.1 BT may be re- unstrained cubic crystal. Let us denote the coordinates ferred to as the moderately high pressure. of a point p in the crystal before strain by (a I) a%,ua) and 417
418
D. H. CHUNG
those in the crystal after a strain by (x1, x2, x3). These points determine then a displacement vector with components (x-a,), (x2- Q), (X,-U,); thus, the components of the displacements are simply y=x,-0,.
(r=
1,2,3).
@=$fJ”+l#JI+~2+&1+dh+.
.
where I#J~ contains the jth order terms of r),j and for a body with no initial stresses 4,, and 4, can be set equal to zero. Thus, &, &, and& are:
(1)
For this reason, it is convenient to discuss the strain according to the Lagrangian scheme. The Lagrangian strain components in Murnaghan’s notation are[2]
(1 I) 44
where the subscripts ranges are 1, 2, 3 and 6, are the Kronecker deltas. Using the Jacobian J of the deformation, we can write equation (2) in a compact form as
[VI= 3( J*J- 13)
( 10)
=& ukrmno~7)i~7)kl~nlnT”lJ
where the subscripts ranges are 1, 2, 3 and the Einstein summation convention for repeated subscripts is assumed. The rigorous relation of the stresses T,, to the strains n.... 1s
(?.a) or in a compact form
where./* is the transpose of./ given by (12a)
db,,%,%)
J=
(3)
Na,, a2. ~1
and /,is the unit matrix. Consider a special strain which is composed of a hydrostatic strain and a subsequent homogeneous strain. By the homogeneous strain, the point p initially at (a,, Q, uY) is brought to the point (x1”,x2”,x9”), given by x,“= (1-(*)a,,
(s=
1,2,3)
Here, pO and p are the crystal densities before and after a strain, respectively, and the compression ratio is
PO
The additional displacements geneous strain are then u:.=x
&..&“,
(13)
det J
and the gradient of @ with respect to nns is
(4)
thus, every line in the crystal is shortened by the factor (I-a),where ru-n($?-1).
1
P
Ha@ -=
(5)
(14)
aqnq t,
resulting from the homo-
J (r,s=
1,2,3)
(6)
where &, = &; thus, & represent the six homogeneous strain components and they are infinitesimal quantities. The final coordinates are then given by X, = x,O+ u:.
(7)
From this relation Birch [ 11, we can set
(12), as was done by
T, = -PS,
where-P
is hydrostatic
t T:,
(19
pressure given by
From these, we find that
(16) z=
(l-(Y)(&?+&A.
(8)
Here, since /ST8are infinitesimal quantities as noted above, only the first powers of & are retained and all others are neglected. In view of the particular strain defined by equation (7), the Lagrangian strains become nrs= &S77+tPrr(L-t2n).
(9)
The strain energy density @ per unit initial volume is usually given by a power series of the Lagrangian strains as
and T:, are expressed in terms of the p’s and ‘a new set’ of elastic constants. This new set of the elastic constants are the effective second-order elastic constants under the hydrostatic pressure and we will denote hereafter to as CI1, C,, and C,,. Since
ELASTIC
CONSTANTS
SUBJECTED
and (18)
T;z = 2C,&
after having gone through a long algebra, we find the expressions for the effective secondorder elastic constants as follows: CM =
Cl1 +
7) (2c11+
+~2(-kll
2c12 -
+kllll + 2cll12
2c12 +
+
Cl11 +
+klll cll22
PRESSURE
419
Ghate[6]. In light of the present analysis, the writer believes that the minus signs of the quantities cI1 and c12 found in the q2 term of Ghate’s equation (23) should have been plus signs. And, as for the expression for C44, the quantity (+QcI14J should be found in the q2 term of Ghate’s equation (24).
&I,)
f5cl12 +
TO HYDROSTATIC
+
cl23
(19)
%23)
3. THE ULTRASONIC EFFECTIVE ELASTIC CONSTANTS
The expressions of the effective elastic constants as given by equations (19-2 1) can +r12(cll+~c12-hill-cl12 be either the adiabatic or isothermal expressions, and the proper designation of these is (20) + clllZ+ C1122+kl123) obviously done be adding the proper superscript either‘s’ or ‘T’ to all the elastic constants. c4, = c44 -t- 7) (Cl1 + 2c12 + c44+ Cl44 + 2%x) The acoustic data resulting from the usual --c,2-k44+3clll + q2(-c11 acoustic experiments with pressure are neither + 3c,,2 + cl23 + cl44 + 2clSS thermodynamically adiabatic nor thermodynamically isothermal quantities, but they (21) +?k1144 +- Cl155 + 2Cl25, + Cl266). are ‘thermodynamically mixed’ isothermalWhere cILy, cLLvxand cfivhCare the second-, adiabatic quantities [7]. Thus, in this section, third- and fourth-order elastic constants of we seek for the expressions of the effective crystal in Voigt’s notation, respectively, and elastic constants that may be resulting from experiments at high they are expressed in accordance with the the ultrasonic-pressure pressures. thermodynamic definition [3]. Recalling the usual behaviors of ultrasonic It may be noted that, in equations (19-2 l), the coefficients of the terms in r) with the wave velocities in the medium of a cubic we note that a longitudinal second- and third-order elastic constants are crystal[8,9], stiffness cI1 and shear stiffness c44 result the conventional expressions for the effective elastic constants[l, 4-61 and they agree with directly from measurements of the longitudithose derived initially by Birch[ l] when the nal and transverse wave velocities in the [OOl] direction of the crystal, respectively. If one third-order elastic constants in Birch’s definimeasures a transverse wave velocity in [ 1lo] tion are converted into those of more general thermodynamic definition.* However, the polarized in the [ 1TO] direction, the resulting coefficients of the terms in q2 in equations (20) stiffness constant is (cl1 - c12)/2. Thus, from and (21) are at variance with ones given by this, one finds immediately the elastic constant cl2 as a typical procedure. Following exactly the same procedure as the above but subjected *The relations between the c,.,“~defined by Briigger (c”,:,) and those by Birch (CS,‘;*)are: tiLI = 6&, cflz to hydrostatic pressure, we find the ultrasonic = 2c7;:‘,, ci%= c%, &6 = Q&, c$ = k$%, and fl& effective elastic constants of cubic crystals as: = +_$&. It is noted that the relation between Birch’s clSR Cl2
=
Cl2 +
rl c-c11
-
Cl2 +
2c112+
Cl231
and Briigger’s cd56 should be as given in this paper, provided cqs6 term in the expression of the strain energy is [?~c~~~(~MMJ~~ + 7)21r)31r)1J I. However, if the term in the expression of the strain energy is [c,W(r)lz~2al)31+ 7)~1~1)13)1as in Birch’s original paper (e.g. equation 12 of [ l]), the relation should be c& = &z$&.
C ll(ultrasonic)
=
a+
7) (CL
+
Ccl”
+
3BT)
+ q2 (-&c;, + C,“” + *Cdm +C,m+tCaT+C*T-33BT)
(22)
420
D. H. CHUNG
C Wultrasonic) = cfz + ,I-/(c& + Chrt’- 3B7’)
+ r/y+& -;ca7c
44tultrasonic)
=
-I-Cb’?’-I- C,” + gcl’lz:$ C&T-t 3W)
C+j + 7) (Cd4 -t C,“”
(23)
coordinates of a material point in two reference states ai and ergaccording to (aJay) = h. The Lagrangian strain tensors corresponding to these two reference states are qij and q$, and they are related by
3B9
-t-
-+qy-+c& + C,” ++c,,n+ Cg7” -i- c,“+ c,“-
3B7‘).
(24)
Where B7‘= (CT,+ 2~:;) 13 and the Cj are c, = c,,,+2c1,, Cb = 2c112+ cl23 c,
=
cs4*
Cd
=
Cl111
c,
=
cl,,,+
c,
=
Cl144
+
(25) 06)
2e*@j +
2&1*12
+
2c,,,5
where e = 4( A”- f ) . Since from thermodynamics (alaP), = -( V/B~)(~/~V)~ and (~A/~~),~= + V”, we find by differentiating equation (32) that
(27) (28)
c,,,,+
IT,,23
(2%
(30) (31)
c, = 2Gz55+ C1266.
The superscripts ‘s’, ‘T’ and ‘MI’ designate thermodynamically adiabatic, thermodynamically isothermal, and thermodynamically mixed elastic constants, respectively. Since the Cj are related to the pressure derivatives of the linear elastic constants c,,, these relationships are to be found. 4. RELATION OF PRESSURE DERIVATIVES EFFECTIVE ELASTIC CONSTANTS
OF THE
(33) Note that the first term in equation (33) is by definition the zero-pressure second-order elastic constants. The second term is, however, thermodynamically mixed third-order elastic constants at zero pressure. Thus, from equation (3 3), it follows that [7]
TO PARTIAL
CONTRACTIONS OF THE HIGHER-ORDER ELASTIC CONSTANTS
The pressure-dependent elastic constants are [7, IO]
second-order
+
PDijki
i-g= rl
(32) where Dukl =I S&l - &$jk - &,&. V denotes the volume of crystal at reference state characterized by the hydrostatic pressure P, and +j is the strain tensor corresponding to an arbitrarily deformed state characterized by that pressure P. V” is defined by the relation (V/v”) = A”, where A is a factor given by the
+3(&&j
]I.
(35)
y(: is the Criineisen constant, j3 is the coefEcient of volume expansion, and A is the ratio of the adiabatic bulk modulus to the isothermal bulk modulus and it is given by A = 1+ T/$y(;. The quantities given by equation (35) are the primary experimentai quantities which result from the usual ultrasonic-pressure experiments at low pressures. For cubic crystals, equation (3 5) reduces to:
ELASTIC
CONSTANTS
SUBJECTED
TO HYDROSTATIC
PRESSURE
421
Therefore, solving equation (39) for C?jjklmmRn, we find explicitly the followings:
(37)
Alternately, the expressions for C,“, Cblfi and C,” can be obtained directly from the effective elastic constants given earlier by taking the pressure derivatives and evaluating them at zero-pressure. Thus, we find
2C,‘n+3C’I:23=
C~+2CBm=
(6BT-CC,T-2CbT)
-1 1
(6R’+2Cgl.)[I+($$]
+c,,-2c,“+
(3B$$‘$T=
C,“.
(43)
Or, from the effective elastic constants given by equations (22-24), we find that
c,~~=-3~~[1+{~)~=~-cf, (364
CA”’= c&- 2C,“+
c.‘fl=-3BT[1+{~}~=,]-c~~. (38a) To obtain the relation of the second pressure derivatives of the elastic constants to partial contractions of the higher-order elastic constants, we start from equation (34). By taking the pressure derivatives of (Ksj,, /#)T and arranging the result,
(6BT - CaT- 2CbT)
(4W P=O
csm = ci2 - 2cb”+
(6BT- C,T- 2&T)
x [-~+E1_,3 (42a) Cc”
=
cq*
-
2c,m + (6BT-
c,7’-
2C*T)
(39) (43a)
where C$klmmnn are certain linear combinations of the fourth-order elastic constants and
Thus, from equations
(28-31) and equations
D. H. CHUNG
422 CB”
=
2c”’ 1112 +
Cc” = cm 1144 +
w%22
‘Wm
+
+
(45)
WL123
4~7255+ 2c%,, + (46)
These are the primary experimental quantities that are resulting from the ultrasonic-pressure experiments at high pressures. 5. SUMMARY
AND CONCLUDING
NOTES
Summarizing the foregone sections, the followings may be stated: (i). Expressions for the effective elastic constants of a cubic crystal subjected to moderately high hydrostatic pressure are derived from the consideration of the rigorous stress-strain relation set by Murnaghan’s theory of finite deformations, and these have been compared with the earlier work. Discrepancies were found in the expressions of the effective second-order elastic constants C,, and C,,. The expressions as ones presented here may be derived from the strain-energy density considerations, and this method may be used to distinguish the observed discrepancies. (ii). Expressions for the effective secondorder elastic constants are C,” = CjL” + C1n+ CI,$ where C, and C,[ are certain linear combinations of the second- and the higher-order elastic constants of crystal and r] is the Lagrangian strain which depends upon pressure. The present expressions are distinguished from the expressions of Birch[ I], Seeger and Buck [4], and Thurston [S] and others mainly by the appearance of the C,& terms in the expressions of the effective elastic constants of cubic crystals. second-order (iii). Ultrasonic effective elastic constants at high hydrostatic pressures
were
derived
and
in terms of elastic constants. The rigorous relationships between the pressure derivatives of the elastic constants and partial contractions of the higher-order elastic constants were presented also. And, then, the primary experimental quantities that may be resulting from the ultrasonic-pressure experiments have been identified in a useful form in terms of the thermodynamically mixed second-, third-, and fourth-order elastic constants of the crystal under the study. thermodynamically
presented
mixed higher-order
Acknowledgements-Sincere appreciation is expressed to Professor Gene Simmons for his support and interest shown in this work. An early stage of this work was done while the author was a Post-Doctoral Research Fellow at The Pennsylvania State University and was supported by the U.S. Office of Naval Research, Contract No. Nonr-656(27). The work was completed at Massachusetts Institute of Technology and was supported by the National Aeronautics and Space Administration, Grant NGR-22-009-176, for which the author gratefully acknowledges. REFERENCES 1. BIRCH F., Phys. Rev. 71,809 (1947). F. D., Am. J. Math. 59,235 (1937); 2. MURNAGHAN and also, Finite Deformation of an Elastic Solid Wiley, New York (195 1). 3. BRUGGERK.,Phys.Rev.l33,A1611(1964). 4. SEEGER A. and BUCK O., Z. Naturf. GA, 1056 (1960). 5. THURSTON R. N.,J. acoust. Sot. Am. 37,348 (1965). 6. GHATE P. B., Phys. St&us Solidi 14,325 (1966). 7. BARSCH G. R., Phys. Status Solidi 19, 129 (1967): see also the references cited by Barsch. 8. See, for example, HUNTINGTON H. B., Solid State Physics (Edited by Seitz F. and Turnbull D.), Vol. 7. Academic Press, New York (1958). 9. THURSTON R. N., Physical Acoustics (Edited by Mason W. P.), Vol. IA. Academic Press, New York (1964). G. and LUDWIG W., Solid St&e 10. LEIBFRIED Physics (Edited by Seitz F. and Turnbull D.), Vol. 12. Academic Press, New York (196 1).