- Email: [email protected]
Surface green function matching for a three-dimensional nonlocal continuum
ht. J. Enasp Sci. Vol. 24, No. 7, pp. 1107-1114, 1986 Rilwd in Grent witin
oo20-7225/&i 53.00 + 80 RIWIlOUJ_Lut
SURFACE GREEN ~N~ION MATCHING FOR A THREE-DIMENSIONAL NONLOCAL CONTINUUM J. 0. A. IDIODIt Intemation~ Centre for Theoretical Physics, 34100 Tries@ Italy Abstract-With a view toward helping to bridge the gap, fmm the continuum side, between discrete and continuum models of crystalline, elastic solids, explicit results a presented for nonlocal stress tensors that describe exactly some lattice dynamical models that haveb&mwidely used in the literattne for cubic lattices. The surface Green function matching (SGFM) method, which has been used suecessfully for a variety of surface problems, is then extended, within a continuum approach, to a nonlocal continuum that models a three-dimensional discrete lattice. The practical use of the method is demonstrated by performing a fairly complete analytical study of the vibrational surface modes of the SCC ~mi-infinite medium. Sonie results are presented for the f 100) direction of the (00 1) surface of the SCC lattice. Rirsum&-Nous pkntons ici des resultats explicites pour des tenseurs de tension nonlocaux que dtkivent d’une faGonexacte plusieurs modZles de dynamique r&iculaire qu’ont ktk appliqucS dans la litt&ratureaux cristaux avec symbrie cubique. La m&ode de raccordement des fonctions de Green de surface (SGFM), qu’a 6t6 appliqke avec sue&s pour 6tudier des nombreux probI&mesde surfke, est appliquk par la suite B un milieu continu et nonlocal simulant un r&au c&&in en tmis dimensions. L’utilisation pratique de la m&hode est demonstr6e en faisant une itude compIZte des modes vibrationelles de surface d’un milieu cubique simple semiinfini. On prkente quelques result&s pour la direction [HO] de la surface (001) du rtkeau cubique simple ~m~nfini. 1. INTRODUCTION
of calculations related to the dynamical properties of solids appear to be more easily carried out on the basis of con~nuum theory at the present time than by lattice theory [ 1, 21. However, the continuum theory that is usually employed rests on the macroscopic picture provided by the theory of classical elasticity and this theory is known to be incomplete [3-61. A more exact (microscopic) picture of the actual processes in crystals may be obtained by the methods of crystal lattice dynamics which takes into account the existence of long-range, cohesive forces and also the effects of microstructure [6-lo]. Efforts to improve the classical theory of elasticity, within the domain of continuum mechanics, have led to numerous theories being proposed f 1 l- 131.These theories are yet to be widely used in solid-state physics. It is now well known that nonlocal theories of elasticity posses the important effects of dispersion that are missing in the local classical theory of elasticity. Thus nonlocal elasticity, if sufficiently well developed, plays an intermediate role between the usual approaches of the local classical elasticity and the lattice dynamics, and it can be used as a bridge between both of them. The purpose of this paper is to extend the surface Green function matching (SGFM) formalism, developed by Garcia-Moliner and co-workers [ 141,to a nonlocal continuum which models a ~~rnen~on~ discrete lattice of point maSSeSconnected by spring forces. This is an extension to three dimensions of a previous work which was restricted to one dimension [ 151. In passing from one dimension to three dimensions, we are confronted by the nontrivial problem of constructing a nonlocal stress tensor. The outline of this paper is as follows. The main features of the nonlocal model employed in this work are presented in Section II. The nonlocal model is based on the works of Kunin [ 16, 171,where further details can be found. In Section III, we present explicit expressions for nonlocal stress tensors that describe exactly some lattice. dynamical models that have been widely used in the literature. We address, in Section IV, the problem of determining surface modes of a semi-infinite three-dimensional medium. The surface problem is handled by employing the SGFM technique, and this leads to a simple formula for the surface mode. The usefulness and practicality of the SGFM method have been widely demonstrated in ~CERTAIN KINDS
t On leave of absence from Imo State University, Owerri, Nigeria. 1107
J. 0. A. IDIODI
1108
several problems in the theory of surface or interface waves [ 141. The theory of surface vibrations, both in continuum and lattice models, has been the subject of a number of reviews (see, for instance, Fame11 f18], and Ref. IS]); these reviews are recommended for general background and for details falling outside the scope of this study. Conclusions are presented in Section V. II.
THE
NONLOCAL
MODEL
Consider a triple of noncoplanar vectors e, (a = 1,2, 3) with a common origin. The set of points obtained through all displacements of the origin by vectors n = n”e, (here and in the following, by identical superscripts and subscripts we contemplate summation, and na is an arbitrary integer) forms a thee-~mension~ lattice with an elementary cell of the form of a pa~lelepi~, constructed on e,. In this work we shall focus on cubic lattices. Now suppose that a pointwise particle of mass m is located at each knot of the lattice at equilibrium, and that the particles interact by means of linear elastic bonds of the most general nature. Then the resulting system is a three-dimensional medium of simple structure and by definition, the only kinematic variable of the medium of simple structure is displacement; it determines the state of the medium completely. In what follows, we shall model this three-dimensional medium by an infinite homogeneous nonlocal elastic continuum of constant mass density p. Let U, ((r = 1, 2, 3) be the displacement of the particles from the equilibrium position. Then, in the harmonic approximation, the equation of motion of the homogeneous medium in the (r, t) and (k, w) representations take the forms [ 171 p
aQJ%-, 8 - adV, at2
t) = qp, t),
-pw2U”(k, w) - ikhrXa(k,w) = q”(k, w).
(2-l) (2.2)
Here qn(r, t) is the external force density at the position I and at time t, and the nonlocal
symmetric stress tensor #Oris defined by T~~(I,t) =
s
CxnaS(r- r’)e,&‘, t) dr’,
i.e. +(k,
w) = CXa“@(k)tp~k,o)
12.31
in terms of the usual strain tensor defined by
or
c,s(k, w) = ik(,U,j)(k, w).
(2.4)
Here, and subsequently, indices which are contained in brackets (parentheses) are assumed to be subject to the operation of alternation (symmetrization) [ 191. The operator of elastic mod&i Chars(k) possesses the symmetry properties (2.5) i .e . Cx*“@(k)is symmetric inside the first and second pairs and is hermitian with respect to the permutation of the pairs. It will be assumed in the following that action at a distance is bounded (i.e. CharB(r - r’) = 0 for Ir - r’1> L, L being some characteristic radius of interaction). When we consider fields which change slowly enough over distances of the order of the radius of interaction L, we can perform a transition to approximate models by replacing the integral operator in (2.3) by a Werential operator. This will be assumed, and the resulting di&rential operator will be considered in the approximation by first roots [ 15- 173.
Surface Green function matching 111. NONLOCAL
STRESS
TENSOR
FOR
1109
SOME
LATTICE
MODELS
In this section we present explicit expressions for the nonlocal stress tensor in the krepresentation for some lattice models that have been widely used in the literature. In view of eqn (2.3) it is sufficient to give the expressions for the operator of elastic moduli Cxn’@(k). Only the nonzero components of CxuN8(k)are presented.
(a) The XC lattice and the model of Gazis et al. [4] The lattice dynamical model of Gazis et al. [4] for the SCC lattice assumes Hooke’s law interaction of nearest and next-nearest neighbours due to central forces characterized by force constants (Yand 8, respectively. In addition, the model takes into account forces between particles that are due to angular stiffness of a system of three consecutive nearest neighbours which form a right angle in the equilibrium configuration. These angular stiIIhess forces are characterized by a force constant y. For this model, we have, after some lengthy tensor algebra [ 171, PAX(k) = $ ((Y + 4@S2(k,) - 2
2 k;S2(k,,)S2(kA),
(3. la)
P
(&A) CxA““(k)= $ [4(8 + y)s(2k,)s(2k,)
- (B + 2r){s2(k,)
+
S2W)l
WI4 n4A
+ 2
(kg + kf)s2(kA)s2(k,,),
(3. lb)
0
CApA’ = g
(3. Ic)
(/3 -i-2r)[s2(k,) + S2(kp) - 11,
Wr) CAA”yk)= $ (@ + 2y)k,u
[ 1 - s2(kdl
k
@+A
kAk,s2(k,)s2(k,),
- $
A
(3.ld)
0
with (3. le)
4kJ = sin(ak,/2)/(&/2) and V. = a3,
A, p, Y = 1, 2, 3.
(3-K)
The elastic constants of local classical elasticity are given by CA”“‘(k)evaluated at k = 0. Thus, in the Voigt notation for the elastic moduli we have CII = C22= C3, = C”““(O) = (a + 4/3)/a, Cl* = Cl3 = C23= CAti’
(3.2a)
= 2/3/a,
(3.2b)
c, = c,, = Cj6 = CAfiAw(0) = (20 + 47)/a ,
(3.2~)
in agreement with Gazis et al. [4]. (b) The BCC lattice and the model of Clark et al. [20, 211 In this model, (r and /I are the central force constants for nearest and next-nearest neighhours, respectively;while y is the angular stiIfnessforce constant. We have taken into account only one type of angle. For this model we have, cAAAA(k)
=
Q’[(b + ‘b)s2(2k)+ (a + 2T)s2(kA)l v.
-(n+Wj$[ktS’(k,,) 0
1
+ kzS2(kv) - $ kiktS2(k,,)S2(kv) S2(kA), (3.3a)
J. 0. A. IDIODI
1110
c?**(k) = 2(a - y) g S(2k,)S(2k,) 1 - $ k;P(k,)
x (k’x + k;)S’(k,)S’(k,) C
1 1
+ (LY+ 2~) &
1 - 2 k2.S2(kJ
0
- (a + 27)
&tS2W +S2Wl
[S2(2kA)+ S2(2k,J],
(3.3b)
[S2(2k,) + S2(2k,,) - I],
(3.3~)
- $ kfS”(k.)] + $
0
C”“Yk) = far + 29 2 [S2(kx) + S2(kp) - t]
- $ ktS’(k”)] - $ CW’(k) = --(a + 27) j&
0
k&,S*(k)
a’ 16 k*S’(k,) v
1 -
1
S2(kJ + $
k,
[S2(2kJ - 11
0
x
1, (3.34
a2
-WW+,
k
0
A, cc,v = 1, 2, 3; and in each equation in (3.3a)-(3.3d), X # P # v.
The elastic constants of classical elasticity are obtained in the same manner as was done for the SCC. (c) The FCC Iattice Calculations of the ~bmtion~ modes of FCC crystals have often employed a force con-
stant model similar to the type considered above for SCC and BCC crystals, i.e. central forces between hrst and second neigbbours together with angle bending forces (see, for instance, Refs. [22-241). The method employed above can be similarly applied here to obtain the operator of elastic moduli P’@(k). We do not find it relevant to again present the results for this case. We have sufficiently demonstrated above that the operator of elastic moduli can be determined for the lattice models of interest. IV.
SURFACE
GREEN FOR
FUNCTION CUBIC
MATCHING
ANALYSIS
CRYSTALS
In order to inquire about the possibility of surface modes, we must solve a boundary problem. We shall apply the formalism developed in the previous sections to the (001) surface of the SCC lattice, and we shall work ~ou~out in the (k, co)representation with sigu convention exp[i(k r - wt)]. In the absence of external forces, the problem is to solve the equations of motion for the displacements Vi, i.e. l
[-pw2Pu + kxk,,C”@(k)]UB(k,w) = 0,
CY,~~J,P= 1,2,3,
(4.1)
with appropriate boundary conditions at the surface. The stresses and strains are given by eqns (2.3) and (2.4). The dynamic Green’s function G corresponding to eqn (4.1) satisfies the equation [-pw2Sacp+ k~k~CA*~(k)]~~~(k, o) = 8,
(4.2)
Surface Green function matching
1111
i.e. the Kernel of the Green’s tensor can be obtained by the purely algebraic operation of inverting the matrix Lap defined by fi@
‘=
-pw2@ + kAk,CAa”@(k).
(4.3)
We take our surface at x3 = 0 and since we shall focus on the [ 1001 direction of the (00 1) surface, we put k = (k,, 0,
i.e. kz = 0.
k3),
(4.4)
For a free surface at ~3 = 0 the boundary conditions are ri3 = 0 which means, by eqns (2.3) and (3.1 a)-(3.1 f), that the surface projections of the Gafl and of their normal derivatives will be involved. These are defined by (4.5) and for the tih normal derivative ‘Gag, we have “Gs
=
lim
no
L 2r
+m exp(Tik3Nk&,,Ak,
w)dk3.
s co _
(4.6)
Related to the stress is the operator 2 extensively discussed in the works of Ga~ia-Moliner and co-workers [ 141. For the [ 1001 direction, a high symme~ axon, our problem factors into a one-dimensional problem for the shear horizontal (SH) mode and a ~~rnension~ problem for the sagittal (S) mode. (a) Shear horizontal (SH) mode along [ 1001 direction of SCC lattice For the SH mode, we solve, in the approximation by first roots [ 15-161 [--pa* + kAk,CX2F2(k)]UZ(k,w) = 0,
(4.7)
subject to the boundary condition 723 = 0, i.e. C23~~(k)~~k, w) = 0,
(4.8)
where E,,#is defined by eqn (2.4) and C“+@ is defined by eqn (3.1). We can immediately apply the results of the surface Green function matching analysis for the one-dimensional nonlocal continuum [ 151. Hence, the surface mode frequency, if any, is given by ao=
-(4a: - 18a,a2) If: [42(a$ - 3aJ31”* 54
(4.9)
with a2 = 30/a*,
al = 360/a4,
and p&,
= C,
k: -$k:+&kf
(4.10)
1.
(4.11)
Since w must be real, it follows from (4.10) and (4.9) that the [ 1001 direction of the SCC lattice does not sustain a shear horizontal surface mode. The SH wave that does exist is not a surface wave but just the T2 bulk wave given in this model by eqn (4.11). This conclusion is in agreement with previous workers [ 18,251.
J. 0. A. IDIODI
1112
(b) Sag&tat(S) mode along [ 1001direction of XC lattice For the S mode, we must solve the coupled equations [-pu* -t k,k$?“‘(k)]U,(k,
w) + [k,k,C”“3(k)]U3(k, w) = 0
and
[kxkPCX3F’(k)]U,(k, o) + [-pw’ + k~k,PW3(k)]U3(k, W)= 0.
(4.12)
Subject to the boundary conditions C33@@(k)c Ps(k, w) = 0.
Ci3@@(k)eJk,W)= 0,
(4.13)
Using eqns (3.1) and (3.2) and carrying out the SGFM analysis, we obtain GI I(&,
4
G&,
4 = Gdhbmho
= G
I ffoAo+r&~o
Wk,, w)= G3dk1, Al:‘&,
w) = A::‘#,
4
, 0)
+ r401/[tP’(iqo)~‘(iqi)lt
(4. I4a)
+ r~VP’&W(i42)1,
(4.14b) (4.14c)
= 0,
(4.M)
&l/2,
=
(4.14e) (4.14f) with HO =
-2C1lGdqoq2,
Ai
=
(4.1w
(4ao - &)(a~ + 2%Yao, C,,
(4.14h)
Cll,
(4.14i) (4.14j) (4.14k)
In the approximation by first roots employed for this calculation, 9(k3, co)is defined by Q(k3,4 = CI ,Cdk:
+ &@)Xk:
(4.15)
+ &N,
where (q&(w):m = 0, 23 are the zeroes of N,
0) = C,&M(~~ + aA2 + aa)
(4.16)
a+&, 0) ak
(4.17)
and @%3&N =
3
I h=iq&)
The surface modes, if any, are obtained from [ 141 det G;’ * det I-A(%-‘I
= 03
Putting eqns (4.14a)-(4.14k) into (4.18) we obtain
i.e.
It - AISASII =FO
GIIGB
’
(4.18)
St&ace Green function matching
1113
where x =
~44hoY40
-
Y40 -
YIO)~G4YSOY10
-
ClZYiO
-
G2Y40).
Equation (4.19) is the surface mode dispersion relation for the sag&al mode. In the transition to the local limit Q - 0, then (4.19) simplifies to the form L4rrw1*rf
-
G&:> h
+
Y4(GG4YT
+
Y4)
-
a2m
= o
(4.20)
with (4.2 1)
‘74 = Y4o/u-o
= Gk:
-
P~21/G1
*
(4.22)
Equation (4.20) is the well-known result of classical elasticity [26]. From eqn (4.19) the surface mode frequency can be obtained as a function of k, . We now make contact with the work of Gazis et al. [4] by exhibiting in Fig. I the graph of the normalized frequency, (4.23) as a function of the wavenumber # (=kt a). The physical parameters were chosen to fit the data for potassium chloride which is nearly monatomic in so far as atomic masses are concerned and which crystallizes in a simple cubic array. The close agreement of our curve (labelled b) with that of Gazis et al. (labelled c) is very encouraging, despite the fact that we have solved our nonlocal equations in the approximation by first roots. Our results show that the surface wave exhibit dispersion, i.e. a variation of the phase velocity with wavelength.
Fig. 1. Dispersion curve of normalized frequency versus the dimensionless wave number #, for waves in the [ lOO]direction in a KCL crystal with elastic constants C,, = 3.980 X 10” Nm-*, C,, = 0.625 X 10” Nm-* and CM = 0.620 X 10” Nm-‘. “a” is the dispersion curve of Iocal continuum theory. “b” is the dispersion curve of nonlocal continuum theory. “c” is the dispersion curve obtained in Lattice dynamics (Gazis d al.).
J. 0. A. IDIODI
1114
V. CONCLUSION
In this work the SGFM formalism has been applied to a nonlocal continuum that models a three-dimensional discrete lattice. This is an extension to three dimensions of a previous work that was restricted to one dimension. Explicit results have been presented here for nonlocal stress tensors that describe exactly some lattice dynamical models that have been widely used in the literature. The practicality of the formalism is demonstrated by investigating SH and S surface modes along the [ 1001 direction of the (001) surface of the SCC lattice. The predictions of this model are in agreement with those of previous workers. One of the motivations for this study is the need to offer a simpler approach to problems of interest in lattice dynamics, exploiting fully a procedure that is more in the analytical spirit than being entirely numerical. In addition, this study has also helped to bridge the gap, from the continuum side, between discrete and continuum models of crystalline, elastic solids. Acknowledgments-The
author is very much indebted to Professor F. Garcia-Moliner and To Dr. V. R. Velasco for suggesting the problem and for many valuable discussions. The author would also like to thank Professors Abdus !&lam and Mario Tosi, the IAEA and UNESCO for an ICTP fellowship and for hospitality at the International Centre for Theoretical Physics, Trieste, Italy. REFERENCES
111A. A. MARADUDIN, In Proceedings
ofthe International Conference on Lattice Dynamics, Paris, 5-9 September
1977 (Edited by M. Balkanski), p. 284. Flammarion (1978). Letl. Appl. Engng Sci. 1, I 1 (1973).
121A. C. ERINGEN,
[41 The Cosserat continuum was historically one of the first models of elastic media which could not be described within the scope of classical elasticity. A Cosserat medium is a continuum, each point of which possesses the six degrees of freedom of a rigid body. See Refs. [ 1I- 131for generalized models of the Cosserat continuum. D. C. GAZIS, R. HERMAN and R. F. WALLIS, Phys. Rev. 119, 533 (1960). tz; A. C. ERINGEN, Int. J. Engng Sci. 19, 1461 (1981); and references therein. WI F. W. de WETTE and G. P. ALLDREDGE, Methods Comput. Phys. 15, 163 (1976); and references therein. I71 A. A. MARADUDIN, E. W. MONTROLL, G. H. WEISS and 1. P. IPATOVA, Theory ofLattice Dynamics in the Harmonic Approximation, 2nd ed. Academic Press, New York (197 I). PI A. A. MARADUDIN, R. F. WALLIS and L. DOBRZYNSKI (Editors), Handbook ofsurfaces and Interfaces Vol. 3. Garland STPM Press, New York/London (1980). [91 P. BRUESCH, Phonons: Theory and Experiments I, Springer Series in Solid-State Sciences. Springer-Verlag, Berlin/Heidelberg/New York (1982). S. K. SINHA, Crit Rev. Solid State Sci. 3, 273 (1973). tryi C. TRUESDELL and R. A. TOUPIN, In Principles of Classical Mechanics and Field Theory, Encyclopedia of Physics, Vol. III/l (Edited by S. R&e), p. 20. Springer-Verlag, Bedin/G6ttingen/Heidelberg (1960). (121 E. KRONER (Editor), Mechanics of GeneralizedContinua. Proc. IUTAM Symposium on the Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart 1967. Springer-Verlag, Berlin/Heidelberg/New York (1968). R. STOJANOVIC. Mechanics ofPolar Continua. CISME. Udine C1969). F. GARCIA-MOLINER, G. PiATERO and V. R. VELkSCO, surf: &i. 136, 601 (1984); and references therein. 1151 J. 0. A. IDIODI, Surface Green Function Matching for a Nonlocal Continuum. One-Dimensional Model, to be published. I161 I. A. KUNIN, Elastic Media with Microstructure. I. One-Dimensional Models, Springer Series in Solid-State Sciences, Vol. 26. Springer-Verlag, Berlin/Heidelberg/New York (1982). 1171 I. A. KUNIN, Elastic Media with Microstructure II. Three-Dimensional Models, Springer Series in SolidState Sciences Vol. 44. Springer-Verlag, Berlin/Heidelberg/New York (1983). iI81 G. W. FARNELL, Phys. Acoust. 6, 109 (1970) (Edited by W. P. Mason and R. N. Thurston), Academic Press, New York. J. A. SCHOUTEN, Tensor Analysis for Physicists, 2nd ed. Oxford Univ. Press, London (1954). B. C. CLARK, D. C. GAZIS and R. F. WALLIS, Phys. Rev. A 134,1486 (1964). L. DOBRZYNSKI and P. MASRI, J. Phys. Chem. So/ids 33, 1603 (1972). P. S. YUEN and Y. P. VARSHNI, Phys. Rev. 164,895 (1967). S. W. MUSSER and K. H. RIEDER, Phys. Rev. B 2,3034 (1970). D. CASTIEL, L. DOBRZYNSKI and D. SPANJAARD, Su$ Sci. 59,252 (1976). G. P. ALLDREDGE, Phys. Left. A 41,281 (1972). V. R. VELASCO and F. GARCIA-MOLINER. J. Phys. C 13,2237 (1980); see also, R. Stoneley, Proc. Roy. Sot. London A 232,447
(1955). (Received 25 June 1985)
oo20-7225/&i 53.00 + 80 RIWIlOUJ_Lut
SURFACE GREEN ~N~ION MATCHING FOR A THREE-DIMENSIONAL NONLOCAL CONTINUUM J. 0. A. IDIODIt Intemation~ Centre for Theoretical Physics, 34100 Tries@ Italy Abstract-With a view toward helping to bridge the gap, fmm the continuum side, between discrete and continuum models of crystalline, elastic solids, explicit results a presented for nonlocal stress tensors that describe exactly some lattice dynamical models that haveb&mwidely used in the literattne for cubic lattices. The surface Green function matching (SGFM) method, which has been used suecessfully for a variety of surface problems, is then extended, within a continuum approach, to a nonlocal continuum that models a three-dimensional discrete lattice. The practical use of the method is demonstrated by performing a fairly complete analytical study of the vibrational surface modes of the SCC ~mi-infinite medium. Sonie results are presented for the f 100) direction of the (00 1) surface of the SCC lattice. Rirsum&-Nous pkntons ici des resultats explicites pour des tenseurs de tension nonlocaux que dtkivent d’une faGonexacte plusieurs modZles de dynamique r&iculaire qu’ont ktk appliqucS dans la litt&ratureaux cristaux avec symbrie cubique. La m&ode de raccordement des fonctions de Green de surface (SGFM), qu’a 6t6 appliqke avec sue&s pour 6tudier des nombreux probI&mesde surfke, est appliquk par la suite B un milieu continu et nonlocal simulant un r&au c&&in en tmis dimensions. L’utilisation pratique de la m&hode est demonstr6e en faisant une itude compIZte des modes vibrationelles de surface d’un milieu cubique simple semiinfini. On prkente quelques result&s pour la direction [HO] de la surface (001) du rtkeau cubique simple ~m~nfini. 1. INTRODUCTION
of calculations related to the dynamical properties of solids appear to be more easily carried out on the basis of con~nuum theory at the present time than by lattice theory [ 1, 21. However, the continuum theory that is usually employed rests on the macroscopic picture provided by the theory of classical elasticity and this theory is known to be incomplete [3-61. A more exact (microscopic) picture of the actual processes in crystals may be obtained by the methods of crystal lattice dynamics which takes into account the existence of long-range, cohesive forces and also the effects of microstructure [6-lo]. Efforts to improve the classical theory of elasticity, within the domain of continuum mechanics, have led to numerous theories being proposed f 1 l- 131.These theories are yet to be widely used in solid-state physics. It is now well known that nonlocal theories of elasticity posses the important effects of dispersion that are missing in the local classical theory of elasticity. Thus nonlocal elasticity, if sufficiently well developed, plays an intermediate role between the usual approaches of the local classical elasticity and the lattice dynamics, and it can be used as a bridge between both of them. The purpose of this paper is to extend the surface Green function matching (SGFM) formalism, developed by Garcia-Moliner and co-workers [ 141,to a nonlocal continuum which models a ~~rnen~on~ discrete lattice of point maSSeSconnected by spring forces. This is an extension to three dimensions of a previous work which was restricted to one dimension [ 151. In passing from one dimension to three dimensions, we are confronted by the nontrivial problem of constructing a nonlocal stress tensor. The outline of this paper is as follows. The main features of the nonlocal model employed in this work are presented in Section II. The nonlocal model is based on the works of Kunin [ 16, 171,where further details can be found. In Section III, we present explicit expressions for nonlocal stress tensors that describe exactly some lattice. dynamical models that have been widely used in the literature. We address, in Section IV, the problem of determining surface modes of a semi-infinite three-dimensional medium. The surface problem is handled by employing the SGFM technique, and this leads to a simple formula for the surface mode. The usefulness and practicality of the SGFM method have been widely demonstrated in ~CERTAIN KINDS
t On leave of absence from Imo State University, Owerri, Nigeria. 1107
J. 0. A. IDIODI
1108
several problems in the theory of surface or interface waves [ 141. The theory of surface vibrations, both in continuum and lattice models, has been the subject of a number of reviews (see, for instance, Fame11 f18], and Ref. IS]); these reviews are recommended for general background and for details falling outside the scope of this study. Conclusions are presented in Section V. II.
THE
NONLOCAL
MODEL
Consider a triple of noncoplanar vectors e, (a = 1,2, 3) with a common origin. The set of points obtained through all displacements of the origin by vectors n = n”e, (here and in the following, by identical superscripts and subscripts we contemplate summation, and na is an arbitrary integer) forms a thee-~mension~ lattice with an elementary cell of the form of a pa~lelepi~, constructed on e,. In this work we shall focus on cubic lattices. Now suppose that a pointwise particle of mass m is located at each knot of the lattice at equilibrium, and that the particles interact by means of linear elastic bonds of the most general nature. Then the resulting system is a three-dimensional medium of simple structure and by definition, the only kinematic variable of the medium of simple structure is displacement; it determines the state of the medium completely. In what follows, we shall model this three-dimensional medium by an infinite homogeneous nonlocal elastic continuum of constant mass density p. Let U, ((r = 1, 2, 3) be the displacement of the particles from the equilibrium position. Then, in the harmonic approximation, the equation of motion of the homogeneous medium in the (r, t) and (k, w) representations take the forms [ 171 p
aQJ%-, 8 - adV, at2
t) = qp, t),
-pw2U”(k, w) - ikhrXa(k,w) = q”(k, w).
(2-l) (2.2)
Here qn(r, t) is the external force density at the position I and at time t, and the nonlocal
symmetric stress tensor #Oris defined by T~~(I,t) =
s
CxnaS(r- r’)e,&‘, t) dr’,
i.e. +(k,
w) = CXa“@(k)tp~k,o)
12.31
in terms of the usual strain tensor defined by
or
c,s(k, w) = ik(,U,j)(k, w).
(2.4)
Here, and subsequently, indices which are contained in brackets (parentheses) are assumed to be subject to the operation of alternation (symmetrization) [ 191. The operator of elastic mod&i Chars(k) possesses the symmetry properties (2.5) i .e . Cx*“@(k)is symmetric inside the first and second pairs and is hermitian with respect to the permutation of the pairs. It will be assumed in the following that action at a distance is bounded (i.e. CharB(r - r’) = 0 for Ir - r’1> L, L being some characteristic radius of interaction). When we consider fields which change slowly enough over distances of the order of the radius of interaction L, we can perform a transition to approximate models by replacing the integral operator in (2.3) by a Werential operator. This will be assumed, and the resulting di&rential operator will be considered in the approximation by first roots [ 15- 173.
Surface Green function matching 111. NONLOCAL
STRESS
TENSOR
FOR
1109
SOME
LATTICE
MODELS
In this section we present explicit expressions for the nonlocal stress tensor in the krepresentation for some lattice models that have been widely used in the literature. In view of eqn (2.3) it is sufficient to give the expressions for the operator of elastic moduli Cxn’@(k). Only the nonzero components of CxuN8(k)are presented.
(a) The XC lattice and the model of Gazis et al. [4] The lattice dynamical model of Gazis et al. [4] for the SCC lattice assumes Hooke’s law interaction of nearest and next-nearest neighbours due to central forces characterized by force constants (Yand 8, respectively. In addition, the model takes into account forces between particles that are due to angular stiffness of a system of three consecutive nearest neighbours which form a right angle in the equilibrium configuration. These angular stiIIhess forces are characterized by a force constant y. For this model, we have, after some lengthy tensor algebra [ 171, PAX(k) = $ ((Y + 4@S2(k,) - 2
2 k;S2(k,,)S2(kA),
(3. la)
P
(&A) CxA““(k)= $ [4(8 + y)s(2k,)s(2k,)
- (B + 2r){s2(k,)
+
S2W)l
WI4 n4A
+ 2
(kg + kf)s2(kA)s2(k,,),
(3. lb)
0
CApA’ = g
(3. Ic)
(/3 -i-2r)[s2(k,) + S2(kp) - 11,
Wr) CAA”yk)= $ (@ + 2y)k,u
[ 1 - s2(kdl
k
@+A
kAk,s2(k,)s2(k,),
- $
A
(3.ld)
0
with (3. le)
4kJ = sin(ak,/2)/(&/2) and V. = a3,
A, p, Y = 1, 2, 3.
(3-K)
The elastic constants of local classical elasticity are given by CA”“‘(k)evaluated at k = 0. Thus, in the Voigt notation for the elastic moduli we have CII = C22= C3, = C”““(O) = (a + 4/3)/a, Cl* = Cl3 = C23= CAti’
(3.2a)
= 2/3/a,
(3.2b)
c, = c,, = Cj6 = CAfiAw(0) = (20 + 47)/a ,
(3.2~)
in agreement with Gazis et al. [4]. (b) The BCC lattice and the model of Clark et al. [20, 211 In this model, (r and /I are the central force constants for nearest and next-nearest neighhours, respectively;while y is the angular stiIfnessforce constant. We have taken into account only one type of angle. For this model we have, cAAAA(k)
=
Q’[(b + ‘b)s2(2k)+ (a + 2T)s2(kA)l v.
-(n+Wj$[ktS’(k,,) 0
1
+ kzS2(kv) - $ kiktS2(k,,)S2(kv) S2(kA), (3.3a)
J. 0. A. IDIODI
1110
c?**(k) = 2(a - y) g S(2k,)S(2k,) 1 - $ k;P(k,)
x (k’x + k;)S’(k,)S’(k,) C
1 1
+ (LY+ 2~) &
1 - 2 k2.S2(kJ
0
- (a + 27)
&tS2W +S2Wl
[S2(2kA)+ S2(2k,J],
(3.3b)
[S2(2k,) + S2(2k,,) - I],
(3.3~)
- $ kfS”(k.)] + $
0
C”“Yk) = far + 29 2 [S2(kx) + S2(kp) - t]
- $ ktS’(k”)] - $ CW’(k) = --(a + 27) j&
0
k&,S*(k)
a’ 16 k*S’(k,) v
1 -
1
S2(kJ + $
k,
[S2(2kJ - 11
0
x
1, (3.34
a2
-WW+,
k
0
A, cc,v = 1, 2, 3; and in each equation in (3.3a)-(3.3d), X # P # v.
The elastic constants of classical elasticity are obtained in the same manner as was done for the SCC. (c) The FCC Iattice Calculations of the ~bmtion~ modes of FCC crystals have often employed a force con-
stant model similar to the type considered above for SCC and BCC crystals, i.e. central forces between hrst and second neigbbours together with angle bending forces (see, for instance, Refs. [22-241). The method employed above can be similarly applied here to obtain the operator of elastic moduli P’@(k). We do not find it relevant to again present the results for this case. We have sufficiently demonstrated above that the operator of elastic moduli can be determined for the lattice models of interest. IV.
SURFACE
GREEN FOR
FUNCTION CUBIC
MATCHING
ANALYSIS
CRYSTALS
In order to inquire about the possibility of surface modes, we must solve a boundary problem. We shall apply the formalism developed in the previous sections to the (001) surface of the SCC lattice, and we shall work ~ou~out in the (k, co)representation with sigu convention exp[i(k r - wt)]. In the absence of external forces, the problem is to solve the equations of motion for the displacements Vi, i.e. l
[-pw2Pu + kxk,,C”@(k)]UB(k,w) = 0,
CY,~~J,P= 1,2,3,
(4.1)
with appropriate boundary conditions at the surface. The stresses and strains are given by eqns (2.3) and (2.4). The dynamic Green’s function G corresponding to eqn (4.1) satisfies the equation [-pw2Sacp+ k~k~CA*~(k)]~~~(k, o) = 8,
(4.2)
Surface Green function matching
1111
i.e. the Kernel of the Green’s tensor can be obtained by the purely algebraic operation of inverting the matrix Lap defined by fi@
‘=
-pw2@ + kAk,CAa”@(k).
(4.3)
We take our surface at x3 = 0 and since we shall focus on the [ 1001 direction of the (00 1) surface, we put k = (k,, 0,
i.e. kz = 0.
k3),
(4.4)
For a free surface at ~3 = 0 the boundary conditions are ri3 = 0 which means, by eqns (2.3) and (3.1 a)-(3.1 f), that the surface projections of the Gafl and of their normal derivatives will be involved. These are defined by (4.5) and for the tih normal derivative ‘Gag, we have “Gs
=
lim
no
L 2r
+m exp(Tik3Nk&,,Ak,
w)dk3.
s co _
(4.6)
Related to the stress is the operator 2 extensively discussed in the works of Ga~ia-Moliner and co-workers [ 141. For the [ 1001 direction, a high symme~ axon, our problem factors into a one-dimensional problem for the shear horizontal (SH) mode and a ~~rnension~ problem for the sagittal (S) mode. (a) Shear horizontal (SH) mode along [ 1001 direction of SCC lattice For the SH mode, we solve, in the approximation by first roots [ 15-161 [--pa* + kAk,CX2F2(k)]UZ(k,w) = 0,
(4.7)
subject to the boundary condition 723 = 0, i.e. C23~~(k)~~k, w) = 0,
(4.8)
where E,,#is defined by eqn (2.4) and C“+@ is defined by eqn (3.1). We can immediately apply the results of the surface Green function matching analysis for the one-dimensional nonlocal continuum [ 151. Hence, the surface mode frequency, if any, is given by ao=
-(4a: - 18a,a2) If: [42(a$ - 3aJ31”* 54
(4.9)
with a2 = 30/a*,
al = 360/a4,
and p&,
= C,
k: -$k:+&kf
(4.10)
1.
(4.11)
Since w must be real, it follows from (4.10) and (4.9) that the [ 1001 direction of the SCC lattice does not sustain a shear horizontal surface mode. The SH wave that does exist is not a surface wave but just the T2 bulk wave given in this model by eqn (4.11). This conclusion is in agreement with previous workers [ 18,251.
J. 0. A. IDIODI
1112
(b) Sag&tat(S) mode along [ 1001direction of XC lattice For the S mode, we must solve the coupled equations [-pu* -t k,k$?“‘(k)]U,(k,
w) + [k,k,C”“3(k)]U3(k, w) = 0
and
[kxkPCX3F’(k)]U,(k, o) + [-pw’ + k~k,PW3(k)]U3(k, W)= 0.
(4.12)
Subject to the boundary conditions C33@@(k)c Ps(k, w) = 0.
Ci3@@(k)eJk,W)= 0,
(4.13)
Using eqns (3.1) and (3.2) and carrying out the SGFM analysis, we obtain GI I(&,
4
G&,
4 = Gdhbmho
= G
I ffoAo+r&~o
Wk,, w)= G3dk1, Al:‘&,
w) = A::‘#,
4
, 0)
+ r401/[tP’(iqo)~‘(iqi)lt
(4. I4a)
+ r~VP’&W(i42)1,
(4.14b) (4.14c)
= 0,
(4.M)
&l/2,
=
(4.14e) (4.14f) with HO =
-2C1lGdqoq2,
Ai
=
(4.1w
(4ao - &)(a~ + 2%Yao, C,,
(4.14h)
Cll,
(4.14i) (4.14j) (4.14k)
In the approximation by first roots employed for this calculation, 9(k3, co)is defined by Q(k3,4 = CI ,Cdk:
+ &@)Xk:
(4.15)
+ &N,
where (q&(w):m = 0, 23 are the zeroes of N,
0) = C,&M(~~ + aA2 + aa)
(4.16)
a+&, 0) ak
(4.17)
and @%3&N =
3
I h=iq&)
The surface modes, if any, are obtained from [ 141 det G;’ * det I-A(%-‘I
= 03
Putting eqns (4.14a)-(4.14k) into (4.18) we obtain
i.e.
It - AISASII =FO
GIIGB
’
(4.18)
St&ace Green function matching
1113
where x =
~44hoY40
-
Y40 -
YIO)~G4YSOY10
-
ClZYiO
-
G2Y40).
Equation (4.19) is the surface mode dispersion relation for the sag&al mode. In the transition to the local limit Q - 0, then (4.19) simplifies to the form L4rrw1*rf
-
G&:> h
+
Y4(GG4YT
+
Y4)
-
a2m
= o
(4.20)
with (4.2 1)
‘74 = Y4o/u-o
= Gk:
-
P~21/G1
*
(4.22)
Equation (4.20) is the well-known result of classical elasticity [26]. From eqn (4.19) the surface mode frequency can be obtained as a function of k, . We now make contact with the work of Gazis et al. [4] by exhibiting in Fig. I the graph of the normalized frequency, (4.23) as a function of the wavenumber # (=kt a). The physical parameters were chosen to fit the data for potassium chloride which is nearly monatomic in so far as atomic masses are concerned and which crystallizes in a simple cubic array. The close agreement of our curve (labelled b) with that of Gazis et al. (labelled c) is very encouraging, despite the fact that we have solved our nonlocal equations in the approximation by first roots. Our results show that the surface wave exhibit dispersion, i.e. a variation of the phase velocity with wavelength.
Fig. 1. Dispersion curve of normalized frequency versus the dimensionless wave number #, for waves in the [ lOO]direction in a KCL crystal with elastic constants C,, = 3.980 X 10” Nm-*, C,, = 0.625 X 10” Nm-* and CM = 0.620 X 10” Nm-‘. “a” is the dispersion curve of Iocal continuum theory. “b” is the dispersion curve of nonlocal continuum theory. “c” is the dispersion curve obtained in Lattice dynamics (Gazis d al.).
J. 0. A. IDIODI
1114
V. CONCLUSION
In this work the SGFM formalism has been applied to a nonlocal continuum that models a three-dimensional discrete lattice. This is an extension to three dimensions of a previous work that was restricted to one dimension. Explicit results have been presented here for nonlocal stress tensors that describe exactly some lattice dynamical models that have been widely used in the literature. The practicality of the formalism is demonstrated by investigating SH and S surface modes along the [ 1001 direction of the (001) surface of the SCC lattice. The predictions of this model are in agreement with those of previous workers. One of the motivations for this study is the need to offer a simpler approach to problems of interest in lattice dynamics, exploiting fully a procedure that is more in the analytical spirit than being entirely numerical. In addition, this study has also helped to bridge the gap, from the continuum side, between discrete and continuum models of crystalline, elastic solids. Acknowledgments-The
author is very much indebted to Professor F. Garcia-Moliner and To Dr. V. R. Velasco for suggesting the problem and for many valuable discussions. The author would also like to thank Professors Abdus !&lam and Mario Tosi, the IAEA and UNESCO for an ICTP fellowship and for hospitality at the International Centre for Theoretical Physics, Trieste, Italy. REFERENCES
111A. A. MARADUDIN, In Proceedings
ofthe International Conference on Lattice Dynamics, Paris, 5-9 September
1977 (Edited by M. Balkanski), p. 284. Flammarion (1978). Letl. Appl. Engng Sci. 1, I 1 (1973).
121A. C. ERINGEN,
[41 The Cosserat continuum was historically one of the first models of elastic media which could not be described within the scope of classical elasticity. A Cosserat medium is a continuum, each point of which possesses the six degrees of freedom of a rigid body. See Refs. [ 1I- 131for generalized models of the Cosserat continuum. D. C. GAZIS, R. HERMAN and R. F. WALLIS, Phys. Rev. 119, 533 (1960). tz; A. C. ERINGEN, Int. J. Engng Sci. 19, 1461 (1981); and references therein. WI F. W. de WETTE and G. P. ALLDREDGE, Methods Comput. Phys. 15, 163 (1976); and references therein. I71 A. A. MARADUDIN, E. W. MONTROLL, G. H. WEISS and 1. P. IPATOVA, Theory ofLattice Dynamics in the Harmonic Approximation, 2nd ed. Academic Press, New York (197 I). PI A. A. MARADUDIN, R. F. WALLIS and L. DOBRZYNSKI (Editors), Handbook ofsurfaces and Interfaces Vol. 3. Garland STPM Press, New York/London (1980). [91 P. BRUESCH, Phonons: Theory and Experiments I, Springer Series in Solid-State Sciences. Springer-Verlag, Berlin/Heidelberg/New York (1982). S. K. SINHA, Crit Rev. Solid State Sci. 3, 273 (1973). tryi C. TRUESDELL and R. A. TOUPIN, In Principles of Classical Mechanics and Field Theory, Encyclopedia of Physics, Vol. III/l (Edited by S. R&e), p. 20. Springer-Verlag, Bedin/G6ttingen/Heidelberg (1960). (121 E. KRONER (Editor), Mechanics of GeneralizedContinua. Proc. IUTAM Symposium on the Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart 1967. Springer-Verlag, Berlin/Heidelberg/New York (1968). R. STOJANOVIC. Mechanics ofPolar Continua. CISME. Udine C1969). F. GARCIA-MOLINER, G. PiATERO and V. R. VELkSCO, surf: &i. 136, 601 (1984); and references therein. 1151 J. 0. A. IDIODI, Surface Green Function Matching for a Nonlocal Continuum. One-Dimensional Model, to be published. I161 I. A. KUNIN, Elastic Media with Microstructure. I. One-Dimensional Models, Springer Series in Solid-State Sciences, Vol. 26. Springer-Verlag, Berlin/Heidelberg/New York (1982). 1171 I. A. KUNIN, Elastic Media with Microstructure II. Three-Dimensional Models, Springer Series in SolidState Sciences Vol. 44. Springer-Verlag, Berlin/Heidelberg/New York (1983). iI81 G. W. FARNELL, Phys. Acoust. 6, 109 (1970) (Edited by W. P. Mason and R. N. Thurston), Academic Press, New York. J. A. SCHOUTEN, Tensor Analysis for Physicists, 2nd ed. Oxford Univ. Press, London (1954). B. C. CLARK, D. C. GAZIS and R. F. WALLIS, Phys. Rev. A 134,1486 (1964). L. DOBRZYNSKI and P. MASRI, J. Phys. Chem. So/ids 33, 1603 (1972). P. S. YUEN and Y. P. VARSHNI, Phys. Rev. 164,895 (1967). S. W. MUSSER and K. H. RIEDER, Phys. Rev. B 2,3034 (1970). D. CASTIEL, L. DOBRZYNSKI and D. SPANJAARD, Su$ Sci. 59,252 (1976). G. P. ALLDREDGE, Phys. Left. A 41,281 (1972). V. R. VELASCO and F. GARCIA-MOLINER. J. Phys. C 13,2237 (1980); see also, R. Stoneley, Proc. Roy. Sot. London A 232,447
(1955). (Received 25 June 1985)