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Some applications of micropolar mechanics to earthquake problems
00?0-7225WI/Ofs@355-l0$02.~/0 Copyright 0 1981 Pergamon PressLtd.
hr. 1. Engng Sci. Vol. 19. pp. 815-W 1981 Printed in Great Britain. All tights reserved
SOME APPLICATIONS OF MICROPOLAR MECHANICS TO EARTHQUAKE PROBLEMS DORIN IE$AN Department of Mathematics, University of Ia$, 6600~Ia$,Romania Abstract-In this paper the theory of micropolar continua has been used to study some earthq~ke problems. In the first part of the paper the changes in the Earth’s inertia tensor due to earthquake faulting are calculated by using a reciprocal relation. The case of a homogeneous and isotropic elastic Earth modef is analyzed. The result is compared to that of the classical theory. In the second part of the paper the problem of body loadings equivalent to a seismic dislocation, in the framework of the linear theory of micropolar thermoelasticity, is studied. I. INTRODUCTION
elasticity theory is believed to be inadequate for the study of a material possesing granular structure. The granular character of the medium requires the use of the micropolar continuum mechanics[l-31. It appears that the micropolar continua possesses promise for ente~ng the Earth sciences because of the fact that most materials of interest are made of granules. This paper is concerned with the application of the theory of micropolar continua to the analysis of some earthquake problems. In the theory of Earth’s rotation, the calculation of changes in the Earth’s inertia tensor accompanying earthquakes is of major interest. The subject was considered in various papers[4-6]. In [6], the changes in the Earth’s inertia tensor are calculated by a directed application of the reciprocal theorem. In the first part of this paper we consider a micropolar Earth model and study the changes in Earth’s inertia tensor due to earthquake fautting. The problem is reduced to the finding of the stress and couple stress distributions which result when the model is steadily rotated about some central axis. In order to compare the result to that of the classical theory a special case is analyzed. In [7],it is shown that the seismic dislocations may be simulated by means of a suitable distribution of body force acting in an unfaulted medium. Formulae for body forces equivalent to a given dislocation are established in the framework of the linear theory of classical elasticity. This result was generalized in various theories& 91. In the second part of this paper we deal with the problem of body loadings equivalent to a seismic dislocation, in the framework of the linear dynamic theory of micropolar thermoelasticity. FundamentaI aspects of dislocation theory in micropolar media have been considered in various papers [ 10-141. THE CLASSICAL
2. INERTIA CHANGES
Throughout this paper a rectangular Cartesian coordinate system Ox, (k = 1, 2, 3) is used, Let (el, e2, e3) be the corresponding unit base vectors. Let V be a bounded regular region with the boundary S. We designate by ni the components of the outwark unit normal of S. We shall empfoy the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (I, 2, 3), summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate. The position vector to any point in space is r = xie+ In this paper we consider the linear theory of micropolar thermoelasticity. Let ui and v3i denote the components of the displacement vector and the components of the microrotation vector, respectively. The strain tensors are given by eij
=
uj.i
f
EjikpkiDkr
Qj
=
qj.1,
t2.u
where is the alternating symbol. Let 0 be the temperature measured from the absolute temperature TOof the natural state. In this section we consider the equilibrium theory of thermoelasticity, In this case the thermal and 6ijk
855
856
DORINIESAN
mechanical fields are not coupled. We assume that the function 0 is known. The constitutive equations are tij = Aijrsers+ Bij,sK,, - Diie, mij
=
Brsijer,
+
C;jrsK,,-
Eije,
(2.2)
where tij is the stress tensor, mij is the couple stress tensor, and A,,, Bij,, Cij,s,Dij, Eij are characteristic coefficients of the material. The equilibrium equations are tji,j + mj,j
+
Pfi
f?ijktjk
O,
=
pli = 0,
+
(2.3)
where p is the density in the reference state, fi are the components of body force vector and Ii are the components of body couple vector. The characteristic coefficients obey the symmetry relations
Aijr.7 = An,, Cijrs = Cnip
(2.4)
We assume that the domain V contains an internal regular surface Z on which the slip displacement are to be prescribed. The sides of Z are denoted by C+ and Z-. Let vi the components of the unit normal vector v of 2, directed from the ( -) to (t ) side. We consider the conditions
t:iVj
=
tJ;Vi,
TTl:iVj
=
t?ljiVj
on C,
(2.5)
where g+ and g- are the limits of g(x) as x approaches a point on Z’ and 2-, respectively, and Ui, @i are prescribed functions. Let us consider the boundary conditions tjinj
=
6,
l?ljflj
=
tii
on S,
(2.6)
where [ and pi are prescribed functions. The eqns (2.lH2.3) are considered in V\Z. Let us consider two &StiC StattX {uy’, @‘I”‘, e$‘, ~(iiu), t$r’, m f’} (a = 1, 2) corresponding to (a = 1, 2), respectively. Taking into the systems of loadings cfp’, II”‘, Uy), a?), iy’, fip’), ~9’~‘) account the relations (2.4) we obtain (t$' t Qj#'))e(i:) t (m(iil)+ E,#‘))K~’ = (tf) + Qj@))ej,) + (mf) + Eij/_#‘))~(ijl). (2.7)
If we denote
Zas =
IV
[($‘t
Dijt3(a))ejf) t (rn$T) t E,e’“‘)K$f’dv,
(2.8)
from (2.7) it follows
112 = z*, .
(2.9)
Some applications of micropolar mechanics to earthquake problems
857
Using the relations (2.1), (2.3), (2.5), (2.6) and the divergence theorem we get
where
7-I”’= vjt:.i”)+, MI”’ = vjm;‘+.
(2.11)
From (2.9) and (2.10) we obtain the following reciprocity relation
Now let us consider the elastic state {Uiy$i, lij, yij, uij, pij} corresponding to the system cfj” = fi, jj” = 0, U$” = 0, cg{‘)= 0, $) = 0, fill’ = 0, 0(l)= 0) and the elastic state {Ui,rpi,eij, Kij, tij, mij} corresponding to the system cfp’ = 0, /f*)= 0, Uf) = Ui, a$*)= Qi, L$*) = 0, fij*) = 0, f$*)= e}. Then, from (2.12) we obtain /vPfiu,dv =I, (Tu + Mi@i)da + I, (Di,eijt Eijyij)edo,
(2.13)
where Mi=pivk
T=aivj,
(2.14)
The moment of inertia about the axis (d), passing through the origin and having the orientation of a unit vector N is 4N) =
I
V
P 6’dv,
(2.15)
where 4 = r - (Nr)N. We can write Zoq = &jNiN, where .lij
=
f p(?Sij
Xgj)du,
(2.16)
V
is the inertia tensor. When the body V is deformed, we have a change in moment of inertia aI(N)= NiN$Jj = 2 to first order. UES Vol.
19. No.
6-H
fV
P @ido,
(2.17)
DORIN IESAN
858
Let us suppose that the body is rotated with angular velocity w about the axis (d). This creates the centrifugal body force (2.18)
F = o=r - (WP)W,
where w = UN. If o = 1, then F = 6. Let us denote by (A) the problem of determining a solution corresponding to the system of loadings A(N)= {f$” = ,yi, I$” = 0, u$‘J = 0, a$‘) = 0, i(l) = 0, fi$“= 0, 8’“= 0). We denote by So.,, = {uIN’, cplN’,@‘I, K$Y),Q’), m$/“)}the elastic state corresponding to the system ACN’.From (2.13) we obtain
where TV”) = t!“’ = m!N’v. ,i V,? M!N) I ,r I’
(2.20)
The relations (2.17) and (2.19) lead to 81(N)= NiNjSIij = 2
+2 I”
Iz
( TIN’U;+ M$N’@i)da
(D..e!i”’ do. II 9 11II + E..K!!‘))~
(2.21)
We, therefore, conclude that the change in moment of inertia is determined by the temperature variation and the characteristics Ui, @i of the seismic dislocation, and the steady rotation solution SCN’. In the case of isotropic and centrosymmetric solids we have 0, = bS,j, Eij = 0, where 6, is Kronecker’s delta. In this case the relation (2.21) becomes
( TIN’U;+ MiN’@i)da+ 2
bBe!N’dv I, .
(2.22)
rJ , t’i,“‘, m(iiN)have the form It is easy to see that the functions u$~‘, cplN’,eiy’, I#‘)
et?) = 1,
e$r’N,N,, . . ., rn$r) = m$r’N,N,,
e(rs) . * 9 rn!r) are symmetrical in r and s. where u@)) p, Since’ (2.22) holds for any choice of N, we can write
(2.23)
where
A similar result can be obtained for anisotropic media. From (2.23) we see that the changes in the inertia tensor can be calculated if we know the steady rotation solution SN’.
859
Some applications of micropolar mechanics to earthquake problems 3. APPLICATION
TO A MICROPOLAR
EARTH MODEL
Let us consider a micropolar Earth model. We assume that the domain V occupied by the Earth is a sphere of radius a and center at 0. We consider that V is occupied by a homogeneous isotropic and centrosymmetric micropolar material. In this case the constitutive equations become tij = he,,6ij + (/A + K)f?;jf pf?ij- bO8ij, Mij =
Ct'K,,Sijt PKji t J'Kij,
(3.1)
where A, j.~,K, CY, p, y, b are constants. In order to compare the change in the moment of inertia calculated in the framework of the micropolar theory to that of the classical theory we will consider the case when N = e3. The problem of deformation of a homogeneous and isotropic micropolar elastic sphere by rotation about the axis Ox3 was studied in[15]. The solution of this problem, using the spherical coordinates (R, 8, $) and physical components, is (l-2v)R uR = 15(2p tK)(lt 3AR (f$
V)
+ 14s2) t 6Bl-‘R-3’21S(lR)
1
SF_(3 cos2 6 - l), -+2(1- v) 3 (1-2v) ‘a = - 2 ( 42(1 - v)(2p t
K)
t AR &-~;IR2+21s2
[
t BR-*‘2[1,j2(1R)t 3(IR)-‘15,2(lR)1t a) ,.,$
=o,
(PR
=o,
$8
1
(3.2)
sin 21%
=o,
cp+= - $7AR2 t Bc-~~R-“~I~,~(IR)]sin 28, where IP is the modified Bessel function of order p, and A = 2;;; $Il,,2(lo)tq~~/2(~~)l[ml7/2(~a)t n4,2(la)l-', B =_2p(3tv)(2-a)s2a3" 3(2cL t K) c _
[m~,n(la)t &,z(W',
4~2(2v + 3)(1 - v) [ml7/2@)+ nl~,2(ru)l-'[pr,,,(ra) t &,2(h)], 3(2~ + K)
s2 =2(2jiyt A V=2At2jLtK'
K)'
c2=&q912/42PtK) Yb‘t'd +,
n=(2-(r)(7+5v)(lu)-‘,
m=7t5vt112(1-V)(2-u)c2(lu)-2, p=lt
6(3 t v)c2 t2(2-o)(15-23v)c2, (2v t 3)(lu)2 (2v t 3)(lu)2
q=2-u ill .
(3.3)
Let T be the unit vector in the slip direction at every point of Z. We have Q = TiU where U is the slip magnitude. We consider the case of a strike-slip fault for then yr = Tr = 0. If we assume that @i= 0, fi = 0, then from (2.22) and (3.2) we obtain Sl,, = 2
IP
PUdu,
(3.4)
860
DORINIESAN
where p =
1
3 (1- 24R2P T~Z.J~ sin* 6 - ,(2~ + K)
(4~ -7)R* _ 42$2 2(1 _ v)
14(1- u)
- 2BRm3’*[17j2(lR) + 3(lR)-‘I,j,(lR)] - 2(1 _ y) T,+Qsin* 8. >
(3.5)
Here we have noted that n, = 0. Since P is a smooth function of position over size scale very much larger than typical fault dimensions, one will generally not have to integrate, but may instead write SI,, = 2PUZ where P is evaluated at a representative point of Z. In this case we find that this Earth model leads to the inertia change
The corresponding relation in the classical theory of elasticity is [6]
$9z3, =- 2(4At
3/.&r*- (SAt 4p)r* pravs sin’ 6, 19A+ 14~
where A and I_Lare the Lame constants. 4.BODYLOADINGSEQUIVALENTTOASEISMICDISLOCATION In this section we consider the dynamic theory of micropolar thermoelasticity. We shall be concerned with the body loadings which would have to be applied in the absence of the fault to produce the same radiation as a given dislocation. The equations of motion are tji.j
+
pfi
=
mji,j
pii
t
~ijktjk
+
pli = Iijcpi,
(4.1)
where lij is the microinertia tensor and a superposed dot denotes partial derivation with respect to the time t. We have =hfi
(4.2)
PTOti = 4i.i + Pr,
(4.3)
hj
The equation of energy is
where n is the entropy per unit mass, 4i are the components of the heat flux vector and r is the heat supply per unit mass. To the constitutive eqns (2.2) we must adjoin the equations PT) =
Dijeij
t
EijKij
+
UB, qi =
&ije,j,
(4.4)
where a and k, are constitutive coefficients. We assume that kij = kj;.
(4.5)
ui = ai, tij = bi, cpi= ci, di = dj, n = no for t = 0.
(4.6)
The initial conditions are
861
Some applications of micropolar mechanics to earthquake problems
For the seismic faulting we may assume [7] that the stress vector and the couple stress vector are continuous in V. To the conditions (2.5) and (2.6) we adjoin the conditions 8+ - K = A, &Vi = q;Pi on C,
(4.7)
qiiii =; q on S,
(4.8)
and
respectively, where A and 4 are prescribed. We denote by u * u the convolution of u and tr I [u * u](x, t) =
I0
u(x, t - T)U(X, T)dr.
From (4.1) (4.3) and (4.6) we obtain
(4.9)
where g(t) = f, h(f) = 1,
E = pg * fj t p(fbi + Uj), (4.10)
Li= pg*ii t fij(td, + cj), Q=ph
* r+pTo7jo.
Let us consider two thermoelastic states I@) = {up), &I, @, qjB))(e = 1,2) corresponding to two different systems of loadings.
e$;),
KIT),
+j,
tj@,
m{;),
Using the relations (2.4) and the properties of the convolution, from (2.2) and (4.4) we get
Adding up these relations we obtain t!!) * et?’ + m$;’ * ,(@’ I/
(I
P77(‘1 * #2) =: t,) * ,$!) + mf’ * #‘-
prl’2’ * e(l).
(4.11)
If we introduce the notations
Lea=:
I
g*(f~~)*e~~)t~~~)~
@‘-- pn’“’ * @@‘)dv, (LY,/3 = 1, 2),
(4.12)
V
from (4.11) we have L 12 =
L,.
(4.13)
862
DORINIESAN
Using the relations (2.1), (4.4), (4.9) and the divergence theorem we get
where we have used the notations (2.11) and
On the basis of (4.2), (4.5) and (4.13) we obtain the reciprocity relation
ij”*U(2’+Al’)*Oi2)_~h*4(‘)*e(2) da
I
0
T$” * u$ + MI” * @,12) _f
H(l) * A(2)
h *
Fj2’
* u$”
+ Lj2’
* @
* ,#
da
1
0
_ $
* Q’2’
* ,j”’
0
1 do
(4.15)
Let us consider homogeneous initial conditions. In this case the relation (4.5) becomes
I [ V
P fl W* u$2)t/j*)
*
q$2)--!-h * +I)*
I
[1;*)* &2) + 5$*) * (p1;2) - -!-h* (i(l) * e(2) da 1 0
t
=
fj2) * up) + /j2) * cp$*)+
f$2)*
up + 5$
* ‘y’
rTo
h *
_+
,.(2) * ($1)
h *
I[ x
7-12’* u$*’ + M{2’ * @,I’)- f
I h *
0
dv
I
$2) * ,#I) da
0
-
e(2) dv
0
da.
H(2) * A”’
I
(4.16)
In the absence of the discontinuities we can obtain the reciprocity relations established in [17, 181. The reciprocity relations constitute powerful theoretical tools in the assessment of the theory of seismic-source mechanism and in the studies connected with seismic-wave propagations [Q-9, 161. Let us calculate the thermomechanical body loadings equivalent with a given dislocation. These loadings will be singular distributions concentrated on 1.
863
Some applications of micropolar mechanics IOearthquake problems
Let us assume that uq” 1 7 cp$, t912)are infinitely derivable functions of xi, t with compact support in V x (0, m). If the functions ur, (pi”, 0(” are given, then ff’, II:“, r(*’are calculated by means of the eqns (2.1), (2.2) (4.1) (4.3) and (4.4). In this case Z.J’*’ = a(*) = H(*) = 0. If the state I’(‘)corresponds to the faulted medium, then from (4.1) we obtain
(4.17)
If we take into account the relations
=-
u(J’(x 1 7 t)& .J.(x - y)du,
where S is the Dirac delta, then the relation (4.17) can be written in the form (fi”+4)*U1*‘t(Ij’)+~,)*(912)
-+I
x (r(l) t 3) *
f$2’]du
0
=
(4.18)
where we have used the notations
From (4.18) it follows that the effect of the discontinuities across Z can be represented by the extra body forces $i, the extra body couples ~ipi,and the extra heat supply 3, given by (4.19). As in [19], the reciprocity relations can be used in order to obtain integral representations of the displacement, microrotation and temperature fields. REFERENCES [I] A. C. ERINGEN and E. S. SUHUBI, 1~. J. Engng Sci. 2, 189,389 (1964). [Z] A. C. ERINGEN, Theory ofMicropo/arE/oslicity, MathematicalFMndamentals o/Fracture (Edited by H. Leibowilz), Vol. 2. Academic Press, New York (1%7). 131V. R. PARFI’IT and A. C. ERINGEN, J. Acousf. Sot. Amer. 45, 1258(1969). I41D. E. WYLIE and L. MANSINHA, Geophys. J. R. astr. SK 23,359 (1971).
864
DORIN IE$AN
[S] F. A. DAHLEN, Geophys. /. R. ask Sot. 25, 157(1971). [6] J. R. RICE and M. A. CHINNERY, Geophys. J. R. astr. Sot. 29,79 (1972). [7] R. BLJRRIDGEand L. KNOPOFF, BUN.Seism. Sot. Amer. 54, 1875(1964). [8] E. BOSCHI, I/ Nuouo Cimenfo, 18B,293 (1973). [9] E. BOSCH1and E. DI CURZIO, II Nuouo Cimento, 28B, 257 (1975). [IO] A. C. ERINGEN and W. D. CLAUS, Jr., Proc. NBS Conf. Fundamental Aspects of Dislocations Theory (1%9). [l I] W. D. CLAUS, Jr. and A. C. ERINGEN, ht. J. EngngSci. 9,605 (1971). [12] W. NOWACKI, Bull. Acad. Polon. Sci., St+. Sci. Techn. 21,431 (1973). [13] J. P. NOWACKI, Bull. Acad. Polon. Sci., St?. Sci. Techn. 21,585 (1973). [14] J. P. NOWACKI, Bull. Acad. Polon. Sci., Sir. Sci. Techn. 22,379 (1974). [IS] N. $ANDRU, Probleme Actuale in Mecanica Solidelor, Vol. I. Editura Academiei R. S. Romania (1975). [16] A. F. GANGI, J. geophys. Res. 75, 2088(1970). [I71 D. IE$AN, C. Rend. Acad. SC.Paris, 265,271 (1%7). [18] D. IESAN, Int. J. Engng Sci. 7, 1213(1%9). [I91 D. IESAN, Reo. Roum. Math. Pures et Appl. 15, 1181(1970). (Received 4 April 1980)
hr. 1. Engng Sci. Vol. 19. pp. 815-W 1981 Printed in Great Britain. All tights reserved
SOME APPLICATIONS OF MICROPOLAR MECHANICS TO EARTHQUAKE PROBLEMS DORIN IE$AN Department of Mathematics, University of Ia$, 6600~Ia$,Romania Abstract-In this paper the theory of micropolar continua has been used to study some earthq~ke problems. In the first part of the paper the changes in the Earth’s inertia tensor due to earthquake faulting are calculated by using a reciprocal relation. The case of a homogeneous and isotropic elastic Earth modef is analyzed. The result is compared to that of the classical theory. In the second part of the paper the problem of body loadings equivalent to a seismic dislocation, in the framework of the linear theory of micropolar thermoelasticity, is studied. I. INTRODUCTION
elasticity theory is believed to be inadequate for the study of a material possesing granular structure. The granular character of the medium requires the use of the micropolar continuum mechanics[l-31. It appears that the micropolar continua possesses promise for ente~ng the Earth sciences because of the fact that most materials of interest are made of granules. This paper is concerned with the application of the theory of micropolar continua to the analysis of some earthquake problems. In the theory of Earth’s rotation, the calculation of changes in the Earth’s inertia tensor accompanying earthquakes is of major interest. The subject was considered in various papers[4-6]. In [6], the changes in the Earth’s inertia tensor are calculated by a directed application of the reciprocal theorem. In the first part of this paper we consider a micropolar Earth model and study the changes in Earth’s inertia tensor due to earthquake fautting. The problem is reduced to the finding of the stress and couple stress distributions which result when the model is steadily rotated about some central axis. In order to compare the result to that of the classical theory a special case is analyzed. In [7],it is shown that the seismic dislocations may be simulated by means of a suitable distribution of body force acting in an unfaulted medium. Formulae for body forces equivalent to a given dislocation are established in the framework of the linear theory of classical elasticity. This result was generalized in various theories& 91. In the second part of this paper we deal with the problem of body loadings equivalent to a seismic dislocation, in the framework of the linear dynamic theory of micropolar thermoelasticity. FundamentaI aspects of dislocation theory in micropolar media have been considered in various papers [ 10-141. THE CLASSICAL
2. INERTIA CHANGES
Throughout this paper a rectangular Cartesian coordinate system Ox, (k = 1, 2, 3) is used, Let (el, e2, e3) be the corresponding unit base vectors. Let V be a bounded regular region with the boundary S. We designate by ni the components of the outwark unit normal of S. We shall empfoy the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (I, 2, 3), summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate. The position vector to any point in space is r = xie+ In this paper we consider the linear theory of micropolar thermoelasticity. Let ui and v3i denote the components of the displacement vector and the components of the microrotation vector, respectively. The strain tensors are given by eij
=
uj.i
f
EjikpkiDkr
Qj
=
qj.1,
t2.u
where is the alternating symbol. Let 0 be the temperature measured from the absolute temperature TOof the natural state. In this section we consider the equilibrium theory of thermoelasticity, In this case the thermal and 6ijk
855
856
DORINIESAN
mechanical fields are not coupled. We assume that the function 0 is known. The constitutive equations are tij = Aijrsers+ Bij,sK,, - Diie, mij
=
Brsijer,
+
C;jrsK,,-
Eije,
(2.2)
where tij is the stress tensor, mij is the couple stress tensor, and A,,, Bij,, Cij,s,Dij, Eij are characteristic coefficients of the material. The equilibrium equations are tji,j + mj,j
+
Pfi
f?ijktjk
O,
=
pli = 0,
+
(2.3)
where p is the density in the reference state, fi are the components of body force vector and Ii are the components of body couple vector. The characteristic coefficients obey the symmetry relations
Aijr.7 = An,, Cijrs = Cnip
(2.4)
We assume that the domain V contains an internal regular surface Z on which the slip displacement are to be prescribed. The sides of Z are denoted by C+ and Z-. Let vi the components of the unit normal vector v of 2, directed from the ( -) to (t ) side. We consider the conditions
t:iVj
=
tJ;Vi,
TTl:iVj
=
t?ljiVj
on C,
(2.5)
where g+ and g- are the limits of g(x) as x approaches a point on Z’ and 2-, respectively, and Ui, @i are prescribed functions. Let us consider the boundary conditions tjinj
=
6,
l?ljflj
=
tii
on S,
(2.6)
where [ and pi are prescribed functions. The eqns (2.lH2.3) are considered in V\Z. Let us consider two &StiC StattX {uy’, @‘I”‘, e$‘, ~(iiu), t$r’, m f’} (a = 1, 2) corresponding to (a = 1, 2), respectively. Taking into the systems of loadings cfp’, II”‘, Uy), a?), iy’, fip’), ~9’~‘) account the relations (2.4) we obtain (t$' t Qj#'))e(i:) t (m(iil)+ E,#‘))K~’ = (tf) + Qj@))ej,) + (mf) + Eij/_#‘))~(ijl). (2.7)
If we denote
Zas =
IV
[($‘t
Dijt3(a))ejf) t (rn$T) t E,e’“‘)K$f’dv,
(2.8)
from (2.7) it follows
112 = z*, .
(2.9)
Some applications of micropolar mechanics to earthquake problems
857
Using the relations (2.1), (2.3), (2.5), (2.6) and the divergence theorem we get
where
7-I”’= vjt:.i”)+, MI”’ = vjm;‘+.
(2.11)
From (2.9) and (2.10) we obtain the following reciprocity relation
Now let us consider the elastic state {Uiy$i, lij, yij, uij, pij} corresponding to the system cfj” = fi, jj” = 0, U$” = 0, cg{‘)= 0, $) = 0, fill’ = 0, 0(l)= 0) and the elastic state {Ui,rpi,eij, Kij, tij, mij} corresponding to the system cfp’ = 0, /f*)= 0, Uf) = Ui, a$*)= Qi, L$*) = 0, fij*) = 0, f$*)= e}. Then, from (2.12) we obtain /vPfiu,dv =I, (Tu + Mi@i)da + I, (Di,eijt Eijyij)edo,
(2.13)
where Mi=pivk
T=aivj,
(2.14)
The moment of inertia about the axis (d), passing through the origin and having the orientation of a unit vector N is 4N) =
I
V
P 6’dv,
(2.15)
where 4 = r - (Nr)N. We can write Zoq = &jNiN, where .lij
=
f p(?Sij
Xgj)du,
(2.16)
V
is the inertia tensor. When the body V is deformed, we have a change in moment of inertia aI(N)= NiN$Jj = 2 to first order. UES Vol.
19. No.
6-H
fV
P @ido,
(2.17)
DORIN IESAN
858
Let us suppose that the body is rotated with angular velocity w about the axis (d). This creates the centrifugal body force (2.18)
F = o=r - (WP)W,
where w = UN. If o = 1, then F = 6. Let us denote by (A) the problem of determining a solution corresponding to the system of loadings A(N)= {f$” = ,yi, I$” = 0, u$‘J = 0, a$‘) = 0, i(l) = 0, fi$“= 0, 8’“= 0). We denote by So.,, = {uIN’, cplN’,@‘I, K$Y),Q’), m$/“)}the elastic state corresponding to the system ACN’.From (2.13) we obtain
where TV”) = t!“’ = m!N’v. ,i V,? M!N) I ,r I’
(2.20)
The relations (2.17) and (2.19) lead to 81(N)= NiNjSIij = 2
+2 I”
Iz
( TIN’U;+ M$N’@i)da
(D..e!i”’ do. II 9 11II + E..K!!‘))~
(2.21)
We, therefore, conclude that the change in moment of inertia is determined by the temperature variation and the characteristics Ui, @i of the seismic dislocation, and the steady rotation solution SCN’. In the case of isotropic and centrosymmetric solids we have 0, = bS,j, Eij = 0, where 6, is Kronecker’s delta. In this case the relation (2.21) becomes
( TIN’U;+ MiN’@i)da+ 2
bBe!N’dv I, .
(2.22)
rJ , t’i,“‘, m(iiN)have the form It is easy to see that the functions u$~‘, cplN’,eiy’, I#‘)
et?) = 1,
e$r’N,N,, . . ., rn$r) = m$r’N,N,,
e(rs) . * 9 rn!r) are symmetrical in r and s. where u@)) p, Since’ (2.22) holds for any choice of N, we can write
(2.23)
where
A similar result can be obtained for anisotropic media. From (2.23) we see that the changes in the inertia tensor can be calculated if we know the steady rotation solution SN’.
859
Some applications of micropolar mechanics to earthquake problems 3. APPLICATION
TO A MICROPOLAR
EARTH MODEL
Let us consider a micropolar Earth model. We assume that the domain V occupied by the Earth is a sphere of radius a and center at 0. We consider that V is occupied by a homogeneous isotropic and centrosymmetric micropolar material. In this case the constitutive equations become tij = he,,6ij + (/A + K)f?;jf pf?ij- bO8ij, Mij =
Ct'K,,Sijt PKji t J'Kij,
(3.1)
where A, j.~,K, CY, p, y, b are constants. In order to compare the change in the moment of inertia calculated in the framework of the micropolar theory to that of the classical theory we will consider the case when N = e3. The problem of deformation of a homogeneous and isotropic micropolar elastic sphere by rotation about the axis Ox3 was studied in[15]. The solution of this problem, using the spherical coordinates (R, 8, $) and physical components, is (l-2v)R uR = 15(2p tK)(lt 3AR (f$
V)
+ 14s2) t 6Bl-‘R-3’21S(lR)
1
SF_(3 cos2 6 - l), -+2(1- v) 3 (1-2v) ‘a = - 2 ( 42(1 - v)(2p t
K)
t AR &-~;IR2+21s2
[
t BR-*‘2[1,j2(1R)t 3(IR)-‘15,2(lR)1t a) ,.,$
=o,
(PR
=o,
$8
1
(3.2)
sin 21%
=o,
cp+= - $7AR2 t Bc-~~R-“~I~,~(IR)]sin 28, where IP is the modified Bessel function of order p, and A = 2;;; $Il,,2(lo)tq~~/2(~~)l[ml7/2(~a)t n4,2(la)l-', B =_2p(3tv)(2-a)s2a3" 3(2cL t K) c _
[m~,n(la)t &,z(W',
4~2(2v + 3)(1 - v) [ml7/2@)+ nl~,2(ru)l-'[pr,,,(ra) t &,2(h)], 3(2~ + K)
s2 =2(2jiyt A V=2At2jLtK'
K)'
c2=&q912/42PtK) Yb‘t'd +,
n=(2-(r)(7+5v)(lu)-‘,
m=7t5vt112(1-V)(2-u)c2(lu)-2, p=lt
6(3 t v)c2 t2(2-o)(15-23v)c2, (2v t 3)(lu)2 (2v t 3)(lu)2
q=2-u ill .
(3.3)
Let T be the unit vector in the slip direction at every point of Z. We have Q = TiU where U is the slip magnitude. We consider the case of a strike-slip fault for then yr = Tr = 0. If we assume that @i= 0, fi = 0, then from (2.22) and (3.2) we obtain Sl,, = 2
IP
PUdu,
(3.4)
860
DORINIESAN
where p =
1
3 (1- 24R2P T~Z.J~ sin* 6 - ,(2~ + K)
(4~ -7)R* _ 42$2 2(1 _ v)
14(1- u)
- 2BRm3’*[17j2(lR) + 3(lR)-‘I,j,(lR)] - 2(1 _ y) T,+Qsin* 8. >
(3.5)
Here we have noted that n, = 0. Since P is a smooth function of position over size scale very much larger than typical fault dimensions, one will generally not have to integrate, but may instead write SI,, = 2PUZ where P is evaluated at a representative point of Z. In this case we find that this Earth model leads to the inertia change
The corresponding relation in the classical theory of elasticity is [6]
$9z3, =- 2(4At
3/.&r*- (SAt 4p)r* pravs sin’ 6, 19A+ 14~
where A and I_Lare the Lame constants. 4.BODYLOADINGSEQUIVALENTTOASEISMICDISLOCATION In this section we consider the dynamic theory of micropolar thermoelasticity. We shall be concerned with the body loadings which would have to be applied in the absence of the fault to produce the same radiation as a given dislocation. The equations of motion are tji.j
+
pfi
=
mji,j
pii
t
~ijktjk
+
pli = Iijcpi,
(4.1)
where lij is the microinertia tensor and a superposed dot denotes partial derivation with respect to the time t. We have =hfi
(4.2)
PTOti = 4i.i + Pr,
(4.3)
hj
The equation of energy is
where n is the entropy per unit mass, 4i are the components of the heat flux vector and r is the heat supply per unit mass. To the constitutive eqns (2.2) we must adjoin the equations PT) =
Dijeij
t
EijKij
+
UB, qi =
&ije,j,
(4.4)
where a and k, are constitutive coefficients. We assume that kij = kj;.
(4.5)
ui = ai, tij = bi, cpi= ci, di = dj, n = no for t = 0.
(4.6)
The initial conditions are
861
Some applications of micropolar mechanics to earthquake problems
For the seismic faulting we may assume [7] that the stress vector and the couple stress vector are continuous in V. To the conditions (2.5) and (2.6) we adjoin the conditions 8+ - K = A, &Vi = q;Pi on C,
(4.7)
qiiii =; q on S,
(4.8)
and
respectively, where A and 4 are prescribed. We denote by u * u the convolution of u and tr I [u * u](x, t) =
I0
u(x, t - T)U(X, T)dr.
From (4.1) (4.3) and (4.6) we obtain
(4.9)
where g(t) = f, h(f) = 1,
E = pg * fj t p(fbi + Uj), (4.10)
Li= pg*ii t fij(td, + cj), Q=ph
* r+pTo7jo.
Let us consider two thermoelastic states I@) = {up), &I, @, qjB))(e = 1,2) corresponding to two different systems of loadings.
e$;),
KIT),
+j,
tj@,
m{;),
Using the relations (2.4) and the properties of the convolution, from (2.2) and (4.4) we get
Adding up these relations we obtain t!!) * et?’ + m$;’ * ,(@’ I/
(I
P77(‘1 * #2) =: t,) * ,$!) + mf’ * #‘-
prl’2’ * e(l).
(4.11)
If we introduce the notations
Lea=:
I
g*(f~~)*e~~)t~~~)~
@‘-- pn’“’ * @@‘)dv, (LY,/3 = 1, 2),
(4.12)
V
from (4.11) we have L 12 =
L,.
(4.13)
862
DORINIESAN
Using the relations (2.1), (4.4), (4.9) and the divergence theorem we get
where we have used the notations (2.11) and
On the basis of (4.2), (4.5) and (4.13) we obtain the reciprocity relation
ij”*U(2’+Al’)*Oi2)_~h*4(‘)*e(2) da
I
0
T$” * u$ + MI” * @,12) _f
H(l) * A(2)
h *
Fj2’
* u$”
+ Lj2’
* @
* ,#
da
1
0
_ $
* Q’2’
* ,j”’
0
1 do
(4.15)
Let us consider homogeneous initial conditions. In this case the relation (4.5) becomes
I [ V
P fl W* u$2)t/j*)
*
q$2)--!-h * +I)*
I
[1;*)* &2) + 5$*) * (p1;2) - -!-h* (i(l) * e(2) da 1 0
t
=
fj2) * up) + /j2) * cp$*)+
f$2)*
up + 5$
* ‘y’
rTo
h *
_+
,.(2) * ($1)
h *
I[ x
7-12’* u$*’ + M{2’ * @,I’)- f
I h *
0
dv
I
$2) * ,#I) da
0
-
e(2) dv
0
da.
H(2) * A”’
I
(4.16)
In the absence of the discontinuities we can obtain the reciprocity relations established in [17, 181. The reciprocity relations constitute powerful theoretical tools in the assessment of the theory of seismic-source mechanism and in the studies connected with seismic-wave propagations [Q-9, 161. Let us calculate the thermomechanical body loadings equivalent with a given dislocation. These loadings will be singular distributions concentrated on 1.
863
Some applications of micropolar mechanics IOearthquake problems
Let us assume that uq” 1 7 cp$, t912)are infinitely derivable functions of xi, t with compact support in V x (0, m). If the functions ur, (pi”, 0(” are given, then ff’, II:“, r(*’are calculated by means of the eqns (2.1), (2.2) (4.1) (4.3) and (4.4). In this case Z.J’*’ = a(*) = H(*) = 0. If the state I’(‘)corresponds to the faulted medium, then from (4.1) we obtain
(4.17)
If we take into account the relations
=-
u(J’(x 1 7 t)& .J.(x - y)du,
where S is the Dirac delta, then the relation (4.17) can be written in the form (fi”+4)*U1*‘t(Ij’)+~,)*(912)
-+I
x (r(l) t 3) *
f$2’]du
0
=
(4.18)
where we have used the notations
From (4.18) it follows that the effect of the discontinuities across Z can be represented by the extra body forces $i, the extra body couples ~ipi,and the extra heat supply 3, given by (4.19). As in [19], the reciprocity relations can be used in order to obtain integral representations of the displacement, microrotation and temperature fields. REFERENCES [I] A. C. ERINGEN and E. S. SUHUBI, 1~. J. Engng Sci. 2, 189,389 (1964). [Z] A. C. ERINGEN, Theory ofMicropo/arE/oslicity, MathematicalFMndamentals o/Fracture (Edited by H. Leibowilz), Vol. 2. Academic Press, New York (1%7). 131V. R. PARFI’IT and A. C. ERINGEN, J. Acousf. Sot. Amer. 45, 1258(1969). I41D. E. WYLIE and L. MANSINHA, Geophys. J. R. astr. SK 23,359 (1971).
864
DORIN IE$AN
[S] F. A. DAHLEN, Geophys. /. R. ask Sot. 25, 157(1971). [6] J. R. RICE and M. A. CHINNERY, Geophys. J. R. astr. Sot. 29,79 (1972). [7] R. BLJRRIDGEand L. KNOPOFF, BUN.Seism. Sot. Amer. 54, 1875(1964). [8] E. BOSCHI, I/ Nuouo Cimenfo, 18B,293 (1973). [9] E. BOSCH1and E. DI CURZIO, II Nuouo Cimento, 28B, 257 (1975). [IO] A. C. ERINGEN and W. D. CLAUS, Jr., Proc. NBS Conf. Fundamental Aspects of Dislocations Theory (1%9). [l I] W. D. CLAUS, Jr. and A. C. ERINGEN, ht. J. EngngSci. 9,605 (1971). [12] W. NOWACKI, Bull. Acad. Polon. Sci., St+. Sci. Techn. 21,431 (1973). [13] J. P. NOWACKI, Bull. Acad. Polon. Sci., St?. Sci. Techn. 21,585 (1973). [14] J. P. NOWACKI, Bull. Acad. Polon. Sci., Sir. Sci. Techn. 22,379 (1974). [IS] N. $ANDRU, Probleme Actuale in Mecanica Solidelor, Vol. I. Editura Academiei R. S. Romania (1975). [16] A. F. GANGI, J. geophys. Res. 75, 2088(1970). [I71 D. IE$AN, C. Rend. Acad. SC.Paris, 265,271 (1%7). [18] D. IESAN, Int. J. Engng Sci. 7, 1213(1%9). [I91 D. IESAN, Reo. Roum. Math. Pures et Appl. 15, 1181(1970). (Received 4 April 1980)