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Effective Linear Response of Random Media to Stochastic Sources.

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Annalen der Physik. 7. Folge, Band 38, Heft 1, 1981, S. 7-23
J. A. Barth, Leipzig
Effective Linear Response of Random Media
to Stochastic Sources
Sektion Physik der Technischen Universitit Dresden
A b s t r a c t . The notion of effective sourcesis introducedinorderto describe the mean field response
of a random medium to stochastic sources. Effective sources and stochastic sources are connected
by a linear functional which depends on the random material properties of the medium. A formal
perturbation series for the functional is given. Further, a single-grain approximation is proposed to
oalculate the effective sources in a strongly heterogeneous phase mixture. The approach is applied
to the electrostatic field in a mixture consisting of approximately spherical grains. Detailed calculations are carried out for stochastic sources of a special type in a two-component mixture.
Effektive lineare Response heterogener Medien auf stochastische Quellen
I n h a l tsubersicht. Zur Beschreibung eines von stochastischen Quellen in einem heterogenen
Material hervorgerufenen mittleren Feldes wird der Begriff der effektiven Quellen eingefiihrt. Die
effektiven Quellen sind mit den stochastischen Quellen iiber ein lineares Fnnktional verkniipft, das
die stochastischen Materialeigenschaften des Mediums enthiilt. Eine formale Stkirungsreihe fur dieses
Funktional wird angegeben. Des weiteren wird eine ,,Ein-Korn"-Naherung z u r Berechnung der
effektiven Quellen in stark heterogenen Phasengemischen vorgeschlagen. Zur Illustration wird die
Methode auf das elektrostatische Feld in einem aus naherungsweise kugelformigen Kornern bestehenden Gemisch angewandt. Detaillierte Rechnungen werden fur stochastische Quellen eines speziellen
Typs in einem zweikomponentigen Gemisch ausgefiihrt.
1. Introduction
I n a previous work [l],in the following referred to as I, the effective linear response,
i.e. the mean field response, and effective constitutive laws for mean fields in stochastically heterogeneous media have been considered under the assumpt,ion of non-randoni
sources. This restriction has generally been adopted in studies on heterogeneous media
(cf. [2--41). On the other hand, there is considerable work done on the problem of stochastic load or sources acting on homogeneous media [5-7]. This case is much easier
t o treat than that of stochastic material properties, a t least if the system is linear. Wave
emission from stochastic sources leads to the problem of coherence [8].
The purpose of the present paper is to join both these problenis and t o deal with the
effective behaviour of a linear heterogeneous mediuiii under stochastic load or source
terms. The considerations will be confined t o the study of the mean field response, as
has been done in I. An almost trivial example of a stochastic load is the force of gravity
acting on a heterogeneous elastic body because it contains the mass density as a randomly
varying parameter. More interesting cases may occur if we consider heat conduction or
diffusion phenomena with random sources due, e.g., t o chemical reactions, sinter processes, mechanical deformations and so on, radiation emission from random sources etc.
In order to develop the general notions which will be used t o treat the problem, we
start from a linear equation for a certain field u(r, t )
L,tu(r, t ) = q d r , 4.
The subscript st indicates that the operator L,,, as well as the source density qat,show
stochastic fluctuations. The inean field (u(r,t ) ) , averaged over an ensemble of samples,
is governed by an equation of the saiiie type
L(u(r. f ) : = q(r,t ) ,
which contains an effective operator L and an effective source term q. The definitions
of the effective quantities are somewhat arbitrary, but it seeins reasonable t o require L
to be conipletely independent of the source teriii qst and to depend merely on the properties of the inaterial described by Lst.Consequently, the operator L in eq. (1.2) must
be identified with the effective operator in the case of non-random sources which has
been studied in I (and which there was denoted by Len).So we are left with the task of
deducing a general expression for the effective source term q.
To this end let u s forinally solve eq. (I .l)by ineans of the stochastic Green’s function
introduced in I
u ( r ,t ) = Jdr’ Jdt‘ gst(r,r’. t , t‘) qsl(r’,t’) = : gstqst(r,t ) .
I n order to avoid boundary effects, we choose infinite samples. The sources are imagined
to be localized in a finite region, and t o produce fields which are vanishing a t infinity.
This boundary condition can be satisfied by a n appropriate choice of the Green’s function which has to tend to zero at great distances.
Splitting up qsl into the average value (qst) and the fluctuation q’
4x1 = (qxi) q’
and averaging eq. (1.3), we obtain
(u> = (981) (qxt?
The brackets always denote ensemble averages. Now we take into account the general
expression for the effective operator found in I
L = 9-1. g : = (g,,).
Applying this operator to eq. (1.5) gives
L ( u ) = (q.Qr> L(g,tq’>*
A comparison with eq. (1.2) yields a n expression for the effective source density
q = (qsr) L(gsiq’) = L(gd7,t) = : QSt.
The operator P is independent of the sources, and only containst he properties of the
inaterial which are characterized by the stochastic Green’s operator gat.Therefore. according to (1.8). the relation between q and qst is a linear one.
If there is no correlation between the fluctuations of the inaterial parameters on the
one hand and the sources on the other hand, the last term in eq. (1.7) vanishes, and the
effective source density reduces to the mean value
q = ‘%gat) (981) = <qSt>’
The stochastic Green’s function being unknown, an exact evaluation of q is, of course,
impossible. I n the following section, a perturbation series for q is given which can be
applied to weakly heterogeneous materials. I n Section 3 the method developed in I is
iiiodified in order to evaluate q in strongly heterogeneous bodies. To illustrate the general
ideas, Section -1 is devoted to the special case of electrostatics, which involves only one
stochastic material parameter and a scalar potential field. Finally Section 5 gives detailed
calculations for a siniple type of stochastic source distributions.
Linear Response of Random Media to Stochastic Sources
2. Perturbation Treatment
Analogously to ( 1 . 4 ) , we split up the stochastic operator L8,into
L,, = Lo L'.
Lo represents a homogeneous operator (which must not necessarily be identical with the
mean value of L8,).Its inversion go = L o 1 is supposed t o be known. The inhomogeneous
deviations L' are considered as small perturbations in coniparison with Lo. Thus the
linear operators L and P may be expanded in terms of L'. If we insert the decomposition
( 2 . 1 ) into eq. ( l . l ) ,we find
Lou = - L u
Averaging leads to
Lo<iLi = -(L'u)
The difference between the last two equations may be written as
Lou = Lo<u) + (1- M ) (q8t- L'u).
Here, the operator M ... = (...) prescribes a n ensemble average of the whole expression
on its right, whereas the projection operator 1 - M singles out the fluctuating part of
this expression. Application of go = L r 1 yields
= <u>
go(1 - M , (q8l - L'u) *
According to the boundary condition discussed above, the Green's function in go has to
vanish a t large distances. Equation (2.5) can be solved by iteration which leads to a n
infinite series
L')n [<u> f go(' - M ,
inserting this result into eq. (2.3), we write all terms acting on ( u ) on the left hand
side and all the other terms acting on qSton the right hand side of the eqaution, and obtain
+ M c (-L'go(l - M))nL'l(u>
= M c [-L'go(l - Wlnqat-
A coniparison with eqs. ( 1 . 2 ) and (1.8) yields the following perturbation series for the
effective quantities L and q
L =Lo+ M
2 [-L'go(l - M)lnL'= M n=O
2 [--L'go(l
- M)]"L,t.
The result for L is well known [9].
I f we use the notation of eq. (1.4) and specialize Lo t o the average operator <L,t),
the first terms of (2.9) are
From eqs. (2.8), (2.9) we can deduce a relation between the operators L and P
L = PL,,.
The perturbation series (2.9) clearly shows P t o be an integral operator
Ppst(r,t ) = J dr' J dt' P ( r , r', t - t') qst(r',t')
because it contains the integral operator go. Moreover, from (2.11), the range of the integral kernels of L and P can be expected t o be of the same order of magnitude. Thus,
recalling the results of I for the effective operator L, we can assume the integrand in
P t o decay rapidly for distances 1 r - r' j beyond the correlation length .2, Along range
behaviour may occur only for extreme heterogeneities.
3. Single-grain approach to q
The perturbation treatment fails in the case of strongly heterogeneous bodies. Therefore, a self-consistent single-grain approximation has been used in I to calculate L.
This method presupposed a body consisting of a statistical arrangement of homogeneous
regions (grains), which differ from one another by their material properties ci. Hence,
the stochastic material parameters involved in L,, could he written as
c(r) = z c i O i ( r ) ,
Oi(r) =
0 outside
the i-th grain,
C Oi(r)= 1 .
Further, L,, had been decomposed into a homogeneous part Lo and deviations Li due
t o the i-th regions, respectively
L, = L, 3-
where Lo was chosen in such a way that, for a special harmonic mean field, the relation
= L(u)
Now, the method applied in I will be carried over t o the calculation of q. To this end
we have to assume that, analogously to (3.2)) the fluctuating part q' of the source term
can be split up into a sum of contributions qi owing to the i-th regions, respectively
q' =
c qi + 9".
The additional term q" contains all the fluctuations which are not correlated with the
structure of the material and which, therefore, are due t o exterior stochastic parameters.
According t o eq. (1.9)) this term does not contribute t o q. Thus we obtain froni (1 3)
and (3.3)
+ L<F
The functions qi obey t o the equations
These equations being practically unsolvable, we introduce a single-grain approximation,
in the sense of I, by dropping the mixed terms j =+ i in eq. (3.3)
+ Li) qi
= Qi*
Linear Responss of Rmdom Media to Stochastic Sources
This equation is of the same type as eq. (2.10) of I, and may be solved in simple cases.
Let us denote the corresponding Green's operator by gi
l;li = (Lo + L p q i = : g iq i.
It depends only on the properties of the i-th grain and the effective properties involved
in Lo.
I n the case of statistical homogeneity, it is convenient t o transform q into the Fourierspace. Then, the integro-differential operator L becomes a number L ( k ) ,as has been
seen in I
q(k)= J dr e-*"
q(r) = (qat(k)) L(k)J dr
'Here, a n eventual time dependence is omitted because it does not influence further considerations. I n case of need, the time can be handled analogously to the space coordinates.
Moreover, it is useful t o refer the coordinates in the Green's function gi t o the center rf
of the i-th grain
7; = giqi = J dr' gi(r - ri, r'
- ri) qi(r').
The Fourier-transformation then yields
Jdr e-ik'r l;li = J d r dr' e-*" gi(r - ri, r' - ri) qi(r')
= J dr dr' e-*'r gi(r, r') e-ik'ri qi(r'
+ ri).
This expression contains the following statistical parameters : the kind of the i-th grain,
which will be denoted by (x, and which comprehends its shape, size and material properties, and the position ri of this grain. Furthermore, qi may depend on the random
surroundings of the i-th grain. The ensemble average of (3.4) will be performed in two
steps. First, we average, for a fixed kind a,with respect to the position ri and, if necessary, t o the surroundings, and define
where N means the total number of grains. Then we introduce the mean number N , of
grains of kind cx andthe probability pa = N,/N of a grain t o beof kind (x. With theaid
of these notions the integral in (3.8) can be written
J d r e - * . r ( 2 l;li(r))= 2 pa J dr dr' e-ik'rga(r, r') qa(r', k).
=: -qa(r,
4. Application to Electrostatics
Further calculations are specified t o the case of electrostatics, which has been chosen
for the seek of simplicity. Then, the various operators take the forms
= - E~ A , Li = - - ( E ~ E ~ Oi(r)
L(k) = k ~ ( kk)
(cf. I),where E , ~ , e0, E~ and E denote the corresponding dielectric parameters. The field u
and the sources qat correspond t o the electrostatic potential and the charge density,
respectively. Let us notice that electrostatics is mathematically equivalent t o other
stationary problems as electric or heat conduction, diffusion and so on. We have only
t o change the interpretation of the notations. It must be emphasized that, in general,
~ ( kdefined
by eq. (4.1) in connection with eq. (1.6) doesnot describe the relationbetween
the mean electric displacement ( D ) and the mean electric field ( E ) as it does in the case
of non-stochastic sources. Prom Maxwell's equations it follows for the mean displacement
ar ( D W >= (qst(r))
or in the Fourier-space
ik ' ( D ) k == (Pbt)kOn the other hand, the mean electric field ( E ) is governed by equation (1.2))where the
operator L is given by the last expression of eq. (4.1)
k - &(k)k ( U ) k = ik . & ( k )( E ) k = q ( k ) .
Comparison of both these equations yields
Therefore, the relation between (D)and ( E ) depends on the stochastic properties of
the source term. A combination of eqs. (4.2) and (4.4) shows i. t o connect also the mean
field with the mean charge
ik * i ( k )( E ) k = <qat>k(4.5)
If the stochastic fluctuations of ESt(r) are small, the perturbation series (2.9)) (2.10) can
be used to evaluate the effective sources q. We obtain
In the case of strong heterogeneity, we use the single-grain approximation (3.8)) (3.12),
and start with the evaluation of the integral
t a ( r ' ,k) := Jdr e--ik'r ga(r, r') = J d r e--ik'r gd(r',r) .
Taking into account the definition of the Green's function (3.7)) (3.9) and its symmetry
with respect t o a permutation of the coordinates r and r') we can see that 6, satisfies
the equation
(Lo La)t a ( r ,k ) = e--ik.r.
The operator Labelongs t o a grain of kind oc centered a t the origin r = 0. Equation (4.8)
resembles very much t o the equation obtained in I for the field fluctuations ui.Therefore,
it can be solved in a similar way. For this aim, we have to restrict ourselves t o a special
grain shape. We assume that all grains may be approximated by spheres of radii R,.
Then, the solution of (4.8) is found analogously t o I. Outside a grain a we have
= e--ik*r,
> R,,
+ 2 A;(?)
I+ 1
Pl([), 5 = cos ( + k , r)
where the rotational symmetry with respect to the axis k is taken into account. Terms
diverging at infinity are omitted. The Pl are Legendre's polynomials. Inside the grain
similar equations hold
- e--ik.r
Linear Response of Random Media to Stochastic Sources
The coefficients A,"and B," are determined by the boundary conditions a t the grain
face r = R,
For this end, we have to expand the exponential term exp (--ik. r ) into a series of
spherical harmonics
- ik.r
c (21 + 1)(-ill
j,W P l ( 0 ,
where the j l are spherical Bessel functions. Inserting eqs. (4.9), (4.10), (4.12) into the
boundary conditions (4.11) we obtain
+ 1)A? + E,IB~= 0 ,
where the abbreviation
is introduced. The first term in eq. (4.14) vanishes because of
2 p a J d r e-ik'r qa(r, k) = (2J d r e-ik'r qi(r))
J d r e--ik." (q' - q") = 0.
Thus, the effective charge density is finally given by
This representation of q(k) using the moments Y r ( k )proves t o be convenient for an expansion in a power series of kR,. Restricting ourselves to a la-th order approxiniation
in kR,, we can neglect all nionients Yr with I > n on account of the factor jl(kRa)in
eq. (4.17), the leading term of which is of I-th order. For k --f 0, eq. (4.17) simplifies t o
I n the ordinary space this result.takes the form
J d r d r ) = s d r <Qst(r)).
The total effective charge is equal to the total mean charge. Thus the operator P only
effects a redistribution of the mean charge. The same result can be obtained by integrating
the perturbation series (2.9), because all ternis except the zeroth-order term are divergences of vector fields which are vanishing a t large distances froin the sources. Consequently, their integrals are equal to zero. The zerot,h-order contribution leads back t o
eq. (4.19), which, therefore, is valid independently of the single-grain approximation.
It follows that the behaviour of the mean field at large distances froni the sources agrees
with the field which would be created by a mean charge distribution. Here, distances
can be considered as large, if they considerably exceed the correlation length of the
niaterial. This results from the fact stated at the end of Section 2 that the operatorP
perfornis displacements of the mean charges which are of the order of magnitude of the
correlation length. Only in the case of extrenie heterogeneities, we can expect larger
displacements in accordance with the non-local properties of P [11.
Finally, it inay be noticed that, for slowly varying fields kR,--f 0, the relation (4.4)
between (D)and ( E ) simplifies t o
i ( k = 0) = &(k = O ) ,
( D ( k = 0)) = &(k = 0) ( E ( k = 0))
as can be seen froni eq. (4.18). Therefore, a t sufficiently large distances froin the charge
distribution, a local, charge-independent dielectric constant can be used.
5. A Simple Example
Now, the method outlined above will be illustrated by an special, relatively simple
example. We assume a charge density
(4edr)) = qo(r),
The factor (ai 1)varies stochastically froni grain to grain. Its value is uniquely determined by the kind of grain 01 (i.e. ai= a&).Equation (5.1) suggests the following definition
of c/i
qdr) = %@iW QO(') *
Averaging over the grain positions, according t o eq. (:4.11),yields
where @,(r)describes a grain of the kind
grains this function is
O,(r) = @ ( R ,- r ) =
located at the origin. For spherically shaped
Linear Response of Random Media to Stochastic Sources
lnserting eq. ( 3 . 3 ) into the definition (4.15) of the nioinents Yf, we obtain
+ 1),<Ha
J dr
= -uJZ
4nR’ qO(k)(!%@ - 1 jl(x)}
Tlic calculation is given in the Appendix. With the use of this result, the effective source
distribution (3.1 7 ) beconies
-ua,az (1
xvherc the abbreviations
S(.c. y) : = 2 41
T(.L..Y) : =
+ 1)(22 + 1)
F 1Z(2l1+ 1)Ey
are introduced. The quantity v, = paN 4nR213V characterizes the volume fraction of
the grains of kind a.The funtion S had already been defined in I, where a good approxinmtion for the infinite sun1 had been found
S ( x , Y) w
+ Y Y ( 2 + Y) + Y(3&
+y) +
+ cos 2x + 2 2 Si(2x) - 2
Y2(X -
B(x) := - -x2 Si(2x)
The suiii T is related t o S b y
T ( z ,y)
= -[(l
+ y) S(n,y) - x2a(x)]
(cf. Appendix). The function W ( x ,y) vanishes in the limiting case of homogeneousinean
fields LR, + 0 in accordance with (4.18). In the opposite case of extremely inhomogeneous fields k R + 00, W reduces t o
Consequently, in this liinit, q ( k ) becomes
The last expression holds because of eq. (4.17) of I. This result is exact, as will be discussed below.
We apply the result (5.6) t o a two-phase mixture whose grainsare ofequalsize R,= R.
Thus, there are only two kinds of grains a = 1, 2 characterized by the dielectric constants + E, and occupying the volume fractions vl, v2, respectively. I n order t o satisfy
the third equation of (5.1), we write
a, = -, a,=--.
-Then, the effective charge density (5.6) takes the form
=: 1
Depending on the choice of a, the effective charge for k
0 can inore or less differ froin
the average charge.
The expression A W comprised in the braces is plotted in Figs. 1 and 2 for different
parameter values. It increases with the heterogeneity of the material, but remains finite
if the heterogeneity tends t o extreme values, El/&2 --f 0,oo.
Now we consider the quantity i defined in eq. (4.4), which connects the inem field
with the mean displacement and the mean charge, respectively and, therefore, represents
a (charge-dependent) dielectric parameter. For k --f 0, itends to E , as has been quoted
in eq. (4.20). I n the opposite limit k + 00, eq. (4.4) together with (5.11) and with (4.17)
of I yield
Fig. 1. Effective source distribution for a two-component mixture with volume fractions 0.8:0.2 and
different heterogenities EJE, according to eq. (5.13)
Linear Response of Random Media to Stochastic Sources
Fig. 2. Effective source distribution for a two-component mixture with volume fractions 0,5: 0,5
according to eq. (5.13)
This result coincides with the exact one which can be found by the following reasoning
(cf. [2]). The mean potential is given by
(u(k))= -J dr e-ik*rJ dr’ <g,,(r, r’) qSt(r’)).
I n the limit k + 00, only small distances dr = I r - r’ 1 < k-1 contribute t o the integral.
But for small distances Ar Q R the stochastic Green’s function may be replaced by
Thus, we obtain for kR + 00
and for the special charge density given in (5.1)
1 a,
in accordance with (5.14). The dielectric parameter ~ ( =
k 0 0 ) follows for nonstochastic
sources a, = 0
= 2v,
, &,
2 Ann. Pliysik. 7. Folge, Bd. 38
(cf. eq. (4.17) of I).Finally, according to eq. (4.4), the effective sources q are connected
with (5.18) and (5.19) by
which, for k -+ 00, leads t o the result already given in (5.11).
For a strongly heterogeneous medium .sg % E ~ expression
(5.14) niay be of the order
of magnitude of .sl (like E(k +00))but it may also attain the order of .s2, if 1 a, tends
to zero. This latter case signifies that there are no charges situated in the first component
with small E ~ Then.
the k-dependence of ;(k) turns out to be completely different from
the behaviour of E discussed in I. Above the percolation threshold v2 > v, = 113, e.g.,
the dielectric parameter ;(k) remains of the order of .s2 for all values of k, whereas ~ ( k )
decays rapidly to values of the order of .sl. Accordingly, the sharp peak at k = 0 observed
in I for ~ ( k )does
, not occur in i ( k ) .Therefore, after transition to the ordinary space, the
kernel function i ( r )does not necessarily show the long range behaviour which had been
obtained for E(r).This example illustrates the great influence that statistical properties
of the source distribution exert on the effective dielectricity i(k).
Fig. 3. Effective charge densities q(7) for a point charge situated in the second component of
two-componcent mixture
Linear Response of Random Media to Stoohastic Sources
To illustrate more in detail the results given in (5.13), we consider the mean field
response to a point charge
qo(r) = Q d(r), qo(k) = - Q .
The effective charge density in the physical space q(r) is given by the three-dimensional
Fourier-transform of eq. (5.13)
dkk sin kr[q(k)- ~(oo)].
q ( r ) = ( 2 ~ d(r)
) ~q(k = oo)
It consists of a point charge
at the center and a surrounding cloud
1 O0
dt t sin-t [ A W ( t )- A W ( o o ) ]
2n2R2 r
Fig. 4. Effective charge densities p(r) for a point charge situated in the second component of a
two-component mixture
A W ( k R ) : = W ( k R ,E ~ / E ( ~ ) ) W(LR,E ~ / E ( ~ ) ) .
This charge distribution is drawn in Figs. 3 and 4 for several two-component phase
mixtures and for a = -vl (no charges in component 1).The extension of the cloud is
about the grain size R except for very strong heterogeneities E & + C G
above the
percolation threshold v2 > 1/3 (cf. Fig. 4).The corresponding mean potential can be obtained froni (4.3) by Fourier transformation
4n Q
( u ( r ) ) = --
" d k 1+ a A W ( k R ) sinkr
r (2n)3$
dt 2 ~ ( 0 )
4na(k = 0 ) r
~ ( t )
+ a A W ( t ) )sin(%
The first factor in the last line describes the potential of a point charge Q in a medium
with a n effective dielectric constant E (k = 0). The integral gives the correction due t o
the dispersion of the effective dielectric parameter e(E) and due t o the effective sources.
This correct,ion factor is plotted in Figs. 5 and 6 for the special choice a = -z+ (no
Fig. 5. Mean potentials ( u ( r ) ) for a point charge situated in the second component of a two.component mixture (full lines) and for a non-random point charge (dashed lines)
Linear Response of Random Media to Stochastir Sourres
charges in component 1-solid lines). For comparison, the mean potential of a non-stochastic point charge (a = 0) is also represented (dashed lines). It shows a completely
different behaviour. This clearly demonstrates the great influence of stochastic properties
of the source distribution on the mean field response a t small distances.
I n general the correction factor tends rapidly t o unity for increasing distances from
the point charge. For distances large compared to the grain size the mean potential is
that of the mean charge produced in an effective material without dispersion. The
= 50, 100) indicates a breakdown of
strange behaviour of some curves in Fig. 6
the approximation scheme for extremely strong heterogeneities. Especially the change
in the sign of the potential is in conflict with physical expectation. This feature is niathematically connected with the maxima in the curves of Fig. 2, which, therefore, also have
t o be attributed to the inaccuracy of the approximation used.
rf R
Fig. G. Mean potentials (u(r)>for a point charge situated in the second component of a two-component mixture (full lines) and for a non-random point charge (dashed lines)
6. Conclusion
I n the present paper, the notion of effective sources has been introduced and procedures for calculating this quantity have been proposed. I n spite of the basic character
of this notion, practical evaluations of effective sources and the corresponding mean fields
near the sources will be, in general, very cumbersome. Moreover, it is to expect that in
most cases a precise knowledge of the stochastic source distribution will not be available.
Fortunately, the problem simplifies rigourously as long as we are only interested in static
fields at large distances from the sources, because in this case the effective sources may be
approximated by the mean sources and the nonlocality of the effective constitutive
laws may be disregarded. On the contrary, in the dynamical case of wave emission from
stochastic sources the stochastic source effects probably remain existent even at large
distances from the sources.
Finally, let us notice that stochastic initial conditions in non-stationary problems may
be treated in the same manner as random sources. I n fact, initial conditions may always
be transformed into supplementary source terms. For time-independent material parameters, for example, this can easely be done by performing a Laplace-transformation
with respect t o the time.
A c k n o w l d g e m e n t . The author is indebted t o W. HASSLERfor numerical calculations.
I n the appendix the calculations leading to the expressions for Yf (5.5) and T (5.9)
will be sketched. The integral in ( 5 . 5 ) may be transformed into
Twofold integration by parts yields
The last contribution vanishes because of
[($3P1(5)]= 0 .
(A 3)
To evaluate the surface integrals, we apply the expansion (4.12), and take into account
the orthogonality of Legrendre's polynomials
j- d5 5 ( 5 ) P245) =
I n this way, we obtain the equation
which has been used in (5.5).
A relation between the sums S and T appearing in (5.7) may be established with the
aid of the identity
1 (1 +Y)W+ 1) 21 1
-- 2
(A 6)
Inserting this equation into the definition of T ( x ,y) (5.7) and taking into account
the exact relations [l, 101
1 sin (2s)
cos 2m
2 s Si (2s) = : 1
Linear Response of Random Nedia to Stochastic Sources
T ( z ,y) = - [(l y) S ( x , y) - x2ar(x)].
If we make use of the approximation (5.8) for S(.u,y) derived in I, eq. (A 8) becomes
Int. J. Solids Struct. 12, 173 (1976).
[2] If. J. BERAN,
J. J. McCoy, Int. J. Solids Struct. 6, 1035 (1970).
[3] 11. J. BERAN,
Phys. Status Solidi (a) 6, 365 (1971).
[A] E. KROXER,H. KOCH,SM Archives 1, 183 (1976).
[5] s. CRANDALL(Ed.), Random Vibrations. M.T.T. Press, Cambridge Mass., Vol. I, 1958; Vol. I1
[6] V. A. LOMAKIN,
Statistical Problems in the Mechanics of Deformable Solids. Moscow, 1970 (in
[7] L. ARNOLD, Stochastic Differential Equations. Wiley-Interscience,New York, 1974.
[8] J. PEBm-4, Coherence of Light. van Nostrand-Reinhold, London, 1973.
[9] U. FRISCH,
Wave Propagation in Random Nedia, in: Probabilistic Methods in Applied Mathematics, Vol. I, Ed. by A. T.BACHURA-REID,
Academic Press, New York, 1968.
[lo] G . N. WATSON,
A Treatise on the Theory of BesseI Functions. Cambridge, 1958.
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