The nonlinear viscoelastic cosserat model for the wave propagation with generation of subharmonics

\

Int[ J[ Solids Structures Vol[ 24\ Nos 23Ð24\ pp[ 3464Ð3475\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9919Ð6572:87:,*see front matter

Pergamon

PII ] S9919Ð6572"87#99973Ð4

THE NONLINEAR VISCOELASTIC COSSERAT MODEL FOR THE WAVE PROPAGATION WITH GENERATION OF SUBHARMONICS O[ YU[ DINARIEV and V[ N[ NIKOLAEVSKII$ United Institute of Physics of the Earth\ Moscow 012709\ Russia "Received 10 April 0886 ^ in revised form 14 November 0886# Abstract*A Cosserat model is developed in nonlinear form to describe the appearance of lower frequencies in seismic waves spectra[ As in the Ericksen theory of liquid crystals\ elastic and viscous rheological elements are combined to get sensible results[ The translational degrees of freedom are described by linear elasticity but the rotational kinematically independent motion is governed by the nonlinear elastic potential and linear viscosity[ The dynamics of linear perturbations show that the longitudinal translations and rotations both decouple completely from other motions[ These rotations represent local oscillations due to the existence of the potential and they do not propagate[ However\ the linear transverse translations and microrotations are coupled[ It is shown that they describe seismic and acoustic waves[ In the nonlinear case these motions demonstrate some new interesting phenomena[ For instance\ if the propagation of the harmonic elastic wave is considered the microrotations behave like nonlinear oscillators excited by the external harmonic force[ Thus the system produces the same e}ects as those obtained recently for the Du.ng type equation with the help of the theory of attractors[ This means that the initial harmonic wave generates secondary waves with lower frequenc! ies[ These secondary frequencies are usually commensurable with the initial ones[ Numerical results show that the phenomenon still takes place if the initial wave consists of continuously distributed harmonics[ The generation of lower frequencies in granular media has been observed and reported but the theoretical explanation was lacking since it was thought conventionally that weak nonlinearities were able to produce higher frequencies only[ Þ 0887 Elsevier Science Ltd[ All rights reserved[

0[ INTRODUCTION

The Cosserat theory "Cosserat and Cosserat\ 0898 ^ Ericksen and Trusdell\ 0847 ^ Trusdell\ 0855# of kinematically independent dynamics of microstructure leads essentially to dynam! ics of two interpenetrating continua with interaction of a moment of momentum type[ This model is close in spirit to the FrenkelÐBiot poroelasticity "Biot\ 0845 ^ Frenkel\ 0833# where the di}erence in translational velocities of solid and ~uid particles is a basic ingredient[ This mathematical construction is extremely useful for the dynamic description of naturally fragmented materials such as rock and soil[ It serves to represent properties of real seismic waves\ of ground noise\ etc[\ which cannot be treated using conventional elasticity[ The signi_cance of a continuum with microstructure in geomechanics must be esti! mated in comparison with other possible models[ However a model with microstructure can be utilised in a wider range because of a multiscale character "in time and space# of geological processes[ The key point in applications is to combine properly the rheological laws at di}erent scales[ The EricksenÐLeslie model of a liquid crystal "Ericksen\ 0856 ^ Leslie\ 0857# can be used as an example of a proper approach[ We should like to stress that nonlinear version of the theory explaining many natural phenomena "Nikolaevskiy\ 0885 ^ Nikolaev and Galkin\ 0876# reduce to equations which are well known in modern mathematical physics[ $ Author to whom correspondence should be addressed[ Tel[ ] 996 984 143 12 14[ Fax ] 996 984 143 89 77[ E!mail ] victorÝuipe!ras[scgis[ru 3464

3465

O[ Yu[ Dinariev and V[ N[ Nikolaevskii 1[ BASIC EQUATIONS

The propagation of the waves in media with microstructure possesses some peculiar features due to the interaction of the translational and rotational degrees of freedom[ In the present work we investigate the wave processes when the macroscopic motion is purely elastic and the microscopic motion is dissipative[ Previously models were considered in which the macroscopic ~ow was viscous and the dynamics of the microrotations were described by equations of the nonlinear elasticity type "Dinariev and Nikolaevskii\ 0884a\ b#[ Here we consider the medium governed by the following sets of equations\ the mass balance ] dr ¦rvi\i  9\ rdt

"0#

dvi  pij\j ¦fi \ dt

"1#

the momentum balance ] r the moment of momentum balance ] d "o x rv ¦JVi #¦"oijk x j rvk ¦JVi #vl\l "oikl xk plj ¦pij # \j ¦oijk x j fk ¦mi \ dt ijk j k

"2#

and the energy balance ] d "K¦rU#¦"K¦rU#vl\l "vi pij # \j ¦"Vi pij # \j ¦fi vi ¦mi Vi ¦qi\i ¦o\ dt

"3#

where the conventional summation is used for repeated indices[ We use an inertial frame of reference with time t and coordinates xi\ and the notation ] r is the mass density\ vi the velocity\ pij the total stress tensor\ fi the body force\ J the scalar " for simplicity# moment of inertia "per unit volume#\ which is a tensor in a general case\ Vi the total angular velocity\ pij the couple stress tensor\ mi the moment of momentum\ K "ovivi¦JViVi#:1 the kinetic energy\ U the internal energy "per unit mass#\ qi the heat ~ux\ o the heat production[ The material time derivative is d 1 1  ¦vi dt 1t 1xi and oijk are components of the alternator tensor[ Introduce the displacement vector ui  ui "t\ xj# and the total rotation vector 8i  8i "t\ xj#[ Then the velocities can be expressed as the derivatives vi 

dui d8i \ Vi  [ dt dt

The system of equations "0#Ð"3# serves to determine the following variables ] r  r"t\ x j #\ ui  ui "t\ x j #\ 8i  8i "t\ x j #\ T  T"t\ x j # where T is the temperature[ In order to have complete formulation of the problem it is necessary to add to the constitutive relations\ which determine the quantities pij\ pij\ qi\ mi\

The nonlinear viscoelastic Cosserat model for the wave propagation

3466

U[ We shall formulate the constitutive relations in accordance with irreversible ther! modynamics "de Groot and Masur\ 0851#[ Assuming in_nitesimal displacement\ the approximations are\ for velocities vi  1t ui \ Vi  1t 8i \ for the strain tensor oij  "0:1#"ui\ j ¦u j\i # and for the vector of relative rotation 8i  8i ¦"0:1#oijk u j\k [ Suppose that the internal energy U depends on T\ oij \ 8i only\ then from "0# the mass density r changes are related to the strain tensor by r  `r9 \ `  det"dij −oij #\ where r9 is the initial density and dij are components of the unit tensor[ The Gibbs equation corresponding to our choice of the thermodynamic parameters of the system is T ds  dU−r−0 "sij doij ¦ni d8i #[

"4#

Here s is the entropy "per unit mass#\ sij is the elastic stress tensor and ni is the elastic moment of momentum generated by the microrotations[ The following expressions are consequences of eqn "4# ] sij 

0 1

0 1

1U 1U \ ni  [ 1oij s 18i s

"5#

The entropy time derivative is given by eqns "1#Ð"5# rT

ds "pij −sij #vi\ j ¦pi\ j Vi\ j ¦oijk Vi p jk ¦o−qi\i −ni Vi \ dt

"6#

where Vi  1t 8i is the relative angular velocity[ The viscous stress tensor tij  pij −sij is de_ned as the di}erence between total and elastic stresses[ In general it is composed of the symmetric tijs  t"ij# and the antisymmetric part tija  tðijŁ [ The entropy production "Truesdell\ 0861# is given by Pr

ds −T −0 o¦"qi T −0 # \i \ dt

which can be expressed with the help of "6# P  T −0 "tijs vi\ j ¦tija "vði\jŁ ¦oijk Vk #¦pij Vi\ j −ni Vi #¦qi "T −0 # \i [

"7#

According to the second law of thermodynamics\ this quantity must not be negative "Trusdell\ 0861# ]

3467

O[ Yu[ Dinariev and V[ N[ Nikolaevskii

P − 9[

"8#

We consider only dynamical processes which have zero dissipation in deformation processes governed solely by translational motions\ that is when tijs  9[ Then introducing the notation ji  t jk oijk −ni \ k\i  −T −0 T\i \ c  Vi\i \ cij  V"ij# −"0:2#dij c\ g  pi\i \ gij  pij −"0:2#dij g\ we reduce expression "7# to the form P  T −0 "jk Vk ¦gc¦gij cij ¦qi ki #[ According to the Onsager principle "de Groot and Masur\ 0851# this expression and the rotational invariance "The Pierre Curie principle# yields the following linear kinetic laws for dissipative ~uxes ] g  a0 c\ gij  a1 cij \ ji  b0 Vi \ qi  b1 ki \ where we introduced the kinetic coe.cients\ a0\ a1\ ba[ According to "8# the inequalities a0 − 9\ a1 − 9\ b0 − 9\ b1 − 9\ have to be valid[ In order to make the next step it is necessary to specify the functional form of the internal energy U[ 2[ LINEAR DYNAMICS CASE

Consider small dynamical perturbations of the initial state r  r9 \ T  T9 \ ui  9\ 8i  9[

"09#

In the lowest second!order approximation for displacements the internal energy is U  U9 "T#¦"1r9 # −0 "l0 "oii # 1 ¦l1 oij oij #¦r9−0 W\

"00#

where the elastic potential W for the microrotations is W  W"8i # "0:1#z8i 8i \

"01#

and l0\ l1\ z are constants[ The initial state "09# is assumed to be stable\ which implies the inequalities l0 \ l1 \ z × 9[ The corresponding elastic stress components are determined by "5#\ "00#\ "01# are the linear approximations sij  l0 okk dij ¦l1 oij \ ni  z8i [ Denote the double Fourier transform of any _eld f  f"t\ xi# as

"02#

The nonlinear viscoelastic Cosserat model for the wave propagation

3468

g

fF "v\ ki #  exp"−ivt−iki xi # f"t\ xi # dt dx j \

where v and ki are\ respectively\ frequency and wave numbers\ and introduce the notation q  T−T9 \ Cv 

dU9 "T# \ k1  ki ki [ dT

Equations "1#Ð"3# and "02# after linearization yield the system of equations for the Fourier transforms ] 9  ivr9 viF −ik j pijF  Aij0 u jF ¦B ij0 8 jF \ 9  ivJViF ¦oijk p jkF −ik j pijF −MiF  Aij1 u jF ¦B ij1 8 jF ¦C i1 qF \ 9  ivr9 CV uF ¦iki qiF  B 2j 8 jF ¦C 2 qF [

"03#

The coe.cients of this system are tensors of the second!\ _rst! and zero!order[ They can be represented in a form which explicitly takes into account SO"2#!invariance by decomposing them into the sum of _xed linear independent tensors ] Aija  Aa0 dij ¦Aa1 ki k j ¦Aa2 oijk ikk \ B ija  B a0 dij ¦B a1 ki k j ¦B a2 oijk ikk \ C i1  iki C 1 \ B i2  iki B 2 \ a  0\ 1[ The scalar coe.cients Aab\ Bab\ C1\ B2\ C2 can be calculated directly from "1#Ð"3# ]

0

1

0 0 0 A00  l1 k1 −v1 r9 ¦ ivb0 ¦ z k1 \ 1 1 1 0 0 A01  l0 ¦ l1 ¦ "ivb0 ¦z#\ 1 3 0 0 B 02  "ivb0 ¦z#\ A12  "ivb0 ¦z#\ 1 1 0 B 10  ivb0 ¦z−Jv1 ¦ a1 ivk1 \ 1 B 11 

0

1

0 0 a0 ¦ a1 iv\ C 1  9\ 2 1

B 2  9\ C 2  k1 b1 T 9−0 ¦ivr9 CV [ To investigate di}erent modes we calculate the determinant of "03# D  P0 P1 P 21 \ where P0  A00 ¦k1 A01 "l0 ¦l1 #k1 −v1 r9 \ P1  B 10 C 2 ¦"B 11 C 2 ¦B 2 C 1 #k1 \

3479

O[ Yu[ Dinariev and V[ N[ Nikolaevskii

P2  A00 B 10 −k1 A12 B 02 [ The general dispersion relation D  9 for the wave velocity can be decomposed into the three equations Pa  9\ a  0\ 1\ 2[ The _rst equation P0  9 evidently describes the longitudinal!translation waves which do not couple with other motions[ The equations P1  9 and P2  9 describe the longitudinal rotational waves and the coupled transverse translational!rotational waves corre! spondingly[ In order to demonstrate this\ _rst consider the dispersion relations in case of zero dissipation "P 0 9# ] P1  ivr9 CV "z−Jv1 #  9[

"04#

v  2v0 \ v0 "z:J# 0:1

"05#

This equation has the solution

which corresponds to the rotational oscillations[ Further\ we have P2 

00

1

1

0 0 0 l ¦ z k1 −v1 r9 "z−Jv1 #− z1 k1  9[ 1 1 1 3

"06#

Assuming a solution in the form k  a0 v\

"07#

eqn "06# becomes a01

00

1

1

0 l1 ¦ z Jv1 −zl1  1r9 "Jv1 −z#[ 1

It follows that there are no wave solutions in the frequency range v1 ¾ =v= ¾ v0 \ v1  v0 "0¦"1l1 # −0 z# −0:1 \ but solutions exist in the range v0 ³ =v=

and

=v= ³ v1 [

The former are called the {{acoustic|| "high frequency# waves[ They correspond to micro! structure " fragment scale# oscillations[ The latter are called the seismic waves and can be observed by conventional tools in seismology[ Dissipation a}ects the solutions of eqns "04# and "06#[ For simplicity assume that there is only one nonzero viscosity b  b0 × 9\

"08#

and the dissipative terms in the dispersion relations are relatively small[ Then the corrections of the _rst!order to "05#\ "07# are the following ]

The nonlinear viscoelastic Cosserat model for the wave propagation

3470

v  2v0 ¦ib"1J# −0 \ k  a0 v¦a1 b\ a1  −ir9 J 1 a0−0

00

1

0 l1 ¦ z Jv1 −zl1 1

−1

1

v5 [

So the friction coe.cient b\ which determines the viscous interaction of translational and rotational motions\ accounts for damping of fragment "microstructure# oscillations and attenuation of the transverse waves[

3[ NONLINEAR SPECTRA TRANSFORMATION

Now we turn our attention to possible nonlinear e}ects[ Let the processes be isothermal and depend only on one space coordinate x  x0\ and let the variables be represented in the following functional form ui  di1 u"t\ x#\ 8i  di2 8"t\ x#[ Assume nonlinearity of the model by taking into account the higher!order terms in eqn "01#\ then 0 0 0 W"8#  z81 ¦ z0 82 ¦ z1 83 [ 1 2 3

"19#

The stability condition of the initial rest state "09# requires that z × 9\ z1 × 9[ Again we consider the model with only one viscous coe.cient "08#[ Equations "1# and "2# give us coupled equations for u"t\ x#\ 8"t\ x# as the unknown variables ]

0

1

0 0 dW \ r9 1t1 u− l1 1x1 u  − 1x b 1t 8¦ 1 1 d8

J 1t1 8¦b 1t 8¦

dW 0  − J 1t1 1x u\ d8 1

"10#

"11#

where dW  z8¦z0 81 ¦z1 82 [ d8 Solutions of "10# and "11# can be found by the following procedure ] "a# construct a solution of "10# with zero right!hand side\ "b# substitute this into the right!hand side of "11# and _nd the solution for 8\ "c# substitute the latter into the right!hand side of "10# and _nd the correction to u\ and so on[ This procedure is essentially one particular way to implement the nonlinear perturbation technique[ It produces the convergent sequence\ if the right!hand side in "10# is su.ciently small in comparison with every term in the left!hand side[ Let us consider what happens when at the _rst step we have a harmonic wave

3471

O[ Yu[ Dinariev and V[ N[ Nikolaevskii

u  −a9 cos"vt−kx#\ v  kV\ V 1  1r9 :l1 [

"12#

Substituting "12# and "11#\ we obtain the Du.ng type equation for the damped nonlinear oscillator with harmonic external force[ After some transformation it can be reduced to the canonical form 1h1 F¦l 1h F¦`9 F¦`0 F1 ¦F2  A9 sin"1p"h−h9 ##\ 8"t\ x#  z1−0:1 J 0:1 "1p# −0 vF"h\ x#\h "1p# −0 vt\ h9 "1p# −0 kx\ l  1pbv−0 J −0 \ `9  1zJ −0:1 v−0 z1−0:1 \ `0  1pz0 J −0:1 v−0 z1−0:1 \ A9  1p1 ka9 vz10:1 J −0:1 [

"13#

Equation "13# has been investigated in the context of the theory of attractors "Feigenbaum\ 0879#\ which determines the general properties of solutions[ Any solution can be determined by the point in the phase space V\ 1hV " for example by the corresponding values at h  9#[ The dynamics of "13# is adequately represented by the Poincare map in the phase space M]

"F"9\ x#\1h F"9\ x## :"F"0\ x#\1h F"0\ x##\

which transforms the phase point into the point given by the solution at the end of the period[ This map is diminishing any phase volume by the factor e−l[ Therefore\ the stable solutions "attractors# are represented in the phase space by the sets of zero phase volume[ A stable solution can consist of a _nite number of phase points "so!called simple attractors# or of in_nite number of points "so!called strange attractors#[ In the _rst case the solution is periodic with an integer period n[ In the second case the stable solution is stochastic!like[ In both cases the response of the medium with microstructure to the initial wave "12# can contain lower frequencies[ In the case of the simple attractor the response can be decomposed into the discrete set of harmonic oscillations with the spectrum "mv:n# where m is an integer[ This is a new phenomenon[ Conventionally the nonlinear terms generate higher frequencies[ Experimentally the appearance of lower frequencies has been observed "Vilchinskaya and Nikolaevskii\ 0873 ^ Guschin et al[\ 0883 ^ Bubnov et al[\ 0883# in the _eld and in the laboratory\ but with great disbelief due to absence of any theoretical background[ Let us discuss the possible values of n[ According to Feigenbaum "0879# if to change the parameters a9\ v of the initial wave "12# one can encounter the period doubling bifurcation sequence in the response of the medium with microstructure[ However\ a _nite readjustment of the solution "not a mere bifurcation# can happen with any change of n[ Consider an attractor solution V  V9 "h−h9# of "13# and substitute it into "10#[ Then the correction to the initial wave "12# is described by

0

1

0 0 dW  F"t−V −0 x# r9 1t1 Du− l1 1x1 Du  − 1x b 1t 8¦ 1 1 d8 which can be solved explicitly ] Du  l1−0 VxF0 "t−V −0 x#\ F0  01 J 1t "8¦01 1x u#[ Thus\ the translational wave can really contain lower frequencies due to microrotations[ Now seismic signals are usually composed of harmonics distributed continuously over some frequency range[ Let Dv be the typical width of this distribution\ then the quantity

The nonlinear viscoelastic Cosserat model for the wave propagation

3472

t "Dv# −0 characterizes the duration of the signal[ The existence of an attractor can still be exhibited by the system until t will be negligible in comparison with the time "1Jb−0# of the rotational  oscillations[ Let the coe.cients of function "19# be such that "19# has the form W  W9 81 "8−a# 1 [ The initial signal is characterized by the strain o  o"t# which is distributed over a frequency range ]

g

0 o  1x u  1o9 o0−0 h" f # sin"1pft# df\ 1 where

g

h" f #  exp ð−"ln" f:f9 #:ln"0¦"j:f9 ### 1 Ł\ o0  h" f #df\

and\ say f9  1999 Hz[ ðThis frequency is chosen for comparison with the laboratory experiments "Guschin et al[ 0883 ^ Bubnov et al[ 0883#Ł[ The results of the computation are represented by the spectral density function\ where the spectral density of any function `  `"t# is de_ned by the expression

S" f # 

bg

b

exp"−i1pft#`"t# dt [

In Fig[ 0 the spectral density of the initial signal is normalized at the maximum value[ The frequency distribution width j is equal to 049 Hz[

Fig[ 0[ The normalized spectrum of the initial signal[

3473

O[ Yu[ Dinariev and V[ N[ Nikolaevskii

Fig[ 1[ The period doubling in the response spectrum[

Fig[ 2[ The response spectrum[ The broad peak looks like period tripling[

In Figs 1Ð3 the spectral density for the rotational response 8"t# is normalized also at the maximum value[ Parameters for these numerical experiments are chosen also in accord! ance with "Guschin et al[\ 0883 ^ Bubnov et al[\ 0883# as follows ] Figure No[ J\ kg:m 1J:b\ s W9\ kg:"ms1# b f9\ Hz j\ Hz o9

1 09−8 09−1 199 09−1 1999 049 09−2

2 09−8 09−1 099 09−1 1999 049 1[4×09−2

3 09−8 09−1 099 09−1 1999 0 1[4×09−2

Evidently Fig[ 1 represents the period doubling[ But period doubling is not the only option as we mentioned earlier\ and as can be seen in Fig[ 2[

The nonlinear viscoelastic Cosserat model for the wave propagation

3474

Fig[ 3[ Period tripling in the response spectrum in the case of a narrow frequency distribution in the initial signal[

Fig[ 4[ Experimental initial signal spectrum measured near the source[ The data was taken from Guschin et al[ "0883#[

Figure 2 corresponds to a more complicated case but the maximum amplitude at 559 Hz means period tripling[ It becomes even more clear if we make the frequency distribution narrower while other values are the same\ as shown in Fig[ 3[ In order to compare the numerical results with experiments we show in Figs 4 and 5 two experimental curves published in "Guschin et al[\ 0883#[ These curves show the spectral density of a signal propagating in soil "wet sand#[ The spectral density is normalized at the maximum value as before[ The transformed signal in Fig[ 5 looks very much like period tripling[ This example is only one of many from "Guschin et al[\ 0883 ^ Bubnov et al[\ 0883# which allow us to conclude\ that the numerical results given in Figs 1 and 2 qualitatively agree with the experiments[ We see that a nonlinear medium with microstructure dem! onstrates new theoretical phenomena which are in good agreement with observations for granular materials[ This provides a stimulus for further theoretical and experimental investigations in this direction[

3475

O[ Yu[ Dinariev and V[ N[ Nikolaevskii

Fig[ 5[ Experimental signal spectrum measured at some distance from the source[ The data was taken from Guschin et al[ "0883#[

REFERENCES Biot\ M[ A[ "0845# Theory of propagation of elastic waves in a ~uid!saturated porous solid[ Journal of the Acoustic Society of America 17\ 057Ð080[ Bubnov E[ Ya[\ Zaslavskii Yu[ M[ and Rubzov S[ N[ "0883# The impulse seismic probe of the undersurface nonhomogenuity[ Preprint N!390[ NIRFI\ Niznii Novgorod[ Cosserat\ E[ and Cosserat\ F[ "0898# Theorie des Corps Deformables[ Hermann\ Paris[ de Groot S[ R[ and Masur P[ "0851# Non!equilibrium Thermodynamics[ North Holland\ Amsterdam[ Dinariev\ O[ Yu[ and Nikolaevskii\ V[ N[ "0884a# The creep of rock as a seismic noise source[ Proc[ Russ[ Acad[ Sci[ "DAN#\ V[225\ 628Ð630[ Dinariev\ O[ Yu[ and Nikolaevskii\ V[ N[ "0884b# Unsteady regime for microrotations[ J[ Appl[ Math[ Mech[ "PMM# 46\ 064Ð079[ Ericksen\ J[ L[ "0856# Continuum theory of liquid crystals[ Applied Mechanics Review 19\ 0918Ð0921[ Ericksen\ J[ L[ and Trusdell\ C[ "0847# Exact theory of stress and strain in rods and shells[ Arch[ Rat[ Mech[ and Anal[ 0\ 184Ð212[ Feigenbaum\ M[ J[ "0879# Universal behaviour in nonlinear systems[ Los Alamos Science 0\ 3Ð16[ Frenkel\ Ya[ I[ "0833# On the theory of seismic and seismoelectric phenomena[ Trans[ USSR Acad[ Sci[ "Izv[ Ser[ Geoph[ Geo`r[# 7\ 023Ð038[ Guschin\ V[ V[\ Zaslavskii Yu[ M[ and Rubzov\ S[ N[ "0883# The transformation of the high frequency impulse spectrum while propagating in the surface layer of soil[ Prepring N284[ NIRFI\ Niznii Novgorod[ Leslie\ F[ M[ "0857# Some constitutive equations for liquid crystals[ Archives for Rational Mechanics and Analysis 17\ 154Ð172[ Nikolaev\ A[ V[\ and Galkin\ I[ N[\ eds[ "0876# Problems in Nonlinear Seismics[ Nauka\ Moscow\ p[ 177[ Nikolaevskiy\ V[ N[ "0885# Geomechanics and Fluidodynamics[ Kluwer\ Dordrecht\ p[ 238[ Trusdell\ C[ "0885# Six Lectures on Modern Natural Philosophy[ Springer!Verlag\ New York[ Trusdell\ C[ "0861# A First in Rational Continuum Mechanics[ The Hopkins University\ Baltimore\ Maryland[ Vilchinskaya and Nikolaevskii\ V[ N[ "0873# The acoustic emission and the spectrum of seismic signals[ Solid Earth Physics "Izv[ USSR Acad[ Sci[ Fiz[ Zem[# N4\ 282Ð399[