Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium
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Інститут геофізики ім. С.I. Субботіна НАН України
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| Zitieren: | Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium / B. Maslov // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 99-101. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1018632016-06-09T03:02:07Z Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium Maslov, B. 2010 Article Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium / B. Maslov // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 99-101. — Бібліогр.: 5 назв. — англ. 0203-3100 http://dspace.nbuv.gov.ua/handle/123456789/101863 en Геофизический журнал Інститут геофізики ім. С.I. Субботіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Article |
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Maslov, B. |
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Maslov, B. Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium Геофизический журнал |
| author_facet |
Maslov, B. |
| author_sort |
Maslov, B. |
| title |
Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium |
| title_short |
Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium |
| title_full |
Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium |
| title_fullStr |
Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium |
| title_full_unstemmed |
Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium |
| title_sort |
dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium |
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Інститут геофізики ім. С.I. Субботіна НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/101863 |
| citation_txt |
Dispersion and scatter of elastic waves in a pre-stressed and fractured geological medium / B. Maslov // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 99-101. — Бібліогр.: 5 назв. — англ. |
| series |
Геофизический журнал |
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AT maslovb dispersionandscatterofelasticwavesinaprestressedandfracturedgeologicalmedium |
| first_indexed |
2025-07-07T11:29:55Z |
| last_indexed |
2025-07-07T11:29:55Z |
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1836987498226515968 |
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Dispersion and scatter of elastic waves in a pre-stressed
and fractured geological medium
B. Maslov, 2010
Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine
maslov@inmech.kiev.ua
It is known [Crampin, Peacock, 2008] shear-
waves propagating in anisotropic rocks split into two
approximately orthogonal polarisations that travel
at different velocities. Such seismic birefringence
aligned azimuthally is widely observed in almost all
igneous, metamorphic, and sedimentary rocks in
the Earth’s crust in almost all geological and tec-
tonic regimes. The causes and interpretation of shear
wave splitting in the Earth’s crust are still not clear-
ly understood [Maslov et al., 2006]. Although, in
principle, shear-wave splitting is simple in concept
and easy to interpret in terms of systems of aniso-
tropic symmetry, in practice there are subtle diffe-
rences from isotropic propagation that make it easy
to make errors in interpretation.
So the solution of such problems is of interest
not only revealing of prominent features of wave move-
ment in the non-heterogeneous media, as that: dis-
persions, attenuations, but also and revealing of the
reasons causing these effects. For long-wave pro-
cesses the geo structure may be considered as
some homogeneous media with the overall elastic
properties [Maslov et al., 2001]. However in such
approach it is impossible to reveal dependence of
speed elastic waves from frequency that is observed
in experiments on frequency of an order of several
thousand hertz. To investigate the phenomena of a
dispersion and attenuation in micro non-uniform
media one uses the so-called dynamic effective
characteristics defined from the equations of move-
ment for representative volume [Maslov, 1982].
Here the problem of the overall dynamic charac-
teristics determination for random geo structure with
initial stress strain state in components is consi-
dered. The basic equations of incremental theory of
elasticity are resulted in [Maslov, 2008]. Basic of
them are the equations of balance of an initial state
, 0R
ma a . (1)
The equations of elastic motion for actual state
2
, 2
A m
ma a
u
t
, A ba
ma mbF s ,
/ba
abs W e . (2)
Boundary and initial conditions
A A
ma a mn P , Sx ,
( ,0) ( ,0)m mu fx x , Sx ,
( , ) ( )m mu tx x , 0t ,
(3)
1 ( )m
m
u
t
x , 0t .
Constitutive law for increments [Crampin, Pea-
cock, 2008]
A ijab
im abL H ,
2
iamb
ik mn im
ak nb ab
W W
L F F
e e e
, (4)
1
2ab ma mb abe F F ,
ma ma maF H , ,ma m aH u .
The frequently used in nonlinear geodynamics
plastic potential is
1 1/
1 2
1
3
2 1
nn
W p p p
n
,
1 mmp e ,
2 mn nmp e e ,
(5)
1/21/22
1 2
2 2
3
3 3
p I I e e .
In this case
."�(,'$�&/$-�*+"'��"'$0
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2 3abcd abcdK æ
3
2abcd abcdJ E ,
2
ˆ ˆ
3abcd ab cdE e e , ˆ /ab abe e p ,
abkn
abmn mkl F , ab abmn mns l H ,
1
2abcd ac bd ad bcI ,
1
3abcd ab cdJ , abcd abcd abcdK I J , (6)
I1, I2 — main invariants of finite deformation tensor
[Maslov, 1982; Crampin, Peacock, 2008].
The state of homogeneous preliminary compres-
sion is considered in detail. Then, having restricted
to a case of small gradients of initial displacements,
the incremental elasticity tensor Labcd is obtained.
Substituting in (4) we gain the equations of motion
which under the form presented here are like the
Lame linear theory of elasticity equations. Fourier
transformation led to equation for fluctuations of dis-
placement, with circular frequency dependency.
Random geo structures in which casual fields are
statistically homogeneous and ergodic concerning
not deformed reference representative volume are
considered. On the boundary of elementary macro
volume the fluctuations of displacements are equal
to zero. So this case provides a perfect analogy of
approach to problems of the linear theory of elasti-
city and incremental theories. As to the solution in
incremental theories we are interested in averaged
gradients of displacements in components while in
the linear theory of micro non-uniform geological
media the problem consists in determination of ave-
raged on components deformations (the symme-
trized gradients). Green's function of the equations
in long-wave approach looks like, similar to a case
of the linear theory. Constants are expressed in
speeds of longitudinal and shear waves.
Substituting the solution of a statistical dyna-
mic problem in the macroscopic equations of mo-
tion, we consider in a detail a case of a plane har-
monic wave. From the dispersive equation it is found
phase and group speeds of waves. Attenuation fac-
tors are obtained as functions of constituent prop-
erties, frequency and initial stress state. The ac-
count of preliminary plastic deformation leads es-
sential variations of speed values. Concrete exam-
ples of calculation of dependence of phase speed
of a shear wave from frequency in a case when com-
ponents are in a state of long term initial plastic
deformation are considered. Natural state of media
as reference example is been modeled too. As re-
sult, influence of an initial plastic state has the same
order, as influence of frequency of a wave spread-
ing.
Azimuthally-aligned shear-wave splitting is widely
observed so the splitting may be used as diagnos-
tic of some form of seismic anisotropy and stres-
saligned fluid-saturated micro cracks as the cause
of azimuthally-aligned shear-wave splitting [Maslov,
1982; Maslov et al., 2001]. Shear-wave splitting is
modeled here by dependence of incremental elas-
ticity (4) from stressed state. And contra versus the
splitting is some kind of diagnostic for some form of
seismic anisotropy. The next idea is possibility to
investigate stress-aligned fluid-saturated micro
cracks [Maslov et al., 2001] as the cause of azi-
muthally-aligned shear-wave splitting.
As we outlined earlier [Maslov et al., 2006] for
the analysis of a long term motion of geo structure
it is useful the continuum damage concept. The
micro cracks distributed in regular intervals or ca-
sually in a material, may be offered as the variable
of a degree degradation of elastic properties. The
measure continuum damage may be considered as
formal reduction of the area of cross-section sec-
tion of the sample. Then it is possible to enter ef-
fective stress and the destruction moment to identi-
fy with achievement damage values. Damage accu-
mulation is stochastic process by the nature there-
fore even at performance of qualitative, well control-
lable field experiments, the big statistical variability
of data is observed. We use further a hypothesis of
equivalence of elastic energy of a material in initial
state and the damaged material. If the free from
stress configuration of elementary volume in a point
has passed in the new form described by a field of
plastic deformations ,T
abe tx then constitutive law
(4) may be rewritten in form
ijab T
ij ab abL e e . (7)
Such a model of incremental elastic behavior of
the geological media does not assume change of elas-
ticity owing to occurrence of a field of deformations of
transformation. Thus, formally some element of a
source is supposed free from initial stress if deforma-
tions are equal in it to the deformations of transforma-
tion mentioned earlier (7). The general models of
cracks or the destructions of ruptures represented as
a surface of rupture of a field of dislocations also in a
limit can be described as field distribution ,T
abe tx in
a narrow zone. If to put a crack of a zone of transfor-
mation aspiring to zero corresponding components
will aspire to infinity so that there is the rupture of
dislocations equivalent to the phenomenon of destruc-
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tion. Thus deformations we present here through Dirac
-function. Therefore the solution of a problem with
some kind of damage degradation also can be pre-
sented through Green dynamic function [Maslov, 1982].
The crack density [Maslov et al., 2006] used for
calculation is approximately equal to one hundredth
Crampin S., Peacock S. A review of the current under-
standing of seismic shear-wave splitting in the
Earth’s crust and common fallacies in interpreta-
tion // Wave Motion. — 2008. — 45��������� ���
��
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Maslov B. P. Overall dynamic characteristics of com-
posite materials with initial stress // Prikl. Mech. —
1982. — 18����������� ��
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Maslov B. P. Thermal-stress concentration near inclu-
sions in visco-elastic random composites // J. En-
of the percentage of shear-wave velocity anisotropy
in aligned cracks in a reference initial medium. Flu-
id-saturated micro cracks model suggested to eva-
luate viscous effects dispersion, scatter and split-
ting of elastic waves in pre stressed and fractured
geological medium.
References
gineering Mathematics. — 2008. — 61. — P. 339—
355.
Maslov B. P., Prodaivoda G. T., Vyzhva S. A.� �� ���
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Maslov B. P., Prodayvoda G. T., Vyzhva S. A. Mathema-
tical modeling of elastic wave velocity anisotropy in
a cracked geological medium // Geophys. J. —
2001. — 20���������P. 191—212 (in Russian).
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