Journal of Applied Mathematics and Physics, 2014, 2, 1039-1046
Published Online November 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.212118
How to cite this paper: Oparin, V.N., Usoltseva, O.M., Tsoi, P.A. and Semenov, V.N. (2014) Evolution of Stress-Strain State
in the Structural Heterogeneities Geomaterials under Uniaxial and Biaxial Loading. Journal of Applied Mathematics and
Physics, 2, 1039-1046. http://dx.doi.org/10.4236/jamp.2014.212118
Evolution of Stress-Strain State in the
Structural Heterogeneities Geomaterials
under Uniaxial and Biaxial Loading
V. N. Oparin, O. M. Usoltseva*, P. A. Tsoi, V. N. Semenov
Institute of Mining, Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russia
Received August 2014
Abstract
The aim of this study was to analyze distribution and development of st ress-stress state in struc-
tured rock specimens subject to uniaxial and biaxial loading to failure using digital speckle corre-
lation method. Within the experimental analysis of wave processes in the block-hierarchy struc-
ture of geomedia (uniaxial and biaxial compression and shearing of prismatic geomaterial speci-
mens), the authors revealed the fact of initiation of low-frequency micro-deformation processes
under slow (quasi-static) disturbances. The estimation of the deformation-wave behavior of geo-
materi als as the “summed” contributions made by elements of the scanned surfaces with differ-
ent-oriented (in-phase and anti-phase) oscillations has been performed using the energy ap-
proach that is based on the scanning function R, analogous to the “center of mass” in the classical
mechanics.
Keywords
Laboratory Exp erime nt, Digital Speckle Correlation Method, Rock Mass, Hierarchical Block
Structure, Microstrains, Deformation-Wave Processes
1. Introduction
Dynamic advance of the theory of pendulum-type waves in stressed geomedia with block-hierarchical structure
[1]-[3] has initiated a new research trend concerned with focal areas/sources of disastrous events (earthquakes,
rock bursts etc.). This trend is called “geomechanical thermodynamics”. In a sense, this is an equivalent of clas-
sical thermodynamics where molecules are replaced by large clusters (from submolecular to macroblocks in
size), taking into account the oscillating motion (translational and rotational) of these clusters approximately as-
sumed as rigid bodies. This statement constitutes the phenomenological basis of the theory of pendulum waves
that are transferred by structural elements of rocks and rock masses at different hierarchical levels [4].
The mechanical model of a self-stressed rock mass was first proposed in [5] [6]. The mechanical conditions of
interaction between structural elements of a constrained rock mass were replaced by nonlinear springs nested
within each other (according to Sadovsky’s concept) [7]. This mechanical model allowed explaining some very
*
Corresponding author.

V. N. Oparin et al.
1040
important experimentally observed features of evolution of harmonic acoustic signals in loaded blocky media
with a cylindrical cavity (simulating stress concentrator or would-be source area), such as [8]:
1) the presence of rigid correlation between the loading stages in a medium with structure and the ampli-
tude-frequency characteristics of harmonic signals recorded in blocks composing geomedium models;
2) the frequencies of resonant acoustic oscillations of blocks of the model, that significantly differ from each
other at early stages of loading and exhibit convergence at the prefailure stage;
3) at the prefailure stage in a blocky geomedium, both the convergence of resonant frequencies of geoblocks
and the amplification of amplitudes of harmonic signals take place due to the transition of elastic potential
energy accumulated by geoblocks into oscillatory (kinetic) energy of acoustic waves radiation; this breaks the
classical law of attenuation of harmonic signals with distance from their source.
In essence, it has for the first time been found experimentally that the modeled would-be destruction source is
able to evolve into acoustically active geomedium that acts as a self-oscillating or a “geomechanical laser” sys-
tem under critical loading. However, it is not a monochromic system as distinguished from classical optical laser
systems, the acoustic radiation spectrum assumes the discrete-canonical form:
( )
( )
00
2, 0,1,2,; 2
i
ip
ff ifV== ±±=∆
where
p
V
is the P-wave velocity in geoblocks and Δ is the diameter of geoblocks.
It is important to mention that [9] presents the first simplest model describing conditions of the canonical
spectrum of acoustic waves assuming that the rigidity of interaction between structural blocks (“springs”) is the
nonlinear function of their relative displacements.
2. Macro- and Microstrains in Artificial Geomaterial Specimens under
Biaxial Loading to Failure
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Please do not revise any of the current designations. The authors have previously performed tests on rock spe-
cimens with structural inhomogeneity [10]. These tests were aimed at studying peculiarities of distribution and
evolution of microlevel stress-strain state on work surfaces of the specimens under uniaxial loading to failure
using speckle photography method.
It was found out that:
1) the uniaxial loading of prismatic specimens made of sandstone, marble and sylvinite by slow (quasistatic)
force ((Instron 8802 testing system, stiff loading, movable grip velocity of 0.02 - 0.2 mm/min) generates
low-frequency inner microstrains at a certain stress level;
2) the amplitude of these deformation–wave processes depends significantly on the level of macroloading; at
the stage of elastic deformation, when stresses are under 0.5 of ultimate strength of the geomaterial, the oscilla-
tions of microdeformation components are almost absent; at the stage of nonlinear elastic deformation, under
stresses in the range from 0.5 to 1 of specimen’s ultimate strength, the amplitudes of microstrain oscillations
significantly increase, including the descending stage; at the stage of residual strength, the oscillation amplitudes
the microdeformation components decrease drastically (3 - 5 times) as compared to the previous stages;
3) in the elements of scanned surface of a rock specimen, which cover the area of a future crack, amplitudes
of the microstrain velocities έу several times exceed the microstrain velocity amplitudes on the surface areas of
an undisturbed geomaterial. Sometimes the deformation velocities increase under load increment.
The objective of this work is to define the influence of the type of loading on the behavior of deformation
processes in artificial geomaterials right down to peak loading.
The artificial specimens were made of alabaster and water: 60% of alabaster, 35% of water, 5% of Neolit glue;
the inclusions were presented by balls 3 - 4 mm in diameter made of silicium dioxide (silica gel) annealed with
additives by stages. For the inclusions, the ultimate compression strength was 10.6 МPа; for the binder, the ul-
timate strength 5.4 MPa under axial compression and 10.5 MPa biaxial compression (under biaxial compression,
2 lim
const 0.5 σσ = =
, where
lim
σ
is the ultimate strength of material). The prismatic specimens were 60 × 60 ×
11 mm in size (Figure 1).

V. N. Oparin et al.
1041
(a) (b)
Figure 1. The artificial specimen (a) ; the loading diagram (b).
The tests of the geomaterial specimens were conducted on servohydraulic press Instron-8802, which allowed
loading at the assigned rate of force and displacement. Biaxial loading was implemented by a special device
creating additional (press independent) lateral loading of the prismatic specimen. The force of the additional lat-
eral compression of the specimen, displacement of the press cross-head and the press force were continuously
recorded during the experiments. The maximum press force was set at a fixed cross-head rate of 1 mm/min. Mi-
crostrains on the work surface of the prismatic specimens were registered with the help of the automated digital
speckle-photography analysis system ALMEC-tv [11].
A series of tests on uniaxial and biaxial stiff compression (to failure) of prismatic specimens made of artificial
geomaterials under uniaxial and biaxial stiff compression (to failure) was conducted. Figure 2 shows the
“macrodeformation” results.
The comparative analysis of microdeformation processes in the specimen was performed for the scanned
areas of various size: 1—areas of failure, 2—areas of undisturbed material. The sizes of the tested areas were
varied in the range from 0.1 to 0.5 of the size of the scanned surface (Figure 3). Below the representative expe-
rimental data on the biaxial compression are analyzed.
On the work surface of the specimen, rectangular areas of the same size were chosen (Figure 3), and total
microstrains were calculated for these areas in longitudinal
( )
x
, transverse
( )
y
and shear
()
xy
directions—
х
ε
,
у
ε
and
ху
ε
, respectively. Figures 4(а)-(c) shows the plots of total microstrains (red color—the failure
area (1), blue color—the undisturbed area (2) of the specimen surface; black color shows the relation of stress
and time under loading up to ultimate strength of the specimen in the dimensionless coordinates
lim
Р Р
and
lim
tt
, where
lim
1tt =
corresponds to the peak loading
( )
lim
1Р Р=
.
The plots exhibit the following behavioral features of the microdeformation components:
• the microstrains
х
ε
in areas 1 and 2 have almost the same values right down to ultimate strength;
• the microstrains
у
ε
in areas 1 and 2 show almost the antiphase change, especially at
lim
0.5tt =
and on-
ward, and differ significantly;
• the plots of
ху
ε
in areas 1 and 2, starting from
lim
0.4tt =
, are significantly cophased. However, the dif-
ference in their absolute values increases up to the ultimate strength: at
lim
1Р Р=
0.1
ху
ε = −
in area 2 and
0.5
ху
ε =
in area 1;
• at
lim
0.5 - 0.6tt ∈
in all deformation components there appear high-frequency oscillations. The amplitude
of oscillations is significantly higher in area 2 as compared to area 1.
The evaluation of deformation-wave behavior of the artificial geomaterials as the “sum” of contributions of
the scanned surface elements with counter-directional (cophase and antiphase) oscillations used the energy ap-
proach as in [10]. The energy approach to evaluation of deformation-wave processes is based on the scanning
function R that is equivalent to the notion of the “center of mass” in the classical mechanics, and describes the
way of determining the reduced center of seismic energy release for a given period of time within the limits of a
given volume of a rock mass.
For these experiments, the coordinates of R and the mechanical trajectory of the reduced center of deforma-
tion energy release are respectively given by:
11
NN
ii i
ii
Rr εε
= =
=
∑∑
(1)
() ()
22
00ii i
r xxyy=− +−
(2)

V. N. Oparin et al.
1042
Figure 2. The uniaxial (1) and biaxial (2) compression; the time
of the crack initiation is t = 130 s under the uniaxial com-
pression and t = 140 s under the biaxial compression.
Figure 3. The scanned specimen surface in the biaxial com-
pression test after failure (time t = 140 s); 1—areas of failure,
2—areas of undisturbed material.
Figure 4. The microstrains
х
ε
,
у
ε
and
ху
ε
(a, b, c) and scanning functions
( )
x
Rt
,
( )
y
Rt
and
( )
xy
Rt
(d,
e, f) for the microstrains
х
ε
,
у
ε
and
ху
ε
versus time in the dimensionless coordinates
lim
tt
,
lim
Р Р
in
the test on biaxial loading of artificial geomaterial specimen.

V. N. Oparin et al.
1043
where
i
ε
is the strain at the point
i
r
( )
,
ii
xy
,
0
x
,
0
y
is the coordinate origin (the geometric center of the
specimen work surface), N is the number of measuring points on the work surface of the specimen; summing is
conducted at the time
i
t
(image
i
) and the known Р.
Figures 4(d)-(f) shows the plots of the scanning functions
( )
x
Rt
and
( )
y
Rt
,
( )
xy
Rt
for (1), separately
for each microdeformation component
( )
,,
хуху
εεε
, under loading of the specimen to ultimate strength, in the
dimensionless coordinates
lim
Р Р
and
lim
tt
, where tlim corresponds to the peak loading
( )
lim
1Р Р=
. The
calculations were performed using the dimensionless coordinates
maxi
rr
, where
max
r
is the maximum size of
i
r
in the longitudinal direction of the specimen.
According to the plots,
( )
x
Rt
,
( )
y
Rt
and
( )
хy
Rt
have the following features:
• the function
( )
x
Rt
in areas 1 and 2 up to
lim
0.5tt =
remains almost unaltered. However, at
lim 0.65tt =
-
0.7 it undergoes significant alternating change. The oscillatory process for
( )
x
Rt
keeps up to the ultimate
strength, and the amplitude of oscillation of
( )
x
Rt
in area 1 essentially exceeds its oscillation amplitude in
area 2;
• the function
( )
y
Rt
in area 1 changes drastically at
lim 0.19tt =
- 0.4, and then takes on a constant value;
in area 2 the oscillatory changes of
( )
y
Rt
appear at
lim
0.6tt =
- 0.87, with further decrease;
• the function
( )
xy
Rt
in area 1 keeps almost constant values, except for the interval
lim
0.6tt =
- 0.73; in
area 2 it oscillates with the growing amplitude up to the ultimate strength.
Based on the analysis of the behavior of the functions
х
ε
,
у
ε
,
ху
ε
,
( )
x
Rt
,
( )
y
Rt
and
( )
xy
Rt
, the fol-
lowing pre-failure features are definable in the specimens:
1) the components of same deformation (the microstrains
у
ε
,
ху
ε
) in areas 1 and 2 differ significantly;
2) the amplitude of oscillation of the microdeformation components in the area of failure significantly exceeds
their oscillation amplitude in the undisturbed area, which implies the higher deformation velocity and the change
of the strain state type (it is also observed for the microstrains
у
ε
,
ху
ε
);
3) the scanning functions have oscillatory character, which gives evidence оf inhomogeneity of the inner
strain state of geomaterial. However, coming to the peak loading
( )
lim
1tt=
before the crack formation
( )
lim
1.5tt =
, the scanning functions
( )
y
Rt
,
( )
xy
Rt
take on the constant values. Such behavior of the scan-
ning functions
( )
y
Rt
,
( )
xy
Rt
in combination with the increase in values of the microstrains
у
ε
,
ху
ε
and
their oscillation amplitudes implies the localization of deformation peak values and the narrowing of the area
where the microstrains are concentrated.
On the basis of the abovesaid, we conclude that the microstrains
у
ε
,
ху
ε
are responsible for failure. The
area of macrofailure formation can clearly be seen. Figure 5 shows the picture of the specimen surface de-
struc t ed under post-limit loading at
lim
1.3tt =
. A cleavage crack has stepped surface and forms angles of 20 -
40˚ with the direction of the axial force, i.е. it creates combination of microcracks in the directions of
у
ε
and
ху
ε
.
3. Test of the Artificial Geomaterial Specimen under Uniaxial Loading to Failur e
The uniaxial compression test of the specimen made of artificial geomaterial was carried out to be compared to
the case described above. On the basis of the described criteria, the authors will make an attempt to evaluate the
type of future failure by the time behavior of the microstrains
х
ε
,
у
ε
and
ху
ε
as well as the behavior of the
scanning functions
( )
x
Rt
,
( )
y
Rt
and
( )
xy
Rt
.
For this type of loading, the behavior of
х
ε
is almost the same in areas 1 and 2 of the scanning surface
(Figure 6(а)) within the limits of the test accuracy. The scanning functions
( )
x
Rt
(Figure 6(а)) also behave
the same way in areas 1 and 2 and coincide at
lim
0.8tt =
; further there appear high-frequency oscillations of
deformations right down to ultimate strength at
lim
1tt =
.
The microstrains
у
ε
behave drastically different in areas 1 and 2 (Figure 6(b)). Their change is practically
antiphase. At
lim
~1tt
their values are more than 10 times different (0.002 and 0.02). The amplitude of
high -frequency oscillations of
у
ε
is much higher than the oscillation amplitudes of
х
ε
and
ху
ε
. The function
( )
y
Rt
in area 2 (Figure 6(e)) oscillates at high frequency right down to ultimate strength, whereas the function
( )
y
Rt
in area 1 takes on a constant value at
lim
0.8tt =
- 1.
The behavior of the scanning functions for
ху
ε
(Figur e 6(c)) significantly differs in areas 1 and 2 (anti-
phase). However, their absolute value decreases under loading up to ultimate strength. The scanning functions
( )
xy
Rt
in areas 1 and 2 (Figure 6( f)) also have oscillatory character but they differ drastically in structure (am-

V. N. Oparin et al.
1044
Figure 5. The picture of the specimen surface after failure
under biaxial compression at the time
lim
1.3tt =
.
Figure 6. The microstrains
х
ε
,
у
ε
and
ху
ε
(a, b, c) and scanning functions
( )
x
Rt
,
( )
y
Rt
and
( )
xy
Rt
(d,
e, f) for the microstrains
х
ε
,
у
ε
and
ху
ε
versus time in the dimensionless coordinates
lim
tt
,
lim
Р Р
in
the test on uniaxial loading of artificial geomaterial specimen.
plitude–frequency spectrum) at
lim
0.4tt >
.
The features found in the behavior of microcharacteristics of deformation indicate the direction
y
is the
most hazardous in terms of failure as higher values of the microstrains
у
ε
and their change velocities are ob-
served in this direction. The steady state of
( )
y
Rt
denotes the narrowing of zone of their localization. Accor-
dingly, macrofailure is expected by the component
у
ε
.
The features found in the behavior of microcharacteristics of deformation indicate the direction
y
is the
most hazardous in terms of failure as higher values of the microstrains
у
ε
and their change velocities are ob-
served in this direction. The steady state of
( )
y
Rt
denotes the narrowing of zone of their localization. Accor-
dingly, macrofailure is expected by the component
у
ε
.
Indeed, Figure 7 shows that the crack formed mainly in the
y
direction of
у
ε
at
lim
1.4tt =
.
The tests of artificial geomaterials under biaxial and uniaxial loading allowed determination of the behavior of
the microstrains
х
ε
,
у
ε
and
ху
ε
and their scanning functions
( )
x
Rt
,
( )
y
Rt
and
( )
xy
Rt
. These behavioral
features of microstrains can be used as the testing parameter for finding the time, location and type of an inci-
pient crack.
It should be mentioned that [9] [12] present the general patterns of deformation localization at the stage of

V. N. Oparin et al.
1045
Figure 7. The picture of the surface of the artificial geomaterial
specimen after its failure under uniaxial compression, the test
time
lim
1.3tt =
.
pre-failure in sylvinite, marble and sandstone specimens obtained using the speckle-photography method. The
autowave behavior of inelastic deformation under compression of rocks has been emphasized. Auto wave veloc-
ity is of the order of 10-5 - 10-4 m/s.
4. Conclusions
The experimental studies on the specimens made of artificial geomaterials with structural inhomogeneity in the
form of inclusions have allowed the detailed testing of the peculiarities of distribution and evolution of their mi-
crolevel stress–strain state on work surfaces of the rock specimens under uniaxial and biaxial loading to failure
using speckle photography method. Within the limits of the experimental study of wave processes in the geome-
dia with structural hierarchy of blocks, the earlier found phenomenon of low-frequency microdeformation gen-
erated by slow (quasistatic) force has been confirmed.
The energy approach has been used to evaluate the deformation-wave behavior of geomaterials as the “sum”
of contributions of scanned surface elements with counter-directional (cophased and antiphase) oscillations. The
energy approach to evaluation of deformation-wave processes is based on the scanning function R that is the
equivalent of the notion of the “center of mass” in the classical mechanics, and describes the way of determining
the reduced center of seismic energy release for a set period of time within the limits of a given volume of rock
mass .
The authors have defined the features of the behavior of the scanning functions for the microstrains
х
ε
,
у
ε
and
ху
ε
(in longitudinal, transverse and shear directions) under loading to failure. These behavioral features
can be used as the testing parameter for locating a nucleating main fracture. The research findings are of practic-
al importance as applied to monitoring of geomechanical-geodynamic safety in mines on the basis of integrated
usage of seismic-deformation data.
Acknowledgem ents
The work was supported by partly the Russian Academy of Sciences, project ONZ-RAN 3.1, grant no.
12-05-01057. The equipment is kindly provided by the Shared Geomechanical, Geophysical and Geodynamic
Measurement Center of the Siberian Branch of the Russian Academy of Sciences.
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