Quasistatic loading of Berea sandstone
Запропонована феноменологiчна модель для опису властивостей напруження-деформацiя пiсковика пiд дiєю повiльного навантаження. Розглянута комбiнацiя трьох механiзмiв, якi пов’язуються з внутрiшнiми обмiнними процесами: механiзм стандартного релаксуючого твердого тiла, пружний механiзм з прилипанням,...
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Видавничий дім "Академперіодика" НАН України
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| Цитувати: | Quasistatic loading of Berea sandstone / V.O. Vakhnenko, O.O. Vakhnenko, J.A. TenCate, T. J. Shankland // Доп. НАН України. — 2008. — № 1. — С. 118-126. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859607630752776192 |
|---|---|
| author | Vakhnenko, V.O. Vakhnenko, O.O. TenCate, J.A. Shankland, T.J. |
| author_facet | Vakhnenko, V.O. Vakhnenko, O.O. TenCate, J.A. Shankland, T.J. |
| citation_txt | Quasistatic loading of Berea sandstone / V.O. Vakhnenko, O.O. Vakhnenko, J.A. TenCate, T. J. Shankland // Доп. НАН України. — 2008. — № 1. — С. 118-126. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| description | Запропонована феноменологiчна модель для опису властивостей напруження-деформацiя пiсковика пiд дiєю повiльного навантаження. Розглянута комбiнацiя трьох механiзмiв, якi пов’язуються з внутрiшнiми обмiнними процесами: механiзм стандартного релаксуючого твердого тiла, пружний механiзм з прилипанням, механiзм залишкової
пластичної деформацiї. З малою кiлькiстю параметрiв модель вiдтворює як якiсно, так
i кiлькiсно головнi експериментальнi данi по напруженню-деформацiї для пiсковика Береа. Модель правильно вiдтворює великi та малi петлi на траєкторiї напруження-деформацiя (пам’ять про кiнцеву точку). Власне запропонована залежнiсть деформацiї
вiд напруження є не чим iншим, як рiвнянням стану пiсковика.
|
| first_indexed | 2025-11-28T06:29:35Z |
| format | Article |
| fulltext |
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western East European Craton // Geoph. Res. Abstracts. – 2006. – 8, No 07385.
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Поступило в редакцию 16.03.2007Институт геохимии, минералогии
и рудообразования НАН Украины, Киев
Институт геологии рудных месторождений,
петрографии, минералогии и геохимии РАН, Москва
UDC 539.3
© 2008
V.O. Vakhnenko, O. O. Vakhnenko, J.A. TenCate, T. J. Shankland
Quasistatic loading of Berea sandstone
(Presented by Corresponding Member of the NAS of Ukraine V.A. Danylenko)
Запропонована феноменологiчна модель для опису властивостей напруження-деформа-
цiя пiсковика пiд дiєю повiльного навантаження. Розглянута комбiнацiя трьох механi-
змiв, якi пов’язуються з внутрiшнiми обмiнними процесами: механiзм стандартного
релаксуючого твердого тiла, пружний механiзм з прилипанням, механiзм залишкової
пластичної деформацiї. З малою кiлькiстю параметрiв модель вiдтворює як якiсно, так
i кiлькiсно головнi експериментальнi данi по напруженню-деформацiї для пiсковика Бе-
реа. Модель правильно вiдтворює великi та малi петлi на траєкторiї напруження-де-
формацiя (пам’ять про кiнцеву точку). Власне запропонована залежнiсть деформацiї
вiд напруження є не чим iншим, як рiвнянням стану пiсковика.
The measurements of typical stress-strain dependences for rocks under quasistatic loading point
out their essentially nonlinear behavior. The results by Boitnott [1], Hilbert et al. [2] and Dar-
ling et al. [3] on repeatable hysteretic loops in stress-strain curves are well known. In essence,
modeling the stress-strain dependences reproduces the sandstone state equation. Dealing only
with macroparameters such as stress and strain while the processes on a microlevel still remain
unknown makes it very difficult to create a model that adequately describes these properties.
Recent experiments [3] showed that the most remarkable stress-strain properties of rocks are
118 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №1
Fig. 1. Experimental results for Berea sandstone: (a) Boitnott [1], (b) Hilbert et al. [2], (c) and Darling et
al. [3]. Stress-strain trajectories with their original coordinate meshes are placed within common coordinates. The
systematic strain shifts caused by the apparatus adjustments are zero for Boitnott data, 0.00042 for Hilbert et al.
data, and 0.00097 for Darling et al. data
determined by a small volume of material at grain contacts. However, it is unclear how in-
terior equilibration processes in rocks under quasistatic loading can be studied in detail. In
the literature there are a number of models that qualitatively describe the relationships between
macroparameters such as stress and strain. First of all, there are two models, the Hertz–Mindlin
model [4] and the Preisach–Mayergoyz model [5]. However, with these approaches, there is some
difficulty in assigning a set of model hysteretic elements to real physical processes. While these
approaches can duplicate experimental observations, their incompletely formulated connection
between the distribution of auxiliary elements and maximum stress levels leads to limited pre-
dictive power.
We suggest three appropriately formalized mechanisms [6] that appear to actually occur in
rocks under quasistatic loading: (i) a “standard solid relaxation” mechanism [7], (ii) a “sticky-
spring” mechanism, and (iii) a “permanent plastic deformation” mechanism. A suitable combi-
nation of these mechanisms enables us to derive some general stress-strain relations, although
without a detailed description of interior equilibration processes. As a result, we can obtain
a phenomenological model that allows us to simulate qualitative and quantitative stress-strain
characteristics and to reproduce the distinctive features typical of the basic experimental obser-
vations by Boitnott [1], Hilbert et al. [2], and Darling et al. [3] for Berea sandstone.
1. Observation of experimental data. In order to facilitate the analysis of three groups
of fundamental experimental data for Berea sandstone by Boitnott (Fig. 1 in [5]), Hilbert et al.
(Fig. 2 in [5]), and Darling et al. (Fig. 1 in [3]), we place them in a common presentation. Because,
in different experiments, the origins of strain coordinates were introduced in different ways, it is
an advantage to combine all experimental stress-strain curves into a single format. We proceed
from the assumption that, for all three experimental curves, the points relating to maximum
stress should be placed somewhere on the longest of available unconditioned curves, i. e., on the
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2008, №1 119
Fig. 2. Features of the stress-strain relations for Meule sandstone [3] (qualitative picture)
bottom curve of Fig. 1c, while the origin for the common strain coordinate should be chosen
from Fig. 1a, where the starting point of the unconditioned (bottom) curve is documented. In
this terminology, the word “unconditioned” refers to an initial curve that starts at zero stress on
a sample that has been undisturbed for a long period (of the order of many hours or a day) as is
the case in Fig. 1a. In contrast, the curves of Fig. 1b are “conditioned”, that is, have undergone
multiple stress cycles. In this case, the starting point is not shown and, except for the final,
highest point, the unconditioned curve is absent. The original experimental figures have different
scales, and, in Fig. 1, we have placed the experimental curves (omitting the scale numbers for
clarity) into common coordinates. In this procedure, Fig. 1a preserves its coordinates, while
Fig. 1b and Fig. 1c are shifted to include zero strain positions; the strain shift is zero for Fig. 1a,
0.00042 for Fig. 1b, and 0.00097 for Fig. 1c.
It is pertinent to note that this approach for introducing common coordinates is not ideal
inasmuch as it essentially treats the actual position of the initial (unconditioned) curve as
independent of the rate of increase of the applied stress, which in general is not the case. However,
experimentally such a rate dependence is mainly detectable at high stresses and can be neglected
in the first (but rather good) approximation without practical consequences within the common
frame of reference.
The first effect that arises from a detailed analysis of experimental data as in Fig. 1 is
the manifestation of some internal relaxation process that appears as a characteristic loop-like
retardation in the strain response upon external loading and unloading stresses. Recently we
discussed the relaxation mechanism for sandstones in terms of a phenomenological standard
solid relaxation mechanism that arises from a nonlinear generalization [8] of the well-established
relaxation modeling in the framework of a standard linear solid [9] (also see [7] and references
therein).
Second, stress cycling gives rise to hysteresis loops in stress-strain curves. Observing the
experimental dependences from [3] (particularly Fig. 1 in [3]) reveals that opposite sides of
each loop are not entirely stuck together even at an infinitely slow loading, that is, the loops
persist independently of loading time. Hence, there must be specific irreversible interior changes
responsible for the loop formation, i. e., ones that can not be attributed simply to relaxation.
We presume that some sort of friction has to be involved in any mechanism responsible for this
effect. Therefore, we are forced to take a second mechanism, referred to here as a sticky-spring
mechanism, into account for describing this aspect of sandstone stress-strain properties.
120 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №1
Finally, the whole treatment would be incomplete without including the third mechanism
termed permanent plastic deformation. This third mechanism is needed to explain the observati-
on that unconditioned and conditioned experimental curves differ from each other due to a
permanent deformation, that is, a strain offset.
The next section provides a comprehensive treatment of these three mechanisms in order to
model the interior processes that arise in rock samples under quasistatic compression.
2. Model approaches. This paper treats the uniaxial compression of a rock sample restri-
cted to quasistatic loading. In this (slow loading) approximation, the stress σ turns out to be
uniform along a sample and is determined by the absolute value of the external loading whi-
ch plays the role of an external governing parameter. For the latter reason, we assign both σ
and strain ε to be positive quantities as they are usually regarded in quasistatic compression
experiments.
The fact that it is sufficient to operate directly with the stress-strain relation for the interpre-
tation of quasistatic experiments provides a good basis for understanding the main mechanisms
of inelasticity and elasticity, especially nonlinear ones, as well as allows one to formalize and ver-
ify these mechanisms. As mentioned above, we consider separately three mechanisms to account
for interior processes in a rock sample under quasistatic loading: (i) standard solid relaxation,
(ii) sticky-spring, and (iii) permanent plastic deformation.
2.1. Standard solid relaxation mechanism. The first part εr of the total strain ε to
be considered is a relaxation mechanism caused by an interior equilibration process. We use the
superscript index r to distinguish εr from other contributions to ε. According to the experimental
curves in Section 1, εr may depend on time not only implicitly through the governing stress σ
but also explicitly through relaxation. Thus, in general, strain can have different values at the
same stress. However, the main hypothesis, which will be confirmed a posteriori, consists in
assuming that strain also responds to the stress variation in time, or more precisely, to its time
derivative σ̇. The relaxation mechanism has been described in detail in [7], from which we use
relations (2)–(5). The model incorporated in the dynamical state equation (2) in [7] is termed
the standard solid relaxation mechanism in view of its generic property of interconnection of two
different nonlinear elastic state equations that are mediated through hidden interior relaxation
processes, just as two linear state equations are interconnected in the theory of a standard
linear solid.
We observe, as is noted in [7], that modeling the Boitnott’s experiments leads to almost ideal
results. In contrast, the Hilbert’s experiment cannot be adequately described solely by relaxation,
inasmuch as it does not close the small loops through the stress-strain cusps for any assignment
of constants in the state equations. The relaxation mechanism by itself does not explain end-
point memory [6, 7].
2.2. Sticky-spring mechanism. In order to develop a means to explain the end-point
memory of the stress-strain curves mentioned above, it helps to examine the stress-strain curves
for Meule sandstone in [3]. For this purpose, we select only important parts of these data and
depict them qualitatively in Fig. 2. Points corresponding to each other in the stress protocol
picture (Fig. 2b) and the stress-strain picture (Fig. 2a) are marked by the same capital letters
and are unprimed and primed, respectively. In the time intervals AB, CD, and EF , stress is
constant. The fact that points A′, C ′, and E′ do not coincide with the respective points B′, D′,
and F ′ could be explained by relaxation alone. However, relaxation by itself should inevitably
lead the experimentally distinct points D′ and F ′ (and even B′) to coincide. To resolve this
problem and understand the discrepancy, it is necessary to include an additional mechanism,
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2008, №1 121
Fig. 3. Analog model to illustrate the sticky-spring mechanism in the upper left corner and its stress protocol in
the lower right corner. Shown is the cork displacement ε
s in response to external loading σ. The curve of medial
equilibrium ε = ε
s
m(σ) is marked by a dash-dotted line
the sticky-spring mechanism, and, for the sake of convenience, to formulate it separately from
the other mechanisms.
A prototype system to illustrate this mechanism is given in the left upper corner of Fig. 3.
The system consists of a cylinder containing a cork plug and an elastic nonlinear element. In the
sticky-spring mechanism, we invoke the friction between a plug and tube walls along with the
elastic restoring force supplied by the elastic nonlinear element. The maximum frictional force
is taken to be proportional to the threshold stresses σ∓ (positive values) that must be overcome
by external stress σ against an internal elastic force (or vice versa) in order for the cork plug to
be pushed from one position into another. In terms of the stress-strain variables, we postulate
the principal features of the sticky-spring mechanism as follows:
dεs
dt
= θ(σ̇)θ(ε−(σ) − εs)
dε−
dσ
σ̇ + θ(−σ̇)θ(εs
− ε+(σ))
dε+
dσ
σ̇. (1)
Here θ(z) is the Heaviside step function. The partial strain εs is associated with the sticky-spring
contribution to the total strain ε, while the functions ε−(σ) and ε+(σ) are determined via the
medial equilibrium state function εs
m
(σ) and two positive threshold stresses σ+ and σ− as follows:
ε−(σ) ≡ εs
m
(σ − σ−), ε+(σ) ≡ εs
m
(σ + σ+). (2)
We note that no restrictions are imposed on the threshold values σ− and σ+ that are responsible
for the friction. In principle, they can be functions of stress σ.
Some aspects of the sticky-spring mechanism are presented in Fig. 3, which illustrates the
dependence of sample strain εs on stress σ. For this purpose, Eq. (1) has been numerically
integrated from the initial condition εs(t = 0) = 0 using the pressure protocol given in the
right lower corner of Fig. 3. The essential feature of the sticky-spring mechanism consists in
producing a stripe-like horizontal continuum of stationary states between the curves εs = ε−(σ)
and εs = ε+(σ) as in Fig. 3. Only in the limit of very small threshold stresses (σ− → +0,
σ+ → +0) do these curves come together and give rise to a single curve εs = εm(p). This medial
curve turns out to bisect the stripe and can be thought as the equilibrium curve of the process
122 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №1
in the limiting case of σ− → +0, σ+ → +0. Another essential part of this mechanism is its
elastic component manifested through the inclination of each envelope curve ε−(σ) and ε+(σ)
with respect to the stress axis.
For the function εs
m
(σ), we assume εs
m
(σ) = εr
e
(σ), where εr
e
(σ)) is determined by Eqs. (2)–(4)
from [7]. In accord with thermodynamic principles, this relation thus requires that the final
position of a true equilibrium state be independent of the origin of the internal processes that
led to this equilibrium.
In Section 3 we show that, in the proper combination, the standard solid relaxation mechani-
sm and the sticky-spring mechanism enable us to model both relaxation steps on conditioned
curves under a fixed load (Fig. 2) and end-point memory, respectively. However, to include the
unconditioned portion of the curves, we must invoke a mechanism that takes plastic deformation
into account.
2.3. Permanent plastic deformation mechanism. Taking the intuitively understandable
features of permanent plastic deformation into account, we postulate that, under compression,
i. e., during increasing initial loading σ̇ > 0, the sample must contract, on the one hand, with
a permanent plastic contribution εp to total strain to obey a linear Hooke-like law εp = σ/Ep
(where the appropriate Young modulus Ep is presumed to be stress-independent). On the other
hand, it must simultaneously experience interior irreversible deformations. Conversely, when
external loading decreases, σ̇ < 0, the plastic component εp must remain fixed. To formalize
the statements above, the state equation for the permanent plastic deformation mechanism can
be described as
dεp
dt
= θ(σ̇)θ
(
σ
Ep
− εp
)
σ̇
Ep
. (3)
According to this mechanism, once a peak loading is achieved, the possible store of plas-
tic deformation in the rock sample becomes saturated. Thereafter, when loading less than the
peak stress, the permanent plastic deformation mechanism does not appear in subsequent cy-
cles. Through the Heaviside function, only an unconditioned curve manifests permanent plastic
deformation; on conditioned curves, it does not contribute.
The separation of the sticky-spring mechanism and the permanent plastic deformation mech-
anism can have a physical interpretation. In the experimental results [1, 2, 3] each incre-
ment of stress (starting at zero stress) beyond the previous highest stress produces irreversible
changes in the rock fabric as crack surfaces slide and asperities are crushed. The permanent
plastic mechanism is a means of incorporating these irreversible changes. In a regime where
stress cycles at stresses less than the maximum previously achieved, the sticky-spring mech-
anism is applied. It may be that the Preisach–Mayergoyz approach [5] can cover the whole
stress range, yet there is the utility in the present approach, where damaging stresses are
separated from a regime in which stress cycles are associated with reversible changes in the
rock.
Some additional details of the suggested mechanisms can be found in [6], in particular, the
connection with the Preisach–Mayergoyz model [5].
3. Simulation of stress-strain relations. In previous sections, we suggested three mechani-
sms by which interior interaction processes in sandstones are assumed to take place. Because the
physical origins of these processes have not been well established, we use a phenomenological
approach in which they are not specifically defined. Taking all three developed mechanisms into
account, we rely upon the minimum number of processes, i. e., only a single process for each
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2008, №1 123
mechanism. For loading by a given stress protocol, we can solve Eqs. (1), (3) and (2)–(5) from
[7] with the initial conditions εr(t = 0) = εs(t = 0) = εp(t = 0) = 0 and find the total strain ε
as a linear combination of partial strains:
ε = b(εr + εp) + (1 − b)(εs + εp). (4)
Here, the constant b is bounded inside the interval 0 6 b 6 1. Thus, at b = 1 we have the
relaxation mechanism with permanent plastic deformation only, while, at b = 0, we retain only
the sticky-spring mechanism plus a permanent plastic deformation. Choosing relation (4) as a
linear combination, we are able to tune the single parameter b to obtain a physical conditi-
on such that the true equilibrium state is independent of any one type of interior relaxation
process.
If the stress is fixed at one moment, then the relaxation mechanism moves the whole system
to some new equilibrium during a characteristic relaxation time. Owing to the sticky-spring
mechanism plus the permanent plastic deformation mechanism, there can be several equilibrium
states at the same stress. The ambiguity of the equilibrium state dependence on stress can be
found in [10].
The best fit of the calculated results as applied to all three groups of experiments on Berea
sandstone [1, 2, 3] (see also Fig. 1) was obtained with the parameters listed in Table 1. The
parameters τ , E−
e
, E+
e
, D, and a are assigned for Eqs. (2)–(5) in [7].
Figure 4 presents the results of numerical simulations. Comparing the calculated curves
(Fig. 4) with experimental data (Fig. 1), we observe an acceptable coincidence of these results
both qualitatively and quantitatively. First, we find the small loops in curve 2 of Fig. 4 that
model the experiment of Hilbert et al. [2]. These loops are closed at the cusps. Although the
time relaxation is small, the relaxation mechanism cannot be completely removed because it
plays an important role in describing the end-point memory; this effect is manifested by the
small loops on the theoretical curve 2 in Fig. 4, which reproduce the experiment of Hilbert et
al. [2]. On the one hand, it is precisely the effect of small but finite relaxation time that enables
one to close a small loop through a cusp (see curve 2 in Fig. 4 once again). On the other hand,
the relaxation provides the means to produce the small loops in the modeling.
4. Conclusion. A phenomenological model to describe the stress-strain properties of Berea
sandstone under quasistatic loading is suggested. Analysis of experimental observations has
demonstrated the need to invoke several mechanisms that are responsible for interior equilibrati-
on processes in sandstone: the standard solid relaxation mechanism, the sticky-spring mechanism,
and the permanent plastic deformation mechanism [6]. To justify these mechanisms, we have used
an approach in which the interior processes in a sample are not explicitly defined, which simpli-
fies drastically the mathematical description. Only by properly combining all three mechanisms
are we able to obtain acceptable models. The resulting treatment reproduces extremely complex
stress-strain trajectories with only nine adjustable parameters.
Because of the proposed treatment of quasi-static stress-strain relations, it becomes possi-
ble to realize an adequate and self-consistent simulation that describes both qualitatively and
quantitatively the principal features of complex experimental data by Boitnott [1], Hilbert et
Table 1. Fitting parameters
τ , s E
−
e , GPa E
+
e , GPa D, MPa−1
σ
−
, MPa σ+, MPa Ep, GPa a b
3 5 23 0.03 4 4 70 0.2 0.8
124 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №1
Fig. 4. Computer modeling of stress-strain trajectories for Berea sandstone. Curve 1 relates to the Boitnott
experiment [1], curve 2 models the Hilbert et al. [2] experiment, and curve 3 reproduces the Darling et al.
experiment [3]
al. [2], and Darling et al. [3] for Berea sandstone and, in particular, the details of end-point
memory.
JAT and TJS thank the Geosciences Research Program, Office of Basic Energy Sciences of the US
Department of Energy for sustained assistance. VOV is grateful to Prof. V.A. Danylenko for the sti-
mulating criticism and helpful discussions.
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contacts to nonlinear stress-strain behavior // Geophys. Res. Lett. – 2004. – 31. – L16604.
4. Nihei K.T., Hilbert jr. L. B., Cook N.G.W. et al. Frictional effects on the volumetric strain of sandstone //
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Received 12.09.2007Subbotin Institute of Geophysics,
Ukrainian Academy of Sciences, Kiev, Ukraine
Bogolyubov Institute for Theoretical Physics,
Ukrainian Academy of Sciences, Kiev, Ukraine
Los Alamos National Laboratory, Los Alamos,
New Mexico, USA
126 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №1
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| id | nasplib_isofts_kiev_ua-123456789-4153 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-11-28T06:29:35Z |
| publishDate | 2008 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Vakhnenko, V.O. Vakhnenko, O.O. TenCate, J.A. Shankland, T.J. 2009-07-16T09:26:08Z 2009-07-16T09:26:08Z 2008 Quasistatic loading of Berea sandstone / V.O. Vakhnenko, O.O. Vakhnenko, J.A. TenCate, T. J. Shankland // Доп. НАН України. — 2008. — № 1. — С. 118-126. — Бібліогр.: 10 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/4153 539.3 Запропонована феноменологiчна модель для опису властивостей напруження-деформацiя пiсковика пiд дiєю повiльного навантаження. Розглянута комбiнацiя трьох механiзмiв, якi пов’язуються з внутрiшнiми обмiнними процесами: механiзм стандартного релаксуючого твердого тiла, пружний механiзм з прилипанням, механiзм залишкової пластичної деформацiї. З малою кiлькiстю параметрiв модель вiдтворює як якiсно, так i кiлькiсно головнi експериментальнi данi по напруженню-деформацiї для пiсковика Береа. Модель правильно вiдтворює великi та малi петлi на траєкторiї напруження-деформацiя (пам’ять про кiнцеву точку). Власне запропонована залежнiсть деформацiї вiд напруження є не чим iншим, як рiвнянням стану пiсковика. en Видавничий дім "Академперіодика" НАН України Науки про Землю Quasistatic loading of Berea sandstone Article published earlier |
| spellingShingle | Quasistatic loading of Berea sandstone Vakhnenko, V.O. Vakhnenko, O.O. TenCate, J.A. Shankland, T.J. Науки про Землю |
| title | Quasistatic loading of Berea sandstone |
| title_full | Quasistatic loading of Berea sandstone |
| title_fullStr | Quasistatic loading of Berea sandstone |
| title_full_unstemmed | Quasistatic loading of Berea sandstone |
| title_short | Quasistatic loading of Berea sandstone |
| title_sort | quasistatic loading of berea sandstone |
| topic | Науки про Землю |
| topic_facet | Науки про Землю |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4153 |
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