ISSN 0010-9525, Cosmic Research, 2009, Vol. 47, No. 3, pp. 226–234. © Pleiades Publishing, Ltd., 2009. Original Russian Text © E.N. Chumachenko, R.R. Nazirov, 2009, published in Kosmicheskie Issledovaniya, 2009, Vol. 47, No. 3, pp. 247–255.
Some Problems Associated with Design of Cryobots E. N. Chumachenko and R. R. Nazirov Space Research Institute, Russian Academy of Sciences, Profsoyuznaya ul. 84/32, Moscow 117997, Russia Received July 11, 2007
Abstract—The probable difficulties, arising when one is involved in design of cryobots, are considered in the paper. The problems of origin of large thicknesses of an ice layer and efficiency of applying the cryobot-type instrumentation for studying the ice surface of Europa are discussed. PACS: 07.05.Tp; 46.25.Hf; 96.30.Id DOI: 10.1134/S0010952509030058
INTRODUCTION The idea to design cryobots is the idea of creating the instrumentation for penetration through the ice thickness into waters of the ocean of Europa (the satellite of Jupiter). The motion of a cryobot should be accomplished by thawing through an ice mass in the vicinity of its position due to maintaining the corresponding temperature on the cryobot’s surface. It was initially supposed that this device should be as compact and efficient as possible. Of course, the restrictions on dimensions and on weight are related with difficulties of instrument transportation to the surface of a corresponding object under study. It was also supposed that, in spite of its small size, such a device could be supplied with a powerful energy source (the atomic mini-reactor, for instance) capable of maintaining a preset temperature on the cryobot’s surface. Such instrumentation could be used on our planet as well. For example, to perform unique investigations of the famous subglacial lake Vostok near the pole of cold on the Earth. The average annual temperature in this Antarctic region equals –55°ë. The round-the-clock polar night lasts about four months. The normal atmospheric pressure is equal to only 460 mm Hg, as at the top of Mount Elbrus, and the humidity of air is lower than that in Sahara. The scientists date the formation of ice structures in the lake vicinity to the pre-Cambrian period (from 570 million to 3.5 billion years ago). The life in the subglacial lake is supposed to exist to a high degree of probability. Probably, it is present in the lake in the most elementary forms, which can exist without the photosynthesis process. There is a hope that these organisms were kept since the times of glacial cover formation and have been isolated from the external world for millions of years. The drilling of ice structures some kilometers thick is not completed yet, and scientists all over the world
vigorously discuss possible consequences of unsuccessful break of the ice cover. The Russian scientists, who perform drilling, attentively study and analyze all the versions advanced. The authors of this paper, on the basis of theoretical estimations of the process of penetration of a conventional object (a hot drop) into highly cooled structure (ice), have attempted to formulate possible difficulties facing the designers developing cryobots1. SIMULATION MODELING Following the data of papers [1–3], in subsequent simulation modeling we have accepted the values of Young modulus (Ö), Poisson coefficient (ν), specific heat capacity (C), heat conductivity (λ), limiting values c p of stresses at compression ( σ s ) and extension ( σ s ), and the linear expansion coefficient (α), which are presented in Table. On the basis of a great number of papers published after flights of the Voyager and Galileo spacecraft, it was accepted that the temperature on the ice cover surface of Europa can reach –170°ë, the thickness of ice cover is about 10 km, and transition of ice structures into the liquid state presumably occurs at a temperature of –5°ë. On the frontal and side surfaces of the object (we called it conventionally “the hot capsule”) the temperature was accepted to be equal to 30°ë, and on the rear horizontal plate it was 15°ë. The temperature fields and stress-strain state have been calculated by means of the SPLEN computing system (www.kommek.ru). Figure 1 presents the obtained temperature fields for three values of temperatures of an ice monolith. The 1 We
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thank Prof. B.I. Rabinovich and L.V. Ksanfomality for their active participation in discussion of our resuls.
SOME PROBLEMS ASSOCIATED WITH DESIGN OF CRYOBOTS
227
Table E, kg/mm2
σ s , kg/mm2
σ s , kg/mm2
ν
C, J/(kg/K)
λ, W/(m/K)
942
3.00
2.7
0.3
2050
0.023
100
Ice (–170°C)
1450
5.29
5.0
0.3
850
0.050
10
Water (–5°C)
250
2.84
2.6
0.45
4200
0.022
120
Material
Ice (–8°C)
c
p
graphic contour marks the zones, in which the medium around the capsule was in the liquid state at a previous step of “thawing-through”. The dark (red) color shows a liquid phase of ice at the current step of “thawingthrough”. For monolith’s temperature of –100°ë the rapid closure (hardening) of a thawed trace is typical. The liquid trace assumes the form of a concave lens. This is a direct consequence of temperature drop on the capsule’s surface. At the ice monolith’s temperature of –50°ë the liquid trace assumes the form of a highly convex lens. That is, the effect of temperature drop on the capsule’s surface noticeably decreases. In Fig. 1c the ice monolith’s temperature equals – 6°ë. The horizontal line on the presented temperature fields shows a conventional interface between the liquid and solid states of a medium in the sector under study. In this case the nonfreezing, liquid droplet-shaped trail is formed behind the capsule. When estimating the stress-strain state in the capsule vicinity we make use of the integrated Schleicher– Nadai parameter for probabilistic estimation of destructions of ice structures around the capsule. In Figure 2 the zones, in which continuity violation of the ice monolith is most probable, are marked by dark color. At temperatures lower than –50°ë the zones of fracture are situated around the capsule, the leading fracture zone being the larger the lower is the ice temperature. At temperatures close to the fusion temperature there are no leading fracture zones, and the continuity violation is most probable in the capsule trail zone. DISCUSSION OF RESULTS The analysis of the stress-strain state of the process of “thawing-through” ice structures makes it possible to perform some numerical estimations, which, in turn, give an COSMIC RESEARCH
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α, 106/K
idea about the problems to be solved by the developers (designers) and users (customers) of a cryobot. Calculations have shown that a rather high hydrostatic pressure can develop in the liquid around a capsule. It equals 220 MPa at surrounding ice mass temperature of –100°ë, 150 MPa at –50°ë, and 40 MPa at –6°ë. That is, along with the problem of maintaining the capsule’s temperature, rather topical is also the problem of capsule protection against high pressure. The second “feature” of the process is its velocity. At a temperature of –100°ë the capsule motion velocity occurred to be ~0.17 · 10–2 mm/s. This is a very low velocity; for example, 1 km of an ice layer can be passedthrough at such a velocity only in about 18 years. At the ice structure temperature of –50°ë the velocity slightly grows and becomes equal to ~0.50 · 10–2 mm/s. Nevertheless, the ice layer one kilometer thick can be overcome, in this case, only for ~6 years. And only at ice temperature of –6°ë the velocity of motion becomes acceptable and equal to ~0.4 mm/s (1 km of ice layer can be passedthrough in about 29 days). Thus, it is obvious that additional studies are necessary to optimize the shape and size of a capsule, to produce the systems for regulating the temperature of its surface with the purpose of increasing the velocity of passing through ice structures. Detailed investigations will be required while designing the system for controlling effects providing for the directed motion of a capsule, essentially differing from the vertical one that is ensured by gravitational forces. Of course, the obtained solutions are only approximate estimates. The temperature-dependent properties of ice structures are not taken into account to a full measure. The influence of various impregnations on the mechanical properties of ice is not clear yet. An essential factor is the formation of ice chip in the capsule vicinity. Continuous change of density, pressure and viscosity in the capsule vicinity makes the problem nonlinear in several parameters at once. The formation
228
CHUMACHENKO, NAZIROV (‡) T ·10–1 –10.00 –8.94 –7.89 –6.83 –5.78 –4.72 –3.67 –2.61 –1.56 –0.50 3.00
(b) T –50 –45 –40 –35 –30 –25 –20 –15 –10 –5 30
Fig. 1. Temperature fields in the ice monolith with temperature: (a) –100°ë, (b) –50°ë, and (c) –6°ë.
of chip can either slow down the process of penetration into the ice thickness, or speed up this process at organizing appropriate methods (technologies) of passage. Still open remains also the question related to downloading the information obtained by a probe. Various possibilities can be considered here. For example, the
variant with throwing mini radio beacons along the probe path for organizing communication with a powerful station on the Europa surface. The most interesting solution would be return of the vehicle with taken samples to the planet surface. This task is, undoubtedly, more complicated, but not impossible. COSMIC RESEARCH
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(c) T –6.00 –5.89 –5.78 –5.67 –5.56 –5.44 –5.33 –5.22 –5.11 –5.00 30.00
Fig. 1. Contd.
In all cases, continuation of investigations in this direction can result in development a unique instrumentation for performing experimental works on the frozen surface of the ocean of Jupiter moon Europa. APPENDIX GENERAL STATEMENT OF THE BOUNDARYVALUE PROBLEM The physical state of ice structures can be described by the equations, which relate the stress tensor deviator with the strain tensor deviator in accordance with the elastoplastic flow theory. 2 σu ˜ σ˜ ij = --- -----ε , 3 ε˙ u ij 1 σ = --- σ ij δ ij , 3
σ˜ ij = σ ij – σδ ij , (1)
ε˜ ij = ε ij – εδ ij ,
1 ε = --- ε ij δ ij . 3
The intensity of stresses nonlinearly depends on the intensity of strains and on temperature σ u = σ u ( ε u, T ),
σu =
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2˜ ˜ --- ε ij ε ij . (2) 3 2009
The components of the stress tensor depend on components of the strain tensor, in a given changeable field of temperatures, as follows: σ ij = ( 2σ u /3ε u )ε ij + 3 ( K – 2σ u /ε u )εδ ij – α ( 2σ u /3ε u )∆T δ ij ,
(3)
where ∆T is the temperature variation at the current step of solution (°ë), and α is the coefficient of linear expansion of a body (1/°ë). For the elastic state of ice (in this case 2σu/3εu = 2G), passing from coefficients G and K, characterizing the shear modulus and the volume compression coefficient, to coefficients Ö and ν (Young’s modulus and Poisson’s coefficient, respectively), we obtain 1+ν E 3ν σ ij = ------------ ⎛ ε ij + --------------- δ ij ε – --------------- δ ij α∆T⎞ . ⎠ 1 – 2ν 1 + ν⎝ 1 – 2ν
(4)
The “thawing-through” process is rather slow, quasistatic; therefore, at each stage of constructing the solution, the equations of balance should be valid: σij, j = 0. (5) Let the ice structure segment we are studying occupy the volume V with boundary S, in the Cartesian coordinate system XYZ. The boundary of a body is separated into three parts, so that S = Sσ ∪ Su ∪ Sσu.
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CHUMACHENKO, NAZIROV (‡)
γ 0 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 1.65
60 50 40 30 20 10 0 (b) 60 50 40 30 20 10 0 (c)
Field of distruction paraameters
80 70 60 50 40 30 20 10 0 0
10 20 30 40 50 60
0
10 20 30 40 50 60
0
10 20 30 40 50 60
Fig. 2. Probabilistic destruction zones in the ice monolith with temperature: (a) –100°ë, (b) –50°ë, and (c) –6°ë.
On the boundary part Sσ the surface forces Pn = P ni ki are acting σ ij ( x 1, x 2, x 3 )n j
Sσ
= P ni ( x 1, x 2, x 3 ).
(6)
On the boundary part Su the displacements u* = u *i ki are specified u i ( x 1, x 2, x 3 )
Su
= u *i ( x 1, x 2, x 3 ).
(7)
On the boundary part Sσu one component of vectors, Pn and u* is specified
σ 1 j ( x 1, x 2, x 3 )n j u 2 ( x 1, x 2, x 3 )
S σu
S σu
= P n1 ( x 1, x 2, x 3 ),
= u *2 ( x 1, x 2, x 3 ).
(8)
The three functions of displacement: u1(x1, x2, x3), u2(x1, x2, x3), and u3(x1, x2, x3) are required quantities. The Cauchy relationships, which relate the strain tensor components with displacements of particles, have the following form: 1 ∂u ∂u ε ij = --- ⎛ -------i + -------j⎞ . 2 ⎝ ∂x j ∂x i⎠ COSMIC RESEARCH
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In order to calculate the stress-strain states of ice structures, we should find the distribution of temperatures in the vicinity of the object under study. Again, in constructing the estimates we suppose the properties of the medium, surrounding the object, to be homogeneous, isotropic, and depending solely on temperature. The general equation of heat conduction has the form ∂T cρ ------ = λ ∂τ
∑ i
∂T -------2- + Q, ∂x i 2
(10)
where T = T(xi, t) is the body’s temperature at point (xi) at time t; c = c(T) is the specific heat capacity (J/(kg°ë)); ρ = ρ(T) is density (kg/m3); λ = λ(T) is the heat conductivity of dimension W/(m °C); Q = Q(xi, T) is the intensity of heat sources inside a body (W/m3), which is considered to be positive, if the body is supplied with heat. At the initial time instant, the initial temperature distribution T(xi, 0) = T0(xi) is specified in the region V. At the boundary of the region S = L1 ∪ L2 ∪ L3 the following conditions of heat exchange with the environment can be specified: ∂T – λ -----∂n ∂T – λ -----∂n T
= q,
(11)
= ψ ( T – T env ),
(12)
L1
L2 L3
(13)
= T bound ,
that is, the constant temperature Tbound can be specified on the part L3 of the boundary, and convective heat exchange ψ(T – Tenv) or the heat flow of intensity q (W/m2) can be specified on the remaining part of the boundary. Convective heat exchange is characterized by the heat exchange coefficient ψ (W/(m2 °ë)) and by the environment temperature Tenv. We should pay attention to the fact that the subdivision of the boundary, which determines the types of boundary conditions both for calculating the stressstrain state and for calculating the evolution of temperature fields, continuously changes and is determined by E(1 – ν) ⎧ ⎪ σ r = ------------------------------------( 1 + ν ) ( 1 – 2ν ) ⎪ ⎪ E(1 – ν) ⎪ σ z = ------------------------------------( 1 + ν ) ( 1 – 2ν ) ⎪ ⎨ ⎪ E(1 – ν) ⎪ σ ϕ = ------------------------------------( 1 + ν ) ( 1 – 2ν ) ⎪ ⎪ E ⎪ σ rz = -----------ε 1 + ν rz ⎩ COSMIC RESEARCH
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the current position and physical state of a medium segment around a cryobot, chosen for studying. Consider now the situation, when a certain smallsize object, on whose surface the constant temperature is maintained exceeding the ice thawing temperature, is situated inside the ice monolith. For analyzing the temperature fields, arising around such an object, and for estimating the stress-strain state of ice structures, it is sufficient to consider some sequence of axi-symmetric neighborhoods of a moving object. After appropriate substitutions our problem is reduced to searching for the solution to Eqs. (5) and (10) with boundary conditions (6–8, 13), which are usually solved in the interrelated sequence [4]. ALGORITHM FOR SOLUTION CONSTRUCTION The approximate solution to the stated problem will be sought by means of the finite element method (FEM), in cylindrical coordinates. We use the circular elements of a triangular cross section as finite elements. The field of displacements is uniquely determined by displacements u(r, z) and v(r, z) in the direction of axes R and Z. The stress-strain state does not depend on the angular coordinate. However, the radial displacements cause strains εϕ. Based on this fact, we can present the strains and stresses by the vector-columns { ε } = ε r ε z ε ϕ ε rz , T
{ σ } = σ r σ z σ ϕ σ rz .
(14)
T
Cauchy relations (9) take on the form ∂u ε r = -----, ∂r
∂v ε z = ------- , ∂z
u ε ϕ = --- , r
1 ∂u ∂v ε rz = --- ⎛ ----- + -------⎞ . 2 ⎝ ∂z ∂r ⎠
(15)
Relationships (4) can be written as follows:
ν ν ε r + ------------ ε z + ------------ ε ϕ + α∆T 1–ν 1–ν ν ν ------------ ε r + ε z + ------------ ε ϕ + α∆T 1–ν 1–ν ν ν ------------ ε r + ------------ εz + ε ϕ + α∆T 1–ν 1–ν
(16)
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In the field of elastic strains, the vector of nodal displacements of an element determines the corresponding stress-strain state of the given element. In accordance with the principle of additivity of elastic and temperature deformations, this stress-strain state is superimposed by the stress-strain state caused by the change of temperature. Here, the temperature field of a body is supposed to be independent of the stress-strain state and is calculated by heat conductivity equations (10). (e)
(e)
(e)
{ ε } = [ B ] { δ }, (e)
(e)
(e)
(17) (e)
{ σ } = [ D ] ( { ε } – { ε T } ),
As is known, among all displacements, satisfying kinematic boundary conditions, the extreme value of potential energy is acquired from those displacements, which satisfy the equations of balance. Making use of this circumstance for deriving the basic resolving FEM equations, we obtain (19)
a a
ci ν ν c i b j 1 – 2ν c j b i - + ---------------- -------------- + -------------------- ---, = ------------ ---------1 – ν 4S ( e ) ( 1 – ν ) 16S ( e ) ( 1 – ν )6 r
12
c ν ν b i c j 1 – 2ν c i b j -) + ---------------- -------------) + -------------------- ----j , = ------------ ---------( e ( e ( 1 – ν ) 16S ( 1 – ν )6 r 1 – ν 4S
21
a
∑ [K
],
(20)
E
[F] = –
∑ [F
(e)
].
(21)
e=1
The element stiffness matrix [K] and the load vector {F} are formed by summing up the contributions of separate elements (e)
(e) T
(e)
[ K ] = [ B ] [ D][ B ]
∫ dV . V
(22)
(e)
The line over [B] denotes the approximate, average value. Taking into account that the volume of a toruslike, triangular (in the cross section) element is given by the formula V(e) = 2π r S(e), where r is the radius-vector of element’s center of gravity, we obtain the final expression for [K(e)] a a E(1 – ν) (e) [ K ij ] = 2πr ------------------------------------- ij ij , ( 1 + ν ) ( 1 – 2ν ) 21 22 a ij a ij 11
12
(23)
c i c j 1 – 2ν b i b j -. - + --------------- ------------= ---------(e) 1 – ν 16S ( e ) 4S
K 2r – 1, 2s – 1 K 2r – 1, 2s
(e)
[ K ] rs
K 2r, 2s – 1
.
(24)
K 2r, 2s
In the general case, the load vector appearing in the right-hand side of the basic FEM relation is determined by thermal expansion, by the effect of concentrated mass and surface forces. (e)
[F ] = – –
∫ V
e=1
22
Each submatrix [K(e)]rs (r and s can assume the values i, j, m) has dimension two by two and, depending on the numbers of element’s nodes, it is sent into corresponding sites of the global stiffness matrix
E
[K] =
b i b j 1 – 2ν c j c i - + --------------- ------------= ---------(e) 1 – ν 16S ( e ) 4S (e)
∫ [B V
(e)
11
1 ν S + --- ⎛ -------------------- ( b i + b j ) + -------⎞ , ⎝ r ( 1 – ν )6 9r ⎠
(18)
where {ε(e)} is the vector of components of strain of an element; [B(e)] is the matrix of coupling the displacements and strains, it is determined from Cauchy rela(e) tions (15); { ε T } is the vector of components of temperature deformation of an element; {σ(e)} is the vector of components of stresses of an element; and [D(e)] is the matrix obtained from relations (16).
[K]{δ} = {F},
a
(e)
(e) T
(e)
(e)
(e)
] [ D ] { ε T } dV – { F c }
(e)
(e) T
(e)
[ N ] { F g } dV –
∫ S
(e) T
(e)
(25)
[ N ] { F S } dV .
(e)
When calculating the extreme states in the ice monoloth it is expedient to make use of some strength theory. The problem of rational choice of a criterion is reduced to determination of some function F(σ1, σ2, σ3), whose value does not depend on the relationship between principal stresses. The values of principal stresses correspond to destruction or to a specified tolerance on the residual strain. If these values correspond to the state preceding the destruction, then the condition of destruction is said to be satisfied: F(σ1, σ2, σ3) ≤ K. (26) In condition (26) quantity ä (the strength criterion) usually has some definite physical interpretation: the normal or tangential stress, the intensity of stresses, the maximum elongation, the distortion energy, etc. Sometimes, however, the strength criterion does not have any direct physical meaning. For more accurate analysis of the processes occurring in ice structures, one must take into account that they resist to compression and extension in a different COSMIC RESEARCH
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τu 1
τu*
τu* 2
4 B
τ1u 3
τutek
σ
σ*
0
0 Fig. 3. Extreme curves of Schleicher–Nadai’s theory of destruction.
manner. The compression strength is higher than the tensile strength. This effect can be taken into account, for example, by assuming that the value of critical tangential stress depends on the value of critical normal stress acting in the same plane. This assumption is basis for Mohr’s theory of failure [5–7], which allows one not only to characterize the stress state at destruction, but also to predict orientation of the fracture plane. In our case we use a more convenient, from our point of view, destruction theory by Schleicher–Nadai [6]. In some sense, it is similar to Mohr’s theory, but it is formulated in terms of intensity of tangential stresses 1 τu and mean stresses σ = --- (σ11 + σ22 + σ33). At the dan3 gerous state the intensity of tangential stresses is a function of hydrostatic pressure that is characteristic for a given material τu = f(σ).
(27)
On the plane (τu, σ) Eq. (27) determines some curve—the destruction boundary (Fig. 3). The dashed straight line 1 corresponds to the condition of greatest intensity of tangential stresses («shear rupture»); the straight line 2 correponds to the condition of greatest volume extension («brittle fracture»). The general case is characterized by some curves 3 and 4. Curve 3 has no COSMIC RESEARCH
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σtek
σ*
σ1
σ
Fig. 4. To calculation of destruction probability.
sharp transition from shear rupture to brittle fracture. On the contrary, curve 4 sharply differentiates two types of destructions. Depending on the stress-strain state, one and the same material can either demonstrate brittle fracture behavior or fail in a viscous manner. The Schleicher–Nadai theory allows one to take into account the duality of the character of destruction and to construct effective probabilistic estimates. One of acceptable approximations of Schleicher–Nadai’s curve is the parabola symmetric about axis Oτu and passing through critical points (0, τ u* ) and (σ*, 0), where τ *u is the maximum value of shear intensity, and σ* is the maximum value of average tensile stresses. Since the values of σ s and σ s are usually obtained from the experiments for single-axis stress–strain, one 1 p 1 can accept that τ u* = ------- σ u* = ------- σ s and, accordingly, 3 3 1 p σ* = --- σ s . 3 c
p
The probability of destruction in the vicinity of an arbitrary point of the considered segment with a cryo-
234
CHUMACHENKO, NAZIROV
bot can be estimated, after calculating the stress-strain state, as the ratio 2.
tek
V p = L /L*,
(28)
tek 2
where (Ltek)2 = (σtek)2 + ( τ u ) , and L* is defined as the distance (along the line passing trough the point (σtek, ( τ u ) ) from the coordinate origin in the system of axes (σ, τu) to the curve τu = f(σ), which has the form tek
τ *u 2 τ u = – -------------σ + τ *u 2 ( σ* )
for
σ > 0.
3.
4.
(29)
Here we suppose that in the case, if σ ≤ 0, then L* = τ *u . In Fig. 4 the value of Ltek is interpreted as segment OA, and L* corresponds to segment OB. Point B on the intersection of beam OA and Schleicher–Nadai’s curve 1 has coordinates (σ1, τ u ). REFERENCES 1. Novikov, O.S., Mekhanicheskie svoistva zhidkikh metallov. Ekstremal’nye svoistva minimal’nykh monokristallov metallov (Mechanical Properties of Liquid Met-
5.
6. 7.
als: Extreme Properties of Metal Minimal Monocrystals), Moscow: Editorial URSS, 2004. Dokuchaev, L.V., Spectrum of Natural Vibrations of the Ice Ocean of Jupiter’s Moon Europa, Kosm. Issled., 2003, vol. 41, no. 3, pp. 277–284. [Cosmic Research, pp. 257–263]. Dokuchaev, L.V., Gravitationally Perturbed Elastic Waves in the Hidden Ocean of Europa, Kosm. Issled., 2005, vol. 43, no. 3, pp. 209–214. [Cosmic Research, pp. 199–204]. Chumachenko E.N. and Pechenkin, D.V., Modelirovanie i raschet termouprugoplasticheskikh deformatsii pri analize lokal’no izotropnykh konstruktsii (Modeling and Calculation of Thermoelastoplastic Deformations when Analyzing Locally Isotropic Constructions), Moscow: MIEM, 2000. Kurlenya, M.V., Mirenkov, V.E., and Shutov, V.A., Osnovy matematicheskogo modelirovaniya razrusheniya (Principles of Mathematical Modeling of Destruction), Novosibirsk: Sibirskoe otdel. RAN, 1998. Kachanov, L.M., Osnovy mekhaniki razrusheniya (Fundamentals of Destruction Mechanics), Moscow: Nauka, 1974. Pisarenko, G.S. and Lebedev, A.A., Deformirovanie i prochnost' materialov pri slozhnom napryazhennom sostoyanii (Deformation and Resistance of Materials under a Complicated State of Stress), Kiev: Naukova dumka, 1976.
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