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Актуальность исследования обусловлена широкой распространенностью проблемы предотвращения прорывов водных масс в подземные горные выработки, успешное решение которой во многом определяет эффективность и безопасность разработки месторождений. Цель: исследование взаимосвязи современных геодинамических движений с опасностью прорыва водных объектов в горные выработки при разработке месторождений в сложных гидрогеологических условиях. Объект: массив месторождений, находящиеся в сложных гидрогеологических условиях, а именно залегающих под мощной осадочной толщей, вмещающей несколько водоносных горизонтов. Методы: аналитический, спутниковое позиционирование двухчастотными геодезическими GPS приемниками, компьютерное моделирование векторного и тензорного деформационного поля. Результаты. На основании анализа произошедших аварийных ситуаций, связанных с прорывами вод на рудниках, их причин и способов предупреждения установлено, что одним из важнейших факторов, вызывающих прорыв водных объектов, являются современные геодинамические движения, формирующие неоднородное мозаичное напряженно-деформированное состояние в массиве пород, препятствующее сохранению целостности водозащитных массивов. Известные способы предотвращения опасности прорыва вод в горные выработки путем закладки выработанного пространства и поддержания налегающей толщи целиками при назначении и применении параметров очистных работ без учета формирования реального напряженно-деформированного состояния под влиянием геодинамических движений, как правило, сопровождаются авариями. В качестве рекомендаций по предупреждению прорывов вод в горные выработки представлен и апробирован метод изучения геодинамических движений и процессов деформирования, протекающих в массиве, на примере Еланского и Ёлкинского полиметаллических месторождений. Установлены векторы трендовых движения пунктов совместно с тектоническим строением исследуемого региона. Зафиксированы максимальные вертикальные поднятия и максимальные горизонтальные смещения. Результаты измерений обработаны и преобразованы в векторное и тензорное представление деформационного поля, где выделены главные компоненты тензора деформаций. Выводы. Исследуемый массив находится в умеренной зоне напряженно-деформированного состояния с характерной разнонаправленностью тензора деформаций, что приводит к развитию сдвиговых деформаций в массиве горных пород, отрицательно влияющих на устойчивость горных выработок, а также способствующих раскрытию естественных структурных нарушений массива горных пород и повышению его фильтрационных свойств. Повышение надежности применяемых способов предотвращения опасности прорывов вод в горные выработки требует проведения специальных исследований формирования реального напряженно-деформированного состояния и учета его параметров при проектировании и ведении очистных работ. Relevance of the study is conditioned by wide spread problem of prevention of water mass breakthroughs into underground mine workings, successful solution of which in many respects determines the efficiency and safety of exploration. The aim of the research is to study the interrelation of modern geodynamic movements with danger of water objects breakthrough into mine workings during development of deposits in difficult hydrogeological conditions. Object: massif of deposits in difficult hydrogeological conditions, located under a powerful sedimentary stratum containing several aquifers. Methods: analytical, satellite positioning with dual-frequency GPS geodetic receivers, computer simulation of vector and tensor deformation field. Results. Based on the analysis of occurred emergency situations related to water breakthroughs at the mines, their causes and methods of prevention, it was found that one of the most important factors causing a breakthrough of water bodies are modern geodynamic movements that form a heterogeneous mosaic stress-strain state in the rock mass, which prevents the preservation of the integrity of waterprotective massifs. The known methods to prevent the danger of water breakthrough in the rock by laying the depleted space and maintaining the overlying thickness of the whole, the appointment and application of parameters of treatment works without taking into account the formation of real tensely-deformed state under the influence of modern geodynamic movements are usually accompanied by accidents. As recommendations for prevention of water breakthroughs into the mine workings the method of studying modern geodynamic movements and the stress-strain state formed by them was presented and tested on the example of the massif Elanskoe and Elkinskoe polymetallic deposits. The vectors of trend movements of points together with the tectonic structure of the investigated region are established. Maximum vertical rises and maximum horizontal displacements were fixed. The obtained results of measurements using mathematical apparatus of solid medium mechanics were transformed into vectorial and tensor representation of deformation field with selection of main components of deformation tensor. Conclusions. The studied array is located in the temperate zone of stress-strain state with a characteristic multi-directionality of the deformation tensor, which leads to the development of shear deformations in an array of rocks, adversely affecting the stability of rock workings, as well as contributing to the disclosure of natural structural disorders of the array of rocks and improvement of its filtration properties. Increase of reliability of applied methods for preventing the danger of water breakthroughs into mine workings demands carrying out of special researches of formation of a real tensely-deformed condition and consideration of its parameters at designing and conducting clearing works.

Some sufficient conditions of the energy conservation for weak solutions of incompressible viscoelastic flows are given in this paper. First, for a periodic domain in ℝ3, and the coefficient of viscosity μ = 0, energy conservation is proved for u and F in certain Besov spaces. Furthermore, in the whole space ℝ3, it is shown that the conditions on the velocity u and the deformation tensor F can be relaxed, that is, $$u \in B_{3,c(\mathbb{N})}^{{1 \over 3}}$$ , and $$F \in B_{3,\infty }^{{1 \over 3}}$$ . Finally, when μ > 0, in a periodic domain in ℝd again, a result independent of the spacial dimension is established. More precisely, it is shown that the energy is conserved for u ∈ Lr (0, T; Ls (Ω)) for any $${1 \over r} + {1 \over s} \leqslant {1 \over 2}$$ , with s ⩾ 4, and F ∈ Lm(0, T; Ln(Ω)) for any $${1 \over m} + {1 \over n} \leqslant {1 \over 2}$$ , with n ⩾ 4.

This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor $$\mathcal {D}(u)$$ D ( u ) to the 3D Navier–Stokes equations in Lorentz spaces. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that the norm $$\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} $$ ‖ λ 2 + ‖ L p , ∞ ( 0 , T ; L q , ∞ ( R 3 ) ) with $$ {2}/{p}+{3}/{q}=2$$ 2 / p + 3 / q = 2 $$( {3}/{2} ( 3 / 2 < q ≤ ∞ ) is small. This generalizes the corresponding works of Neustupa–Penel and Miller.

In this paper, we are concerned with the regularity of suitable weak solutions to the 3D Navier–Stokes equations in Lorentz spaces. We obtain $$\varepsilon $$ -regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity or deformation tensor in Lorentz spaces, which generalizes the corresponding results derived by Gustafson et al. (Commun Math Phys 273:161–176, 2007) in Lebesgue spaces. As applications, this allows us to extend recent results involving Leray’s blow up rate in time, and to show that the number of singular points of weak solutions belonging to $$ L^{p,\infty }(-1,0;L^{q,l}({\mathbb {R}}^{3})) $$ is finite, where the pair (p, q) satisfies $$ {2}/{p}+{3}/{q}=1$$ with $$3

Invariants for geodesic and F-planar mappings defined on a generalized Riemannian space are obtained in this paper. These invariants are generalizations of the Thomas projective parameter and the W...

From solar supergranulation to salt flat in Bolivia, from veins on leaves to cells on Drosophila wing discs, polygon-based networks exhibit great complexities, yet similarities and consistent patterns emerge. Based on analysis of 99 polygonal tessellations of a wide variety of physical origins, this work demonstrates the ubiquity of an exponential distribution in the squared norm of the deformation tensor, E(2), which directly leads to the ubiquitous presence of Gamma distributions in polygon aspect ratio as recently demonstrated by Atia et al. [Nat. Phys. 14, 613 (2018)]. In turn an analytical approach is developed to illustrate its origin. E(2) relates to most energy forms, and its Boltzmann-like feature allows the definition of a pseudo-temperature that promises utility in a thermodynamic ensemble framework.

Analysis of systems and structures from their cross-sectional images finds applications in many branches. Therefore, the question of content, quantity, and accuracy of information obtained from various techniques based on cross-sectional views of structures is particularly important. Application of conventional techniques for two-dimensional imaging on the analysis of structure from a cross-sectional image is limited. The reason for this limitation is the fact that these techniques use a fixed cross-sectional plane and therefore cannot check the 3D structural changes caused by deformation. Geometric orientation of a grained structure must be considered when data, scanned from a cross section, is processed in order to obtain information about local deformation in this structure. The so-called degree of structure orientation in 3D can be estimated experimentally from the cross-sectional image of the structure by the statistical (Saltykov) method of oriented testing lines. Subsequently if the correlation between orientation and deformation were to be known a detailed map of local deformation in the structure could be revealed. Unfortunately, exact theoretical works dealing with the assessment of local deformation by means of change of structure orientation in 3D are still missing. Our work seeks to partially remove this shortcoming. In our work we are interested in how the transformation of the image of a grained structure in a cross-sectional plane reflects structure deformation. An initial shape of grains is assumed which is transformed into a deformed shape by analytic calculation. We present brief mathematical derivations aimed at the problem of single grain-surface area deformation. The main goal of this work led to the design of a computationally low consuming procedure for quantification of local deformation in a grained structure based on the distortion of the image of this structure in a cross-sectional view.

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

In this paper, we are concerned with the compressible viscoelastic flows in whole space \begin{document}$ \mathbb{R}^n $\end{document} with \begin{document}$ n\geq2 $\end{document} . We aim at extending the global existence in energy spaces (see [ 18 ] by Hu & Wang and [ 30 ] by Qian & Zhang) such that it holds in more general \begin{document}$ L^p $\end{document} critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define "two effective velocities" which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the \begin{document}$ L^p $\end{document} critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the \begin{document}$ L^p $\end{document} framework, which improve recent decay efforts for compressible viscoelastic flows.

Shear bands are common in dense quasi-static granular flows. They can appear in the interior of the flowing material or at confining boundaries and are typically of the order of ten particle diameters in thickness. Deformation tends to be localized in shear bands separating non-deforming or weakly deforming regions. Dilatancy and sharp velocity variation are typical in these shear layers. Much work has been reported in the literature concerning the development of non-local quasi-static rheological models to predict the flow behaviour in shear layers. In a recent article, Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, R3) derive a non-local extension to a classical plasticity model by postulating that some local quantities appearing in the yield function, which stipulates the relationship between different components of the stress for the material to undergo sustained yielding, and the flow rule which provides information on the rate of deformation tensor to within an arbitrary multiplicative constant, should be replaced by their local averages. They then obtain an explicit non-local model which does not involve new microstructural variables and they show that the model captures velocity and volume fraction fields in simple shear flows, although some model parameters must be fitted to achieve quantitative agreement. This article discusses the work of Dsouza & Nott (2020) and comments on work ahead for further testing and developing the model.

The strain intensity as well as the stress-strain state of grains has been experimentally estimated for aluminum sample in strip rolling. The level of correlation between the intensity of deformation and stress-strain state of individual grains in the sample has been calculated. An experimental two-dimensional function of its joint distribution has been constructed. The parameters of two-dimensional function of probability densities have been defined. A relative number of grains with deformation twice as high as the average deformation of the sample has been obtained. The results allow predicting the plasticity exhaustion in individual grains of the sample during rolling and can be used to estimate the likelihood of defects (micro pores and micro cracks) arising during the rolling of aluminum strips.

This paper develops a joint deformation tensor (JD), which considers all of the joint's mechanical and geometrical parameters that affect the deformability of the rock mass. The method based on JD (JD method) and an elastic deformation anisotropy index (EDAI) are deduced for estimating the spatial anisotropy deformation of a jointed rock mass. The numerical modeling and in situ true triaxial compressive experiments well verified the effectiveness of the EDAI and JD method for the rock mass containing one joint set, orthogonal joint sets or the rock mass containing any types of joint network with unity stiffness ratio.

This work showcases a method to map the full deformation tensor in a single micro-sized crystal. It is shown that measuring the position of two Bragg reflections in reciprocal space is sufficient to obtain the full deformation tensor, if the condition of incompressibility of the material is imposed. This method is used to reveal the surface tension induced deformation at the edges of an as-grown single-crystal VO2 microwire. All components of the deformation tensor of the microwire were measured down to an absolute value of 10−4 in an 8 × 14 µm projected area of the wire. With a beam-defined spatial resolution of 150 × 150 nm, the measurement time was merely 2.5 h.

Purpose. The CuInP2S6 and CuInP2Se6 crystals are characterized by an order-disorder phase transition mechanism, which is caused by electron-phonon interaction and the Jan-Teller cooperative effect. Vibronic instability is related to the electronic configuration of copper d10 in octahedral symmetry, and the disorder in the paraelectric phases can be explained by the jumping movements of Cu ions. It is of interest to study the conditions of this instability at the pressure. The main aim of our paper is to carry out the investigation of the matrix of the vibronic potential energy and the adiabatic potentials at the pressure.Methods. Adiabatic potentials concept based on the group -theoretical analysis and Pikus’ method of invariants.Results. For the case of hydrostatic pressure for crystals with D3d symmetry, an adiabatic potential depending on the components of the strain tensor and the coordinates of normal vibrations active in the Jahn-Teller effect has been obtained. We carried out the numerical simulations of the pressure effect on the shape of the adiabatic potentials From our qualitative analysis follows that the changing of the adiabatic potential depth (EJT ) and the value of the minimum displacement ρ0 takes place. Especially the shape of the adiabatic potential energy is changed at the shift strain εxy.Conclusions. Using the symmetry methods, we constructed the matrix of the vibronic potential energy and found the adiabatic potentials. It is shown that in a consequence of the phase transition the splitting of the equivalent minima (disorder) of the adiabatic potential on the group of the nonequivalent minima (order) is occurred. The shape of these adiabatic potentials shows that in their concept one can confirm about the order-disorder phase transition and also to consider the pressure influence on these potentials., Предложена новая методика исследования влияния давления на адиабатический потенциал, который базируется на теории групп и методе инвариантов Г.Е.Пикуса. Для случая гидростатического давления для кристаллов с D3d симметрией получено адиабатический потенциал, зависимый от компонентов тензора деформаций и координат нормальных колебаний, активных в эффекте Яна-Теллера. В результате численного моделирования показано, что при учете деформаций наблюдается расщепление эквивалентных минимумов (что соответствует беспорядку) адиабатического потенциала на группу неэквивалентных минимумов (порядок). Представленное исследование дает возможность рассмотреть механизм фазовых переходов типа порядок - беспорядок, который связывается с электрон - фононным взаимодействием и кооперативным эффектом Яна-Теллера в слоистых кристаллах CuInP2S6 и CuInP2Se6., Запропонована нова методика дослідження впливу тиску на адіабатичний потенціал, яка базується на теорії груп та методі інваріантів Г.Є. Пікуса. Для випадку гідростатичного тиску для кристалів з D3d симетрією одержано адіабатичний потенціал, залежний від компонентів тензора деформацій і координат нормальних коливань, активних в ефекті Яна-Теллера. У результаті чисельного моделювання показано, що при врахуванні деформацій має місце розщеплення еквівалентних мінімумів (що відповідає безладу) адіабатичного потенціалу на групу нееквівалентних мінімумів (лад). Представлене дослідження дає можливість розглянути механізм фазових переходів типу лад - безлад, який пов’язується з електрон-фононною взаємодією і кооперативним ефектом Яна-Теллера в шаруавтих кристалах CuInP2S6 і CuInP2Se6.

Global organization of three-dimensional (3D) Lagrangian chaotic transport is difficult to infer without extensive computation. For 3D time-periodic flows with one invariant, we show how constraints on deformation that arise from volume-preservation and periodic lines result in resonant degenerate points that periodically have zero net deformation. These points organize all Lagrangian transport in such flows through coordination of lower-order and higher-order periodic lines and prefigure unique transport structures that arise after perturbation and breaking of the invariant. Degenerate points of periodic lines and the extended 3D structures associated with them are easily identified through the trace of the deformation tensor calculated along periodic lines. These results reveal the importance of degenerate points in understanding transport in one-invariant fluid flows.

Horizontal strains related to mining-induced subsidence may endanger infrastructure and surface users’ safety. While directional horizontal strains should be well determined, appropriate solutions for a complete assessment of the terrain surface deformation field are still required. As a result, the presented study examined a new method for calculating horizontal strain tensor based on the decomposition of satellite radar interferometry (InSAR) observations into vertical and azimuth look direction (ALD) displacements. Based on a geometric integral model, we tested our method on experimental data before applying it to an underground copper ore mine in Poland. In the case study, the displacement field was determined using the Multi-Temporal InSAR method on Sentinel-1 data. The model data relative error did not exceed 0.02 at σ = ±0.003. For the case study, land subsidence of up to −167 mm and ALD displacements ranging from −110 mm to +62 mm was obtained, whereas the extreme values of horizontal strains ranged from −0.52 mm/m to +0.36 mm/m at σ = ±0.050 mm/m. Our results demonstrate the high accuracy of the method in determining the horizontal strain tensor. As a result, the approach can broaden the assessment of the environmental impact of land subsidence worldwide.

In this paper, we are concerned with the local regularity of suitable weak solutions to the 3D Navier–Stokes equations and its generalized case with fractional power of the Laplacian ( − Δ ) α ( 3 / 4 ≤ α 1 ). In the first part, we derive some ϵ -regularity criteria in terms of the deformation tensor D ( u ) for the classical Navier–Stokes equations. In the second part, for the fractional case, we obtain some regularity conditions for suitable weak solutions including the velocity u , the gradient of the velocity ∇ u , the rotation tensor Ω ( u ) and the deformation tensor D ( u ) . This generalizes the results obtained by Gustafson et al. (2007).

In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible breakdown of such solutions at critical time in terms of both the L ∞ ( 0 , T ; L 6 ) -norm of the density of particles and the L 1 ( 0 , T ; L ∞ ) -norm of the deformation tensor of velocity gradient.

Oxygen plays a critical role in human cell survival and proliferation; hence, it is an essential factor in cancer tumor propagation. In the literature, many studies have investigated the oxygen contribution to cell proliferation and tumor growth; however, lack of the studies considering the effect of blood oxygen solubility on the tumor growth is remarkably sensed among the literature. Accordingly, in this study, we investigated the effect of oxygen Bunsen solubility coefficient on the brain glioma growth, following continuum mechanics principles. A biomechanical simulation was conducted by assuming a compressible hyperelastic model of growth for spherical glioma tissue. Patient-specific MRI dataset was utilized to develop the geometry of the brain. An isotropic growth deformation tensor alongside oxygen and cell number concentration relations were applied to the simulation. The results demonstrated that the tumor maximum volume at the end of 90 days decreased from 1315.59 mm3 to 1236.43 mm3, as the increase of oxygen Bunsen solubility coefficient for 20 %. Thus, we concluded that the accessibility of oxygen with higher values of blood solubility lowers the risk of tumor sudden growth, which was verified in comparison to literature experiments.

We show that the covariant Raychaudhuri identity describing kinematic characteristics of space-time admits a representation involving a geometrical scalar $\xi$ which, depending on circumstances, might be related to, e.g., relativistic temperature or cosmological scalar field. With an appropriately chosen spacetime deformation tensor (fixing the symmetry of a problem under consideration), such scalarization opens a wide scope for physical applications. We consider few such applications including dynamics of cosmological (anti)scalar background, non-variational deduction of the field equations, scalar and black-hole thermodynamics and the reshaping of the Einstein equations into the Klein-Gordon equation in thermodynamic Killing space., Comment: Accepted in PRD

In this paper, we consider the short time strong solutions to the ideal compressible magnetohydrodynamic equations with initial vacuum, where the velocity field satisfies the Navier-slip condition. Inspired by Kato and Lax's idea, we use the contraction mapping argument to prove the local existence. Moreover, under the Navier-slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L ∞ norm of the deformation tensor D ( u ) .

The objective of this paper is to develop a new unified theoretical structure for modeling interstitial growth and muscle activation in soft tissues. The model assumes a simple continuum with a single velocity field. In contrast with many other formulations, evolution equations are proposed directly for a scalar measure of elastic dilatation and a tensorial measure of elastic distortional deformation. The evolution equation for elastic dilatation includes a rate of mass supply or removal that controls volumetric growth and causes the elastic dilatation to evolve towards its homeostatic value. Similarly, the evolution equation for elastic distortional deformation includes a rate of growth that causes the elastic distortional deformation tensor to evolve towards its homeostatic value. Specific forms for these inelastic rates of growth and the associated homeostatic values have been considered for volumetric, area and fiber growth processes, as well as for muscle activation. Since the rate of growth appears in the evolution equations and not a growth tensor it is possible to model the combined effects of multiple growth and muscle activation mechanisms simultaneously. Also, robust, strongly objective, numerical algorithms have been developed to integrate the evolution equations.

The paper shows that the regularity up to the boundary of a weak solution v of the Navier–Stokes equation with generalized Navier’s slip boundary conditions follows from certain rate of integrability of at least one of the functions ζ1, (ζ2)+ (the positive part of ζ2), and ζ3, where ζ1≤ζ2≤ζ3 are the eigenvalues of the rate of deformation tensor D(v). A regularity criterion in terms of the principal invariants of tensor D(v) is also formulated.

The response to stress delivered by a non-isochoric hollow cylinder is critical to its future life and integrity. The design of cylinders must emphasize durability; therefore, the knowledge of the yield stress of the constituent material is essential. The stress component of a pressurized compressible circular cylinder made of non-linearly elastic material was considered here and the yield stress was determined.

The possibility of inheritance of the elastic field in the zone of appearance of the initial excited state by the wave process controlling growth of the martensite crystal is shown. The case when the wave vectors of the control waves belong to plane {110}c of the original cubic phase is analyzed. Implementability of such inheritance is demonstrated with the use of elastic moduli for single-crystal bcc titanium, which advances the predictive capacity of the dynamic theory. © 2017, Springer Science+Business Media, LLC, part of Springer Nature. The work has been performed with financial support of the Russian Foundation for Basic Research (Project Number 14-08-00734).

This paper investigates the global existence and the decay rate in time of a solution to the Cauchy problem for an incompressible Oldroyd model with a deformation tensor damping term. There are three major results. The first is the global existence of the solution for small initial data. Second, we derive the sharp time decay of the solution in $L^{2}-$norm. Finally, the sharp time decay of the solution of higher order Sobolev norms is obtained., Comment: 13 pages