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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.

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.

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.

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.

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 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.

We consider the three-dimensional Boussinesq equations and obtain some regularity criteria via the velocity gradient (or the vorticity, or the deformation tensor) and the temperature gradient.

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

An adaptive meshless method based on the multi-scale Reproducing Kernel Particle Method (RKPM) for analysis of nonlinear elasto-plastic deformation is proposed in this research. In the proposed method, the equivalent strain, stress, and the second invariant of the Cauchy–Green deformation tensor are decomposed into two scale components, viz., high- and low-scale components by deriving them from the multi-scale decomposed displacement. Through combining the high-scale components of strain and the stress update algorithm, the equivalent stress is decomposed into two scale components. An adaptive algorithm is proposed to locate the high gradient region and enrich the nodes in the region to improve the computational accuracy of RKPM. Using the algorithm, the high-scale components of strain and stress and the second invariant of the Cauchy–Green deformation tensor are normalized and used as criteria to implement the adaptive analysis. To verify the validity of the proposed adaptive meshless method in nonlinear elasto-plastic deformation, four case studies are calculated by the multi-scale RKPM. The patch test results show that the used multi-scale RKPM is reliable in analysis of the regular and irregular nodal distribution. The results of other three cases show that the proposed adaptive algorithm can not only locate the high gradient region well, but also improve the computational accuracy in analysis of the nonlinear elasto-plastic deformation.

In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in $R^n(n=2,3)$. Firstly, we establish a regularity criterion in terms of the $BMO$ norm of the gradient of columns of the deformation tensor in two space dimensions; secondly, we obtain a Beale-Kato-Majda-type criterion in terms of vorticity with the $BMO$ norm in two and three space dimensions.

We consider a hyperbolic quasilinear fluid model, that arises from a delayed version for the constitutive law for the deformation tensor in the incompressible Navier-Stokes equation. We prove global existence of small solutions and asymptotic results in R and the half-space with slip boundary conditions. Futhermore we show that this relaxed system is close to the classical Navier-Stokes equation in the sense that for small times t the solutions converge in high Sobolev norms to the solution of the incompressible Navier-Stokes equation.

This paper studies the classical problem of extension, inflation, and circumferential shearing of an annular cylinder for a new class of compressible elastic bodies wherein the left Cauchy–Green deformation tensor is given as a function of the Cauchy stress tensor. We use a semi-inverse method to study the problem by assuming forms for both the deformation field and the stress field. Focusing our attention on to three specific constitutive relations and two geometries corresponding to thick and thin annular cylinders, we study the qualitative features of the governing differential equations. The models are chosen so that they exhibit qualitatively different response features, one of them displaying a limiting stretch. The classical assumption that the hoop and axial stresses are nearly constant through the thickness of the thin annular cylinder subjected to inflation holds for this class of elastic bodies too. However, for thick-walled annular cylinders subjected to inflation at constant length and for a class of models that exhibits limiting stretch we find that stress “boundary layers” form and that the radial stretch is not monotonic.

An analysis of symmetry changes upon the structural B2-ISS2 phase transition with subsequent martensitic transformations into the B19 and B19′ phases has been performed. An interrelation between the atomic displacements and the lattice deformation has been shown to exist. It has been established that the structures with longitudinal and transverse displacements have the same symmetry and can pass into one another via a twinning operation. A sequence of transformations of the tensor of lattice deformations consistent with the transformations of the atomic displacements has been obtained. It has been found that the operations that provide these transformations are the operations of twinning.

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.

Formation of spatially non-uniform mechanical tension in a transparent solid due to the lattice ionization under the intensive ultrashort laser pulses action is predicted and theoretically described. Within the framework of the continual theory of elasticity the estimation of both average values of deformation tensor and its variations in solid dielectric with non-uniform random distribution of ions was carried out. It was shown that three possible types of medium modiflcation can arise in solid depending on its ionization degree and the thickness of ionized layer. Since the Coulomb flelds inside ionized layer can be non-uniform, all three types of structuring can arise in solid simultaneously. PACS numbers: 81.07.{b, 81.16.{c

The synthesis of K2BiF5 by a tribochemical reaction is reported. This compound crystallises in a K2SmF5-type arrangement with the lattice parameters a = 11.3862(2), b = 7.5480(1), c = 6.6008(1) Å and space group Pnma. The effect of substituting Bi into the K2SmF5-type structure is discussed in comparison with other compounds considering the effect of the lone-pair activity of Bi3+.

International audience; A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero but equal to the density of the dislo-cations, a measure concentrated on the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Carte-sian maps allow us to consider deformations in L p with 1 ≤ p < 2, as required for dislocation-induced strain singularities. Firstly we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislo-cations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic, and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation , with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

The Zeldovich approximation (ZA) predicts the formation of a web of singularities. While these singularities may only exist in the most formal interpretation of the ZA, they provide a powerful tool for the analysis of initial conditions. We present a novel method to find the skeleton of the resulting cosmic web based on singularities in the primordial deformation tensor and its higher order derivatives. We show that the A_3-lines predict the formation of filaments in a two-dimensional model. We continue with applications of the adhesion model to visualise structures in the local (z < 0.03) universe., 9 pages, 8 figures, Proceedings of IAU Symposium 308 "The Zeldovich Universe: Genesis and Growth of the Cosmic Web", 23-28 June 2014, Tallinn, Estonia

The Press-Schechter theory of the cosmological mass function and its modifications allow to constraint cosmological scenarios of the structure formation. Recently a few new models have been suggested that explored the influence of anisotropic collapse on the shape of the mass function. I discuss in more detail a particular model that assumes that a fluid particle becomes a part of a gravitationally bound halo when the smallest eigenvalue of the deformation tensor of the filtered initial density field reaches a certain threshold (like the filtered density contrast reaches the threshold in the Press-Schechter formalism). Choosing the smallest eigenvalue guarantees that the fluid particle in question experiences collapse along all three axes. The model shows a better agreement with the N-body simulations than the standard Press-Schechter model., Talk at 15th Florida Workshop in Nonlinear Astronomy and Physics "The Onset of Nonlinearity", 17-19 February 2000, Gainesville, FL (12 pages, 5 figures)

The dependence of the glass transition on the local volumetric strain of each atom has been shown by Egami and others. The atomistic strain defined by the deformation tensor of the Voronoi polyhedra exhibits a different tendency from that of the total strain of the whole system. Even below the glass transition temperature, some percent of the atoms have strains that are already over the critical strain predicted by theory. In addition, the atomic deviatoric distortion has a non-zero value, while the total deviatoric distortion remains zero at the glass transition in the heat cycle. In this paper, the strain states at the glass transition, such as the volumetric strain and deviatoric distortion, are investigated for Cu and Zr based amorphous metals with three different compositions. We found that the local glass transition depends on the deviatoric distortion of atoms and, in these cases, starts at 600 K, a lower temperature, than that of the global glass transition at 1100 K.

In the context of finite deformations, a purely kinematical treatment of Volterra dislocations is presented. These dislocations correspond to deformations that have a jump discontinuity across a singular surface in a fixed reference configuration of the continuum, but whose right Cauchy-Green deformation tensor field is continuous and has second partial derivatives that are continuous. Analytical and geometrical proofs of Weingarten's theorem for finite deformations are given. Tune-dependent Volterra dislocations are also considered.

In this note we extend integrability conditions for the symmetric stretch tensor $U$ in the polar decomposition of the deformation gradient $\nabla\varphi=F=R\,U$ to the non-symmetric case. In doing so we recover integrability conditions for the first Cosserat deformation tensor. Let $F=\bar R\,\bar U$ with $\bar R:\Omega\subset\mathbb{R}^3\longrightarrow\mathrm{SO}(3)$ and $\bar U:\Omega\subset\mathbb{R}^3\longrightarrow \mathrm{GL}(3)$. Then $\mathfrak{K}:={\bar R}^T\mathrm{Grad}\,{\bar R}=\mathrm{Anti}\Big( \frac{1}{\mathrm{det} \bar U}\Big[\bar U(\mathrm{Curl} \bar U)^T-\frac{1}{2} \mathrm{tr}(\bar U(\mathrm{Curl} \bar U)^T) 1\!\!1 \Big]\bar U\Big),$ giving a connection between the first Cosserat deformation tensor $\bar U$ and the second Cosserat tensor ${\mathfrak{K}}$. (Here, Anti denotes an isomorphism between $\mathbb{R}^{3\times 3}$ and $\mathfrak{So}(3):=\{\,\mathfrak{A}\in\mathbb{R}^{3\times 3\times 3}\,|\,\mathfrak{A}.u\in\mathfrak{so}(3)\;\forall u\in \mathbb{R}^3\}$.) The formula shows that it is not possible to prescribe $\bar U$ and $\mathfrak{K}$ independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy) we show existence of minimizers in Sobolev-spaces., Comment: 17 pages. arXiv admin note: text overlap with arXiv:1410.4225

In this paper a stationary action principle is proven to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments, for instance by Cahn and Hilliard. Remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen. In general continua whose deformation energy depend on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second gradient) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both material and spatial description and the corresponding Euler-Lagrange bulk and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and grad C or on C^-1 and grad C^-1 ; where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal or Seppecher for an alternative deduction based on thermodynamic arguments) are recovered. A version of Bernoulli law valid for capillary fluids is found and, in the Appendix B, useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to continuum analytical mechanics are also presented. In this context the reader is also referred to Capecchi and Ruta., Comment: 52 pages

This study presents results of numerical simulations of the flow of fibre suspensions in a Newtonian fluid through a 4:1 planar contraction. Two approaches are adopted to determine the fibre orientation. The first one uses orientation tensors defined as dyadic products of the orientation vector, while the second one is based on the fibre aligned assumption. An implicit time discretization scheme and a mixed finite element method based on the introduction of the rate of deformation tensor as an additional unknown are used to obtain the steady-state flow. The numerical technique we use allows us to examine the flow of fibre suspensions in both dilute and semi-dilute regimes at high values of the parameters controlling inertial and fibre effects. The predicted flow patterns and fibre orientation are discussed, and a systematic comparison between the predictions of the two approaches is presented. On presente dans cette etude les resultats de simulations numeriques de l'ecoulement de suspensions de fibres dans un fluide newtonien dans une contraction planaire 4:1. Deux approches sont adoptees pour determiner l'orientation des fibres. La premiere recourt aux tenseurs d'orientation definis comme des produits dyadiques du vecteur d'orientation, tandis que la seconde repose sur l'hypothese des fibres alignees. Pour obtenir l'ecoulement a l'etat permanent, on emploie un schema de discretisation du temps implicite et une methode d'elements finis mixte s'appuyant sur l'introduction de la vitesse du tenseur de deformation comme inconnue supplementaire. La technique numerique que nous utilisons permet d'examiner l'ecoulement des suspensions de fibres autant en regime dilue que semi-dilue a des valeurs elevees des parametres controlant les effets de l'inertie et des fibres. Les profils d'ecoulement et l'orientation des fibres predits sont etudies, et une comparaison systematique entre les predictions de ces deux approches est presentee.

Nonspherical collapse is modelled, under the Zeldovich approximation, by six-dimensional random walks of the initial deformation tensor field. The collapse boundary adopted here is a slightly-modified version of that proposed by Chiueh and Lee (2001). Not only the mass function agrees with the fitting formula of Sheth and Tormen (1999), but the bias function and conditional mass function constructed by this model are also found to agree reasonably well with the simulation results of Jing (1998) and Somerville et al. (2000), respectively. In particular, by introducing a small mass gap, we find a fitting formula for the conditional mass function, which works well even at small time intervals between parent and progenitor halos during the merging history., Comment: 28 pages, 11 figures, section 3.2 complemented, Fig.1 and Fig. 4 modified, version accepted for publication in ApJ

We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motions of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of the displacement gradient and the existence of stress functions on non-contractible bodies. We also derive the local compatibility equations in terms of the Green deformation tensor for motions of $2$D and $3$D bodies, and shells in curved ambient spaces with constant curvatures.

Horizontalieji ir vertikalieji Žemės plutos judesiai tarpusavyje susiję, nes jie yra tų pačių Žemės plutą veikiančių jėgų pasekmė. Taip pat yra abipusis ryšys tarp Žemės plutos deformacijų ir tektoninių įtampų sklaidos. Todėl pagal išmatuotas vertikaliųjų Žemės plutos deformacijų reikšmes galima įvertinti horizontaliųjų deformacijų rodiklius, nustatyti jų kaitos dėsningumus ir atvirkščiai. Straipsnyje pateikta nustatytoji analitinė sąsaja tarp horizontaliųjų bei vertikaliųjų Žemės plutos deformacijų. Sąsajoms nustatyti taikyta vektorinė bei tenzorinė analizė. Pateiktas algoritmas horizontaliųjų Žemės plutos deformacijų rodikliams įvertinti pagal vertikaliųjų judesių matavimo rezultatus. Deformacijų rodikliai įvertinti baigtinių elementų metodu. Pateiktas parengto algoritmo taikymo skaitinis pavyzdys. Horizontal and vertical movements of the Earth crust are interrelated, because the same forces affecting the Earth crust cause them. There is a both-side interrelation between deformations of the Earth crust and spreading of tectonic tensions. So, it is possible to assess the indicators of horizontal deformations and to establish the regularities of their changes by examining the measured values of vertical deformations of the Earth crust, and vice versa. The established analytical interrelation between horizontal and vertical deformations of the Earth crust is described in the paper. The interrelation has been found by application the vector and tensor analysis. An algorithm for assessing the indicators of horizontal deformations of the Earth crust according to the results of measuring the vertical movements is provided. The method of finite elements is applied for assessment the indicators of deformations. A numerical example of application of the developed algorithm is provided as well. Горизонтальные и вертикальные движения земной коры междусобой тесно связаны, ��аккак они являются следствием воздействия на земную кору тех же сил. Поэтому должна быть возможность по результатам измеренных вертикальных движений земной коры определять также показатели горизонтальных деформаций, их закономерности. В статье приведена полученная аналитическая связь между горизонтальными и вертикальными движениями земной коры. Для определения связи применены векторный и тензорный анализы. Приведен алгоритм оценки показателей горизонтальных движений земной коры по результатам измерения вертикальных движений. Для оценки показателей деформаций применен метод конечных элементов. Приведен численный пример.

We discuss the relation between three recent approaches of describing the dynamics and the spatial distribution of particles suspended in turbulent flows: phase-space singularities in the inertial particle dynamics (caustics), real-space singularities of the deformation tensor, and random uncorrelated motion. We discuss how the phase- and real-space singularities are related. Their formation is well understood in terms of a local theory. We discuss implications for random uncorrelated motion. Our results are supported by results of direct numerical simulations of inertial particles in model flows., 20 pages, 9 figures, as published