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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.

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.

Tensor-based morphometry (TBM) studies encode the anatomical information in spatial deformations which are locally characterized by Jacobian matrices. Current methods perform voxel-wise statistical analysis on some features, such as the Jacobian determinant or the Cauchy---Green deformation tensor, which are not complete descriptors of the local deformation. This article introduces a right-invariant Riemannian distance on the GL+(n) group of Jacobian matrices making use of the complete geometrical information of the local deformation. A numerical recipe for the computation of the proposed distance is given. Additionally, experiments are performed on both a synthetic deformation study and a cross-sectional brain MRI study.

Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term.

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

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.

The crystal chemistry of many AB2O6 compounds is summarized in family trees comparing the crystal symmetry of several structures in group-subgroup relations. In this second part of a series the largest family derived from a hexagonally close packed arrangement of anions is described. On reducing the symmetry, different choices of octahedral sites for A and B cations and voids lead to four branches – the Rosiaite-, the ZnTa2O6-, the columbite- and the rutile-branch. We trace the reasons for the specific distribution patterns of the cations in the individual compounds by comparing long range contributions, i.e. the Madelung Part of Lattice Energy (MAPLE), and local effects represented by bond valence calculations according to I. D. Brown. Long range effects largely determine the choice of sites in the respective space groups, however, small local adjustments play a decisive role for the stability of a specific structural modification. Such mechanisms ultimately impose how the structure of an individual compound deviates from an ideally packed arrangement. We discuss the difference between the ideal structures and the real examples given in the family tree by defining a sort of deformation tensor and a measure of similarity, and we describe and depict the anisotropic deformation by a strain tensor.

In this paper, we get a blow-up criterion for the strong solutions to the 3D viscous liquid–gas two-phase flow model with vacuum. More precisely, we prove that the bound of L t 1 L x ∞ norm of the deformation tensor of velocity gradient controls the possible breakdown of the strong solutions.

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.

In the present paper, the local well-posedness of the incompressible viscoelastic fluid system in the whole space is proved under the following assumption on the initial data: the deformation tensor is Holder continuous and the velocity is Lp integrable, p > d, where d is the space dimension.

Problems in elasticity are described by relations between the displacement field u, the strain or deformation tensor e, and the stress tensor σ. In 2D elasticity and if volume forces are absent, one may also consider an elastic potential. We explain these settings for a domain Ω with finitely many holes. On such domains the problem for the displacement and the potential problem are not necessarily equivalent. We will also recall the relation between these quantities. We conclude the paper with the potential setting for some specific problems in semi-periodic domains with infinite many holes. Also the setting for non-smooth domains is addressed.

The average stretching direction, local rotation angular, and stretching ratio parameters of molecular trains were used to express the rheology deformation. Based on this micro geometrical deformation, the macro deformation of medium was expressed. Then, using intrinsic elasticity concept, the stress—strain relation was obtained. In this theoretic formulation, the response functions of extension ratio and rotation angular were used to express the rheology feature of medium. For medium composed by incompressible molecular trains, the local rotation angular divides rheology deformation into three kinds: viscoelastic deformation or elasticity enhancement, viscoplastic deformation or elasticity degenerate and constant elasticity range. These results explain the experimental features of rheology deformation well.

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

Wrinkle simulation can reflect the texture and simulation effect of clothes. In this paper, we realize a new cloth wrinkle enhancement method based on the coarse mesh simulation. Firstly, we extend the deformation tensor to generate the mesh vertex stretch tensor, so as to obtain one measuring factor for the three-dimensional cloth mesh vertex deformation. Then, we introduce a new method to extract the wrinkle baseline in the 2D mesh which is stretching from 3D coarse mesh and mapping it to 3D coarse mesh. Finally, we use the convolution surface as framework to simulate the wrinkle shape. We also propose and realize the wrinkle surface adjustment method. Experiment result proves the effectiveness of this method, namely this method can significantly improve the cloth wrinkle effect based on the coarse mesh simulation.

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)

This work is concerned with the study of steady flows of an incompressible viscoelastic fluid of Oldroyd type, with viscosity depending on the second invariant of the rate of deformation tensor in an exterior domain. We establish a result of existence and uniqueness of strong solutions for sufficiently small data and give estimates relating these solutions to those of the corresponding generalized Newtonian fluid.

This work is concerned with the study of steady flows of an incompressible quasi-Newtonian fluid, with viscosity depending on the second invariant of the rate of deformation tensor, in bounded domains. We first establish a result of existence of measure-valued solutions. Next, we prove that under some natural monotonicity conditions, these solutions are weak.

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 this paper, we will study the Oldroyd model describing fluids with viscoelastic properties. Global classical solutions for the two-dimensional incompressible Oldroyd model with small initial displacements are shown to exist via the incompressible limit. The main difficulty is the lack of the damping mechanism on the deformation tensor.

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 paper the intrinsic role of the size of the microstructural representative volume element (RVE) in a second-order computational homogenization is investigated. The presented second-order computational homogenization is an extension of the classical first-order computational homogenization scheme and is based on a proper incorporation of the macroscopic gradient of the deformation tensor and the associated higher-order stress measure into the multiscale framework. The macroscopic homogenized continuum obtained through this scheme is the full second gradient continuum. It is demonstrated with several examples that the size of the microstructural RVE used in a second-order computational homogenization scheme may be related to the length scale of the associated macroscopic homogenized higher-order continuum. It is shown that the analytical second-order homogenization of a microstructurally homogeneous linearly elastic material leads to the second gradient elastic Mindlin's continuum on the macroscale, where the resulting macroscopic length scale parameter is proportional to the RVE size. Several numerical microstructural and multiscale analyses reveal the significance of the contribution of the physical and geometrical nonlinearities in the relation between the RVE size and the calculated macroscopic response. Based on the obtained results, some conclusions are drawn with respect to the choice of the microstructural RVE in the second-order computational homogenization analysis.