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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.

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.

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.

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.

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

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 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)

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

It is shown that the optical excitation of impurities produces a homogeneous deformation of crystal lattice which induces a macroscopic polarization in polar crystals due to the direct piezoelectric effect. The deformation tensor and corresponding macroscopic polarization of some polar crystals upon the optical excitation of impurities are calculated.

Turbulent flows are considered as random motions, the velocity v and the pressure p being random variables. After having defined a clear probabilistic framework, we look for equations satisfied by the expectations \(\overline{\mathbf{v}}\) and \(\overline{p}\) of v and p, which yield the Reynolds stress \(\boldsymbol{\sigma }^{(R)} = \overline{\mathbf{v}^{\prime} \otimes \mathbf{v}^{\prime}}\). In keeping with the Boussinesq assumption, we write \(\boldsymbol{\sigma }^{(R)}\) in terms of the mean deformation tensor \(D\overline{\mathbf{v}}\) and the eddy viscosity ν t , which must be modeled. The modeling process leads to the assumption that ν t depends on the turbulent kinetic energy \(k = (1/2)\overline{\vert \mathbf{v}^{\prime}\vert ^{2}}\) and the turbulent diffusion \(\mathcal{E} = 2\nu \overline{\vert D\mathbf{v}^{\prime}\vert ^{2}}\), addressing the issue of finding equations for k and \(\mathcal{E}\) to compute ν t , and therefore \((\overline{\mathbf{v}},\overline{p})\). To realize this, hypotheses about turbulence are necessary, such as local homogeneity, expressed by the local invariance of the correlation tensors under translations. The concept of mild homogeneity is introduced, which is the minimal hypothesis about correlations that allows the derivation of the \(k -\mathcal{E}\) model carried out in this chapter, using additional standard closure assumptions.

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.

We shall now proceed to a theoretical study in which we shall emphasize geometrical aspects of the problems; these problems will be dealt for a fixed t or taking into account the time too [7].

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

In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field., 22 pages

Let us consider a continuum \(\mathcal{B}\). Due to Axiom2, at time t=0 there is a one-to-one correspondence between every material point \(\mathcal{M}\in \mathcal{B}\)and its radius-vector \(\mathop{\mathbf{x}}^{\circ } =\overrightarrow{ O\mathcal{M}}\)in a Cartesian coordinate system \(O\bar{{\mathbf{e}}}_{i}\). Denote Cartesian coordinates of the radius-vector by \(\mathop{x}^ {\circ }{}^{i}\)(\(\mathop{\mathbf{x}}^{\circ } =\mathop{ x}^ {\circ }{}^{i}\bar{{\mathbf{e}}}_{i}\)) and introduce curvilinear coordinates X i of the same material point \(\mathcal{M}\)in the form of some differentiable one-to-one functions $$\mathop{x}^ {\circ }{}^{i} =\mathop{ x}^ {\circ }{}^{i}({X}^{k}).$$ (2.1)

We prove that the stress tensor, tau^{ab}, of a molecular system with arbitrary, short-range interactions can be point-wisely expressed as the functional derivative of the partition function with respect to the local deformation tensor. In this approach the set of components of tau^{ab} has a simple interpretation as the set of Lagrangian multipliers which need to be introduced to enforce the conditions relating point particle displacements to the body local deformation tensor. The question of the non-uniqueness of the formula for tau^{ab} is discussed., Comment: 8 pages

In the current section an overview of different Finite Element (FE) formulations used for sheet metal forming simulation, now and in the past, will be given. The theories of FE-simulation of large deformation problems will be briefly touched upon herein, but for thorough presentations the reader is referred to the many existing text books on the subject, e.g. Belytschko et al. [1], Zienkiewicz and Taylor [2] and Crisfield [3]. The object is rather to give a general understanding of the advantages and disadvantages of the various methods in use. Review articles on sheet forming simulation and comparative studies of different FE-procedures are found in e.g. Honecker and Mattiasson [4], Onate and Agelet de Saracibar [5], Onate et al. [6], Mattiasson [7], Kawka et al. [8], Wenner [9], Wang et al. [10], Mattiasson [11], Makinouchi [12] and Wenner [13]. State-of-the-art articles on the utilization of sheet forming simulation today, and outlooks against the future are given in Banabic and Tekkaya [14] and Roll and Weigand [15].

We generalize Doroshkevich's celebrated formulae for the eigenvalues of the initial shear field associated with Gaussian statistics to the local non-Gaussian f_{nl} model. This is possible because, to at least second order in f_{nl}, distributions at fixed overdensity are unchanged from the case f_{nl}=0. We use this generalization to estimate the effect of f_{nl}\ne 0 on the abundance of virialized halos. Halo abundances are expected to be related to the probability that a certain quantity in the initial fluctuation field exceeds a threshold value, and we study two choices for this variable: it can either be the sum of the eigenvalues of the initial deformation tensor (the initial overdensity), or its smallest eigenvalue. The approach based on a critical overdensity yields results which are in excellent agreement with numerical measurements. We then use these same methods to develop approximations describing the sensitivity of void abundances on f_{nl}. While a positive f_{nl} produces more extremely massive halos, it makes fewer extremely large voids. Its effect thus is qualitatively different from a simple rescaling of the normalisation of the density fluctuation field \sigma_8. Therefore, void abundances furnish complementary information to cluster abundances, and a joint comparison of both might provide interesting constraints on primordial non-Gaussianity., Comment: 16 pages, 8 figures, match version accepted by MNRAS

We study properties of stable, strictly stable and locally outermost marginally outer trapped surfaces in spacelike hypersurfaces of spacetimes possessing certain symmetries such as isometries, homotheties and conformal Killings. We first obtain results for general diffeomorphisms in terms of the so-called metric deformation tensor and then particularize to different types of symmetries. In particular, we find restrictions at the surfaces on the vector field generating the symmetry. Some consequences are discussed. As an application we present a result on non-existence of stable marginally outer trapped surfaces in slices of FLRW., Comment: 23 pages, 3 figures

Whereas the order parameter of a phase transition is transformed either as the components of the electric or magnetic polarisation or as a linear combination of the deformation tensor components, the velocities of the electromagnetic waves and of sound, which propagate in a definite symmetry direction, become zero when approaching a critical point. A rather large class of such phase transitions of the second kind, the order parameter of which is transformed simultaneously as the electric and magnetic polarisations or as both polarisation and the components of the deformation tensor, is studied. In the latter case the bound electromagnetic and sound waves stand for softening hydrodynamic modes. It is shown that at such 'combined' phase transitions qualitatively new effects arise in the behaviour of the softening hydrodynamic waves. The strongest anomalies are observed at magnetoelectric phase transitions.

Now let us consider the application of tensors and tensor functions to present-day continuum mechanics. This science so widely uses the tensor theory that practically every text-book on continuum mechanics starts with introducing into tensor algebra and analysis.

Relations for the temperature and frequency dependence of the linear elastic constants for crystals of arbitrary symmetry near a second-order structural phase transition are derived on the basis of a thermodynamic potential including time and space (one-, two-, or three-dimensional) fluctuations of the order parameter. The calculation is based on a perturbation expansion with respect to the small deformation tensor components and is given in detail for a two-component order parameter. The results are used for the explanation of the elastic constant anomalies of GMO and Hg2Cl2. The influence of the higher-order interaction terms above or below the phase transition is investigated with an application to the case of Hg2Cl2. Des expressions pour les variations des constants elastiques des cristaux n'import quelle symetrie en fonction de la temperature et de la frequence au voisinage d'une transition structurelle du second ordre sont derivees a la base des fluctuations de temps et d'espace (d'une, de deux ou de trois dimensions) du parametre d'ordre avec utilisation du developpement de perturbation d'apres des elements du tenseur de deformation. Ces expressions sont liees avec des coefficients de developpement du potentiel thermodynamique. Le calcul detaille est fait pour le parametre d'ordre de deux elements et les resultats sont utilises pour l'explication des variations critiques des constantes elastiques de GMO et de Hg2Cl2. L'influence des termes de couplage d'ordre plus haut (au-dessus ou au-dessous de la transition) est examinee et elle est appliquee dans le cas de Hg2Cl2.

Approximate wavefunctions and energies of the ground state of the shallow acceptor in Ge were obtained by solving the effective mass equation possessing terms proportional to the deformation tensor components. The variational approach was used involving trial wavefunctions similar to those of D Schechter, but possessing transformation properties required by the symmetry of the uniaxially deformed crystal. Results were obtained for the stresses X in the (100) directions, varying from -600 MPa to +600 MPa. At X to 0, where the energies of the impurity and the band states are expressible through the deformation potential constants (respectively b' and b), it is found that the value of b'/b is 0.55 and for sufficiently large X, b'/b to 1, in agreement with experiments. The calculated energies were further used for finding the solutions of the effective mass equation for large distances from the impurity nucleus. The solutions have two components with different spatial extent and anisotropies, and they are traceable to the light and heavy hole bands of Ge. Their behaviour with stress explains some of the observed features of the hopping piezoresistance in p-type Ge.

In the present work we want to show how to get in an operative way the rela tive standard quantities already obtained geometrically by means of the projection method. Methods of measurement of the foregoing standard quantities are therefore shown: the spatial metric tensor, the deformation tensor, the vortex spatial tensor, the gravitational standard field vector, the standard time, the relative standard velocity, the standard mass, the whole standard energy.

Summary. Each particle of a continuum is assigned a second order tensor which is taken as a measure of the deformation of some neighborhood of the particle, and which is determined by a functional depending on the configurations of that neighborhood. Two invariance restrictions are imposed on the functional whose values are spatial strain tensors, that is, associated with the deformed configuration. The first requirement is that a time shift and rigid transformation of the deformed configuration leave the spatial deformation tensor unaltered relative to it. The second requires that if particles of distinct continua undergo the same deformation, the corresponding deformation tensors should be the same. For the special case in which the functional depends on the deformation in the smallest neighborhood of a particle, the restrictions imply that the deformation tensors associated with the deformed and reference configurations arc isotropic functions of the ]eft and right CAITClVZGREEN tensors, respectively. Zusammenfassung. Jedem Teilchen eines Kontinuums wird ein Tensor zwciter Stufe als Mal3 fiir die Deformation einer gewissen Nachbarschaft dieses Teilchen zugeordnet, der durch ein Funktional bestimmt wird, das yon der I~onfiguration dieser Nachbarschaft abh~Lng~. Zwei Invarianzbcdingu ngen werden diesem Funktional, dessen Werte raumliche Verzerrungstenso ren darstellen, auferlegt, und zwar im Hinblick auf die deformierte Konfiguration. Die erste Forderung besagt, dal3 eine Zeitverschiebung und eine starre Transformation der deformierten ]~onfiguration den rgumlichen Verzerrungstenso r im Hinblick auf diese unge~ndert lassen. Die zweite Einschr~nkung besag% dal3 entsprechende Deformationstens oren von Partikcln verschiedener Kontinua, die dieselbe Verformung erlitten haben, gleich scin sollen. Im Spezialfall, dal3 die Funktionale nur yon der Deformation in der n~ichsten Umgebung des Partikels abh~Lngen, beinhalten die Einschrgnkungcn die Aussage, daf3 die mit dem deformierten ned dem undcformierten Zustand verkniipften Deformationstensoren nur isotrope Funktionen des linken und des rechten CAIse~u GREEN Tensors sein k6nnen.

A suitable framework allowing to describe yielding and failure of anisotropic materials with the required generality and pertinence is furnished by the theory of representations for tensor functions. In this Chapter, invariant formulations of the plastic behavior and failure of transversely isotropic solids are developed. For other material symmetries, the corresponding constitutive relations can be derived in a similar manner.

In this introduction we shall be concerned with a simple approach to tensor algebra, and the tensor formulation of stress, deformation and energy, leading to Hooke’s law, the properties of anisotropy, and considerations of elastic symmetry.

Consider a tetrahedron, centroid P, of continuous material, isolated in thought from the material which surrounds it. Let the lengths of the edges be infinitesimal of order l.

The deformation of a plane elastic flat body is characterized by a tensor (deformation tensor): $${\varepsilon _{Ij}} = \frac{1}{2}\left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} = \frac{{\partial {u_j}}}{{\partial {x_i}}}} \right),i,j = 1,2. $$ (5.1.1)

A body is deformed when forces are applied to it; the distances between points in the body subject to the forces are somewhat different from those between the same points in the absence of the forces . The change in distance is due to displacement of one relative to another during the deformation. Let r0 be the radius vector of some point in the body (relative to some fixed point in space) before the deformation, and let r(xi) be the same after deformation. The difference between these vectors is the displacement vector of the point and is (1.1) or in coordinate form (1.2)

The paper deals with the determination of tensor of the so-called small deformation of an elementary area of an object surface by means of an optical experimental method, which makes use of properties of the speckle field in an optically free space or in an image field. The term “small deformation tensor” is introduced and a relationship between a small deformation tensor of surface and a speckle field in free space or image field is analyzed.

A component of the deformation tensor has been measured for part of the Fermi surface of tin and the corresponding attenuation coefficient for longitudinal ultrasonic waves has been deduced.