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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.

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.

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.

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.

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).

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.

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.

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.

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.

A complete set of f.c.c. rolling texture predictions, obtained with the use of a relaxed constraint model, is presented in this paper. The whole spectrum of textures between the copper and brass types can be found if some components of reaction stresses (between grains and the matrix) are relaxed. The best prediction of the copper type texture is found if ϵ13 is a free parameter and the other components of grain deformation tensor are controlled by reaction stresses. The best prediction of the brass type texture is obtained if ϵ12, ϵ13 and ϵ23 are free parameters. However, a decisive factor for the texture transition from the copper to the brass type is the relaxation of the ϵ12 component. We present for all the considered cases the orientation distribution functions (O.D.F.) and also the pole figures for some selected cases. Nous presentons pour tous les cas les fonctions de distribution d'orientations cristallines ainsi que dans certains cas — les figures de pǒles. Nous presentons un ensemble des previsions des textures c.f.c. de laminage obtenues par le modele avec une relaxation partielle des contraintes de reaction. Une gamme de textures contenues entre celles type cuivre et type laiton peut ětre trouvee si certaines composantes des contraintes de reaction (quiagissent entre les grains et la matrice) sont relaxees. La meilleure prevision de la texture type cuivre est obtenue si ϵ13 est laisse libre et les autres composantes du tenseur de deformation d'un grain sont controllees (par les contraintes de reaction). D'autre part le meilleure prevision de la texture type laiton est trouve si les composantes ϵ12, ϵ13 et ϵ23 sont libres. Neanmoins, un facteur decisif pour la transition de la texture entre celle type cuivre et type laiton est une relaxation de la composante ϵ12.

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.

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.

A phenomenological theory of improper ferroelastic Phase Transition, that superionic hydrogen-bonded Rb3H(SeO4)2, crystals undergo at Tc = 447 K, is presented. The temperature dependences of components of the spontaneous deformation tensor, elastic coefficient tensor, and electric conductivity in the vicinity of phase transition are calculated. A possible microscopic model of order–disorder type of the phase transition in the proton subsystem is discussed briefly. [Russian Text Ignored].

The magnitude of the elasticity tensor of a comparison body remains unclarified if we use a singular approximation [1] to estimate the effective values of the elasticity tensor. Below we will use a parametrix method [2] to determine the first approximation of the random component of the deformation tensor and the effective values of the elasticity tensor, and will also compare the exact solution for one particular heterogeneous and a previously used approximation.

That the statistical method of conditional distributions was productively used to describe uniform and, subsequently, nonuniform systems is indicative of its efficiency, which allows more and more complicated physical effects to be investigated with respect adjustments of the basic idea of the method to every particular situation. So, in this paper the earlier developed statistical theory of nonuniform systems is extended to the case of irregular deformable crystalline media with the aim of concerted account of the contributions due to the real lattice deformation within the framework of the statistical conditional distribution method. Various kinds of defects resulting in the relaxation of the lattice parameters near the defects may be a source of the lattice regularity disturbances. The statistical expression obtained for crystal free energy depends parametrically on the deformation tensor of the medium.

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.

The stress relaxation properties of SBR vulcanizates, those unfilled and those filled with carbon, under orthogonal-biaxial deformation, were subjected to observation in a region from 3.4 to 10 of I1 values where I1 was invariant of deformation tensor, and the contour maps of ∂W/∂I1 and ∂W/∂I2 were obtained as functions of time t. The following results have been obtained.The time dependence of both ∂W/∂I1 and ∂W/∂I2 of the unfiilled SBR vulcanizate are very similar within experimental error, but in the case of the filled SBR, those are quite different. In the filled SBR, the relaxation rate of ∂W/∂I1 decreases with increase in I1, and on the contrary, the rate of ∂W/∂I2 increases with increase in I1. The shape of ∂W/∂I1 surface of the filled SBR is similar to that of the unfilled SBR, but the former is on a considerably higher level than the latter, while the shape of ∂W/∂I2 surface of the filled SBR is different from that of the unfilled SBR.The values of ∂W/∂I1 and ∂W/∂I2 in the uniaxial deformation are obtained by the extrapolating method using the biaxial data. It is shown that the values of ∂W/∂I1 and ∂W/∂I2 in the uniaxial deformation decrease with increase in deformation. In this fact is suggested that strain energy function W cannot be estimated by Mooney-Rivlin plot, and that the constants C1 and C2 obtained from the plot are almost of no significance.

The Taylor theory of polycrystal plasticity was applied to three-axial deformation accomplished by glide on (111) [110] or (110) [111] glide systems. The orientation dependence of the Taylor factor was used to calculate the angular dependence of the relative strength of a textured material. The angular dependence of the strain ratio R — calculated from the minimum value of the Taylor factor — was compared with the measured strain ratio. The orientation changes of the crystallites of a polycrystalline aggregate after 1% plastic deformation were calculated and compared with experimental values. The orientations which are deformed without orientation changes were calculated and compared with those of maximum orientation density found by a three-dimensional texture analysis of cold rolled copper and iron. The dependence of the rotationless orientations on the axis ratio of the deformation tensor was calculated. This allows suggestions to be made on the dependence of the rolling textures on the lateral broadening of the sheet during rolling. Die Taylor-Theorie der Vielkristallplastizitat wurde auf dreiachsige Verformungszustande und Gleitung auf (111) [110]- oder (110) [111]-Gleitsystemen angewandt. Aus der Orientierungsabhangigkeit des Taylor-Faktors wurde die Winkelabhangigkeit der relativen Festigkeit eines Texturmaterials berechnet. Die Winkelabhangigkeit des R-Wertes der plastischen Verformung — berechnet aus dem Minimalwert des Taylor-Faktors — wurde mit den gemessenen R-Werten verglichen. Die Orientierungsanderungen der Kristallite eines vielkristallinen Aggregates nach 1% plastischer Verformung wurden berechnet und mit experimentellen Werten verglichen. Diejenigen Orientierungen, die sich ohne Orientierungsanderung verformen, wurden berechnet und mit den Orientierungen maximaler Orientierungsdichte verglichen, die sich aus der dreidimensionalen Texturanalyse kaltgewalzter Kupfer- und Eisenbleche ergaben. Die Abhangigkeit der drehungsfreien Orientierungen vom Achsenverhaltnis des Deformationstensors wurde berechnet. Daraus ergeben sich Schlusfolgerungen uber die Abhangigkeit der Walztexturen von der relativen Breitung des Bleches wahrend der Walzung.

The effect is considered of non-uniform deformations satisfying the condition λ|∇u αβ |≪|u αβ |, where λ is the free path length of an electron in the undeformed crystal and, u αβ is the deformation tensor, on the electrical conductivity of a metal crystal with alloying elements. Account is taken of the local change of the dispersion law for conductivity electrons and also of the dependence on the coordinates of the alloying element of the electron scattering cross-section on the alloying element in the non-uniformly deformed crystal. An explicit expression is obtained for the tensor of the effective conductivity, first introduced by Herring and connecting the average volume density of the electrical current with the average electrical field in the crystal. The local electron scattering cross-section on the charged alloying elements is evaluated on the Thomas-Fermi model taking into account the local change of the constant shielding. Account is also taken of the possibility of redistribution of the alloying elements in the field of the deformations. The dependence is found and discussed of the conductivity on the sign of the constant, introduced in the continuous approximation, of the interaction of the alloying element with the deformation field. By way of illustration of the results obtained the simple case of unidimensional deformation is considered.

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.