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

This 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.

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

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

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.

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 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.

A structure-based constitutive equation for rubberlike networks is proposed. It is obtained by combining the Langevin-statistics-based theory of Arruda and Boyce (AB) with a term based on the first invariant of the generalized deformation tensor which follows from some theoretical treatments of the constraint effect. The combined (ABGI) four-parameter strain–energy function has been found to give (with the exception of the very-low strain region) a very good description (deviations

Five molecular models of rubber elasticity which employ different treatments of entanglement effects (the Kloczkowski−Mark−Erman diffused-constraint model, the Edwards−Vilgis (E−V) slip−link model, the tube models of Gaylord−Douglas (G−D), Kaliske−Heinrich, Rubinstein−Panyukov versions) are assessed using biaxial deformation data for an entanglement-dominated network of end-linked poly(dimethylsiloxane) (PDMS) in which trapped entanglements are dominant in number relative to chemical cross-links. The theoretical stress−strain relations were calculated from the elastic free energy (W) of each model. Using the reduced stress (the nominal stress divided by equilibrium modulus Go), the strain-dependent predictions of each model were tested from two different viewpoints, i.e., the dependence of the reduced stresses on the principal ratio and the Ii dependence of (∂W/∂Ij)/Go (i,j = 1,2), where I1 and I2 are the first and second invariants of deformation tensor (the Rivlin−Saunders method). The diffused-constrai...

The phenomenological strain energy density function (W) for the elastomeric networks of end-linked poly(dimethylsiloxane) (PDMS) has been investigated as a function of the first and second invariants I1 and I2 of the Green's deformation tensor on the basis of the quasi-equilibrium stress−strain relationships of general biaxial deformations varying independently each of two principal strains. The Ii dependence of ∂W/∂Ij (i,j = 1,2) was obtained from the biaxial stress−strain data using the Rivlin−Saunders method. In the 3-dimensional plots of ∂W/∂Ii (i = 1,2) against both the (I1 − 3)- and (I2 − 3)-axes, the data points of each derivative at large deformations appear to fall on a plane inclining against the (I1,I2) plane, which suggests that both the derivatives linearly depend on each of I1 and I2. The formula of W is reasonably deduced from such linear dependence of ∂W/∂Ii on Ij (i,j = 1,2) as W = C10(I1 − 3) + C01(I2 − 3) + C11(I1 − 3)(I2 − 3) + C20(I1 − 3)2 + C02(I2 − 3)2. Each of the numerical coeffic...

We construct a Ginzburg-Landau model for gels undergoing spinodal decomposition in terms of the polymer volume fraction and a deformation tensor. We numerically demonstrate that the domain growth is extremely slowed down in late stages, where the surface tension force, which drives the coarsening in usual fluids, is cancelled by the elastic force. The patterns closely resemble those observed in highly viscoelastic, asymmetric binary mixtures.

Elastic properties of real elastomers were examined by using partial derivatives of the strain energy density function (W) with respect to the invariant (I i , i=1,2,3) of deformation tensor. The values of the derivatives at zero strain limit for styrene-butadiene rubber (SBR), nitrile-butadiene rubber (NBR) and butadiene rubber (BR) were compared with those derived from the theory shown in the previous paper, where compressibility of the materials was taken into account. The asymptotic behavior of the derivatives for isoprene rubber (IR) and segmented polyurethaneurea (SPU) was also compared with those of the three types of rubbers. It was shown that the theoretical predictions for the limiting value agreed fairly well with the experimental values of the elastomers.

In Chapter I, we described properties of the stressed state of a medium and indicated two reasons for appearing or changing stresses. The first reason is the reaction of the medium to the change of shape of its elements. Such stresses are said to be elastic (cf. Sections 4 and 8). The second reason is the internal friction between neighboring elements of the medium. Such stresses expressed through the strain rate (cf. Section 5) are said to be viscous.

It has so far been believed that phase separation in isotropic condensed matter can be classified into either solid or fluid models, including some modification due to elastic effects. Contrary to this common belief, we recently found unusual phase separation, which can be described by neither of the above models, in a mixture whose components have a large difference in their dynamics. Thus, we proposed that a new model of phase separation, namely, “viscoelastic model”, is necessary to describe phase separation in such dynamically asymmetric mixtures. This model is likely a general model that can describe all types of isotropic phase separation including solid and fluid model as special cases. Viscoelastic phase separation in dynamically asymmetric mixtures can be characterized by the order-parameter switching phenomena: The primary order parameter switches from the composition to the deformation tensor as in gel, and back to the composition again, reflecting viscoelastic relaxation between a characteristic deformation time of phase separation and the slowest rheological time of the system. This unusual behavior can be explained by the above general nature of a viscoelastic model. It is remarkable that orderparameter switching can occur even during an ordering process driven by a ‘single’ thermodynamic driving force. We argue that spongelike patterns observed in phase separation of many materials may be produced by a common physical mechanism of viscoelastic phase separation. For example, spongelike patterns formed during polymerization-induced phase separation can also be explained by our viscoelastic model.

The mechanical relaxation of strained semiconducting stripes is studied. The deformation tensor is calculated using a classical approach of elasticity problem with predeformations; the electronic band structure is then simulated using an 8 band kp model including strain. To confirm the models developed, compressively and tensely strained stripes were fabricated and characterized by photoluminescence measurements. Theoretical and experimental results are in very good agreement and show the importance of mechanical anisotropic relaxation phenomena in optoelectronic devices like waveguide structures, modulators, or lasers.

It has frequently been observed that some elastomeric networks show an abrupt increase in the nominal stress, and consequently the modulus [f*], at high elongations.1 Molecular interpretation of this upturn has generally been attributed to strain-induced crystallization.

As analysis of crystal structure stability against homogeneous deformation is carried out for simple and transition metals with body-centered cubic (bcc) and face-centered cubic (fcc) lattices under pressure. It is shown that at high compression ratios a crystal lattice can turn unstable against shear deformations, leading to new lower-symmetry structures. To analyze these structures, the Landau theory of phase transitions with finite deformation tensor components as order parameters is applied. A comparison is made with experimental data on high-pressure phases. Prospects are discussed for using the suggested approach to explain structural transformations in solids at high pressures.

The behaviour of Oldroyd and Walters-Pre dries on modified models is analysed. To provide the siutable description for shearing and elongational flows it is convinient to use viscosity, relaxation time and retardation time that depend on the second and third invariants of the rate of deformation tensor. The introduction of the third invariant of the rate of deformation tensor in rheological equations is substantiated.

加硫SBRの二軸クリープ試験を行い, その力学的特性について実験的に考察した. その結果, 物質特性値であるひずみエネルギ密度関数を変形テンソルの不変量の関数としたとき線形形式の近似式を誘導し, その有効性を確認するとともに定数の意味についても検討した. またひずみエネルギ密度こう配関数を時間の関数としたとき加硫度のみに依存する指数関数形式の近似式を誘導し, 任意の時間での変形量が推定できることを示した.

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].

A noncontacting experimental procedure has been used for characterizing the high-temperature large-strain uniaxial behavior of polycarbonate (PC), polyetherimide (PEI), and poly(butylene terephthalate) (PBT), through controlled tests on flat specimens cut from extruded sheet. While each of these polymers exhibits stable necking over a broad temperature range, the transition to the necked state is shown to be more gradual at higher temperatures, resulting in a less sharply defined neck transition region during the drawing process. These trends in necking behavior, and other phenomena such as double necking, which only occur at high temperatures, are illustrated through contour plots of the Cauchy-Green deformation tensor. Elevated temperature true-stress versus stretch data are given for each of these polymers.

General relations were derived by expanding the strain-energy density function in terms of the invariants of the deformation tensor. Some constitutive equations obtained by keeping a third term in the expanson in addition to the two terms retained in the Mooney-Rivlin equation were tested in the light of currently available experimental data. It is shown that by the retention of the third term the upswing in the Mooney stress at low values of γ−1 is successfully predicted, and the stress—strain behavior can be described with excellent accuracy up to break, even in carbon black-filled rubber which is difficult to describe by the Mooney-Rivlin equation.

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 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.

For a long time researchers have shown interest in the measurements of large deformations on the surface of an object.

The molecular model of an elastomeric network with local intermolecular correlations, given by Flory, is used to calculate the components of the molecular deformation tensor and molecular orientation. Effects of molecular parameters such as severity of entanglements, network inhomogeneities and conditions during cross-linking are discussed. Components of molecular deformation and orientation are calculated for a network under uniaxial stress.

It is well-known that the constitutive equation of elastic materials not supporting tension has a unique solution ([2],[5]). For isotropic materials it is easy to calculate the solution for the constitutive equation, as the material’s elastic constants and the deformation tensor vary. On the contrary,the study of anisotropic materials is more complex. The aim of this paper is to propose a method for calculating the solution for transversely isotropic materials,which can be used in the case of plane strain state.

Determination of segmental orientation in stretched rubbery networks may serve to test the validity of elaborated models of rubberlike elasticity, in conjunction with stress-strain measurements.