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Journal articles on the topic 'Strain tensor'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 March 2023

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1

Morawiec, A. "On accounting for preferred crystallite orientations in determination of average elastic strain by diffraction." Journal of Applied Crystallography 51, no. 1 (February 1, 2018): 148–56. http://dx.doi.org/10.1107/s1600576718000079.

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Standard diffraction-based measurements of elastic strains in polycrystalline materials rely on shifts of Bragg peaks. Measurement results are usually given in the form of a single tensor assumed to represent the average stress in the material, but the question about the true relationship between the tensor and the average stress generally goes without notice. This paper describes a novel procedure for analysis of data obtained from such measurements. It is applicable in cases when spatial correlations in the material are ignored and statistical information about the polycrystalline specimen i

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2

Bazˇant, Zdeneˇk P. "Easy-to-Compute Tensors With Symmetric Inverse Approximating Hencky Finite Strain and Its Rate." Journal of Engineering Materials and Technology 120, no. 2 (April 1, 1998): 131–36. http://dx.doi.org/10.1115/1.2807001.

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It is shown that there exist approximations of the Hencky (logarithmic) finite strain tensor of various degrees of accuracy, having the following characteristics: (1) The tensors are close enough to the Hencky strain tensor for most practical purposes and coincide with it up to the quadratic term of the Taylor series expansion; (2) are easy to compute (the spectral representation being unnecessary); and (3) exhibit tension-compression symmetry (i.e., the strain tensor of the inverse transformation is minus the original strain tensor). Furthermore, an additive decomposition of the proposed stra

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3

Chandra, N., and Zhiyum Xie. "Development of Generalized Plane-Strain Tensors for the Concentric Cylinder." Journal of Applied Mechanics 62, no. 3 (September 1, 1995): 590–94. http://dx.doi.org/10.1115/1.2895986.

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A pair of two new tensors called GPS tensors S and D is proposed for the concentric cylindrical inclusion problem. GPS tensor S relates the strain in the inclusion constrained by the matrix of finite radius to the uniform transformation strain (eigenstrain), whereas tensor D relates the strain in the matrix to the same eigenstrain. When the cylindrical matrix is of infinite radius, tensor S reduces to the appropriate Eshelby’s tensor. Explicit expressions to evaluate thermal residual stresses σr, σθ and σz in the matrix and the fiber using tensor D and tensor S, respectively, are developed. Si

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4

Boutelier, David, Christoph Schrank, and Klaus Regenauer-Lieb. "2-D finite displacements and strain from particle imaging velocimetry (PIV) analysis of tectonic analogue models with TecPIV." Solid Earth 10, no. 4 (July 15, 2019): 1123–39. http://dx.doi.org/10.5194/se-10-1123-2019.

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Abstract. Image correlation techniques have provided new ways to analyse the distribution of deformation in analogue models of tectonics in space and time. Here, we demonstrate, using a new version of our software package (TecPIV), how the correlation of successive time-lapse images of a deforming model allows not only to evaluate the components of the strain-rate tensor at any time in the model but also to calculate the finite displacements and finite strain tensor. We illustrate with synthetic images how the algorithm produces maps of the velocity gradients, small-strain tensor components, i

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5

Surana, Karan S., and Stephen W. Long. "Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality." Entropy 22, no. 4 (April 14, 2020): 443. http://dx.doi.org/10.3390/e22040443.

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This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders n

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6

Theocaris, P. S., and D. P. Sokolis. "Linear elastic eigenstates of the compliance tensor for trigonal crystals." Zeitschrift für Kristallographie - Crystalline Materials 215, no. 1 (January 1, 2000): 1–9. http://dx.doi.org/10.1524/zkri.2000.215.1.01.

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The spectral decomposition of the compliance fourth-rank tensor, representative of a trigonal crystalline or other anisotropic medium, is offered in this paper, and its characteristic values and idempotent fourth-rank tensors are established, with respect to the Cartesian tensor components. Consequently, it is proven that the idempotent tensors serve to analyse the second-rank symmetric tensor space into orthogonal subspaces, resolving the stress and strain tensors for the trigonal medium into their eigentensors, and, finally, decomposing the total elastic strain energy density into distinct,

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7

Sokolova, M. Yu, and D. V. Khristich. "FINITE STRAINS OF NONLINEAR ELASTIC ANISOTROPIC MATERIALS." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 70 (2021): 103–16. http://dx.doi.org/10.17223/19988621/70/9.

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Anisotropic materials with the symmetry of elastic properties inherent in crystals of cubic syngony are considered. Cubic materials are close to isotropic ones by their mechanical properties. For a cubic material, the elasticity tensor written in an arbitrary (laboratory) coordinate system, in the general case, has 21 non-zero components that are not independent. An experimental method is proposed for determining such a coordinate system, called canonical, in which a tensor of elastic properties includes only three nonzero independent constants. The nonlinear model of the mechanical behavior o

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8

Theocaris, Pericles S., and Dimitrios P. Sokolis. "Spectral decomposition of the linear elastic tensor for monoclinic symmetry." Acta Crystallographica Section A Foundations of Crystallography 55, no. 4 (July 1, 1999): 635–47. http://dx.doi.org/10.1107/s0108767398016766.

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The compliance fourth-rank tensor related to crystalline or other anisotropic media belonging to the monoclinic crystal system is spectrally decomposed for the first time, and its characteristic values and idempotent fourth-rank tensors are established. Further, it is proven that the idempotent tensors resolve the stress and strain second-rank tensors into eigentensors, thus giving rise to a decomposition of the total elastic strain-energy density into non-interacting strain-energy parts. Several examples of representative inorganic crystals of the monoclinic system illustrate the results of t

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9

Moore, J. G., S. A. Schorn, and J. Moore. "Education Committee Best Paper of 1995 Award: Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex." Journal of Turbomachinery 118, no. 4 (October 1, 1996): 622–29. http://dx.doi.org/10.1115/1.2840917.

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Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or three-dimensional glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modeled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal dire

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10

Elata, D., and M. B. Rubin. "Isotropy of Strain Energy Functions Which Depend Only on a Finite Number of Directional Strain Measures." Journal of Applied Mechanics 61, no. 2 (June 1, 1994): 284–89. http://dx.doi.org/10.1115/1.2901442.

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Motivated by studies of low-density materials and fiber-dominated composites, we consider an elastic material whose strain energy function depends only on a finite number of directional strain measures, which correspond to the strains of material fibers in specific material directions. It is shown that an arbitrary set of six distinct direction vectors can be used to define six symmetric base tensors which span the space of all symmetric second-order tensors. Using these base tensors and their reciprocal tensors we develop a representation for the strain tensor in terms of six directional stra

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11

Le Quang, H., and Q. C. He. "The number and types of all possible rotational symmetries for flexoelectric tensors." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2132 (March 2, 2011): 2369–86. http://dx.doi.org/10.1098/rspa.2010.0521.

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Flexoelectricity is due to the electric polarization generated by a non-zero strain gradient in a dielectric material without or with centrosymmetric microstructure. It is characterized by a fourth-order tensor, referred to as flexoelectric tensor, which relates the electric polarization vector to the gradient of the second-order strain tensor. This paper solves the fundamental problem of determining the number and types of all possible rotational symmetries for flexoelectric tensors and specifies the number of independent material parameters contained in a flexoelectric tensor belonging to a

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12

Kozlov, V. V., and A. A. Markin. "FINITE DEFORMATIONS OF A TOROIDAL SHELL." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 71 (2021): 106–20. http://dx.doi.org/10.17223/19988621/71/9.

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The stress-strain state of a nonlinear elastic shell exposed to the internal pressure is considered. A surface of the shell is toroidal in shape in the initial state. The Lagrangian coordinates of the shell are assigned to a cylindrical system. The kinematic characteristics of the process are shown: a law of the motion of points, vectors of a material basis, a strain affinor and its polar decomposition, the Cauchy-Green strain measure and tensor, the Finger measure, and the “left” and the“right” Hencky strain tensors. Neglecting the shear components of the stress tensor, a constitutive relatio

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13

Markin, Alexey, Marina Sokolova, Dmitrii Khristich, and Yuri Astapov. "The Physically Nonlinear Model of an Elastic Material and Its Identification." International Journal of Applied Mechanics 11, no. 07 (August 2019): 1950064. http://dx.doi.org/10.1142/s1758825119500649.

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This work is devoted to the new variant of relations between the energetically conjugated Hencky strain tensor and corotational Kirchhoff stress tensor. The elastic energy is represented as a third-order polynomial of the Hencky tensor containing five material constants. Unlike the Almansi tensor in the Murnaghan model, the Hencky tensor allows assigning a clear physical meaning to material constants. Linear part of the constitutive relation represents the Hencky model and contains the bulk modulus and the shear modulus. The two extra constants express nonlinear effects at a purely volumetric

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14

Mott, Peter H., Ali S. Argon, and Ulrich W. Suter. "The atomic strain tensor." Journal of Computational Physics 101, no. 1 (July 1992): 140–50. http://dx.doi.org/10.1016/0021-9991(92)90048-4.

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15

Freed, Alan D. "Natural Strain." Journal of Engineering Materials and Technology 117, no. 4 (October 1, 1995): 379–85. http://dx.doi.org/10.1115/1.2804729.

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The purpose of this paper is to present a consistent and thorough development of the strain and strain-rate measures affiliated with Hencky. Natural measures for strain and strain-rate, as I refer to them, are first expressed in terms of the fundamental body-metric tensors of Lodge. These strain and strain-rate measures are mixed tensor fields. They are mapped from the body to space in both the Eulerian and Lagrangian configurations, and then transformed from general to Cartesian fields. There they are compared with the various strain and strain-rate measures found in the literature. A simple

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16

Cowin, S. C. "Wolff’s Law of Trabecular Architecture at Remodeling Equilibrium." Journal of Biomechanical Engineering 108, no. 1 (February 1, 1986): 83–88. http://dx.doi.org/10.1115/1.3138584.

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An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff’s law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the stat

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17

Pereverzev, Andrey, та Tommy Sewell. "Elastic Coefficients of β-HMX as Functions of Pressure and Temperature from Molecular Dynamics". Crystals 10, № 12 (10 грудня 2020): 1123. http://dx.doi.org/10.3390/cryst10121123.

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The isothermal second-order elastic stiffness tensor and isotropic moduli of β-1,3,5,7- tetranitro-1,3,5,7-tetrazoctane (β-HMX) were calculated, using the P21/n space group convention, from molecular dynamics for hydrostatic pressures ranging from 10−4 to 30 GPa and temperatures ranging from 300 to 1100 K using a validated all-atom flexible-molecule force field. The elastic stiffness tensor components were calculated as derivatives of the Cauchy stress tensor components with respect to linear strain components. These derivatives were evaluated numerically by imposing small, prescribed finite s

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18

Rahmani, M., V. Nafisi, J. Asgari, and A. Nadimi. "CRUSTAL DEFORMATION IN NW IRAN: INSIGHTS FROM DIFFERENT INVARIANT AND VARIANT COMPONENTS OF GEODETIC STRAIN RATE TENSORS." ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences X-4/W1-2022 (January 14, 2023): 631–37. http://dx.doi.org/10.5194/isprs-annals-x-4-w1-2022-631-2023.

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Abstract. Northwest of Iran, as a tectonically active region, has experienced numerous devastating earthquakes. That is why it is so important to study the earth deformation in this area and to provide more precise insights. So far, most researchers have had the preference of using the invariant component of strain rate tensor for investigating the Earth's shape deformation in the region. However, to examine the efficiency of the variant components of the geodesic strain rate tensor in interpreting deformations of north-western Iran, we have in this article maps of variant components of the ge

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19

Kozlova, O. V., E. P. Zharikova, and A. I. Khromov. "Fields of a Finite Strain Tensor in the Neighborhood of Discontinuity of the Velocity Field of Displacements under Axisymmetric Strain." Materials Science Forum 945 (February 2019): 873–78. http://dx.doi.org/10.4028/www.scientific.net/msf.945.873.

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The problem of the distribution fields of a finite strain tensor in the neighborhood of points of discontinuities of speeds of movements under axisymmetric strain conditions is considered. The Almansi finite strain tensor is a measure of deformation, the motion of points of discontinuities is assumed to be given from the solution of the problems strain bodies taking into account change geometry of the free surface. The relations defining fields of a tensor the finite strains are obtained by integrating the system of equations, binding components of The Almansi finite strain tensor and strain r

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20

Ostrowska-Maciejewska, J., and D. Harris. "Three-dimensional constitutive equations for rigid/perfectly plastic granular materials." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 1 (July 1990): 153–69. http://dx.doi.org/10.1017/s0305004100069024.

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AbstractA three-dimensional constitutive equation governing the flow of an isotropic rigid/perfectly plastic granular material is presented. The equation relates the strain-rate tensor to the Cauchy stress tensor and to the co-rotational rate of the Cauchy stress. It contains scalar functions of the scalar invariants involving the stress, stress-rate and strain-rate tensors together with parameters which characterize the material. The model generalizes the double-shearing model and its relationship to existing theories is demonstrated.

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21

Felício Fuck, Rodrigo, and Ilya Tsvankin. "Analysis of the symmetry of a stressed medium using nonlinear elasticity." GEOPHYSICS 74, no. 5 (September 2009): WB79—WB87. http://dx.doi.org/10.1190/1.3157251.

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Velocity variations caused by subsurface stress changes play an important role in monitoring compacting reservoirs and in several other applications of seismic methods. A general way to describe stress- or strain-induced velocity fields is by employing the theory of nonlinear elasticity, which operates with third-order elastic (TOE) tensors. These sixth-rank strain-sensitivity tensors, however, are difficult to manipulate because of the large number of terms involved in the algebraic operations. Thus, even evaluation of the anisotropic symmetry of a medium under stress/strain proves to be a ch

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22

Абишев, Н. М., Э. И. Байбеков, Б. З. Малкин, М. Н. Попова, Д. С. Пыталев та С. А. Климин. "Деформационное уширение и тонкая структура спектральных линий в оптических спектрах диэлектрических кристаллов, содержащих редкоземельные ионы". Физика твердого тела 61, № 5 (2019): 898. http://dx.doi.org/10.21883/ftt.2019.05.47589.22f.

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AbstractThe procedure of calculation of the spectral line shape in optical spectra of rare-earth ions in crystals with the inclusion of random deformations of an elastically anisotropic crystal lattice caused by point defects is developed. The distribution function of components of the random strain tensor in the case of a low defect concentration is obtained as the generalized six-dimensional Lorentz distribution. The distribution function parameters are represented by the integral functional of the strain tensor components on a sphere of unit radius containing an isotropic point defect in it

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23

Neff, Patrizio, Kai Graban, Eva Schweickert, and Robert J. Martin. "The axiomatic introduction of arbitrary strain tensors by Hans Richter – a commented translation of ‘Strain tensor, strain deviator and stress tensor for finite deformations’." Mathematics and Mechanics of Solids 25, no. 5 (January 28, 2020): 1060–80. http://dx.doi.org/10.1177/1081286519880594.

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We provide a faithful translation of Hans Richter’s important 1949 paper ‘Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Formänderungen’ from its original German version into English, complemented by an introduction summarizing Richter’s achievements.

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24

Yang, Zixuan, and Bing-Chen Wang. "On the topology of the eigenframe of the subgrid-scale stress tensor." Journal of Fluid Mechanics 798 (June 7, 2016): 598–627. http://dx.doi.org/10.1017/jfm.2016.336.

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In this paper, the geometrical properties of the subgrid-scale (SGS) stress tensor are investigated through its eigenvalues and eigenvectors. The concepts of Euler rotation angle and axis are utilized to investigate the relative rotation of the eigenframe of the SGS stress tensor with respect to that of the resolved strain rate tensor. Both Euler rotation angle and axis are natural invariants of the rotation matrix, which uniquely describe the topological relation between the eigenframes of these two tensors. Different from the reference frame fixed to a rigid body, the eigenframe of a tensor

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25

Park, Taehyo, and G. Z. Voyiadjis. "Kinematic Description of Damage." Journal of Applied Mechanics 65, no. 1 (March 1, 1998): 93–98. http://dx.doi.org/10.1115/1.2789052.

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In this paper the kinematics of damage for finite elastic deformations is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. However, the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses. One uses either the hypothesis of strain equivalence or the hypothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a relation between the effective strain and the

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26

Micunovic, Milan. "Some thermodynamic aspects of viscoplasticity of ferromagnetic." Facta universitatis - series: Electronics and Energetics 15, no. 2 (2002): 195–204. http://dx.doi.org/10.2298/fuee0202195m.

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The paper deals with viscoplasticity of ferromagnetic materials. Tensor representation is applied to a set of evolution equations comprising the plastic stretching and residual magnetization tensors. Small magneto elastic strains of isotropic insulators are considered in detail in two special cases of finite as well as small plastic strain. A special emphasis is given to piezomagnetism effects in the case of uniaxial cycling strain. Vakulenko's irreversible thermodynamics is applied to irreversible magnetic phenomena by means of a hereditary evolution function. Purely mechanical as well as pur

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27

Demidov, Dmitry N., Alexander B. Sivak, and Polina A. Sivak. "New Method for Calculation of Radiation Defect Dipole Tensor and Its Application to Di-Interstitials in Copper." Symmetry 13, no. 7 (June 27, 2021): 1154. http://dx.doi.org/10.3390/sym13071154.

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The effect of external and internal elastic strain fields on the anisotropic diffusion of radiation defects (RDs) can be taken into account if one knows the dipole tensor of saddle-point configurations of the diffusing RDs. It is usually calculated by molecular statics, since the insufficient accuracy of the available experimental techniques makes determining it experimentally difficult. However, for an RD with multiple crystallographically non-equivalent metastable and saddle-point configurations (as in the case of di-interstitials), the problem becomes practically unsolvable due to its compl

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28

Xu, You Liang. "Description of Large Deformation Problem Using Means of Visual Strain." Applied Mechanics and Materials 275-277 (January 2013): 16–22. http://dx.doi.org/10.4028/www.scientific.net/amm.275-277.16.

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The constitutive equation of large deformation problem is closely related to geometric description. Nowadays, linear strain tensor is no longer unsuitable to describe large deformation. However, the existing non-linear strain tensors have complicated forms as well as no apparent geometric or physical meaning. While, the increment method is used to solve, however, convergence and efficiency are low sometimes. Thus the idea of visual strain tensor is proposed, with distinct meaning and visual image. Beside, it is likely to be used in engineering measurement, and it can be connected with measured

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29

Gullett, P. M., M. F. Horstemeyer, M. I. Baskes, and H. Fang. "A deformation gradient tensor and strain tensors for atomistic simulations." Modelling and Simulation in Materials Science and Engineering 16, no. 1 (December 13, 2007): 015001. http://dx.doi.org/10.1088/0965-0393/16/1/015001.

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30

Lazar, Markus, and Giacomo Po. "On Mindlin’s isotropic strain gradient elasticity: Green tensors, regularization, and operator-split." Journal of Micromechanics and Molecular Physics 03, no. 03n04 (September 2018): 1840008. http://dx.doi.org/10.1142/s2424913018400088.

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The theory of Mindlin’s isotropic strain gradient elasticity of form II is reviewed. Three-dimensional and two-dimensional Green tensors and their first and second derivatives are derived for an unbounded medium. Using an operator-split in Mindlin’s strain gradient elasticity, three-dimensional and two-dimensional regularization function tensors are computed, which are the three-dimensional and two-dimensional Green tensors of a tensorial Helmholtz equation. In addition, a length scale tensor is introduced, which is responsible for the characteristic material lengths of strain gradient elastic

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31

Yamato, Takahisa. "Strain tensor field in proteins." Journal of Molecular Graphics 14, no. 2 (April 1996): 105–7. http://dx.doi.org/10.1016/0263-7855(96)00022-7.

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Englund, Erin K., Christopher P. Elder, Qing Xu, Zhaohua Ding, and Bruce M. Damon. "Combined diffusion and strain tensor MRI reveals a heterogeneous, planar pattern of strain development during isometric muscle contraction." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 300, no. 5 (May 2011): R1079—R1090. http://dx.doi.org/10.1152/ajpregu.00474.2010.

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The purposes of this study were to create a three-dimensional representation of strain during isometric contraction in vivo and to interpret it with respect to the muscle fiber direction. Diffusion tensor MRI was used to measure the muscle fiber direction of the tibialis anterior (TA) muscle of seven healthy volunteers. Spatial-tagging MRI was used to measure linear strains in six directions during separate 50% maximal isometric contractions of the TA. The strain tensor (E) was computed in the TA's deep and superficial compartments and compared with the respective diffusion tensors. Diagonaliz

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33

Tang, M. X., J. W. Huang, J. C. E, Y. Y. Zhang, and S. N. Luo. "Full strain tensor measurements with X-ray diffraction and strain field mapping: a simulation study." Journal of Synchrotron Radiation 27, no. 3 (April 15, 2020): 646–52. http://dx.doi.org/10.1107/s1600577520003926.

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Strain tensor measurements are important for understanding elastic and plastic deformation, but full bulk strain tensor measurement techniques are still lacking, in particular for dynamic loading. Here, such a methodology is reported, combining imaging-based strain field mapping and simultaneous X-ray diffraction for four typical loading modes: one-dimensional strain/stress compression/tension. Strain field mapping resolves two in-plane principal strains, and X-ray diffraction analysis yields volumetric strain, and thus the out-of-plane principal strain. This methodology is validated against d

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34

Voyiadjis, George Z., and Peter I. Kattan. "Finite Strain Plasticity and Damage in Constitutive Modeling of Metals With Spin Tensors." Applied Mechanics Reviews 45, no. 3S (March 1, 1992): S95—S109. http://dx.doi.org/10.1115/1.3121396.

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The analysis of damage and plastic deformation in metals is very important towards the full understanding of the various damage mechanisms in these materials. A coupled theory of damage mechanics and finite strain plasticity is proposed. The theory is based on a sound mathematical and mechanical background and is thermodynamically consistent. It is formulated using spatial coordinates utilizing a von Mises type yield criterion with both isotropic and kinematic hardening. The derivation is based on the concept of effective stress that was originally proposed by Kachanov [1] for the case of unia

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35

Monchiet, Vincent, and Guy Bonnet. "Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2126 (June 2, 2010): 314–32. http://dx.doi.org/10.1098/rspa.2010.0149.

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In this paper, the derivation of irreducible bases for a class of isotropic 2 n th-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an n th-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2 n th-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain f

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Srivastava, Ankit, and Sia Nemat-Nasser. "Overall dynamic properties of three-dimensional periodic elastic composites." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2137 (September 21, 2011): 269–87. http://dx.doi.org/10.1098/rspa.2011.0440.

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This article presents a method for the homogenization of three-dimensional periodic elastic composites. It allows for the evaluation of the averaged overall frequency-dependent dynamic material constitutive tensors relating the averaged dynamic field variable tensors of velocity, strain, stress and linear momentum. Although the form of the dynamic constitutive relation for three-dimensional elastodynamic wave propagation has been known, this is the first time that explicit calculations of the effective parameters (for three dimensions) are presented. We show that for three-dimensional periodic

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Le Quang, Hung, Qi-Chang He, and Nicolas Auffray. "Classification of first strain-gradient elasticity tensors by symmetry planes." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2251 (July 2021): 20210165. http://dx.doi.org/10.1098/rspa.2021.0165.

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First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with

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38

Palacios, Angel Fierros. "The Small Deformation Strain Tensor as a Fundamental Metric Tensor." Journal of High Energy Physics, Gravitation and Cosmology 01, no. 01 (2015): 35–47. http://dx.doi.org/10.4236/jhepgc.2015.11004.

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Gao, X. L., and H. M. Ma. "Strain gradient solution for Eshelby’s ellipsoidal inclusion problem." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2120 (March 17, 2010): 2425–46. http://dx.doi.org/10.1098/rspa.2009.0631.

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Eshelby’s problem of an ellipsoidal inclusion embedded in an infinite homogeneous isotropic elastic material and prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is analytically solved. The solution is based on a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The fourth-order Eshelby tensor is obtained in analytical expressions for both the regions inside and outside the inclusion in terms of two line integrals and two surface integrals. This non-classical Eshelby ten

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Astapov, Yuri, and Dmitrii Khristich. "Finite Deformations of an Elastic Cylinder During Indentation." International Journal of Applied Mechanics 10, no. 03 (April 2018): 1850026. http://dx.doi.org/10.1142/s1758825118500266.

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The problem about the indentation of the rigid spherical stamp into the cylindrical specimen was considered. The material of the specimen was assumed to be weakly compressible. The formulation of the problem was performed for the case of finite deformations. The method of construction of the constitutive relations in terms of logarithmic strain tensor for elastic media and the variant of the algorithm to take into account the variation of the contact zone were proposed. The expansion of Hencky tensor and its time derivative into the series in powers of Cauchy strain tensor were used to calcula

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Keylock, Christopher J. "The Schur decomposition of the velocity gradient tensor for turbulent flows." Journal of Fluid Mechanics 848 (June 13, 2018): 876–905. http://dx.doi.org/10.1017/jfm.2018.344.

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The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal par

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Кульчин, Ю. Н., В. Е. Рагозина та О. В. Дудко. "Учет влияния полей остаточных деформаций в современных физико-механических технологиях обработки конструкционных материалов". Письма в журнал технической физики 45, № 1 (2019): 27. http://dx.doi.org/10.21883/pjtf.2019.01.47152.17487.

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AbstractA strict determination of the mechanisms of redistribution of previously accumulated irreversible strains as a result of additional elastic shock actions on the material is given for a nonlinear gradient model of large elastic–plastic strain. It is shown that this redistribution is limited by rigid transport and rotation of the plastic strain tensor. Formulas for a change in the initial components of the plastic strain tensor in elastic waves are derived. It is shown that the preliminary plastic field affects the dynamics of further reversible strain as one of the factors of formation

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Auffray, N., B. Kolev, and M. Olive. "Handbook of bi-dimensional tensors: Part I: Harmonic decomposition and symmetry classes." Mathematics and Mechanics of Solids 22, no. 9 (May 23, 2016): 1847–65. http://dx.doi.org/10.1177/1081286516649017.

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To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in [Formula: see text] using mathematical tools coming from representation theory. The power of this met

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Zhavoronok, Sergey. "ON DIFFERENT DEFINITIONS OF STRAIN TENSORS IN GENERAL SHELL THEORIES OF VEKUA-AMOSOV TYPE." International Journal for Computational Civil and Structural Engineering 17, no. 1 (March 24, 2021): 117–26. http://dx.doi.org/10.22337/2587-9618-2021-17-1-117-126.

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Several possible definiions of strains in a general shell theory of I.N. Vekua – A.A. Amosov type are considered. The higher-order shell model is definedon a two-dimensional manifold within a set of fieldvariables of the firstkind determined by the expansion factors of the spatial vector fieldof the translation. Two base vector systems are introduced, the firs one so-called concomitant corresponds to the cotangent fibrtion of the modelling surface while the other is defind on a surface equidistant to the modelling one. The distortion appears as a two-point tensor referred to both base systems

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Roy, George. "Complementarity of experimental and numerical methods for determining residual stress states." Powder Diffraction 24, S1 (June 2009): S3—S10. http://dx.doi.org/10.1154/1.3139050.

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Residual stress states in engineering structures are usually determined by measuring components of stress tensors with depth below the material surface. There are destructive and nondestructive methods to measure strain tensor components and convert them into stress tensor components by a variety of techniques derived from constitutive (material) equations. In this study, four methods for determining the strain tensor components are presented: X-ray diffraction method (XRDM), magnetic Barkhausen noise method (MBNM), hole drilling method (HDM), and cut-and-section method (CSM); the first two ar

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Gladwin, Michael T., Ross L. Gwyther, Rhodes Hart, Max Francis, and M. J. S. Johnston. "Borehole tensor strain measurements in California." Journal of Geophysical Research 92, B8 (1987): 7981. http://dx.doi.org/10.1029/jb092ib08p07981.

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de Prunelé, E. "Linear strain tensor and differential geometry." American Journal of Physics 75, no. 10 (October 2007): 881–87. http://dx.doi.org/10.1119/1.2750376.

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Levine, Lyle E., Chukwudi Okoro, and Ruqing Xu. "Full elastic strain and stress tensor measurements from individual dislocation cells in copper through-Si vias." IUCrJ 2, no. 6 (September 30, 2015): 635–42. http://dx.doi.org/10.1107/s2052252515015031.

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Nondestructive measurements of the full elastic strain and stress tensors from individual dislocation cells distributed along the full extent of a 50 µm-long polycrystalline copper via in Si is reported. Determining all of the components of these tensors from sub-micrometre regions within deformed metals presents considerable challenges. The primary issues are ensuring that different diffraction peaks originate from the same sample volume and that accurate determination is made of the peak positions from plastically deformed samples. For these measurements, three widely separated reflections w

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Karas, I., R. Gálik, Š. Pogran, and M. Šesták. "Computer simulation of the teat-cup liner stress and strain tensor." Research in Agricultural Engineering 59, No. 3 (September 18, 2013): 114–19. http://dx.doi.org/10.17221/23/2012-rae.

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The three-notch black pressed teat-cup liners were selected into the set of the analysed teat-cup liners, and, for comparison purposes, teat-cup liners produced from silicone mixture. Wall thicknesses of the analysed teat-cup liners were adjusted by smoothing of the middle working part in length of 30 mm. The working part thickness thus changed from the original 2.37 to 0.41 mm for a teat-cup liner produced from black mixture and from the original thickness of 2.30 to 0.40 mm for a silicone teat-cup liner. A possibility of maximum closure of teat-cup liners in the pressing tact was assessed at

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Pyrz, Ryszard. "Atomistic/Continuum Transition – the Concept of Atomic Strain Tensor." Key Engineering Materials 312 (June 2006): 193–98. http://dx.doi.org/10.4028/www.scientific.net/kem.312.193.

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A new atomic strain concept is formulated that allows calculation of continuum quantities directly within a discrete atomic (molecular) system. The concept is based on the Voronoi tessellation of the molecular system and calculation of atomic site strains, which connects continuum variables and atomic quantities when the later are averaged over a sufficiently large volume treated as a point of the continuum body. The atomic strain tensor is applied to investigate interfacial properties of polymer based nanocomposites.

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