Artykuły w czasopismach: „Strain tensor”

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 is limited to texture-related intensity pole figures and strain pole figures. A tensor closest to auxiliary strain tensors linked to the results of measurements in particular specimen directions is considered to represent the strain state. This tensor is shown to be a good approximation of the average strain tensor. A closed-form expression allowing for its direct computation from experimental pole figures is given. The performance of the procedure is illustrated using simulated data.

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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 strain tensor into volumetric and deviatoric (isochoric) parts is given. The deviatoric part depends on the volume change, but this dependence is negligible for materials that are incapable of large volume changes. A general relationship between the rate of the approximate Hencky strain tensor and the deformation rate tensor can be easily established.

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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. Since the geometry of the present problem is of finite radius, the effect of fiber volume fraction on the stress distribution can be easily studied. Results for the thermal residual stress distributions are compared with Eshelby’s infinite domain solution and finite element results for a specified fiber volume fraction.

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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, incremental or instantaneous principal strains and maximum shear. The incremental displacements can then be summed up with Eulerian or Lagrangian summation, and the components of the 2-D finite strain tensor can be calculated together with the finite principal strain and maximum finite shear. We benchmark the measures of finite displacements using specific synthetic tests for each summation mode. The deformation gradient tensor is calculated from the deformed state and decomposed into the finite rigid-body rotation and left or right finite-stretch tensors, allowing the deformation ellipsoids to be drawn. The finite strain has long been the only quantified measure of strain in analogue models. The presented software package allows producing these finite strain measures while also accessing incremental measures of strain. The more complete characterisation of the deformation of tectonic analogue models will facilitate the comparison with numerical simulations and geological data and help produce conceptual mechanical models.

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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 ε as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders n Θ . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.

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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, autonomous components. Finally, bounds on the values of the compliance tensor components for the trigonal system, dictated by the classical thermodynamical argument for the elastic potential to be positive definite, are estimated by imposing the characteristic values of the compliance tensor to be strictly positive.

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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 of cubic materials is developed, taking into account geometric and physical nonlinearities. The specific potential strain energy for a hyperelastic cubic material is written as a function of the tensor invariants, which are projections of the Cauchy-Green strain tensor into eigensubspaces of the cubic material. Expansions of elasticity tensors of the fourth and sixth ranks in tensor bases in eigensubspaces are determined for the cubic material. Relations between stresses and finite strains containing the second degree of deformations are obtained. The expressions for the stress tensor reflect the mutual influence of the processes occurring in various eigensubspaces of the material under consideration.

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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 the theoretical analysis. It is also proven that the essential parameters required for a coordinate-invariant characterization of the elastic properties of a crystal exhibiting monoclinic symmetry are both the six characteristic values of the compliance tensor and seven dimensionless parameters. These material constants, referred to as the eigenangles, are shown to be accountable for the orientation of the stress and strain eigentensors, when represented in a stress coordinate system. Finally, the restrictions dictated by the classical thermodynamical argument on the elements of the compliance tensor, which are necessary and sufficient for the elastic strain-energy density to be positive definite, are investigated for the monoclinic symmetry.

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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 directions and the three-dimensional graphic representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.

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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 strain measures. The functional form of the strain energy of a general anisotropic nonlinear elastic material may then be expressed in terms of these directional strain measures. Next, we consider general nonlinear isotropic response by developing explicit functional forms for three independent strain invariants in terms of these directional strain measures. Finally, with reference to previous work, we discuss isotropy of specific functional forms of directional strain measures associated with up to 15 directions in space.

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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 given symmetry class. These results are useful and even indispensable for experimentally identifying or theoretically/numerically estimating the flexoelectric coefficients of a dielectric material.

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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 relation is obtained as a quasilinear relation between true stress tensor and the Hencky corotation tensor. A system of equilibrium equations is presented in terms of physical components of the true stress tensor in the Lagrangian coordinates. Using the equilibrium equations and the incompressibility condition, a closed system of nonlinear ordinary differential equations is obtained to determine six unknown functions, depending on the angle indicating a position of the points along the cross-section in the initial state. The method of successive approximations is applied to estimate stress tensor components and to derive logarithms of the elongations of material fibers.

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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 strain and a purely isochoric strain, whereas the third constant takes into account the possible deviation from the similarity of the deviators of the Hencky stress and strain tensors. The resulting relations are naturally generalized for incompressible materials. In this case, the overall number of constants decreases from five to two. The designed test unit was used for a compression test of prismatic specimens made of incompressible material. The proposed version of the relations is in good agreement with the experimental data on the compression of rubber samples.

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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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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 Cartesian description for Hencky strain-rate in the Lagrangian state is obtained.

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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 statement of Wolff’s law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T* H* = H* T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E* H* = H* E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.

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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 strains on the equilibrated β-HMX crystal at a given pressure and temperature and using the equilibrium stress tensors of the strained cells to obtain the derivatives of stress with respect to strain. For a fixed temperature, the elastic coefficients increase substantially with increasing pressure, whereas, for a fixed pressure, the elastic coefficients decrease as temperature increases, in accordance with physical expectations. Comparisons to previous experimental and computational results are provided where possible.

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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 geodetic strain rate tensor (normal strain rate along north and eastbound). Using the velocity field gathered from a previous article, and also using a simple and straightforward method, the strain rate tensors were calculated. The obtained contraction along the north direction (from the normal strain along this axis) confirms the Eurasia-Arabia collision. Besides, the obtained extension along the east direction and the derived expansion of the dilatation, show the effect of Anatolian motion to the west and eastward movement of the central Iran plateau on the tectonic structure of the studied area. These two results showed that the variant component of strain rate tensor also provides us with useful information about a region shape deformation.

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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 rate tensor along the trajectory of the movement of the material particles. At the same time features of the displacement velocity field are considered in the form of cross points of characteristics of indicial equations which define displacement velocity field (center of the fan of characteristics for a deformation case in axisymmetric deformation of ideal rigid-plastic bodies conditions). The limiting trajectories of the motion of particles contracting to the discontinuity point are considered.

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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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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 challenging task. We employ a matrix representation of TOE tensors that allows computation of strain-related stiffness perturbations from a linear combination of [Formula: see text] matrices scaled by the components of the strain tensor. In addition to streamlining the numerical algorithm, this approach helps to predict strain-induced symmetry using relatively straightforward algebraic considerations. For example, our analysis shows that a transversely isotropic (TI) medium acquires orthorhombic symmetry if one of the principal directions of the strain tensor is aligned with the symmetry axis. Otherwise, the strained TI medium can become monoclinic or even triclinic.

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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 its center. The numerical calculations of the strain tensors induced by point defects and the parameters of the distribution functions of random strains in LiLuF_4 and LaAlO_3 crystals have been performed. The calculated envelope with the doublet structure corresponding to the Γ_2(^3 H _4) → Γ_34(^3 H _5) singlet–doublet transition in the absorption spectrum of Pr^3+ ions in the LiLuF_4 crystal agrees well with the data of the measurements.

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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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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 consists of three orthonormal eigenvectors, which by their nature are subjected to directional aliasing. In order to describe the geometric relationship between the SGS stress and resolved strain rate tensors, an effective method is proposed to uniquely determine the topology of the eigenframes. The proposed method has been used for testing three SGS stress models in the context of homogeneous isotropic turbulence at three Reynolds numbers, using both a priori and a posteriori approaches.

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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 damage elastic strain that is also applicable to finite strains. This is accomplished in this work by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. The proposed approach shows that it is equivalent to the hypothesis of energy equivalence for finite strains. In this work, the damage is described kinematically in the elastic domain using the fourth-order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. The constitutive equations of the elastic-damage behavior are derived through the kinematics of damage using the simple mapping instead of the other two hypotheses.

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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 purely magnetic irreversible phenomena are considered in detail.

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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 complexity. In this paper, we used a different approach to solving this problem. The molecular dynamics (MD) method was used to calculate the strain dependences of the RD diffusion tensor for various types of strain states. These dependences were used to determine the dipole tensor of the effective RD saddle-point configuration, which takes into account the contributions of all real saddle-point configurations. The proposed approach was used for studying the diffusion characteristics of RDs, such as di-interstitials in FCC copper (used in plasma-facing components of fusion reactors under development). The effect of the external elastic field on the MD-calculated normalized diffusion tensor (ratio of the diffusion tensor to a third of its trace) of di-interstitials was fully consistent with analytical predictions based on the kinetic theory, the parameters of which were the components of the dipole tensors, including the range of non-linear dependence of the diffusion tensor on strains. The results obtained allowed for one to simulate the anisotropic diffusion of di-interstitials in external and internal elastic fields, and to take into account the contribution of di-interstitials to the radiation deformation of crystals. This contribution can be significant, as MD data on the primary radiation damage in copper shows that ~20% of self-interstitial atoms produced by cascades of atomic collisions are combined into di-interstitials.

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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 constitutive equation directly. Thus this research provides a new idea and method for solving large-deformation problems in practical engineering.

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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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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 elasticity. Moreover, based on the Green tensors of Mindlin’s strain gradient elasticity, point, line and double forces are studied.

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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. Diagonalization of E revealed a planar strain pattern, with one nonzero negative strain (εN) and one nonzero positive strain (εP); both strains were larger in magnitude ( P < 0.05) in the deep compartment [εN = −40.4 ± 4.3%, εP = 35.1 ± 3.5% (means ± SE)] than in the superficial compartment (εN = −24.3 ± 3.9%, εP = 6.3 ± 4.9%). The principal shortening direction deviated from the fiber direction by 24.0 ± 1.3° and 39.8 ± 6.1° in the deep and superficial compartments, respectively ( P < 0.05, deep vs. superficial). The deviation of the shortening direction from the fiber direction was due primarily to the lower angle of elevation of the shortening direction over the axial plane than that of the fiber direction. It is concluded that three-dimensional analyses of strain interpreted with respect to the fiber architecture are necessary to characterize skeletal muscle contraction in vivo. The deviation of the principal shortening direction from the fiber direction may relate to intramuscle variations in fiber length and pennation angle.

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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 direct molecular dynamics simulations on nanocrystalline tantalum. This methodology can be implemented with simultaneous X-ray diffraction and digital image correlation in synchrotron radiation or free-electron laser experiments.

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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 uniaxial tension. The plasticity model is first formulated in a fictitious undamaged configuration of the body. Then certain transformation equations are derived to transform this model into a damage-plasticity model in the damaged configuration of the body. Certain assumptions are made in order to make this transformation possible. These assumptions include small elastic strains and the hypothesis of elastic energy equivalence of Ref 17. The corotational stress rate equations are also discussed since they are used extensively in the constitutive relations. Therefore, the use of spin tensors is also discussed since they play a major role in the definition of the corotational rates. In addition, a modified spin tensor is proposed to be used in the coupled model. Furthermore, the nature of the fourth-rank damage effect tensor is discussed for a general state of deformation and damage. Also, the explicit matrix representation of this tensor is rigorously derived and can be used in future applications to solve plane stress and plane strain problems involving damage. Finally, the problem of finite simple shear is investigated using the proposed model. The resulting equations are solved using a Runge-Kutta-Verner fifth order and sixth order method. The stress-strain curves are obtained for a certain expression of the modified spin tensor and are compared with other spin tensors. Also, the evolution of the backstress and damage variables is presented. The results obtained compare favorably with previous results.

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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 field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

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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 composites, the overall compliance (stiffness) tensor, as produced directly by our formulation, is Hermitian, regardless of whether the corresponding unit cell is geometrically or materially symmetric. Overall, mass density is shown to be a tensor and, like the overall compliance tensor, always Hermitian. The average strain and linear momentum tensors are, however, coupled, and the coupling tensors are shown to be each others' Hermitian transpose. Finally, we present a numerical example of a three-dimensional periodic composite composed of elastic cubes periodically distributed in an elastic matrix. The presented results corroborate the predictions of the theoretical treatment illustrating the frequency dependence of the constitutive parameters. We also show that the effective properties calculated in this paper satisfy the dispersion relation of the composite.

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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 respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

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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 tensor consists of a classical part and a gradient part. The former involves Poisson’s ratio only, while the latter includes the length scale parameter additionally, which enables the newly obtained Eshelby tensor to capture the inclusion size effect, unlike its counterpart based on classical elasticity. The accompanying fifth-order Eshelby-like tensor relates the prescribed eigenstrain gradient to the disturbed strain and has only a gradient part. When the strain gradient effect is not considered, the new Eshelby tensor reduces to the classical Eshelby tensor, and the Eshelby-like tensor vanishes. In addition, the current Eshelby tensor for the ellipsoidal inclusion problem includes those for the spherical and cylindrical inclusion problems based on the SSGET as two limiting cases. The non-classical Eshelby tensor depends on the position and is non-uniform even inside the inclusion, which differ from its classical counterpart. For homogenization applications, the volume average of the new Eshelby tensor over the ellipsoidal inclusion is analytically obtained. The numerical results quantitatively show that the components of the newly derived Eshelby tensor vary with both the position and the inclusion size, unlike their classical counterparts. When the inclusion size is small, it is found that the contribution of the gradient part is significantly large. It is also seen that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. Moreover, these components are observed to approach the values of their classical counterparts from below when the inclusion size becomes sufficiently large.

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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 calculate correctly the components of these tensors. Within the indentation problem, we used the model of nonlinear elastic material which provides the best agreement between numerical solution and experimental data among other used types of constitutive relations including various elastic and hypoelastic models.

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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 part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in$\unicode[STIX]{x1D64C}{-}\unicode[STIX]{x1D64D}$space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions.

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42

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 of the initial quasi-static elastic field, which cannot be obtained in a purely elastic process.

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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 method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.

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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 after covariant differentiationof the translation vector feld. Thus, two main definition of the strain tensor become possible, the firstone referred to the main basis whereas the second to the concomitant one. Some possible simplificationsof these tensors are considered, and the interrelation between the general theory of A.A. Amosov type and the classical ones is shown.

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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 are nondestructive, and the third and fourth are semidestructive and destructive, respectively. A complementarity of the experimental and two numerical methods such as boundary element method and finite element method is explained. An application of the experimental and numerical methods to measure residual stress states in an industrial component, an L-shaped part of a supporting column in a high voltage structure, is presented.

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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 were examined from selected, individual grains along the via. The lattice spacings and peak positions were measured for multiple dislocation cell interiors within each grain and the cell-interior peaks were sorted out using the measured included angles. A comprehensive uncertainty analysis using a Monte Carlo uncertainty algorithm provided uncertainties for the elastic strain tensor and stress tensor components.

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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 the working suction of 50 kPa, the flow of distilled water through a measurement device reached the value of 4.4 l/min. Under laboratory conditions, with the above criteria fulfilled, the following stress relations were detected at teat tips: black teat-cup liner 30.25 kPa, silicone teat-cup liner 23.14 kPa. From the acquired results follows that the silicone teat-cup liner showed, from the aspect of suction loss, a more favourable value by 7.11 kPa. Physical-mechanical qualities of the analysed teat-cup liners were further used for the computer simulation of the teat-cup liners stress and strain tensor. &nbsp; &nbsp;

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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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