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Journal articles on the topic 'Second order tensor'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Hesselink, Lambertus, Thierry Delmarcelle, James L. Helman, and Steve Bryson. "Topology of Second-Order Tensor Fields." Computers in Physics 9, no. 3 (1995): 304. http://dx.doi.org/10.1063/1.4823408.

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2

Smith, G. F., and B. A. Younis. "Isotropic tensor-valued polynomial function of second and third-order tensors." International Journal of Engineering Science 43, no. 5-6 (March 2005): 447–56. http://dx.doi.org/10.1016/j.ijengsci.2004.12.004.

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3

Bejan, Cornelia Livia, and Mircea Crasmareanu. "Parallel second-order tensors on Vaisman manifolds." International Journal of Geometric Methods in Modern Physics 14, no. 02 (January 18, 2017): 1750023. http://dx.doi.org/10.1142/s0219887817500232.

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The aim of this paper is to study the class of parallel tensor fields [Formula: see text] of [Formula: see text]-type in a Vaisman geometry [Formula: see text]. A sufficient condition for the reduction of such symmetric tensors [Formula: see text] to a constant multiple of [Formula: see text] is given by the skew-symmetry of [Formula: see text] with respect to the complex structure [Formula: see text]. As an application of the main result, we prove that certain vector fields on a [Formula: see text]-manifold turn out to be Killing. Also, we connect our main result with the Weyl connection of conformal geometry as well as with possible Ricci solitons in [Formula: see text] manifolds.

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4

Heras, José A. "Decomposition of a symmetric second-order tensor." European Journal of Physics 39, no. 3 (March 9, 2018): 035202. http://dx.doi.org/10.1088/1361-6404/aa9c9d.

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5

Delmarcelle, T., and L. Hesselink. "Visualizing second-order tensor fields with hyperstreamlines." IEEE Computer Graphics and Applications 13, no. 4 (July 1993): 25–33. http://dx.doi.org/10.1109/38.219447.

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6

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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7

Xiao, H. "Two General Representation Theorems for Arbitrary-Order-Tensor-Valued Isotropic and Anisotropic Tensor Functions of Vectors and Second Order Tensors." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 76, no. 3 (1996): 155–65. http://dx.doi.org/10.1002/zamm.19960760310.

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8

Zheng, Q. S. "On Tensor Functions of Second-Order Tensors and Vectors UnderN-Gonal ClassesDnh." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 77, no. 8 (1997): 595–608. http://dx.doi.org/10.1002/zamm.19970770812.

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9

Šprlák, Michal, and Pavel Novák. "Spherical integral transforms of second-order gravitational tensor components onto third-order gravitational tensor components." Journal of Geodesy 91, no. 2 (October 5, 2016): 167–94. http://dx.doi.org/10.1007/s00190-016-0951-4.

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10

Calisti, V., A. Lebée, A. A. Novotny, and J. Sokolowski. "Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes." Journal of Elasticity 144, no. 2 (May 2021): 141–67. http://dx.doi.org/10.1007/s10659-021-09836-6.

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AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

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11

Oster, T., C. Rössl, and H. Theisel. "Core Lines in 3D Second-Order Tensor Fields." Computer Graphics Forum 37, no. 3 (June 2018): 327–37. http://dx.doi.org/10.1111/cgf.13423.

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12

Qiang, Yang, Li Zhongkui, and L. G. Tham. "An explicit expression of second-order fabric-tensor dependent elastic compliance tensor." Mechanics Research Communications 28, no. 3 (May 2001): 255–60. http://dx.doi.org/10.1016/s0093-6413(01)00170-7.

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13

Sharma, Ramesh. "Second order parallel tensor in real and complex space forms." International Journal of Mathematics and Mathematical Sciences 12, no. 4 (1989): 787–90. http://dx.doi.org/10.1155/s0161171289000967.

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Levy's theorem ‘A second order parallel symmetric non-singular tensor in a real space form is proportional to the metric tensor’ has been generalized by showing that it holds even if one assumes the second order tensor to be parallel (not necessarily symmetric and non-singular) in a real space form of dimension greater than two. Analogous result has been established for a complex space form.It has been shown that an affine Killing vector field in a non-flat complex space form is Killing and analytic.

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14

O'Connell, Daniel R. H., and Lane R. Johnson. "Second-order moment tensors of microearthquakes at The Geysers geothermal field, California." Bulletin of the Seismological Society of America 78, no. 5 (October 1, 1988): 1674–92. http://dx.doi.org/10.1785/bssa0780051674.

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Abstract The Geysers geothermal field is the site of intense microseismicity which appears to be associated with steam production. It seems that focal mechanisms of earthquakes at The Geysers vary systematically with depth, but P-wave first-motion focal mechanism studies have been hampered by inadequate resolution. In this study an unconstrained frequency domain moment tensor inversion method is used to over come P-wave first-motion focal sphere distribution problems and to investigate microearthquake source properties. A goal was to investigate the feasibility of using waveforms to invert for the second-order moment tensor of microearthquakes in the complex setting of The Geysers. Derived frequency-domain moment tensors for two earthquakes were verified by mechanisms estimated from P-wave first motions and required far fewer stations. For one event, 19 P-wave first motions were insufficient to distinguish between normal-slip and strike-slip focal mechanisms, but a well-constrained strike-slip solution was obtained from the waveform principal moment inversion using data from six stations. Improved waveform focal mechanism resolution was a direct consequence of using P- and S-wave data together in a progressive velocity-hypocenter inversion to minimize Green function errors. The effects of hypocenter mislocation and velocity model Green function errors on moment tensor estimates were investigated. Synthetic tests indicate that these errors can introduce spurious isotropic and compensated linear vector dipole components as large as 26 per cent for these events, whereas principal moment orientations errors were <8°. In spite of unfavorable recording geometries and large (0.6 km) station elevation differences, the results indicate that waveform moment tensor estimates for microearthquake sources can be robust and constrain source mechanisms using data from a relatively small number of stations.

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15

Chen, Zhongming, Yannan Chen, Liqun Qi, and Wennan Zou. "Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor." Mathematics and Mechanics of Solids 24, no. 10 (March 8, 2019): 3092–102. http://dx.doi.org/10.1177/1081286519835246.

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The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith–Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.

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16

Wang, Yaning, and Ximin Liu. "Second order parallel tensors on almost Kenmotsu manifolds satisfying the nullity distributions." Filomat 28, no. 4 (2014): 839–47. http://dx.doi.org/10.2298/fil1404839w.

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In this paper, we prove that if there exists a second order symmetric parallel tensor on an almost Kenmotsu manifold (M2n+1, ?, ?, ?, g) whose characteristic vector field ? belongs to the (k,?)'-nullity distribution, then either M2n+1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, or the second order parallel tensor is a constant multiple of the associated metric tensor of M2n+1. Furthermore, some properties of an almost Kenmotsu manifold admitting a second order parallel tensor with ? belonging to the (k,?)-nullity distribution are also obtained.

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17

Yang, Jun, Yuki Kurashige, Frederick R. Manby, and Garnet K. L. Chan. "Tensor factorizations of local second-order Møller–Plesset theory." Journal of Chemical Physics 134, no. 4 (January 27, 2011): 044123. http://dx.doi.org/10.1063/1.3528935.

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18

Wang, Su-Jing, Chun-Guang Zhou, Na Zhang, Xu-Jun Peng, Yu-Hsin Chen, and Xiaohua Liu. "Face recognition using second-order discriminant tensor subspace analysis." Neurocomputing 74, no. 12-13 (June 2011): 2142–56. http://dx.doi.org/10.1016/j.neucom.2011.01.024.

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19

Hesselink, L., Y. Levy, and Y. Lavin. "The topology of symmetric, second-order 3D tensor fields." IEEE Transactions on Visualization and Computer Graphics 3, no. 1 (1997): 1–11. http://dx.doi.org/10.1109/2945.582332.

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20

He, Q. C. "A Remarkable Tensor in Plane Linear Elasticity." Journal of Applied Mechanics 64, no. 3 (September 1, 1997): 704–7. http://dx.doi.org/10.1115/1.2788952.

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It is shown that any two-dimensional elastic tensor can be orthogonally and uniquely decomposed into a symmetric tensor and an antisymmetric tensor. To within a scalar multiplier, the latter turns out to be equal to the right-angle rotation on the space of two-dimensional second-order symmetric tensors. On the basis of these facts, several useful results are derived for the traction boundary value problem of plane linear elasticity.

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21

Yurduşen, İsmet, O. Oğulcan Tuncer, and Pavel Winternitz. "Superintegrable systems with spin and second-order tensor and pseudo-tensor integrals of motion." Journal of Physics A: Mathematical and Theoretical 54, no. 30 (June 30, 2021): 305201. http://dx.doi.org/10.1088/1751-8121/ac0a9e.

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22

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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23

CATTANEO, STEFANO, MIKAEL SILTANEN, and MARTTI KAURANEN. "NEW POLARIZATION TECHNIQUES FOR PRECISE CHARACTERIZATION OF SECOND-ORDER THIN FILMS." Journal of Nonlinear Optical Physics & Materials 12, no. 04 (December 2003): 513–23. http://dx.doi.org/10.1142/s0218863503001614.

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We present two new techniques with increased precision to determine the second-harmonic susceptibility tensor of thin films. The tensor is associated with the macroscopic structure of such samples but cannot be measured directly. Instead, it must be extracted from experimental results using a theoretical model. The first technique is based on regression analysis of a large number of independent measurements. The second technique relies on the use of two, instead of one, input beams at the fundamental frequency. Both techniques allow for addressing the quality of the theoretical models used to describe the experiment.

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24

Stepanov, S., and I. Tsyganok. "Vanishing theorems for higher-order Killing and Codazzi." Differential Geometry of Manifolds of Figures, no. 50 (2019): 141–47. http://dx.doi.org/10.5922/0321-4796-2019-50-16.

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A Killing p-tensor (for an arbitrary natural number p ≥ 2) is a symmetric p-tensor with vanishing symmetrized covariant derivative. On the other hand, Codazzi p-tensor is a symmetric p-tensor with symmetric covariant derivative. Let M be a complete and simply connected Riemannian manifold of nonpositive (resp. non-negative) sectional curvature. In the first case we prove that an arbitrary symmetric traceless Killing p-tensor is parallel on M if its norm is a Lq -function for some q > 0. If in addition the volume of this manifold is infinite, then this tensor is equal to zero. In the second case we prove that an arbitrary traceless Codazzi p-tensor is equal to zero on a noncompact manifold M if its norm is a Lq -function for some q  1 .

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25

Xiao, B., and J. Feng. "Higher order elastic tensors of crystal structure under non-linear deformation." Journal of Micromechanics and Molecular Physics 04, no. 04 (December 2019): 1950007. http://dx.doi.org/10.1142/s2424913019500073.

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The higher-order elastic tensors can be used to characterize the linear and non-linear mechanical properties of crystals at ultra-high pressures. Besides the widely studied second-order elastic constants, the third- and fourth-order elastic constants are sixth and eighth tensors, respectively. The independent tensor components of them are completely determined by the symmetry of the crystal. Using the relations between elastic constants and sound velocity in solid, the independent elastic constants can be measured experimentally. The anisotropy in elasticity of crystal structures is directly determined by the independent elastic constants.

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26

Mantica, Carlo Alberto, and Luca Guido Molinari. "A second-order identity for the Riemann tensor and applications." Colloquium Mathematicum 122, no. 1 (2011): 69–82. http://dx.doi.org/10.4064/cm122-1-7.

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27

Sharif, M., and Saira Waheed. "Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity." Advances in High Energy Physics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/253985.

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The study of energy conditions has many significant applications in general relativistic and cosmological contexts. This paper explores the energy conditions in the framework of the most general scalar-tensor theory with field equations involving second-order derivatives. For this purpose, we use flat FRW universe model with perfect fluid matter contents. By taking power law ansatz for scalar field, we discuss the strong, weak, null, and dominant energy conditions in terms of deceleration, jerk, and snap parameters. Some particular cases of this theory likek-essence model, modified gravity theories and so forth. are analyzed with the help of the derived energy conditions, and the possible constraints on the free parameters of the presented models are determined.

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28

Zhou, Yujian, Liang Bao, and Yiqin Lin. "Fast Second-Order Orthogonal Tensor Subspace Analysis for Face Recognition." Journal of Applied Mathematics 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/871565.

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Tensor subspace analysis (TSA) and discriminant TSA (DTSA) are two effective two-sided projection methods for dimensionality reduction and feature extraction of face image matrices. However, they have two serious drawbacks. Firstly, TSA and DTSA iteratively compute the left and right projection matrices. At each iteration, two generalized eigenvalue problems are required to solve, which makes them inapplicable for high dimensional image data. Secondly, the metric structure of the facial image space cannot be preserved since the left and right projection matrices are not usually orthonormal. In this paper, we propose the orthogonal TSA (OTSA) and orthogonal DTSA (ODTSA). In contrast to TSA and DTSA, two trace ratio optimization problems are required to be solved at each iteration. Thus, OTSA and ODTSA have much less computational cost than their nonorthogonal counterparts since the trace ratio optimization problem can be solved by the inexpensive Newton-Lanczos method. Experimental results show that the proposed methods achieve much higher recognition accuracy and have much lower training cost.

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29

Zhong-heng, Guo. "Derivatives of the principal invariants of a second-order tensor." Journal of Elasticity 22, no. 2-3 (December 1989): 185–91. http://dx.doi.org/10.1007/bf00041110.

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30

Senchuk, A., and G. C. Tabisz. "Second-order collision-induced light scattering: a spherical tensor approach." Journal of Raman Spectroscopy 42, no. 5 (January 19, 2011): 1049–54. http://dx.doi.org/10.1002/jrs.2833.

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31

Mazėtis, E. "On intrinsic tensor structures of a second order tangent bundle." Lithuanian Mathematical Journal 36, no. 4 (October 1996): 409–19. http://dx.doi.org/10.1007/bf02986864.

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32

Balendran, B., and Sia Nemat-Nasser. "Derivative of a function of a nonsymmetric second-order tensor." Quarterly of Applied Mathematics 54, no. 3 (September 1, 1996): 583–600. http://dx.doi.org/10.1090/qam/1402412.

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33

Medeiros, Paul de, and Christopher M. Hull. "Geometric second order field equations for general tensor gauge fields." Journal of High Energy Physics 2003, no. 05 (May 10, 2003): 019. http://dx.doi.org/10.1088/1126-6708/2003/05/019.

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34

Su, Ping-Jung, Wei-Liang Chen, Yang-Fang Chen, and Chen-Yuan Dong. "Determination of Collagen Nanostructure from Second-Order Susceptibility Tensor Analysis." Biophysical Journal 100, no. 8 (April 2011): 2053–62. http://dx.doi.org/10.1016/j.bpj.2011.02.015.

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35

Haussühl, S., and H. Buchen. "Second-order electroelastic tensor of cubic KAl(SO4)2·12H2O." Solid State Communications 60, no. 9 (December 1986): 729–33. http://dx.doi.org/10.1016/0038-1098(86)90431-x.

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36

Wang, Bo, Guoquan Ren, Zhining Li, and Qingzhu Li. "The stability optimization algorithm of second-order magnetic gradient tensor." AIP Advances 11, no. 7 (July 1, 2021): 075322. http://dx.doi.org/10.1063/5.0056361.

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37

Georgievskii, D. V. "Second Order Linear Differential Operators over High Rank Tensor Fields." Mechanics of Solids 55, no. 6 (November 2020): 808–12. http://dx.doi.org/10.3103/s0025654420060060.

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38

KAURANEN, MARTTI, THIERRY VERBIEST, and ANDRÉ PERSOONS. "CHIRAL MATERIALS IN SECOND-ORDER NONLINEAR OPTICS." Journal of Nonlinear Optical Physics & Materials 08, no. 02 (June 1999): 171–89. http://dx.doi.org/10.1142/s0218863599000138.

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We review how chirality can lead to new types of second-order nonlinear optical materials. Chiral molecules are noncentrosymmetric with a nonvanishing electric-dipole-allowed second-order response, which persists in macroscopic samples with high rotational symmetry such as isotropic solutions. On the other hand, contributions of magnetic-dipole interactions, which can be strong in chiral materials, allow second-order processes in centrosymmetric materials. The magnetic contributions of thin films of chiral polyisocyanides and polythiophenes are comparable to electric contributions. Components of the electric-dipole susceptibility tensor associated with chirality dominate the response of thin films of a chiral helicenebisquinone.

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39

Magden, Abdullah, Kubra Karaca, and Aydin Gezer. "The second-order tangent bundle with deformed 2nd lift metric." International Journal of Geometric Methods in Modern Physics 16, no. 04 (April 2019): 1950062. http://dx.doi.org/10.1142/s0219887819500622.

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Let [Formula: see text] be a pseudo-Riemannian manifold and [Formula: see text] be its second-order tangent bundle equipped with the deformed [Formula: see text]nd lift metric [Formula: see text] which is obtained from the [Formula: see text]nd lift metric by deforming the horizontal part with a symmetric [Formula: see text]-tensor field [Formula: see text]. In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of [Formula: see text]. We give necessary and sufficient conditions for [Formula: see text] to be semi-symmetric. Secondly, we show that [Formula: see text] is a plural-holomorphic [Formula: see text]-manifold with the natural integrable nilpotent structure. Finally, we get the conditions under which [Formula: see text] with the [Formula: see text]nd lift of an almost complex structure is an anti-Kähler manifold.

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40

He, Ruijian, and Xinlong Feng. "Second Order Convergence of the Interpolation based on-Element." Numerical Mathematics: Theory, Methods and Applications 9, no. 4 (November 2016): 595–618. http://dx.doi.org/10.4208/nmtma.2016.m1503.

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AbstractIn this paper, the second order convergence of the interpolation based on-element is derived in the case ofd=1, 2 and 3. Using the integral average on each element, the new basis functions of tensor product type is builded up and we can easily extend it to the higher dimensional case. Finally, some numerical tests are made to show the analytical results of the interpolation errors.

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41

Arrigo, Francesca, Desmond J. Higham, and Francesco Tudisco. "A framework for second-order eigenvector centralities and clustering coefficients." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2236 (April 2020): 20190724. http://dx.doi.org/10.1098/rspa.2019.0724.

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We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron–Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.

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42

Smentek, Lidia. "Magnetic Linear Birefringence in Rare Earth Systems. Second-Order Approach." Collection of Czechoslovak Chemical Communications 68, no. 2 (2003): 253–64. http://dx.doi.org/10.1135/cccc20030253.

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An analysis defines the improved version of the theoretical model of magnetic linear birefringence in rare earth doped materials developed and applied in several papers of Kolmakova et al. The approach presented here is formulated in the language of tensor operators instead of equivalent Stevens objects, and in addition the radial terms of the effective operators contributing to the polarizability tensor are all defined for the complete radial basis sets of one-electron functions. The approach is based on perturbation theory. The effective operators represent the perturbing influence of singly excited configurations 4fN-1n' l' for l' = d, g and all n'. The model is defined within the single configuration approximation, and it addresses especially the problem of evaluation of the radial integrals. The discussion of the tensorial form of the effective operators is completed by the numerical values of appropriate radial integrals evaluated for the ions across the lanthanide series to be used in future theoretical analysis.

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43

Voyiadjis, G. Z., and T. Park. "Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor." Journal of Applied Mechanics 64, no. 1 (March 1, 1997): 106–10. http://dx.doi.org/10.1115/1.2787259.

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Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

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44

Wu, Gang, Roderick E. Wasylishen, and Ronald D. Curtis. "A phosphorus-31 NMR study of solid carbonylhydridotris(triphenylphosphine)rhodium(I). Unusual MAS sideband intensities in second-order NMR spin systems." Canadian Journal of Chemistry 70, no. 3 (March 1, 1992): 863–69. http://dx.doi.org/10.1139/v92-114.

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The CP/MAS 31P NMR spectrum of carbonylhydridotris(triphenylphosphine)rhodium(I), RhH(CO)(PPh3)3 (1), can be described as a tightly coupled ABMX spin system (X = 103Rh). In contrast to the solution 31P NMR spectrum, the three equatorial phosphorus nuclei are nonequivalent in the solid state and this structural feature allows us to measure the two-bond spin–spin couplings, 2J(31P,31P). A new method is proposed for extracting the principal components of the chemical shift tensor from slow MAS NMR spectra in a tightly J-coupled two-spin system. For compound 1, the principal components of the 31P chemical shift tensors calculated using this method are in good agreement with those obtained from NMR spectra of a static sample. The principal components of the 31P chemical shift tensors determined for 1 are compared with those reported previously for Wilkinson's catalyst, RhCl(PPh3)3. The δ22 component of the 31P chemical shift tensors in 1 shows the largest variation with structure. This is ascribed to differences in the orientation of the P—Cipso bond about the equatorial plane of the trigonal bipyramidal structure. Keywords: rhodium–phosphine compounds, AB spin system, 31P chemical shift tensor, magic-angle spinning, molecular structure.

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45

Verwoerd, Wynand S. "Iterative solution of quadratic tensor equations for mutual polarisation." International Journal of Mathematics and Mathematical Sciences 32, no. 5 (2002): 301–12. http://dx.doi.org/10.1155/s0161171202110246.

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To describe mutual polarisation in bulk materials containing high polarisability molecules, local fields beyond the linear approximation need to be included. A second order tensor equation is formulated, and it describes this in the case of crystalline or at least locally ordered materials such as an idealised polymer. It is shown that this equation is solved by a set of recursion equations that relate the induced dipole moment, linear polarisability, and first hyperpolarisability in the material to the intrinsic values of the same properties of isolated molecules. From these, macroscopic susceptibility tensors up to second order can be calculated for the material.

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46

Basak, Anup, and Valery I. Levitas. "An exact formulation for exponential-logarithmic transformation stretches in a multiphase phase field approach to martensitic transformations." Mathematics and Mechanics of Solids 25, no. 6 (February 14, 2020): 1219–46. http://dx.doi.org/10.1177/1081286520905352.

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A general theoretical and computational procedure for dealing with an exponential-logarithmic kinematic model for transformation stretch tensor in a multiphase phase field approach to stress- and temperature-induced martensitic transformations with N martensitic variants is developed for transformations between all possible crystal lattices. This kinematic model, where the natural logarithm of transformation stretch tensor is a linear combination of natural logarithm of the Bain tensors, yields isochoric variant–variant transformations for the entire transformation path. Such a condition is plausible and cannot be satisfied by the widely used kinematic model where the transformation stretch tensor is linear in Bain tensors. Earlier general multiphase phase field studies can handle commutative Bain tensors only. In the present treatment, the exact expressions for the first and second derivatives of the transformation stretch tensor with respect to the order parameters are obtained. Using these relations, the transformation work for austenite ↔ martensite and variant ↔ variant transformations is analyzed and the thermodynamic instability criteria for all homogeneous phases are expressed explicitly. The finite element procedure with an emphasis on the derivation of the tangent matrix for the phase field equations, which involves second derivatives of the transformation deformation gradients with respect to the order parameters, is developed. Change in anisotropic elastic properties during austenite–martensitic variants and variant–variant transformations is taken into account. The numerical results exhibiting twinned microstructures for cubic to orthorhombic and cubic to monoclinic-I transformations are presented.

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47

Kuzyk, M. G., K. D. Singer, H. E. Zahn, and L. A. King. "Second-order nonlinear-optical tensor properties of poled films under stress." Journal of the Optical Society of America B 6, no. 4 (April 1, 1989): 742. http://dx.doi.org/10.1364/josab.6.000742.

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48

Hlawatsch, M., J. E. Vollrath, F. Sadlo, and D. Weiskopf. "Coherent Structures of Characteristic Curves in Symmetric Second Order Tensor Fields." IEEE Transactions on Visualization and Computer Graphics 17, no. 6 (June 2011): 781–94. http://dx.doi.org/10.1109/tvcg.2010.107.

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Habisohn, Chris X. "Calculation of radiated gravitational energy using the second‐order Einstein tensor." Journal of Mathematical Physics 27, no. 11 (November 1986): 2759–69. http://dx.doi.org/10.1063/1.527300.

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Carrilho, Pedro, and Karim A. Malik. "Vector and tensor contributions to the curvature perturbation at second order." Journal of Cosmology and Astroparticle Physics 2016, no. 02 (February 8, 2016): 021. http://dx.doi.org/10.1088/1475-7516/2016/02/021.

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