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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Ares de Parga, G., R. E. González-Narvaez, and R. Mares. "Conservation of the Energy-Momentum." International Journal of Theoretical Physics 56, no. 10 (August 1, 2017): 3213–31. http://dx.doi.org/10.1007/s10773-017-3489-1.

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2

Wu, Zhao-Yan. "Gravitational Energy-Momentum and Conservation of Energy-Momentum in General Relativity." Communications in Theoretical Physics 65, no. 6 (June 1, 2016): 716–30. http://dx.doi.org/10.1088/0253-6102/65/6/716.

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3

van den Heuvel, B. M. "Energy‐momentum conservation in gauge theories." Journal of Mathematical Physics 35, no. 4 (April 1994): 1668–87. http://dx.doi.org/10.1063/1.530563.

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4

Bak, Dongsu, D. Cangemi, and R. Jackiw. "Energy-momentum conservation in gravity theories." Physical Review D 49, no. 10 (May 15, 1994): 5173–81. http://dx.doi.org/10.1103/physrevd.49.5173.

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5

GOLDMAN, T. "NEUTRINO OSCILLATIONS AND ENERGY–MOMENTUM CONSERVATION." Modern Physics Letters A 25, no. 07 (March 7, 2010): 479–87. http://dx.doi.org/10.1142/s0217732310032706.

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A description of neutrino oscillation phenomena is presented which is based on relativistic quantum mechanics with four-momentum conservation. This is different from both conventional approaches which arbitrarily use either equal energies or equal momenta for the different neutrino mass eigenstates. Both entangled state and source dependence aspects are also included. The time dependence of the wave function is found to be crucial to recovering the conventional result to second order in the neutrino masses. An ambiguity appears at fourth order which generally leads to source dependence, but the standard formula can be promoted to this order by a plausible convention.

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6

Li, Miao. "Energy–momentum conservation and holographic S-matrix." Nuclear Physics B 568, no. 1-2 (February 2000): 195–207. http://dx.doi.org/10.1016/s0550-3213(99)00656-2.

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7

Guo, D., T. E. Knight, and J. K. McCusker. "Angular Momentum Conservation in Dipolar Energy Transfer." Science 334, no. 6063 (December 22, 2011): 1684–87. http://dx.doi.org/10.1126/science.1211459.

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8

Zandi, Omid, Zahra Atlasbaf, and Mohammad Sadegh Abrishamian. "Combined Electromagnetic Energy and Momentum Conservation Equation." IEEE Transactions on Antennas and Propagation 58, no. 11 (November 2010): 3585–92. http://dx.doi.org/10.1109/tap.2010.2071340.

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9

Nissani, Noah, and Elhanan Leibowitz. "Global energy-momentum conservation in general relativity." International Journal of Theoretical Physics 28, no. 2 (February 1989): 235–45. http://dx.doi.org/10.1007/bf00669815.

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10

Moradpour, H., J. P. Morais Graça, I. P. Lobo, and I. G. Salako. "Energy Definition and Dark Energy: A Thermodynamic Analysis." Advances in High Energy Physics 2018 (August 9, 2018): 1–8. http://dx.doi.org/10.1155/2018/7124730.

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Accepting the Komar mass definition of a source with energy-momentum tensor Tμν and using the thermodynamic pressure definition, we find a relaxed energy-momentum conservation law. Thereinafter, we study some cosmological consequences of the obtained energy-momentum conservation law. It has been found out that the dark sectors of cosmos are unifiable into one cosmic fluid in our setup. While this cosmic fluid impels the universe to enter an accelerated expansion phase, it may even show a baryonic behavior by itself during the cosmos evolution. Indeed, in this manner, while Tμν behaves baryonically, a part of it, namely, Tμν(e) which is satisfying the ordinary energy-momentum conservation law, is responsible for the current accelerated expansion.

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11

Velten, Hermano, and Thiago R. P. Caramês. "To Conserve, or Not to Conserve: A Review of Nonconservative Theories of Gravity." Universe 7, no. 2 (February 4, 2021): 38. http://dx.doi.org/10.3390/universe7020038.

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Apart from the familiar structure firmly-rooted in the general relativistic field equations where the energy–momentum tensor has a null divergence i.e., it conserves, there exists a considerable number of extended theories of gravity allowing departures from the usual conservative framework. Many of these theories became popular in the last few years, aiming to describe the phenomenology behind dark matter and dark energy. However, within these scenarios, it is common to see attempts to preserve the conservative property of the energy–momentum tensor. Most of the time, it is done by means of some additional constraint that ensures the validity of the standard conservation law, as long as this option is available in the theory. However, if no such extra constraint is available, the theory will inevitably carry a non-trivial conservation law as part of its structure. In this work, we review some of such proposals discussing the theoretical construction leading to the non-conservation of the energy–momentum tensor.

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12

Sugama, H., T. H. Watanabe, and M. Nunami. "Conservation of energy and momentum in nonrelativistic plasmas." Physics of Plasmas 20, no. 2 (February 2013): 024503. http://dx.doi.org/10.1063/1.4789869.

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13

Moyssides, P. G., C. Patrinos, and P. Hatzikonstantinou. "Electromagnetic energy and momentum conservation in pendulum experiments." IEEE Transactions on Magnetics 39, no. 4 (July 2003): 2024–29. http://dx.doi.org/10.1109/tmag.2003.808598.

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14

Chubykalo, Andrew E., Augusto Espinoza, and B. P. Kosyakov. "The origin of the energy–momentum conservation law." Annals of Physics 384 (September 2017): 85–104. http://dx.doi.org/10.1016/j.aop.2017.06.018.

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15

Akhmedov, E. K., and A. Y. Smirnov. "Neutrino Oscillations: Entanglement, Energy-Momentum Conservation and QFT." Foundations of Physics 41, no. 8 (February 25, 2011): 1279–306. http://dx.doi.org/10.1007/s10701-011-9545-4.

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16

Al-Rawaf, A. S., and M. O. Taha. "Cosmology of general relativity without energy-momentum conservation." General Relativity and Gravitation 28, no. 8 (August 1996): 935–52. http://dx.doi.org/10.1007/bf02113090.

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17

Mereš, Michal, Ivan Melo, Boris Tomášik, Vladimír Balek, and Vladimír Černý. "Generating heavy particles with energy and momentum conservation." Computer Physics Communications 182, no. 12 (December 2011): 2561–66. http://dx.doi.org/10.1016/j.cpc.2011.06.015.

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18

HUO, YUNJIE, TIANJUN LI, YI LIAO, DIMITRI V. NANOPOULOS, YONGHUI QI, and FEI WANG. "THE SUPERLUMINAL NEUTRINOS FROM DEFORMED LORENTZ INVARIANCE." Modern Physics Letters A 27, no. 33 (October 24, 2012): 1250196. http://dx.doi.org/10.1142/s0217732312501969.

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We study two superluminal neutrino scenarios where [Formula: see text] is a constant. To be consistent with the OPERA, Borexino and ICARUS experiments and with the SN1987a observations, we assume that δvν on the Earth is about three-order larger than that on the interstellar scale. To explain the theoretical challenges from the Bremsstrahlung effects and pion decays, we consider the deformed Lorentz invariance, and show that the superluminal neutrino dispersion relations can be realized properly while the modifications to the dispersion relations of the other Standard Model particles can be negligible. In addition, we propose the deformed energy and momentum conservation laws for a generic physical process. In Scenario I the momentum conservation law is preserved while the energy conservation law is deformed. In Scenario II the energy conservation law is preserved while the momentum conservation law is deformed. We present the energy and momentum conservation laws in terms of neutrino momentum in Scenario I and in terms of neutrino energy in Scenario II. In such formats, the energy and momentum conservation laws are exactly the same as those in the traditional quantum field theory with Lorentz symmetry. Thus, all the above theoretical challenges can be automatically solved. We show explicitly that the Bremsstrahlung processes are forbidden and there is no problem for pion decays.

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19

Shaw, Tiffany A., and Theodore G. Shepherd. "A Theoretical Framework for Energy and Momentum Consistency in Subgrid-Scale Parameterization for Climate Models." Journal of the Atmospheric Sciences 66, no. 10 (October 1, 2009): 3095–114. http://dx.doi.org/10.1175/2009jas3051.1.

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Abstract A theoretical framework for the joint conservation of energy and momentum in the parameterization of subgrid-scale processes in climate models is presented. The framework couples a hydrostatic resolved (planetary scale) flow to a nonhydrostatic subgrid-scale (mesoscale) flow. The temporal and horizontal spatial scale separation between the planetary scale and mesoscale is imposed using multiple-scale asymptotics. Energy and momentum are exchanged through subgrid-scale flux convergences of heat, pressure, and momentum. The generation and dissipation of subgrid-scale energy and momentum is understood using wave-activity conservation laws that are derived by exploiting the (mesoscale) temporal and horizontal spatial homogeneities in the planetary-scale flow. The relations between these conservation laws and the planetary-scale dynamics represent generalized nonacceleration theorems. A derived relationship between the wave-activity fluxes—which represents a generalization of the second Eliassen–Palm theorem—is key to ensuring consistency between energy and momentum conservation. The framework includes a consistent formulation of heating and entropy production due to kinetic energy dissipation.

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20

LIU, YU-XIAO, LI-JIE ZHANG, YONG-QIANG WANG, and YI-SHI DUAN. "ENERGY–MOMENTUM FOR RANDALL–SUNDRUM MODELS." Modern Physics Letters A 23, no. 10 (March 28, 2008): 769–79. http://dx.doi.org/10.1142/s0217732308024110.

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Анотація:

We investigate the conservation law of energy–momentum for Randall–Sundrum models by the general displacement transform. The energy–momentum current has a superpotential and are therefore identically conserved. It is shown that for Randall–Sundrum solution, the momentum vanishes and most of the bulk energy is localized near the Planck brane. The energy density is ε = ε0 e-3k|y|.

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21

Caviglia, G., and A. Morro. "Conservation laws for incompressible fluids." International Journal of Mathematics and Mathematical Sciences 12, no. 2 (1989): 377–84. http://dx.doi.org/10.1155/s0161171289000438.

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By means of a direct approach, a complete set of conservation laws for incompressible fluids is determined. The problem is solved in the material (Lagrangian) description and the results are eventually rewritten in the spatial (Eulerian) formulation. A new infinite family of conservation laws is determined, besides those for linear momentum, angular momentum, energy and helicity.

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22

López-Bonilla, J., J. Morales, and G. Ovando. "Conservation laws for energy and momentum in curved spaces." Journal of Zhejiang University-SCIENCE A 8, no. 4 (April 2007): 665–68. http://dx.doi.org/10.1631/jzus.2007.a0665.

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23

Bock, N. "Energy momentum conservation effects on two-particle correlation functions." Physics of Particles and Nuclei Letters 8, no. 9 (December 2011): 955–58. http://dx.doi.org/10.1134/s1547477111090068.

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24

Sardanashvily, Gennadi. "Stress - energy - momentum conservation law in gauge gravitation theory." Classical and Quantum Gravity 14, no. 5 (May 1, 1997): 1357–70. http://dx.doi.org/10.1088/0264-9381/14/5/034.

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25

Ootsuka, Takayoshi, Ryoko Yahagi, Muneyuki Ishida, and Erico Tanaka. "Energy-momentum conservation laws in Finsler/Kawaguchi Lagrangian formulation." Classical and Quantum Gravity 32, no. 16 (July 30, 2015): 165016. http://dx.doi.org/10.1088/0264-9381/32/16/165016.

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26

Duan, Yi-Shi, Ji-Cheng Liu, and Xue-Geng Dong. "General covariant energy-momentum conservation law in general spacetime." General Relativity and Gravitation 20, no. 5 (May 1988): 485–96. http://dx.doi.org/10.1007/bf00758123.

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27

Lechner, Kurt. "Dynamics of self-interacting strings and energy-momentum conservation." Annals of Physics 383 (August 2017): 357–88. http://dx.doi.org/10.1016/j.aop.2017.05.024.

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28

Carmona, J. M., J. L. Cortés, and B. Romeo. "Modified energy–momentum conservation laws and vacuum Cherenkov radiation." Astroparticle Physics 71 (December 2015): 21–30. http://dx.doi.org/10.1016/j.astropartphys.2015.04.009.

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29

Gratus, Jonathan. "Maxwell–Lorentz without self-interactions: conservation of energy and momentum." Journal of Physics A: Mathematical and Theoretical 55, no. 6 (January 21, 2022): 065202. http://dx.doi.org/10.1088/1751-8121/ac48ee.

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Abstract Since a classical charged point particle radiates energy and momentum it is argued that there must be a radiation reaction force. Here we present an action for the Maxwell–Lorentz without self-interactions model, where each particle only responds to the fields of the other charged particles. The corresponding stress–energy tensor automatically conserves energy and momentum in Minkowski and other appropriate spacetimes. Hence there is no need for any radiation reaction.

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30

Ripa, P. "Wave energy-momentum and pseudoenergy-momentum conservation for the layered quasi-geostrophic instability problem." Journal of Fluid Mechanics 235, no. -1 (February 1992): 379. http://dx.doi.org/10.1017/s0022112092001150.

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31

Bamba, Kazuharu, and Katsutaro Shimizu. "Construction of energy–momentum tensor of gravitation." International Journal of Geometric Methods in Modern Physics 13, no. 01 (January 2016): 1650001. http://dx.doi.org/10.1142/s0219887816500018.

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We construct the gravitational energy–momentum tensor in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy–momentum conservation is satisfied because one of the two indices of the energy–momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.

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32

Zhou, Yinqiu, and Xiuming Wang. "A methodology for formulating dynamical equations in analytical mechanics based on the principle of energy conservation." Journal of Physics Communications 6, no. 3 (March 1, 2022): 035006. http://dx.doi.org/10.1088/2399-6528/ac57f8.

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Abstract In this work, a methodology is proposed for formulating general dynamical equations in mechanics under the umbrella of the principle of energy conservation. It is shown that Lagrange’s equation, Hamilton’s canonical equations, and Hamilton-Jacobi’s equation are all formulated based on the principle of energy conservation with a simple energy conservation equation, i.e., the rate of kinetic and potential energy with time is equal to the rate of work with time done by external forces; while D’Alembert’s principle is a special case of the law of the conservation of energy, with either the virtual displacements (‘frozen’ time) or the virtual displacement (‘frozen’ generalized coordinates). It is argued that all of the formulations for characterizing the dynamical behaviors of a system can be derived from the principle of energy conservation, and the principle of energy conservation is an underlying guide for constructing mechanics in a broad sense. The proposed methodology provides an efficient way to tackle the dynamical problems in general mechanics, including dissipation continuum systems, especially for those with multi-physical field interactions and couplings. It is pointed out that, on the contrary to the classical analytical mechanics, especially to existing Hamiltonian mechanics, the physics essences of Hamilton’s variational principle, Lagrange’s equation, and the Newtonian second law of motion, including their derivatives such as momentum and angular momentum conservations, are the consequences of the law of conservation of energy. In addition, our proposed methodology is easier to understand with clear physical meanings and can be used for explaining the existing mechanical principles or theorems. Finally, as an application example, the methodology is applied in fluid mechanics to derive Cauchy’s first law of motion.

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33

Barriga-Carrasco, Manuel D. "Applying full conserving dielectric function to the energy loss straggling." Laser and Particle Beams 29, no. 1 (February 10, 2011): 81–86. http://dx.doi.org/10.1017/s0263034610000789.

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AbstractThe purpose of this paper is to calculate proton energy loss straggling using a full conserving dielectric function (FCDF) for plasmas at any degeneracy. This dielectric function takes into account plasma electron-electron collision considering density, momentum, and energy conservation. When only momentum conservation law is accomplished, the FCDF reproduces the well known Mermin dielectric function, when none of the conservations laws are obeyed, the random phase approximation (RPA) is recovered. Then, the FCDF is applied for the first time to the determination of the energy loss straggling. Differences among diverse dielectric functions to determine straggling follow the same behavior for all kind of plasmas then, they do not depend on the plasma degeneracy but essentially do on the value of the collision frequency. These discrepancies can rise up to 5% between FCDF values and the Mermin ones, and 2% between the FCDF ones and RPA ones for plasma with high enough collision frequency. The similarity between FCDF and RPA results is not surprising, as all conservation laws are also considered in RPA dielectric function. The fact that FCDF and RPA give similar results and the fact that FCDF considers electron-electron collisions and RPA does not, means that latter collisions are not significant for energy loss straggling calculations.

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34

Huang, Zaixing. "Noether’s theorem in peridynamics." Mathematics and Mechanics of Solids 24, no. 11 (November 12, 2018): 3394–402. http://dx.doi.org/10.1177/1081286518812931.

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By introducing a new nonlocal argument, the Lagrangian formulation of peridynamics is investigated. The peridynamic Euler–Lagrange equation is derived from Hamilton’s principle, and Noether’s theorem is extended into peridynamics. With the help of the peridynamic Noether’s theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby integral are determined. The results show that the peridynamic conservation laws exist only in a spatial integral form rather than in a pointwise form due to nonlocality. In bond-based peridynamics, energy conservation requires that the influence function is independent of the relative displacement field, or energy dissipation will occur. In state-based peridynamics, the angular momentum conservation causes a constraint on the constitutive relation between the force vector-state and the deformation vector-state. The Eshelby integral of peridynamics is given, which can be used to judge nucleation of defects and to calculate the energy release rates caused by damage, fracture and phase transition.

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35

Ducloué, B., L. Szymanowski, and S. Wallon. "Violation of energy–momentum conservation in Mueller–Navelet jets production." Physics Letters B 738 (November 2014): 311–16. http://dx.doi.org/10.1016/j.physletb.2014.09.025.

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36

Sfarti, Adrian. "Relativistic energy-momentum: the concepts of conservation versus frame invariance." International Journal of Nuclear Energy Science and Technology 5, no. 2 (2010): 127. http://dx.doi.org/10.1504/ijnest.2010.030554.

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37

Kaiser, Gerald. "Energy–momentum conservation in pre-metric electrodynamics with magnetic charges." Journal of Physics A: Mathematical and General 37, no. 28 (July 1, 2004): 7163–68. http://dx.doi.org/10.1088/0305-4470/37/28/007.

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38

Prosen, Tomaž, and David K. Campbell. "Momentum Conservation Implies Anomalous Energy Transport in 1D Classical Lattices." Physical Review Letters 84, no. 13 (March 27, 2000): 2857–60. http://dx.doi.org/10.1103/physrevlett.84.2857.

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39

Azoff, E. M. "Generalized energy-momentum conservation equations in the relaxation time approximation." Solid-State Electronics 30, no. 9 (September 1987): 913–17. http://dx.doi.org/10.1016/0038-1101(87)90127-4.

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40

Oltean, Marius, Hossein Bazrafshan Moghaddam, and Richard J. Epp. "Quasilocal conservation laws in cosmology: A first look." International Journal of Modern Physics D 29, no. 14 (October 2020): 2043029. http://dx.doi.org/10.1142/s0218271820430294.

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Анотація:

Quasilocal definitions of stress-energy–momentum—that is, in the form of boundary densities (in lieu of local volume densities) — have proven generally very useful in formulating and applying conservation laws in general relativity. In this Essay, we take a basic look into applying these to cosmology, specifically using the Brown–York quasilocal stress-energy–momentum tensor for matter and gravity combined. We compute this tensor and present some simple results for a flat FLRW spacetime with a perfect fluid matter source. We emphasize the importance of the vacuum energy, which is almost universally underappreciated (and usually “subtracted”), and discuss the quasilocal interpretation of the cosmological constant.

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41

PODIO-GUIDUGLI, P., S. SELLERS, and G. VERGARA CAFFARELLI. "ON THE REPRESENTATION OF ENERGY AND MOMENTUM IN ELASTICITY." Mathematical Models and Methods in Applied Sciences 10, no. 02 (March 2000): 203–16. http://dx.doi.org/10.1142/s0218202500000136.

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In order to clarify common assumptions on the form of energy and momentum in elasticity, a generalized conservation format is proposed for finite elasticity, in which total energy and momentum are not specified a priori. Velocity, stress, and total energy are assumed to depend constitutively on deformation gradient and momentum in a manner restricted by a dissipation principle and certain mild invariance requirements. Under these assumptions, representations are obtained for energy and momentum, demonstrating that (i) the total energy splits into separate internal and kinetic contributions, and (ii) the momentum is linear in the velocity. It is further shown that, if the stress response is strongly elliptic, the classical specifications for kinetic energy and momentum are sufficient to give elasticity the standard format of a quasilinear hyperbolic system.

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42

Arminjon, Mayeul. "On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor." Advances in Mathematical Physics 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/9679460.

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We review the energy concept in the case of a continuum or a system of fields. First, we analyze the emergence of a true local conservation equation for the energy of a continuous medium, taking the example of an isentropic continuum in Newtonian gravity. Next, we consider a continuum or a system of fields in special relativity: we recall that the conservation of the energy-momentum tensor contains two local conservation equations of the same kind as before. We show that both of these equations depend on the reference frame and that, however, they can be given a rigorous meaning. Then, we review the definitions of the canonical and Hilbert energy-momentum tensors from a Lagrangian through the principle of stationary action in general space-time. Using relatively elementary mathematics, we prove precise results regarding the definition of the Hilbert tensor field, its uniqueness, and its tensoriality. We recall the meaning of its covariant conservation equation. We end with a proof of uniqueness of the energy density and flux, when both depend polynomially on the fields.

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43

Staniforth, Andrew, and Nigel Wood. "The Deep-Atmosphere Euler Equations in a Generalized Vertical Coordinate." Monthly Weather Review 131, no. 8 (August 1, 2003): 1931–38. http://dx.doi.org/10.1175//2564.1.

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Abstract Previous analysis of the hydrostatic primitive equations using a generalized vertical coordinate is extended to the deep-atmosphere nonhydrostatic Euler equations, and some special vertical coordinates of interest are noted. Energy and axial angular momentum budgets are also derived. This would facilitate the development of conserving finite-difference schemes for deep-atmosphere models. It is found that the implied principles of energy and axial angular momentum conservation depend on the form of the upper boundary. In particular, for a modeled atmosphere of finite extent, global energy conservation is only obtained for a rigid lid, fixed in space and time. To additionally conserve global axial angular momentum, the height of the lid cannot vary with longitude.

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44

Hirose, A., and R. Dick. "Fresnel’s formulae and the Minkowski momentum." Canadian Journal of Physics 87, no. 4 (April 2009): 407–10. http://dx.doi.org/10.1139/p09-033.

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Анотація:

It is shown that the momentum of an electromagnetic wave in a dielectric medium can be uniquely determined to be the Minkowski momentum by considering oblique incidence of electromagnetic wave on a flat dielectric boundary. The Minkowski momentum is consistent with the Fresnel’s formulae and satifies the energy and momentum conservation laws.

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45

Shimizu, Katsutaro. "Proposal for the proper gravitational energy–momentum tensor." Modern Physics Letters A 31, no. 26 (August 17, 2016): 1650151. http://dx.doi.org/10.1142/s0217732316501510.

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We propose a gravitational energy–momentum (GEMT) tensor of the general relativity obtained using Noether’s theorem. It transforms as a tensor under general coordinate transformations. One of the two indices of the GEMT labels a local Lorentz frame that satisfies the energy–momentum conservation law. The energies for a gravitational wave, a Schwarzschild black hole and a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe are calculated as examples. The gravitational energy of the Schwarzschild black hole exists only outside the horizon, its value being the negative of the black hole mass.

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46

Daszkiewicz, Marcin. "The energy–momentum conservation law in two-particle system for twist-deformed Galilei Hopf algebras." Modern Physics Letters A 34, no. 03 (January 30, 2019): 1950024. http://dx.doi.org/10.1142/s021773231950024x.

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Анотація:

In this paper, we discuss the energy–momentum conservation principle for two-particle system in the case of canonically and Lie-algebraically twist-deformed Galilei Hopf algebra. Particularly, we provide consistency with the coproducts energy and momentum addition law as well as its symmetric with respect to the exchange of particles counterpart. Besides, we show that the vanishing of total four momentum for two Lie-algebraically deformed kinematical models leads to the discrete values of energies and momenta only in the case of the symmetrized addition rules.

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47

Kholmetskii, Alexander, and Tolga Yarman. "Retraction: Conservative relativity principle and energy-momentum conservation in a superimposed gravitational and electric field." Canadian Journal of Physics 95, no. 10 (October 2017): 1030. http://dx.doi.org/10.1139/cjp-2017-0531.

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48

Yu, Pengfei, Weifeng Leng, and Yaohong Suo. "Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity." Applied Sciences 11, no. 2 (January 12, 2021): 681. http://dx.doi.org/10.3390/app11020681.

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Анотація:

The flexoelectricity, which is a new electromechanical coupling phenomenon between strain gradients and electric polarization, has a great influence on the fracture analysis of flexoelectric solids due to the large gradients near the cracks. On the other hand, although the flexoelectricity has been extensively investigated in recent decades, the study on flexoelectricity in nonhomogeneous materials is still rare, especially the fracture problems. Therefore, in this manuscript, the conservation integrals for nonhomogeneous flexoelectric materials are obtained to solve the fracture problem. Application of operators such as grad, div, and curl to electric Gibbs free energy and internal energy, the energy-momentum tensor, angular momentum tensor, and dilatation flux can also be derived. We examine the correctness of the conservation integrals by comparing with the previous work and discuss the operator method here and Noether theorem in the previous work. Finally, considering the flexoelectric effect, a nonhomogeneous beam problem with crack is solved to show the application of the conservation integrals.

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49

Kholmetskii, A. L., O. V. Missevitch, and T. Yarman. "Energy–momentum conservation in classical electrodynamics and electrically bound quantum systems." Physica Scripta 82, no. 4 (September 14, 2010): 045301. http://dx.doi.org/10.1088/0031-8949/82/04/045301.

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50

Koivisto, Tomi. "A note on covariant conservation of energy–momentum in modified gravities." Classical and Quantum Gravity 23, no. 12 (June 5, 2006): 4289–96. http://dx.doi.org/10.1088/0264-9381/23/12/n01.

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