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Scalars and Vectors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The permutation symbol in N dimensions, also called the Levi-Civita symbol, has N indices and is denoted εij…. It is defined in such a way that ε12…N = 1 and it is anti-symmetric in all indices. This means that if any two indices change positions, the permutation symbol changes sign. In particular, this means that if any two indices are the same, the permutation symbol takes the value zero. In three dimensions, the permutation symbol εijk has three indices and six non-zero components () ε123=ε231=ε312=−ε132=−ε321=−ε213=1,
A microstructure-dependent Kirchhoff plate model based on a reformulated strain gradient elasticity theory
Published in Mechanics of Advanced Materials and Structures, 2022
Gongye Zhang, Chenyi Zheng, Changwen Mi, Xin-Lin Gao
According to the reformulated strain gradient elasticity theory, the total strain energy U in an elastically deformed isotropic linear elastic material occupying region Ω has the form: where dV is a volume element, and are, respectively, the components of the Cauchy stress, symmetric part of the double stress and symmetric part of the couple stress, and and are, respectively, the components of ε, ηS and χS defined by with ui being the displacement components and θi being the components of the rotation vector given by in which is the permutation (Levi-Civita) symbol.
Hopfions, heliknotons, skyrmions, torons and both abelian and nonabelian vortices in chiral liquid crystals
Published in Liquid Crystals Reviews, 2022
Jin-Sheng Wu, Ivan I. Smalyukh
The elastic part, determined by the spatial derivatives of the tensorial order parameter, reads: with x being the spatial coordinates and ε the antisymmetric tensor (Levi-Civita symbol). Summation over all indices is assumed in Equation (14). These elastic distortion modes can be related to those in Frank-Oseen free energy (Equation (3)), as derived in Ref. [60] by neglecting the spatial gradient of the scalar order parameter and enforcing that S = Seq: