On some problems of tensor calculus. I

Nikabadze, M. U.

doi:10.1007/s10958-009-9595-8

Cited by 14 publications

(4 citation statements)

References 7 publications

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“…Here, P and µ µ µ are tensors of stresses and couple stresses, C is the discriminant tensor (third-rank tensor) [22], u is the vector of displacements, ϕ ϕ ϕ is the vector of (inner) rotation, ρ is the material density, F is the mass force density, m is the mass moment density, and 2 ⊗ is the inner 2-product (for example, C 2 ⊗ P = r i C ijk P jk ). The definition of inner r-product and the problems related to it are considered in [1][2][3]22,29]. It is easy to see that by Equation (27), Equation (28) can be rewritten in the form…”

Section: Representations Of Motion Equationsmentioning

confidence: 99%

“…Note that Equation 29is called the motion equation of the micropolar deformable rigid thin body under the new parameterization of thin body domain. Taking into account the first relation in Equation 23, we can rewrite Equation (29) in the form…”

Section: Representations Of Motion Equationsmentioning

confidence: 99%

“…The very important direction is the study of eigenvalue problems for the tensor and tensor-block matrix of any even rank, since the constitutive relations for most classical and micropolar media of different rheology (which here, of course, also include porous and multilayer textile media) are written using a tensor and tensor-block matrix of even rank, respectively. Some questions concerning these problems, as well as tensor calculus, have been studied in some detail in [4,29,[45][46][47][48][49]. A very important direction is also the investigation of internal structures of differential tensors-operators and tensor-block matrix operators of even rank.…”

Section: Conditions Under Relative Displacement Of Points Of Uneven Cmentioning

confidence: 99%

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On the Modeling of Five-Layer Thin Prismatic Bodies

Nikabadze

Ulukhanyan

Moseshvili

et al. 2019

MCA

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Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies.

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Section: Representations Of Motion Equationsmentioning

confidence: 99%

Section: Representations Of Motion Equationsmentioning

confidence: 99%

Section: Conditions Under Relative Displacement Of Points Of Uneven Cmentioning

confidence: 99%

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On the Modeling of Five-Layer Thin Prismatic Bodies

Nikabadze

Ulukhanyan

Moseshvili

et al. 2019

MCA

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“…In [1,2], differential geometric formulae of three-dimensional (3D) domains and two-dimensional (2D) surface are defined in curvilinear ordinates, respectively. Besides, there are some scientists, such as Pobedrya [3], Vekua [4], and Nikabadze [5], who have some contributions in this field. In this paper, we assume that the three-dimensional elastic shell with equal thickness comprises a series of overlying surfaces like middle surface.…”

Section: Introductionmentioning

confidence: 99%

Some Differential Geometric Relations in the Elastic Shell

Shen

et al. 2016

Mathematical Problems in Engineering

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The theory of the elastic shells is one of the most important parts of the theory of solid mechanics. The elastic shell can be described with its middle surface; that is, the three-dimensional elastic shell with equal thickness comprises a series of overlying surfaces like middle surface. In this paper, the differential geometric relations between elastic shell and its middle surface are provided under the curvilinear coordinate systems, which are very important for forming two-dimensional linear and nonlinear elastic shell models. Concretely, the metric tensors, the determinant of metric matrix field, the Christoffel symbols, and Riemann tensors on the three-dimensional elasticity are expressed by those on the two-dimensional middle surface, which are featured by the asymptotic expressions with respect to the variable in the direction of thickness of the shell. Thus, the novelty of this work is that we can further split three-dimensional mechanics equations into two-dimensional variation problems. Finally, two kinds of special shells, hemispherical shell and semicylindrical shell, are provided as the examples.

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Eigenvalue Problems of a Tensor and a Tensor-Block Matrix (TMB) of Any Even Rank with Some Applications in Mechanics

Nikabadze

2016

Advanced Structured Materials

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On some problems of tensor calculus. I

Cited by 14 publications

References 7 publications

On the Modeling of Five-Layer Thin Prismatic Bodies

On the Modeling of Five-Layer Thin Prismatic Bodies

Some Differential Geometric Relations in the Elastic Shell

Eigenvalue Problems of a Tensor and a Tensor-Block Matrix (TMB) of Any Even Rank with Some Applications in Mechanics

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