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Some variational principles in thethree-dimensional micropolar theories of solids and thin solidsстатья
- Авторы: Nikabadze Mikhail, Ulukhanyan Armine
- Сборник: Theoretical Analyses, Computations, and Experiments of Multiscale Materials
- Серия: Advanced Structured Materials
- Том: 175
- Глава в коллективной монографии
- Год издания: 2022
- Издательство: Springer Nature
- Местоположение издательства: Switzerland
- Первая страница: 193
- Последняя страница: 251
- DOI: 10.1007/978-3-031-04548-6_11
- Аннотация: In this work, we formulated the variational principles of Lagrange, Castigliano, the generalized Reissner-type variational principles, as well as the principle of virtual work and the principle of complementary virtual work of three-dimensional micropolar mechanics of solids of some rheologies (for example, elastic, viscoelastic, etc.) in the case of potentiality, as well as nonpotentiality of stress and couple stress tensors. For this purpose, we have derived the necessary integral relations. We have proved the concept of living force and the generalized work theorem from which Clapeyron's theorem is derived. We have constructed Lagrangian, Castiglianian and generalized Reissner-type operators. We have given the definition of the generalized Legendre transform and obtained the Legendre-type identity. We have proved Lagrange and Castigliano's theorems concerning variational principles. We have written down the compatibility conditions with respect to tensors of strain and bending-torsion in various forms. We have presented equations of the Beltrami–Michell type for the stress and couple stress tensors with both asymmetric and symmetric differential tensor operators. We have given statements of mixed boundary value problem and initial boundary value problem with respect to vectors of displacements and rotations, as well as statements of the mixed boundary value problem and the new statement of the boundary value problem with respect to stress and couple stress tensors. We have given the definition of the generalized solution of the boundary value problem. We have proved the theorem of the minimum at the stationary point of the Lagrangian at the stationary point and the theorem of the maximum of the Castiglianian at the stationary point, as well as the theorem on the uniqueness of the generalized solution of boundary value problems. The generalized Reissner-type operator of three-dimensional micropolar mechanics of solids is presented, on the basis of which the generalized Reissner-type operator of three-dimensional micropolar mechanics of thin solids with one small size is obtained under the new parameterization of the domains of these bodies.From the last Reissner-type operator, in turn, the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size is derived under the new parametrization of the domains of these bodies. It should be noted that the advantage of the new parameterization is that it is experimentally more accessible than other parameterizations . Further, applying the method of orthogonal polynomials (expansion of unknown quantities in series in terms of a system of orthogonal polynomials), from the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size under the new parameterization of the domains of these bodies, the Reissner variational principle of micropolar mechanics of thin solids with one small size in the moments with respect to the system of Legendre polynomials is derived. In addition, the method is described for obtaining the variational principles of Lagrange and Castigliano of micropolar mechanics of thin solid with one small size under the new parametrization of the domains of these bodies in moments with respect to systems of the first and second kind Chebyshev polynomials. The effective parametrization of a multilayer thin domain, called a new parametrization, is considered and consists in using, in contrast to the classical approaches, several base surfaces. In addition, the new parameterization is characterized by the fact that it is experimentally more accessible than other parameterizations used in the scientific literature, since the front surfaces are used as basic ones. Also, when obtaining any relation (a system of equations, constitutive relations, boundary and initial conditions, variational principles, etc.) in the moments of the theory of multilayer thin bodies under the new parametrization of the domain of a thin body, it is sufficient in the corresponding relation of the theory of a single-layer thin body under the root letters of the quantities to supply the index $\alpha$, which denotes the number of the layer $\alpha$ and give this index values from 1 to $K$, where $K$ is the number of layers. Therefore, for the correct statement of the initial-boundary value problems to the equations of motion and the boundary and initial conditions in the moments, it is also necessary to add interlayer contact conditions, which must also be taken into account when writing the variational operators and formulating the variational principles. What has been said above can be called the rule of obtaining the desired relation in the theory of multilayer thin bodies from the corresponding relation in the theory of single-layer thin bodies. Applying this rule, below we give the representation of the generalized Reissner-type operator and formulate the generalized Reissner-type variational principle both in the case of full contact of adjacent layers of a multilayer structure and in the presence of zones of weakened adhesion. The description of obtaining of dual operators and variational principles of Reissner-type, as well as of Lagrangian and Castiglianian and variational principles of Lagrange and Castigliano is given. In the presence of domains of weakened adhesion at interphase boundaries in a multilayer thin body, one of the main problems is the problem of modeling the interface (interphase boundary). In this paper, the jump-type model (description of the interface by a surface of zero thickness) is presented in comparative detail.
- Добавил в систему: Никабадзе Михаил Ушангиевич