Papers by Armine Ulukhanyan
Continuum Mechanics and Thermodynamics, 2022
Continuum Mechanics and Thermodynamics, Mar 27, 2019
In this work, we considered the new parametrization of a multilayer thin domain. In particular, i... more In this work, we considered the new parametrization of a multilayer thin domain. In particular, in contrast to classic approaches, we used several base surfaces and an analytic method with the application of orthogonal polynomial systems. We gave the vector parametric equation of each layer and the system of vector parametric equations of a multilayer thin domain and introduced the geometric characteristics for the proposed parametrization. We also derived the expressions for the transfer components of the second-rank identity tensor and the relations connecting the various families of bases and presented some differential operators, the system of equations of motion, the heat flow equation, the constitutive relations of the theory of the micropolar elasticity, and the Fourier heat conduction law under this parametrization of the thin-body domain. Finally, we gave the classification and statements of boundary value problems in the theory of thin bodies.
Mechanics of Solids, 2013
ABSTRACT Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being de... more ABSTRACT Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1–14] and by many others, as well as by the symposiumdedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held inParis in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17–24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21].It should be noted that, in a majority of cases, the structure strength calculations are based on the classical theory of elasticity. But there are materials such as animal bones, graphite, several polymers, polyurethane films, porous materials (pumice), various synthetic materials, and materials with inclusions which, under certain conditions, exhibit micropolar properties. There are effects which cannot be prescribed by the classical theory. In statics, nonclassical behavior can be observed in bending of thin films and cantilevers, in torsion of thin and thin-walled rods, and in the case of stress concentration near holes, corner points, cracks, and inclusions. For example, thin specimens are more rigid in bending and torsion as is prescribed by the classical theory [25–27]. The stress concentration near holes decreases, and the concentration factor depends on the radius [28]. The stress concentration near cracks also becomes lower. Conversely, the stress concentration near inclusions is higher than predicted by the classical theory [29–31]. If the material has no center of symmetry of elastic properties, then calculations according to the micropolar theory shows that the specimen is twisted in tension [32]. In dynamical problems, several phenomena also differ from the classical concepts. For example, shear waves propagate with dispersion, microrotation waves arise, and the vibration natural modes differ from the classical ones [2, 7, 11–13, 33]. All these phenomena are used to determine material constants of the micropolar theory of elasticity. There are many methods for determining such constants [2, 34].Since thin bodies (one-, two-, three-, and multilayer structures) are widely used, it is necessary to create new refined microcontinual theories of thin bodies and advanced methods for their computations. In the present paper, various representations of the system of equations of motion are obtained in the micropolar theory of thin bodies with two small parameters in momenta with respect to a system of Legendre polynomials in the case where an arbitrary line is taken for the base. In this connection, a vector parametric equation of the region of a thin body is given for the parametrization under study, different families of bases (frames) are introduced, and expressions for components of the unit tensor of rank two (UTRT) are obtained. Representations of gradient, tensor divergence, equations of motion, and boundary conditions for the considered parametrization are given. Definitions of (m, n)th-order moment of a variable with respect to an arbitrary system of orthogonal polynomials and a system of Legendre polynomials is given. Expressions for themoments of partial derivatives and several expressions with respect to a system of Legendre polynomials and boundary conditions in moments are obtained.
The general solutions to hyperbolic equations of fourth and sixth order are obtained using Vekua’... more The general solutions to hyperbolic equations of fourth and sixth order are obtained using Vekua’s method for the representation of the general solutions to elliptic equations of order 2n with the aid of n analytic functions. It is assumed that the right-hand sides of the hyperbolic equations can be expanded in time series of sines. The systems of equations of various approximations for a prismatic thin body in terms of moments with respect to the system of Legendre polynomials can be reduced to these equations and to some hyperbolic-type equations of higher order.
Some questions about the new parameterization of three-dimensional thin body with one small size ... more Some questions about the new parameterization of three-dimensional thin body with one small size are considered. The vector parametric equations of the multilayered thin body are given. The geometric characteristics are considered. The representations of several differential operators, the equations of motion, the boundary conditions and the constitutive relations (CR) of classic and micropolar theories of elasticity are given. Some interlayer contact conditions are obtained. The definition of the k-order moment of any quantity with respect to the system of Legendre polynomials is given. The problems of the classic and micropolar theories of one-, two- and three-layered prismatic elastic thin bodies in moments of displacement and rotation vectors with respect to the system of Legendre polynomials are formulated. The problem of one-layered two-dimensional rectangular plate with two pinched edges under the distributed load are solved.
Advanced Structured Materials
Some questions about the parametrization of three-dimensional thin body with one small size under... more Some questions about the parametrization of three-dimensional thin body with one small size under an arbitrary base surface and the changing of transverse coordinate from –1 to 1 are considered. The vector parametric equation of the thin body domain is given. In particular, we have defined the various families of bases and geometric characteristics generated by them. Expressions for the components of the second rank isotropic tensor are obtained. The representations of some differential operators, the equations of motion, and the constitutive relations of micropolar elasticity theory under the considered parametrization of the thin body domain are given. The inverse tensor operators to a tensor operator of the equations of motion in terms of displacements for an isotropic homogeneous material and to a stress operator are found. They allow decomposing equations and boundary conditions. The inverse matrix differential tensor operator to the matrix differential tensor operator of the equations of motion in displacements and rotations of the micropolar theory of elasticity is constructed for isotropic homogeneous materials with a symmetry center as well as for materials without a symmetry center. We obtain the equations with respect to displacement vector and rotation vector individually. As a special case, a reduced continuum is considered. Cases in which it is easy to invert the stress and the couple stress operator are found out. From the decomposed equations of classical (micropolar) theory of elasticity, the corresponding decomposed equations of quasistatic problems of theory of prismatic bodies with constant thickness in displacements (in displacements and rotations) are obtained. From these systems of equations, we derive the equations in moments of unknown vector functions with respect to any system of orthogonal polynomials. We obtain the systems of equations of various approximations (from zero to eighth order) in moments with respect to the systems of Legendre and second kind Chebyshev polynomials. The system splits and for each moment of unknown vector function we, obtain a high order elliptic type equation (the system order depends on the order of approximation), the characteristic roots of which can be easily found. Using the method of Vekua, their analytical solution is obtained. For micropolar theory of thin prismatic bodies with two small sizes and a the rectangular cross-section, the decomposed equations in moments of displacement and rotation vectors via an arbitrary system of polynomials (Legendre, Chebyshev) are obtained. Similar equations are also deduced for the reduced medium containing classical equation. The decomposed systems of equations of eight approximations for micropolar theory of multilayer prismatic bodies of constant thickness in moments of displacement and rotation vectors are obtained. Using Vekua method, we can find the analytical solutions for this system and for equations for the reduced medium.
Higher Gradient Materials and Related Generalized Continua
The wave velocities in some media under different types of anisotropy are estimated applying the ... more The wave velocities in some media under different types of anisotropy are estimated applying the eigenvalue problem of material tensors. In this connection, the canonical representations of material tensors, as well as kinematic and dynamic conditions on the strong discontinuity surface are given. Using them the problem of finding the wave velocities is reduced to the eigenvalue problem for the corresponding dispersion tensor. In addition, dispersion equations for determining the wave velocities are obtained. In particular, classical and micropolar materials with different symbols of anisotropy (structure) are considered and the wave velocities through the eigenvalues of material tensors are found.
Mathematical and Computational Applications
Proceeding from three-dimensional formulations of initial boundary value problems of the three-di... more 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.
Journal of Physics: Conference Series
It is known that the eigenvalues of the tensor and the tensor-block matrix are invariant quantiti... more It is known that the eigenvalues of the tensor and the tensor-block matrix are invariant quantities. Therefore, in this work, our goal is to find the expression for the velocities of wave propagation of certain media through the eigenvalues of the material tensors. In particular, we consider materials with the anisotropy symbol {1.5} and {5.1}, as well as isotropic materials, and for them we determine the expressions for the velocities of wave propagation. In addition, we obtained expressions for the velocities of wave propagation for materials of cubic syngony with the anisotropy symbol {1,2,3} (the matrix of the elastic modulus tensor components has three independent components), hexagonal system (transversal isotropy) with anisotropy symbol {1,1,2,2} (the matrix of the elastic modulus tensor components has five independent components), trigonal system with anisotropy symbol {1,1,2,2} (the matrix of the elastic modulus tensor components has six independent components), tetragonal system with anisotropy symbol {1,1,1,2,1} (the matrix of the elastic modulus tensor components has six independent components). We also obtained the expressions for the velocities of wave propagation for a micro-polar medium with the anisotropy symbol {1.5.3} and {5.1.3}, and for an isotropic micro-polar material.
Mathematical and Computational Applications
The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any ev... more The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elastic...
Mechanics of Solids
For a thin anisotropic body that is inhomogeneous with respect to curvilinear coordinates x 1 and... more For a thin anisotropic body that is inhomogeneous with respect to curvilinear coordinates x 1 and x 2 and for an arbitrary homogeneous prismatic anisotropic elastic body of variable thickness with one small dimension in the case of the classical parametrization of its domain, we obtain the equations of motion of the Cosserat theory of elasticity in terms of moments with the kinematic boundary conditions of kinematic meaning and with boundary conditions of physical meaning taken into account. For prismatic bodies of constant thickness, these equations are used to obtain the system of zeroorder approximation equations for an isotropic medium. For the zero-order moments of the third components of the displacement and rotation vectors, fourth-order wave equations are obtained. In the classical theory, such equations are obtained in the first-order approximation for the zeroand first-order moments of the first invariant of the plane strain and for the third components of the displacement vector. In contrast to the Timoshenko type wave equation, in the equation for the zero-order moment of the third component of the displacement vector, the shear coefficient is k = 1. Moreover, the plate cylindrical rigidity coincides with the rigidity obtained by I. N. Vekua, and for Poisson's ratio equal to 0.5, the coefficient multiplying the acceleration is zero. In the case of a transversally isotropic medium, in the first-and second-order approximations, the fourth-and sixth-order hyperbolic equations are obtained for the zero-, first-, and second-order moments of the first invariant of the plane strain and of the third component of the displacement vector, respectively. The matrix of velocities of the wave propagation in an infinitely transversally isotropic elastic medium is written in the principal directions, and this matrix shows that the coefficients of these equations can be expressed in terms of these velocities. Most of this paper was published in [1].
Higher Gradient Materials and Related Generalized Continua, 2019
Some questions on parametrization with an arbitrary base surface of a thin-body domain with one s... more Some questions on parametrization with an arbitrary base surface of a thin-body domain with one small size are considered. This parametrization is convenient to use in those cases when the domain of the thin body does not have symmetry with respect to any surface. In addition, it is more convenient to find the moments of mechanical quantities than the classical. Various families of bases (frames) and the corresponding families of parameterizations generated by them are considered. Expressions for the components of the second rank unit tensor are obtained. Representations of some differential operators, the system of motion equations, and the constitutive relation (CR) of the micropolar theory of elasticity are given for the considered parametrization of a thin body domain. The main recurrence formulas of system of orthogonal Legendre polynomials are written out and some additional recurrence relations are obtained, which play an important role in the construction of various variants...
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Papers by Armine Ulukhanyan