The problem on natural vibrations of a flat strip of anisotropic two-dimensional Cosserat medium under the assumption of small deformations and in the absence of external forces and moments is investigated. It is shown that two natural frequencies correspond to each wave number. The natural forms of oscillations and the relation between them are found. It is concluded that at oscillations with the lower of the two frequencies the inclusion rotations accompany the longitudinal displacement of the strip, and at oscillations with a higher frequency they prevent it. The obtained results are illustrated on the example of a medium model with specific parameter values. The plots show the dependences of natural frequencies, phase and group velocities on the wave number, and their asymptotic behavior is studied.

Рассмотрены новые голономные тензорные меры деформации и напряжений. Построены модели нелинейной упругости, с учетом которых выведены решения задач о растяжении тонкой широкой пластины и одноосном растяжении стержня из несжимаемых материалов. Полученные модели совпадают с классическими в случае малых деформаций и проявляют существенно различные свойства при больших деформациях.

We consider new holonomic tensor measures of strain and stresses and build nonlinear elasticity models for which the problems of stretching of a thin wide plate and uniaxial stretching of a rod from incompressible materials are solved. These models are congruent with classical ones when deformation is small, and they essentially demonstrate various properties at large deformations.

The fundamentals of the theory of objective tensor mechanical characteristics are presented: the general concept of objectivity, types of objectivity, simple tensor analogs of mechanical characteristics and simple commutative diagrams connecting them. For objective tensors of the second rank, a general classification of simple diagrams is given, examples of diagrams are given. The mappings connecting objective mechanical tensor processes of various ranks and types of objectivity are considered. The concept of mappings independent of the frame of reference is introduced, which form the basis of the theory of constitutive relations of media, and a theorem on the necessary and sufficient conditions for such independence is established. Examples are given. For two arbitrary diagrams of objective tensor processes (generally speaking, of different ranks), a complete set (package) of related mappings (conductors) of all possible analogs of one diagram into all possible analogs of another diagram is considered—a package of conductors, or a mapping of diagrams. For coinciding diagrams (of the same rank with the same intertwining operators), a generalizing concept of objective derivatives is introduced that make up a subpackage of conductors connecting tensors of the same (all possible) types of objectivity. General equations for the objective derivatives of scalars, vectors, and second-rank tensors are presented, presented in Lagrangian and Eulerian forms. It is shown that the well-known derivatives of Zaremba–Jaumann, Oldroyd, Cotter–Rivlin, Truesdell, Hill, Sedov, Dienes, Gordon–Schowalter are covered by these equations. The objective integration operator is constructed as the inverse operator to the objective derivative. Examples of the use of objective derivatives and objective integrals in the construction of the constitutive relations of media at finite deformations are given.

It is assumed that a certain reference frame is inertial for a system of moving and interacting bodies called a large system. In the framework of classical continuum mechanics, some necessary and sufficient conditions are obtained for the existence of a reference frame for a subsystem of this large system considered as an independent large system. The motion of such a new reference frame with respect to the old reference frame (with the accuracy up to the Galilean transformations) is specified.

Some approaches to the axiomatic formulation of theoretical fundamentals in continuum mechanics are considered. The basic concepts, laws, and hypotheses in the classical theory of continuum mechanics and their modifications for its nonclassical versions are discussed. In the framework of the classical version of the rational theory, a number of axioms are proposed for the general theory of constitutive relations. For the media of nonclassical type, the approaches to axiomatic formulation are studied by the example of the rational mechanics of moment media (Cosserat continuum): some specific notions of bodies with their attributes and the forms of their interactions and motions are introduced, the appropriate generalizations of the main laws and hypotheses are given, and the general forms of constitutive relations are analyzed for arbitrary and small strains. The approaches to the construction of medium models are discussed in accordance with the method of mechanical modeling proposed by A. A. Il’yushin.

In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate ("analogous") relations between tensors of one and the same (any) rank, and of various types of objectivity. Mappings expressing dependence between objective tensor processes of various ranks and types are considered. The fundamental concept of frame-independence of such mappings is introduced as being inherent to constitutive relations of various physical and mechanical properties in the Newtonian approach. The criteria are established for such frame-independence. The mathematical restrictions imposed on the frame-independent mappings by the objectivity types of connected tensors are simultaneously revealed. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time, and a new concept of objective integration, are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin's isotropy are generalized on the range of finite strains.

A theory of constitutive relations describing the resistance of bodies to deformation is developed. This theory takes into account the presence of internal body forces and internal kinematic constraints. A number of axioms and a general reduced form of constitutive relations for classical media are proposed. For simple bodies it is proved that the Il'yushin and Noll constitutive relations are equivalent.

A generalized theory of stress and strain tensor measures in the classical continuum mechanics is discussed: the main axioms of the theory are proposed, the general formulas for new tensor measures are derived, arid an energy conjugate theorem is formulated to distinguish the complete Lagrangian class of measures. As a subclass, a simple Lagrangian class of energy conjugate measures of stresses and finite strains is constructed in which the families of holonomic and corotational measures are distinguished. The characteristics of holonomic and corotational measures are studied by comparing the tensor measures of the simple Lagrangian class with one another and with logarithmic measures. For the simple Lagrangian class and its families, their completeness and closure are shown with respect to the choice of a generating pair of energetically conjugate measures. The applications of the new tensor measures in modeling the properties of plasticity, viscoelasticity, and shape memory are mentioned.

The theoretical advancement in the field of modern nonlinear continuum mechanics is discussed. The paper includes elements of the mathematical apparatus, development of the foundations of a General tensor theory of mechanical processes and their representations, including generalization of concepts of objective derivatives and integrals, concepts of tensor measures of stresses and deformations, new approaches in theory of the resistance of solids to deformation.

The article presents approaches to the formulation and methods of solving initial boundary value problems of solid mechanics. Classical formulations of problems are considered. Principal scheme of the generalized formulation of problems in the form of operator equations in function spaces is presented and illustrated on the example of boundary value problems in the theory of small elastic deformations. The mathematical structure of iterative methods (method of elastic solutions and its modifications) and incremental approaches are stated in the article. Theorems on the existence and uniqueness of solutions, the convergence of the methods are given. The author discusses specific issues of formulating initial boundary value problems for finite deformations. The difficulties of the Lagrangian and finiteness of Eulerian descriptions are noticed. The conditions of possibility of effective application of the Euler formulation of problems that lead to substantial restrictions on the mechanical properties of material are displayed. The examples of the lack of the solutions at finite strains are given and unreasonableness of a requirement of uniqueness of solutions of problems of statics is shown. For evolutionary problems the author proposes a hypothesis on uniqueness of solutions as continuous-time field processes.

The traditional principles of the theory of constitutive relations in classical continuum mechanics are discussed. In light of the approaches by A. A. Ilyushin and by W. Noll, the equivalence and the completeness of their general reduced forms of constitutive relations for simple classical media are noted. First, it is mentioned that a possible presence of internal kinematic constraints is rarely taken into account and needs special modifications in formulating the principles and relations. Secondly, a systematic study of internal body forces in constitutive relations has not been done before. Here a unified approach to the theory of constitutive relations is proposed to describe properties of deformation resistance of a body including both internal kinematic constraints and internal body forces. The general reduced forms of the system of constitutive relations are derived in terms of different definitions of a dynamical process in a body. The case of a simple body is considered in detail in view of the theory of objective tensors, their diagrams and frame-independent relations between objective tensors. The completeness of Ilyushin’s and Noll’s types of relations for the most general constitutive formulations is noted and confirmed by examples.

We present a survey of the main results and scientific ideas due to the 20th century prominent scientist Aleksey Antonovich Il’yushin, which can be regarded today, at his hundredth birthday anniversary, as his scientific heritage. The survey material is arranged (mainly, chronologically) in the key directions of Il’yushin’s activities such as the theory of viscoplastic flow, hydrodynamic stability, dynamics of deformable media, supersonic aerodynamics and related flatter problems, theory of elastoplastic processes, theory of plastic flow, thermoviscoelasticity and thermodynamics, strength of polymer bodies and structures, general theory of constitutive relations in classical continuum mechanics, and nonclassical models of continuum.

Представлен обзор основных результатов и научных идей выдающегося ученого XX века Алексея Антоновича Ильюшина, которые сейчас, в дни его столетнего юбилея, можно трактовать как научное наследие. Материал обзора структурирован (преимущественно хронологически) по ключевым направлениям деятельности А.А. Ильюшина: теория вязкопластического течения, гидродинамическая устойчивость, динамика деформируемых сред, сверхзвуковая аэродинамика и связанные с этим проблемы флаттера, теория упругопластических процессов, теория пластического течения, термовязкоупругость и термодинамика, прочность полимерных тел и конструкций, общая теория определяющих соотношений в классической механике сплошной среды, а также неклассические модели сплошных сред

A multiphase model of a liquid- and gas-saturated porous medium is proposed. The model takes into account the finite deformations of a skeleton, arbitrary flows of liquids and gases, and the phase mass transfer between the skeleton and the liquids. Some basic relations are given for the corresponding boundary value problems in the cases of arbitrary and small motions. The constitutive relations describing the properties of the skeleton resistance to deformation (skeleton stresses) and the mutual resistance of components (internal interaction) are considered in detail. The internal actions exerted on the skeleton by a moving liquid (gas) are discussed when these actions may take the form of drag forces, lift (displacing) forces, overturning moments, and rotational (screw) moments.

A model of an equipped elastic rod is considered. In the average sense, this model shows the properties of the one-dimensional Cosserat continuum during longitudinal and torsional motions. Natural and forced torsional vibrations are studied in the case of flow loading. Several conditions for vibration stability and for the end of vibrations are formulated. The following distinctive features of motion are found: each vibration mode has two different shapes and two different frequencies and the onset of the divergence regime is observed when the external loads become more intensive.

The aim of this work is to touch some analytical and geometrical aspects in formulations of mathematical problems in classic and non-classic continuum mechanics and to demonstrate the connection of these aspects in generalized theory of stress and strain tensor measures, in finite plasticity, in application of the method of mechanical modeling to building-up models of Cosserat type structures and saturated porous media.

The rod models of longitudinal, torsional, and bending vibrations are used to find the natural vibration spectra of a carbon nanotube. The spectrum of natural radial vibrations is found using the membrane theory of cylindrical shells. The coefficients of these models are chosen by comparing the results obtained on the basis of the micromodel with the Keating interaction potential in the framework of the long-wave approximation and on the basis of a continuous model. It is shown that the spectra of longitudinal, radial, and torsional vibrations of the carbon nanotube are of the same order of magnitude (the minimum frequency is about 1011 Hz), whereas the natural frequency spectrum for the bending vibrations is of two orders of magnitude less (the minimum frequency is about 109 Hz). These spectra belong to the super-high frequency range.

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.

Approaches to mechanical (constructive) modeling of nonhomogeneous media with complicated structure are presented. The constitutive and motion equations of special type discrete systems are obtained as average equations demonstrating the Cosserat type properties of the systems. The equations for saturated porous media are proposed with special attention to the different types of inner interactions. The invariance properties of these interactions and their quantitative contributions are analyzed using the principle of material objectivity and the methods of measurement theory. The invariances of compound (force, moment) internal interactions are studied.