A variant of the mechanistic gravitation theory is considered as a 4D theory of elasticity of the event space. A 4D displacement vector is introduced, where the fourth component is the local uneven time of the physical process generating the gravitational field. An analysis of the kinematic model of the mechanistic theory of gravity is presented. It is shown that Einstein’s gravity is a particular theory of the 4D theory of elasticity of the event space with a field of defects. Kinematic models of 4D space-time continuum are proposed, allowing to formulate the variational mechanistic models of gravity. Lagrangian of gravitational field models are formulated for the kinematic variables of a defect-free space–time continuum, continuum with conserved dislocations and a 4D space-time continuum with generated dislocations and conserved disclinations.

The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated.

We consider the classical problem of elasticity theory concerning the conditions of strain compatibility, which ensure the determination of a continuous field of displacements of an elastic body by the strain field. We construct generalized Cesàro representations that allow defining the displacement field through integrodifferential operators on the components of the strain tensor deviator with an accuracy up to quadratic polynomials. It has been established that the quadratures both for the pseudovector of local rotations and for the bulk strain are completely determined by the strain deviator field. We present the conditions for the existence of the listed quadratures, which are written in the form of five third differential order compatibility equations for the five components of the strain deviator tensor.

In the article we conside the general model of the gradient theory of elasticity, in which the deformed state energy is determined in addition to classical deformations by two additional modules with a dilatation gradient and a double displacement rotor. It is shown that the general displacement vector in this model can be represented as a superposition of classical displacements and a cohesive field that satisfies the Helmholtz-type scaling equation with two scaling parameters. Based on this expansion a generalized Papkovich–Neuber representation is proved, which expresses displacements in the gradient elasticity in terms of auxiliary potentials that satisfy the Helmholtz, Laplace, and Poisson equations. With its help fundamental systems of solutions in the gradient elasticity are constructed, which are equivalent to a system of harmonic polynomials. These systems have the property of completeness and minimality and are used to approximate solutions in problems of the gradient theory of elasticity. For the case of inhomogeneous media with multilayer spherical inclusions these solutions analytically exactly satisfy all contact conditions at the interphase boundaries. As an example, an exact solution of the Eshelby problem for a multilayer spherical inclusion in the uniform strain field is given. This solution can be used to evaluate the effective properties of the scale-effect composites using the Christensen–Eshelby method.

A method is proposed for constructing variational models of continuous media for reversible and irreversible processes based on the generalized Hamilton–Ostrogradskii principle, which reduces to the principle of stationarity for a non-integrable variational form of a spatial-temporal continuum. For dissipative processes, the corresponding linear variational form is constructed as a sum of variation of the Lagrangian of the reversible part and a linear combination of dissipation channels of physically nonlinear processes. Examples of using the variational approach to the description of hydrodynamic models are considered. The corresponding variational models of the Darcy hydrodynamics, linear Navier–Stokes hydrodynamics, Brinkman hydrodynamics, gradient hydrodynamics, and some generalization of the classical nonlinear Navier–Stokes hydrodynamics are constructed. For modeling irreversible processes of hydrodynamics with allowance for coupling of deformation with the associated physical processes of heat transfer, it is proposed to use variational formalism for the spatial-temporal continuum, where the spatial and temporal processes are considered simultaneously and consistently because the normalized time is a coordinate.

Two formulations of problems in the theory of elasticity in stresses are considered. The first one is based on Papkovich’s compatibility equations. The second one is based on the Saint-Venant compatibility equations. It is shown that the Cesaro formulas in both formulations make it possible to introduce as a vector of indefinite Lagrange multipliers the vector of partial solutions of inhomogeneous equilibrium equations satisfying the Neumann vector problem. On the other hand, it is shown that the compatibility equations introduced as links between distortions (Papkovich compatibility) or deformations (Saint-Venant compatibility) allow one to introduce the corresponding tensors of indefinite Lagrange multipliers. It is shown that these tensors can be considered as functions of stresses. In the first setting, the stress function tensor of the second rank has nine components, since it is generally asymmetric. In the second formulation, the stress function tensor is symmetrical and has six components. In particular, the possibility of introducing three stress functions is also discussed.

Abstract—Three material lifetime models are compared within the framework of the hypothesis of accumulation of residual strains. The models are based on the Ramberg–Osgood law, an alternative empirical law, and a so-called theoretical law constructed from the solutions of various differential equations in linear and nonlinear sections in a stress–strain curve. Postulating a linear section in a stress–strain curve is shown to automatically leads to the fact that the endurance limit is determined by the point of proportionality limit.

The problem of Brinkman hydrodynamics is considered. A variational generalized Sedov model is constructed to simulate the problem of filtration of a viscous and, in the general case, compressible fluid in a porous medium, taking into account the adhesion interactions between the fluid and the porous structure, considered as a solid deformable body. It is proposed to use a variant of the model of surface interactions, characterized by a reversible component in the dynamic process and a dissipative component, for the introduction of which the corresponding dissipation channel is formulated in the Sedov variational principle. An analytical solution of the problem of the flow of an incompressible fluid around a solitary inclusion of a spherical or cylindrical shape is constructed under the condition of a polynomial behavior of the velocity at infinity. As an application, the problem of determining the effective properties of filtration in a flow around a periodic system of spheres or cylinders is considered. To solve this problem, it is proposed to use the Trefftz approximation scheme developed earlier for the problems of the gradient theory of elasticity. As an approximating basis, it is proposed to use an analytically constructed system of solutions for the flow of an incompressible fluid with polynomial behavior at infinity around a solitary inclusion. A constructive method is proposed for constructing this system of functions in a closed finite form using the generalized Papkovich–Neiber representation and a system of canonical potentials constructed using radial multipliers (functions that depend only on the radius) based on harmonic polynomials. In this case, the Gauss theorem on the representation of an arbitrary homogeneous polynomial in the form of an expansion in terms of a system of homogeneous harmonic polynomials is used. It is shown that due to the analytical construction of the used functions, the generalized Trefftz scheme can be effectively applied in the cell problem when we determine the effective permeability.

A variant of the model of coupled thermohydrodynamics and heat balance is proposed. We use Sedov’s variational principle and develop (a closed energy–consistent mathematical gradient model for a four-dimensional generalized space–time continuum). In accordance with Sedov’s method for modeling dissipative processes, the principle of stationarity of a functional is generalized to the principle of stationarity of a non-integrable variational form. The physical properties of the transversely-isotropic four-dimensional continuum are determined by generalized properties tensors of the third, fourth and fifth ranks, constructed so that the internal forces and moments depend only on distortions. In general, the presented model contains gradient and scale parameters, which determine relaxation effects in both thermomechanical and thermal processes. Variants of Brinkman-type fluid models of heat transfer processes for compressible and incompressible media are obtained as a result of the analysis of the general equations of motion of the space–time continuum. In particular, the equations of thermoelasticity, thermohydrodynamics, Navier-Stokes equations for compressible and incompressible media, Darcy equations, Brinkman equations are obtained.

We consider the coupled processes of thermohydrodynamics and heat conduction and construct a variational model of such coupled problems using four-dimensional space-time continuum where time is an equal coordinate along with spatial coordinates. In this consideration 3D subspace of generalized four-dimensional pseudo-continuum is associated with three-dimension deformed media. In the general case we consider irreversible processes and use the variational principle of possible displacements and the variational Sedov’s principle to construct variation model. It is assumed that the 4D pseudo-continuum considered below is transversely isotropic in the direction of the unit vector of time. Thus, an asymmetric 4D stress tensor preserves the symmetry properties with respect to spatial tangential stresses in 3D subspaces. It is shown that the proposed version of the model allows us to formulate the full range of consistent thermomechanical and thermodynamic physical relations, and the system of governing equations includes, as special cases, the equations of thermoelasticity, thermohydrodynamics, the linear Navier–Stokes equations for compressible and incompressible media, the equations of heat balance with the laws of thermal conductivity of Fourier, Maxwell–Cattaneo.

We present the theory of space–time elasticity and demonstrate that it is the extended reversible thermodynamics and gives the coupled model of thermoelasticity and heat conductivity and involves traditional thermoelasticity. We formulate the generally covariant variational model’s dynamic thermoelasticity and heat conductivity in which the basic kinematic and static variables are unified tensor objects (subject, matter). Variation statement defines the whole set of the initial-boundary problems for the 4D vector governing equation (Euler equation), the spatial projections of which define motion equations and the time projection gives the heat conductivity equation. We show that space–time elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, space–time elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity. Moreover, we establish that the Maxwell–Cattaneo law and Fourier law can be defined for the reversible processes as compatibility equations without introducing dissipation. We argue that the present framework of space–time elasticity should prove adequate to describe the thermoelastic phenomena at low temperatures for interpreting the results of molecular simulations of heat conduction in solids and for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.

It is considered a continuum theory of the adhesion properties of the surface of elastic bodies, which can be considered as theory of surface elasticity. We consider the surface of the body as the set of all the boundary points of the elastic body and believe that upon deformation, this surface is endowed with its own density of surface energy in the case of an adhesion-active surface. The definition of the “ideal” and gradient theory of elasticity of surface interactions is given, and it is shown that the ideal adhesion theory constructed by Gurtin and Murdoch, taking into account the properties of symmetry and material indifference, is far from complete. The work gives a fairly broad generalization of the surface-related theory of elastic bodies. The statements of the problems of propagation of surface waves on the adhesion-active surface of the classical elastic half-space are considered. We considered five types of surface waves that are attractive from the point of view of experimental determination of the characteristics of adhesive interactions and found that these types of surface waves could not be existed for the classical theory of elasticity with adhesion-passive surfaces, where moduli of the adhesion interactions are equal zero. The first three of these types of waves are associated separately with each of the three components of the surface displacement vector. The fourth and fifth types of surface waves are associated, respectively, with the field of local changes in the surface area and with the field of its local rotations with a vector that coincides with the normal to the surface.

We consider generalized variational non-local models of media with fields of defects and show that the methods of continuum mechanics are very effective in modeling connected reversible and irreversible thermomechanical processes. It is postulated that the tensor of free distortions is determined only by the spherical tensor, which is interpreted as a dilatation associated with a change in temperature. A variational model of coupled thermoelasticity and hyperbolic thermal conductivity is under construction. It describes the general case of non-locality, when gradient properties are determined by scale parameters that are responsible for both mechanical and temperature effects. The analysis of boundary value problems is given, the physical interpretation of all model parameters is given through known thermomechanical parameters. We also offer a variation model of irreversible thermodynamic processes, which is based on the principle of L. I. Sedov. In this case, the variation form for the dissipative part of the change in energy is based on the non-integrability condition proposed by the authors.

A gradient theory of the coupled theory of elasticity, thermoelasticity and thermal conductivity based on a generalized model of media with fields of defects is developed. Defectiveness is defined only by free dilatation and the tensor of incompatible distortions is determined by the spherical tensor. In the general case, the unified model includes scale parameters that are responsible for mechanical and temperature scale effects. In the particular case the proposed model describes the gradient thermoelasticity, in which effects are controlled by a mechanical scale parameter, and in the limit case, it describes the classical thermoelasticity, when this parameter tends to zero. The analysis of boundary problems of the general model is given. Particular cases are considered, and it is shown that gradient thermal conductivity and thermoelasticity make it possible to simulate thermal resistance and size effects.

In this work, the methods of continuum mechanics are transferred to the four-dimensional continuum, in which the normalized four-dimensional vector-potential of the electromagnetic field is playing the role of 4D-displacement. Classical electrodynamics is presented as a theory of elasticity of 4D-medium with anti-symmetric stress tensor, where Faraday equations are the equations of compatibility, and Ampere equations are the equations of equilibrium. It is shown, that for the considered continuum, the tensor of the fourth rank modules in defining ratios formally allows not only anti-symmetric structure. Uncertain symmetry of electromagnetic “stresses” allows to build a version of the non-anti-symmetric electrodynamics and to predict new effects of the interaction of electromagnetic field with spatially-isotropic material: dynamic, thermal and striction.

Краткий проблемно-ориентированный анализ угроз и вызовов национальной безопасности России, а также категориально-понятийного, методологического, политико-правового и научно-образовательного обеспечения деятельности по их парированию

A sequential presentation of the theory of media by conserved dislocations as a variant of the theory of media with a microstructure (according to Mindlin's definition) is given as well as a rather complete description of the particular variants of the theory, relevant from an applied point of view: Cosserat and Aero-Kuvshinskii media, porous media, media with "twinning". The correctness of the formulation of models is determined by the use of a "kinematic" variational principle based on a formal description of the kinematics of media, the formulation of kinematic constraints for media of different complexity, and the construction of the corresponding potential energy of deformation using the Lagrange multiplier procedure. A system of defining relations is established and an agreed formulation of the boundary value problem is formulated. In this paper, much attention is paid to the analysis of the physical side of models of the media studied. The interpretation of all physical characteristics responsible for nonclassical effects is proposed, and a description of the spectrum of adhesive mechanical parameters is given. The generalized Aero-Kuvshinskii hypothesis about the proportionality of free and constrained distortions is proposed.

A general covariance variational model of reversible thermodynamics is developed in which the kinematic and force variables are the components of unified tensor objects in the space−time continuum, and the resolving equations of the dynamic thermoelasticity and heat-conduction of an ideal (defect-free) media are described by the 4D-vector equation. It is shown that the formulations of relations of the generalized Duhamel−Neumann representation and the Maxwell−Cattaneo law follow directly from the constitutive relations of the space−time-continuum model without additional hypotheses and assumptions. It is proved that the Maxwell−Cattaneo and Fourier generalized heat-conduction laws are unambiguously characterized by well-known thermomechanical parameters determined under isothermal and adiabatic conditions for reversible coupled deformation processes and heat-conduction despite the fact that one usually relates both the Fourier law and the relaxation time in the Maxwell−Cattaneo law with the dissipative processes.

We consider the generic gradient elasticity theory of Mindlin-Tupin and try to establish a class of applied models of gradient elasticity, for which the boundary value problems of the gradient theory with static boundary conditions are divided into a sequence of two subtasks, one of which is classical. Such applied models are very effective in applications, because their solutions reduce exactly to a consistent solution of boundary value problems of the second and not of the fourth order. We consider gradient theories with a general structure of tensors of gradient modules that satisfy potentiality conditions and additional symmetry conditions, which is considered as a criterion of correctness. It is shown that their gradient tensors of the elastic modules are represented in the form of an expansion with respect to the tensor basis of five sixth-rank tensors, three of which satisfy a special property. Each of these basis tensors is represented as a convolution of fourth-rank tensors, and the corresponding quadratic form is a convolution of vectors. It is shown that for the traditional gradient Mindlin-Tupin theory, the "classical" static conditions on the body surface are not satisfied locally. However, if the gradient modules are represented as a convolution of the "classical" tensors of elastic moduli, then the set of the boundary value problems of such gradient theory admits a full fractionation of the initial boundary value problem into two: the "classical" boundary value problem and the "cohesive" boundary value problem. It is established the structure of the applied gradient models with such property of separating boundary value problems. They are particular cases of gradient elasticity theories with gradient modulus tensors, representable in the form of an expansion in three basis tensors of the sixth rank, satisfying the properties of the representation in the form of convolution via fourth-rank tensors. We formulated "vector" gradient Mindlin-Tupin model that preserves the classical form of static boundary conditions. Such a model leads to a specific variant of the gradient theory with a single non-classical modulus, or one-parametrical model. It is shown that the obtained gradient model can be considered as some generalization of the well-known applied theory GradEla providing for it the separation of boundary value problems.