For the first time, an asymptotic solution has been constructed for the contact problem of friction of a rigid half-strip stamp of an anisotropic multilayer composite material. The relative width of the half-strip is assumed to be a large parameter that determines the asymptotic expansion. The method is based on generalizing the approach to constructing asymptotic solutions for simpler contact problems. Previously, the asymptotic method has been developed to solve the contact problem in case of a strip-stamp with a large relative width. The method proved to be effective because it provided a satisfactory agreement with the solution constructed by the counter asymptotic expansion for the stamps in the form of a strip of a small relative width. In this paper, it is applied in a much more complex and previously unsolved contact problem for a stamp in the form of a semi-infinite strip. The complexity of this problem lies in the fact that in order to apply the asymptotic approach, it is necessary to develop a method for solving two-dimensional Wiener-Hopf equations, which has already been done by the authors and it is already used in this work. Similar problems occur in engineering practice and construction when creating various objects, when developing an electronic element base, in seismology, when assessing the state of seismicity in the transition zone of a mountain range into a plain. By using the existing numerical methods, it is possible to describe the behavior of the concentration of contact stresses at the stamp boundary, and especially at the corner points of the boundary, where the most vulnerable parts of the structure are located. However, it is not possible to construct a complete solution of the distribution of contact stresses under the half-strip stamp together with the features at the boundary, due to the area infiniteness. In this paper, a solution is constructed that correctly reflects the real distribution of the contact stresses under the stamp and aims at an accurate solution with an increasing strip width parameter.

In this paper, for the first time, an exact solution is constructed to the contact problem of the action of a rigid wedge-shaped die with an obtuse angle on a layer of composite material having arbitrary anisotropy. The study is based on the application of the exact solution of the two-dimensional Wiener-Hopfi integral equation for a wedge-shaped die with a right angle, previously constructed by the block element method. This made it possible, thanks to the homeomorphism of stamp carriers as topological spaces, to construct exact solutions to contact problems for wedge-shaped, obtuse-angled stamps. In comparison with strip stamps, the solution contains an additively additional term describing the concentration of contact stresses at the corner point, that is, at the top of the stamp. The calculation of the characteristic of the contact stress concentration at this point is close to the values obtained by approximate numerical methods in a number of studies. In the area considered away from the top of the stamp, the exact solution becomes the solution for the case of a semi-infinite stamp. The developed method is applicable to composites of arbitrary anisotropies occurring in linearly elastic materials and crystals of any cross-section. It can be useful in engineering practice and in seismology in assessing contact stress concentrations in zones of lithospheric plates having similar angles.

For the first time, an exact solution is given to the contact problem of the non-stationary action of a wedge-shaped, right-angled stamp occupying the first quadrant, which act on a deformable multilayer base. The base, which is affected by a rigid stamp in the shape of a quarter plane, can be a multilayer anisotropic composite material. It is assumed that it is possible to construct a Green’s function for it, which makes it possible to construct an integral equation of the contact problem. The geometric Cartesian coordinates of the first quadrant and the time parameter, which varies along the entire axis, are taken as parameters describing the integral equation. It is assumed that time in the boundary value problem under consideration follows from negative infinity, crosses the origin and grows to infinity, covering the entire time interval. Thus, there is no requirement in the formulation of the Cochet problem when it is necessary to set initial conditions. In this formulation, the problem is reduced to solving the three-dimensional Wiener–Hopf integral equation. The authors are not aware of any attempts to solve this problem analytically or numerically. The investigation and solution of the contact problem was carried out using block elements in a variant applicable to integral equations. It is proved that the constructed solution exactly satisfies the integral equation. The properties of the constructed solution are studied. In particular, it is shown that the solution of the non-stationary contact problem has a higher concentration of contact stresses at the edges of the stamps and at the angular point of the stamp, compared with a static case. This corresponds to the observed in practice more effective non-stationary effect of rigid bodies on deformable media, for their destruction, compared with static. The results may be useful in engineering practice, seismology, in assessing the impact of incoming waves on foundations, in the areas of using Wiener–Hopf integral equations in probability theory and statistics, and other areas.

In this work, for the first time, an exact solution of the contact problem of interaction with a multilayer base of two semi-infinite stamps, the ends of which are parallel to each other, is constructed. The stamps are assumed to be absolutely rigid, and the distance between them can have any finite value. The task is an important stage in the algorithm for constructing models of a new type of crack in materials of different rheologies. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been little studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The problem considered in this article can be considered as the result of the formation of a new type of crack with absolutely rigid banks and a deformable lower boundary. Thanks to it, after the solution, it becomes possible to switch to deformable stamps and a crack of a new type in the rheological medium. The solution of this problem turned out to be possible due to the construction of exact solutions of the Wiener–Hopf integral equations on a finite segment. This paper shows how the solution of one of the previously unsolved problems allows us to investigate and solve exactly other problems and to identify previously unknown properties and resonances. As a result of constructing an exact solution to the problem, the fact that the solution of dynamic contact problems for stamp systems is not unique was confirmed and a dispersion equation for finding resonant frequencies was constructed.

This paper is the first ever rigorous investigation of a two-dimensional dynamic contact problem of the action of a deformable die on a quadrant of a multilayer medium. Compared to an absolutely rigid die, its deformable counterpart introduces additional features. They consist in the potential occurrence of discrete resonances predicted by Academician I. I. Vorovich. The paper shows that a method based on using brick-level elements makes it possible to derive an equation describing resonant frequencies to study contact problems with a deformable die made of complex-rheology materials, including smart materials. First of all, the case of a deformable die made of a simple-rheology material described by the Helmholtz equations is considered. Solutions of the boundary problems for complex-rheology dies are represented as a combination of solutions of boundary problems for simple-rheology dies.

For the first time, the exact solution of the dynamic contact problem of the frictionless action of a rigid die in the form of a half-strip on an anisotropic multilayer composite base is constructed by the block element method. It is assumed that the stamp is subjected to a harmonic effect in time, causing a wave process outside the contact zone. Thus, for the first time, the two-dimensional Wiener – Hopf integral equation with a difference kernel in the region representing the half-band is precisely solved. Using known numerical methods, it is possible to describe the behavior of the concentration of contact stresses at the stamp boundary in cases of isotropic materials. However, it was not possible to construct an accurate solution for the distribution of contact stresses in the anisotropic case under a half-strip stamp, together with features at the boundary. For the first time, a solution was constructed reflecting the real distribution of contact stresses and their concentrations under the stamp. The solution obtained in the work tends to the solutions obtained for a strip or a quarter of the plane when the half-strip degenerates into these areas. To ensure the correct formulation of the problem, the principle of Mandelstam's marginal absorption is applied. The block element method, factorization methods, and the Newton – Kantorovich method are used in the research process. The constructed exact solution of the contact problem makes it possible to distinguish functions describing the concentration of contact stresses at the boundaries of the stamp, including at the corner points of the half-strip stamp. The constructed indicators of the concentration of contact stresses in the angular zones of the half-strip stamp are close to the values constructed earlier in publications by approximate methods. The result of this work can be useful in engineering practice, seismology, as well as in other fields.

В работе впервые дается точное решение двумерного интегрального уравнения Винера - Хопфа, широко применяемого в смешанных, в том числе контактных, задачах. Рассматривается уравнение, описывающее контактную задачу о действии жесткого штампа в четверти плоскости на анизотропную деформируемую среду произвольной реологии. Попытки аналитического решения этой задачи предпринимались многими авторами, однако они завершались получением тех или иных приближенных решений. Разработанный авторами универсальный метод моделирования, основанный на свойствах блочных элементов, а также решения двумерных интегральных уравнений с мероморфной функцией в ядре подсказали подход, позволивший построить точное решение двумерного интегрального уравнения методом факторизации.

The paper for the first time develops a method for solving dynamic contact problems on the effect of a rigid die in the form of a strip of finite width on a layered anisotropic composite. By applying Mandelstam's principle of marginal absorption, the initial boundary value problem is reduced to a boundary value problem with absorption having a single solution. The symbol of the integral equation has no singularities on the real axis. The contact problem is reduced to solving a two-dimensional integral equation with a difference kernel. The application of the Fourier transform along the coordinate along the strip reduces the integral equation to a one-dimensional one containing a free real parameter of the Fourier transform. Integral equations with an exact solution are introduced, with symbols majoring above and below the symbol of the integral equation. Using the factorization method, the initial integral equation is reduced to two integral equations of the second type, the operator of which turns out to be compressive with a sufficiently large bandwidth. By applying the Newton–Kantorovich method to this integral equation, an exact solution of the integral equation is constructed in an operator form. The presence of the majorant symbol of the integral equation allows us to obtain an upper and lower estimate of the constructed exact solution of the integral equation containing the parameter of the Maldenstam limit absorption principle. After that, in the constructed solution, this parameter rushes to zero from above. The solution of the integral equation is obtained in an analytical form and allows us to identify all its singular features. The result is important when searching for harbingers of an increase in seismicity in mountainous areas. The method is applicable in all cases when it is possible to construct a Green function for a non-mixed boundary value problem in a layered anisotropic composite.

The paper presents for the first time one of the methods for studying and solving contact problems with a deformed stamp for those cases when there is a need to change the rheology of the stamp material. It is based on a new universal modeling method previously published by the authors, which is used in boundary-value problems for systems of partial differential equations. With its help, solutions of complex-vector boundary-value problems for systems of differential equations can be decomposed into solutions of scalar boundary-value problems for individual differential equations. Among them, the Helmholtz equations are the simplest. The solutions to the scalar boundary-value problems are represented as fractals, self-similar mathematical objects, first introduced by the American mathematician B. Mandelbrot. The role of fractals is performed by packed block elements. The transition from systems of differential equations in partial derivatives to individual equations is carried out using the transformation of Academician B.G. Galerkin or representation by potentials. It is known that the solutions of dynamic contact problems with a deformable stamp of complex rheology are cumbersome and their study is always difficult. The problem is complicated by the presence of discrete resonant frequencies in such problems, which were once discovered by Academician I.I. Vorovich. A contact problem with a deformable punch admits the construction of a solution if it is possible to solve the contact problem for an absolutely rigid punch and construct a solution to the boundary problem for a deformable punch. In earlier works of the authors, the deformable stamp was described by a separate Helmholtz equation. In this paper, we consider a contact problem on the action of a semiinfinite stamp on a multilayer base, described by the system of Lame equations. One of the methods of transition to other rheologies is shown when describing the properties of a deformable stamp in contact problems.

Актуальность работы состоит в необходимости дальнейшего развития применения высокоточных механико-математических методов в проблеме прогноза нарастания сейсмичности. В частности, строгих математических подходов исследования сейсмичности в горных территориях крайне мало. Целью проведенных исследованийявилось решение задачи выявления условий резонансного поведения гармонически колеблющихся литосферных плит, а также горных массивов, вызываемого периодическими приливными воздействиями Луны, атмосферными и иными источниками. Методы работы. Применение новейших математических разработок в области механики деформируемых штампов, опубликованных в высокорейтинговых журналах. Изучается тот случай, когда разлом может иметь любую ширину и литосферные плиты могут приближаться торцами. Такая же ситуация возникает на достаточно узких горных дорогах, окруженных скальными образованиями, а также при приближении долин, где горные гряды оказываются достаточно удаленными. Применяется новейшая разработка, опирающаяся на метод блочного элемента, а также теория контактных задач с деформируемым штампом. При исследовании, использованы методы блочного элемента. Результаты исследования. Разработан метод учета разнотипности горных рельефов и пород за счет возможности перехода, при описании берегов трещин нового типа, к материалам изменяемых реологий и установлены дисперсионные соотношения для определения резонансных частот. Таким образом, с помощью применяемых новых методов в статье показана возможность получения соотношений, позволяющих оценивать степень опасности разрушения литосферных плит. The relevance of the work lies in the need for further development of the use of high-precision mechanical and mathematical methods in the problem of forecasting the increase in seismicity. In particular, there are very few rigorous mathematical approaches to studying seismicity in mountainous areas. The purpose of the research was to solve the problem of identifying the conditions of resonant behavior of harmoniously vibrating lithospheric plates, as well as mountain ranges caused by periodic tidal influences of the Moon, atmospheric and other sources. Methods.Application of the latest mathematical developments in the field of deformable stamp mechanics, published in highly rated journals. The case is being studied when a fault can have any width and lithospheric plates can approach with their ends. The same situation occurs on fairly narrow mountain roads surrounded by rock formations, as well as when approaching valleys where mountain ranges are quite remote. The latest development is applied, based on the block element method, as well as the theory of contact problems with a deformable stamp. In the study, block element methods were used. Results. A method has been developed to account for the diversity of mountain reliefs and rocks due to the possibility of transition, when describing the crack edges of a new type, to materials of variable rheologies and dispersion relations have been established to determine resonant frequencies. Thus, using the applied new methods, the article shows the possibility of obtaining ratios that allow assessing the degree of destruction danger for lithospheric plates.

This paper presents one of the methods for studying the behavior of deformable stamps on a deformable base. It is based on a new universal modeling method previously published by the authors, used in boundary value problems for systems of partial differential equations. Its advantage is the possibility of avoiding the need to solve complex boundary value problems for systems of differential equations by replacing them with separate differential equations, among which the Helmholtz equations are the simplest. With the help of combinations of solutions of boundary value problems for this equation, it is possible to describe the behavior of complex solutions of multicomponent boundary value problems, including for cracks of a new type formed by objects on a deformable base, and models of nano particles located on deformable multicomponent bases. However, without the ability to solve contact problems for deformable stamps, these models are not built. The mixed problem is reduced to the solution of the Wiener–Hopf integral equation. Two cases are considered: the case of a large-width strip stamp and the case of a semi-infinite stamp. A packed block element is accepted as a deformable stamp, as a solution of the Helmholtz equation in the specified area. Mechanically, it can be imitated as a membrane that is located on a multilayer medium occupying the contact area. A combination of such objects can be used to describe solutions to the contact problem for flat deformable objects of more complex rheology, as well as for three-dimensional ones. Along with the proof of constructing an exact solution to the contact problem under consideration, the appearance of unknown functionals is noted in the course of the study. In problems with an absolutely solid stamp, they do not occur. The paper finds a way to determine them and an analytical representation of them is obtained. It is suggested that their appearance will be typical for solving other contact problems with deformable stamps. The features of the method and the results obtained are discussed.

In this paper, for the first time, a two-dimensional dynamic contact problem on the action of a deformable stamp on a quarter of the plane of a multilayer medium is strictly mathematically investigated. In contrast to the case of an absolutely solid stamp, a deformable stamp introduces additional features, consisting in the possibility of the occurrence of discrete resonances predicted by academician I.I. Vorovich. The paper shows that the use of a method based on the use of block elements makes it possible to obtain an equation describing resonant frequencies. To study contact problems with a deformable stamp made of materials of complex rheology, including smart materials, it is proposed in the paper to first conduct a study for the case of a deformable stamp made of a material of simple rheology described by Helmholtz equations. Solutions of boundary value problems for stamps of complex rheology, after that, are represented by a combination of solutions of boundary value problems for stamps of simple rheology.

In this paper, for the first time, methods for obtaining resonant frequencies in plane and spatial contact prob-lems for deformable stamps are presented in a complex. Their existence was predicted by academician I.I. Vorovich. He constructed one-dimensional models of de-formable stamps consisting of a rigid stamp and a spring, illustrating the occurrence of resonances. Currently, thanks to the use of block element methods, it has become possible to investigate real contact problems with a deformable stamp. Earlier in the works of the authors, the existence of a new type of earthquakes, called starting, was established. In the course of the study, some functionals remained uncomputed, the role of which was not clear. It is shown that they serve to construct dispersion curves giving resonant frequencies predicted by academician I.I. Vorovich. A study is carried out for different types of contact problems and shows how the method of constructing dispersion equations becomes more complicated, both due to the complication of the shape of stamps and the complication of the rheologies of deformable stamps. Semi-infinite and deformable stamps of simple rheology in the form of a quarter plane are considered. In addition, a semi-infinite stamp with the rheology of Kirchhoff plates is analyzed. The obtained results provide an answer to one of the methods of constructing dispersion equations for I.I.Vorovich resonances in contact problems for different deformable stamps. The study is closely related to the identification of new precursors of seismicity, as well as to solve some problems in the theory of strength.

In this paper, for the first time, an exact solution to the contact problem posed on the surface of a multilayer medium in a quarter-plane is constructed. This is achieved by applying a new universal modeling method developed for studying and solving boundary value problems for partial differential equations. In this paper, the method is applied to two-dimensional Wiener–Hopf integral equations in the quarter-plane arising in mixed problems of deformable solid mechanics, in contact problems. A feature of mixed problems for layered media is the presence of meromorphic functions in Fourier transformations of the kernels of integral equations. As in differential equations, this made it possible to find fragments of differential equations in the representation of Wiener–Hopf integral equations and to find a way to reduce a two-dimensional integral equation to one-dimensional. This makes it possible to reduce integral equations in this field to infinite systems of linear algebraic equations having an inverse infinite matrix. This type of integral equations is not available for numerical solution, due to the unlimited scope of the equation, and has not been studied analytically before. The exact solution to the two-dimensional Wiener–Hopf equation in a quarter-plane makes it possible to construct high-precision solutions to contact problems in limited domains, just like in the one-dimensional case. Wiener–Hopf integral equations are widely used in various fields for materials with complex rheology, including strength theory, diffraction, flaw detection, and tribology.

This paper presents an approach that allows us for the first time to construct an exact solution of the Wiener–Hopf integral equations on a finite segment for the case of meromorphic functions in Fourier transforms of the kernel. The Wiener–Hopf integral equation is traditionally considered set on a semi-infinite segment. However, in applications, there are often cases of their application specified on a finite segment. For these purposes, approximate methods of applying these integral equations have been developed. However, when considering the Wiener–Hopf integral equations generated by mixed problems of continuum mechanics and mathematical physics in a multilayer medium of finite thickness, it turned out that these integral equations are solved exactly both on semi-infinite and finite segments. The approach is based on a new modeling method in differential equations and in some types of integral equations. It allows the reduction of Wiener–Hopf integral equations to infinite systems of linear algebraic equations that are solved exactly. The obtained result opens up the possibility of constructing exact solutions to boundary value problems for deformable stamps and cracks of a new type in bounded bodies.

For the first time, an accurate analytical solution of mixed or contact problems for multicomponent multilayer materials has been constructed. It is assumed that the contact problem is formulated at the boundary of a multilayer multicomponent material in a semi-infinite region. These can be contact problems for a multilayer medium that simultaneously includes thermoelectroelastic, magnetoelastic, piezoelastic, water-saturated, nanomaterials and other layers described by linear partial differential equations. In the contact area, there can be any conditions of mechanical, physical or chemical properties that lead the boundary problem to a system of arbitrary finite number of Wiener-Hopf integral equations with a meromorphic matrix in the core. The article uses a new universal modeling method that allowed factorizing the operator of an infinite system of linear algebraic equations.

In this paper, for the first time, an exact solution of contact problems for rigid or deformable stamps in a strip of finite width is obtained. It is assumed that the strip is located on a multilayer base of finite thickness. The universal modeling method previously developed by the authors is used. With its help, the solutions of complex boundary value problems for systems of partial differential equations are reduced, using the Galerkin transform, to solving individual differential equations, among which the Helmholtz equations are the simplest. In an earlier work of the authors published, when studying the problem for a deformable stamp in a strip, we had to limit ourselves to an asymptotic solution that is valid only for strips of large relative width. The solution for a strip of any finite size was constrained by the impossibility of constructing an exact solution to the contact problem for a rigid stamp in a strip of any finite width. As a result of the exact solution of the Wiener-Hopf integral equation in a finite-width band for the case of a multilayer medium, this contact problem was solved. The approach applied to the solution consists in constructing an exact operator equation of an infinite system of algebraic equations for large-width bands, using the operator formula of functions from matrices and investigating the constructed solution in the range of small relative bandwidth. The solution obtained in this way coincides with the solution obtained another method, namely, the singular integral method for the case of a small relative bandwidth. The constructed solution brings closer to the problems of research for materials of complex rheologies of contact problems with a deformable stamp, the description of cracks of a new type in limited bodies, modeling of nano particles, the study of tectonic plates of limited dimensions.

The article presents the mechanical concept of self-assembly of nanoparticles. It is assumed that nanoparticles are deformable stamps in a plane dynamic contact problem, lying on the boundary of a multilayer deformable medium. The constant vibration in the microcosm is caused by the oscillatory mode by the energy of phonons and magnons. Earlier, in the works of the authors, the mechanical concept of self-organization of nanoparticles was presented. It is based on high-frequency resonance, which causes the formation of standing waves. They localize the available aggregates of nanoparticles on the crest of standing waves. The self-assembly of nanoparticles is based on resonance, previously predicted by Academician I. I. Vorovich and inherent only in deformable dies in contact problems on a multilayer medium. Deformable nanoparticles are modeled by fractals representing packed block elements described by the Helmholtz equation. The resonance of the deformable dies allows the capture of nanoparticles, dictated by the Coulomb forces of attraction. It is shown that the combination of two fractals generates a new fractal with a combined carrier, and in the case of multiple association, a fragment of a nanomaterial is obtained. To implement the study, for the first time it was possible to construct a high-precision approximate solution of a plane contact problem on the action of a stamp of any finite size on a multilayer base. This result is dictated by the need for an analytical construction of the theory of self-assembly of nanomaterials.

In this paper, for the first time, mathematically strictly, models of Griffiths cracks and cracks of a new type are constructed simultaneously. Accurate models are constructed as a result of the convergence of the ends of two semi-infinite deformable stamps located on a deformable multilayer base. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been poorly studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The construction of models in the work is based precisely on these descriptions of the origin of cracks formed by the ends of approaching deformable stamps. When creating models, the results of previously performed studies on the construction of exact solutions to a number of integral equations of contact problems, published in this journal, are used. The paper found that the destruction of the medium, in the case of cracks of a new type, occurs earlier than in the case of Griffith cracks, which, according to the authors, explains the reason for the destruction of Griffith cracks in the experiment earlier than the theory predicts. Griffiths explained this by the presence of microcracks on the border of the main crack. The authors tend to believe that the cause is the transformation of one type of crack into another, under certain conditions. The results obtained make it possible to compare and quantify the behavior of the medium in the presence of both one and another type of cracks. It is applicable both for the study of mutual transitions of Griffith cracks and cracks of a new type, and for the construction of models of adhesion of wetted nanoparticles.