System Of Singular Integro-differential Equations Research Articles

Iterative Method of Determining Stress Intensity Coefficients Under Dynamic Loading of the Crack System

An elastic isotropic body in a state of plane deformation, which contains a system of randomly placed cracks under the action of a dynamic (harmonic) loading, is considered. The authors set the problem of determining the stress field around the cracks under the conditions of their wave interaction. The solution method is based on the introduction of displacements in the body in the form of a superposition of discontinuous solutions of the equations of motion constructed for each crack. With this in mind, the initial problem is reduced to a system of singular integro-differential equations with respect to unknown displacement jumps on the crack surfaces. To solve this system, a new iterative method, which involves solving a set of independent integro-differential equations that differ only in their right-hand parts at each iteration, is proposed. For the zero approximation, solutions that correspond to individual cracks under the action of dynamic loading are chosen. Such a new approach allows to avoid the difficulties associated with the need to solve systems of integro-differential equations of large dimensions that arise when traditional methods are used. Based on the results of the iterations, formulas for calculating the stress intensity coefficients for each crack were obtained. In the partial case of four cracks, a good agreement between the results obtained during the direct solution of the system of eight integro-differential equations by the mechanical quadrature method and the results obtained by the iterative method was established. In general, numerical examples demonstrate the convergence and stability of the proposed method in the case of systems with a fairly large number of densely located cracks. The influence of the interaction between cracks on the stress intensity factor (SIF) value under dynamic loading conditions was studied. An important and new result for fracture mechanics is the detection of the absolute maximum of the normal stresses at certain frequencies of the oscillating normal loading. The number of interacting cracks and the configuration of the crack system itself affect the values of the frequencies at which SIF reaches a maximum and the maximum values. These maximum values significantly (by several times) exceed the SIF values of single cracks under a similar loading. At the same time, under conditions of static or low-frequency loading, it is possible to reduce the SIF values compared to the SIF for individual cracks. When cracks are sheared, the values of the tangential stresses have a tendency to decrease with increasing frequency, and their values do not significantly differ from the values of the tangential stress for an individual crack.

Вплив поздовжньої і поперечної теплопровідності міжфазних щілин на ефективні параметри біматеріалу

The effective parameters of the bi-material with a periodic system of interfacial cracks are studied, taking into account their longitudinal and transverse thermal conductivity. The bi-material is subjected to tensile forces and uniform heat flow. The transverse and longitudinal thermal conductivity of the cracks is taken into account by the thermal resistance of the filler and the thermal conductivity of the surface films, respectively. The thermal resistance of the filler is directly proportional to the opening of the cracks and inversely proportional to the thermal conductivity of the filler. Thermal conductivity of surface films does not change under the influence of load. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for an opening cracks and a temperature jump between the cracks faces. An analytical-numerical iterative procedure for solving this system is proposed. Based on the obtained solution, the effective temperature jump and the effective thermal resistance of the bi-material are determined. The dependences of the effective parameters of the bi-material on the applied load and thermal conductivity coefficients of the filler and the surface films of the cracks are analyzed.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established.

A Novel Projection Method for Cauchy-Type Systems of Singular Integro-Differential Equations

This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests provide theoretical results.

System of cracks under the impact of plane elastic waves

The problem of determining the two-dimensional diffraction field and the stress state arising from the interaction of elastic waves with a system of arbitrarily located cracks is solved. Cracks are located in an unbounded elastic medium which is in a plane deformation state. The method of solution is based on the use of discontinuous solutions of the equations of motion of elastic media and the reduction of the original problem to a system of singular integro-differential equations. The obtained systems are solved numerically the mechanical quadrature method.

Параметр термічної ректифікації у біматеріалі з міжфазною тріщинуватістю

The phenomenon of thermal rectification in bi-material with a periodic system of gas-filled interfacial cracks has been studied. The change of thermal resistance and pressure of the crack filler under the action of tensile forces and uniform heat flow is taken into account. The gas pressure is determined from the Clapeyron-Mendeleev equation. The thermal resistance of the gas is directly proportional to the opening of the cracks and inversely proportional to the thermal conductivity of the gas. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for an opening cracks and a temperature jump between the cracks faces. An analytical-numerical iterative procedure is proposed to solve the system of singular integro-differential equations. The parameter of thermal rectification is determined and its dependence on the applied on the thermal and mechanical loads, thermal conductivity and mass of the gas and the interfacial cracking factor of the bi-material is analyzed. The nonlinear dependence of the thermal rectification parameter on the heat flow density and the interfacial cracking factor is established. The thermal rectification is manifested to a greater extent for higher values of heat flow and lower thermal conductivity of gas.

Modeling of Crack Self-Healing Kinetics

A model based on the concept of interface crack bridging and thermofluctuation theory is proposed for analyzing the efficiency of crack self-healing in composite materials in view of three main stages: (1) crack initiation and growth under external load, (2) self-healing activation, and (3) self-healing with partial or complete bond recovery between crack faces. A system of singular integro-differential equations is derived for numerical estimations of bond tractions in bridging zones during their formation and stress intensity factors (major characteristics of crack self-healing efficiency) with account of external loads and bond tractions. Kinetic estimates are presented for the time of bond recovery during the formation of a crack bridging zone. The model is applied to an interface crack between two dissimilar materials with a self-healing adhesive.

A new B-spline collocation method for singular integro-differential equations of higher orders

Two numerical schemes based on B-spline collocation method are presented for approximate solution of the systems of singular integro-differential equations of higher orders appearing in fracture problems with Steigmann–Ogden surface energy. It is shown that the resulting systems of linear equations are well-conditioned and the numerical examples are in good agreement with known solutions. Although this paper is concerned with the numerical solution to the specific equations, the proposed method can be easily generalized to a large class of singular integro-differential equations.

Modeling Partial Closure of a Variable-Width Slot with Cohesion End Zones in Rock Mass

The article considers the problem on compression of an isotropic medium with a variable-width slot with cohesive zones at the ends. Under the action of tectonic and gravitational forces in rock mass, contact zones appear between the slot surfaces. The author analyzes the case of a number of the contact zones between the slot surfaces. In a part of the contact zone, cohesion of the slot edges takes place, while sliding is possible in the other part. The unknown parameters of the partial closure of the slot are found by solving a system of singular integro-differential equations. The contact stresses, the cohesion forces, as well as the sizes of the contact areas, cohesion zones and pre-fracture end zones are determined.

Partial closure of the periodic system of variable width bridged slits in an isotropic medium under compression

The problem of compression of an isotropic medium by a periodic system of variable width slits comparable with elastic deformations is considered. The slit faces are under internal pressure. It is considered that the slits have end zones wherein material's tractions act. It is accepted that interaction of the slit's surface under applied loads may reduce the formation of contact zones of their surfaces. The formation of several contact areas of faces of each slit is studied. Therewith, it is considered that on some parts of a contact area there happens stick of the faces, on the remaining part there may happen slippage. The problem of equilibrium of a periodic system of slits with partially contacting faces under compressing load is reduced to the problem of linear conjugation of analytic functions. Determination of unknown parameters characterizing partial closing of a periodic system of variable width slits is reduced to the solution of the system of singular integro-differential equations. The contact stresses, the tractions in bonds between the slit faces in the end zones, the sizes of the contact areas, the adhesion and end zones are studied.

Термопружна поведінка біматеріалу з міжфазною щілиною, заповненою нестисливою рідиною

Thermo-stressed state of the bimaterial with the interfacial crack filled with an incompressible fluid is investigated. The bimaterial with zero Dundurs parameter is considered. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for a temperature jump between the cracks faces and an opening crack. The pressure of the fluid is determined from the equation of the incompressible fluid. Using the method of successive approximations a numerical solution of the problem is constructed. The opening crack and temperature jump are determined by the finite sum for Chebyshev polynomials of the second kind. The coefficients for Chebyshev polynomials are determined from the system of linear algebraic equations. Having made numerical calculations, the dependences of the pressure of the fluid and the opening crack from the mechanical loads and density and direction of the heat flow have been constructed. The influence of fluid pressure on the stress intensity factor mode I are analyzed.

An Adhesive Contact Problem for a Semi-plane with a Surface Elasticity in the Steigmann-Ogden Form

The main goal of the present paper is to study a two-dimensional problem of adhesive contact of a rigid stamp with an elastic upper semi-plane. Surface elasticity in the Steigmann-Ogden form acts on the free boundary of the semi-plane. The mechanical problem is solved using integral representations of the stresses and the displacements. The resulting system of singular integro-differential equations is regularized using Fourier transform method. The numerical procedure of the solution of the system of singular integro-differential equations is presented, and the numerical results are obtained for different values of the mechanical parameters. The size-dependency of the solutions of the problem is studied. It can be seen that taking into account curvature-dependency of the surface energy is increasingly important at the very small scales (characteristic length of the contact interval is under $20~\mbox{nm}$ ).

The boundary value contact problem of electroelasticity for piecewise-homogeneous piezo-electric plate with elastic inclusion and cut

In this paper a contact problem of the theory of electroelasticity for piecewise-homogeneous plate of piezo-electric material with infinite cut and elastic finite inclusion of variable bending rigidity is considered. By using methods of the theory of analytic function, the problem is reduced to a system of singular integro-differential equations with fixed singularity. Using an integral transformation, a Riemann problem is obtained, the solution of which is presented in explicit form.

The thermoelastic contact problem for wavy surfaces with a heat-conducting medium in interface gaps

This paper presents a study on the thermoelastic contact between a wavy surface and a flat surface in the presence of a heat-conducting interstitial medium in interface gaps. The influence of applied mechanical and thermal loads on the deformation of the gaps is taken into account. The contact problem is reduced to a system of singular integro-differential equations for a temperature jump across the gaps and the height of the gaps. Solutions are obtained for the cases of thermoinsulated and heat-conducting gaps. It is shown that, in contrast to the thermoinsulated gap model, the use of the heat-conducting gap model makes it possible to construct a physically correct solution of the contact problem. It is revealed that the wavy interface with heat-conducting gaps exhibits thermal rectification. The effects of the medium’s thermal conductivity, the pressure and heat flow magnitudes and the waviness amplitude on the effective thermal contact resistance and the level of thermal rectification are analysed.

Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann–Ogden form

In this paper, the surface elasticity in the form proposed by Steigmann and Ogden is applied to study a plane problem of frictionless contact of a rigid stamp with an elastic upper semi-plane. The results of this work generalize the results for contact problems with Gurtin–Murdoch elasticity by including additional dependency on the curvature of the surface. The mechanical problem is reduced to a system of singular integro-differential equations, which is further regularized using the Fourier transform. The size dependency of the solutions of the problem is highlighted. It is observed that the curvature dependence of the surface energy is increasingly important at small scales. The numerical procedure of the solution of the system of singular integro-differential equations is presented, and numerical results are obtained for different values of the mechanical parameters.

A straight mixed mode fracture with the steigmann–ogden boundary condition

A problem of a straight mixed mode non-interface fracture in an infinite plane is treated analytically with the help of complex analysis techniques. The surfaces of the fracture are subjected to surface elasticity in the form proposed by Steigmann and Ogden (Steigmann and Ogden, Proc. R. Soc. A453 (1997); Steigmann and Ogden, Proc. R. Soc. A455 (1999)). The boundary conditions on the banks of the fracture connect the stresses and the derivatives of the displacements. The mechanical problem is reduced to two systems of singular integro-differential equations which are further reduced to the systems of equations with logarithmic singularities. It is shown that modeling of the fracture with the Steigmann–Ogden elasticity produces the stress and strain fields, which are bounded at the crack tips. The existence and uniqueness of the solution for almost all the values of the parameters is proved. Additionally, it is shown that introduction of the surface mechanics into the modeling of fracture leads to the size-dependent equations. A numerical scheme of the solution of the systems of singular integro-differential equations is suggested, and the numerical results are presented for different values of the mechanical and the geometric parameters.

Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary

A new model of fracture mechanics considered previously by Sendova and Walton \cite{SendovaWalton2010}, Zemlyanova \cite{Zemlyanova2013}, and Zemlyanova and Walton \cite{Zemlyanova2012} is further developed on the example of a mixed mode curvilinear interface fracture located on the boundary of a partially debonded thin elastic inclusion embedded in an infinite thin elastic matrix. The effect of the nano-structure of the material is incorporated into the model in the form of a curvature-depended surface tension acting on the boundary of the fracture. It is shown that the introduction of the surface tension allows to eliminate the classical oscillating and power singularities of the order $1/2$ present in the linear elastic fracture mechanics. The mathematical methods used to solve the problem are based on the Muskhelishvili's complex potentials and the Savruk's integral representations. The mechanical problem is reduced to the system of singular integro-differential equations which is further reduced to a system of weakly-singular integral equations. The numerical computations and comparison with known results are presented.

The boundary contact problem of electroelasticity and related integral differential equations

The problem of finding mechanical and electrical fields in a homogeneous piezoelectric plate supported by a thin wedged-shaped cover plate is considered. Using the methods of the theory of analytic functions, the problem is reduced to a system of singular integro-differential equations. With the help of integral transformation, the exact solution of the posed by us problem is obtained.

A Rigid Stamp Indentation into a Semiplane with a Curvature-Dependent Surface Tension on the Boundary

It has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience, fracture propagation, and contact mechanics. This paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature-dependent surface tension acting on the boundary of the semiplane. Cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered. Using the method of integral transforms, each problem is reduced to a system of singular integro-differential equations, which is further reduced to one or two weakly singular integral equations. It has been shown that the introduction of the curvature-dependent surface tension eliminates the classical singularities of the order 1/2 of the stresses and strains at the end-points of the contact interval. The numerical solution of the problem is obtained by approximation of unknown functions with Taylor polynomials.

Numerical Solutions of the System of Singular Integro-Differential Equations in Classical Hölder Spaces

New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integro-differential equations that are defined on arbitrary smooth closed contours of the complex plane. We carry out the convergence analysis in classical Holder spaces. A numerical example is also presented.