Dynamical problems of the theory of elasticity for solids with double porosity

In this paper the dynamical theory of elasticity for solids with double porosity is presented. The single-layer and double-layer potentials are constructed and basic properties are established. The uniqueness theorems of the internal and external boundary value problems (BVPs) of steady vibrations are proved. The existence theorems of classical solution of the external BVPs by means of the boundary integral method and the theory of multidimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plane Waves and Boundary Value Problems in the Theory of Elasticity for Solids with Double Porosity

This paper concerns with the dynamical theory of elasticity for solids with double porosity. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation with double porosity. The basic properties of plane waves are established. The radiation conditions of regular vectors are given. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness theorems are proved. The basic properties of elastopotentials are given. The existence of regular (classical) solution of the external BVP by means of the potential method (boundary integral method) and the theory of singular integral equations are proved.     

External boundary value problems in the quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity

In the present paper the linear quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity is considered. The basic external boundary value problems (BVPs) of steady vibrations in this theory are formulated. The uniqueness and existence theorems for regular (classical) solutions of the BVPs are proved by using of the potential method (boundary integral equations method) and the theory of singular integral equations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The boundary value problems of the full coupled theory of poroelasticity for materials with double porosity

In this paper the full coupled quasi-static theory of poroelasticity for materials with double porosity is considered. The basic boundary value problems (BVPs) of the steady vibrations are investigated. The uniqueness theorems of the internal BVPs of steady vibrations are proved. The basic properties of elastopotentials are established. The existence of regular solutions of the BVPs by means of the boundary integral equations method and the theory of singular integral equations is proved. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity

In this paper, the fully coupled theory of elasticity for solids with double porosity is considered. The explicit solutions of the basic boundary value problems (BVPs) in the fully coupled linear equilibrium theory of elasticity for the space with double porosity and spherical cavity are constructed. The solutions of these BVPs are represented by means of absolutely and uniformly convergent series. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary value problems of steady vibrations in the theory of thermoelasticity with microtemperatures

In this paper the linear theory of steady vibrations of thermoelasticity with microtemperatures for isotropic solids with microstructure is considered. The uniqueness and existence theorems of solutions of the internal and external second boundary value problems (BVPs) by means of the boundary integral method (potential method) and the theory of singular integral equations are proved. The existence of eigenfrequencies of the internal homogeneous BVP of steady vibrations is studied. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

External boundary value problems of steady vibrations in the theory of rigid bodies with a double porosity structure

This paper concerns with the linear 3D theory of rigid solids with a double porosity structure. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Potential method in the steady vibrations problems of the theory of thermoviscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness and existence theorems of solutions of the boundary value problems of the theory of consolidation with double porosity

In this paper we consider the Aifantis' theory of consolidation with double porosity and we prove the uniqueness and existence theorems of solutions of basic boundary value problems (BVPs) of statics for the two-dimensional finite and infinite domains. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems (BVPs) for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original BVPs is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik–Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.     

Fundamental solutions in the linear theory of consolidation for elastic solids with double porosity

This paper concerns the theory of consolidation for elastic solids with double porosity, and the governing fully coupled linear quasi-static equations are considered. The system of these equations is based on the equilibrium equations for a solid, conservation of fluid mass, the effective stress concept, and Darcy’s law for material with double porosity. Two levels of spatial cases of consolidation theory for a solid with double porosity are considered: equations of steady vibrations and equations of equilibrium. The fundamental solutions of these equations are constructed by means of elementary functions. Finally, the basic properties of these solutions are established.     

Boundary value problems of the theory of viscoelasticity for Kelvin-Voight materials

In this paper, the classical Kelvin-Voight model of the linear theory of viscoelasticity is considered and the following results are obtained: the fundamental solution of the equation of steady vibrations is constructed, the basic properties of plane waves and elastopotentials are established, the uniqueness theorem of the internal and external boundary value problems (BVPs) are proved, the existence theorems for classical solutions of the BVPs by means of the potential method and the theory of two-dimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

External boundary value problems in the quasi static theory of elasticity for triple porosity materials

In this paper the quasi static linear theory of elasticity for materials with triple porosity is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of Poisson's equation with global,local and nonlocal boundary conditions

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.     

General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)

Here we discuss three-dimensional so-called basic and mixed boundary value problems (BVP) for steady state oscillations of piecewise homogeneous anisotropic bodies imbedded into an infinite elastic continuum. Uniqueness is shown with the help of generalized Sommerfeld–Kupradze radiation conditions, while existence follows for arbitrary values of the oscillation parameter by the reduction of the original interface transmission BVPs to equivalent uniquely solvable boundary integral or pseudodifferential equations on the interfaces. For the basic BVPs, we show classical regularity and, in addition for the mixed BVPs that the solutions are Hölder continuous with exponent α ∈ (0, 1/2) in the neighbourhood of the curves of discontinuity of the boundary and transmission conditions.     

Solving boundary value problems,integral, and integro-differential equations using Gegenbauer integration matrices

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.     

Explicit solution of the boundary value problems of the theory of elasticity for solids with double porosity

In this paper the Aifantis' theory of elasticity for solids with double porosity is considered and the 2D boundary value problem (BVP) of static is investigated. The uniqueness theorem of the internal BVP is proved. The explicit solution the BVP is constructed in the form of absolutely and uniform convergent series for a circle. The numerical solution of the BVP for a circle is obtained. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries

The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). The fast multipole method (FMM) known from the literature was included into modified PIES. To consider complex curvilinear shapes of a boundary, the modification of a binary tree used by the FMM is proposed. The FMM combined with the PIES, called the fast PIES, also allows a significant reduction of random access memory (RAM) utilization. Therefore, it is possible to solve complex engineering problems on a standard personal computer (PC). The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions of complex BVPs described by the curvilinear boundary at a reasonable time on the PC.

     

Propagación de ondas de Rayleigh en medios con grietas

This work is focused on the finding of numerical results for detection and characterization of sub-surface cracks in solids under the incidence of Rayleigh's elastic waves. The results are obtained from boundary integral equations, which belong to the field of dynamics of elasticity. Once applied the boundary conditions, a system of Fredholm's integral equations of second kind and zero order is obtained, which is solved using Gaussian elimination. The method that is used for the solution of such integral equations is known as the Indirect Boundary Element Method, which can be seen as a derivation of the Somigliana's classic theorem. On the basis of the analysis made in the frequency domain, resonance peaks emerge and allow us to infer the presence of cracks through the spectral ratios. Several models of cracked media were analyzed, where analyses reveal the great utility that displays the use of spectral ratios to identify cracks. We studied the effects of orientation and location of cracks. The results show good agreement with the previously published.     

带Riemann-Stieltjes积分条件的三阶边值问题的单调正解

研究了一类带Riemann-Stieltjes积分条件的非线性三阶非局部边值问题,将边值问题正解存在性的研究转化为扰动Hammerstein积分方程的研究,通过构造Green(格林)函数及讨论其性质,运用推广的Leggett-Williams型不动点定理,得到了至少存在3个和2n-1个正解的存在性准则, 所得结果推广和改进了最近文献中的结果,并充分反映了非线性项含导数对正解存在性研究的影响.主要结果由实例加以阐述.