The Interface Crack Problem for Anisotropic Bodies

The two-dimensional interface crack problem is investigated for anisotropic bodies in the Comninou formulation. It is established that, as in the isotropic case, properly incorporating contact zones at the crack tips avoids contradictions connected with the oscillating asymptotic behaviour of physical and mechanical characteristics leading to the overlapping of material. Applying the special integral representation formulae for the displacement field the problem in question is reduced to the scalar singular integral equation with the index equal to -1. The analysis of this equation is given. The comparison with the results of previous authors shows that the integral equations corresponding to the interface crack problems in the anisotropic and isotropic cases are actually the same from the point of view of the theoretical and numerical analysis

International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering

This book presents eleven peer-reviewed papers from the 3rd International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering (AMINSE2017) held in Tbilisi, Georgia in December 2017. Written by researchers from the region (Georgia, Russia, Turkey) and from Western countries (France, Germany, Italy, Luxemburg, Spain, USA), it discusses key aspects of mathematics and informatics, and their applications in natural sciences and engineering. Featuring theoretical, practical and numerical contributions, the book appeals to scientists from various disciplines interested in applications of mathematics and informatics in natural sciences and engineering

International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering

This book presents peer-reviewed papers from the 4th International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering (AMINSE2019), held in Tbilisi, Georgia, in September 2019. Written by leading researchers from Austria, France, Germany, Georgia, Hungary, Romania, South Korea and the UK, the book discusses important aspects of mathematics, and informatics, and their applications in natural sciences and engineering. It particularly focuses on Lie algebras and applications, strategic graph rewriting, interactive modeling frameworks, rule-based frameworks, elastic composites, piezoelectrics, electromagnetic force models, limiting distribution, degenerate Ito-SDEs, induced operators, subgaussian random elements, transmission problems, pseudo-differential equations, and degenerate partial differential equations. Featuring theoretical, practical and numerical contributions, the book will appeal to scientists from various disciplines interested in applications of mathematics and informatics in natural sciences and engineering.

Three-dimensional mathematical Problems of thermoelasticity of anisotropic Bodies

CHAPTER I. Basic Equations. Fundamental Matrices. Thermo-Radiation Conditions 1. Basic differential equations of thermoelasticity theory 2. Fundamental matrices 3. Thermo-radiating conditions. Somigliana type integral representations CHAPTER II. Formulation of Boundary Value and Interface Problems 4. Functional spaces 5. Formulation of basic and mixed BVPs 6. Formulation of crack type problems 7. Formulation of basic and mixed interface problems CHAPTER III. Uniqueness Theorems 8. Uniqueness theorems in pseudo-oscillation problems 9. Uniqueness theorems in steady state oscillation problems CHAPTER IV. Potentials and Boundary Integral Operators 10. Thermoelastic steady state oscillation potentials 11. Pseudo-oscillation potentials CHAPTER V. Regular Boundary Value and Interface Problems 12. Basic BVPs of pseudo-oscillations 13. Basic exterior BVPs of steady state oscillations 14. Basic interface problems of pseudo-oscillations 15. Basic interface problems of steady state oscillations CHAPTER VI. Mixed and Crack Type Problems 16. Basic mixed BVPs 17. Crack type problems 18. Mixed interface problems of steady state oscillations 19. Mixed interface problems of pseudo-oscillation

International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering

This book presents eleven peer-reviewed papers from the 3rd International Conference on Applications of Mathematics and Informatics in Natural Sciences and Engineering (AMINSE2017) held in Tbilisi, Georgia in December 2017. Written by researchers from the region (Georgia, Russia, Turkey) and from Western countries (France, Germany, Italy, Luxemburg, Spain, USA), it discusses key aspects of mathematics and informatics, and their applications in natural sciences and engineering. Featuring theoretical, practical and numerical contributions, the book appeals to scientists from various disciplines interested in applications of mathematics and informatics in natural sciences and engineering

Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body (Ω+) is embedded in an unbounded fluid domain (Ω-). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω+, we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω-, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev-Slobodetskii spaces.</p

Mixed Interface Problems Of Thermoelastic Pseudo-Oscillations

Three--dimensional basic and mixed interface problems of the mathematical theory of thermoelastic pseudo--oscillations are considered for piecewise homogeneous anisotropic bodies. Applying the method of boundary potentials and the theory of pseudodifferential equations existence and uniqueness theorems of solutions are proved in the the space of regular functions C k+ff and in the Bessel--potential (H s p ) and Besov (B s p;q ) spaces. In addition to the classical regularity results for solutions to the basic interface problems, it is shown that in the mixed interface problems the displacement vector and the temperature are Holder continuous with exponent 0 ! ff ! 1=2. Introduction The paper deals with the three--dimensional interface problems of the mathematical theory of thermoelastic pseudo--oscillations for piecewise homogeneous anisotropic bodies. The most general case of the structure of a piecewise homogeneous elastic body under consideration can be mathematicaly described ..

Method of fundamental solutions for mixed and crack type problems in the classical theory of elasticity

We analyse some new aspects concerning application of the fundamental solution method to the basic three-dimensional boundary value problems, mixed transmission problems, and also interior and interfacial crack type problems for steady state oscillation equations of the elasticity theory. First we present existence and uniqueness theorems of weak solutions and derive the corresponding norm estimates in appropriate function spaces. Afterwards, by means of the columns of Kupradze’s fundamental solution matrix special systems of vector functions are constructed explicitly. The linear independence and completeness of these systems are proved in appropriate Sobolev–Slobodetskii and Besov function spaces. It is shown that the problem of construction of approximate solutions to the basic and mixed boundary value problems and to the interior and interfacial crack problems can be reduced to the problems of approximation of the given boundary vector functions by elements of the linear spans of the corresponding complete systems constructed by the fundamental solution vectors. By this approach the approximate solutions of the boundary value and transmission problems are represented in the form of linear combinations of the columns of the fundamental solution matrix with appropriately chosen poles distributed outside the domain under consideration. The unknown coefficients of the linear combinations are defined by the approximation conditions of the corresponding boundary and transmission data. Keywords: Method of fundamental solutions, Theory of elasticity, Elastic vibrations, Mixed boundary value problem, Mixed transmission problem, Crack problem, Approximate solution

Recent trends in operator theory and partial differential equations: the Roland Duduchava anniversary volume

This volume is dedicated to the eminent Georgian mathematician Roland Duduchava on the occasion of his 70th birthday. It presents recent results on Toeplitz, Wiener-Hopf, and pseudodifferential operators, boundary value problems, operator theory, approximation theory, and reflects the broad spectrum of Roland Duduchava's research. The book is addressed to a wide audience of pure and applied mathematicians