Two-dimensional Green’s functions for semi-infinite isotropic thermoelastic plane

Based on the two-dimensional steady-state governing equations of isotropic thermoelastic material and the compact general solution expressed in three harmonic functions, the corresponding three harmonic functions contain nine undetermined constants are constructed for a line heat source applied in the interior of a semi-infinite thermoelastic plane. All components of thermoelastic field in the semi-infinite plane can be derived by substituting the harmonic functions into the general solution. And the undetermined constants can be obtained by the compatibility conditions, equilibrium conditions and the different boundary conditions for extended Mindlin problem and extended Lorentz problem. Thus, the Green’s functions in above two cases are obtained, and the numerical results are given graphically by contours.     

2D steady-state general solution and fundamental solution for fluid-saturated,orthotropic, poroelastic materials

The 2D steady-state solutions regarding the expressions of stress and strain for fluid-saturated, orthotropic, poroelastic plane are derived in this paper. For this object, the general solutions of the corresponding governing equation are first obtained and expressed in harmonic functions. Based on these compact general solutions, the suitable harmonic functions with undetermined constants for line fluid source in the interior of infinite poroelastic body and a line fluid source on the surface of semi-infinite poroelastic body are presented, respectively. The fundamental solutions can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions, equilibrium conditions and boundary conditions.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. I: General solutions and fundamental solutions

Four steady-state general solutions are derived in this paper for the two-dimensional equation of isotropic thermoelastic materials. Using the differential operator theory, three general solutions can be derived and expressed in terms of one function, which satisfies a six-order partial differential equation. By virtue of the Almansi’s theorem, the three general solutions can be transferred to three general solutions which are expressed in terms of two harmonic functions, respectively. At last, a integrate general solution expressed in three harmonic functions is obtained by superposing the obtained two general solutions. The proposed general solution is very simple in form and can be used easily in certain boundary problems. As two examples, the fundamental solutions for both a line heat source in the interior of infinite plane and a line heat source on the surface of semi-infinite plane are presented by virtue of the obtained general solutions.     

2D Green’s functions for semi-infinite orthotropic thermoelastic plane

Based on the 2D general solutions of orthotropic thermoelastic material, the Green’s function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic plane is constructed by three newly introduced harmonic functions. All components of coupled field in semi-infinite thermoelastic plane are expressed in terms of elementary functions. Numerical results are given graphically by contours.     

ON THE FUNDAMENTAL PROBLEM FOR AN INFINITE ELASTIC PLANE BONDED BY DIFFERENT ANISOTROPIC MATERIALS WITH CRACKS

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3D Green’s functions for a transversely isotropic thermoelastic cone

We use the compact harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional Green’s functions of a steady point heat source on the apex of a transversely isotropic thermoelastic cone by three newly introduced harmonic functions. All components of thermoelastic field are expressed in terms of elementary functions and are convenient to use. When the apex angle 2α equals to π, the solution reduce to the important solution of semi-infinite body with a surface point heat source. Numerical results are given graphically by contours.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

Harmonic functions on locally finite networks

Are there nonconstant bounded harmonic functions on an infinite locally finite network under natural transition conditions as continuity at the ramification nodes and classical Kirchhoff conditions at all vertices? We present sufficient criteria for such a network to be a Liouville space, while we show that a large class of infinite trees admit infinitely many linearly independent bounded harmonic functions. Finally, we show that the standard unit cube grid graphs and some of Kepler’s plane tiling graphs are Liouville spaces.     

Temperature and displacement fields in brake and clutch systems with thermoelastic instabilities

In brake and clutch systems kinetic energy is converted into thermal energy. Experiments show that the corresponding temperature field can develop unstable periodic structures. The temperature field couples to the displacement field by thermal expansion. Local pressure maxima in the frictional plane and the corresponding maxima in heat generated cause thermoelastic instabilities (TEI). A model describing both effects covers layers of thermoelastic materials for all necessary mechanical components of the system. The set of field equations of each layer can analytically be solved by separation of constants. These solutions must fulfill the boundary conditions e.g. in the sliding plane. A stability discussion yields whether TEI appear or not. As a study a brake system is analyzed comprising two pads pressed against a rotating disk with cooling channels inside. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

带裂缝的不同材料拼接平面混合问题

本文讨论了带若干条任意形状裂缝的不同材料拼接平面的混合问题，即已知裂缝一侧的位移和另一侧的外应力求弹性平衡，给出了混合问题的正确提法，问题是用复就方法求解的，并归结为求解某种正则型奇异积分方程组，证明了适当且唯一地选择某些待定常数的值，该方程组有唯一解。     

On the Cauchy problem for thermoelastic plates

The Cauchy problem for an infinite thermoelastic plate with a non‐homogeneous governing system and homogeneous initial conditions is solved by means of an area potential. This is the first step in the construction of a potential theory for time‐dependent problems for thermoelastic plates, enabling the reduction of various initial‐boundary value problems to their versions for the homogeneous system of equations with homogeneous initial conditions, which, in turn, may then be solved by means of dynamic potentials. Copyright © 2006 John Wiley & Sons, Ltd.     

Green's functions for a point force and dislocation outside an elliptic inclusion in plane elasticity

The plane elasticity problem of an infinite plate containing an elliptic inclusion is considered. The Green's functions for a point force and/or a dislocation located outside the inclusion are derived. By using the complex potential approach of Muskhelishvili, the general solutions are obtained in a form of carefully selected functions plus an infinite series. The numerical convergence of the solutions is better than that of Stagni-Lizzio's solutions. The proposed solutions can also be applied to the case of a point force and dislocation acting at a point right on the interface.     

The scattering of an harmonic elastic wave by a volume inclusion with a thin interlayer

The boundary element method is used to investigate the propagation of harmonic elastic waves in an infinite matrix with a volume inclusion with a thin interlayer between the inclusion and the matrix. A boundary-integral formulation of the problem is based on a consideration of a two-phase medium, consisting of the matrix and the inclusion, on the contact surface of which conditions of proportional dependence between the forces and jumps in the displacements, which model the interlayer, are satisfied. These conditions are taken into account implicitly in the boundary integral equations obtained, which are subsequently regularized and discretized on the grid of boundary elements introduced. The numerical results obtained demonstrate the effect of the interlayer on the dynamic contact stresses for a spherical inclusion in the field of a plane longitudinal wave.     

Supercyclic sequences of differential operators

Summary Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff--MacLane--Godefroy--Shapiro&apos;s theorems. A new, general eigenvalue criterium for supercyclicity is also provided.     

Excessiveness of harmonic functions for certain diffusions

In an earlier paper⁽⁴⁾ the author has shown that a diffusion process whose potential kernel satisfies certain analytic conditions has all of its excessive harmonic functions, which are not identically infinite, continuous. This paper shows that under these conditions (concerning its potential kernel), the excessiveness of its nonnegative harmonic functions isautomatic.     

Backward Uniqueness for Thermoelastic Plates with Rotational Forces

This paper provides a backward uniqueness theorem for thermoelastic plate models which account for rotational forces, under all sets of canonical boundary conditions, including the most challenging case of so-called free boundary conditions. The proof is abstract and accomodates space variable coefficients in the model. This result is derived in two steps. First, in Section 3, a new backward uniqueness theorem for strongly continuous semigroups is given, which is of interest in itself. It is based on the assumption that the resolvent operator of the generator be bounded on suitable rays of the complex plane. Its proof uses the Phragmen-Lindelof Theorem. Next, the paper verifies a fortiori the required resolvent conditions, under all sets of canonical boundary conditions. The explicit proof (in Section 4) considers the most demanding case of free boundary conditions. An abstract version of this proof, and a corresponding backward uniqueness result, are then noted in Section 5, which gives the most general result of this paper. It covers thermoelastic wave equations as well. The results here presented were motivated by, and hence have important implications in, continuous observability/exact controllability problems for thermoelastic plates, and boundary observations/controls, see [8]. March 12, 1999     

Supply of harmonic functions of an infinite number of variables. II

There are exceptionally many harmonic functions of an infinite number of variables. Using for the estimate of the infinite-dimensional Laplacian introduced by P. Levy, estimates of the germ of sums of orthogonal random variables, there are obtained optimal (in a certain sense) conditions of the harmonicity of the functions in a Hilbert space. Along with harmonicity conditions obtained earlier based on estimates of the germ of sums of dependent random variables, they allow one to encompass the manifold of harmonic functions of an infinite number of variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1687–1693, December, 1990.     

Supply of harmonic functions of an infinite number of variables. I

The infinite-dimensional Laplacian (introduced by P. Levy in 1922) has a number of unusual properties. In particular, the supply of harmonic functions of an infinite number of variables connected with this Laplacian is exceptionally large. In this paper, with the help of estimates of the growth of sums of dependent random variables we get (in a certain sense) optimal conditions for functions on a Hilbert space to be harmonic.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1576–1579, November, 1990.     

Surface waves in micropolar thermoelasticity

Free surface waves of arbitrary form in a homogeneous and isotropic linear micropolar thermoelastic half-space with stress-free plane boundary are investigated. It is found that all physical quantities associated with the waves are derivable from two scalar functions and that there exist two families of waves in general. One of these is the classical thermoelastic wave modified under the influence of the microelastic field and the other is a new surface wave not encountered in classical elasticity. The waves are not necessarily plane waves and even when these are assumed to propagate in a fixed direction parallel to the boundary, unlike in classical elasticity, the problem is not one of plane strain. Explicit expressions for the displacement vector, microrotation vector and the temperature are obtained and the nature of deformation has been analysed. Several earlier results are deduced as particular cases of the more general results obtained here.     

A general solution for three dimensional steady-state transversely isotropic hygro-thermo-magneto-piezoelectric media

In this article, we extract the general solution of three dimensional (3D) equations using potential theory method (PTM) for steady-state, transversely isotropic, hygro-thermo-magneto-piezoelectric media (HTMPM). The governing equations are simplified by introducing the displacement functions. A general solution is completely determined by advantage of the superposition principle and operator theory, which is connected in terms of two functions, fulfilling a second-order and twelfth-order homogeneous partial differential equation (PDE), separately. With the help of Almansi’s theorem, the general solution can be further shortened, which is stated by seven harmonic functions only. The acquired general solutions are straightforward structure and helpful in boundary value problems of HTMPM. Further, we apply the 3D fundamental solutions inside an infinite and on the surface of semi-infinite of a steady point heat source united with a steady point moisture source transversely isotropic HTMPM. Comprehensive and exact solutions are given in the form of elementary functions, which appear as a standard for various types of approximate solutions and numerical codes. Some numerical simulation is conducted based on the obtained general solutions.