Degeneracy, Derivative Rule, and Green’s Functions of Anisotropic Elasticity

2004 ◽  
Vol 71 (2) ◽  
pp. 273-282 ◽  
Author(s):  
Wan-Lee Yin
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Direct Proof ◽  
Anisotropic Elasticity ◽  
Higher Order ◽  
Zeroth Order ◽  
Isotropic Elasticity ◽  
Anisotropic Elastic ◽  
Analytical Expressions ◽  
Basic Elasticity

Degenerate and extra-degenerate anisotropic elastic materials have repeated material eigenvalues whose multiplicity is greater than the number of independent eigensolutions. Using basic elasticity relations, a simple, direct proof is given to show that higher-order eigensolutions may be obtained from the analytical expressions of the zeroth-order eigensolutions according to the derivative rule. These higher-order eigensolutions contribute to the complexity of the general solutions of degenerate and extra-degenerate materials, and to the analytical difficulties inherent in such cases including isotropic elasticity. For all types of anisotropic materials, the general solution is given specific forms to obtain Green’s functions of several domains with straight or elliptical boundaries. These results, presented in fully explicit expressions, extend Green’s functions of nondegenerate materials to degenerate and extra-degenerate cases that have not been explored previously.

Three-Dimensional Green’s Functions in Anisotropic Elastic Bimaterials With Imperfect Interfaces

2003 ◽  
Vol 70 (2) ◽  
pp. 180-190 ◽  
Author(s):  
E. Pan
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Imperfect Interface ◽  
Boundary Integral ◽  
Stress Fields ◽  
Superposition Method ◽  
Interface Conditions ◽  
Complementary Part ◽  
Anisotropic Elastic

In this paper, three-dimensional Green’s functions in anisotropic elastic bimaterials with imperfect interface conditions are derived based on the extended Stroh formalism and the Mindlin’s superposition method. Four different interface models are considered: perfect-bond, smooth-bond, dislocation-like, and force-like. While the first one is for a perfect interface, other three models are for imperfect ones. By introducing certain modified eigenmatrices, it is shown that the bimaterial Green’s functions for the three imperfect interface conditions have mathematically similar concise expressions as those for the perfect-bond interface. That is, the physical-domain bimaterial Green’s functions can be obtained as a sum of a homogeneous full-space Green’s function in an explicit form and a complementary part in terms of simple line-integrals over [0,π] suitable for standard numerical integration. Furthermore, the corresponding two-dimensional bimaterial Green’s functions have been also derived analytically for the three imperfect interface conditions. Based on the bimaterial Green’s functions, the effects of different interface conditions on the displacement and stress fields are discussed. It is shown that only the complementary part of the solution contributes to the difference of the displacement and stress fields due to different interface conditions. Numerical examples are given for the Green’s functions in the bimaterials made of two anisotropic half-spaces. It is observed that different interface conditions can produce substantially different results for some Green’s stress components in the vicinity of the interface, which should be of great interest to the design of interface. Finally, we remark that these bimaterial Green’s functions can be implemented into the boundary integral formulation for the analysis of layered structures where imperfect bond may exist.

Green’s Functions for Infinite and Semi-infinite Anisotropic Thin Plates

2003 ◽  
Vol 70 (2) ◽  
pp. 260-267 ◽  
Author(s):  
Z.-Q. Cheng ◽  
J. N. Reddy
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Infinite Plate ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Green Functions ◽  
Real Form ◽  
Concentrated Forces ◽  
Anisotropic Elastic ◽  
Kirchhoff Theory

This paper presents fundamental solutions of an anisotropic elastic thin plate within the context of the Kirchhoff theory. The plate material is inhomogeneous in the thickness direction. Two systems of problems with non-self-equilibrated loads are solved. The first is concerned with in-plane concentrated forces and moments and in-plane discontinuous displacements and slopes, and the second with transverse concentrated forces. Exact closed-form Green’s functions for infinite and semi-infinite plates are obtained using the recently established octet formalism by the authors for coupled stretching and bending deformations of a plate. The Green functions for an infinite plate and the surface Green functions for a semi-infinite plate are presented in a real form. The hoop stress resultants are also presented in a real form for a semi-infinite plate.

scholarly journals Renormalized spectrum of quasiparticle in limited number of states, strongly interacting with two-mode polarization phonons at T=0 K

2021 ◽  
Vol 24 (1) ◽  
pp. 13705
Author(s):  
M.V. Tkach ◽  
Ju.O. Seti ◽  
O.M. Voitsekhivska
Keyword(s):  
Energy Spectrum ◽  
Ground State ◽  
Green's Functions ◽  
Green’S Functions ◽  
Phonon Modes ◽  
Strongly Interacting ◽  
Renormalized Energy ◽  
Initial States ◽  
Analytical Expressions

Within unitary transformed Hamiltonian of Fröhlich type, using the Green's functions method, exact renormalized energy spectrum of quasiparticle strongly interacting with two-mode polarization phonons is obtained at T=0 K in a model of the system with limited number of its initial states. Exact analytical expressions for the average number of phonons in ground state and in all satellite states of the system are presented. Their dependences on a magnitude of interaction between quasiparticle and both phonon modes are analyzed.

POSITIVITY OF GREEN'S FUNCTIONS TO VOLTERRA INTEGRAL AND HIGHER ORDER INTEGRO-DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 07 (04) ◽  
pp. 405-418 ◽  
Author(s):  
M. I. GIL'
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Green's Functions ◽  
Arbitrary Order ◽  
Green’S Functions ◽  
Higher Order ◽  
Volterra Integral Equations ◽  
Differential Delay

We consider Volterra integral equations and arbitrary order integro-differential equations. We establish positivity conditions and two-sided estimates for Green's functions. These results are then applied to obtain stability and positivity conditions for equations with nonlinear causal mappings (operators) and linear integro-differential parts. Such equations include differential, difference, differential-delay, integro-differential and other traditional equations.

Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solids

1994 ◽  
Vol 447 (1929) ◽  
pp. 175-188 ◽  
Keyword(s):  
Time Domain ◽  
Unit Sphere ◽  
Green's Functions ◽  
Green’S Functions ◽  
Fundamental Solutions ◽  
Anisotropic Elasticity ◽  
Field Equations ◽  
One Dimensional ◽  
Time Harmonic ◽  
Point Forces

A procedure is described to generate fundamental solutions or Green’s functions for time harmonic point forces and sources. The linearity of the field equations permits the Green’s function to be represented as an integral over the surface of a unit sphere, where the integrand is the solution of a one-dimensional impulse response problem. The method is demonstrated for the theories of piezoelectricity, thermoelasticity, and poroelasticity. Time domain analogues are discussed and compared with known expressions for anisotropic elasticity.

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