Three-Dimensional Stress Singularities at Conical Notches and Inclusions in Transversely Isotropic Materials

1986 ◽  
Vol 53 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Nihal Somaratna ◽  
T. C. T. Ting
Keyword(s):  
General Solution ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Procedure ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Stress Singularities ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials

This study examines analytically the possible existence of stress singularities of the form σ = ρδf(θ,φ) at the apex of axisymmetric conical boundaries in transversely isotropic materials. (ρ, θ, φ) refer to spherical coordinates with the origin at the apex. The problems of one conical boundary and of two conical boundaries with a common apex are considered. The boundaries are either rigidly clamped or traction free. Separation of variables enables the general solution to be expressed in terms of Legendre functions of the first and second kind. Imposition of boundary conditions leads to an eigenequation that would determine possible values of δ. The degenerate case that arises when the eigenvalues of the elasiticity constants are identical is also discussed. Isotropic materials constitute only a particular case in this class of degenerate materials and previously reported eigenequations corresponding to isotropic materials are shown to be recoverable from the present results. Numerical results corresponding to a few selected cases are also presented to illustrate the solution procedure.

Eigenfunctions at a Singular Point in Transversely Isotropic Materials Under Axisymmetric Deformations

1985 ◽  
Vol 52 (3) ◽  
pp. 565-570 ◽  
Author(s):  
T. C. T. Ting ◽  
Yijian Jin ◽  
S. C. Chou
Keyword(s):  
Asymptotic Solution ◽  
Three Dimensional ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Axisymmetric Deformation ◽  
The Body ◽  
Isotropic Materials ◽  
Plane Strain Deformation ◽  
Polar Coordinate ◽  
Transversely Isotropic Materials

When a two-dimensional elastic body that contains a notch or a crack is under a plane stress or plane strain deformation, the asymptotic solution of the stress near the apex of the notch or crack is simply a series of eigenfunctions of the form ρδf (ψ,δ) in which (ρ,ψ) is the polar coordinate with origin at the apex and δ is the eigenvalue. If the body is a three-dimensional elastic solid that contains axisymmetric notches or cracks and subjected to an axisymmetric deformation, the eigenfunctions associated with an eigenvalue contains not only the ρδ term, but also the ρδ+1, ρδ+2… terms. Therefore, the second and higher-order terms of the asymptotic solution are not simply the second and subsequent eigenfunctions. We present the eigenfunctions for transversely isotropic materials under an axisymmetric deformation. The degenerate case in which the eigenvalues p1 and p2 of the elasticity constants are identical is also considered. The latter includes the isotropic material as a special case.

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
Suneet Singh ◽  
Prashant K. Jain ◽  
Rizwan-uddin
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Coordinate System ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Polar Coordinate System ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Azimuthal Direction ◽  
Spherical Coordinate

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all the three types of boundary conditions in the outer and inner surfaces of the sphere. The solution procedure is demonstrated on an illustrative problem for which results are obtained.

THREE-DIMENSIONAL INFINITE ELEMENTS BASED ON A TREFFTZ FORMULATION

2001 ◽  
Vol 09 (02) ◽  
pp. 381-394 ◽  
Author(s):  
ISAAC HARARI ◽  
PARAMA BARAI ◽  
PAUL E. BARBONE ◽  
MICHAEL SLAVUTIN
Keyword(s):  
Separation Of Variables ◽  
Outer Boundary ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Singular Behavior ◽  
Associated Legendre Functions ◽  
Exterior Problems ◽  
Infinite Elements ◽  
Time Harmonic

Three-dimensional infinite elements for exterior problems of time-harmonic acoustics are developed. The infinite elements mesh only the outer boundary of the finite element domain and need not match the finite elements on the interface. A four-noded infinite element, based on separation of variables in spherical coordinates, is presented. Singular behavior of associated Legendre functions at the poles is circumvented. Numerical results validate the good performance of this approach.

scholarly journals Effective elastic properties of transversely isotropic materials with concave pores

2021 ◽  
Vol 153 ◽  
pp. 103665
Author(s):  
K. Du ◽  
L. Cheng ◽  
J.F. Barthélémy ◽  
I. Sevostianov ◽  
A. Giraud ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Transversely Isotropic ◽  
Effective Elastic Properties ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials
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