Stresses in a Stretched Slab Having a Spherical Cavity

1959 ◽  
Vol 26 (2) ◽  
pp. 235-240
Author(s):  
Chih-Bing Ling
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Spherical Cavity ◽  
Analytic Solution ◽  
Linear Equations ◽  
Hankel Transform ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Biharmonic Functions ◽  
Infinite Slab

Abstract This paper presents an analytic solution for an infinite slab having a symmetrically located spherical cavity when it is stretched by an all-round tension. The required stress function is constructed by combining linearly two sets of periodic biharmonic functions and a biharmonic integral. The sets of biharmonic functions are derived from two fundamental functions specially built up for the purpose. The arbitrary functions involved in the biharmonic integral are first adjusted to satisfy the boundary conditions on the surfaces of the slab by applying the Hankel transform of zero order. Then the stress function is expanded in spherical co-ordinates and the boundary conditions on the surface of the cavity are satisfied by adjusting the coefficients of superposition attached to the sets of biharmonic functions. The resulting system of linear equations is solved by the method of successive approximations. The solution is finally illustrated by numerical examples for two radii of the cavity.

Stresses in a Transversely Isotropic Slab Having a Spherical Cavity

1973 ◽  
Vol 40 (3) ◽  
pp. 752-758 ◽  
Author(s):  
A. Atsumi ◽  
S. Itou
Keyword(s):  
Stress Distribution ◽  
Boundary Conditions ◽  
Spherical Cavity ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Isotropic Slab ◽  
Infinite Slab

This paper deals with the analysis of the stress distribution arising in a transversely isotropic infinite slab with a symmetrically located spherical cavity under all-around tension. Difficulties in satisfying both boundary conditions on the surfaces of the slab and the surface of the cavity are successfully overcome by using the methods of Hankel transform and Schmidt-orthogonormalization. For some practical materials the influence of transverse isotropy upon stress distribution is presented in the form of curves.

scholarly journals Application of the successive approximation method to the analysis of bending plates

2018 ◽  
Vol 251 ◽  
pp. 04058
Author(s):  
Radek Gabbasov ◽  
Vladimir Filatov ◽  
Nikita Ryasny
Keyword(s):  
Boundary Conditions ◽  
Successive Approximation ◽  
Approximation Method ◽  
High Accuracy ◽  
Numerical Results ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Successive Approximation Method ◽  
Medium Thickness ◽  
Bending Plates

This work presents an algorithm for calculating the bending plates of medium thickness according to the Reissner’. To obtain numerical results, the method of successive approximations (MSA) is used. This method has high accuracy and fast convergence, which was confirmed by the solution of a range of tasks. Publication of the results of the calculation of plates of medium thickness with the boundary conditions revised here is supposed to be in the following articles.

scholarly journals THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS

World Science ◽  
2019 ◽  
Vol 1 (11(51)) ◽  
pp. 31-39
Author(s):  
Zelensky A. G.
Keyword(s):  
Boundary Conditions ◽  
High Efficiency ◽  
Boundary Value ◽  
Three Dimensional ◽  
Median Plane ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shallow Shells ◽  
Common Solution ◽  
Arbitrary Thickness

The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two- dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.

Dynamic response of an elastic half-space to time-dependent surface tractions over an embedded spherical cavity

1965 ◽  
Vol 287 (1411) ◽  
pp. 549-567 ◽  
Keyword(s):  
General Solution ◽  
Point Source ◽  
Half Space ◽  
Dynamic Problem ◽  
Spherical Cavity ◽  
Elastic Half Space ◽  
Time Dependent ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Limit Case

The dynamic problem of the deformation of a homogeneous, perfectly elastic and isotropic half space due to harmonically time-dependent tractions over the boundary of an embedded spherical cavity is discussed. The solution is developed completely and rigorously by a method of successive approximations. Lamb’s solution for a point source in a half-space is derived as a limit case of the general solution. The problem is suggested by its applications in the theory of underground explosions and in seismology.

scholarly journals Приближенные граничные условия для задачи нахождения оптических коэффициентов ультратонких металлических пленок в СВЧ и ТГЦ диапазонах

2020 ◽  
Vol 128 (9) ◽  
pp. 1327
Author(s):  
П.С. Глазунов ◽  
В.А. Вдовин ◽  
В.Г. Андреев
Keyword(s):  
Boundary Conditions ◽  
Maximum Error ◽  
Dielectric Substrate ◽  
Thickness Dependence ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Transmission Coefficients ◽  
Fabry Perot ◽  
Approximate Boundary Conditions ◽  
Analytical Expressions

Approximate boundary conditions for a problem of calculating the optical coefficients of a system composed of inhomogeneous ultrathin metallic film with an arbitrary thickness dependence of conductivity deposited on dielectric substrate are obtained. The derivation of the boundary conditions is based on the Picard method of successive approximations. Analytical expressions for the errors in calculating the optical coefficients with use of the proposed approximate boundary conditions are presented. It is shown that the error increases with the frequency and the film thickness increasing. The maximum error for films of 10 nm-thickness does not exceed 10.7% at 1 THz. As an example, the complex optical coefficients of a system similar to Fabry-Perot etalon and a metal film without a substrate with model thickness dependence of conductivity are calculated. The coincidence between the results of numerical simulation and calculations performed with approximate boundary conditions is shown. The possibility of direct calculating the average conductivity of a film from experimentally measured reflection and transmission coefficients is demonstrated.

Simplified Approach to Solution of Mixed Boundary Value Problems on Homogeneous Circular Domain in Elasticity

2018 ◽  
Vol 86 (2) ◽  
Author(s):  
Gaurav Singh ◽  
Tanmay K. Bhandakkar
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Mixed Boundary ◽  
Linear Equations ◽  
Mixed Boundary Conditions ◽  
Circular Domain ◽  
Infinite Plane ◽  
Airy Stress Function ◽  
System Of Linear Equations ◽  
Starting Point

This work proposes a novel strategy to render mixed boundary conditions on circular linear elastic homogeneous domain to displacement-based condition all along the surface. With Michell solution as the starting point, the boundary conditions and extent of the domain are used to associate the appropriate type and number of terms in the Airy stress function. Using the orthogonality of sine and cosine functions, the modified boundary conditions lead to a system of linear equations for the unknown coefficients in the Airy stress function. Solution of the system of linear equations provides the Airy stress function and subsequently stresses and displacement. The effectiveness of the present approach in terms of ease of implementation, accuracy, and versatility to model variants of circular domain is demonstrated through excellent comparison of the solution of following problems: (i) annulus with mixed boundary conditions on outer radius and prescribed traction on the inner radius, (ii) cavity surface with mixed boundary conditions in an infinite plane subjected to far-field uniaxial loading, and (iii) circular disc constrained on part of the surface and subjected to uniform pressure on rest of the surface.

Stresses in a Transversely Isotropic Circular Cylinder Having a Spherical Cavity

1974 ◽  
Vol 41 (2) ◽  
pp. 507-511 ◽  
Author(s):  
A. Atsumi ◽  
S. Itou
Keyword(s):  
Stress Distribution ◽  
Circular Cylinder ◽  
Boundary Conditions ◽  
Spherical Cavity ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Numerical Calculations ◽  
Infinite Cylinder ◽  
Longitudinal Tension

This paper concerns the analysis of the stress distribution arising in a transversely isotropic infinite cylinder having a spherical cavity under longitudinal tension. Boundary conditions on the surface of the cylinder and the cavity are well satisfied by using the methods of Hankel transform and Schmidt-orthonormalization. Numerical calculations for some practical materials are carried out and the influence of transverse isotropy upon stress distribution is clarified.

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