Explicit Solutions of Plane Elastostatics Problems in Heterogeneous Solids

1998 ◽  
Vol 14 (2) ◽  
pp. 67-73 ◽  
Author(s):  
C. K. Chao ◽  
B. Gao
Keyword(s):  
Elastic Moduli ◽  
Homogeneous Problem ◽  
Circular Inclusion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Infinite Matrix ◽  
Heterogeneous Solids ◽  
Arbitrary Loading ◽  
Accuracy Result ◽  
Plane Elastostatics

AbstractThe problem of two circular inclusions of arbitrary radii and of different elastic moduli, which are perfectly bonded to an infinite matrix subjected to arbitrary loading, is solved by the heterogenization technique. This implies that the solution of the heterogeneous problem can be readily obtained from that of the corresponding homogeneous problem by a simple algebraic substitution. Based on the method of successive approximations and the technique of analytical continuation, the solution is formulated in a manner which leads to an approximate, but arbitrary accuracy, result. The present derived solution can be also applied to the problem with straight boundaries. Both the problem of two circular inclusions embedded in an infinite matrix and the problem of a circular inclusion embedded in a half-plane matrix are considered as our examples to demonstrate the use of the present approach.

A Circular Inclusion With Circumferentially Inhomogeneous Sliding Interface in Plane Elastostatics

1998 ◽  
Vol 65 (1) ◽  
pp. 30-38 ◽  
Author(s):  
C. Q. Ru
Keyword(s):  
Circular Inclusion ◽  
Form Solution ◽  
Infinite Matrix ◽  
Rigorous Solution ◽  
First Order ◽  
Sliding Interface ◽  
The Mean ◽  
Plane Elastostatics ◽  
Basic Boundary ◽  
General Method

A general method is presented to obtain the rigorous solution for a circular inclusion embedded within an infinite matrix with a circumferentially inhomogeneous sliding interface in plane elastostatics. By virtue of analytic continuation, the basic boundary value problem for four analytic functions is reduced to a first-order differential equation for a single analytic function inside the circular inclusion. The finite form solution is obtained that includes a finite number of unknown constants determined by the analyticity of the solution and certain other auxiliary conditions. With this method, the exact values of the average stresses within the circular inclusion can be calculated without solving the full problem. Several specific examples are used to illustrate the method. The effects of the circumferential variation of the interface parameter on the mean stress at the interface and the average stresses within the inclusion are discussed.

Multiple Circular Inclusion Problems in Plane Elastostatics

1974 ◽  
Vol 41 (1) ◽  
pp. 215-221 ◽  
Author(s):  
I.-W. Yu ◽  
G. P. Sendeckyj
Keyword(s):  
Uniaxial Tension ◽  
Elastic Moduli ◽  
Circular Inclusion ◽  
Elastic Matrix ◽  
Schwarz Alternating Method ◽  
Inclusion Problems ◽  
Alternating Method ◽  
Elastic Inclusions ◽  
Plane Bending ◽  
Plane Elastostatics

The problem of an unbounded elastic matrix containing any number of elastic inclusions is considered. The inclusions can have any radii and elastic moduli. Furthermore, the spacing of the inclusions can be arbitrary. The solution for the cases of uniaxial tension and in-plane bending is found by the Schwarz alternating method. Graphical results are presented for a number of examples.

On Bonded Inclusions With Circular or Straight Boundaries in Plane Elastostatics

1990 ◽  
Vol 57 (4) ◽  
pp. 850-856 ◽  
Author(s):  
T. Honein ◽  
G. Herrmann
Keyword(s):  
Homogeneous Problem ◽  
Dissimilar Materials ◽  
Circular Inclusion ◽  
Algebraic Expression ◽  
Infinite Domain ◽  
Limiting Case ◽  
Plane Elastostatics

It is shown that the solution, in plane elastostatics, for an infinite domain with a bonded circular inclusion (heterogeneous problem), may be obtained from the solution of the corresponding homogeneous problem merely by substitution into a simple algebraic expression (heterogenization). This relation is universal in the sense of being independent of the loading considered. The case of two half-planes occupied by two dissimilar materials and bonded along a straight boundary is obtained as a limiting case.

scholarly journals A circular inclusion with inhomogeneous non-slip imperfect interface in harmonic materials

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160285 ◽  
Author(s):  
D. R. McArthur ◽  
L. J. Sudak
Keyword(s):  
Tangential Direction ◽  
Circular Inclusion ◽  
Imperfect Interface ◽  
Variable Coefficients ◽  
Interface Condition ◽  
Infinite Matrix ◽  
Linear Spring ◽  
Spring Type ◽  
Repeated Use ◽  
Plane Elastostatics

In this work, a rigorous study is presented for the problem associated with a circular inclusion embedded in an infinite matrix in finite plane elastostatics where both the inclusion and matrix are comprised a harmonic material. The inclusion/matrix boundary is treated as a circumferentially inhomogeneous imperfect interface that is described by a linear spring-type imperfect interface model where in the tangential direction, the interface parameter is infinite in magnitude and in the normal direction, the interface parameter is finite in magnitude (the so-called non-slip interface condition). Through the repeated use of the technique of analytic continuation, the boundary value problem for four analytic functions is reduced to solve a single first-order linear ordinary differential equation with variable coefficients for a single analytic function defined within the inclusion. The unknown coefficients of said function are then found via various analyticity requirements. The method is illustrated, using a specific example of a particular class of inhomogeneous non-slip imperfect interface. The results from these calculations are then contrasted with the results from the homogeneous imperfect interface. These comparisons indicate that the circumferential variation of interface damage has a pronounced effect on the average boundary stress.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

On a sphere performing linear and torsional oscillations in a viscous fluid

1988 ◽  
Vol 66 (7) ◽  
pp. 576-579
Author(s):  
G. T. Karahalios ◽  
C. Sfetsos
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Time Varying ◽  
Torsional Oscillations

A sphere executes small-amplitude linear and torsional oscillations in a fluid at rest. The equations of motion of the fluid are solved by the method of successive approximations. Outside the boundary layer, a steady secondary flow is induced in addition to the time-varying motion.

scholarly journals Green’s Function In Free Axisymmetric Vibration Analysis Of Annular Thin Plates With Different Boundary Conditions

2015 ◽  
Vol 20 (4) ◽  
pp. 939-951
Author(s):  
K.K. Żur
Keyword(s):  
Power Series ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Thin Plates ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Axisymmetric Vibration ◽  
Annular Plates ◽  
The Core ◽  
Analytical Frequency

Abstract Free vibration analysis of homogeneous and isotropic annular thin plates by using Green’s functions is considered. The formula of the influence function for uniform thin circular and annular plates is presented in closed-form. The limited independent solutions of differential Euler equation were expanded in the Neumann power series based on properties of integral equations. The analytical frequency equations as power series were obtained using the method of successive approximations. The natural axisymmetric frequencies for singularities when the core radius approaches zero are calculated. The results are compared with selected results presented in the literature.

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