Interactions Between Dislocations and Anisotropic Elastic Elliptical Inclusions

1994 ◽  
Vol 61 (3) ◽  
pp. 548-554 ◽  
Author(s):  
Wen J. Yen ◽  
Chyanbin Hwu
Keyword(s):  
Anisotropic Media ◽  
Analytical Continuation ◽  
Two Dimensional ◽  
Closed Form Solutions ◽  
Special Cases ◽  
General Field ◽  
Isotropic Media ◽  
In Series ◽  
Elastic Inclusions ◽  
Anisotropic Elastic

A general field solution for the stresses and displacements of the interactions between dislocations and inclusions has been derived in this paper by applying the Stroh’s formalism and the Muskhelishvili’s method of analytical continuation. The solutions are valid for general elastic anisotropic media under two-dimensional deformation. The interaction energy between dislocations and elastic inclusions is obtained explicitly. The solutions in general are expressed in series form for elastic inclusions. However, for the special cases when the elastic inclusions are replaced by a hole or rigid inclusion, simple closed-form solutions are derived. The general solutions are verified by considering the isotropic media since it is the only solution available in the literature. For the general anisotropic media, a series of contour diagrams for the glide component of the force on a dislocation are provided in this paper to study the effects of inclusion hardness, shape, and matrix anisotropy.

On the Anisotropic Elastic Inclusions in Plane Elastostatics

1993 ◽  
Vol 60 (3) ◽  
pp. 626-632 ◽  
Author(s):  
Chyanbin Hwu ◽  
Wen J. Yen
Keyword(s):  
Major Axis ◽  
Analytical Continuation ◽  
Complex Matrix ◽  
Arbitrary Loading ◽  
Special Cases ◽  
The Matrix ◽  
Point Forces ◽  
Elastic Inclusions ◽  
Anisotropic Elastic ◽  
Plane Elastostatics

By combining the method of Stroh’s formalism, the concept of perturbation, the technique of conformal mapping and the method of analytical continuation, a general analytical solution for the elliptical anisotropic elastic inclusions embedded in an infinite anisotropic matrix subjected to an arbitrary loading has been obtained in this paper. The inclusion as well as the matrix are of general anisotropic elastic materials which do not imply any material symmetry. The special cases when the inclusion is rigid or a hole are also studied. The arbitrary loadings include in-plane and antiplane loadings. The shapes of ellipses cover the lines or circles when the minor axis is taken to be zero or equal to the major axis. The solutions of the stresses and deformations in the entire domain are expressed in complex matrix notation. Simplified results are provided for the interfacial stresses along the inclusion boundary. Some interesting examples are solved explicitly, such as point forces or dislocations in the matrix and uniform loadings at infinity. Since the general solutions have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our results are exact and universal.

Contact Problems of Two Dissimilar Anisotropic Elastic Bodies

1998 ◽  
Vol 65 (3) ◽  
pp. 580-587 ◽  
Author(s):  
Chyanbin Hwu ◽  
C. W. Fan
Keyword(s):  
Boundary Conditions ◽  
Contact Region ◽  
Complex Function ◽  
Contact Problems ◽  
Anisotropic Elasticity ◽  
Analytical Continuation ◽  
Two Dimensional ◽  
Different Boundary Conditions ◽  
Elastic Bodies ◽  
Anisotropic Elastic

In this paper, a two-dimensional contact problem of two dissimilar anisotropic elastic bodies is studied. The shapes of the boundaries of these two elastic bodies have been assumed to be approximately straight, but the contact region is not necessary to be small and the contact surface can be nonsmooth. Base upon these assumptions, three different boundary conditions are considered and solved. They are: the contact in the presence of friction, the contact in the absence of friction, and the contact in complete adhesion. By applying the Stroh’s formalism for anisotropic elasticity and the method of analytical continuation for complex function manipulation, general solutions satisfying these different boundary conditions are obtained in analytical forms. When one of the elastic bodies is rigid and the boundary shape of the other elastic body is considered to be fiat, the reduced solutions can be proved to be identical to those presented in the literature for the problems of rigid punches indenting into (or sliding along) the anisotropic elastic halfplane. For the purpose of illustration, examples are also given when the shapes of the boundaries of the elastic bodies are approximated by the parabolic curves.

Electromagnetic plane waves in anisotropic media: an approach using bivectors

1990 ◽  
Vol 330 (1614) ◽  
pp. 335-393 ◽  
Keyword(s):  
Plane Waves ◽  
Anisotropic Media ◽  
Circularly Polarized ◽  
Homogeneous Wave ◽  
Special Cases ◽  
Inhomogeneous Plane ◽  
Isotropic Media ◽  
Electromagnetic Plane Waves ◽  
Qualitative Changes ◽  
Electromagnetic Plane

The propagation of time-harmonic electromagnetic plane waves in non-absorbing, non-optically active, electrically and magnetically anisotropic media is considered. Both homogeneous and inhomogeneous plane waves are considered. All such solutions to Maxwell equations are obtained for crystals with arbitrary uniform magnetic anisotropy to the electrical anisotropy introduces qualitative changes. For example, for homogeneous linearly polarized waves in magnetically isotropic media the electric displacement vector D and the magnetic induction vector B are always orthogonal, whereas for magnetically anisotropic media these vectors are generally along the common conjugate radii of pairs of ellipses and are only orthogonal in special cases. Also in magnetically isotropic media a homogeneous wave with D and B both circularly polarized may propagate along an optic axis. However, for magnetically and electrically anisotropic media there is in general no homogeneous wave for which D and B are both circularly polarized. For inhomogeneous waves there are similar qualitative changes for magnetically anisotropic media. The description of an inhomogeneous plane wave involves two complex vectors, or bivectors: the amplitude and slowness bivectors. By a systematic use of the properties of bivectors and their associated directional ellipses, many of the results obtained are given a direct geometrical interpretation.

Thermoelastic Problems for the Anisotropic Elastic Half-Plane

2004 ◽  
Vol 126 (3) ◽  
pp. 459-465 ◽  
Author(s):  
Yuan Lin ◽  
Timothy C. Ovaert
Keyword(s):  
Half Plane ◽  
Hilbert Problem ◽  
Thermoelastic Problem ◽  
Two Dimensional ◽  
Transfer Condition ◽  
Isotropic Media ◽  
Anisotropic Elastic ◽  
Steady State Heat ◽  
Steady State Heat Transfer ◽  
Thermoelastic Problems

By applying the extended version of Stroh’s formalism, the two-dimensional thermoelastic problem for a semi-infinite anisotropic elastic half-plane is formulated. The steady-state heat transfer condition is assumed and the technique of analytical continuation is employed; the formulation leads to the Hilbert problem, which can be solved in closed form. The general solutions due to different kinds of thermal and mechanical boundary conditions are obtained. The results show that unlike the two-dimensional thermoelastic problem for an isotropic media, where a simply-connected elastic body in a state of plane strain or plane stress remains stress free if the temperature distribution is harmonic and the boundaries are free of traction, the stress within the semi-infinite anisotropic media will generally not equal zero even if the boundary is free of traction.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

Quasi‐shear waves in inhomogeneous weakly anisotropic media by the quasi‐isotropic approach: A model study

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Ivan Pšenčík ◽  
Joe A. Dellinger
Keyword(s):  
Order Approximation ◽  
Anisotropic Media ◽  
Ray Method ◽  
Weak Anisotropy ◽  
S Waves ◽  
Model Study ◽  
Arbitrary Strength ◽  
Isotropic Media ◽  
Synthetic Models ◽  
Ray Methods

In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.

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