Elastic Fields Resulting From Concentrated Loading on a Three-Dimensional Incompressible Wedge

1995 ◽  
Vol 62 (3) ◽  
pp. 557-565 ◽  
Author(s):  
M. T. Hanson
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Elastic Field ◽  
Point Force ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Elementary Functions ◽  
Normal Loading ◽  
Direction Parallel ◽  
Elastic Fields

This paper considers point force or point moment loading applied to the surface of a three-dimensional wedge. The wedge is two-dimensional in geometry but the loading may vary in a direction parallel to the wedge apex, thus creating a three-dimensional problem within the realm of linear elasticity. The wedge is homogeneous, isotropic, and the assumption of incompressibility is taken in order for solutions to be obtained. The loading cases considered presently are as follows: point normal loading on the wedge face, point moment loading on the wedge face, and an arbitrarily directed force or moment applied at a point on the apex of the wedge. The solutions given here are closed-form expressions. For point force or point moment loading on the wedge face, the elastic field is given in terms of a single integral containing associated Legendre functions. When the point force or moment is at the wedge tip, closed-form (nonintegral) expressions are obtained in terms of elementary functions. An interesting result of the present research indicates that the wedge paradox in two-dimensional elasticity also exists in the three-dimensional case for a concentrated moment at the wedge apex applied in one direction, but that it does not exist for a moment applied in the other two directions.

The Elastic Field of an Elliptic Inclusion With a Slipping Interface

1986 ◽  
Vol 53 (1) ◽  
pp. 103-107 ◽  
Author(s):  
E. Tsuchida ◽  
T. Mura ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Elastic Field ◽  
Elliptic Inclusion ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Bonded Interface ◽  
The Matrix ◽  
Displacement Potentials ◽  
Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

Equivalent Circuits of the Elastic Field

1944 ◽  
Vol 11 (3) ◽  
pp. A149-A161
Author(s):  
Gabriel Kron
Keyword(s):  
Steady State ◽  
Reference Frames ◽  
Three Dimensional ◽  
Elastic Field ◽  
Companion Paper ◽  
Theory Of Elasticity ◽  
Equivalent Circuits ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Nonlinear Stress

Abstract This paper presents equivalent circuits representing the partial differential equations of the theory of elasticity for bodies of arbitrary shapes. Transient, steady-state, or sinusoidally oscillating elastic-field phenomena may now be studied, within any desired degree of accuracy, either by a “network analyzer,” or by numerical- and analytical-circuit methods. Such problems are the propagation of elastic waves, determination of the natural frequencies of vibration of elastic bodies, or of stresses and strains in steady-stressed states. The elastic body may be non-homogeneous, may have arbitrary shape and arbitrary boundary conditions, it may rotate at a uniform angular velocity and may, for representation, be divided into blocks of uneven length in different directions. The circuits are developed to handle both two- and three-dimensional phenomena. They are expressed in all types of orthogonal curvilinear reference frames in order to simplify the boundary relations and to allow the solution of three-dimensional problems with axial and other symmetry by the use of only a two-dimensional network. Detailed circuits are given for the important cases of axial symmetry, cylindrical co-ordinates (two-dimensional) and rectangular co-ordinates (two- and three-dimensional). Nonlinear stress-strain relations in the plastic range may be handled by a step-by-step variation of the circuit constants. Nonisotropic bodies and nonorthogonal reference frames, however, require an extension of the circuits given. The circuits for steady-state stress and small oscillation phenomena require only inductances and capacitors, while the circuits for transients require also standard (not ideal) transformers. A companion paper deals in detail with numerical and experimental methods to solve the equivalent circuits.

Three-Dimensional Simulations of Phase Separation in Model Binary Alloy Systems with Elasticity

1997 ◽  
Vol 481 ◽  
Author(s):  
D. Orlikowski ◽  
C. Sagui ◽  
A. S. Somoza ◽  
C. Roland
Keyword(s):  
Phase Separation ◽  
Binary Alloy ◽  
Large Scale ◽  
Three Dimensional ◽  
Elastic Field ◽  
Dimensional System ◽  
Elastic Fields ◽  
Slowing Down ◽  
Three Dimensional Simulations ◽  
Alloy Systems

ABSTRACTWe report on large-scale three-dimensional simulations of phase separation in model binary alloy systems in the presence of elastic fields. The elastic field has several important effects on the morphology of the system: the ordered domains are subject to shape transformations, and spatial ordering. In contrast to two-dimensional system, no significant slowing down in the growth is observed. There is also no evidence of any “reverse coarsening” of the domains.

scholarly journals Surface Displacements due to a Steadily Moving Point Force

1996 ◽  
Vol 63 (2) ◽  
pp. 245-251 ◽  
Author(s):  
J. R. Barber
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Elastic Half Space ◽  
Point Force ◽  
Two Dimensional ◽  
Normal Surface ◽  
Displacement Fields ◽  
Linear Superposition ◽  
Surface Displacements ◽  
Stress And Displacement Fields

Closed-form expressions are obtained for the normal surface displacements due to a normal point force moving at constant speed over the surface of an elastic half-space. The Smirnov-Sobolev technique is used to reduce the problem to a linear superposition of two-dimensional stress and displacement fields.

Three-dimensional thermo-electro-elastic field in one-dimensional hexagonal piezoelectric quasi-crystal weakened by an elliptical crack

2021 ◽  
pp. 108128652110592
Author(s):  
Yuwei Liu ◽  
Xuesong Tang ◽  
Peiliang Duan ◽  
Tao Wang ◽  
Peidong Li
Keyword(s):  
Analytical Solution ◽  
Elliptic Integral ◽  
Electric Displacement ◽  
Three Dimensional ◽  
Elastic Field ◽  
Elliptical Crack ◽  
Elementary Functions ◽  
Surface Displacements ◽  
Electric Displacement Intensity Factor ◽  
Generalized Potential Theory

In this paper, an analytical solution is developed for the problem of an infinite 1D hexagonal piezoelectric quasi-crystal medium weakened by an elliptical crack and subjected to mixed loads on the crack surfaces. The mixed loads comprise the phonon pressure, phason pressure, electric displacement, and temperature increment, and the crack surfaces can be electrically permeable or impermeable. Based on a general solution, combined with the generalized potential theory, the steady-state 3D thermo-electro-elastic field variables in the quasi-crystal are obtained in terms of elliptic integral functions and elementary functions. Several important physical quantities on the cracked plane, such as the generalized crack surface displacements, normal stresses, and stress intensity factors, are derived in closed forms. An illustrative numerical calculation verifies the presented analytical solution and shows the distribution of the 3D thermo-electro-elastic field. It is indicated that the influence of the phason field on the result is pronounced, especially for the electric field variables, and the electric permeability of crack surfaces has a significant effect on the electric displacement intensity factor at the crack tip.

Elastic Fields in a Polyhedral Inclusion With Uniform Eigenstrains and Related Problems

2000 ◽  
Vol 68 (3) ◽  
pp. 441-452 ◽  
Author(s):  
H. Nozaki ◽  
M. Taya
Keyword(s):  
Stress Field ◽  
Strain Energy ◽  
Dimensional Space ◽  
Spherical Inclusion ◽  
Three Dimensional ◽  
Elastic Field ◽  
Bounded Region ◽  
Symmetrical Structure ◽  
Elastic Fields ◽  
Three Dimensional Space

In this paper, the elastic field in an infinite elastic body containing a polyhedral inclusion with uniform eigenstrains is investigated. Exact solutions are obtained for the stress field in and around a fully general polyhedron, i.e., an arbitrary bounded region of three-dimensional space with a piecewise planner boundary. Numerical results are presented for the stress field and the strain energy for several major polyhedra and the effective stiffness of a composite with regular polyhedral inhomogeneities. It is found that the stresses at the center of a polyhedral inclusion with uniaxial eigenstrain do not coincide with those for a spherical inclusion (Eshelby’s solution) except for dodecahedron and icosahedron which belong to icosidodeca family, i.e., highly symmetrical structure.

The Elastic Field Resulting From Elliptical Hertzian Contact of Transversely Isotropic Bodies: Closed-Form Solutions for Normal and Shear Loading

1997 ◽  
Vol 64 (3) ◽  
pp. 457-465 ◽  
Author(s):  
M. T. Hanson ◽  
I. W. Puja
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Normal Pressure ◽  
Shear Loading ◽  
Potential Functions ◽  
Hertzian Contact ◽  
Elementary Functions ◽  
Shear Traction

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

Stream Functions and Streamlines for Visualizing and Quantifying Side Flows in EHL of Elliptical Contacts

2000 ◽  
Vol 123 (3) ◽  
pp. 603-607 ◽  
Author(s):  
Hsing-Sen S. Hsiao ◽  
Bernard J. Hamrock ◽  
John H. Tripp
Keyword(s):  
Closed Form ◽  
Contact Zone ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Hertzian Contact ◽  
Two Dimensional ◽  
Three Dimensional Flow ◽  
Flow Rates ◽  
Stream Functions ◽  
Pass Through

Three-dimensional stream functions for flows in elastohydrodynamically lubricated elliptical conjunctions are formulated. Closed-form and numerical solutions for the stream functions on special planes are obtained. Streamlines on these special planes are plotted to reveal the trajectories of the lubricant particles that pass by, pass through, or flow back from the Hertzian contact zone. Furthermore, a conceptual column stream function and column streamlines are introduced to present the three-dimensional flow in a two-dimensional manner. Thereby, the column streamlines can be plotted to visualize and quantify the flow rates of the lubricant that passes by or passes through the Hertzian zone.

On the Displacement of a Two-Dimensional Eshelby Inclusion of Elliptic Cylindrical Shape

2017 ◽  
Vol 84 (7) ◽  
Author(s):  
Xiaoqing Jin ◽  
Xiangning Zhang ◽  
Pu Li ◽  
Zheng Xu ◽  
Yumei Hu ◽  
...  
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Companion Paper ◽  
Stress And Strain ◽  
Cylindrical Shape ◽  
Two Dimensional ◽  
Elasticity Solution ◽  
Strain Fields ◽  
Closed Form Solutions ◽  
Eshelby Inclusion

In a companion paper, we have obtained the closed-form solutions to the stress and strain fields of a two-dimensional Eshelby inclusion. The current work is concerned with the complementary formulation of the displacement. All the formulae are derived in explicit closed-form, based on the degenerate case of a three-dimensional (3D) ellipsoidal inclusion. A benchmark example is provided to validate the present analytical solutions. In conjunction with our previous study, a complete elasticity solution to the classical elliptic cylindrical inclusion is hence documented in Cartesian coordinates for the convenience of engineering applications.

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