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.

Fast Fourier Transform Based Numerical Methods for Elasto-Plastic Contacts of Nominally Flat Surfaces

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
W. Wayne Chen ◽  
Shuangbiao Liu ◽  
Q. Jane Wang
Keyword(s):  
Fourier Transform ◽  
Numerical Methods ◽  
Fast Fourier Transform ◽  
Analytical Solutions ◽  
Three Dimensional ◽  
Stress And Strain ◽  
Two Dimensional ◽  
Response Functions ◽  
Strain Fields ◽  
Flat Surfaces

This paper presents a three-dimensional numerical elasto-plastic model for the contact of nominally flat surfaces based on the periodic expandability of surface topography. This model is built on two algorithms: the continuous convolution and Fourier transform (CC-FT) and discrete convolution and fast Fourier transform (DC-FFT), modified with duplicated padding. This model considers the effect of asperity interactions and gives a detailed description of subsurface stress and strain fields caused by the contact of elasto-plastic solids with rough surfaces. Formulas of the frequency response functions (FRF) for elastic/plastic stresses and residual displacement are given in this paper. The model is verified by comparing the numerical results to several analytical solutions. The model is utilized to simulate the contacts involving a two-dimensional wavy surface and an engineering rough surface in order to examine its capability of evaluating the elasto-plastic contact behaviors of nominally flat surfaces.

Three-Dimensional Non-Linear Shell Theory for Flexible Multibody Dynamics

2015 ◽  
Author(s):  
S. L. Han ◽  
O. A. Bauchau
Keyword(s):  
Shell Theory ◽  
Three Dimensional ◽  
Reduction Process ◽  
Flexible Multibody Dynamics ◽  
Stress And Strain ◽  
Two Dimensional ◽  
Governing Equations ◽  
Strain Fields ◽  
Flexible Multibody ◽  
Material Line

In flexible multibody systems, many components are approximated as shells. Classical shell theories, such as Kirchhoff or Reissner-Mindlin shell theory, form the basis of the analytical development for shell dynamics. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite shells, leading to three-dimensional deformations that generate complex stress states. To overcome this problem, a novel three-dimensional shell theory is proposed in this paper. Kinematically, the problem is decomposed into an arbitrarily large rigid-normal-material-line motion and a warping field. The sectional strains associated with the rigid-normal-material-line motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the global equations describing geometrically exact shells and the local equations describing local deformations. The governing equations for geometrically exact shells are nonlinear, two-dimensional equations, whereas the local equations are linear, one dimensional, provide the detailed distribution of three-dimensional stress and strain fields. Based on a set of approximated solutions, the local equations is reduced to the corresponding global equations. In the reduction process, a 9 × 9 sectional stiffness matrix can be found, which takes into account the warping effects due to material heterogeneity. In the recovery process, three-dimensional stress and strain fields at any point in the shell can be recovered from the two-dimensional shell solution. The proposed method proposed is valid for anisotropic shells with arbitrarily complex through-the-thickness lay-up configuration.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

scholarly journals Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration

2019 ◽  
Vol 484 (6) ◽  
pp. 672-677
Author(s):  
A. V. Vokhmintcev ◽  
A. V. Melnikov ◽  
K. V. Mironov ◽  
V. V. Burlutskiy
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Three Dimensional ◽  
Dynamic Environment ◽  
Closed Form Solution ◽  
Point Clouds ◽  
Accurate Method ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Reconstruction Methods

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and meansquare distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

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.

Approximate Analysis of Penetration of a Cylindrical Roller Into a Thin Elastic Coating

1990 ◽  
Vol 112 (3) ◽  
pp. 460-468 ◽  
Author(s):  
Tsung-Ju Gwo ◽  
Thomas J. Lardner
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Approximate Analytical Solution ◽  
Experimental Studies ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Rigid Substrate ◽  
Closed Form Solutions ◽  
Finite Element Calculations ◽  
Cylindrical Roller

An approximate analytical solution to the problem of two-dimensional indentation of a frictionless cylinder into a thin elastic coating bonded to a rigid substrate has been obtained using the approach introduced by Matthewson for axisymmetric indentation. We show by comparing the results of the approximate solution to the exact solutions and to finite element calculations that the approximate solution is accurate for a/h> 2. The advantage of this approach is that the results are expressed in closed form and the accuracy of the approximate solution improves with increasing values of a/h. For a/h>2, for a given load, the theory overestimates the value of a/h compared to the exact solution by less than 10 percent. In many experimental studies and in preliminary design, it is convenient to have closed-form solutions exhibiting the dependence of the parameters.

scholarly journals Motility of spermatozoa at surfaces

Reproduction ◽  
2003 ◽  
pp. 259-270 ◽  
Author(s):  
DM Woolley
Keyword(s):  
Three Dimensional ◽  
Neck Region ◽  
Sperm Head ◽  
Solid Boundary ◽  
Cylindrical Shape ◽  
Two Dimensional ◽  
Flagellar Beat ◽  
Planar Shape ◽  
Planar Wave

The hydrodynamic basis for the accumulation of spermatozoa at surfaces has been investigated. The general conclusion is that when spermatozoa arrive at a surface, they will remain there if the vector of the time-averaged thrust is directed towards that surface. This can arise in two basic ways. First, consider spermatozoa that maintain a three-dimensional waveform and roll (spin) as they progress: in this case, it is argued that the conical (rather than cylindrical) shape of the flagellar envelope will establish the direction-of-thrust necessary for capture by the surface. (Additional findings, for spermatozoa of this type, are that the swim-trajectory is curved and that the direction of its curvature reveals the roll-direction of the cell.) Second, consider spermatozoa that maintain a strictly two-dimensional waveform at the surface: in this case, spermatozoa can be captured because the plane-of-flattening of the sperm head is tilted slightly relative to the plane of the flagellar beat. The sperm head is acting as a hydrofoil and, in one orientation only, it comes to exert a pressure against the surface. (This pressure may possibly, in vivo, aid the penetration of the zona pellucida.) The hydrofoil action of sperm heads may explain any bias in the circling direction of spermatozoa that execute two-dimensional waves at surfaces. Finally, a more complex phenomenon is where interaction of the spermatozoa with the surface appears to induce a three-dimensional to two-dimensional conversion of the flagellar wave (thus permitting the hydrofoil effect described). This is characteristic of sperm with 'twisted planar' rather than helical waves. In mammalian spermatozoa, approximately half the beat cycle is planar and the other half generates a pattern of torque causing the head to roll clockwise (seen from ahead), producing a torsion of the neck region of the flagellum. It is the gradual suppression of this torsion, by either impedance at the solid boundary or by raised viscosity, that converts the 'twisted planar' shape into a planar wave.

A Nonlinear Elastic Model for Isotropic Materials With Different Behavior in Tension and Compression

1982 ◽  
Vol 104 (1) ◽  
pp. 26-28 ◽  
Author(s):  
Gianluca Medri
Keyword(s):  
Mechanical Characterization ◽  
Three Dimensional ◽  
Small Deformation ◽  
Stress And Strain ◽  
Elastic Model ◽  
Strain Fields ◽  
Isotropic Materials ◽  
Tension And Compression ◽  
Nonlinear Elastic

This note presents a model suitable for the mechanical characterization of isotropic materials with different behavior in tension and compression. The model has been derived from the nonlinear elastic theory and elaborated to adapt it to the small deformation field; the constitutive relation may reliably correlate stress and strain fields even in three-dimensional elastic problems.

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