scholarly journals Elastic Analysis for Nanocontact Problem with Surface Stress Effects under Shear Load

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
D. X. Lei ◽  
L. Y. Wang ◽  
Z. Y. Ou
Keyword(s):  
Surface Stress ◽  
Solid Mechanics ◽  
Integral Transform ◽  
Surface Elasticity ◽  
Elastic Field ◽  
Fourier Integral ◽  
Shear Load ◽  
Transform Method ◽  
Stress Effects ◽  
Special Cases

Consideration of surface stress effects on the elastic field of nanocontact problem has extensive applications in several modern problems of solid mechanics. In this paper, the effects of surface stress on the contact problem at nanometers are studied in the frame of surface elasticity theory. Fourier integral transform method is adopted to derive the fundamental solution of the nanocontact problem under shear load. As two special cases, the deformations induced by a uniformly distributed shear load and a concentrated shear force are discussed in detail, respectively. The results indicate some interesting characteristics in nanocontact mechanics, which are distinctly different from those in macrocontact problem. At nanoscale, both the contact stresses and the displacements on the deformed surface transit continuously across the uniform distributed shear load boundary as a result of surface stress. In addition, the indent depth and the contact stress depend strongly on the surface stress for nanoindentation.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

Thermal Response of Rolling Components Under Mixed Boundary Conditions: An Analytical Approach

1993 ◽  
Vol 115 (4) ◽  
pp. 857-865 ◽  
Author(s):  
P. Ulysse ◽  
M. M. Khonsari
Keyword(s):  
Temperature Distribution ◽  
Peclet Number ◽  
Integral Transform ◽  
Mixed Boundary ◽  
Order Approximation ◽  
Thermal Response ◽  
Fourier Integral ◽  
Cylinder Surface ◽  
Péclet Number ◽  
Special Cases

An analytical solution for the steady-state temperature distribution in a cylinder undergoing uniform heating and nonuniform cooling is presented. The method of solution is a Fourier integral transform technique. The analysis shows that the Neumann series resulting from an integral equation can be well represented by a first-order approximation when the Peclet number is large. Furthermore, it is shown that the ratio of the Biot number to the square root of the Peclet number of the cooling zones is found to play an important role in governing the thermal response of the cylinder surface. The predicted results for the circumferential temperature distribution are compared to published experimental measurements for hot rolling and also existing analytical solutions for special cases. The agreement is found to be very good. By an appropriate superposition technique, the analysis presented may be easily extended to various heat sources and convective cooling zones at different locations of the cylinder surface.

Contact Analysis Considering Surface Stress and Surface Elasticity: Increase of Indentation Hardness and Yield Stress

2012 ◽  
Author(s):  
Takao Hayashi ◽  
Hideo Koguchi
Keyword(s):  
Yield Stress ◽  
Surface Stress ◽  
Indentation Depth ◽  
Surface Elasticity ◽  
Elastic Half Space ◽  
Contact Analysis ◽  
Indentation Hardness ◽  
Indentation Modulus ◽  
Transform Method ◽  
Anisotropic Elastic

An increase in indentation hardness with decreasing indentation depth has been observed in nanoindentation studies. It is known as the indentation size effect. The indentation modulus in Molecular Dynamics (MD) contact analysis is larger than that in theoretical analysis (Hertz contact theory). In this paper, elasto-plastic contact analysis for an anisotropic elastic half-space is performed using the surface Green’s function considering surface stress and surface elasticity. A contact analysis is conducted to investigate the effect of surface stress on yield stress and indentation hardness. The discrete convolution, fast Fourier transform method and conjugate gradient method are applied to the contact analysis. The hardening model of the elasto-perfect plastic law is used in this study. The yield stress is determined so that a contact area considering surface stress is agreed with the one ignoring surface stress. Then, the yield stress ignoring surface stress and surface elasticity fixed at a constant. It is found that the yield stress considering surface stress and surface elasticity increases with decreasing the indentation depth. The indentation hardness considering surface stress and surface elasticity is calculated using the determined yield stress. The effects of surface stress and surface elasticity on the indentation hardness and the yield stress is discussed.

scholarly journals Solusi Model Perubahan Garis Pantai dengan Metode Transformasi Elzaki

2021 ◽  
Vol 4 (2) ◽  
pp. 66
Author(s):  
Maya Sari Wahyuni ◽  
S. Sukarna ◽  
Muh. Irham Rosadi
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Human Activities ◽  
Integral Transform ◽  
Research Result ◽  
Shoreline Change ◽  
Fourier Integral ◽  
Change Model ◽  
Transform Method

. Pantai merupakan kawasan yang sering dimanfaatkan untuk berbagai kegiatan manusia, namun seringkali upaya pemanfaatan tersebut menyebabkan permasalahan pantai sehingga garis pantai berubah. Salah satu cara yang dapat digunakan untuk mengetahui perubahan garis pantai yaitu dengan membuat model matematika. Model perubahan garis pantai berbentuk persamaan diferensial parsial dapat diselesaikan secara analitik dengan menggunakan metode transformasi Elazki. Metode transformasi Elzaki merupakan salah satu bentuk transformasi integral yang diperoleh dari integral Fourier sehingga didapatkan transformasi Elzaki dan sifat-sifat dasarnya. Perubahan garis pantai pada penelitian ini dipengaruhi oleh adanya groin. Penyelesaian model perubahan garis pantai dengan metode transformasi Elzaki dilakukan dengan menerapkan transformasi Elzaki pada model perubahan garis pantai untuk memperoleh model perubahan garis pantai yang baru, kemudian menerapkan syarat batas, kemudian menerapkan invers transformasi Elzaki sehingga diperoleh solusi model perubahan garis pantai. Berdasarkan hasil penelitian, diperoleh bahwa terdapat kesamaan antara pola grafik yang dihasilkan dari solusi model perubahan garis pantai dengan metode transformasi Elzaki dan solusi model perubahan garis pantai dengan metode numerik.Kata Kunci: Perubahan garis pantai, Groin, Analitik, Transformasi Elzaki.The beach is a region that is often used for various human activities, however often these utilization efforts cause beach problems so that the shoreline changes. One way that can be used to determine changes in shoreline is to make a mathematical model. The shoreline change model shaped of partial differential equation can be solved analytically by using the Elzaki transform method. The Elzaki transform method is a form of integral transform obtained from the Fourier integral so that the Elzaki transform and its basic properties are obtained. Shoreline change in this research were affected by groyne. Solution of shoreline change model using Elzaki transform method is carried by applying the Elzaki transform to the shoreline change model to obtain a new shoreline change model, then applying the boundary value, then applying the inverse of Elzaki transform so obtained a solution shoreline change model. Based on the research result, it was found that there was a similiarity between the graphic patterns generated from the solution of shoreline change model using Elzaki transform method and the solution of shoreline change model using numerical method.Keywords: Shoreline change, Groyne, Analitic, Elzaki transform

Interaction between an Embedded Crack and an Interface Crack in Nonhomogeneous Coating System

2005 ◽  
Vol 492-493 ◽  
pp. 397-402
Author(s):  
E.E. Theotokoglou ◽  
Glaucio H. Paulino
Keyword(s):  
Coordinate System ◽  
Half Plane ◽  
Stress Condition ◽  
Integral Transform ◽  
Local Coordinate System ◽  
Fourier Integral ◽  
Plane Stress Condition ◽  
Transform Method ◽  
Fourier Integral Transform ◽  
Embedded Crack

A general methodology is constructed for the fundamental solution of a crack in the homogeneous half-plane interacting with a crack at the interface between the homogeneous elastic half-plane and the nonhomogeneous elastic coating in which the shear modulus varies exponentially with one coordinate. The problem is solved under plane strain or generalized plane stress condition using the Fourier integral transform method. The stress field in the homogeneous half plane is evaluated by the superposition of two states of stresses, one is associated with a local coordinate system in the infinite fractured plate, while the other in the infinite half plane defined in a structural coordinate system.

Effect of Surface Elasticity on the Interaction Between Steps

2006 ◽  
Vol 74 (4) ◽  
pp. 821-823 ◽  
Author(s):  
Gan-Yun Huang ◽  
Shou-Wen Yu
Keyword(s):  
Integral Transform ◽  
Surface Elasticity ◽  
Elastic Interaction ◽  
Fourier Integral ◽  
Shear Loads ◽  
Surface Steps ◽  
Fourier Integral Transform ◽  
Intrinsic Length ◽  
Force Components ◽  
Effect Of Surface

By taking into account the effect of surface elasticity, the problem of a half plane under concentrated normal or shear loads is first considered. The solutions for the displacements or alternatively named surface Green’s functions can be obtained by using the Fourier integral transform technique. Based on such solutions, the elastic interaction between two surface steps that are modeled as force dipoles is further investigated. The results show that the effect of surface elasticity on the interaction energy is significant when the distance between the two steps is in the range of several times the intrinsic length scale of the system. Further, surface elasticity seems to influence the interaction between steps with force components parallel to the surface more strongly than that when the steps exhibit force components only normal to the surface.

scholarly journals Nonlinear Vibrational Analysis of Nanobeams Embedded in an Elastic Medium including Surface Stress Effects

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
S. Azizi ◽  
B. Safaei ◽  
A. M. Fattahi ◽  
M. Tekere
Keyword(s):  
Surface Stress ◽  
High Surface ◽  
Surface Elasticity ◽  
Vibrational Analysis ◽  
Harmonic Balance Method ◽  
Volume Ratio ◽  
Stress Effect ◽  
Nanoscale Structures ◽  
Stress Effects ◽  
Surface Stress Effect

Due to size-dependent behavior of nanostructures, the classical continuum models are not applicable for the analyses at this submicron size. Surface stress effect is one of the most important matters which make the nanoscale structures have different properties compared to the conventional structures due to high surface to volume ratio. In the present study, nonlinear free vibrational characteristics of embedded nanobeams are investigated including surface stress effects. To this end, a thin surface layer is assumed on the upper and lower surfaces of the cross section to separate the surface and bulk of nanobeams with their own different material properties. Based on harmonic balance method, closed-form analytical solution is conducted for nonlinear vibrations to obtain natural frequencies of embedded nanobeams with and without considerations of surface elasticity and residual surface tension effects corresponding to the various values of nondimensional amplitude, elastic foundation modulus, and geometrical variables of the system. Selected numerical results are given to indicate the influence of each one in detail.

The Calculation of Ship Wave in Shallow Water by Finite Difference Method

2013 ◽  
Vol 409-410 ◽  
pp. 1461-1464
Author(s):  
Deng Hui ◽  
Zhi Hong Zhang ◽  
Jian Nong Gu
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Shallow Water ◽  
Difference Method ◽  
Integral Transform ◽  
Fourier Integral ◽  
Transform Method ◽  
The Mathematical Model ◽  
Supercritical Speed

Based on the shallow water wave potential flow theory and slender ship assumption, the mathematical model is established for calculating wave caused by ship moving at supercritical speed. The wave pattern caused by ship moving at supercritical speed in shallow water was calculated by using the finite difference method. The effects of channel wall were analyzed. The computed results were compared with the ones calculated by Fourier integral transform method and experiment. A good agreement exists between the calculated with experimental results. The mathematical model and the calculation method were validated.

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