On some steady state thermoelastic stress distributions in a slab

1965 ◽  
Vol 14 (4) ◽  
pp. 303-310 ◽  
Author(s):  
R. Shail
Keyword(s):  
Steady State ◽  
Thermal Stresses ◽  
Integral Transform ◽  
Displacement Vector ◽  
Scalar Potential ◽  
Thermoelastic Stress ◽  
Fourier Integral ◽  
Elastic Half Space ◽  
Transform Method ◽  
Plane State

The calculation of the steady state thermal stresses in an isotropic elastic half space or slab with traction free faces has been the subject of several investigations. Steinberg and McDowell (1), using an extension of the Bousinesq-Papkowitch method of isothermal elasticity, first derived the now well-known result that in such a body which contains no heat sources there exists a plane state of stress parallel to the boundary planes. Sneddon and Lockett (2) approached this class of problems by direct solution of the equations of thermoelasticity using a double Fourier integral transform method, the results being transformed to Hankel type integrals in the case of axial symmetry. A further approach due to Nowinski (3) exploits the fact that in steady state thermoelasticity each component of the displacement vector is a biharmonic function which can be expressed as a combination of harmonics. However, possibly the most economical method of solution of this type of problem is that of Williams (4) who expressed the displacement vector in terms of two scalar potential functions, one of which is directly related to the temperature field. The same principle has also been used by Fox (5) in treating thermoelastic distributions in a slab containing a spherical cavity.

Steady-State Thermal Stresses in Wedge-Shaped Cutting Tools

1975 ◽  
Vol 97 (3) ◽  
pp. 1060-1066
Author(s):  
P. F. Thomason
Keyword(s):  
Thermal Stress ◽  
Steady State ◽  
Metal Cutting ◽  
Thermal Stresses ◽  
Heat Conduction Problem ◽  
Equivalent Stress ◽  
Cutting Tools ◽  
Thermoelastic Stress ◽  
Plastic State ◽  
Steady State Heat

Closed form expressions for the steady-state thermal stresses in a π/2 wedge, subject to constant-temperature heat sources on the rake and flank contact segments, are obtained from a conformal mapping solution to the steady-state heat conduction problem. It is shown, following a theorem of Muskhelishvili, that the only nonzero thermal stress in the plane-strain wedge is that acting normal to the wedge plane. The thermal stress solutions are superimposed on a previously published isothermal cutting-load solution, to give the complete thermoelastic stress distribution at the wedge surfaces. The thermoelastic stresses are then used to determine the distribution of the equivalent stress, and this gives an indication of the regions on a cutting tool which are likely to be in the plastic state. The results are discussed in relation to the problems of flank wear and rakeface crater wear in metal cutting tools.

Transient Thermoelastic Stress Fields in a Half-Space

2002 ◽  
Vol 125 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Shuangbiao Liu ◽  
Qian Wang
Keyword(s):  
Stress Field ◽  
Half Space ◽  
Frictional Heating ◽  
Three Dimensional ◽  
Thermoelastic Stress ◽  
Parabolic Type ◽  
Elastic Half Space ◽  
Transform Method ◽  
Rough Surface Contact ◽  
Thermoelastic Stress Field

Computing the thermoelastic stress field of a material subjected to frictional heating is essential for component failure prevention and life prediction. However, the analysis for three-dimensional thermoelastic stress field for tribological problems is not well developed. Furthermore, the pressure distribution due to rough surface contact is irregular; hence the frictional heating can hardly be described by an analytical expression. This paper presents a novel set of frequency-domain expressions (frequency response functions) of the thermoelastic stress field of a uniformly moving three-dimensional elastic half-space subjected to arbitrary transient frictional heating, where the velocity of the half-space, its magnitude and direction, can be an arbitrary function of time. General formulas are expressed in the form of time integrals, and important expressions for constant velocities are given for the transient-instantaneous, transient-continuous, and steady-state cases. The thermoelastic stress field inside a translating half-space with constant velocities are illustrated and discussed by using the discrete convolution and fast Fourier transform method when a parabolic type or an irregularly distributed heat source is applied.

Dynamic Response of a Rigid Foundation Subjected to a Distance Blast

2012 ◽  
Author(s):  
Deji Ojetola ◽  
Hamid R. Hamidzadeh
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Integral Transform ◽  
Numerical Technique ◽  
Formal Solution ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Rigid Foundation ◽  
Transform Method

Blasts and explosions occur in many activities that are either man-made or nature induced. The effect of the blasts could have a residual or devastating effect on the buildings at some distance within the vicinity of the explosion. In this investigation, an analytical solution for the time response of a rigid foundation subjected to a distant blast is considered. The medium is considered to be an elastic half space. A formal solution to the wave propagations on the medium is obtained by the integral transform method. To achieve numerical results for this case, an effective numerical technique has been developed for calculation of the integrals represented in the inversion of the transformed relations. Time functions for the vertical and radial displacements of the surface of the elastic half space due to a distant blast load are determined. Mathematical procedures for determination of the dynamic response of the surface of an elastic half-space subjected to the blast along with numerical results for displacements of a rigid foundation are provided.

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.

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

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.

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.

Displacement fields of a Cuboid crystal in a Photoacoustic Cell: Mathematical aspects

2021 ◽  
Vol 101 (1) ◽  
pp. 6-11
Author(s):  
A.P. Sarode ◽  
◽  
O.H. Mahajan ◽  
Keyword(s):  
Displacement Field ◽  
Thermal Stresses ◽  
Integral Transform ◽  
Solid State Physics ◽  
Photoacoustic Cell ◽  
Displacement Fields ◽  
Transform Method ◽  
Premature Failure ◽  
Acoustic Effect ◽  
Photoacoustic Effect

Photo acoustic effect is popular due to a minimal sample preparation during execution, the ability to examine scattering and opaque sample along with the capability to access depth profile. These features enable Photoacoustic spectroscopy to be used in depth-resolved characterization of solids. Thermal interaction is a basic perspective in solid state physics research regarding industrial devices and components. It is a key factor of fabrication and performance of such devices and components. Today, crystalline solids are widely studied due to their wide scientific and industrial applications. Displacement field resulting in thermal stresses is one of the important aspects of premature failure of industrial components and devices. In this paper, displacement fields in photoacoustic effect with solid cuboid crystal are mathematically presented. According to our opinion, displacement fields in photoacoustic effect in three dimensional analysis are not reported earlier. Hence that will be a major contribution of this paper. For a simple cuboid homogeneous crystal kept in a photoacoustic cell, an airy stress function is determined based on laser interaction with surface of the crystal. By applying the finite Marchi-Fasulo integral transform method within the crystal size limitations, displacement field is exactly determined.

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