scholarly journals On modeling bodies with delaminating coatings taking into account the fields of prestresses

2020 ◽  
pp. 5-16 ◽  
Author(s):  
I V Bogachev ◽  
A O Vatulyan
Keyword(s):  
Original Problem ◽  
Dimensional Structure ◽  
Inhomogeneous Body ◽  
Displacement Fields ◽  
Longitudinal Coordinate ◽  
Stress Functions ◽  
Mathematical Section ◽  
Steady Oscillations ◽  
The Fourier Transform ◽  
Upper Boundary

The paper presents a model of steady-state oscillations of an inhomogeneous body with a prestressed exfoliating coating based on a general linearized statement of the problem of the motion of a prestressed-strained elastic body. On its basis, the statement of the problem of oscillations of an inhomogeneous strip consisting of a substrate and a prestressed coating is formulated, between which there is a delamination in a certain region. Steady oscillations are caused by a load applied to the upper boundary of the coating. To calculate the oscillations of the two-dimensional structure under consideration, the Fourier transform in the longitudinal coordinate was used and the original problem was reduced to solving a number of auxiliary boundary value problems with respect to transformants of the desired functions. From the conditions that the stress functions vanish (the cover is modeled as a mathematical section) of the substrate and the coating, the operator relations are constructed in the area of delamination to calculate the opening functions. The kernels of these operator relations are singular and are integrals over an infinite interval. A study was made of the behavior of their integrands at infinity, on the basis of which special approaches were used to calculate the kernels. As a result of solving the obtained hypersingular equations with difference kernels, for which the collocation method is used, the originals of the disclosure functions are constructed. Using a similar approach for inverting the Fourier transformations, we constructed relations to calculate the originals of the displacement functions at the upper boundary of the coverage. Based on the computational experiments, an analysis is made of the influence of the initial geometric and mechanical parameters of the substrate and coating on the values of the disclosure functions in the delamination region and the displacement functions at the upper boundary of the layer. The influence of the prestress level on the amplitude-frequency characteristics (AFC) was also investigated. It was found that the most significant effect on the frequency response is in the vicinity of the frequencies of the thick resonances. Based on the information on the displacement fields, it is possible to construct schemes for identifying delamination characteristics.

scholarly journals Fourier Sine Transform Method for Solving the Cerrutti Problem of the Elastic Half Plane in Plane Strain

2018 ◽  
Vol 14 (1) ◽  
pp. 1-11
Author(s):  
Charles Chinwuba Ike
Keyword(s):  
Boundary Conditions ◽  
Half Plane ◽  
Point Load ◽  
Displacement Fields ◽  
Transform Method ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Stress Functions ◽  
The Fourier Transform

Abstract The Fourier sine transform method was implemented in this study to obtain general solutions for stress and displacement fields in homogeneous, isotropic, linear elastic soil of semi-infinite extent subject to a point load applied tangentially at a point considered the origin of the half plane. The study adopted a stress based formulation of the elasticity problem. Fourier transformation of the biharmonic stress compatibility equation was done to obtain bounded stress functions for the elastic half plane problem. Stresses and boundary conditions expressed in terms of the Boussinesq-Papkovich potential functions were transformed using Fourier sine transforms. Boundary conditions were used to obtain the unknown constants of the stress functions for the Cerrutti problem considered; and the complete determination of the stress fields in the Fourier transform space. Inversion of the Fourier sine transforms for the stresses yielded the general expressions for the stresses in the physical domain space variables. The strain fields were obtained from the kinematic relations. The displacement fields were obtained by integration of the strain-displacement relations. The solutions obtained were identical with solutions in literature obtained using Cerrutti stress functions.

scholarly journals Surface contact behavior of functionally graded thermoelectric materials indented by a conducting punch

2021 ◽  
Author(s):  
Xiaojuan Tian ◽  
Yueting Zhou ◽  
Lihua Wang ◽  
Shenghu Ding
Keyword(s):  
Thermoelectric Materials ◽  
Singular Integral Equations ◽  
Damage Mechanism ◽  
Functionally Graded ◽  
Deep Insight ◽  
Displacement Fields ◽  
Thermoelectric Effects ◽  
Homogeneous Solutions ◽  
The Fourier Transform ◽  
Insight Into

AbstractThe contact problem for thermoelectric materials with functionally graded properties is considered. The material properties, such as the electric conductivity, the thermal conductivity, the shear modulus, and the thermal expansion coefficient, vary in an exponential function. Using the Fourier transform technique, the electro-thermo-elastic problems are transformed into three sets of singular integral equations which are solved numerically in terms of the unknown normal electric current density, the normal energy flux, and the contact pressure. Meanwhile, the complex homogeneous solutions of the displacement fields caused by the gradient parameters are simplified with the help of Euler’s formula. After addressing the non-linearity excited by thermoelectric effects, the particular solutions of the displacement fields can be assessed. The effects of various combinations of material gradient parameters and thermoelectric loads on the contact behaviors of thermoelectric materials are presented. The results give a deep insight into the contact damage mechanism of functionally graded thermoelectric materials (FGTEMs).

Stress Field of Semi-Infinite Interface Crack Tip of Double Dissimilar Orthotropic Composite Materials

2010 ◽  
Vol 97-101 ◽  
pp. 1223-1226
Author(s):  
Jun Lin Li ◽  
Shao Qin Zhang
Keyword(s):  
Composite Material ◽  
Composite Materials ◽  
Interface Crack ◽  
Interfacial Crack ◽  
Crack Tip ◽  
Stress Fields ◽  
Displacement Fields ◽  
Orthotropic Composite ◽  
Stress Functions ◽  
Secular Equations

The problem of orthotropic composite materials semi-infinite interfacial crack was studied, by constructing new stress functions and employing the method of composite material complex. In the case that the secular equations’ discriminates the and theoretical solutions to the stress fields and the displacement fields near semi-infinite interface crack tip without oscillation and inter-embedding between the interfaces of the crack are obtained, a comparison with finite element example was done to verify the correction of theoretical solution.

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

The Analysis of Plane Flat Lap Interface End of Orthotropic Bi-Material

2011 ◽  
Vol 233-235 ◽  
pp. 1950-1953
Author(s):  
Cai Xia Ren ◽  
Jun Lin Li
Keyword(s):  
Stress Intensity Factor ◽  
Function Method ◽  
Complex Function ◽  
Stress Fields ◽  
Stress Singularities ◽  
Displacement Fields ◽  
Multiple Stress ◽  
Stress Functions ◽  
Material Fracture ◽  
Material Plane

The orthotropic bi-material plane interface end of a flat lap is studied by constructing new stress functions and using the composite complex function method of material fracture. When the characteristic equations’ discriminates and, the theoretical formulas of stress fields, displacement fields and the stress intensity factor around the flat lap interface end are derived, indicating that there is no oscillatory singularity. There are multiple stress singularities of the orthotropic bi-material plane flat lap interface end.

The Stress Field of Plane Flat Lap Interface End of Orthotropic Bi-Material

2010 ◽  
Vol 44-47 ◽  
pp. 2827-2831
Author(s):  
Jun Lin Li ◽  
Cai Xia Ren ◽  
Wen Ting Zhao ◽  
Jing Zhao
Keyword(s):  
Stress Intensity Factor ◽  
Function Method ◽  
Complex Function ◽  
Stress Fields ◽  
Stress Singularities ◽  
Displacement Fields ◽  
Multiple Stress ◽  
Stress Functions ◽  
Material Fracture ◽  
Material Plane

The orthotropic bi-material plane interface end of a flat lap is studied by constructing new stress functions and using the composite complex function method of material fracture. When the characteristic equations’ discriminates and , the theoretical formulas of stress fields, displacement fields and the stress intensity factor around the flat lap interface end are derived, indicating that there is no oscillatory singularity. There are multiple stress singularities of the orthotropic bi-material plane flat lap interface end.

Analysis of Stress on Interface Crack of Orthotropic and Isotropic Bi-Materials

2011 ◽  
Vol 284-286 ◽  
pp. 19-24
Author(s):  
Jun Lin Li ◽  
Shao Qin Zhang ◽  
Wei Yang Yang ◽  
Jing Zhao
Keyword(s):  
Interface Crack ◽  
Complex Function ◽  
Dissimilar Materials ◽  
Displacement Fields ◽  
Singularity Index ◽  
Stress And Displacement Fields ◽  
Special Stress ◽  
Stress Functions ◽  
Real Singularity ◽  
Material Fracture

This paper is concerned in semi-infinite interface crack of orthotropic and isotropic bi-materials and using the composite material fracture complex function method. By means of constructing special stress functions with two real singularity index and solving the problem of a class of generalized bi-harmonic equations , the stress and displacement fields of two dissimilar materials are obtained .Results demonstrate that the stress and displacement fields near the crack tip show mixed crack characteristics without oscillation.

Image contrast of dislocation in anisotropic cubic crystals

1978 ◽  
Vol 36 (1) ◽  
pp. 326-327
Author(s):  
S.M. Ohr
Keyword(s):  
Displacement Field ◽  
Finite Size ◽  
Anisotropic Elasticity ◽  
Dynamical Theory ◽  
Image Contrast ◽  
Dislocation Loops ◽  
Displacement Fields ◽  
Cubic Crystals ◽  
Circular Loop ◽  
The Fourier Transform

The calculations of the image contrast of dislocation loops reported in the past have made use of the displacement fields which are derived under the assumptions of elastic isotropy. The only exception is the work of Yoffe, but it is based on the asymptotic field of an infinitesimal loop applied to the first order analytical theory of image contrast. It is therefore desirable to calculate the image contrast by exact numerical integration of the equations of the dynamical theory of electron diffraction making use of the displacement field of a finite loop derived from the anisotropic elasticity theory. Recently, we have presented a scheme by which the displacement field of a circular loop of finite size can be calculated for anisotropic cubic crystals from the Fourier transform of the elastic Green's function. In the present study, this numerical scheme has been combined with the image simulation technique developed earlier to calculate the image contrast of dislocation loops in anisotropic cubic crystals.

scholarly journals Numerical factorization of a matrix-function with exponential factors in an anti-plane problem for a crack with process zone

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190109 ◽  
Author(s):  
P. Livasov ◽  
G. Mishuris
Keyword(s):  
Original Problem ◽  
Process Zone ◽  
Hopf Equation ◽  
Mode Iii ◽  
Theme Issue ◽  
Structured Media ◽  
Exponential Factors ◽  
The Fourier Transform ◽  
Dynamic Phenomena ◽  
Interface Mode

In this paper, we consider an interface mode III crack with a process zone located in front of the fracture tip. The zone is described by imperfect transmission conditions. After application of the Fourier transform, the original problem is reduced to a vectorial Wiener–Hopf equation whose kernel contains oscillatory factors. We perform the factorization numerically using an iterative algorithm and discuss convergence of the method depending on the problem parameters. In the analysis of the solution, special attention is paid to its behaviour near both ends of the process zone. Qualitative analysis was performed to determine admissible values of the process zone's length for which equilibrium cracks exist. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

Fault impedance and earthquake energy in the Fourier transform domain

1980 ◽  
Vol 70 (5) ◽  
pp. 1683-1698
Author(s):  
D. J. Andrews
Keyword(s):  
Elastic Medium ◽  
Time Domain ◽  
Fourier Transformation ◽  
Slip Rate ◽  
Transform Domain ◽  
Impedance Function ◽  
Stress Functions ◽  
Two Space Dimensions ◽  
The Fourier Transform ◽  
Fourier Transform Domain

abstract The response of the elastic medium in which a fault is contained determines one relation between the slip function and the stress function on the fault. In the space-time domain, the slip and stress functions are related by a singular integral equation. If a Fourier transformation is performed over the two space dimensions on the fault and over time, the stress transform equals the slip rate transform times an impedance function. Using this impedance function, we may determine the energy radiated or stored in the elastic medium from either the slip or the stress transform.

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