Dynamic T-Stress for a Mode-I Crack in an Infinite Elastic Plane

2006 ◽  
Vol 74 (2) ◽  
pp. 378-381 ◽  
Author(s):  
Xian-Fang Li
Keyword(s):  
Integral Equation ◽  
Laplace Transform ◽  
Fourier Transforms ◽  
Integral Equation Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Elastic Plane ◽  
Time Space ◽  
T Stress

An integral equation method is presented to determine dynamic elastic T-stress. Special attention is paid to a single crack in an infinite elastic plane subjected to impact loading. By using the Laplace and Fourier transforms, the associated initial-boundary value problem is transformed to a Fredholm integral equation. The dynamic T-stress in the Laplace transform domain can be expressed in terms of its solution. Moreover, an explicit expression for initial T-stress is derived in closed form. Numerically solving the resulting equation and performing the inverse Laplace transform, the transient response of T-stress is determined in the time space, and the response history of the T-stress is shown graphically. Results indicate that T-stress exhibits apparent transient characteristic.

scholarly journals An Initial-Boundary Value Problem for a Viscous Compressible Flow

2007 ◽  
Vol 14 (1) ◽  
pp. 123-134
Author(s):  
Friedrich-Karl Hebeker ◽  
George C. Hsiao
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Stokes Equations ◽  
Boundary Value ◽  
Integral Equation Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Integral ◽  
Viscous Compressible Flow ◽  
Initial Boundary Value

Abstract A constructive approach is presented to treat an initial boundary value problem for isothermal Navier–Stokes equations. It is based on a characteristics (Lagrangean) approximation locally in time and a boundary integral equation method via nonstationary potentials. As a basic problem, the latter leads to a Volterra integral equation of first kind which is proved to be uniquely solvable and even coercive in some anisotropic Sobolev spaces. The solution depends continuously upon the data and can be constructed by a quasioptimal Galerkin procedure.

ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE

2006 ◽  
Vol 08 (02) ◽  
pp. 189-217 ◽  
Author(s):  
ROSA E. CARDIEL ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Pseudo Differential Operator ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ > 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

Landau–Ginzburg-type equations on the half-line in the critical case

2005 ◽  
Vol 135 (6) ◽  
pp. 1241-1262 ◽  
Author(s):  
Elena I. Kaikina ◽  
Hector F. Ruiz-Paredes
Keyword(s):  
Critical Case ◽  
Linear Part ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Image Position ◽  
Initial Boundary Value

We study nonlinear Landau–Ginzburg-type equations on the half-line in the critical case where β ∈ C, ρ > 2. The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K(p) = αpρ, M = [1/2ρ]. The aim of this paper is to prove the global existence of solutions to the initial–boundary-value problem and to find the main term of the asymptotic representation of solutions in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

scholarly journals OBTAINING AND INVESTIGATION OF THE INTEGRAL REPRESENTATION OF SOLUTION AND BOUNDARY INTEGRAL EQUATION FOR THE NON-STATIONARY PROBLEM OF THERMAL CONDUCTIVITY

2016 ◽  
Vol 6 ◽  
pp. 53-58 ◽  
Author(s):  
Grigoriy Zrazhevsky ◽  
Vera Zrazhevska
Keyword(s):  
Thermal Conductivity ◽  
Integral Equation ◽  
Fundamental Solution ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Boundary Integral Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Integral ◽  
Initial Boundary Value

Technological processes in the energy sector and engineering require the calculation of temperature regime of functioning of different constructions. Mathematical model of thermal loading of constructions is reduced to a non-stationary initial-boundary value problem of thermal conductivity. The article examines the formulation of the non-stationary initial-boundary value problem of thermal conductivity in the form of a boundary integral equation, analyzes the singular equation and builds the fundamental solution. To build the integral representation of the solution the method of weighted residuals is used. The correctness of the obtained integral representation of the solution in Minkowski space is confirmed. Singularity of the fundamental solution is investigated. The boundary integral equation and fundamental solution for axially symmetric domain for internal problem is built. The results of the article can be useful for numerical implementation of boundary element method.

RADIATION FROM A FINITE CYLINDRICAL EXPLOSIVE SOURCE

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1384-1393 ◽  
Author(s):  
Anas M. Abo‐Zena
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Applied Pressure ◽  
Inverse Laplace Transform ◽  
Laplace And Fourier Transforms ◽  
Explosion Pressure ◽  
Explosive Source ◽  
The Laplace Transform

For an elastic material with an infinite circular cylindrical hole, the exact solution due to a pressure on a finite length of the cylinder is obtained as a function of the Laplace transform parameter on time and Fourier transform parameter on the z-coordinate (the axis of the cylinder). The applied pressure is a function of the time and the position z. Numerical inversion of the Laplace and Fourier transforms are required to determine the field quantities in the time and space parameters. In the far field, the inverse Fourier transform can be obtained by an asymptotic expansion. It remains to obtain the inverse Laplace transform numerically. We have found that for cylinders whose radius is small compared with the smallest wavelength of interest, an analytical solution can be obtained. Graphical results for the cases of instantaneous explosion and progression of the detonation with constant velocity are given. In both cases an exponential decay of the explosion pressure is assumed.

Sign in / Sign up

Export Citation Format

Share Document