scholarly journals Longitudinal waves in an elastic rod caused by sudden damage to the foundation

2021 ◽  
pp. 1-1
Author(s):  
Ivan Shatskyi ◽  
Vasyl Perepichka ◽  
Maksym Vaskovskyi
Keyword(s):  
Gordon Equation ◽  
Integral Transformation ◽  
Initial Boundary ◽  
Transformation Method ◽  
Elastic Rod ◽  
Longitudinal Waves ◽  
Initial Boundary Problem ◽  
Klein Gordon ◽  
Integral Transformation Method ◽  
Thin Elastic Layer

We study the problem of propagating longitudinal waves in an elastic rod connected to a locally damaged foundation through a thin elastic layer. The motion of the rigid foundation blocks is considered predetermined. We formulated the initial-boundary problem for the Klein-Gordon equation with a discontinuous right-hand side. The nonstationary fields of displacements, velocities, and deformations were investigated by the Laplace integral transformation method. Examples of sudden divergence of fragments of the foundation by a given value and their mutual separation at a constant speed are considered.

scholarly journals Trigonometric Cubic B-spline Collocation Method for Solitons of the Klein-Gordon Equation

2016 ◽  
Author(s):  
Alper Korkmaz ◽  
Ozlem Ersoy ◽  
Idiris Dag
Keyword(s):  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Coupled System ◽  
Initial Boundary ◽  
Spline Collocation ◽  
B Spline ◽  
Klein Gordon ◽  
Initial Boundary Value ◽  
Klein Gordon Equation

In the present study, we derive a new B-spline technique namely trigonometric B-spline collocation algorithm to solve some initial boundary value problems for the nonlinear Klein-Gordon equation. In order to carry out the time integration with Crank-Nicolson implicit method, the order of the equation is reduced to give a coupled system of nonlinear partial differential equations. The collocation approximation based on trigonometric cubic B-splines for spatial discretization is followed by the linearization of the nonlinear term. The efficiency and accuracy of the present method are validated by measuring the error between the numerical and analytical solutions when exist. The conservation laws representing momentum and energy are also computed for all problems.

scholarly journals The Solution of Mitchell's Problem for the Elastic Infinite Cone with a Spherical Crack

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
G. Ya. Popov ◽  
N. D. Vaysfel'd
Keyword(s):  
Integral Transformation ◽  
Tensile Force ◽  
Equation System ◽  
Transformation Method ◽  
Polynomial Method ◽  
Geometrical Parameters ◽  
Equilibrium Equations ◽  
Discontinuous Solutions ◽  
Elastic Cone ◽  
Integral Transformation Method

The new problem about the stress concentration around a spherical crack inside of an elastic cone is solved for the point tensile force enclosed to a cone's edge. The constructed discontinuous solutions of the equilibrium equations have allowed to express the displacements and stress in a cone through their jumps and the jumps of their normal derivatives across the crack's surface. The application of the integral transformation method under the generalized scheme has reduced the problem solving to the solving of the integrodifferential equation system with regard to the displacements' jumps. This system was solved approximately by the orthogonal polynomial method. The use of this method has allowed to take into consideration the order of the solution's singularities at the ends of an integral interval. The correlation between the crack's geometrical parameters, its distance from an edge, and the SIF values is established after the numerical analysis. The limit of the proposed method applicability is specified.

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