A note on the wave equation for a class of constitutive relations for nonlinear elastic bodies that are not Green elastic

2016 ◽  
Vol 23 (2) ◽  
pp. 148-158 ◽  
Author(s):  
R Meneses ◽  
O Orellana ◽  
R Bustamante
Keyword(s):  
Wave Equation ◽  
Stress Tensor ◽  
Constitutive Relations ◽  
Strain Tensor ◽  
Nonlinear Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Elastic Bodies ◽  
New Type

A class of constitutive relations for elastic bodies has been proposed recently, where the linearized strain tensor is expressed as a nonlinear function of the stress tensor. Considering this new type of constitutive equation, the initial boundary value problem for such elastic bodies has been expressed only in terms of the stress tensor. In this communication, this new type of nonlinear wave equation is studied for the case of a one-dimensional straight bar. Conditions for the existence of the travelling wave solutions are given and some self-similar solutions are obtained.

scholarly journals Application of Initial Boundary Value Problem on Logarithmic Wave Equation in Dynamics

2018 ◽  
Author(s):  
Shkelqim Hajrulla ◽  
Leonard Bezati ◽  
Fatmir Hoxha
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Sobolev Inequality ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Initial Boundary Value ◽  
The Galerkin Method

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

2015 ◽  
Vol 725-726 ◽  
pp. 863-868
Author(s):  
Vladimir Lalin ◽  
Elizaveta Zdanchuk
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Stress Tensor ◽  
Granular Medium ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cosserat Continuum ◽  
Suitable Model ◽  
Initial Boundary Value

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

scholarly journals Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1283
Author(s):  
Karel Van Bockstal
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Differential Operator ◽  
Unique Weak Solution ◽  
Fractional Wave Equation ◽  
Equation Of Time ◽  
Initial Boundary Value ◽  
Distributed Order

We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative.

scholarly journals Blow-Up and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Fosheng Wang ◽  
Chengqiang Wang
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Viscoelastic Wave Equation ◽  
Frictional Damping ◽  
Viscoelastic Wave ◽  
Initial Boundary Value

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.

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