D’Alembert Method in Dynamic Problems of Thermoelasticity

Synonyms

Transient dynamic response of a thermoelastic rod, the features of which are described by the Green-Naghdi theory of thermoelasticity without energy dissipation, under shock interaction

Overview

The d’Alembert solution for the problem of the longitudinal collinear collision of two finite elastic plane-ended bars of the same cross section and material was solved for the first time in [1] by means of the one-dimensional wave equation, which postulates a uniform stress distribution across any section of the rods. The impact is accompanied by the generation of two transient longitudinal waves of strain which propagate in the positive and negative x-directions relative to the moving bars with the velocity $c = \sqrt {{E/\rho }}$, where E and ρ are the elastic modulus and the density, respectively. The displacement function u was selected as the main function, which is assigned in the form

$$ u = f(\xi ) + g(\eta ) $$

where $\xi = x - ct$, $\eta = x + ct$, $f(\xi )$,...