Asymptotic solution of a quasistatic thermoelaticity problem for a slender rod
The authors consider a boundary value problem of quasistatic linear thermoelasticity for a thin round rod whose radius is equal to \(\epsilon\) b and whose length is equal to a. Cylindrical coordinates (\(\rho\),\(\psi\),z) are used. Conditions on the lateral boundary of the rod are: \(v=\epsilon \psi (z,t,\epsilon)\), \(\partial W/\partial \rho =0\), \((\partial \theta /\partial \rho)+A\epsilon \theta =0\); and on the edges of the rod are: \(z=ia:\) \((\partial W/\partial \rho)+(\partial v/\partial z)=0\), \(W=\psi_ i(\rho,t)\), \(\theta =\mu_ i(\rho,t)\), \(i=0;1.\) The asymptotic form of the solution, as \(\epsilon\) \(\to 0\), is constructed using a traditional method. Boundary-layers solutions are used which are combined with the regular part of the solution. An asymptotic form of the solution, as time t tends to 0, is taken into account, too. The asymptotic is justified. Under some conditions the existence theorem for the problem is proven. By using the previous results a one-dimensional model of the thermoelastic rod is proposed.
scientific article; zbMATH DE number 4110222