Dynamical problem of thermo-viscoelasticity for piecewise homogeneous half-bounded rods (Q2703271)

Using the method of integral Fourier transformation the authors find in the region NEWLINE\[NEWLINE\left\{t\in(0,\infty),\quad y\in \bigcup_{i=1}^{n+1} (l_{i-1},l_{i}); \quad l_{0}\geq 0, \quad l_{n+1}=\infty\right\}NEWLINE\]NEWLINE the bounded solution \(T(t,y)= \{T_{j}(t,y), j=1,\dots, n+1\}\) to the problem NEWLINE\[NEWLINE\begin{aligned}{1\over a_{j}^{2}}{\partial T_{j}(t,y)\over\partial t}+ \chi_{j}^{2}T_{j}(t,y)-{\partial^{2}T_{j}(t,y)\over\partial y^{2}} &=f_{j}(t,y),\\ T_{j}(t,y)|_{t=0}&=g_{j}(y),\end{aligned} \quad y\in (l_{j-1},l_{j}); \quad j=1,\dots, n+1;NEWLINE\]NEWLINE with boundary conditions \(\left.\left(\alpha_{11}^{0}{\partial\over\partial y}+\beta_{11}^{0}\right) T_{1}\right|_{y=l_{0}}=g_{0}(t)\); \(\left.{\partial\over\partial y}T_{n+1}\right|_{y=\infty}=0\); and conditions of imperfect thermal contacts NEWLINE\[NEWLINE\begin{aligned} \left.\left[\left(R_{k}{\partial\over\partial y}+1\right)T_{k}(t,y)- T_{k+1}(t,y)\right]\right|_{y=l_{k}}&=0, \\ \left.\left[{\partial\over\partial y}T_{k}(t,y)- \nu_{k}{\partial\over\partial y}T_{k+1}(t,y)\right]\right|_{y=l_{k}}&=0,\end{aligned} \qquad k=1,\dots,n.NEWLINE\]NEWLINE Thermal viscoelastic displacements and fields of stresses in a half-bounded piecewise homogeneous rod are also found.