In this very important and useful paper the small longitudinal deformation along the \(x\)-axis of a one-dimensional inhomogeneous thermoviscoelastic body fixed at \(x=0\) and unilaterally constrained at \(x=L\), is considered. Here the expansion and contraction are due to thermal effects and body forces. Mathematically, this problem is expressed by the hyperbolic-parabolic variational inequality (a constraint is imposed on the unknown function): \[ \begin{cases} u_{tt}-M \left( \int^L_0|u_x|^2 dx\right) u_{xx}+ \alpha\theta_x -\gamma u_{xxt}=0 \quad &\text{in }(0,L) \times(0,\infty)\\ \theta_t -\kappa\theta_{xx} +\alpha u_{xt}=0 \quad &\text{in }(0,L)\times (0, \infty)\\ u(0,t)=\theta (0,t)=\theta (L,t)=0\quad & t>0\\ u(x,0)=u_0(x),\quad u_t(x, 0) =u_1(x),\quad & \theta(x,0) =\theta_0(x) \end{cases} \] with the contact conditions \[ \begin{cases} u(L,t)\leq g,\;M\left(\int^L_0|u_x|^2 dx\right) u_x (L,t)+ \gamma u_{xt} (L,t)\leq 0\\ \left\{M\left( \int^L_0|u_x|^2 dx \right) u_x(L,t)+ \gamma u_{xt}(L,t) \right\} \bigl(u(L,t)-g\bigr)=0.\end{cases} \] All results are presented in systematic and self-contained manner, proofs being sufficiently detailed. The authors establish an existence result for the thermoviscoelastic degenerated contact problem. The nonlinear stress-strain relation has the form: \(\sigma=M (\int^L_0|u_x|^2 dx)u_x-\alpha\theta + \gamma u_{xt}\), where \(M\) is a function satisfying: \(M\in C^1((0, \infty))\cap C([0,\infty))\), \(M(s)\geq C|s|^p\). Main result: For the first-order energy associated to the equation, the authors show that there exists a positive constant \(C\) such that the following estimate \(E(t)\leq C(E(0)) (1+t)^{-(p+2)/p} \) holds. Here \(p\) is a positive number which depends on nonlinear terms of the system.