Stability Of Elastic Systems With Local Viscoelasticity

The main work done in this dissertation is organized as follows:In Chapter 1, we consider a vibrating non-homogeneous beam with one segment made of viscoelastic material of Kelvin-Voigt type, and other parts made of elastic material by means of Timoshenko model. (It is well known that if the cross-section dimensions of the beam is not negligible compared with its length, it is necessary to consider the effect of the rotatory inertia. Further more, if the deflection due to shear is not negligible also, Timoshenko beam model should be introduced.) We have deduced mathematical equations modeling its vibration and studied the stability of the semigroup associated with the equation system describing the transverse and shearing vibration of the beam. We obtain the exponential stability under certain hypotheses of smoothness and structural condition of the viscelasticity coefficients of the system and obtain the strong asymptotic stability under weaker hypotheses of the coefficients.In Chapter 2, we consider a vibrating Timoshenko beam with local viscoelasticity of Boltzmann type. We have deduced mathematical equations modeling its vibration and studied the stability of the semigroup associated with the equation system. We obtain the exponential stability under certain hypotheses of smoothness and structural condition of the coefficients of the system, applying the relaxation function decays exponentially. This result does not need the continuity of the damping coefficient at the interface.In Chapter 3, we consider a vibrating Euler-Bonoulli beam with local viscoelasticity of Boltzmann type. We obtain the exponential stability under certain hypotheses of smoothness of the coefficients of the system by. and also obtain the strong asymptotic stability under weaker hypotheses of the coefficients by means of the result in [30].In Chapter 4, we consider a high dimensional non-homogeneous wave equation with localviscoelasticity of Boltzmann type, distributed near the boundary of a bounded simple-connected domain We have studied the stability of the semigroup associated with the equation system. We obtain the exponential stability under certain hypotheses of smoothness and structural condition of the coefficients of the system and obtain the strong asymptotic stability under weaker hypotheses of the coefficients. The structural condition obtained here is similar to that given in [52]. We do not assume zero datum in history which used in [63].