Quasi-static And Dynamical Analysis For Viscoelastic Structures With Fractional Derivative Constitutive Relation

In this dissertation, the fractional derivative constitutive relations are applied to study the static and dynamical problems of viscoelastic structures in theoretical analysis and numerical simulation. The main results contain as follows;(1) A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro-differential equation included fractional integral or fractional derivative in a long history and the difficulty of storing all history data is overcome and the error can be controlled. Two numerical examples are presented. The numerical results are compared with the exact solutions, it is shown that the new numerical method has high accuracy.(2) The dynamic stability of simple supported viscoelastic column subjected to a periodic axial force is investigated. The governing equation of motion is derived as a weakly singular Volterra integro-partial-differential equation, and it is simplified into a weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. By using the averaging method, the dynamical stability is analyzed and the stability criteria of viscoelastic column are achieved. The numerical method for fractional integral presented in this dissertation is applied to solve the reduced equations. Thelong-time dynamical responses of the system with different parameters are simulated and the numerical results agree with the analytical ones. It is also shown the numerical method is reliable.(3) The equations governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitutive relation. The quasi-static behavior of the viscoelastic Timoshenko beam under step loading is analyzed and the analytical solution is obtained. The influence of material parameters on the deflection is investigated. It is indicated that the effect of fractional derivative parameter on the behavior of the beam is distinct. The dynamical response of the viscoelastic Timoshenko beam subjected to a periodic excitation is studied by means of mode shape functions. And the effect of both transverse shear and rotational inertia on the vibration of the beam is discussed. The effect relates to the slenderness ratio of the beam. For large values of the ratio, the effect is significant in both the quasi-static case and dynamical cases. The dynamical responses of the beam are simulated by using the numerical method presented in this dissertation. The numerical results agree with the analytical ones.(4) The motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with finite deformation are derived and simplified by 1-order and 2-order Galerkin method. The viscoelastic material is assumed to obey the three-dimensional fractional derivative constitutive relation. The equations are nonlinear integro-differential ones with weakly singularity about thedeflection and rotational angle. The numerical method for fractional integral presented in this dissertation is applied to solve the equations. The influences of the load parameter and the material parameter on the deflection of the beam are considered respectively. With the increasing of the load parameter, the motion states of the 1-order and 2-order Galerkin truncation systems are both changed from periodic motion with period 1 to complex motions, such as mult-periodicity, quasi-periodicity or chaos. With the increasing of the material parameter, namely the damping of the viscoelastic material is adding, the motions of the systems are changed from muti-periodicity, quasi-periodicity or chaos to one-periodicity. So the increasing of the material parameter benefits the stability of the structures. The numerical methods in nonlinear dynamics are synthetically applied to reveal plenty dynamical behaviors of the beam. The dynamica...