A periodic problem in viscoelasticity with variable coefficient
Résumé
A numerical method is proposed for solving a periodic problem for a material subjected to dynamic and thermal loading. Because of the considerable variations involved, a law of viscoelasticity of the fluid type with variable coefficients is adopted: secular terms of displacement and strain are brought into evidence. A theoretical asymptotic stability result of the related Cauchy problem makes it possible to measure a priori the rate of convergence of this solution towards the solution the periodic problem. The behaviour predicted for the solutions is in fact observed in the examples presented. A Finite Element Method is used in combination with a Runge-Kutta method. In the case of the Maxwell law, a theoretical condition is explicitly provided for numerical stability that is independent of the space discretization, and error estimates are given for the approached solution. These theoretical results are tested numerically on the examples provided. Lastly, the method proposed is applied to an example where the loading comes within the scope of research on an aircraft turbine blade during flight or at overspeed. The results lead to a better understanding of stress development and distribution, and indicate areas of high stress concentration that correspond to the real cases of damage observed.