Axial wave propagation in viscoelastic bars using a new finite-element-based method

• Published in: Journal of engineering mathematics Vol. 77; no. 1; pp. 105 - 117
• Main Authors: Keramat, Alireza, Ahmadi, Ahmad
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2012
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Theoretical and Applied Mechanics; Viscoelasticity; Finite-element method; Correspondence principle; Convolution integral; Laplace transform; Longitudinal wave propagation
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-012-9556-y

A new solution for the finite-element implementation of the axial vibration of viscoelastic bars is proposed. The effect of viscoelasticity on the dynamic response is accounted for by a convolution integral, avoiding the difficulties associated with the high-order time derivatives used in conventional models for viscoelasticity. A convenient but excellent numerical approximation for the obtained convolution integral is proposed. This numerical approximation allows easy implementation of the finite-element procedure in the time domain as it only introduces additional terms to the mass matrix and the force vector. The additional terms are calculated from quantities obtained in the previous time step. To validate the proposed numerical procedure, a few simple examples are presented and solved by the existing direct methods as well as the new alternative method. The examples include the computation of the dynamic axial response of a viscoelastic bar fixed at one end and subject to a step or sinusoidally varying load at the other end. It is concluded that the new method is valid and works satisfactorily.