Kelvin-Voigt model 2026-10-14
The Kelvin-Voigt model puts a spring and dashpot in parallel. Their common displacement makes their forces add. With the convention , the complex stiffness is , giving storage stiffness and loss stiffness . Under a force step, with . The series spring–dashpot is a Maxwell fluid analogue and has different relaxation and creep behavior.
Past exam of the mathematics course of the University of Cambridge 2003 iii Paper 71 6 a Solution Created 2026-10-03 Updated 2026-10-14
The printed diagram and the word “parallel” describe a Kelvin-Voigt model, not a series Maxwell fluid. In the depicted model both elements have displacement and their forces add:For and the convention ,The storage modulus and loss modulus here are element stiffnesses with units ; conversion to stress/strain moduli requires a geometric factor. If “response” instead denotes displacement per force, it is the compliance . Specifying which ratio is used removes that convention ambiguity. The mean dissipated power is , confirming the positive loss sign for .
For completeness, the actual series Maxwell fluid analogue has a common force and additive spring/dashpot extensions. Thus . With ,These different functions cannot both describe the printed parallel arrangement. The distinction also changes the elastic beam's response to a constant load.