Equilibrium Violation from the Complex Stiffness

Evidence from cyclic tests on metals, elastomers and sandy soils reveals that damping forces are nearly rate-independent and structural (hysteretic or rate-independent) damping was widely adopted since the 1940s. While there is no time-domain constitutive equation for a linear spring connected in parallel with a rate-independent dashpot, the dynamic stiffness (transfer function) of this mechanical network can be constructed in the frequency-domain; and it was known since the early 1960s that this mechanical network exhibits a non-causal response. In view of its simplicity in association with the wide practical need to model rate-independent dissipation, this mechanical network was also implemented in time-domain formulations with the label complex stiffness where the force output, P(t) is related in the time-domain to the displacement input, u(t), with P(t) = k(1 + i η)u(t). This paper shows that the complex stiffness, as expressed in the time-domain by various scholars, is a fundamentally flawed construct since in addition to causality it violates equilibrium.