The loss tangent of visco-elastic models

Abstract

When modelling visco-elasticity, be it linearly (standard linear solid with two springs and one damper) or non-linearly (power and log models), it is important to know on which model parameters the viscous energy loss depends on. In this paper, the dependency of the loss tangent (tan δ, ratio of loss modulus to storage modulus) and the phase angle δ on elasticity E and viscosity Ƞ parameters and on the excitation frequency f is derived and evaluated in three visco-elastic models. In the Zener model (standard linear solid of Voight form), tan δ and δ depend on E, Ƞ, and f. f and Ƞ are linked together and always occur as the product fȠ. Tan δ is smaller than π/2. The transient part of the stress function is an exponential function; the steady state part comprises of sine and cosine functions. In the power model, tan δ and δ depend on Ƞ only; (0≤Ƞ<1). Ƞ has no relationship with f in tan δ. Tan δ is smaller than π/2. The transient part of the stress function is a Maclaurin series; the steady state part is a sine function with Ƞπ/2 phase shift. In the log model, tan δ and δ depend on E, Ƞ, and f; but at the same f, larger E/Ƞ have larger tan δ and δ. The viscosity constant appears as a stand alone Ƞ, and as the product of Ƞ and log 2πf. Tan δ can be larger than π/2 at small E/Ƞ (high viscosity) and small frequencies (large cycle periods with small strain rates). The transient part of the stress function comprises of cosine and sine integrals; steady state part consists of sine and cosine functions.