Modeling of the dissipative properties of a material by means of a generalized fractional SLS approach

A physically sound material model must satisfy some strict conditions. It must be causal, i.e. strain vs stress frequency response function must be consistent, where is the complex conjugate operator. Moreover, the model should be consistent with experimental results obtained from any homogeneous structure made of the modeled material. Standard material tests range from quasi-static (stress and strain relaxation) to low to high frequency behavior (free and forced vibrations). The dissipative material properties strongly influence the value of the stress and strain relaxation time and of the damping ratio associated to any natural vibration mode of a tested specimen. The simplest standard material models known from literature are the Hook, Newton, Maxwell and Kelvin material models, and it will be shown that they are not consistent with the previously outlined conditions. Nevertheless, such simple models can be combined to obtain more effective models [1]. The classic standard linear solid (SLS) model is made up of a series arrangement of Kelvin model blocks ( being the model order) where can be high if many conditions deriving from test measurements must be satisfied. The dissipative behavior of such models depends on the viscosity parameter associated to the viscous elements of each Kelvin block constituting the SLS model, but it will be shown that high viscosity values typically lend to both high relaxation times and overdamped free response too. While the former result is generally consistent with experimental results, the latter result is typically not consistent with tests done with specimens in medium to high frequency vibrational conditions. Newtonian material models with fractional derivative operators were proposed in the past [2]. The dissipative behavior of such models depends on both the viscosity parameter and the associated fractional derivative order, typically allowing to model quasi static and free vibrational behavior of material specimens with a low order generalized fractional SLS model.