Characterization of three-dimensional fractional viscoelastic models through complex modulus analysis and polar decomposition

Abstract

Soft materials such as gels, elastomers, and biological tissues have diverse applications in nature and technology due to their viscoelastic nature. These soft materials often exhibit complex rheology and display elastic and viscous characteristics when undergoing deformation. In recent years, fractional calculus has emerged as a promising tool to explain the viscoelastic behavior of soft materials. Scalar constants are primarily used to quantify viscoelastic elements such as springs and dashpots. However, in three-dimensional (3D) space, not all materials show the same elastic or viscoelastic properties in all directions, especially under elastic/viscoelastic wave propagation (or anisotropy). Though previously reported studies on viscoelastic models have explained a power-law decay of the memory functions, none of them explicitly explained the 3D complex modulus through a matrix notation. In this paper, we present a mathematical formulation that employs tensor algebra and fractional calculus to derive the 3D complex modulus of Kelvin–Voigt, Maxwell, and other arrangements of viscoelastic models. The 3D complex modulus provides information about the elastic wave propagation in a media and can be used to explain anisotropy in different viscoelastic materials. Additionally, an advanced formulation of the moduli can improve the modeling in finite element analysis of 3D viscoelastic materials where discretization is vital for studying media of asymmetric shapes. Finally, we demonstrated a polar decomposition method to visualize viscoelastic tensors using the Green–Christoffel tensor and surface plots to represent the degrees of anisotropy and viscoelasticity in the Fourier domain when the medium is probed by a time-harmonic homogeneous plane wave.