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

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