Extracting Dynamic Shear Moduli Functions from the Upper-Convected Maxwell Model Using the Semi-Inverse Approach

This study presents a workflow to extract the frequency (ω) dependent storage modulus G′(ω) and loss modulus G″(ω) – which are commonly used viscoelastic functions to describe small amplitude oscillatory shear (SAOS) flow of complex fluids – from the frame-invariant upper-convected Maxwell (UCM) model using the semi-inverse approach. The SAOS flow is set up between two parallel circular plates, and this resembles a conventional rheometric apparatus normally utilized to experimentally obtain the dynamic moduli. The differential equations governing the shear stress and normal stress as functions of shear rate are derived using the semi-inverse approach, and then solved analytically. Explicit expressions for G′(ω) and G″(ω) are derived from the analytical solution: these exactly match those derived using the phenomenological spring-dashpot Maxwell element. The equations are also solved numerically; here G′(ω) and G″(ω) are extracted from the spectral phase lag between shear stress and shear rate which is obtained using a Fast Fourier Transform. The numerically derived values for G′,G″ show excellent consistency with the UCM derived analytical solutions, thus validating the numerical procedure. These results provide a continuum mechanics derivation for the spring-dashpot framework: a result that keeps with intuitive expectation. A framework to extract dynamic moduli for a generalized multi-mode UCM is also provided. The utility of the workflow is demonstrated by successfully describing in-house experimental SAOS data for a variety of viscoelastic materials ranging from polymer solutions to composite hydrogels.