Fast and Accurate Numerical Methods for Evaluating Constitutive Models under Large Amplitude Oscillatory Shear

Large amplitude oscillatory shear (LAOS) flow is a promising rheological protocol for understanding the nonlinear viscoelastic properties of complex fluids. Experimental LAOS data is especially useful when it can be associated with constitutive theories that describe the underlying physics. Constitutive models (CMs) are nonlinear systems of integral or differential equations in stresses and strain, and cannot be solved analytically to obtain closed-form solutions. We tackle this problem by presenting two new algorithms rooted in Fourier transform rheology for numerically determining the LAOS response of CMs, viz. (i) the spectral method for integral CMs, and (ii) the harmonic balance method for differential CMs. These methods are compared with the conventional quadrature and time-stepping algorithms. They are observed to outperform conventional methods both in terms of accuracy and speed, often by several orders of magnitude. These methods open up new possibilities for rapid model calibration and model selection, when provided with large datasets of LAOS measurements.