The frequency dependence of third-harmonic medium amplitude oscillatory shear (MAOS) modulus [Formula: see text] provides insight into material behavior and microstructure in the asymptotically nonlinear regime. Motivated by the difficulty in the measurement of MAOS moduli, we propose a test for data validation based on nonlinear Kramers–Kronig relations. We extend the approach used to assess the consistency of linear viscoelastic data by expressing the real and imaginary parts of [Formula: see text] as a linear combination of Maxwell elements: the functional form for the MAOS kernels is inspired by time-strain separability (TSS). We propose a statistical test based on fitting a sum of Maxwell elements using LASSO (least absolute shrinkage and selection operator) regression, and call it the SMEL test. It works well on a broad range of materials and models including those that do not obey TSS. It successfully copes with experimental data that are noisy or confined to a limited frequency range. When Maxwell modes obtained from the SMEL test are used to predict the first-harmonic MAOS modulus [Formula: see text], it is possible to identify the range of time scales over which a material exhibits TSS.

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Kramers–Kronig relations for nonlinear rheology. Part II: Validation of medium amplitude oscillatory shear (MAOS) measurements

A system undergoing sol-gel transition passes through a unique point, known as the critical gel state, where it forms the weakest space spanning percolated network. We investigate the nonlinear viscoelastic behavior of a colloidal dispersion at the critical gel state using large amplitude oscillatory shear rheology. The colloidal gel at the critical point is subjected to oscillatory shear flow with increasing strain amplitude at different frequencies. We observe that the first harmonic of the elastic and viscous moduli exhibits a monotonic decrease as the material undergoes a linear to nonlinear transition. We analyze the stress waveform across this transition and obtain the nonlinear moduli and viscosity as a function of frequency and strain amplitude. The analysis of the nonlinear moduli and viscosities suggests intracycle strain stiffening and intracycle shear thinning in the colloidal dispersion. Based on the insights obtained from the nonlinear analysis, we propose a potential scenario of the microstructural changes occurring in the nonlinear region. We also develop an integral model using the time-strain separable Kaye-Bernstein-Kearsley-Zapas constitutive equation with a power-law relaxation modulus and damping function obtained from experiments. The proposed model with a slight adjustment of the damping function inferred using a spectral method, compares well with experimental data at all frequencies.

Large amplitude oscillatory shear study of a colloidal gel near the critical state.

With the increasing demand for underwater detection, interest in the acoustic field of range-dependent ocean waveguides is also growing. For weakly range-dependent ocean waveguides, adiabatic modes represent a compromise between accuracy and computational cost and occupy an important place in the simulation of numerical sound fields. However, either existing adiabatic-mode programs consider too few layers of media or the root-finder tends to miss roots. In addition, none of the programs can solve the acoustic field excited by a line sound source located anywhere in the plane. In this paper, we first derive an expression for the acoustic field excited by a line source by adiabatic modes and then introduce a high-precision spectral method to solve the local eigenmodes. For the lower boundary condition of the acoustic half-space, we use the eigenvalue transformation technique to transform the transcendental algebra system formed by spectral discretization into a generalized eigenvalue problem. Several representative numerical experiments are designed to verify the accuracy of the algorithm. After analyzing the parallelism, the multiprocess and multithread hybrid strategy is adopted to further accelerate the algorithm in parallel, and parallel numerical simulation is carried out on the Tianhe–2 multicore supercomputer; favorable acceleration is achieved.

Parallel numerical simulation of weakly range-dependent ocean acoustic waveguides by adiabatic modes based on a spectral method

The wavenumber integration model is the most precise approach for assessing arbitrary horizontally stratified media within the sphere of computational ocean acoustics. Unlike the normal-mode approach, it considers not only discrete spectra but also continuous spectral components, resulting in fewer model approximation errors for horizontally stratified media. Traditionally, the depth-separated wave equation in the wavenumber integration model has been solved using analytical and semianalytical methods, and numerical solutions have been primarily based on the finite difference and finite element methods. This paper proposes an algorithm for solving the depth equation via the Chebyshev–Tau spectral method, combined with a domain decomposition strategy, resulting in the development of a numerical program named WISpec. The algorithm can simulate the sound field excitation not only from a point source but also from an infinite line source. To that end, the depth equations for each layer are first discretized through the Chebyshev–Tau spectral method and subsequently solved simultaneously by incorporating boundary and interface conditions. Representative numerical experiments are presented to validate the accuracy and speed of WISpec. The high degree of consistency of results obtained from different software tools running the same configuration provides ample evidence that the numerical algorithm described in this paper is accurate, reliable, and numerically stable.

https://pubs.aip.org/aip/pof/article-pdf/doi/10.1063/5.0150221/17795532/057127_1_5.0150221.pdf

A spectral method for the depth-separated solution of a wavenumber integration model for horizontally stratified fluid acoustic waveguides

The method of harmonic balance (HB) is a spectrally accurate method used to obtain periodic steady state solutions to dynamical systems subjected to periodic perturbations. We adapt HB to solve for the stress response of the Giesekus model under large amplitude oscillatory shear (LAOS) deformation. HB transforms the system of differential equations to a set of nonlinear algebraic equations in the Fourier coefficients. Convergence studies find that the difference between the HB and true solutions decays exponentially with the number of harmonics ($H$) included in the ansatz as $e^{-m H}$. The decay coefficient $m$ decreases with increasing strain amplitude, and exhibits a"U"shaped dependence on applied frequency. The computational cost of HB increases slightly faster than linearly with $H$. The net result of rapid convergence and modest increase in computational cost with increasing $H$ implies that HB outperforms the conventional method of using numerical integration to solve differential constitutive equations under oscillatory shear. Numerical experiments find that HB is simultaneously about three orders of magnitude cheaper, and several orders of magnitude more accurate than numerical integration. Thus, it offers a compelling value proposition for parameter estimation or model selection.

The Method of Harmonic Balance for the Giesekus Model under Oscillatory Shear

The time-strain separable Kaye–Bernstein–Kearsley–Zappas model (tssKBKZM) is a popular integral constitutive equation that is used to model the nonlinear response of time-strain separable materials using only their linear viscoelastic properties and damping function. In oscillatory shear, numerical evaluation of tssKBKZM is complicated by the infinite domain of integration, and the oscillatory nature of the integrand. To avoid these problems, a spectrally accurate method is proposed. It approximates the oscillatory portion of the integrand using a discrete Fourier series, which enables analytical evaluation of the resulting integrals for the Maxwell model. The spectral method is generalized for arbitrary discrete and continuous relaxation spectra. Upper bounds for quadrature error, which can often be driven to machine precision, are presented. The Doi–Edwards model with independent-alignment approximation (DE-IA) is a special case of tssKBKZM; for DE-IA, the spectral method is compared with trapezoidal rule to highlight its accuracy and efficiency. The superiority of the proposed method is particularly evident at large strain amplitude and frequency. For continuous relaxation spectra, the spectral method transforms the double integral corresponding to the tssKBKZM to a single integral. Solutions computed to a specified level of accuracy using standard numerical libraries show that the spectral method is typically two to three orders of magnitude faster. Extensions to fractional rheological models, materials with nonzero equilibrium modulus, stretched exponential models, etc., are also discussed.

Shivangi Mittal

Papers

Spectral method for time-strain separable integral constitutive models in oscillatory shear