Stress overshoot, hysteresis, and the Bauschinger effect in sheared dense colloidal suspensions

Dmytro Kushnir, Céline Ruscher, Eckhard Bartsch, Fabrice Thalmann, and Pascal Hébraud

Phys. Rev. E 106, 034611 – Published 15 September 2022

Abstract

The mechanical nonlinear response of dense Brownian suspensions of polymer gel particles is studied experimentally and by means of numerical simulations. It is shown that the response to the application of a constant shear rate depends on the previous history of the suspension. When the flow starts from a suspension at rest, it exhibits an elastic response followed by a stress overshoot and then a plastic flow regime. Conversely, after flow reversal, the stress overshoot does not occur, and the apparent elastic modulus is reduced while numerical simulations reveal that the anisotropy of the local microstructure is delayed relative to the macroscopic stress.

Received 18 February 2022
Accepted 11 July 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034611

Physics Subject Headings (PhySH)

Creep Plastic deformation Rheological properties

Colloidal glass Hard sphere colloids

Molecular dynamics Rheology techniques

Polymers & Soft Matter

Authors & Affiliations

Dmytro Kushnir

IPCMS/CNRS, 23 rue du Loess 67034 Strasbourg, France

Céline Ruscher

ICS/CNRS 23 rue du Loess 67034 Strasbourg, France

Eckhard Bartsch

Institut für Physikalische Chemie and Institut für Makromolekulare Chemie, Albert-Ludwigs-Universität, D-79104 Freiburg, Germany

Fabrice Thalmann

ICS/CNRS 23 rue du Loess 67034 Strasbourg, France

Pascal Hébraud^*

IPCMS/CNRS, 23 rue du Loess 67034 Strasbourg, France

^*Corresponding author: pascal.hebraud@ipcms.unistra.fr

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Vol. 106, Iss. 3 — September 2022

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Figure 1
(a) Schematic illustration of the shear deformation protocol. A series of strain steps is applied to the sample, of constant amplitude $δ γ = 5 \times 10^{- 3}$ . The step duration $δ t$ is varied between $5 \times 10^{- 1}$ and 20 s in order to control the effective shear rate between $10^{- 2}$ and $2.5 \times 10^{- 4} s^{- 1}$ . (b) Stress evolution between $10^{- 1}$ and 10 s after the application of a step strain of amplitude $5 \times 10^{- 3}$ . The horizontal continuous line indicates the stress average over the time interval $[5 \times 10^{- 1}, 10]$ s. The dashed lines indicate the standard deviation of stress over this time interval. (c, d) Evolution of the stress as a function of the deformation, during the startup flow. (c) Experimental data for $ϕ = 0.62$ and for different shear rate values, increasing from bottom to top: $\dot{γ} = 2.5 \times 10^{- 4}, 5 \times 10^{- 4}, 10^{- 3}, 2.5 \times 10^{- 3}, 5 \times 10^{- 3}, 10^{- 2} s^{- 1}$ . Continuous curves are adjustment with the Generalized Maxwell Model. (d) Numerical data for $ϕ = 0.627$ and $\dot{γ} = 3 \times 10^{- 4}, 6 \times 10^{- 4}, 1.5 \times 10^{- 3}, 3 \times 10^{- 3}, 6 \times 10^{- 3}, 1.5 \times 10^{- 2}, 3 \times 10^{- 2}, 6 \times 10^{- 2}, 1.5 \times 10^{- 1}$ , and $3 \times 10^{- 1} s^{- 1}$ . Inset of (c) and (d): Evolution of the elastic modulus as a function of the volume fraction. The dashed lines are power-law adjustments, leading to $G_{0} \propto ϕ^{m}$ , with $m = 14$ for the experimental data and $m = 16$ for the numerical data.
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Figure 2
Evolution of the stress maximum at the overshoot (a) and of the overshoot stress relative to the stress at high shear rate (b) as function of the applied shear rate for $ϕ = 0.62$ (experimental data, $•$ ) and for $ϕ = 0.627$ (numerical data, $▪$ ). (a) The lines are power-law adjustments of the data, leading to $σ_{over} \propto {\dot{γ}}^{α}$ with $α = 0.14$ (continuous line, experimental data) and 0.16 (dashed line, numerical data).
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Figure 3
Stress as a function of the deformation during the first cycle of deformation: the deformation is increased from 0 to $γ_{\max}$ , then decreased from $γ_{\max}$ to $- γ_{\max}$ , and increased back from $- γ_{\max}$ to 0 at a constant absolute value of the shear rate. (a) Experimental data for $ϕ = 0.62$ and $γ_{\max} = 0.03$ , 0.05, 0.07, 0.09, 0.11, 0.13, 0.15, 0.17, and 0. 20. The continuous black lines are adjustment of the data with the Generalized Maxwell Model using a single set of parameter values for all cycles. The thick segments are linear adjustment of the $σ (γ)$ curves around zero stress, after flow reversal, from which the effective modulus, $G_{back}$ is computed. (b) Numerical data for $ϕ = 0.627$ and for the corresponding set of $γ_{\max}$ values as experimental data.
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Figure 4
(a, b) Evolution of the effective modulus, measured as the slope of the $σ$ vs $γ$ curve when the stress is null, after flow reversal [see Fig. 3 for experimental systems (a) and numerical systems (b)] as a function of the maximum deformation applied, $γ_{\max}$ . In (a) dashed lines are the evolutions of $G_{back}$ obtained from the adjustment of the experimental data with the Generalized Maxwell Model (see also Fig. 3). The horizontal thick lines correspond to the elastic modulus at rest, defined as the slope of the startup flow curves when $γ ⟶ 0$ [in (a), continuous lines: experimental data, dotted lines: GMM data). (c, d) Evolution of the area of the hysteresis cycles as a function of their maximum amplitude $γ_{\max}$ . Vertical dashed line indicates the $γ$ value at the stress overshoot, $γ_{over}$ . Continuous lines are linear adjustments of the data for $γ$ values larger and smaller than the deformation at stress overshoot. Three volume fractions are studied, increasing from black to gray: $ϕ = 0.60$ , 0.61, and 0.62 for the experimental system and $ϕ = 0.607$ , 0.617, and 0.627 for the numerical system.
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Figure 5
Results obtained from the numerical simulations. (a) Evolution of the stress (left axis, continuous lines) and of the $x y$ component of the fabric tensor (right axis, dashed line) for a suspensions of volume fraction 0.627 submitted to cycles of maximum amplitudes $γ_{\max} = 0.03$ , 0.05, 0.07, 0.09, 0.11, 0.13, 0.15, 0.17, and 0.20. Both quantities, stress and fabric tensor, are normalized to their maximal values.(b) Stress as a function of the $x y$ component of the fabric tensor, $F_{x y}$ for $γ_{\max} = 0.2$ . The continuous line is an elliptic adjustment of the data. Inset: Stress as a function of $F_{x y}$ during the initial stress increase. The continuous line is a linear adjustment of the curve for small stress values, up to $σ = 3$ Pa. (c) Evolution of the phase $φ$ (in degrees) between the stress and $F_{x y}$ , deduced from the elliptical adjustment of the $σ (F_{x y})$ curve.
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