Emergence of Cooperativity in Plasticity of Soft Glassy Materials

Antoine Le Bouil, Axelle Amon, Sean McNamara, and Jérôme Crassous

Phys. Rev. Lett. 112, 246001 – Published 18 June 2014

Abstract

The elastic coupling between plastic events is generally invoked to interpret plastic properties and the failure of amorphous soft glassy materials. We report an experiment where the emergence of a self-organized plastic flow is observed well before the failure. For this we impose an homogeneous stress on a granular material, and measure local deformations for very small strain increments using a light scattering setup. We observe a nonhomogeneous strain that appears as transient bands of mesoscopic size and a well-defined orientation, which is different from the angle of the macroscopic frictional shear band that appears at the failure. The presence and the orientation of those microbands may be understood by considering how localized plastic reorganizations redistribute stresses in a surrounding continuous elastic medium. We characterize the length scale and persistence of the structure. The presence of plastic events and the mesostructure of the plastic flow are compared to numerical simulations.

Received 23 July 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.246001

Authors & Affiliations

Antoine Le Bouil, Axelle Amon, Sean McNamara, and Jérôme Crassous

Université de Rennes 1, Institut de Physique de Rennes (UMR UR1-CNRS 6251), Bât. 11A, Campus de Beaulieu, F-35042 Rennes, France

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Vol. 112, Iss. 24 — 20 June 2014

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Images

Figure 1
(a) Schematic representation of the biaxial setup. The granular material is enclosed between a latex membrane and a glass plate (not represented here). A partial vacuum inside the membrane creates a confining stress $- σ_{x x}$ . The sample is compressed at a fixed velocity along the $y$ axis through a moving plate (upper plate, dark gray). The light gray back plate as well as the glass plate at the front forbid displacements along the $z$ direction ensuring plane-strain conditions. For compression, $- σ_{x x}, - σ_{y y} > 0$ . (b) A map of correlation $g_{I} (ε, r)$ with a color scale. The dashed area of side $l ≃ 270 d$ is the region of interest for the spatial correlation calculation.
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Figure 2
(a) Applied stresses difference versus deformation ( $- σ_{x x} = 30 kPa$ ). Insets: Left, notations; right: maps of $g_{I} (ε, r)$ before failure ( $ε = - ε_{y y} = 0.91 %$ ) and after failure ( $ε = 5.82 %$ ). (b) Enlargement of the region of interest of the deformation map before failure ( $ε = 3.30 %$ ) showing the mesoscale strain heterogeneities. (c) Correlation function $Ψ^{(0)} (ε, r)$ of $g_{I}$ at $ε = 3.30 %$ showing the plastic flow structure in a square of size $l ≃ 270 d$ in the $r$ plane.
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Figure 3
(a) Schematic representation of a local plastic event specifying the tensors $e^{*}$ (linked to the deformation of the inclusion) and $\tilde{σ}$ (stress redistribution in the surrounding medium due to the plastic event). (b) Angular distribution of ${\tilde{σ}}_{x x} - {\tilde{σ}}_{y y} \propto f (θ)$ in the case of an isovolumic transformation of the inclusion ( $ν = 0.33$ ). (c) Boundary conditions of the numerical simulations. (d) Example of a deformation map from numerical simulation displaying a local event and microbands. (e) Synthetic local reorganization obtained numerically by a modification of the elastic constants of few grains.
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Figure 4
(a) $Ψ^{(0)}$ as function of $r$ for $ε = 2.7 %$ , showing a fast decay at a short distance ( $r / d ≲ 10$ ) followed by a slow decay. (b) $χ^{(0)} (ε, r)$ versus $r / d$ for increasing values of deformations $ε = 1.6 %$ (open triangle), $ε = 2.3 %$ (filled square), $ε = 3.7 %$ (open square), $ε = 4.0 %$ (filled circle), $ε = 4.4 %$ (open circle). Lines are quadratic fits around maximum. (c) Length $ξ / d$ (filled square) and mean amplitude $A$ (open circle) (see text) as functions of the deformation $ε$ . Error bars are given by the uncertainty of the quadratic fit of $χ^{(0)}$ around maximum. The black dotted line indicates the deformation at rupture $ε_{c} \approx 4.66 %$ . The plain line is the cooperative length [28] expected from the nonlocal flow rule of granular material [29]. (d) Relaxation of $(χ^{(Δ ε)} / χ^{(0)}) (ε, ξ (ε))$ as a function of the deformation increment $Δ ε$ for $ξ (ε = 3.3 %) = 33 d$ (filled circle), $ξ (ε = 4.0 %) = 70 d$ (open square), and $ξ (ε = 4.4 %) = 85 d$ (filled square).
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