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Register a new userStart-up shear of spherocylinder packings: effect of friction
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Claus Heussinger
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We study the response to shear deformations of packings of long spherocylindrical particles that interact via frictional forces with friction coefficient $mu$. The packings are produced and deformed with the help of molecular dynamics simulations combined with minimization techniques performed on a GPU. We calculate the linear shear modulus $g_infty$, which is orders of magnitude larger than the modulus $g_0$ in the corresponding frictionless system. The motion of the particles responsible for these large frictional forces is governed by and increases with the length $ell$ of the spherocylinders. One consequence of this motion is that the shear modulus $g_infty$ approaches a finite value in the limit $elltoinfty$, even though the density of the packings vanishes, $rhoproptoell^{-2}$. By way of contrast, the frictionless modulus decreases to zero, $g_0simell^{-2}$, in accordance with the behavior of density. Increasing the strain beyond a value $gamma_csim mu$, the packing undergoes a shear-thinning transition from the large frictional to the smaller frictionless modulus when contacts saturate at the Coulomb inequality and start to slide. In this regime, sliding friction contributes a yield stress $sigma_y=g_inftygamma_c$ and the stress behaves as $sigma=sigma_y+g_0gamma$. The interplay between static and sliding friction gives rise to hysteresis in oscillatory shear simulations.
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Yield stress fluids (YSFs) display a dual nature highlighted by the existence of a yield stress such that YSFs are solid below the yield stress, whereas they flow like liquids above it. Under an applied shear rate $dotgamma$, the solid-to-liquid transition is associated with a complex spatiotemporal scenario. Still, the general phenomenology reported in the literature boils down to a simple sequence that can be divided into a short-time response characterized by the so-called stress overshoot, followed by stress relaxation towards a steady state. Such relaxation can be either long-lasting, which usually involves the growth of a shear band that can be only transient or that may persist at steady-state, or abrupt, in which case the solid-to-liquid transition resembles the failure of a brittle material, involving avalanches. Here we use a continuum model based on a spatially-resolved fluidity approach to rationalize the complete scenario associated with the shear-induced yielding of YSFs. Our model provides a scaling for the coordinates of the stress maximum as a function of $dotgamma$, which shows excellent agreement with experimental and numerical data extracted from the literature. Moreover, our approach shows that such a scaling is intimately linked to the growth dynamics of a fluidized boundary layer in the vicinity of the moving boundary. Yet, such scaling is independent of the fate of that layer, and of the long-term behavior of the YSF. Finally, when including the presence of long-range correlations, we show that our model displays a ductile to brittle transition, i.e., the stress overshoot reduces into a sharp stress drop associated with avalanches, which impacts the scaling of the stress maximum with $dotgamma$. Our work offers a unified picture of shear-induced yielding in YSFs, whose complex spatiotemporal dynamics are deeply connected to non-local effects.
The transient response of model hard sphere glasses is examined during the application of steady rate start-up shear using Brownian Dynamics (BD) simulations, experimental rheology and confocal microscopy. With increasing strain the glass initially exhibits an almost linear elastic stress increase, a stress peak at the yield point and then reaches a constant steady state. The stress overshoot has a non-monotonic dependence with Peclet number, Pe, and volume fraction, {phi}, determined by the available free volume and a competition between structural relaxation and shear advection. Examination of the structural properties under shear revealed an increasing anisotropic radial distribution function, g(r), mostly in the velocity - gradient (xy) plane, which decreases after the stress peak with considerable anisotropy remaining in the steady-state. Low rates minimally distort the structure, while high rates show distortion with signatures of transient elongation. As a mechanism of storing energy, particles are trapped within a cage distorted more than Brownian relaxation allows, while at larger strains, stresses are relaxed as particles are forced out of the cage due to advection. Even in the steady state, intermediate super diffusion is observed at high rates and is a signature of the continuous breaking and reformation of cages under shear.
We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain $gamma$ at fixed packing fraction $phi$. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the $phi$-$gamma$ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, $dphi/dgamma >0$ and $dphi/dgamma <0$. In contrast, MS packings from protocol 2 populate segments with $dphi/dgamma <0$ only. For both simple and pure shear, we derive a relationship between the stress anisotropy and dilatancy $dphi/dgamma$ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1. These results emphasize that for repulsive frictionless disks, different packing-generation protocols give rise to different MS packing probabilities, which lead to differences in macroscopic properties of MS packings.
We investigate the mechanical response of jammed packings of repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations of the soft-particle model, where the repulsive interactions scale as a power-law in the interparticle overlap with exponent $alpha$, have found that the ensemble-averaged shear modulus $langle G rangle$ increases with pressure $P$ as $sim P^{(alpha-3/2)/(alpha-1)}$ at large pressures. However, a deep theoretical understanding of this scaling behavior is lacking. We show that the shear modulus of jammed packings of frictionless, spherical particles has two key contributions: 1) continuous variations as a function of pressure along geometrical families, for which the interparticle contact network does not change, and 2) discontinuous jumps during compression that arise from changes in the contact network. We show that the shear modulus of the first geometrical family for jammed packings can be collapsed onto a master curve: $G^{(1)}/G_0 = (P/P_0)^{(alpha-2)/(alpha-1)} - P/P_0$, where $P_0 sim N^{-2(alpha-1)}$ is a characteristic pressure that separates the two power-law scaling regions and $G_0 sim N^{-2(alpha-3/2)}$. Deviations from this form can occur when there is significant non-affine particle motion near changes in the contact network. We further show that $langle G (P)rangle$ is not simply a sum of two power-laws, but $langle G rangle sim (P/P_c)^a$, where $a approx (alpha -2)/(alpha-1)$ in the $P rightarrow 0$ limit and $langle G rangle sim (P/P_c)^b$, where $b gtrsim (alpha -3/2)/(alpha-1)$ above a characteristic pressure $P_c$. In addition, the magnitudes of both contributions to $langle Grangle$ from geometrical families and changes in the contact network remain comparable in the large-system limit for $P >P_c$.
The mechanical response of packings of purely repulsive, spherical particles to athermal, quasistatic simple shear near jamming onset is highly nonlinear. Previous studies have shown that, at small pressure $p$, the ensemble-averaged static shear modulus $langle G-G_0 rangle$ scales with $p^alpha$, where $alpha approx 1$, but above a characteristic pressure $p^{**}$, $langle G-G_0 rangle sim p^beta$, where $beta approx 0.5$. However, we find that the shear modulus $G^i$ for an individual packing typically decreases linearly with $p$ along a geometrical family where the contact network does not change. We resolve this discrepancy by showing that, while the shear modulus does decrease linearly within geometrical families, $langle G rangle$ also depends on a contribution from discontinuous jumps in $langle G rangle$ that occur at the transitions between geometrical families. For $p > p^{**}$, geometrical-family and rearrangement contributions to $langle G rangle$ are of opposite signs and remain comparable for all system sizes. $langle G rangle$ can be described by a scaling function that smoothly transitions between the two power-law exponents $alpha$ and $beta$. We also demonstrate the phenomenon of {it compression unjamming}, where a jammed packing can unjam via isotropic compression.
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