Connecting shear localization with the long-range correlated polarized stress fields in granular materials

Abstract

One long-lasting puzzle in amorphous solids is shear localization, where local plastic deformation involves cooperative particle rearrangements in small regions of a few inter-particle distances, self-organizing into shear bands and eventually leading to the material failure. Understanding the connection between the structure and dynamics of amorphous solids is essential in physics, material sciences, geotechnical and civil engineering, and geophysics. Here we show a deep connection between shear localization and the intrinsic structures of internal stresses in an isotropically jammed granular material subject to shear. Specifically, we find strong (anti)correlations between the micro shear bands and two polarized stress fields along two directions of maximal shear. By exploring the tensorial characteristics and the rotational symmetry of force network, we reveal that such profound connection is a result of symmetry breaking by shear. Finally, we provide the solid experimental evidence of long-range correlated inherent shear stress in an isotropically jammed granular system.

Understanding the behavior of jammed granular matter is important for a range of phenomena, from materials science to geology. Wang et al. uncover relations between stress correlations and emergence of localized shear bands due to external shear stress, which breaks the rotational symmetry.

Introduction

Shear localization, in which local particle rearrangements appear in narrow regions of a few inter-particle distances, is a fascinating feature that not only appears in granular materials^1–4 but also is shared by many other amorphous solids⁵, such as molecular glass⁶, metallic glass⁷, colloids^8–10, emulsions¹¹, and foams¹². It is not only crucial to material research but also vital to the catastrophic failure of soils in geotechnical and civil engineering¹³, and the control of geo-hazards¹⁴. Shear localization in amorphous solids remains under intense debate due to the disordered nature of materials; its ubiquity requires a general explanation. One important issue is regarding the initial shear localization when a homogeneous and isotropic amorphous solid is subject to shear. In particular, experiments on sand² and two-dimensional (2D) granular materials⁴ show that when a strain much smaller than the yield strain is applied, it appears immediately, showing self-organized spatial structures. The nature of cooperation remains elusive.

The cooperative particle rearrangement implies long-range correlations, which are incorporated in microscopic theories^15–18 in two different means. Both theories assume that local plastic particle rearrangements act like Eshelby inclusions¹⁹, causing anisotropic and long-range effects to their surrounding elastic media. The first theory is formulated on a dynamical basis: an avalanche of Eshelby inclusions leads to the shear-band formation^15–17. However, the recent granular experiment⁴ shows no compelling evidence of the correlation between local stress drop and particle rearrangement, raising concerns of the relevance of the theory to granular materials, especially at the beginning of shear. The second theory is a mean-field theory based on an energy-minimization principle¹⁸. However, it is unclear how the force chains and dissipation in granular materials would corporate with the theory. On the other hand, numerical works^20–22 observe that despite that the correlation of pressure is short range, the spatial correlation of inherent shear stress has an intrinsic quadrupolar anisotropy and a long-range power-law decay  ∝ r^−d in d dimensions, which is explained by a scaling argument assuming mechanical equilibrium and isotropy of amorphous solids²³. Moreover, this observation is explained using field theories^24–26. However, experimental evidence is still missing. Besides, it is equally missing which role and to what extent the long-range stress correlation may play in the shear dynamics. Considering that the spatial correlation of stresses provides a quantification of force network in granular materials^27,28, it is natural to conjecture the missing role played by force network in the shear dynamics of isotropically jammed granular materials.

In this article, we address experimentally the connection between the shear localization and long-range correlated pseudo stress-chains at the beginning of shear. To this purpose, we study isotropically jammed granular materials, consisting of bi-disperse photoelastic disks, subject to pure shear. We observe that micro shear bands occur right at the start of shear, generating those self-organized cooperative particle rearrangements. We find that there is a strong connection between the particle rearrangements and the pseudo stress-chains, as can be quantified by the correlation between the particle-rotation field and the polarized stress field τ₁(α) along the directions of shear, α = ±45°. Further analysis on τ₁(α) for arbitrary angles α verify the infinite degeneracy of these long-range correlated polarized stress, showing a continuous rotational symmetry. Hence, we understand the deep connection between the dynamics of shear localization and the structure of pseudo stress-chains as symmetry breaking by shear. Moreover, we observe that the chain-like characteristics of the polarized stress τ₁ is quantitatively captured in the scaling anisotropy of its autocorrelation $⟨C_{{\tau }_1}\left(r,\theta \right)⟩$, which shows a power law decay slower than r⁻² along the chain direction and faster than r⁻² perpendicular to the chain direction. Surprisingly, the azimuthal averaged harmonic projection of the autocorrelation function

$$⟨{\overline{C}}_{{\tau }_1}\left(r\right)⟩={\pi }^{-1}{\int }_0^{2\pi }\mathrm{d}\theta \cos \left(2\theta \right)⟨C_{{\tau }_1}\left(r,\theta \right)⟩$$ 1

decays as a power law of r⁻², which is still consistent with the continuum description of the stress chains in the theories^23–26. Finally, we verify that the autocorrelations of inherent shear stress τ₂ show a quadrupole-like pattern with a power law decay of r⁻², providing the experimental evidence of the theoretical predictions^23–26.

Results

Initial shear localization

We first prepare an isotropic jammed packing using a biaxial apparatus shown in Fig. 1a, then apply pure shear quasi-statically in steps by compressing along x axis and expanding along y axis while keeping the area fixed. More experimental details can be seen in “Methods”. Surprisingly, shear localization appear at a strain γ = 0.75% much smaller than the yield strain γ_y ~ 3%, as shown in Fig. 1g. Here we use particle rotations to characterize shear localization, whose pattern is similar with local shear strain and $D_{\min }^2$²⁹, as shown in the Supplementary Fig. 1. Local shear strain describes the affine deformation of the nearest neighbors of a particle, and $D_{\min }^2$ quantifies the degree of the nonaffine displacement of the particle relative to its nearest neighbors by subtracting its affine displacement. Particles rotating clockwise tend to align cooperatively in bands along the 45° direction, whereas those rotating counterclockwise tend to align in bands along the  −45° direction, which leads to a quadrupole-like long-range correlator of local strain, as shown in the Supplementary Fig. 2, consistent with previous experiments of colloids^8–10 and 3D granular materials².

Fig. 1. Schematic of the experimental setup and different types of images recorded.

a Experimental setup. b A snapshot of force chains in an isotropically jammed state (an initial state before shear). A yellow rectangle draws a region that is amplified in c. d A corresponding computer reconstructed image using the measured vector contact forces. e A corresponding normal image of particle configuration. f A corresponding UV image for tracking particle rotations. g Spatial distribution of particle rotations, measured from the initial sate γ = 0 in b to γ = 0.75%. The sizes of red/blue circles are proportional to the magnitudes of counterclockwise/clockwise rotations. Scale bar = 10d, where d = 1 cm, is the diameter of a small particle.

In contrast, spatial distributions of local stress changes are rather homogeneous without obvious correlation with particle rearrangements, as shown in Supplementary Fig. 1. As discussed in detail in the introduction, microscopic theories^15–18 can not fully explain the emergent behaviors of shear localization seen in Fig. 1g, which shows organized spatial structures of particle rotation with long-range correlation. It is a great challenge to understand the emergence of those self-organized, cooperative particle rearrangements in the seemingly disordered granular material. Recall that recent simulations and theories show the exhibition of long-range correlated shear stress of inherent structures in amorphous solids, including granular materials^23–26, though no experimental evidence has been found yet, in particular, not in granular materials. If it does exist, such as in the force-chain network shown in Fig.1b, it is urgent to unravel the possible connection between the force network structure and the particle dynamics.

Polarized stress fields

To reveal this connection, we first define the Cauchy stress tensor of an individual particle in the initial state,

$${\mathit{\sigma }}_i\equiv \left[\begin{array}{cc}\hfill {\sigma }_{i,xx}\hfill & \hfill {\sigma }_{i,xy}\hfill \\ \hfill {\sigma }_{i,yx}\hfill & \hfill {\sigma }_{i,yy}\hfill \end{array}\right]\equiv \frac{1}{S_i}\sum _j{\mathit{r}}_{ij}\otimes {\mathit{f}}_{ij}$$ 2

here r_ij is the position vector from the center of disk i to the contact point between disks i and j, f_ij is the contact-force vector between disks i and j, S_i is the area of the Voronoi cell of disk i, the operator ’ ⊗ ’ represents the dyadic product of two vectors, and the index j runs over all disks j in contact with disk i. In an arbitrary Cartesian axes $\left(x^{\prime },y^{\prime }\right)$, of an angle α with respect to the laboratory axes (x, y), the stress tensor is given by,

$${\mathit{\sigma }}_i^{\prime }=\left[\begin{array}{cc}\hfill \cos \alpha \hfill & \hfill \sin \alpha \hfill \\ \hfill -\sin \alpha \hfill & \hfill \cos \alpha \hfill \end{array}\right]{\mathit{\sigma }}_i\left[\begin{array}{cc}\hfill \cos \alpha \hfill & \hfill -\sin \alpha \hfill \\ \hfill \sin \alpha \hfill & \hfill \cos \alpha \hfill \end{array}\right].$$ 3

Correlation between structure and dynamics

Remarkably, we found strong correlations between the micro-bands and the spatial distributions of normal stress along  ±45°, as shown in Fig. 2a. We denote the normal stress along α direction as the polarized stress ${\tau }_{1,i}\left(\alpha \right)={\sigma }_{i,x^{\prime }{x}^{\prime }}$, showing chain-like structures, which we call pseudo stress-chains. We denote the field of τ_1,i(α) of all disks i as τ₁(α). The micro bands, i.e. those cooperative particle motions, just locate within the inter-spaces of pseudo stress-chains in the two polarized stress fields τ₁(α) with angles α = ±45° along directions of maximal shear. At the same time, other quantities, including free volume, contact number and deviatoric shear stress, show weak correlations with the micro-bands, as shown in Supplementary Fig. 3. To quantify the correlations, we compute the correlation function C(τ_1,i(α), R_±) following ref. ³⁰. We choose the particles of top 10% (counterclockwise R₊ and clockwise R₋) rotations, then the median value ${\tau }_1^{\mathrm{m},R_{\pm }}\left(\alpha \right)$ of the polarized stress of these particles and the cumulative distribution functions (CDF) of τ_1,i(α) give the correlation as $C\left({\tau }_{1,i}\left(\alpha \right),R_{\pm }\right)=1-2\cdot \mathrm{CDF}\left({\tau }_1^{\mathrm{m},R_{\pm }}\left(\alpha \right)\right)$. The results are shown in Fig. 2b, where the peak values are around α = ±45°, as expected. In the above calculation of the correlation functions, changing the cutoff (top 10%) of the particle rotation has little influence on the results, as shown in Supplementary Fig. 4.

Fig. 2. Anti-correlation between particle rotations and polarized stress fields.

a A superposition of the polarized stress fields τ₁(α) with α = ±45° and the counterclockwise(red circles)/clockwise(blue circles) particle rotations, the same as in Fig. 1g. In background, the Voronoi cells of disks are painted according to the gray scales of τ_1,i(α) of each disk i. Scale bar = 10d. b The correlations C(τ_1,i(α), R_±) between particle rotations (R₊: counterclockwise, R₋: clockwise) and τ_1,i(α) versus angle α. The bands indicate standard errors of six independent runs.

Symmetry and long-range characteristics of stress fields

From the symmetry perspective, the two polarized stress fields of τ₁(α) along α = ±45° are by no means special owing to the rotational symmetry of the isotropically jammed packing as shown in Fig. 1b. Six more polarized stresses τ₁(α) of α = 0°, 30°, 60°, 90°, 120°, 150° are shown in Fig. 3a, showing filamentary, pseudo stress-chains preferentially aligned along the orientation of polarized angle α. This long-range character is quantified using an autocorrelation function $C_{{\tau }_1}\left(r,\theta \right)$ of coarse-grained polarized stress^21,31, showing a dipolar signature in Fig. 3b, similar with the pressure or force magnitude autocorrelations in shear-jammed granular systems^27,32. Compared to the short-range correlation of local pressure, as shown in Fig. 3c, $C_{{\tau }_1}\left(r,\theta \right)$ decays much slower along a cut in the dipolar direction. One prominent feature of the correlator of polarized stress is the anisotropy of the scaling property, as shown in the log-log plot in Fig. 3d: the correlation function decays slower than r⁻² along the direction of the stress chains, i.e. θ = 0; it decays much faster than r⁻² along the direction perpendicular to the direction of stress chains, i.e. θ = π/2. Nonetheless, a power-law r⁻ⁿ fit of $⟨{\overline{C}}_{{\tau }_1}\left(r\right)⟩$, yields an exponent n = 1.98 ± 0.03 as shown in Fig. 3e, which indicates that an azimuthally averaged harmonic projection of the correlation function does show the scaling consistent with the field theoretical predictions of n = 2 in 2D systems for the description of stress chains in the continuum limit^23–26. A finite-size analysis of $⟨{\overline{C}}_{{\tau }_1}\left(r\right)⟩$ is given in the Supplementary Fig. 5a. By definition, $⟨{\tau }_{1,i}\left(\alpha \right)⟩=\left({\sum }_i{\tau }_{1,i}\left(\alpha \right)S_i\right)\cdot {\left({\sum }_i{S}_i\right)}^{-1}=p$ for an isotropically jammed packing, where S_i is the Voronoi area of disks i. Here p is the pressure of whole system. Owing to the continuous rotational symmetry, there is an infinite degeneracy of such long-range correlated polarized stress fields τ₁(α). Thus, the emergence of shear localization, i.e. micro bands along  ±45°, can be understood as the breaking of the continuous rotational symmetry by shear.

Fig. 3. Spatial distributions of polarized stress fields and their autocorrelation functions.

a Spatial distributions of six polarized stress fields τ₁(α). Scale bar = 10d. b Spatial autocorrelation map $⟨C_{{\tau }_1}\left(r,\theta \right)⟩$ of coarse-grained polarized stress τ₁(α), 〈 ⋯ 〉 denotes ensemble average over 100 configurations and different polarized angles α. Here, 〈C〉 denotes correlation functions. c Cuts of $⟨C_{{\tau }_1}\left(r,\theta \right)⟩$ along θ = 0, π/2 and autocorrelation function of coarse-grained pressure $⟨{\overline{C}}_p\left(r\right)⟩$, the over bar denotes average over θ. d Log-log plots of $⟨C_{{\tau }_1}\left(r,\theta =0\right)⟩$ and $-⟨C_{{\tau }_1}\left(r,\theta =\pi /2\right)⟩$. The solid line  ∝ r⁻² is guide to the eye. e The azimuthal averaged autocorrelation function of τ₁, $⟨{\overline{C}}_{{\tau }_1}\left(r\right)⟩$. The error bars represent the standard errors. The black dashed line indicates a power law fit of $⟨{\overline{C}}_{{\tau }_1}\left(r\right)⟩$ for r > 3d.

Compared to the polarized stress τ_1,i(α), in literature^23–26, much attention has been paid to the quantity τ_2,i(α), the so-called inherent shear stress, due to the anisotropy and long-range characteristics in its spatial autocorrelation despite that no experimental evidence has been reported thus far^23–26. First note that the τ_2,i(α) is directly related to the polarized stress τ_1,i(α) at individual particles by τ_2,i(α) = τ_1,i(α) − τ_1,i(α + (π/2)). Next, we present the spatial autocorrelation map of shear stress τ_2,i(α), which indeed supports the theories^23–26. The autocorrelation map clearly shows a quadrupole-like pattern with $\cos \left(4\theta \right)$ symmetry, as shown in Fig. 4a, consistent with theories^23–26. To verify the power-law decay of the correlation function, cuts along the θ = 0, π/4 lobes and an appropriate spherical harmonics projection

$$⟨{\overline{C}}_{{\tau }_2}\left(r\right)⟩={\pi }^{-1}{\int }_0^{2\pi }\mathrm{d}\theta \cos \left(4\theta \right)⟨C_{{\tau }_2}\left(r,\theta \right)⟩$$ 4

are shown in Fig. 4b. A power-law r⁻ⁿ fit of $⟨{\overline{C}}_{{\tau }_2}\left(r\right)⟩$ give the exponent n = 1.97 ± 0.08, consistent with the theoretical prediction n = 2 in 2D system^23–26. A finite-size analysis of $⟨{\overline{C}}_{{\tau }_2}\left(r\right)⟩$ is given in Supplementary Fig. 5b. The deviation within r ≈ 6d, corresponds to the breakdown of continuum medium^33,34.

Fig. 4. Autocorrelation functions of shear stress τ₂.

a Spatial autocorrelation map $⟨C_{{\tau }_2}\left(r,\theta \right)⟩$ of coarse-grained shear stress τ₂(α), 〈 ⋯ 〉 denotes ensemble average over 100 configurations and different polarized angles α. b Cuts of $⟨C_{{\tau }_2}\left(r,\theta \right)⟩$ along θ = 0 and θ = π/4, and their azimuthal averaged autocorrelation function $⟨{\overline{C}}_{{\tau }_2}\left(r\right)⟩$. The error bars represent the standard errors. The black dashed line indicates a power law fit of $⟨{\overline{C}}_{{\tau }_2}\left(r\right)⟩$ for r > 6d.

Discussion

To conclude, we find the collective particle rearrangements of the emergent behavior of shear localization in an isotropically jammed granular material are closely related to the pseudo stress-chains in the polarized stress fields τ₁(α) for α along the directions of shear. The emergence of shear localization is associated with the breaking of the continuous rotational symmetry by shear. This mechanism is based on symmetry and the long-range character of internal stress without invoking the Eshelby mechanisms, which could also be applied to other amorphous solids. Statistically, the long-range characteristic of the internal stress are revealed from either the spatial correlations of polarized stress τ₁ or the inherent shear stress τ₂, which provides the direct experimental evidence of theoretical predictions^23–26. The present work serves as a starting point to understand more complicated dynamical processes of the evolution of shear localization. As strain increases, especially near the yielding, we suspect that Eshelby processes may contribute to the evolutions of shear localization and the development of global shear bands, which will be an important subject in future studies.

Methods

Experimental details

In this experiment, we use a biaxial apparatus to apply isotropic compression or pure shear on a two-dimensional granular system. The apparatus mainly consists of a rectangular frame mounted on top of a powder-lubricated glass plate with four walls that can move symmetrically with a motion precision of 0.1 mm while keeping the center of mass fixed. The basal friction coefficient is around 0.3. We estimate that the force magnitude of the basal friction is ~36 times smaller compared to the typical contact-force magnitude. Hence the basal friction is negligible. The rectangular area is filled with a random mixture of 2680 bi-disperse photoelastic disks (Vishay PSM-4) of with diameters of 1.4 cm and 1.0 cm and a number ratio of 1:1 to create various unjammed random initial configurations. The four narrow bands between these photoelastic disks and mobile walls are padded using a set of 300 small and 300 large bi-disperse Teflon-taped metal disks of the same sizes of the photoelastic disks. The friction coefficient is <0.1 between the Teflon-taped metal disks and the Teflon-taped mobile walls, which eliminates substantially the collective rotational motion of individual disks near the boundaries. Next, we apply isotropic compression to achieve packing at particular pressure levels. To minimize the potential inhomogeneity of force chains in the jammed packing, we constantly apply mechanical vibrations in random directions of the horizontal plane before the packing fraction ϕ (the ratio between the area of disks and that of the system) exceeds the jamming point ϕ_J ≈ 84.0% of frictionless particles³⁵. At the top, an array of 2 times 2 high-resolution (100 pixel per cm) cameras are aligned and synchronized. Fig. 1b shows one merged image of force-chain network of an isotropic jammed packing based on the pre-calibration of four cameras. The packing in Fig. 1b is confined in a square domain of 67.2 cm times 67.2 cm. Here, ϕ ≈ 84.4%, the mean coordination number is around 4.1, and the pressure is around 11 N m⁻¹.

We then apply pure shear quasi-statically in steps by compressing along x axis and expanding along y axis while keeping the area fixed. The step size is 0.5 mm, resulting in the  ~0.15% change of the strain. At each step, three different images are recorded as shown in Fig. 1. Disk positions are obtained using the normal image. Hough transformation is used to detect the particle position with a sub-pixel resolution. A UV image is taken for tracking individual particle rotation during shear, whose uncertainty is less than 0.02 rad. Contact forces are analyzed from the force-chain image using force-inverse algorithm, which generates a computed force-chain image based on an initial guess of contact forces, and then iterate contact forces to minimize the difference between experimental and computed force-chain image^27,36,37. The relative error of contact force measurement is around 3–4% for the typical force magnitude, and the accuracy of contact forces is checked by plotting a computed image for comparison as shown in Fig. 1d.

Author contributions

Y.Q.W performed the experiment and carried out the data analyses. J.Z. designed the experimental project. Y.Q.W., Y.J.W., and J.Z. contributed in writing the paper and interpreting the data.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

Supplementary information is available for this paper at 10.1038/s41467-020-18217-x.