Dynamic imaging of force chains in 3D granular media

Significance

Since the first medical X-ray radiograph in 1896, tomographic imaging techniques have extended our understanding of the geometry, density, phase composition, and physical processes inside a three-dimensional (3D) body. Here, we introduce interference optical projection tomography to visualize and quantify force chains in 3D granular media—the most abundant form of solid matter on Earth and beyond. By combining the principles of photoelasticity and tomography, our technique provides direct visualization of the particles’ force-chain network and provides the microscopic explanation for why a pack of angular particles is stronger than one of round particles. This particle-level understanding will help forecast geologic phenomena like landslides and earthquakes and better engineer man-made structures like railway ballast and robotic grippers.

Abstract

Granular media constitute the most abundant form of solid matter on Earth and beyond. When external forces are applied to a granular medium, the forces are transmitted through it via chains of contacts among grains—force chains. Understanding the spatial structure and temporal evolution of force chains constitutes a fundamental goal of granular mechanics. Here, we introduce an experimental technique, interference optical projection tomography, to study force chains in three-dimensional (3D) granular packs under triaxial shear loads and illustrate the technique with random assemblies of spheres and icosahedra. We find that, in response to an increasing vertical load, the pack of spheres forms intensifying vertical force chains, while the pack of icosahedra forms more interconnected force-chain networks. This provides microscopic insights into why particles with more angularity are more resistant to shear failure—the interconnected force-chain network is stronger (that is, more resilient to topological collapse) than the isolated force chains in round particles. The longer force chains with less branching in the pack of round particles are more likely to buckle, which leads to the macroscopic failure of the pack. This work paves the way for understanding the grain-scale underpinning of localized failure of 3D granular media, such as shear localization in landslides and stick–slip frictional motion in tectonic and induced earthquakes.

Granular media are a system composed of particles of various sizes and shapes. Compacted or loose, granular media bear the static and dynamic loads of our infrastructure and space explorations (1–5), while their failure can have devastating consequences, such as landslides and earthquakes (6–13). The particle shape greatly determines the behavior of granular media. Round particles, like soybeans, coffee beans, foam, and emulsions, form granular media that deform and flow easily (14–16). Angular and rough particles, such as crushed stones and coffee grounds, form solid-like granular media for mechanical functions, such as railway ballast and robotic gripper (17–20). Studies on three-dimensional (3D) assemblies of spheres have shown that force chains—the chains of contacts among grains where forces are transmitted—control the granular medium’s macroscopic behavior (21–25). However, force chains in 3D assemblies of angular particles remain inaccessible. It is thus still unclear how the force chains in 3D assemblies of angular particles could lead to their vastly different macroscopic behaviors from those of spheres.

For decades, our understanding of force chains has been derived from 2D experiments, in which photoelastic disks of various shapes are packed in a monolayer to simulate granular media (26–33). These disks become birefringent when they are under stresses, with the intensity of birefringence as a function of the stresses. When viewed under a polariscope, the stressed photoelastic disks show visual patterns of fringes and colors, which can be directly interpreted as stress conditions and force chains. As an alternative to photoelasticity, the strain of opaque disks can also be used to calculate the contact force and force chains in 2D granular media (34–37). In the past two decades, other experimental techniques have been developed to study force chains in 3D granular media (25, 38–42). These experiments obtain high-resolution 3D reconstructions of the deformed granular media using 3D refractive index–matching tomography or X-ray computed tomography (CT). The deformation of every particle is incorporated to compute the stress conditions and force chains in 3D. These techniques depend not only on the experimental measurements but also on the computational models, such as the material constitutive models, contact models, and momentum balance equations. Because of the simplicity of 3D spherical contact models (43), only 3D assemblies of round particles have been studied using these techniques.

Here, we introduce a technique, interference optical projection tomography (IOPT), to study the spatial structure and temporal evolution of force chains in 3D granular media composed of spherical and angular particles. This technique combines 3D photoelasticity (44–49), index-matching imaging (39, 45, 48, 50) and CT (51–53) to directly visualizeand quantify the force chains, without the need for solving momentum balance equations coupled with complex contact models of angular particles. We use this technique to study the evolution of force chains in 3D packs of spheres and icosahedra under triaxial shear conditions. We find that, in response to an increasing vertical load, the pack of spheres forms intensifying vertical force chains, while the pack of icosahedra forms more interconnected force-chain networks. The pack of angular particles is more resistant to deformation and shear failure because the interconnected force-chain network is stronger than the isolated force chains in round particles.

Imaging Stresses in 3D Granular Media

Projectional Interferometry: From 3D Stresses to 2D Projections.

Photoelasticity is an experimental method to visualize and quantify the internal stress fields within solid bodies (44, 46, 49, 54). Over the past few decades, photoelasticity has been applied to structural mechanics (49), continuum mechanics (54), fracture mechanics (55), granular mechanics (56), geomechanics (27) and poromechanics (57, 58). When a load is applied to a photoelastic specimen, its internal stress at position (x, y, z) can be expressed as $\mathit{\sigma }(x,y,z)$:

$$\begin{array}{c}\hfill \mathit{\sigma }=\left[\begin{array}{ccc}{\sigma }_x\hfill & {\tau }_{\mathit{xy}}\hfill & {\tau }_{\mathit{xz}}\hfill \\ {\tau }_{\mathit{yx}}\hfill & {\sigma }_y\hfill & {\tau }_{\mathit{yz}}\hfill \\ {\tau }_{\mathit{zx}}\hfill & {\tau }_{\mathit{zy}}\hfill & {\sigma }_z\hfill \end{array}\right].\end{array}$$ [1]

This internal stress makes the photoelastic specimen birefringent—when light transmits along the z direction through the specimen, it is split into two beams with different refractive indices. When exiting the specimen, these two beams have a relative path difference (54), which is known as retardation ($R_t$):

$$\begin{array}{c}\hfill R_t(x,y)=C_{\mathit{so}}{\int }_0^{h(x,y)}({\sigma }_1^{\perp }(x,y,z)-{\sigma }_2^{\perp }(x,y,z))dz,\end{array}$$ [2]

where $C_{\mathit{so}}$ [Pa⁻¹] is the stress-optic coefficient quantifying the intensity of bireferengence as a result of anisotropic stresses; and $h(x,y)$ is the thickness of the specimen along the direction of the light (z direction in Figs. 1D and 2A). ${\sigma }_1^{\perp }$ and ${\sigma }_2^{\perp }$ are the secondary principal stresses, defined as the maximum and minimum principal stresses on the plane perpendicular to the light (x–y plane), respectively (Figs. 1 E and F and 2A). ${\sigma }_1^{\perp }-{\sigma }_2^{\perp }$ can be expressed as

$$\begin{array}{c}\hfill {\sigma }_1^{\perp }-{\sigma }_2^{\perp }=\sqrt{{({\sigma }_x-{\sigma }_y)}^2+4{\tau }_{\mathit{xy}}^2}.\end{array}$$ [3]

Fig. 1.

Photoelastic particles and imaging setup to obtain a projectional interferogram. (A) Photoelastic particles produced by squeeze casting (57). 3.2 mm, 4 mm spheres and 4 mm icosahedra are shown. (B) and (C) Photoelastic particles without and with index-matching fluid, respectively. The index-matching fluid cancels the refraction caused by the particles, forming a transparent mixture. (D) Top view of the dark-field circular polariscope with the tank of index-matching fluid to obtain a projectional interferogram. (E) and (F) Uniaxial compression on a photoelastic sphere without and with index-matching fluid, respectively, to obtain the projectional interferograms.

Fig. 2.

Triaxial setup to produce projectional interferograms of the granular media. (A) Schematic of the triaxial system with the index-matching fluid tank and dark-field circular polariscope. The dark blue parts are stationary, while the light blue parts rotate in the same direction at the same rate. The linear-rotary shaft moves up and down frictionlessly to apply vertical load while maintaining the same rotation as the pack. (B) Picture of the 3D force chain scanner with triaxial stress control.

Therefore, only the components ${\sigma }_x,{\sigma }_y$ and ${\tau }_{\mathit{xy}}$ contribute to the retardation (44). Given its integral form, $R_t(x,y)$ (Eq. 2) is referred to as the retardation integral, which has units of length (usually on the order of 100 nm). We refer to the integral in Eq. 2, ${\int }_0^{h(x,y)}({\sigma }_1^{\perp }(x,y,z)-{\sigma }_2^{\perp }(x,y,z))dz$, as the stress anisotropy integral (SAI).

When viewed through a dark-field circular polariscope, the retardation results in the interference image (54):

$$\begin{array}{c}\hfill I_0(x,y)=I_{\text{max}}{\text{sin}}^2\left(\frac{\pi }{\lambda }{R}_t(x,y)\right),\end{array}$$ [4]

where $I_0$ is the light intensity map of the image; $I_{\text{max}}$ is the maximum light intensity observed in the test; and λ is the wavelength of the light source. When a monochrome light source (light with a single wavelength λ) is used, Eq. 4 shows that the light intensity changes with SAI. When a white light source is used, the light with different wavelengths will consecutively brighten and dim, resulting in a color sequence that is similar to those on the Michel-Lévy chart (57), as shown in Fig. 1 E and F.

In practice, the light source, filters, and the transparency of the material determine the maximum light intensity and color sequence, which can be different from that in the Michel-Lévy chart. In particular, when the material itself is not 100% transparent, it absorbs light and, according to the Beer–Lambert law, results in a reduced intensity (59):

$$\begin{array}{c}\hfill I_c(x,y)=I_0(x,y){\text{e}}^{-C_{\mathit{bl}}h(x,y)},\end{array}$$ [5]

where $I_c$ is the resulting light intensity to be recorded by a camera, $I_0$ is the light intensity before entering the material, and $C_{\mathit{bl}}$ is the light absorption coefficient, which depends on the material. The image observed through the dark-field circular polariscope $I_c(x,y)$ is a 2D projection of the 3D anisotropic stress field in the specimen. We refer to $I_c$ as the projectional interferogram. The projectional interferogram is similar to the projectional radiography in X-ray CT (52) and the projectional images in optical projection tomography (OPT) (60). These images map the integral of a 3D field along the direction of the penetrating beam: projectional interferogram for stress anisotropy, X-ray CT for X-ray density, and OPT for light absorption or fluorescent light.

The widely used 2D photoelasticity is a special case of projectional interferogram, in which specimens of uniform thickness are used to experimentally model structures, fractures, or particles, and visualize the stresses (44, 49, 61). When the material thickness is constant; and the stresses are independent of z, $I_c(x,y)$ directly maps the stress anisotropy. When the fringe patterns appear in a dark-field circular polariscope, each bright or dark line is then the contour of the stress anisotropy. The reason why 3D photoelasticity is less common is that the resulting interferogram visualizes SAI instead of stresses. In addition, a 3D photoelastic specimen is a lens itself, refracting the light shining through and resulting in a distorted image Fig. 1 B and E. To obtain the projectional interferogram, index-matching imaging needs to be used to cancel refraction caused by the 3D photoelastic specimen. This technique is also referred to as integrated photoelasticity, where the photoelastic specimen is observed in an index-matching immersion bath through polarized light (61).

We fabricate high-precision, residual-stress-free 3D photoelastic particles (Fig. 1A) using a squeeze casting technique (57). VytaFlex^TM 20 from Smooth-On, Inc. is used to cast the particles, which produces soft polyurethane particles with an amber color. This material has a refractive index (RI) of 1.476 based on a Brix refractometer measurement. This material is chemically compatible with Dowsel® 550 (RI = 1.49) and Dowsel® 556 (RI = 1.46) oils. By mixing the two oils with a mass ratio of 2:1, we create an oil that has the same RI as the photoelastic particles (Fig. 1 B and C). When the photoelastic material is under stress, its refractive index changes. However, this change is very small and has a negligible effect on the resulting image.

A dark-field circular polariscope with a tank of index-matching fluid is used to obtain the projectional interferogram of a sphere under uniaxial compression (Fig. 1D). To compare, uniaxial tests are also conducted without the index-matching fluid. Fig. 1E shows the uniaxial compression tests without the index-matching fluid. When viewed without the polarizers, the sphere refracts the light from the Top and Bottom platens and shows its profile distinctly (Fig. 1E, i). When viewed through the dark-field circular polariscope, the sphere first appears dark because it is residual-stress-free (Fig. 1E, ii and F, ii). When the sphere is loaded vertically without index-matching fluid (Fig. 1E, ii–viii), a horizontal band of light shows up on the sphere because of the refraction caused by the curved surfaces. The contact surface is not clear in this case. In comparison, the projectional interferograms (Fig. 1F, ii–viii) show that only the center of the sphere lights up first, with a clear view of the contact surface at the top and bottom. The left and right sides of the sphere light up at a higher load because of the lower stress and thinner material contributing to the SAI. Because white light is used, colors appear in sequence along the middle line of the sphere (Fig. 1E, vi–viii and F, vi–viii).

IOPT: From 2D Projections to 3D Stresses.

The projectional interferogram captured by the imaging system ($I_c(x,y)$) is the result of both light interference and light absorption. To obtain the stress information from these images, the light intensity needs to be corrected for light absorption first. In Eq. 5, the parameter $C_{\mathit{bl}}$ for the material is measured by aligning multiple spheres in the direction of light and correlating the light intensity with the number of spheres. For a pack of particles, $C_{\mathit{bl}}$ is measured by correlating the thickness of the pack (such as the pack shown in Fig. 2A) with the light intensity. Our measurements show that $C_{\mathit{bl}}$ is approximately 0.013 mm⁻¹ for the pack of spheres immersed in the index-matching fluid with a packing density of approximately 0.70 (SI Appendix, Fig. S2). Based on the Beer–Lambert law (Eq. 5), the image can be corrected to obtain the interference image, $I_0$.

Eq. 4 provides the mapping between the corrected interference image ($I_0$) and the retardation integral $R_t$. This mapping is monotonic when

$$\begin{array}{c}\hfill \frac{\pi }{\lambda }{C}_{\mathit{so}}{\int }_0^{h(x,y)}({\sigma }_1^{\perp }(x,y,z)-{\sigma }_2^{\perp }(x,y,z))dz<\frac{\pi }{2}.\end{array}$$ [6]

This can be achieved by using materials with lower stress-optic coefficient, and limiting the size of the specimen ($h(x,y)$) and the applied loads. The VytaFlex^TM 20 used in the experiments has a relatively low stress-optic coefficient of $4.1\times {10}^{-10}$ Pa⁻¹, measured with a green light source ($\lambda =515$ nm) (SI Appendix, Fig. S1). The maximum light intensity $I_{\text{max}}$ is obtained by overloading the pack until the pack darkens with the increasing load. With the parameters $C_{\mathit{so}}$, λ and $I_{\text{max}}$ known, we obtain SAI based on Eqs. 2 and 4.

SAI is the projection of the 3D stress tensor field in the photoelastic pack. In this projection, the 3D stress tensor ($\mathit{\sigma }(x,y,z)$) is first projected onto the x–y plane (Fig. 2A) as secondary principal stresses (${\sigma }_1^{\perp }$ and ${\sigma }_2^{\perp }$). Their difference, the secondary stress anisotropy (${\sigma }_1^{\perp }-{\sigma }_2^{\perp }$), is then integrated along the direction of light as the projection. Because of the rotational dependence of the stress tensor and the geometry of the pack, SAI is dependent of the view angle. Thus, SAI should be expressed as a function of the rotation angle θ:

$$\begin{array}{c}\hfill \text{SAI}(x,y,\theta )={\int }_0^{h(x,y,\theta )}({\sigma }_1^{\perp }(x,y,z,\theta )-{\sigma }_2^{\perp }(x,y,z,\theta ))dz.\end{array}$$ [7]

The projectional interferogram differs from the projectional radiography of CT and projectional images of OPT (60) by being the projection of a 3D tensor field instead of a scalar field (61). Tomographic reconstruction of 3D tensor field, tensor tomography (51), is needed to reconstruct the 3D stress field in the photoelastic pack. Tensor tomography constitutes an evolving area of research in biomedical imaging using magnetic resonance imaging (MRI) (51, 62, 63) and experimental mechanics using photoelasticity and index-matching imaging (64–68). With the help of tensor tomography, we can use the experimental technique to study the 3D stress distribution in a solid or 3D force chains in granular media. We call this experimental technique IOPT.

Currently, no tensor tomography algorithm is available to reconstruct the full 3D stress tensor based on the projectional interferogram. However, in some special cases, reconstructing the full 3D tensor is not necessary; for example, under axisymmetric stress conditions (66, 69). In triaxial tests (70), the secondary principal stresses equal the 3D principal stresses when the specimen rotates about the axis of symmetry:

$${\sigma }_1^{\perp }={\sigma }_1={\sigma }_v,$$ [8a]

$${\sigma }_2^{\perp }={\sigma }_2={\sigma }_3={\sigma }_h,$$ [8b]

where ${\sigma }_v$ and ${\sigma }_h$ are the axial (vertical) and confining (horizontal) stresses. In this case, the 3D stress tensor field ($\mathit{\sigma }(x,y,z)$) can be reduced to a 3D scalar field of stress anisotropy (${\sigma }_v-{\sigma }_h$). Based on the projectional interferogram, reconstruction algorithms used for CT and OPT, such as inverse Radon transform or iterative reconstruction (52), can be used to reconstruct the 3D axisymmetric stress anisotropy. This is especially useful to study the interparticle forces that are transmitted heterogeneously through force chains in 3D granular media. We apply this technique to study 3D force chains in packs of angular and round particles under axisymmetric boundary condition known as triaxial shearing.

3D Force Chains under Triaxial Shearing

Experimental Apparatus.

The ability of granular media to bear applied loads depends chiefly on the three-dimensional stress state. To measure the bearing capacity of granular media, a triaxial system is often used to control the confining (horizontal) stress, pore pressure and axial (vertical) stress. We design an experimental apparatus that applies such stress condition while obtaining the projectional interferograms of the pack of photoelastic particles (Fig. 2).

A rotation stage is constructed with a stepper motor to precisely control the rotation of the pack of photoelastic particles (Fig. 2). An index-matching fluid tank is built around this rotation stage. A hollow shaft is used to transmit the rotation to the pack of photoelastic particles in the tank. This hollow shaft is sealed by a rotary joint at the bottom of the tank so that the fluid does not leak out of the tank. The photoelastic particles are packed into a transparent flexible pouch that is not photoelastic. The pouch containing the pack of photoelastic particles filled with index-matched fluid is sealed to the bottom endcap, which tightly fits the hollow shaft to create a fluid body that is isolated from the tank. The bottom of the hollow shaft connects to an index-matching fluid reservoir (not shown in the figure). The fluid in the reservoir, hollow shaft and pore space is connected, so that we can change the pore pressure by changing the elevation of the reservoir.

We apply a negative pore pressure in the granular pack by lowering the reservoir relative to the pouch. This pressure difference between the outside and inside of the pouch imposes a uniform confining effective stress on the granular pack. In fluid-filled granular media, the part of the stress transmitted through the solid particles is effective stress, which is equal to the difference between the applied external stress and the pore pressure (57, 71). Because of the relatively large particles and high permeability of the pack, all the loading can be treated as occurring under constant pore pressure (drained condition). On top of the index-matching fluid tank, the load frame is mounted to apply vertical stresses. A shaft passing through a linear-rotary bearing is used to apply a vertical load at the top of the pack. This shaft transmits a constant load from the gravity of the weights while remaining free to rotate when the projectional interferograms are taken at different angles.

We conduct triaxial tests on 3D packs of 3.2 mm spheres, a mixture of 3.2 mm and 4 mm spheres and icosahedra with 4 mm equivalent diameter. After the 3D pack is saturated and installed on the hollow shaft, a constant confining stress of 2.70 kPa is first applied to the pack by lowering the pore pressure reservoir. Then an increasing vertical stress is applied to the pack by first applying the weight of the linear-rotary shaft (0.62 kPa), then adding weights on the top of the shaft in each load step (0.39 kPa). The pack undergoes compaction as the vertical load increases, until it reaches shear failure. After the pack fails, deforming substantially, the linear-rotary shaft reaches its stroke limit, which in turn causes the pack to relax. Between two loading steps, the rotation stage is programmed to rotate 360^°, while images are taken at a rate of 120 frames per second. A full rotation takes 20 s and yields 2,400 frames of projectional interferograms (Movies S1–S3). These images are processed for the reconstruction and quantitative analysis of the 3D force chains.

3D Force Chain Reconstruction and Analysis.

The steps to reconstruct and quantitatively analyze the 3D force chains are illustrated in Fig. 3. First, the images captured by the camera are corrected for light absorption. The corrected image is the light intensity map $I_0$ resulting from light interference only (Fig. 3B). When the stress anisotropy is within a bounded range, we can obtain an SAI (Eq. 7) from the intensity map (Fig. 3C). With the 2,400 SAIs for 360^° of rotation, tensor tomography can be used to invert for the 3D stress tensor, which reconstructs the stresses in each particle due to interparticle forces (effective stresses). Taking advantage of the macroscopic axisymmetric stress conditions of the 3D pack, we use the same reconstruction algorithms as CT and OPT inverse Radon transform for our case (52) for the 3D scalar field of stress anisotropy. This neglects the rotational dependence of the local 3D stress tensor in the particles, which is a source of error, but is sufficient for resolving the chains of particles under stresses—the 3D force chains.

Fig. 3.

Image processing for quantitative analysis of 3D force chains. (A) Image from the green channel of the camera. (B) Image filtered by a Beer–Lambert filter to correct for light absorption. (C) Stress anisotropy integral mapped from the image intensity. (D) Sinogram for horizontal slice $y=200$. This is constructed from the 2,400 images as the specimen rotates ${360}^{°}$. (E) Reconstruction of horizontal cross-section $y=200$ using the inverse Radon transform. The bright spots are the cross-sections of the force chains. (F) 3D reconstruction of the stress anisotropy field by stacking all the horizontal layers. The color indicates the intensity of stress anisotropy. The force chains are mostly vertical, indicating that the maximum principal stress is vertical. The different colors on the force chains reflect the stress heterogeneity in the granular pack. (G) Skeletonized force chains. (H) Branch points indicated in blue. (I) Isolated branches for length, intensity, and orientation analysis. The branches are isolated by removing the branch points.

We reconstruct the 3D stress anisotropy field one horizontal layer at a time (illustrated here with layer $y=200$), by combining the 360^° rotation of one horizontal layer (2,400 images of one layer) of the 3D scalar field into a sinogram (52) (Fig. 3D). The sinogram is used to reconstruct each horizontal cross-section (here $y=200$) of the stress anisotropy field (Fig. 3E). The same process is repeated for all the horizontal layers in Fig. 3C to reconstruct all the cross-sections. These horizontal cross-sections are stacked vertically to form a 3D scalar matrix, representing the 3D stress anisotropy field of the pack.

The 3D stress anisotropy matrix is then used to visualize and analyze the force chains, with the terminology employed in classic experiments using 2D photoelastic disks (1, 23, 26, 27, 30). Fig. 3F visualizes the brightest 2.5% of the voxels in the 3D stress anisotropy matrix by global histogram thresholding. After the 3D stress anisotropy matrix is binarized with such threshold, each of the connected volumes in the binary matrix is color-mapped with the maximum stress anisotropy in this volume. The resulting 3D image shows that the forces are transmitted in 3D granular media through force chains. Among the force chains, the forces are heterogeneous, with some force chains bearing higher forces than others (Fig. 3F).

We then quantitatively study the evolution of these force chains by analyzing the force chains’ lengths, orientations, and stresses. The visualized force chains (Fig. 3F) are skeletonized with the help of MATLAB built-in function, bwskel. The skeletonized force chains have widths of a single voxel (Fig. 3G), so the number of voxels represents the length of the force chain. The branch points in the skeleton are detected using bwmorph3, ‘branchpoints’ and then deleted to isolate each force-chain segment. The mean stress anisotropy of each skeletonized force-chain segment represents the stress on this segment.

Evolution of 3D Force Chains under Triaxial Shearing

Combining the reconstructed 3D force chains in each load step, we study the evolution of the 3D force chains under triaxial shearing (Movies S4–S6). The results of the three triaxial tests are presented in three groups: 3.2 mm spheres (Fig. 4A, a, α), a mixture of 3.2 mm and 4 mm spheres (Fig. 4B, b, β) and 4 mm icosahedra (Fig. 4C, c, γ). Fig. 4A–C are the stress–strain curves for the three triaxial tests. Fig. 4a–c show the projectional interferograms taken at angle $0^{°}$ of each load step (i) to (ix), while (Fig. 4 α, β, γ) show the reconstructed 3D force chains of the corresponding load step. We first apply an isotropic effective stress of 2.70 kPa, then increase the vertical effective stress ${\sigma }_v$ by 0.39 kPa in each load step. The resulting strain is calculated from the projectional interferogram of each step, by normalizing the change of pack height to its height at the initial isotropic stress condition where ${\sigma }_h={\sigma }_v=2.70$ kPa. Fig. 4 A and B show that the two packs of spheres undergo large deformation when the load increases from 5.2 to 5.6 kPa, indicating the mechanical failure of the granular pack. In comparison, the pack of icosahedra is stronger and more rigid, as indicated by the steeper stress–strain curve in Fig. 4C. These macroscopic behaviors of the packs of spherical and angular particles agree with the experimental studies on round and angular gouge materials (8, 13, 72, 73). The pack of icosahedra undergoes large deformation and fails when the load increases from 5.6 to 6.0 kPa. After failure, the vertical stress on the packs is no longer 6.0 kPa due to the large deformation and the loading shaft’s stroke limit.

Fig. 4.

Evolution of 3D force chains under triaxial shearing. (A–C) Stress–strain curves during triaxial loading. (a–c) Projectional interferograms at angle $0^{°}$ of each load step. (α, β, γ) Reconstructions of 3D force chains in each load step.

Fig. 4a, b, c show the projectional interferograms taken at angle $0^{°}$ of each load step (i) to (ix), which exhibit increasing brightness and intensifying filamentary patterns as the vertical stress increases. Although distorted by the increasing vertical stresses, the filamentary patterns maintain their geometries in each experiment when the stresses are low (Fig. 4i to vi). Further increase of the vertical stress causes the packs to fail and deform substantially (Fig. 4vii to ix). The postfailure filamentary patterns become completely different from the prefailure ones, indicating a global reconfiguration of the 3D force chains in the pack.

The reconstructed 3D force chains in each load step are shown in Fig. 4α, β and γ. In each triaxial test, a global threshold is used to visualize the brightest 2.5% of the voxels under intermediate vertical stresses (4.5 kPa, column (v) in Fig. 4). For triaxial tests with spheres (Fig. 4 A and B), the force chains extend and intensify as the vertical stress increases before shear failure. The force chains are predominantly vertical, aligning with the maximum principal stress. When the pack fails, the force chains tilt, buckle, and reconfigure into different geometries (Fig. 4α and β, viii). When the pack relaxes after failure, the force chains dim and their intensity falls below the threshold (Fig. 4α and β, ix). The force chains in the two packs of spheres exhibit similar behavior even though the pack of a mixture of 3.2 mm and 4 mm spheres deforms less, as indicated by the stress–strain curves (Fig. 4 A and B).

The evolution of the 3D force chains in the pack of icosahedra exhibits a very different behavior. The force chains form more interconnected networks, which grow larger as the vertical stress increases before shear failure. When the pack fails, the networks tilt altogether and form new networks with different geometries (Fig. 4γ, vii, ix). When the pack relaxes, the networks dim and shrink in size. By forming these interconnected 3D networks, the pack of icosahedra is stiffer and stronger than the pack of spheres, as also evidenced by the stress–strain curve (Fig. 4A–C).

To quantitatively compare the 3D force chains in packs of spheres and icosahedra, we use the analysis illustrated in Fig. 3. The force chains in the load steps (v, vi, vii and ix) in Fig. 4α and γ are analyzed. These load steps correspond to the vertical stresses of 4.5, 4.9, 5.2 kPa, and the final failure in the packs. All force-chain segments are isolated to measure their lengths, stress anisotropy intensities, and orientations. We use the dip of the force chains to study their orientation, which is the angle of inclination measured downward from the horizontal. The bivariate histograms of the force-chain segments’ dip and intensity distribution are shown in Fig. 5 B and E. The bivariate histograms of the force-chain segments’ length and intensity distribution are shown in Fig. 5 C and F.

Fig. 5.

Analysis of the force chain evolution leading to shear failure. (A) Skeletonized 3D force chains in the pack of 3.2 mm spheres. (B) Bivariate histogram of force chain dip and intensity for the pack of 3.2 mm spheres. (C) Bivariate histogram of force chain length and intensity for the pack of 3.2 mm spheres. (D) Skeletonized 3D force chains in the pack of 4 mm icosahedra. (E) Bivariate histogram of force chain dip and intensity for the pack of 4 mm icosahedra. (F) Bivariate histogram of force chain length and intensity for the pack of 4 mm icosahedra.

As the vertical stress increases, the force chains in the pack of spheres align vertically, then buckle before failure, as indicated by the increase, then decrease, in the number of force chains with a dip close to 90^° (Fig. 5B). The increasing vertical stress also induces the formation of longer and stronger force chains before the pack fails (Fig. 5C). In contrast, in the pack of icosahedra, the force chains grow into a larger network with more force-chain segments as the vertical stress increases. Because of the greater number of branch points in a network than that in isolated chains, the force-chain segments in the pack of icosahedra have a greater number but are shorter than those in the pack of spheres. When the pack of icosahedra fails, the whole network buckles before failure with no distinct changes in the length and dip of the force chains.

This quantitative comparison shows how differently a pack of spherical particles and a pack of angular particles transmit and bear loads: the former forms intensifying vertical chains, while the latter forms larger force-chain networks to counter the increasing load. This provides microscopic insights into why packs of particles with more angularity are more resistant to shear failure: The interparticle force-chain network is more interconnected, hierarchical, and robust than that in packs of round particles. The longer chains with less branching in the pack of spheres are also prone to buckling, a direct precursor to the global failure of the pack.

Conclusions

We develop a type of tomography, IOPT, to visualize and quantify the force chains in 3D packs of round and angular particles. First, we show that the projectional interferogram is a projection of the 3D stress tensor to a 2D image and is equivalent to the projectional radiography in X-ray CT if the stress is axisymmetric. We build a 3D force chain scanner to obtain projectional interferograms, while the externally loaded 3D pack of particles rotates 360^°. We then use these projectional interferograms to reconstruct the 3D scalar field of stress anisotropy in the pack, which is then visualized as force chains.

We conduct triaxial shear tests with the 3D force chain scanner to observe the spatial structure and temporal evolution of force chains in packs of spheres and icosahedra. We find that, in response to an increasing vertical load, the pack of spheres forms intensifying vertical force chains, while the pack of icosahedra forms more interconnected force-chain networks. The pack of icosahedra is more resistant to deformation and shear failure, because the interconnected force-chain network is more robust and resilient than the isolated force chains in the pack of spheres. The longer chains with less branching in the pack of round particles are more prone to buckling, which leads to an earlier failure of the pack. The behavior of packs made of other angular particles could be investigated using our experimental techniques, and compared quantitatively to that of spheres and icosahedra.

Our technique provides a different way to study force chains in 3D. Instead of inferring forces from deformations (25, 39, 42), IOPT directly visualizes and quantifies the force chains, circumventing the need for constitutive models, contact models and momentum balance equations. No longer constrained by particle shape, IOPT’s rapid scanning of 3D interparticle forces unveils the grain-scale underpinning that causes granular media to jam (1), deform (74), fail (27), and flow with and without fluid coupling (58, 75–77). This work paves the way for advancing our understanding of granular media in landslides (11), coastal erosion (78), avalanches (79), earthquakes (80–82), and liquefactions (83), and may also help us better engineer the granular media for breakwaters, railway ballast, hoppers, and robotic grippers.

With the advancements in material science and digital fabrication (49), IOPT can use a broader range of photoelastic materials and more complex particle geometries in the experimental modeling system for biostructures (84), microcolonies (85), and fabrics (20). Currently, IOPT is limited to axisymmetric stress conditions due to the lack of full tensor tomography algorithms. Machine learning techniques (86) may facilitate the reconstruction of the 3D tensor field, which, in turn, will enable the study of contact mechanics, fracture mechanics, and granular mechanics in 3D systems under general loading conditions.

Materials and Methods

The 3D photoelastic particles are squeeze cast with a two-part resin, VytaFlex^TM 20 from Smooth-On, Inc., which produces soft polyurethane rubber with an amber color. We conduct uniaxial compression test in the circular polariscope to measure the material’s mechanical properties and find the Young’s modulus to be approximately 820 kPa and the stress-optic coefficient to be $4.1\times {10}^{-10}$ Pa⁻¹ (SI Appendix, Fig. S1). When packed in the triaxial system, the granular media have a packing density of approximately 0.70 based on phase relationships and a light absorption coefficient of 0.32 inch⁻¹ (0.013 mm⁻¹) for both sphere and icosahedron particles (SI Appendix, Fig. S2). Based on Brix refractometer measurements, the photoelastic material has a refractive index (RI) of 1.476. We create a chemically compatible index-matching fluid for this material by mixing two oils, Dowsel® 550 (RI = 1.49) and Dowsel® 556 (RI = 1.46), with a mass ratio of 2:1.

The experimental apparatus was designed and fabricated in-house with laser-cut and 3D-printed parts. The rotation motion was provided by a 28BYJ-48 stepper motor controlled by a ULN2003 chip and an Arduino microcontroller. The rotation stage completes a 360^° rotation in 20 s, while a Sony α7R III camera records the video at 120 frames per second for each scan. The rotation stage stops for 30 s between two scans to allow step loading and deformation.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

360° projectional interferogram of a pack of spheres (3.2 mm) at loading step 6 (σ_h = 2.7 kPa, σ_v = 4.9 kPa). 2400 projectional interferograms are taken for 3D reconstruction of this load step.

Movie S2.

360° projectional interferogram of a pack of spheres (3.2 & 4 mm) at loading step 6 (σ_h = 2.7 kPa, σ_v = 4.9 kPa). 2400 projectional interferograms are taken for 3D reconstruction of this load step.

Movie S3.

360° projectional interferogram of a pack of icosahedra at loading step 6 (σ_h = 2.7 kPa, σ_v = 4.9 kPa). 2400 projectional interferograms are taken for 3D reconstruction of this load step.

Movie S4.

Dynamic imaging of force chains in a pack of spheres (3.2 mm) during triaxial shearing.

Movie S5.

Dynamic imaging of force chains in a pack of spheres (3.2 mm & 4 mm) during triaxial shearing.

Movie S6.

Dynamic imaging of force chains in a pack of icosahedra (4 mm) during triaxial shearing.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.