# Failure of an effective stress approach in polydisperse wet granular materials

David Cantor, Emilien Azéma, Carlos Ovalle

► To cite this version:

David Cantor, Emilien Azéma, Carlos Ovalle. Failure of an effective stress approach in polydisperse wet granular materials. Physical Review Research, 2024, 6 (2), pp.L022008. 10.1103/PhysRevResearch.6.L022008. hal-04537651

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# Failure of an effective stress approach in polydisperse wet granular materials

David Cantor, Emilien Azéma, Carlos Ovalle

► To cite this version:

David Cantor, Emilien Azéma, Carlos Ovalle. Failure of an effective stress approach in polydisperse wet granular materials. Physical Review Research, 2024, 6 (2), pp.L022008. 10.1103/PhysRevResearch.6.L022008. hal-04537651

HAL Id: hal-04537651

https://hal.science/hal-04537651v1

Submitted on 8 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Failure of an effective stress approach in polydisperse wet granular materials

David Cantor<sup>1,2,*</sup> Emilien Azéma<sup>1,3,4,†</sup> and Carlos Ovalle<sup>1,2,‡</sup>

<sup>1</sup>Department of Civil, Geological, and Mining Engineering, Polytechnique Montréal H3T 1J4, Canada

<sup>2</sup>Research Institute on Mining and Environment, RIME UQAT-Polytechnique, Montréal H3T 1J4, Canada

<sup>3</sup>LMGC, Université de Montpellier, CNRS, Montpellier 34090, France

<sup>4</sup>Institut Universitaire de France (IUF), Paris 75005, France

<figure><img src="image_2.png" /></figure>

(Received 17 June 2023; accepted 23 January 2024; published 8 April 2024)

In this Letter, the effective stress principle (ESP)—originally developed for granular materials saturated with water and recently extended to unsaturated ones via simulations—is shown to fail once the material presents wide grain size distributions. We demonstrate that the current ESP approaches cannot capture the Mohr-Coulomb strength parameters as soon as the grain size span exceeds $d_{max}/d_{min} \sim 4$. This failure is attributed to significant differences in the fabric generated by solid interactions in the wet material, which are supposedly capable of matching the characteristics of the dry material. We show that a generalization of the ESP requires not only macroscopic considerations but also direct attention to the nature of contact and force networks.

DOI: 10.1103/PhysRevResearch.6.L022008

Take a spoonful of any granular material—sand from a beach, soil from a garden, gravel next to a road. Those grains may display significant variation in sizes, shapes, and geological origins. The whole may also contain water or air within the pores, making it a complex mixture whose mechanical behavior is challenging to characterize.

While geotechnical engineering and physics of granular media have made significant progress over the last century, their theories still need meticulous scrutiny to prove themselves as cornerstones in these fields. Among these theories, the effective stress principle (ESP) stands out as a simple—yet robust—framework for addressing the mechanics of soils whose pores are filled with water (i.e., saturated soils). The ESP, attributed to Karl Terzaghi in the early 20th century [1,2], became common ground for studying soils. The principle states that the stresses on the solid phase of a granular material, coined effective stresses $\sigma_{ij}^{\text{eff}}$, control its shear strength and deformation, and can be estimated after subtracting the water pressure $u_w$ from the total external stresses $\sigma_{ij}$ (i.e., $\sigma_{ij}^{\text{eff}} = \sigma_{ij} - u_w\delta_{ij}$, with $\delta_{ij}$ the Kronecker delta operator).

The ESP proved highly successful, so it was natural to try extending it to unsaturated materials, where the pores are partially filled with water. These mixtures are also called wet materials when the water content is low. The first attempts to generalize the ESP are often attributed to Croney et al. [3], Aitchison [4], and Jennings [5], who proposed the relationship $\sigma_{ij}^{\text{eff}} = \sigma_{ij} - \chi u_w \delta_{ij}$, with $u_w$ the water pressure derived from

capillary bridges and $\chi$ an empirical parameter. Bishop et al. [6,7] extended this approach by including an air pressure $u_a$ different than the atmospheric one, as $\sigma_{ij}^{\text{eff}} = (\sigma_{ij} - u_a\delta_{ij}) - \chi(u_a - u_w)\delta_{ij}$. However, these approaches were proven inaccurate as $\chi$ was identified as a parameter dependent on stresses and strains.

Alternatives were then developed based on two independent variables: the net stress ($\sigma_{ij} - u_a\delta_{ij}$) and matric suction ($u_a - u_w$) [8]. Based on this strategy, extensive work on unsaturated material modeling was developed [9–11]. Yet, the search for an effective stress has never stopped and attempts to find a single stress variable are still reported [12–17]. To date, none is considered to be a complete generalization of the ESP.

More recently, works developed in the frame of the discrete-element method (DEM) have suggested that an effective stress approach could still be used for the description of shear strength in unsaturated materials [18–24]. Since DEM simulations can precisely define the liquid bridge forces between grain pairs, a capillary stress tensor $\sigma_{ij}^{\text{cap}}$ can be directly computed, allowing for an extension of Terzaghi's ESP as $\sigma_{ij}^{\text{cont}} = \sigma_{ij} - \sigma_{ij}^{\text{cap}}$, with the solid contact stress $\sigma_{ij}^{\text{cont}}$ the candidate for the effective stress. This proposal has been received, however, with skepticism [20,25,26], since it only validates the ESP through numerical experiments. In addition, these works employed grain assemblies of similar sizes or with a relatively narrow size distribution. Experimental tests have also employed same-size particle assemblies [27]. It is important to note that these works focused on the shear strength properties. The elastic properties of these materials are still not captured by the solid contact stress definition $\sigma_{ij}^{\text{cont}}$ [22,28].

While several experimental and numerical works have shown that particle size distribution (PSD) does not affect the macroscopic shear strength in dry granular materials [19,29–38], understanding the effect of the PSD on wet granular materials has remained seldom explored. In particular, we

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

<footnote>*david.cantor@polymtl.ca</footnote>

<footnote>†emilien.azema@umontpellier.fr</footnote>

<footnote>‡carlos.ovalle@polymtl.ca</footnote>
# Failure of an effective stress approach in polydisperse wet granular materials

David Cantor<sup>1,2,*</sup> Emilien Azéma<sup>1,3,4,†</sup> and Carlos Ovalle<sup>1,2,‡</sup>

<sup>1</sup>Department of Civil, Geological, and Mining Engineering, Polytechnique Montréal H3T 1J4, Canada

<sup>2</sup>Research Institute on Mining and Environment, RIME UQAT-Polytechnique, Montréal H3T 1J4, Canada

<sup>3</sup>LMGC, Université de Montpellier, CNRS, Montpellier 34090, France

<sup>4</sup>Institut Universitaire de France (IUF), Paris 75005, France

<figure><img src="image_2.png" /></figure>

(Received 17 June 2023; accepted 23 January 2024; published 8 April 2024)

In this Letter, the effective stress principle (ESP)—originally developed for granular materials saturated with water and recently extended to unsaturated ones via simulations—is shown to fail once the material presents wide grain size distributions. We demonstrate that the current ESP approaches cannot capture the Mohr-Coulomb strength parameters as soon as the grain size span exceeds $d_{max}/d_{min} \sim 4$. This failure is attributed to significant differences in the fabric generated by solid interactions in the wet material, which are supposedly capable of matching the characteristics of the dry material. We show that a generalization of the ESP requires not only macroscopic considerations but also direct attention to the nature of contact and force networks.

DOI: 10.1103/PhysRevResearch.6.L022008

Take a spoonful of any granular material—sand from a beach, soil from a garden, gravel next to a road. Those grains may display significant variation in sizes, shapes, and geological origins. The whole may also contain water or air within the pores, making it a complex mixture whose mechanical behavior is challenging to characterize.

While geotechnical engineering and physics of granular media have made significant progress over the last century, their theories still need meticulous scrutiny to prove themselves as cornerstones in these fields. Among these theories, the effective stress principle (ESP) stands out as a simple—yet robust—framework for addressing the mechanics of soils whose pores are filled with water (i.e., saturated soils). The ESP, attributed to Karl Terzaghi in the early 20th century [1,2], became common ground for studying soils. The principle states that the stresses on the solid phase of a granular material, coined effective stresses $\sigma_{ij}^{\text{eff}}$, control its shear strength and deformation, and can be estimated after subtracting the water pressure $u_w$ from the total external stresses $\sigma_{ij}$ (i.e., $\sigma_{ij}^{\text{eff}} = \sigma_{ij} - u_w\delta_{ij}$, with $\delta_{ij}$ the Kronecker delta operator).

The ESP proved highly successful, so it was natural to try extending it to unsaturated materials, where the pores are partially filled with water. These mixtures are also called wet materials when the water content is low. The first attempts to generalize the ESP are often attributed to Croney et al. [3], Aitchison [4], and Jennings [5], who proposed the relationship $\sigma_{ij}^{\text{eff}} = \sigma_{ij} - \chi u_w \delta_{ij}$, with $u_w$ the water pressure derived from

capillary bridges and $\chi$ an empirical parameter. Bishop et al. [6,7] extended this approach by including an air pressure $u_a$ different than the atmospheric one, as $\sigma_{ij}^{\text{eff}} = (\sigma_{ij} - u_a\delta_{ij}) - \chi(u_a - u_w)\delta_{ij}$. However, these approaches were proven inaccurate as $\chi$ was identified as a parameter dependent on stresses and strains.

Alternatives were then developed based on two independent variables: the net stress ($\sigma_{ij} - u_a\delta_{ij}$) and matric suction ($u_a - u_w$) [8]. Based on this strategy, extensive work on unsaturated material modeling was developed [9–11]. Yet, the search for an effective stress has never stopped and attempts to find a single stress variable are still reported [12–17]. To date, none is considered to be a complete generalization of the ESP.

More recently, works developed in the frame of the discrete-element method (DEM) have suggested that an effective stress approach could still be used for the description of shear strength in unsaturated materials [18–24]. Since DEM simulations can precisely define the liquid bridge forces between grain pairs, a capillary stress tensor $\sigma_{ij}^{\text{cap}}$ can be directly computed, allowing for an extension of Terzaghi's ESP as $\sigma_{ij}^{\text{cont}} = \sigma_{ij} - \sigma_{ij}^{\text{cap}}$, with the solid contact stress $\sigma_{ij}^{\text{cont}}$ the candidate for the effective stress. This proposal has been received, however, with skepticism [20,25,26], since it only validates the ESP through numerical experiments. In addition, these works employed grain assemblies of similar sizes or with a relatively narrow size distribution. Experimental tests have also employed same-size particle assemblies [27]. It is important to note that these works focused on the shear strength properties. The elastic properties of these materials are still not captured by the solid contact stress definition $\sigma_{ij}^{\text{cont}}$ [22,28].

While several experimental and numerical works have shown that particle size distribution (PSD) does not affect the macroscopic shear strength in dry granular materials [19,29–38], understanding the effect of the PSD on wet granular materials has remained seldom explored. In particular, we

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

<footnote>*david.cantor@polymtl.ca</footnote>

<footnote>†emilien.azema@umontpellier.fr</footnote>

<footnote>‡carlos.ovalle@polymtl.ca</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1. Evolution of the macroscopic friction angle $\sin(\phi)$ and (inset) the cohesion $c$ for the dry material (black circles), the constant force interaction law (green triangles), the wet law (red rectangles), and the solid contact stress tensor (orange crosses) as a function of particle size span $S$.</figcaption></figure>

can summarize the questions that remain still unresolved in this topic as follows: (1) Is the macroscopic friction in wet granular materials independent of particle size polydispersity as it is in dry materials? (2) What are the effects of particle size polydispersity on macroscopic cohesion? (3) Is an effective stress approach still valid in wet size-dispersed granular materials? Advances in this field would not only benefit the soil mechanics community but also many other applications and industrial processes, such as particle mixing, segregation, flowability, and clustering [39–45].

In this work, we use a three-dimensional DEM approach called contact dynamics (CD) to investigate the shear strength of unsaturated granular materials interacting through pendular water bridges over a wide range of PSDs. The CD method is well suited to handle assemblies of rigid particles with diverse interaction laws, allowing body kinematics and contact forces to be resolved in a time-stepping scheme. For details about this method, please see Refs. [46–48]. We test the ESP as recently proposed in the literature and analyze whether or not it still holds once the dispersion of particle sizes is considered. Finally, a detailed micromechanical description of wet and dry samples allows us to unveil the elements that invalidate the current ESP approach after a certain grain size span. Note that this work does not attempt to propose a generalized version of the ESP notion, but rather aims to build a solid case to show that this theory needs to be reevaluated under the light of more realistic material conditions. More importantly, we show that a paradigm shift may be necessary to extend many of the observations and laws that have been developed for the mechanics of granular media.

The particle size span in a granular material can be characterized using the parameter $S = (d_{max} - d_{min})/(d_{max} + d_{min})$, where $d_{max}$ and $d_{min}$ are the maximal and minimal particle diameters in a sample. We built numerical samples using the free open-source simulation platform LMGC90 [49], varying $S$ in the range [0, 0.9], in steps of 0.1 (i.e., ten samples from monosize to highly polydisperse). The PSD followed a uniform distribution by volume fractions between $d_{min}$ and $d_{max}$ (see Fig. 1 in Supplemental Material [50]). The initial configuration of the samples is obtained after an isotropic

compression of frictionless dry spheres that were placed in boxes using simple geometrical rules [51].

We considered two scenarios to simulate granular materials interacting with attractive capillary forces. First, we used a simple model with a constant attractive force $f_0$ acting between particles as long as they remain at a distance inferior to $\delta_{max}$ to each other. The second model emulates water menisci as proposed by Soulié et al. [52] and Richefeu et al. [53] after experimental tests, where the capillary force $f^{\text{cap}}$ and the distance $\delta_{max}$ depend on the size of the particles and a volume of water at the capillary bridge. In this model, the force $f^{\text{cap}}$ decays exponentially as the particles move apart. We set the water surface tension to $7 \times 10^{-2} \text{ Jm}^{-2}$ and the water content $w = m_w/m_s = 3\%$, with $m_w$ and $m_s$ the water and the solid masses, respectively, which reproduce pendular bridge conditions [53]. We considered a perfectly wetting liquid with a contact angle $\theta = 0^\circ$ [50] (the Supplemental Material contains detailed information about the contact laws). To differentiate the two laws, parameters computed for the first law carry the subscript “cst” for constant attractive force, while for the second law, “wet,” is used for wet contact law.

To characterize the shear strength, we employ the Mohr-Coulomb failure criterion as $q = \mu p + C^\dagger$, with the deviatoric stress $q = (\sigma_1 - \sigma_3)$ and the mean pressure $p = (\sigma_1 + 2\sigma_3)/3$, considering triaxial conditions (i.e., $\sigma_2 = \sigma_3$). The stresses $\sigma_1$ and $\sigma_3$ are the major and minor principal stresses of tensor $\sigma_{ij}$, computed as $\sigma_{ij} = (1/V) \sum_{\nu=c} f_i \ell_j$ [54,55], with $V$ the volume of the sample, $f$ the contact force vectors, and $\ell$ the branch vectors (i.e., vectors joining the center of particles in interaction). The macroscopic strength parameters (i.e., friction angle $\phi$ and cohesion $c$) are defined through the relations $\mu = 6 \sin(\phi)/[3 - \sin(\phi)]$ and $C^\dagger = c 6 \cos(\phi)/[3 - \sin(\phi)]$.

We simulated triaxial tests under six different confinement pressures and a constant vertical velocity $v$ on the top and bottom walls, reproducing quasistatic shearing. This was ensured by using the inertial number $I = \dot{e}d\sqrt{\rho/\sigma_c} \ll 1$, with $\dot{e} = v/z_0$ the rate of vertical deformation, $z_0$ the initial height of the sample, and $\rho$ the grains’ density [56,57]. We set $I$ to $1 \times 10^{-3}$. The tests were performed up to a cumulative vertical deformation $\varepsilon = \Delta/z_0 = 0.6$, with $\Delta$ the added displacement of the upper and bottom walls. We also simulated samples with dry frictional interactions between grains. The friction coefficient was set to 0.4 between particles and zero between walls and particles. These friction coefficient conditions ensure that the principal directions of stresses align with the loading orientation, as done experimentally. Additionally, the response of the numerical simulations is comparable with many other works in the literature dealing with similar triaxial loading conditions. No gravity was used to avoid pressure gradients.

We found that the steady state was reached in all cases after $\varepsilon \sim 0.3$ (i.e., stable shear strength and sample volume under continuous deformation) and computed averaged values of stresses $q$ and $p$ in the range $\varepsilon \in [0.35, 0.5]$ (see the strength and solid fraction curves as a function of $\varepsilon$ in Fig. 3 of the Supplemental Material [50]). We then computed the shear strength parameters after fitting these $q$ and $p$ values to the Mohr-Coulomb envelope.

We defined a characteristic capillary stress $\lambda = f^*/\langle d \rangle^2$, with $f^*$ the average capillary force and $\langle d \rangle$ the average
<figure><img src="image_3.png" /><figcaption>FIG. 1. Evolution of the macroscopic friction angle $\sin(\phi)$ and (inset) the cohesion $c$ for the dry material (black circles), the constant force interaction law (green triangles), the wet law (red rectangles), and the solid contact stress tensor (orange crosses) as a function of particle size span $S$.</figcaption></figure>

can summarize the questions that remain still unresolved in this topic as follows: (1) Is the macroscopic friction in wet granular materials independent of particle size polydispersity as it is in dry materials? (2) What are the effects of particle size polydispersity on macroscopic cohesion? (3) Is an effective stress approach still valid in wet size-dispersed granular materials? Advances in this field would not only benefit the soil mechanics community but also many other applications and industrial processes, such as particle mixing, segregation, flowability, and clustering [39–45].

In this work, we use a three-dimensional DEM approach called contact dynamics (CD) to investigate the shear strength of unsaturated granular materials interacting through pendular water bridges over a wide range of PSDs. The CD method is well suited to handle assemblies of rigid particles with diverse interaction laws, allowing body kinematics and contact forces to be resolved in a time-stepping scheme. For details about this method, please see Refs. [46–48]. We test the ESP as recently proposed in the literature and analyze whether or not it still holds once the dispersion of particle sizes is considered. Finally, a detailed micromechanical description of wet and dry samples allows us to unveil the elements that invalidate the current ESP approach after a certain grain size span. Note that this work does not attempt to propose a generalized version of the ESP notion, but rather aims to build a solid case to show that this theory needs to be reevaluated under the light of more realistic material conditions. More importantly, we show that a paradigm shift may be necessary to extend many of the observations and laws that have been developed for the mechanics of granular media.

The particle size span in a granular material can be characterized using the parameter $S = (d_{max} - d_{min})/(d_{max} + d_{min})$, where $d_{max}$ and $d_{min}$ are the maximal and minimal particle diameters in a sample. We built numerical samples using the free open-source simulation platform LMGC90 [49], varying $S$ in the range [0, 0.9], in steps of 0.1 (i.e., ten samples from monosize to highly polydisperse). The PSD followed a uniform distribution by volume fractions between $d_{min}$ and $d_{max}$ (see Fig. 1 in Supplemental Material [50]). The initial configuration of the samples is obtained after an isotropic

compression of frictionless dry spheres that were placed in boxes using simple geometrical rules [51].

We considered two scenarios to simulate granular materials interacting with attractive capillary forces. First, we used a simple model with a constant attractive force $f_0$ acting between particles as long as they remain at a distance inferior to $\delta_{max}$ to each other. The second model emulates water menisci as proposed by Soulié et al. [52] and Richefeu et al. [53] after experimental tests, where the capillary force $f^{\text{cap}}$ and the distance $\delta_{max}$ depend on the size of the particles and a volume of water at the capillary bridge. In this model, the force $f^{\text{cap}}$ decays exponentially as the particles move apart. We set the water surface tension to $7 \times 10^{-2} \text{ Jm}^{-2}$ and the water content $w = m_w/m_s = 3\%$, with $m_w$ and $m_s$ the water and the solid masses, respectively, which reproduce pendular bridge conditions [53]. We considered a perfectly wetting liquid with a contact angle $\theta = 0^\circ$ [50] (the Supplemental Material contains detailed information about the contact laws). To differentiate the two laws, parameters computed for the first law carry the subscript “cst” for constant attractive force, while for the second law, “wet,” is used for wet contact law.

To characterize the shear strength, we employ the Mohr-Coulomb failure criterion as $q = \mu p + C^\dagger$, with the deviatoric stress $q = (\sigma_1 - \sigma_3)$ and the mean pressure $p = (\sigma_1 + 2\sigma_3)/3$, considering triaxial conditions (i.e., $\sigma_2 = \sigma_3$). The stresses $\sigma_1$ and $\sigma_3$ are the major and minor principal stresses of tensor $\sigma_{ij}$, computed as $\sigma_{ij} = (1/V) \sum_{\nu=c} f_i \ell_j$ [54,55], with $V$ the volume of the sample, $f$ the contact force vectors, and $\ell$ the branch vectors (i.e., vectors joining the center of particles in interaction). The macroscopic strength parameters (i.e., friction angle $\phi$ and cohesion $c$) are defined through the relations $\mu = 6 \sin(\phi)/[3 - \sin(\phi)]$ and $C^\dagger = c 6 \cos(\phi)/[3 - \sin(\phi)]$.

We simulated triaxial tests under six different confinement pressures and a constant vertical velocity $v$ on the top and bottom walls, reproducing quasistatic shearing. This was ensured by using the inertial number $I = \dot{e}d\sqrt{\rho/\sigma_c} \ll 1$, with $\dot{e} = v/z_0$ the rate of vertical deformation, $z_0$ the initial height of the sample, and $\rho$ the grains’ density [56,57]. We set $I$ to $1 \times 10^{-3}$. The tests were performed up to a cumulative vertical deformation $\varepsilon = \Delta/z_0 = 0.6$, with $\Delta$ the added displacement of the upper and bottom walls. We also simulated samples with dry frictional interactions between grains. The friction coefficient was set to 0.4 between particles and zero between walls and particles. These friction coefficient conditions ensure that the principal directions of stresses align with the loading orientation, as done experimentally. Additionally, the response of the numerical simulations is comparable with many other works in the literature dealing with similar triaxial loading conditions. No gravity was used to avoid pressure gradients.

We found that the steady state was reached in all cases after $\varepsilon \sim 0.3$ (i.e., stable shear strength and sample volume under continuous deformation) and computed averaged values of stresses $q$ and $p$ in the range $\varepsilon \in [0.35, 0.5]$ (see the strength and solid fraction curves as a function of $\varepsilon$ in Fig. 3 of the Supplemental Material [50]). We then computed the shear strength parameters after fitting these $q$ and $p$ values to the Mohr-Coulomb envelope.

We defined a characteristic capillary stress $\lambda = f^*/\langle d \rangle^2$, with $f^*$ the average capillary force and $\langle d \rangle$ the average
<figure><img src="image_4.png" /><figcaption>FIG. 2. Evolution of the capillary mean pressure $p^{cap}$ and deviatoric stress $q^{cap}$ as a function of $S$. Averaged values are shown since the different levels of $\eta$ (shown in transparency) do not affect the tensor $\sigma_{ij}^{cap}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 3. Evolution of the macroscopic cohesion $c$ and its prediction using the capillary stress tensor and the ESP as a function of the grain size span $S$. Inset: comparison between the ratio $q/p$ for the dry material and “cont” network as a function of $S$.</figcaption></figure>

particle diameter in each sample, and a dimensionless parameter $\eta = \lambda/p$, which provides a cohesion scale in each test. Clearly, $\eta = 0$ corresponds to the dry material, and as $\eta$ increases either the capillary stress scale grows or lower confinement pressures are applied. Previous numerical tests have observed that values of $\eta > 2.5$ produce shear band formation [23], so to avoid those localization phenomena we considered values no larger than $\eta = 1.6$.

Figure 1 gathers the strength parameters as a function of the particle size span $S$ for the two laws tested and the dry material. For the case with constant attractive force, both the macroscopic friction and cohesion increase with $S$, with increments in friction angle only remarkable after $S \sim 0.4$. For mono- or quasimonodisperse granular materials, the macroscopic friction $\sin(\phi_{cst})$ is indistinguishable from that in the dry material $\sin(\phi_{dry})$. However, macroscopic enhancements of shear strength with increasing particle polydispersity cannot only be attributed to increments in the macroscopic cohesion “c” as suggested in the literature [19]. Another implication of this behavior is that an effective stress approach cannot be applied as no connection between its shear strength and that of the dry material can be established. In turn, the wet material shows that only the macroscopic cohesion is affected by $S$. The macroscopic friction $\sin(\phi_{wet})$ is practically independent of the grain size span and remains fairly close to that of the dry material.

For the wet contact law, and following Refs. [18,23,58], the total stress tensor can be split into contributions by solid contact interactions and capillary bridges as $\sigma_{ij} = \sigma_{ij}^{cont} + \sigma_{ij}^{cap}$. Note that the total contact force between a particle pair is $f = f^{cont} + f^{cap}$. For interactions where the grains are separated a distance $\delta > 0$, $f^{cont}$ is respectively zero.

The stress $\sigma_{ij}^{cont}$ is reminiscent of the effective stress because it represents the stresses in the solid phase after discounting the water contributions. However, $\sigma_{ij}^{cap}$ is not necessarily a spherical tensor as shown in Refs. [59–61]—which debunks the early attempts to extend the ESP that considered a scalar ($u_w$) for the water contributions. Despite this difference, a macroscopic friction coefficient for solid contact interactions $\sin(\phi_{cont})$ can be identified by using the values of $p^{cont}$ and $q^{cont}$ from the tensor $\sigma_{ij}^{cont}$ (see Fig. 1). The values

for $\sin(\phi_{cont})$ match the macroscopic friction angle of the dry material very well, suggesting that the solid contact stress $\sigma^{cont}$ could indeed be a good candidate for the effective stress. Although this result is not new for mono- and quasimonosize grains, the fact that it still holds up for large grain size spans is shown here for the first time.

Regarding the capillary tensor $\sigma_{ij}^{cap}$, Fig. 2 displays the evolution of the mean pressure and deviatoric components, $p^{cap}$ and $q^{cap}$, respectively, as a function of $S$. For grain size spans $S < 0.4$, the deviatoric component of $\sigma_{ij}^{cap}$ remains relatively small and $p^{cap}$ does not show remarkable variations. As larger particle size spans are considered, amplifications of mean pressure and deviatoric capillary stresses are observed, highlighting that $\sigma_{ij}^{cap}$ has an anisotropic fabric that is accentuated with particle size polydispersity.

Based on the observations in Fig. 1, and imposing $q^{cont}/p^{cont} \equiv q^{dry}/p^{dry}$—i.e., considering $\sigma_{ij}^{cont}$ as the effective stress measure—it is possible to show that the macroscopic cohesion obeys the relationship

$$ \frac{c}{\tan(\phi_{dry})} = \frac{q^{cap}}{\mu^{dry}} - p^{cap}, \quad (1) $$

with $\mu^{dry} = q^{dry}/p^{dry}$. The deduction of this equation appears in the Supplemental Material [50].

Figure 3 presents both sides of Eq. (1), i.e., the measured macroscopic cohesion and its estimation via the capillary stress tensor. The matching between these two curves is remarkable up to $S = 0.4$. Beyond this $S$, the right side of the equation progressively drifts away from the macroscopic measures. This mismatch no longer allows us to consider the stress partition in $\sigma_{ij}^{cont}$ and $\sigma_{ij}^{cap}$ as a valid route for defining the effective stresses. In other words, our results show that the equivalency $\sigma_{ij}^{cont} \equiv \sigma_{ij}^{dry}$ is broken. When analyzing the components of these tensors, we detect a subtle but growing mismatch between the ratio $q/p$ of the dry material and the “cont” network (see inset of Fig. 3). This discrepancy occurs despite the excellent agreement between the macroscopic friction angles $\sin(\phi_{dry})$ and $\sin(\phi_{cont})$ previously observed.

To understand the origins of this behavior, let us analyze the contact and force networks in the dry material, and the
<figure><img src="image_4.png" /><figcaption>FIG. 2. Evolution of the capillary mean pressure $p^{\text{cap}}$ and deviatoric stress $q^{\text{cap}}$ as a function of $S$. Averaged values are shown since the different levels of $\eta$ (shown in transparency) do not affect the tensor $\sigma_{ij}^{\text{cap}}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 3. Evolution of the macroscopic cohesion $c$ and its prediction using the capillary stress tensor and the ESP as a function of the grain size span $S$. Inset: comparison between the ratio $q/p$ for the dry material and “cont” network as a function of $S$.</figcaption></figure>

particle diameter in each sample, and a dimensionless parameter $\eta = \lambda/p$, which provides a cohesion scale in each test. Clearly, $\eta = 0$ corresponds to the dry material, and as $\eta$ increases either the capillary stress scale grows or lower confinement pressures are applied. Previous numerical tests have observed that values of $\eta > 2.5$ produce shear band formation [23], so to avoid those localization phenomena we considered values no larger than $\eta = 1.6$.

Figure 1 gathers the strength parameters as a function of the particle size span $S$ for the two laws tested and the dry material. For the case with constant attractive force, both the macroscopic friction and cohesion increase with $S$, with increments in friction angle only remarkable after $S \sim 0.4$. For mono- or quasimonodisperse granular materials, the macroscopic friction $\sin(\phi_{\text{cst}})$ is indistinguishable from that in the dry material $\sin(\phi_{\text{dry}})$. However, macroscopic enhancements of shear strength with increasing particle polydispersity cannot only be attributed to increments in the macroscopic cohesion “c” as suggested in the literature [19]. Another implication of this behavior is that an effective stress approach cannot be applied as no connection between its shear strength and that of the dry material can be established. In turn, the wet material shows that only the macroscopic cohesion is affected by $S$. The macroscopic friction $\sin(\phi_{\text{wet}})$ is practically independent of the grain size span and remains fairly close to that of the dry material.

For the wet contact law, and following Refs. [18,23,58], the total stress tensor can be split into contributions by solid contact interactions and capillary bridges as $\sigma_{ij} = \sigma_{ij}^{\text{cont}} + \sigma_{ij}^{\text{cap}}$. Note that the total contact force between a particle pair is $f = f^{\text{cont}} + f^{\text{cap}}$. For interactions where the grains are separated a distance $\delta > 0$, $f^{\text{cont}}$ is respectively zero.

The stress $\sigma_{ij}^{\text{cont}}$ is reminiscent of the effective stress because it represents the stresses in the solid phase after discounting the water contributions. However, $\sigma_{ij}^{\text{cap}}$ is not necessarily a spherical tensor as shown in Refs. [59–61]—which debunks the early attempts to extend the ESP that considered a scalar ($u_w$) for the water contributions. Despite this difference, a macroscopic friction coefficient for solid contact interactions $\sin(\phi_{\text{cont}})$ can be identified by using the values of $p^{\text{cont}}$ and $q^{\text{cont}}$ from the tensor $\sigma_{ij}^{\text{cont}}$ (see Fig. 1). The values

for $\sin(\phi_{\text{cont}})$ match the macroscopic friction angle of the dry material very well, suggesting that the solid contact stress $\sigma^{\text{cont}}$ could indeed be a good candidate for the effective stress. Although this result is not new for mono- and quasimonosize grains, the fact that it still holds up for large grain size spans is shown here for the first time.

Regarding the capillary tensor $\sigma_{ij}^{\text{cap}}$, Fig. 2 displays the evolution of the mean pressure and deviatoric components, $p^{\text{cap}}$ and $q^{\text{cap}}$, respectively, as a function of $S$. For grain size spans $S < 0.4$, the deviatoric component of $\sigma_{ij}^{\text{cap}}$ remains relatively small and $p^{\text{cap}}$ does not show remarkable variations. As larger particle size spans are considered, amplifications of mean pressure and deviatoric capillary stresses are observed, highlighting that $\sigma_{ij}^{\text{cap}}$ has an anisotropic fabric that is accentuated with particle size polydispersity.

Based on the observations in Fig. 1, and imposing $q^{\text{cont}}/p^{\text{cont}} \equiv q^{\text{dry}}/p^{\text{dry}}$—i.e., considering $\sigma_{ij}^{\text{cont}}$ as the effective stress measure—it is possible to show that the macroscopic cohesion obeys the relationship

$$ \frac{c}{\tan(\phi_{\text{dry}})} = \frac{q^{\text{cap}}}{\mu^{\text{dry}}} - p^{\text{cap}}, \quad (1) $$

with $\mu^{\text{dry}} = q^{\text{dry}}/p^{\text{dry}}$. The deduction of this equation appears in the Supplemental Material [50].

Figure 3 presents both sides of Eq. (1), i.e., the measured macroscopic cohesion and its estimation via the capillary stress tensor. The matching between these two curves is remarkable up to $S = 0.4$. Beyond this $S$, the right side of the equation progressively drifts away from the macroscopic measures. This mismatch no longer allows us to consider the stress partition in $\sigma_{ij}^{\text{cont}}$ and $\sigma_{ij}^{\text{cap}}$ as a valid route for defining the effective stresses. In other words, our results show that the equivalency $\sigma_{ij}^{\text{cont}} \equiv \sigma_{ij}^{\text{dry}}$ is broken. When analyzing the components of these tensors, we detect a subtle but growing mismatch between the ratio $q/p$ of the dry material and the “cont” network (see inset of Fig. 3). This discrepancy occurs despite the excellent agreement between the macroscopic friction angles $\sin(\phi_{\text{dry}})$ and $\sin(\phi_{\text{cont}})$ previously observed.

To understand the origins of this behavior, let us analyze the contact and force networks in the dry material, and the
<figure><img src="image_6.png" /><figcaption>FIG. 4. Evolution of anisotropies linked to (a) contact orientations $a_c$, (b) normal forces $a_n$, (c) tangential forces $a_t$, and (d) branch lengths for the dry material (displayed with a black line) and the “cont” network for increasing level of cohesive scale $\eta$.</figcaption></figure>

solid contact network in the wet material. Under triaxial conditions, the macroscopic strength can be estimated as $q/p \sim 2/5(a_c + a_n + a_t + a_\ell)$, where the right side of the equation is linked to anisotropies of contact orientations $a_c$, normal contact forces $a_n$, tangential contact forces $a_t$, and branch lengths $a_\ell$. This expression is deduced after an analytical development of contact, force, and branch distributions in space and their mathematical description via spherical harmonics [62–64]. The validity of this approach has been well covered in the literature [19,34,65–68].

Figure 4 presents the evolution of these anisotropies, revealing that, even though the dry and “cont” components of the wet material exhibit equivalent macroscopic friction angles, this might be a fortunate coincidence. While for small particle size spans these anisotropies are relatively close to each other, the dry and solid contact fabrics are in reality different for all particle size spans. For instance, in a dry material it is acknowledged that the independence of shear strength on $S$ is due to compensations between a drop of contact anisotropy $a_c$ and increments of normal force anisotropy $a_n$ (as $S$ increases more contacts are found in all directions while the normal contact forces have a broader range and a more defined spatial orientation [19,31]). In the solid contact network, not only are there larger variations of $a_c$ and $a_n$ with $S$, but $a_\ell$ and $a_t$ also evolve with different trends. The fact that $a_c$ and $a_\ell$ decrease faster in the “cont” network suggests that (1) the grains within this network are better connected as $S$ increases and (2) the preferential orientation of solid contacts differ from the dry case.

Dry granular materials are also known to contain an increasing amount of particles that do not contribute to the force transmission (i.e., floaters) as $S$ increases. For instance, while the proportion of floaters $P_0$ is around 4% of the total number of grains in a monodisperse sample, it can reach more than 50% when $S = 0.9$ [34]. In contrast, $P_0$ never exceeds 0.5% in a wet material (see Fig. 5). It is thus unsurprising to find a better match between the macroscopic friction of the

<figure><img src="image_7.png" /><figcaption>FIG. 5. Slices of the samples with particle size span $S = 0.8$ at level of deformation $\epsilon = 0.3$, highlighting the grains carrying forces in gray and the floating particles in red for the (a) dry and (b) wet materials. For the latter case, grains that do not participate in the solid contact network are shown in green.</figcaption></figure>

dry and wet materials for low values of $S$, as the number of floaters is relatively small in such cases. Further inspection of average stresses $q^{\text{cont}}$ and $p^{\text{cont}}$ also reveals that while for small $S$ they are very well aligned, with $q^{\text{dry}}$ and $p^{\text{dry}}$ an increasing mismatching is noticeable beyond $S \sim 0.6$ (see inset in Fig. 3). This suggests that the mathematical construct behind $\sigma_{ij}^{\text{cont}}$—which is only supported by a simple contact force partition—is not fully capable of representing the effective stress in wet granular materials after a particle size equivalent to $d_{\text{max}}/d_{\text{min}} \sim 4$.

This series of analyses and observations allow us to conclude that (1) the macroscopic coefficient of friction remains independent of grain size polydispersity in wet granular materials where interactions between grains mimic capillary bridges; (2) size polydispersity enhances macroscopic cohesion; and, more importantly, (3) the current commonly accepted construction of the effective and capillary stress tensors for wet granular assemblies is no longer valid in a polydisperse granular material. While the decades-long quest to find an expression for effective stresses in unsaturated materials has yielded important insights into the mechanics of these materials, it has yet to fully succeed. Our observations show that this is due to the fact that current approaches do not consider different PSDs and the nature of granular networks that are assumed equivalent. At the same time, these differences open up avenues for repairing and generalizing the ESP. Future attempts need to focus on the role of the class of
<figure><img src="image_6.png" /><figcaption>FIG. 4. Evolution of anisotropies linked to (a) contact orientations $a_c$, (b) normal forces $a_n$, (c) tangential forces $a_t$, and (d) branch lengths for the dry material (displayed with a black line) and the “cont” network for increasing level of cohesive scale $\eta$.</figcaption></figure>

solid contact network in the wet material. Under triaxial conditions, the macroscopic strength can be estimated as $q/p \sim 2/5(a_c + a_n + a_t + a_\ell)$, where the right side of the equation is linked to anisotropies of contact orientations $a_c$, normal contact forces $a_n$, tangential contact forces $a_t$, and branch lengths $a_\ell$. This expression is deduced after an analytical development of contact, force, and branch distributions in space and their mathematical description via spherical harmonics [62–64]. The validity of this approach has been well covered in the literature [19,34,65–68].

Figure 4 presents the evolution of these anisotropies, revealing that, even though the dry and “cont” components of the wet material exhibit equivalent macroscopic friction angles, this might be a fortunate coincidence. While for small particle size spans these anisotropies are relatively close to each other, the dry and solid contact fabrics are in reality different for all particle size spans. For instance, in a dry material it is acknowledged that the independence of shear strength on $S$ is due to compensations between a drop of contact anisotropy $a_c$ and increments of normal force anisotropy $a_n$ (as $S$ increases more contacts are found in all directions while the normal contact forces have a broader range and a more defined spatial orientation [19,31]). In the solid contact network, not only are there larger variations of $a_c$ and $a_n$ with $S$, but $a_\ell$ and $a_t$ also evolve with different trends. The fact that $a_c$ and $a_\ell$ decrease faster in the “cont” network suggests that (1) the grains within this network are better connected as $S$ increases and (2) the preferential orientation of solid contacts differ from the dry case.

Dry granular materials are also known to contain an increasing amount of particles that do not contribute to the force transmission (i.e., floaters) as $S$ increases. For instance, while the proportion of floaters $P_0$ is around 4% of the total number of grains in a monodisperse sample, it can reach more than 50% when $S = 0.9$ [34]. In contrast, $P_0$ never exceeds 0.5% in a wet material (see Fig. 5). It is thus unsurprising to find a better match between the macroscopic friction of the

<figure><img src="image_7.png" /><figcaption>FIG. 5. Slices of the samples with particle size span $S = 0.8$ at level of deformation $\epsilon = 0.3$, highlighting the grains carrying forces in gray and the floating particles in red for the (a) dry and (b) wet materials. For the latter case, grains that do not participate in the solid contact network are shown in green.</figcaption></figure>

dry and wet materials for low values of $S$, as the number of floaters is relatively small in such cases. Further inspection of average stresses $q^{\text{cont}}$ and $p^{\text{cont}}$ also reveals that while for small $S$ they are very well aligned, with $q^{\text{dry}}$ and $p^{\text{dry}}$ an increasing mismatching is noticeable beyond $S \sim 0.6$ (see inset in Fig. 3). This suggests that the mathematical construct behind $\sigma_{ij}^{\text{cont}}$—which is only supported by a simple contact force partition—is not fully capable of representing the effective stress in wet granular materials after a particle size equivalent to $d_{\text{max}}/d_{\text{min}} \sim 4$.

This series of analyses and observations allow us to conclude that (1) the macroscopic coefficient of friction remains independent of grain size polydispersity in wet granular materials where interactions between grains mimic capillary bridges; (2) size polydispersity enhances macroscopic cohesion; and, more importantly, (3) the current commonly accepted construction of the effective and capillary stress tensors for wet granular assemblies is no longer valid in a polydisperse granular material. While the decades-long quest to find an expression for effective stresses in unsaturated materials has yielded important insights into the mechanics of these materials, it has yet to fully succeed. Our observations show that this is due to the fact that current approaches do not consider different PSDs and the nature of granular networks that are assumed equivalent. At the same time, these differences open up avenues for repairing and generalizing the ESP. Future attempts need to focus on the role of the class of
floating particles in the dry material that are no longer rattling
in the pore space of a wet material. It would also be interesting
to revisit these results for different shapes of particle size dis-
tribution. Additionally, the preferential orientation of contact
and force networks for solid contacts and capillary bridges
are key elements to consider when evaluating the components
of an effective stress network. However, it is important to
acknowledge that these considerations quickly extend beyond
the wet granular materials considered in this work. Recent and
highlighted contributions in the fields of physics of granular
media and soil mechanics which included attractive forces
between grains (see, for instance, Refs. [69,70]) have dealt
with materials with particle size spans below $d_{max}/d_{min} = 4$.
Consequently, this study strongly suggests that those works
should be reconsidered in the context of more realistic particle
size distributions to confirm their universal character.

Further progress in this topic will also benefit from the
development of three-phasic simulations (soil-water-air) that
allow one to consider large grain size variability and the shape
evolution of water bridges as the material is sheared. X-ray to-
mography and digital image correlations of grain-water mix-
tures [71] are also promising paths for improving our under-
standing on the mechanics of unsaturated granular materials.

This research benefited from the financial support of
the Natural Sciences and Engineering Research Council of
Canada (NSERC) (Project RGPIN-2019-06118) and the com-
putational resources provided by Calcul Québec and Digital
Research Alliance of Canada (Project RRG ID3604). We
thank Mathieu Renouf and Fabien Soulié for their support
in the development of the numerical simulations. We thank
Franny McGill for copy-editing the manuscript.

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[12] M. Nuth and L. Laloui, Effective stress concept in unsaturated soils: Clarification and validation of a unified framework, *Int. J. Numer. Anal. Methods Geomech.* **32**, 771 (2008).

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floating particles in the dry material that are no longer rattling
in the pore space of a wet material. It would also be interesting
to revisit these results for different shapes of particle size dis-
tribution. Additionally, the preferential orientation of contact
and force networks for solid contacts and capillary bridges
are key elements to consider when evaluating the components
of an effective stress network. However, it is important to
acknowledge that these considerations quickly extend beyond
the wet granular materials considered in this work. Recent and
highlighted contributions in the fields of physics of granular
media and soil mechanics which included attractive forces
between grains (see, for instance, Refs. [69,70]) have dealt
with materials with particle size spans below $d_{max}/d_{min} = 4$.
Consequently, this study strongly suggests that those works
should be reconsidered in the context of more realistic particle
size distributions to confirm their universal character.

Further progress in this topic will also benefit from the
development of three-phasic simulations (soil-water-air) that
allow one to consider large grain size variability and the shape
evolution of water bridges as the material is sheared. X-ray to-
mography and digital image correlations of grain-water mix-
tures [71] are also promising paths for improving our under-
standing on the mechanics of unsaturated granular materials.

This research benefited from the financial support of
the Natural Sciences and Engineering Research Council of
Canada (NSERC) (Project RGPIN-2019-06118) and the com-
putational resources provided by Calcul Québec and Digital
Research Alliance of Canada (Project RRG ID3604). We
thank Mathieu Renouf and Fabien Soulié for their support
in the development of the numerical simulations. We thank
Franny McGill for copy-editing the manuscript.

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[2] K. Terzaghi, *Theoretical Soil Mechanics* (Wiley, New York, 1943).

[3] D. Croney, J. D. Coleman, and W. P. M. Black, Movement and distribution of water in soil in relation to highway design and performance, Highway Research Board Special Report (1958).

[4] G. D. Aitchison, Relationships of moisture stress and effective stress functions in unsaturated soils, in *Proceedings Pore Pressure and Suction in Soils* (Butterworths, London, England, 1961), pp. 47–52.

[5] J. E. B. Jennings, A revised effective stress law for use in the prediction of the behavior of unsaturated soils, in *Proceedings of the Pore Pressure and Suction in Soils* (Butterworths, London, England, 1961), pp. 26–30.

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[19] C. Voivret, F. Radjai, J. Y. Delenne, and M. S. El Youssoufi, Multiscale force networks in highly polydisperse granular media, *Phys. Rev. Lett.* **102**, 178001 (2009).

[20] S. Khamseh, J.-N. Roux, and F. Chevoir, Flow of wet granular materials: A numerical study, *Phys. Rev. E* **92**, 022201 (2015).

[21] V. D. Than, S. Khamseh, A. M. Tang, J.-M. Pereira, F. Chevoir, and J.-N. Roux, Basic mechanical properties of wet granular materials: A DEM study, *J. Eng. Mech.* **143**, C4016001 (2017).

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