# Configurational Energy as a Microstructural Descriptor of Failure Precursors in 2D Frictional Granular Materials

Zhennan Zhu, Francois Nicot, Antoine Wautier, Klaus Regenauer-Lieb, Manman Hu

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Zhennan Zhu, Francois Nicot, Antoine Wautier, Klaus Regenauer-Lieb, Manman Hu. Configurational Energy as a Microstructural Descriptor of Failure Precursors in 2D Frictional Granular Materials. International Journal for Numerical and Analytical Methods in Geomechanics, 2026, 10.1002/nag.70209. hal-05461239

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RESEARCH ARTICLE

OPEN ACCESS

# Configurational Energy as a Microstructural Descriptor of Failure Precursors in 2D Frictional Granular Materials

Zhennan Zhu<fnref n="1" />,²,<fnref n="3" /> ID | Francois Nicot² ID | Antoine Wautier<fnref n="3" /> ID | Klaus Regenauer-lieb⁴ ID | Manman Hu<fnref n="1" /> ID

**Correspondence:** Manman Hu (mmhu@hku.hk)

**Received:** 8 July 2025 | **Revised:** 15 November 2025 | **Accepted:** 6 December 2025

**Keywords:** configurational energy | failure precursors | pattern formation | strain localization

## ABSTRACT

Localized deformation in dense granular materials, often culminating in the formation of shear bands, is a key failure mechanism in geotechnical and material systems. However, predicting the onset of such localization remains a fundamental challenge due to the system's inherent inelasticity and microstructural complexity. In this study, we propose that the evolution of internal configuration—characterized by changes in contact topology and stored potential energy—governs the collective mechanical response and encodes precursors to material failure. To quantify this evolving internal state, we introduce the notion of configurational energy, defined as the change in contact-scale elastic potential energy resulting from a controlled loading—unloading probe. This metric is first formulated at the contact level and subsequently analyzed at the specimen scale using Discrete Element Method (DEM) simulations of biaxial compression. Our results demonstrate that configurational energy captures the system's sensitivity to perturbations and reflects local instability: both positive and negative values emerge at the contact level, with large magnitudes concentrated near regions of active rearrangement. Despite this local variability, the specimen-scale configurational response remains strictly negative, and its magnitude increases systematically as the material approaches failure. Notably, spatial localization of configurational energy precedes the formation of macroscopic shear bands with an evolving internal length scale, offering a mesoscale energetic signature of incipient failure. These findings establish configurational energy as a physically grounded descriptor of microstructural evolution and a promising tool for anticipating failure in frictional granular systems.

## 1 | Introduction

Localized deformation is a fundamental energy dissipation mechanism and a precursor to failure in granular materials, impacting diverse geological phenomena and engineering applications [1]. This phenomenon typically manifests as the material transitions from a predominantly elastic response to a plastic regime, where the capacity for reversible elastic energy storage is exceeded by external work input. This transition is characterized by the emergence of plastic slip lines or strain localization zones, wherein

particle rearrangements become spatially concentrated, often evolving into persistent, narrow shear bands. These mesoscale structures represent pathways of irreversible deformation and are critical indicators of mechanical failure in granular systems [2]. A comprehensive understanding of the onset and evolution of such localized deformation is paramount for predicting and mitigating failure in natural and engineered granular assemblies, including scenarios like fault reactivation, time-dependent degradation of geomaterials, and landslide initiation [3, 4].

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2026 The Author(s). *International Journal for Numerical and Analytical Methods in Geomechanics* published by John Wiley & Sons Ltd.

<footnote n="1">¹Department of Civil Engineering, The University of Hong Kong, Hong Kong, China | ²ISTerre, Université Savoie Mont-Blanc, Le Bourget-du-Lac, France |</footnote>

<footnote n="3">³Unité de Recherche RECOVER, INRAE, Aix-Marseille Université, Aix-en-Provence, France | ⁴WASM: Minerals, Energy and Chemical Engineering, Curtin University, Perth, Australia</footnote>
In dense, cohesionless granular packings, macroscopic mechanical behavior is governed by the complex interplay of elastic deformation and interparticle friction at the contact level. These interactions give rise to a highly heterogeneous distribution of contact forces, forming a complex “contact network”. This network is widely recognized [5, 6] to be bimodal: a relatively small fraction of “strong” contacts (those bearing normal forces exceeding the mean) supports the majority of the applied stress, forming a load-bearing skeleton. This skeleton is stabilized by a percolating network of “weak” contacts. Experimental and numerical investigations show that in some cases, fewer than 40% of contacts in dense packings can sustain as much as 90% of the total deviatoric stress [5, 7, 8].

The evolution of this force-bearing network under external loading is intrinsically linked to the loading history of the material and is path-dependent. The network configuration may be different even when the macroscopic stress is the same. Local force chains, the quasi-linear paths of stress transmission, may buckle, fragment, or reform, thereby altering both the contact network topology and the elastic energy stored at individual contacts [9–11]. Various descriptors, such as contact orientation, coordination number, fabric anisotropy, and topological connectivity metrics, have been employed to characterize the evolving microstructural fabric of granular assemblies [12–15]. However, these geometric and topological measures, while informative, do not fully quantify the progressive redistribution of stored potential energy at the microscale, which is crucial for understanding the internal energy process for structural rearrangements.

This study introduces a complementary, energy-based perspective on granular systems through the concept of configurational energy. Configurational energy is defined as the portion of elastic potential energy stored within a granular system that remains “frozen” in the contact network due to interparticle friction, anchoring the system in a metastable configuration. This energy is not immediately dissipated but is instead retained within the structure as a result of frictional constraints that prevent immediate release. When the system is perturbed, e.g., through external loading or unloading, the configurational energy associated with locally asymmetric contact force can exceed the frictional resistance barrier, triggering contact reconfigurations. The release of configurational energy manifests primarily as frictional dissipation, thereby contributing to irreversible deformation and entropy production. This energy quantifies the topological and energetic heterogeneity within the contact network, serving as a scalar state variable that indicates the structural memory of the granular assembly. While the total configurational energy of a system is not directly measurable and remains a subject of ongoing investigation, its production rate can be robustly evaluated using small, quasi-static cyclic loading perturbations, referred to here as “stress probes”, applied to the bulk material. Production of configurational energy after a stress probe captures the elastic energy that is previously locked within the metastable state, which can be partly released and contributes to entropy production. Although elastic materials typically recover their global stress state following small perturbations, where a closed incremental stress cycle yields zero net dissipation, granular materials violate this expectation. Despite macroscopic reversibility, irreversible processes such as local grain rotations, micro-slips at frictional contacts, and topological rearrangements redistribute

the internal elastic energy landscape. These mechanisms reflect the inherently inelastic and path-dependent nature of granular systems, which continuously self-organize even under quasi-static loading. As a result, the granular assembly departs from thermodynamic equilibrium.

In this work, we employ controlled stress probes along biaxial loading paths to quantify changes in configurational energy—defined as the elastic energy released or restructured due to internal microstructural adjustments—despite the recovery of global stress state. While it remains an open question whether a global, path-independent configurational energy function can be formulated through integration of these incremental changes, our findings show that configurational energy production offers a robust and measurable indicator of microstructural evolution and internal inelasticity.

This paper aims to demonstrate that the evolution of configurational energy provides critical insight into the mechanical state of granular systems and the early signs of failure. High configurational energy production reflects structural heterogeneity and signals that the system is actively undergoing internal reconfiguration. The presence of friction gives rise to a heterogeneous energy landscape in which part of the elastic energy becomes trapped in metastable configurations. Configurational energy thus captures the structural readiness of the system to dissipate energy, though its conversion into frictional dissipation may be delayed. In this sense, configurational energy offers an energetic perspective on internal structural and energetic “disorder”, emphasizing the role of evolving microstructure in driving irreversible material behavior.

The investigation herein employs Discrete Element Method (DEM) simulations to explore this energy-based framework. DEM simulations of biaxial compression are used to investigate how configurational energy evolves across different stages of loading. Our results reveal that configurational energy not only captures the sensitivity of the system to perturbations but also exhibits early spatial localization preceding the formation of macroscopic shear bands. Moreover, the spatial decay of configurational energy fluctuations provides an intrinsic length scale that characterizes the extent of nonlocal mechanical interactions, offering a new mesoscopic basis for modeling the onset of shear localization.

Section 2 elaborates on the theoretical underpinnings of configurational energy, contrasting it with traditional energy partitioning in continuum mechanics and linking it to concepts from configurational mechanics. Section 3 details the numerical setup, including the DEM model parameters and the stress probe protocol used to quantify the production of configurational energy. Section 4 presents results on the contact-scale response, examining correlations between configurational energy and various contact properties. It also focuses on the macroscopic and spatial evolution of configurational energy, demonstrating its localization as a precursor to shear banding. Section 5 discusses the implications of these findings, particularly regarding nonlocal effects [16–19] and the potential for defining an energy-based length scale. Finally, Section 6 summarizes the key contributions and outlines future research directions. This energy-based approach enriches our understanding of granular deformation
<figure><img src="image_3.png" /><figcaption>FIGURE 1 | A stress probe following Cᵢ produces configurational energy ΔE<sup>Cᵢ</sup><sub>conf</sub>. Given two opposite directions following biaxial loading paths, C₁ and C₂, the ΔE<sup>Cᵢ</sup><sub>conf</sub> will be different. A Cyclic stress probe is also applied in this study.</figcaption></figure>

beyond classical stress—strain descriptions, offering a new lens through which to interpret plasticity, memory, and the incipient stages of failure.

## 2 | Conceptual Framework

In classical continuum mechanics, the incremental work input (δw) during material deformation is typically partitioned into recoverable (free) energy (δψ) and dissipated energy (δφ): δw = δψ + δφ. The term ψ usually represents stored elastic energy, while φ accounts for energy dissipated through irreversible processes like frictional sliding. However, discrete granular materials exhibit complex mechanical behavior that necessitates a multiscale perspective, particularly when microstructural features such as force chains and voids exist at length scales distinct from those of macroscopic loading [20].

During the deformation of a granular assembly, work input stores potential energy at inter-particle contacts. This part of energy is often regarded as elastic, i.e., reversible. At the same time, plastic work arises primarily from frictional slip, while irreversible reorganization of the contact network (e.g., contact breakage, formation, or rolling) also contributes to entropy production and energy evolution in the system. Crucially, as noted by Collins [21] and others [9, 22–24], a portion of the input energy in granular materials can become structurally “frozen” in metastable configurations, contributing to delayed frictional dissipation. This retained energy exists because frictional resistance immobilizes contact sliding below the Coulomb threshold and prevents immediate energy release. If the Coulomb criterion is eventually reached, such energy can be released through sliding, becoming frictional dissipation. This portion of energy will be referred to as configurational energy. Such metastable configurations, including highly strained force chains or locked particle clusters, may persist even after the removal of the external deviatoric load. In this study, we define the production of configurational energy as the change in stored elastic energy at contacts following a closed stress probe—i.e., a loading—unloading cycle that restores the macroscopic stress state (see Figure 1). For stress probe C₁, after loading, the potential energy of contact c will have a change

ΔE<sup>+</sup><sub>c,C₁</sub> from initial state. After unloading, it changes with ΔE<sup>-</sup><sub>c,C₁</sub> from the loaded state. The production of configurational energy for this contact will be defined by:

$$ \Delta E_{\text{conf}}^{c,C_1} = \Delta E_{c,C_1}^{+} + \Delta E_{c,C_1}^{-} \quad (1) $$

ΔE<sup>+</sup><sub>c,C₁</sub> is defined by ΔE<sup>+</sup><sub>c,C₁</sub> = E<sup>after</sup><sub>c,e</sub> - E<sup>initial</sup><sub>c,e</sub>, E<sup>after</sup><sub>c,e</sub> and E<sup>initial</sup><sub>c,e</sub> represent the elastic potential energy of a contact c after this loading process and the initial value, respectively. Note that all three values can be either positive or negative for a contact with the unit of J, showing how potential energy for this contact evolves during a stress probe. Production of configurational energy of the system is the sum over all contacts, i.e., ΔE<sup>C₁</sup><sub>conf</sub> = Σ<sub>c</sub> ΔE<sup>C₁</sup><sub>conf</sub>. For stress probe C₂, the production of configurational energy will not be the same.

The notion of configurational energy resonates with the broader framework of configurational mechanics [25], particularly the concept of configurational forces introduced by Gurtin and others [26, 27]. Configurational forces arise from the variation of the total energy of a system with respect to changes in the material configuration(e.g., evolution of defects, phase boundaries), rather than spatial displacement. In granular materials, these forces can be associated with the energetic drive for the evolution of the contact network’s topology and the distribution of stored elastic energy, thereby influencing the system’s path-dependent behavior and its response to further loading. Even if a small probe cycle returns the system to the same macroscopic stress state, local irreversible rearrangements (particle rolling, microslips) modify the granular fabric and redistribute stored elastic energy at contacts. A positive value signifies net storage due to reconfigurations into higher local energy states, while a negative value would imply a net release into more relaxed local states.

To provide a micromechanical intuition for configurational energy, consider a simplified model of a block (representing a grain) constrained by springs and subject to friction, as depicted in Figure 2. Block 4 is elastically constrained by three springs with stiffness k₁, k₂, and k₃, exerting forces F₁, F₂, and F₃, respectively. Springs 1 and 2 are anchored at one end, while Spring 3 can be displaced vertically at its free end, simulating an external perturbation. The block is also in contact with a surface characterized by a friction coefficient μ. This system is a highly simplified representation of the frictional and elastic constraints experienced by a grain within a granular assembly.

While the current definition of configurational energy emphasizes translational force changes at contacts, we acknowledge that rotational rearrangements, including grain spin and contact normal reorientation, can also contribute to the redistribution of stored elastic energy. These effects were not explicitly tracked in the present study due to the use of circular particles and a linear contact model. However, including rotational energy contributions would refine the configurational energy metric, particularly in systems where rolling and angular motion play a dominant role in fabric evolution.

A simplified analysis of this model (detailed in Appendix A) shows that after an unloading-loading cycle of displacement Δ applied to Spring 3, the change in energy stored in Springs 1 and 2
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**FIGURE 2** | (a) A conceptual block-spring-friction system illustrating frictional contact interactions. An unloading with displacement perturbation $\Delta$ is applied to the free end of Spring 3 (for elastic springs, controlling displacement or force makes no difference). The block will leave with an unrecoverable displacement $\Delta_s$. Energy redistribution occurs in Springs 1 and 2 due to potential slip at the frictional contact when the jump of $|F_1 - F_2|$ becomes larger than $\mu F_3$. The diagram illustrates how a vertical perturbation on Spring 3 can induce changes in horizontal forces and potential energy stored in Springs 1 and 2, mediated by friction; (b) Energy landscape for the springs. After a perturbation, the forces of the two springs move towards $F_1 = F_2$, and the total energy of two springs reaches a local minimum.</figcaption></figure>

(representing the configurational energy change in this part of the system) is influenced by the initial pre-compression $F_3$ in Spring 3, the mismatch between $F_1$ and $F_2$, and the frictional properties. Specifically, if Spring 3 is heavily pre-compressed (analogous to a strong contact), the displacement upon unloading might be small (compared to energy release in Spring 3), potentially leading to a small change in stored energy in Springs 1 and 2 if slip occurred. Conversely, if $F_3$ is small (analogous to a weak contact), force that constrains the sliding is small. With a relatively large unloading displacement, potentially a net increase in energy stored in Springs 1 and 2 would occur.

The configurational force, defined as the variation of the configurational energy $U$ with respect to changes in the material configuration, is given by:

$$F_{\text{conf}} = \frac{\Delta U}{\Delta_s} = -\mu F_3 + \frac{1}{2}\mu k_3 \Delta \quad (2)$$

differs from the change in the constraining force of the system, $-\mu k_3 \Delta$ (see Appendix A). The configurational force is a generalized thermodynamic force, whose conjugate generalized thermodynamic variable is the displacement $\Delta_s$. The configurational force acts as a resistance to reconfiguration, which would transmit to the interconnected block-spring-friction systems, initiating energy propagation within the contact network. As a result, local rearrangements in the granular system cascade through the contact network, leading to a complex macroscopic response to perturbations.

Analytically deriving the distribution and evolution of configurational energy in a full granular system is a formidable task due to inherent non-linearity, high dimensionality, and the discrete nature of interactions (contact opening/closing, grain movements). For example, $\Delta$ and $\Delta_s$ are coupled for every contact. To overcome these challenges, we employ the Discrete Element

Method (DEM) to directly simulate and quantify energy redistribution and configurational changes within granular assemblies.

## 3 | Numerical Methodology

Numerical simulations were performed using YADE (Yet Another Dynamic Engine), an open-source DEM framework [28]. The study focuses on a 2D assembly of 5000 circular particles with a uniform distribution with diameter between 0.0043 and 0.0047 m. The initial specimen dimensions were 0.75 m × 0.5 m. Particle density was set to 3000 kg/m³, elastic modulus to $E = 300$ MPa, and the contact normal stiffness is given by $k_n = E \frac{r_1 r_2}{r_1 + r_2}$, given the radius of the contact particles as $r_1$ and $r_2$. The shear stiffness $k_t = 0.5 k_n$. The inter-particle friction angle was 35°. Stresses are reported in kPa under the plane-strain assumption with out-of-plane thickness set as the maximum of grain diameter, consistent with standard practice in 2D DEM studies. The loading rate is carefully controlled to maintain quasi-static conditions; as a result, the energy dissipated by numerical damping remains more than an order of magnitude smaller than the measured configurational energy changes, confirming that the latter reflect physical microstructural processes rather than numerical artifacts.

The granular assembly was first subjected to a biaxial compression loading path under a constant confining pressure of 100 kPa. During this primary loading, representative grain configurations were extracted at selected global stress states (instantaneous configuration of all particles, including their positions, rotations, and contact forces, at particular strains during the loading path). At each chosen state, a small cyclic loading perturbation “stress probe” was applied to the system. The stress probe consisted of further biaxial loading by 0.5 kPa (deviatoric stress increase) followed by unloading back to the original stress state before the probe. The probe amplitude was chosen to be sufficiently small (0.5% of confining pressure) to induce measurable internal
rearrangements and energy redistribution without causing large-scale plastic failure or significantly altering the macroscopic stress state from which the probe was initiated.

Inter-particle contact forces were governed by a linear elastic-frictional model. The elastic potential energy $E_e^c$ stored at an individual contact is calculated as:

$$E_e^c = \frac{\|\mathbf{F}_n\|^2}{2k_n} + \frac{\|\mathbf{F}_s\|^2}{2k_s} \qquad (3)$$

where $\mathbf{F}_n$ and $\mathbf{F}_s$ are the normal and tangential contact force vectors, and $k_n$ and $k_s$ are the normal and tangential contact stiffness, respectively. And the configurational energy for a specific contact after one complete stress probe cycle is thus defined. The notation of $\delta E_{conf}^c$ will be used hereafter to represent the production of configurational energy for contact, and $\delta E_{conf}$ for the system.

This definition captures how microstructural rearrangements during the probe lead to irreversible redistribution of stored elastic energy at contacts. A positive value indicates local energy storage due to reconfiguration into higher-energy states, while a negative value reflects local energy release that transit the system towards the minimal energy configuration.

The dissipation for a contact is defined by the work done by the frictional force on the relative displacement between the contact. Rotation $\delta r$ and sliding $\delta s$ at each contact produce a relative displacement between grains, given by $\delta_c = u_i^n - u_i^p + e_{ijk}(\omega_j^n r_k^n - \omega_j^p r_k^p)$, where $u$, $\omega$, and $r$ represent the displacement, the rotation, and the radius of the grains that form the contact [29]. Thus, the dissipation at the contact will be $\delta E_{fric}^c = \mu^{frie} F_c^n \delta_c$. We propose that configurational energy drives frictional dissipation in granular systems, as its irreversible release during microstructural adjustments contributes to localized entropy production at frictional contacts.

The macroscopic mechanical response of the material followed typical behavior for dense frictional granular materials, exhibiting initial contractive volumetric strain, followed by yielding and a transition to dilative behavior accompanied by strain-softening post-peak stress. The stress states chosen for detailed analysis of configurational energy are indicated in Figure 3.

Due to the large number of interparticle contacts (around tens of thousands), direct visualization of all individual contact-level data can obscure underlying trends. To address this, contact-level data are grouped within geometrical cell as in Figure 4. Each group is typically chosen to contain a similar number of data points (e.g., 100 contacts). In the resulting plots, each data point then represents the mean value for that group, and the shaded region around it denotes a confidence interval (e.g., 95%) with standard deviation of the distribution within that group, reflecting statistical variability.

## 4 | Evolution of Configurational Energy

The analytical framework presented in Section 2, based on a simplified spring-frictional block-spring system, highlights

<figure><img src="image_6.png" /><figcaption>FIGURE 3 | Macroscopic mechanical response: volumetric strain (red) and deviatoric stress (black) versus axial strain during biaxial compression. Stress states in Figure 8 are marked as blue, showing two reference states with axial strain 0.11% and 0.98%. Red dashed lines represent three reference states where the specimen reaches maximum volumetric contraction, maximum deviatoric stress, and critical state marked by the end of softening, respectively. The same reference lines are used throughout this manuscript.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 4 | Spatial coarse-graining of contact-level configurational energy production. Data are taken from the snapshot in Figure 8e and provide data point in Figure 7. (a) Each line represents a contact, colored by its local rate of configurational energy production, where red represents positive values and blue for negative ones. (b) The same data averaged over adaptive spatial cells; each pixel shows the mean value within the corresponding cell.</figcaption></figure>

that the origin of contact-level configurational energy ($\delta E_{conf}^c$) is intrinsically linked to the presence of frictional contacts near Coulomb threshold. In a granular assembly, however, the complexity increases significantly. Relative movements between grains not only activate frictional mechanisms but also dynamically alter the local spring-like constraints on each contact. These inter-grain movements also serve as internal perturbations, propagating through the contact network and generating a series of configurational energy responses. This section investigates the evolution of $\delta E_{conf}^c$ by dissecting its characteristics at the
<figure><img src="image_8.png" /><figcaption>FIGURE 5 | Distribution of normal force and potential energy for contacts, main values are shown as vertical dashed lines. Data shown is for an early loading stage (e.g., axial strain 0.11%) and another one near the peak deviatoric stress(e.g., axial strain 0.98%). The distribution is typical for frictional granular materials, as discussed in [30]. Plots are normalized by the mean value of normal contact force/potential energy of contacts for stress state at axial strain 0.11%, thus the red reference line is located at 1 in x-axis. The energy distribution has a right shift approaching the peak deviatoric stress, given by a more than three times bigger average potential energy, shown as the green dashed line.</figcaption></figure>

contact scale and then examining its collective manifestation and spatial organization at the specimen scale. An analogy to entropy is drawn to interpret the fact that configurational energy is a measure of disorder of the energy distribution within the contact network.

## 4.1 | Contact-Scale Results

In the DEM simulations, the application of a stress probe induces internal stress redistributions that drive local reconfigurations. Although the simplified model (Figure 2) suggested that normal unloading of weakly loaded springs may be more prone to generate positive configurational energy upon perturbation, reality in a granular system involves a complex interplay of normal and tangential constraint changes for each contact and the coupling between perturbation $\Delta$ and induced displacement $\Delta_s$. Consequently, both positive $\delta E_{conf}^c$ (indicating a local increase in stored elastic energy at a contact after the probe) and negative $\delta E_{conf}^c$ (a local release of stored elastic energy) are observed throughout the assembly.

The distinction between “strong” and “weak” contacts, traditionally based on normal force magnitudes [5], can be extended by considering their initial stored elastic potential energy, $E_e^c$ (Equation 3). Figure 5 confirms that the potential energy $E_e^c$ and normal contact force $F_n$ for a contact share a distribution similar to exponential decay for strong contacts and a power-law-like behavior for weak contacts.

We extend the block-spring model to granular assemblies by considering grain-scale contact interactions and their energetic response under mechanical perturbations. Contacts with high initial elastic energy $E_e^c$, referred to as energetically “strong,” behave analogously to heavily pre-compressed springs and form the skeleton of the force-bearing network. Energetically “weak” contacts, with lower $E_e^c$, are more susceptible to rearrangement. The total elastic energy increment $\delta E_{el}$ for the system can be decomposed into two components:

$$ \delta E_{el} = \delta E_{conf} + \delta E_{rec}, \quad (4) $$

where $\delta E_{rec}$ is the recoverable elastic energy, and $\delta E_{conf}$ is the unrecoverable configurational energy. When these constraints of the frictional resistance are mobilized—e.g., via contact sliding induced by external perturbation—reconfiguration may occur, causing a release or accumulation of elastic energy $E_{el}$ at individual contacts. If two contacts on the same grain have significantly different magnitudes, the strong one has the potential to release configurational energy during perturbation, and will release when sliding occurs in the neighborhood of the grain considered. Therefore, the energy balance at contact scale will be written:

$$ \delta_c \cdot f_c = \delta E_{rec}^c + \delta E_{conf}^c + dE_{fric}^c \quad (5) $$

$\delta_c$ and $f_c$ represent the local configurational evolution and the configurational force, respectively. The generalized work provided by this work can be decomposed into three parts, where $\delta E_{rec}^c$ and $\delta E_{conf}^c$ might not be zero at contact scale, but $dE_{fric}^c$ is always non-negative. This equation shows that configurational evolution is always associated with friction dissipation and redistribution of elastic energy within the contact network. Figure 6 provides distribution of potential energy as well as the production of configurational energy with a granular system. When local rearrangement takes place, the local stored configurational energy will be released and generate frictional dissipation.

Although the volume of the specimen, the stress ensemble, and the distribution of the contact network have been considered as descriptions for the entropy of a granular system [31, 32], each of them alone is insufficient [33]. We propose that configurational energy of a granular system is directly linked to the configurational entropy of the system through $\delta E_{conf} = -T_{conf}dS_{conf}$. An increase in configurational energy corresponds to a decrease in configurational entropy—signaling a gain in structural order. And total entropy increment of the system is given by $dS = dS_{fric} + dS_{conf} \cdot dS_{fric} > 0$ is due to frictional dissipation. If $dS_{conf} < 0$, the total entropy production is sub-maximal, highlighting that the system is thermodynamically out of equilibrium. This condition—enabled by the existence of frozen energy—suggests that granular materials delay full relaxation, and instead evolve through partially arrested internal states.

As Figure 6 shows, strong contacts typically exhibit $\delta E_{conf}^c < 0$ as they are perturbed, releasing frozen energy and introducing local disorder. In contrast, weak contacts frequently exhibit positive increments in configurational energy, $\delta E_{conf}^c > 0$, as they absorb part of the energy released by stronger contacts. This duality governs the microstructural evolution of the granular network and drives emergent behavior, wherein the redistribution of energy is inseparable from structural rearrangements.
<figure><img src="image_9.png" /><figcaption>**FIGURE 6** | Local distribution of production of configurational energy at axial strain 0.98% (red represents positive $\delta E_{conf}^c$, blue represents negative ones, and gray represents values close to 0). Local configurational evolution $\delta_c$ initiates change of configurational energy, which is associated with frictional dissipation. On the other hand, such evolution will be restrained within a limited length by the friction among grains. Providing a length scale that will be discussed in Figure 14.</figcaption></figure>

This can be formulated in a thermodynamic framework where configurational energy and entropy are inversely related:

Thus, configurational energy production serves as a scalar descriptor linking contact-scale irreversibility to system-wide mechanical behavior, bridging the gap between structural memory and entropy production. It reveals how energy is trapped and subsequently released across a contact network, shaping the granular assembly's progression toward mechanical instability.

Figure 7 explores correlations between $\delta E_{conf}^c$ and other contact-level properties, the contact potential energy $E_e^c$, the displacement between two grains forming the contact $\Delta_c$, and the frictional dissipation between two grains during relative sliding $\delta E_{fric}^c$, and $dS_{fric} \sim \sum_c \delta E_{fric}^c$. At early loading stages (axial strain 0.11%), Figure 7a shows that the magnitude of $\delta E_{conf}^c$ (both positive and negative) correlates with the initial $E_e^c$ of the contact; more energized contacts tend to experience larger changes. This suggests their active involvement in microstructural adjustments. Figure 7b reveals a strong positive correlation between the magnitude of relative inter-particle displacement during the probe and the magnitude of $\delta E_{conf}^c$. The results are consistent with the frictional block-spring system, though $\Delta$ and $\Delta_s$ are coupled. However, with same displacement, the $\delta E_{conf}^c$ in strong contact is bigger than in weak contacts. This agrees with Equation (2), that strong contacts also process a bigger configurational force.

Figure 7c reveals a complex correlation between the incremental configurational energy, $\delta E_{conf}^c$, and the dissipated energy, $\delta E_{fric}^c$, in granular contacts. Generally, the production of $\delta E_{conf}^c$ leads to local frictional dissipation. In the elastic regime, frictional

dissipation primarily results from the reduction in $\delta E_{conf}^c$ at contacts (with equivalent magnitude), consistent with the frictional block-spring system. Here, strong and weak contacts exhibit distinct behaviors, with strong contacts dissipating more energy. However, as the material approaches peak deviatoric stress, as shown in Figure 7f, such distinction diminishes, with strong and weak contacts contributing comparably to the dissipation process and the magnitude of dissipation is bigger than that of the configurational energy release. Additionally, Figure 7d indicates a weakening correlation between a contact's initial configurational energy, $E_e^c$, and its subsequent $\delta E_{conf}^c$.

This diminishing local correlation suggests a transition: the system's response becomes less dictated by local contact states and more governed by collective evolution of the network system. Rearrangements may propagate over longer distances, hinting at an increasing characteristic length scale of these energetic interactions. Such propagation implies the emergence of non-local effects, where local perturbations influence distant regions through the evolving contact network. This supports the idea of a mesoscopic length scale associated with configurational energy, which we explore further in Section 5.

As the system stores more elastic energy, particularly in shear components, and energetic heterogeneity grows, local instabilities become more probable. This can lead to more frequent and significant instances of negative $\delta E_{conf}^c$ (local energy release), even as other contacts accumulate positive $\delta E_{conf}^c$, signaling an internal structural evolution towards larger-scale yielding.

## 4.2 | Specimen-Scale Results

To investigate the broader implications of contact-scale dynamics and its emergent behavior, we now shift our analysis to the specimen scale, focusing on how the production of configurational energy is distributed in the system and how it evolves over different macroscopic stress.

Under a stress probe of constant magnitude, the contact network topology remains relatively stable during loading, yet the spatial distribution of configurational energy production evolves dynamically. As shown in Figure 8b,e, the change in configurational energy field is initially broadly distributed and correlated with the global contact structure. At the specimen scale, this incremental energy distribution later has a distribution similar to micro shear bands [34–36] and later becomes increasingly concentrated in regions where a shear band forms at a later stage, even before macroscopic localization or visible shear bands emerge, i.e., the spatial distribution of local dissipation (Figure 8e,f). As the material approaches peak deviatoric stress, intensified contact formation and disconnection occur predominantly in regions of significant configurational energy evolution, indicating a strong correlation between local energy dynamics and the emergence of specimen-scale instabilities.

The spatial distribution of the initial potential energy, $E_e^c$, maps the local energetic variations throughout the granular contact network, as illustrated by the heterogeneous distribution in Figure 8a,c. As the stress state increases during loading, the total elastic energy stored in the granular system rises, reflected in
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>**FIGURE 7** | Relationship between the configurational energy $\delta E_{conf}^c$ developed at a contact after a 0.5kPa unloading-loading stress probe along the biaxial loading path, with (a,d) its initial elastic potential energy $E_c^e$, (b,e) the magnitude of the relative displacement between the contacting particles during the probe $\Delta_c$, and (c,f) the frictional dissipation during the unloading process. (a, b, c) for stress probe at axial strain 0.11%, while (d, e, f) at axial strain 0.98%. Both stress states are marked in blue in Figure 3. Each point represents group means (around 100 contacts per group within the cell shown in Figure 4); shaded areas are 95% confidence intervals of the dataset for a given cell. Fitted curves are given to show the linear and power-law-like relations. Units for both axes of all figures will be J.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**FIGURE 8** | (a, b, c) for stress probe at axial strain 0.11%, while (d, e, f) at axial strain 0.98%. Both stress states are marked in blue in Figure 3. The (a, d), (b, e) and (c, f) represent the potential energy $E_e^c$, the production of configurational energy $\delta E_{conf}^c$, and the dissipated $\delta E_{fric}^c$ for a contact. (a,d) depict the value of current state, whereas (b, c, e, f) depict incremental values before and after a stress probe. Purple and green plots represents the location where new contact forms or existing contact disappears. There is a strong correlation between the distribution of frictional dissipation and the production of configurational energy. All values in the figure are shown with unit of J.</figcaption></figure>

a rightward shift of the energy distribution (see Figure 5). This increase corresponds to a gradual buildup of potential energy in contacts forming a percolation network aligned with the deviatoric stress direction.

Building on the specimen-scale concentration of configurational energy in regions of incipient shear band formation, we now analyze how this energy evolves and its implications for shear band development. As shown in Figure 1, stress probes can be applied in various directions, including cyclic probes, which

means several loading/unloading cycles along the same loading path. Figures 9 and 10 illustrate that the production of configurational energy (positive magnitude means that configurational energy is stored within the system) stabilizes after approximately 20 stress probe cycles, with the rate of production gradually decreasing thereafter, indicating a steady-state response where additional configurational energy cannot be produced.

Building on the observed stabilization of configurational energy production under cyclic stress probes, we now examine the effect
<figure><img src="image_17.png" /><figcaption>**FIGURE 9** | Production of configurational energy with a stress probe 0.5 kPa at axial strain 0.98%. Unloading first produces smaller configurational energy. Either direction, the production becomes minimal after 10 cycles. The value after 10 cycles will be used as the production of configurational energy at given stress state.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**FIGURE 10** | $\delta E_{conf}$ for different stress state and probe magnitude. $\delta E_{conf}$ increases approaching the peak deviatoric stress; $\delta E_{conf}$ converges to a consistent value regardless of the probe magnitude.</figcaption></figure>

of varying probe magnitude along an unloading-first path. Three different magnitudes are used in the simulation(0.25, 0.5, and 1 kPa). When the probe magnitude is altered, the incremental configurational energy, $\delta E_{conf}$, increases significantly as the material approaches peak deviatoric stress. Notably, the probe magnitude affects only the first several cycles; after approximately 10 cycles, $\delta E_{conf}$ converges to a consistent value regardless of the probe magnitude, indicating a robust steady-state response in the configurational energy evolution.

Following the observed convergence of $\delta E_{conf}$ under varying probe magnitudes, we now contrast the behavior of granular materials with an ideal elastic system. In a perfectly elastic, ideal material, a stress probe yields $\delta E_{conf} = 0$, as the total configurational energy, $E_{conf}$, is zero, analogous to a spring-block system where no frictional sliding occurs, and elastic energy is fully

recovered post-probe. Thus, a smaller $\delta E_{conf}$ indicates a material closer to ideal elastic behavior. Granular materials, however, are inherently dissipative due to irreversible rearrangements. Figure 11 illustrates the evolution of configurational energy production across different stages of a biaxial loading test, where each point represents a metastable state selected as a snapshot from the test, subjected to cyclic stress probes.

To further elucidate the configurational energy dynamics observed in the biaxial loading test, we expand our analysis of Figure 11, which illustrates the evolution of incremental configurational energy, $\delta E_{conf}$, across various stress states. First, $\delta E_{conf}$ is consistently negative with a stress-controlled probe, reflecting the irreversible nature of configurational energy production and its link to entropy production. As shown in Figure 11b, $\delta E_{conf}$ converges to a steady-state value after approximately 10 cycles for all stress states, indicating a stabilization of entropy production. Second, even with a single cycle, the magnitude of the stress probe has a negligible effect on $\delta E_{conf}$, suggesting that the configurational response is determined by the configuration of the contact structure rather than probe intensity. Third, $\delta E_{conf}$ exhibits a pronounced increase, or burst, as the material approaches the peak deviatoric stress, but remains relatively low at the critical state, highlighting distinct energetic behavior at these stages. Fourth, the buildup of $\delta E_{conf}$ is minimal in both the elastic regime and the critical state.

This negative total configurational energy production can be interpreted in terms of an increase in a form of “configurational order” for the system as a whole. As illustrated in Section 2, a local probe applied to a frictional spring-block system reduces the potential energy difference between adjacent springs, leading to the release of configurational energy and the production of entropy through frictional dissipation.

Building on the observed evolution of $\delta E_{conf}$ across stress states, we now return to the formation of shear bands, focusing on when and where they would take place. Our prior analysis established a correlation between $\delta E_{conf}^c$ and local dissipation. As the material approaches failure, $\delta E_{conf}$ of the system gradually increases, reflecting heightened irreversible rearrangements. Recalling the frictional block-spring system, negative $\delta E_{conf}^c$ corresponds to the generation of frictional dissipation. To further explore this relationship, we analyze the correlation between $\delta E_{conf}$ and dissipation at the specimen scale. Figure 12 illustrates the distribution of $\delta E_{conf}$ inside and outside the shear band, where the shear band domain is defined by regions exhibiting significant strain after reaching the critical state. This figure includes not only $\delta E_{conf}$ and dissipation but also the negative fraction of configurational energy, $\delta E_{conf}^-$, defined as the sum of all negative $\delta E_{conf}^c$ contributions, which, in the frictional block-spring system, is associated with frictional dissipation. These plots reveal that regions within the shear band exhibit pronounced $\delta E_{conf}$ and $\delta E_{conf}^-$, underscoring their role in driving dissipative processes and shear band development.

Following the specimen-scale investigation between $\delta E_{conf}$ and dissipation inside and outside shear band region, we observe a change in the configurational energy response prior to material failure, signaling the onset of shear band formation. After failure, configurational energy production outside the shear
<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>FIGURE 11 | $\delta E_{conf}$ for different stress states with different cyclic stress probes. (a) $\delta E_{conf}$ after one unloading-loading cycle; (b) after 10 cycles, and (c) the buildup from cycle 1 to cycle 10. Reference lines are the same as Figure 3. The production of configurational energy gradually increases as the material approaches peak deviatoric stress. The magnitude of the stress probe has negligible influence on the configurational response of the material.</figcaption></figure>

band becomes negative, while dissipation outside the band remains minimal, indicating that energy from outside the shear band is transferred into the band, driving its continued deformation. Figure 13 further illustrates that the negative fraction of configurational energy, $\delta E_{conf}^-$, predominantly promotes frictional dissipation within the shear band.

Importantly, the spatial localization of configurational energy precedes the onset of shear banding. This localization acts as a mesoscale precursor to failure, detectable long before the formation of macroscopic strain localization. The spatial decay of $\delta E_{conf}^c$ fluctuations defines a characteristic length scale $L_S^{conf}$, which characterizes the extent of nonlocal mechanical interaction. This intrinsic length scale emerges from the tendency of the system to store energy irreversibly during microstructural evolution and offers a new route to quantify non-locality in granular mechanics.

5 | Discussion

The results demonstrate that the production of configurational energy, $\delta E_{conf}$, serves as a sensitive indicator of microstructural evolution and incipient localization in dense granular materials. A negative $\delta E_{conf}^c$ at a specific contact signifies that this particular contact has released some of its frozen energy and settled into a locally more stable and “less strained” state. Negative $\delta E_{conf}$ for the system suggests that configurational energy is consistently released from the system, contributing to an irreversible structural evolution toward stability.

Although individual contacts may undergo both increases and decreases in potential energy following a stress perturbation, the specimen-scale response consistently exhibits a net release of configurational energy. We find that the microstructural relaxation associated with configurational energy is spatially localized: neither positive nor negative fluctuations propagate throughout the entire contact network. Local sliding events, which are initiated by release of configurational energy with production of frictional entropy, can only percolate along the contact network with a limited range confined by frictional resistance, and sequential grain compression is similarly restricted. Accordingly, a characteristic length scale, $L_{Sconf}$, can be defined to quantify the spatial influence of local configurational entropy variations. Beyond this scale, the energetic impact of local rearrangements becomes negligible.

That is, while smaller regions might exhibit a net “ordering” (summed negative $dS_{conf}$), $L_{Sconf}$ would be the scale at which the system’s tendency towards overall positive (or at least non-negative) “configurational entropy production” due to irreversible rearrangements becomes dominant. This aligns with the broader concept of non-locality in granular matters, where the response at a point depends not only on the local mechanical state but also on surrounding structural adjustments. This thereby gives the length scale beyond which the influence of local “ordering” (negative $dS_{conf}^c$ events) is statistically averaged out by the broader tendency of the perturbed system to explore new configurations.

A more systematic investigation is needed to precisely quantify this length scale; however, measurements can currently be
<figure><img src="image_22.png" /><figcaption>**FIGURE 12** | (a) Intuitive diagram to show the concentration of configurational energy within the shear band. Different color shows the magnitude of local average of configurational energy. Red represents positive and blue for negative. Diffusion of such energy can be observed, indicating a possible gradient-flow mechanism of energy propagation. Evolution of (b) $\delta E_{conf}$; (c) its negative fraction $\delta E_{conf}^-$; and (d) frictional dissipation, in different stages. Reference lines are the same as Figure 3.</figcaption></figure>

conducted at the contact scale by investigating how negative and positive $dS_{conf}^c$ events percolate in the contact network. Figures 8 and 6 show that neither negative nor positive $\delta E_{conf}^c$ contributions percolate through the entire system, indicating a localized configurational response. Consistent with the micromechanical concepts described in the frictional block-spring system, perturbations have a finite range of influence, limiting their influence to local contact interactions and constraining the propagation of configurational energy changes across the granular assembly.

To quantify the localized configurational response observed in Figure 6, we track the mean length of percolation chains for positive and negative $\delta E_{conf}^c$ contributions by separating the contact network into two subnetworks, one consisting of contacts with positive $\delta E_{conf}^c$ and the other with negative $\delta E_{conf}^c$. Therefore, each subnetwork consists of a number of isolated chains in which all contacts are with either positive or negative $\delta E_{conf}^c$. The length of each contact chain can be obtained as the isolate chain length.

The average chain length in each sub-network serves as a measure of the configurational length scale, $L_{S_{conf}}$, reflecting the spatial extent of correlated configurational energy changes.

Figure 14 shows that $L_{S_{conf}}$ exists. This length scale represents how far the influence of a local configurational event might propagate. Quantitatively, our observations suggest that the influence of local negative configurational energy starts at sub-particle scales during early loading stages but evolves to span several particle diameters as the system approaches failure. In advanced loading states, this influence extends to approximately five times the average grain diameter, a range that overlaps with typical shear band thicknesses reported in the literature (5d – 15d) [2, 37–39].

The local positive $\delta E_{conf}^c$ initially exhibits a broader spatial extent, suggesting that the storage of the configurational energy propagates further along the contact network compared to the negative $\delta E_{conf}^c$. However, this extended length scale converges
<figure><img src="image_23.png" /><figcaption>**FIGURE 13** | Correlation between negative fraction of configurational energy production $\delta E_{conf}^-$ and frictional dissipation. Reference lines are the same as Figure 3.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**FIGURE 14** | The evolution of average chain length for positive and negative $\delta E_{conf}^c$, the length is given by the average diameter of grain. Reference lines are the same with Figure 3. It can be observed that as the material approaches failure, there is a decrease in chain length for positive $\delta E_{conf}^c$ while an increase for negative ones. Both of them reach a limit before material failure, and are constrained by an upper and lower bound afterwards. The length scale overlaps with typical shear band thicknesses reported [37].</figcaption></figure>

to a value comparable to that of negative $\delta E_{conf}^c$ prior to shear band formation and thereafter. This convergence indicates the emergence of a unified length scale for configurational energy changes, occurring before the onset of shear bands, and highlights the dynamic interplay between energy storage and release in shaping the response of the granular system. The emergence of dual length scales prior to material failure may be attributed to the presence of persistent force chains, wherein chains in positive

configurational energy sub-networks can intersect and interact, amplifying spatial connectivity.

This self-organization observation opens the door to defining $L_{Sconf}$ as a measurable internal length scale that could serve as a precursor to shear localization. Future work could explore this self-organization feature in continuum-level modeling frameworks by incorporating $L_{Sconf}$ into non-local constitutive laws, particularly those based on granular elasticity or gradient plasticity analogies.

Such a length scale offers a new way to quantify non-locality in granular deformation. Unlike classical definitions based on correlation lengths or force chain coherence, $L_{Sconf}$ emerges directly from the energetic response of the system and reflects both the heterogeneity and irreversibility of microstructural evolution. Moreover, $L_{Sconf}$ can be systematically extracted through coarse-graining of the spatial $\delta E_{conf}^c$ field, making it computationally accessible and potentially measurable in experimental settings. This makes it a promising candidate for integration into predictive models of granular failure.

This energy- and entropy-analogous perspective aligns with efforts to understand granular mechanics via force chains, energy pathways, and non-local constitutive models [16, 18, 19, 40]. The spatial field of $\delta E_{conf}^c$ provides a rich descriptor of the evolving internal state, with its localization signaling critical transitions.

In summary, the proposed configurational energy framework introduces a physically grounded, energy-based framework through which we examine and analyze granular material behavior. By linking localized energy fluctuations to a characteristic length scale, we provide a quantitative measure of the onset of failure precursors, offering a pathway toward predictive modeling of granular system incipient failure.

## 6 | Conclusion

This study has introduced and investigated “configurational energy” (this work focuses on its production rate, $\delta E_{conf}$) as a novel internal state variable for characterizing microstructural evolution and predicting the onset of localization in dense frictional granular materials. Defined through small quasi-static stress probes, $\delta E_{conf}^c$ quantifies irreversible energy redistribution at contacts.

Key findings in this study include:

1. **Contact-Scale Dynamics:** $\delta E_{conf}^c$ at a contact correlates with its initial stored energy and relative inter-particle displacement. Both positive (local energy storage) and negative (local energy release) $\delta E_{conf}^c$ occur, with their prevalence and magnitude evolving with global stress. Notably, weak contacts, those with low initial stored energy, are more likely to accumulate configurational energy during a probe, while strong contacts often exhibit energy release due to frictional relaxation. This dual behavior highlights the heterogeneous nature of granular deformation at the microscale.
2. **Specimen-Scale Signatures:** Although individual contacts may exhibit both positive and negative configurational energy changes ($\delta E_{conf}^c$) in response to a stress probe, the specimen always releases configurational energy. This indicates that the configurational entropy production, $dS_{conf}$, is generally non-negative—reflecting a global tendency toward “configurational disorder” despite possible local ordering. Under a given stress state, a small-amplitude cyclic stress probe would facilitate the system’s transition to a metastable configuration associated with a local minimum in the configurational energy landscape. This process is analogous to classical entropy evolution, where perturbations drive the system toward energetically broader and more disordered states. The magnitude of configurational energy production, on the other hand, serves as a mesoscale indicator of the system’s proximity to failure, capturing the underlying microstructural evolution even when macroscopic deformation remains minimal.

3. **Characteristic Length Scale:** A new length scale, $L_{Sconf}$, was proposed based on the spatial decay of configurational energy fluctuations. It represents the length scale at which the production of negative configurational energy can propagate, effectively capturing the mesoscopic extent of non-local mechanical interactions. This scale grows with increasing loading and aligns with typical shear band thicknesses reported in the literature, offering a quantitative measure of the onset of cooperative failure mechanisms. Importantly, $L_{Sconf}$ provides a physically grounded basis for coarse-graining approaches in continuum modeling and opens a new pathway for incorporating non-local effects into predictive granular modeling.

4. **Spatial Localization as a Precursor:** The spatial field of $\delta E_{conf}^c$ localizes into distinct zones of energy accumulation and release well before macroscopic shear band formation, serving as a sensitive indicator of impending material yielding or failure. Significant configurational events occur predominantly within the location that shear band later forms, and as the configurational response of the material diverges, it approaches failure. Notably, the initially distinct internal length scales associated with elastic and irreversible configurational responses converge before material failure. Thus, configurational energy provides a unique perspective for detecting the onset of shear band formation and acts as a robust precursor to granular material instability.

The configurational energy framework, particularly when interpreted with an entropy analogy, offers a nuanced understanding of granular deformation. It complements existing descriptors by providing a mesoscale energetic field that captures the system's evolving internal state and its progression towards critical transitions. Future research will focus on validating and extending these concepts in 3D, with complex particle shapes, and exploring connections to predictive failure models, aiming to advance the fundamental understanding of failure mechanics in granular media. Extending the configurational energy framework to include rotational energy storage, for instance, through tangential bending moments or orientation-dependent stiffness, could be a future addition, allowing its applicability to nonspherical or polygonal particles. Such an extension would align the metric more closely with real-world granular behavior,

where rotational modes significantly affect mechanical stability and failure precursors. In summary, configurational energy bridges the gap between discrete contact-level dynamics and macroscopic mechanical response, offering both a diagnostic tool and a conceptual foundation for next-generation modeling of granular systems.

## Author Contributions

Zhennan Zhu: conceptualization, numerical investigation, writing – original draft, ethodology, analysis, writing – review and editing. Frangois Nicot: conceptualization, methodology, analysis, writing – review and editing. Antoine Wautier: numerical investigation, methodology, analysis, writing – review and editing. Klaus Regenauer-lieb: methodology, analysis, writing – review and editing, funding acquisition. Manman Hu: methodology, analysis, writing – review and editing, funding acquisition.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Appendix A: Derivation of Configurational Energy at Contact Scale

In the initial equilibrium of a frictional block-spring system, suppose that Springs 1, 2, and 3 are compressed, with forces satisfying $F_1 - F_2 = \mu F_3$, indicating that the system is at Coulomb friction threshold. An unloading displacement $\Delta$ is applied to the system, causing block movement $\Delta_s$. The system is then reloaded to recover the initial configuration. Here we discuss perturbation that is perpendicular to friction, as shown in Figure 2a.

An unloading displacement $\Delta$ is applied at point c (reducing compression in Spring 3). The new force in Spring 3 after unloading is: $F'_3 = F_3 - k_3\Delta < F_3$, which makes $F_1 - F_2 > \mu F'_3$. This imbalance causes the block to move rightward by displacement $\Delta_s$. Suppose that $F_1 > F_2$, it leads to reduced compression in Spring 1 and increased compression in Spring 2. The new equilibrium is:

$$F_1 - k_1 \Delta_s = F_2 + k_2 \Delta_s + \mu F'_3 \quad (A.1)$$

Substitute $F'_3 = F_3 - k_3\Delta$ and $F_1 - F_2 = \mu F_3$, we have

$$\Delta_s = \frac{\mu k_3 \Delta}{k_1 + k_2} \quad (A.2)$$
Point c is then reloaded to restore the force in Spring 3 to $F_3$. The friction force during reloading is assumed to remain $\mu F'_3$, indicating the block does not slide further (static friction holds).

If we assume that the unloading-loading process is stress controlled and the average force is $F_3 - \frac{k_3\Delta}{2}$, then the external work done on Spring 3 during the unloading-loading cycle is:

$$ 
\begin{aligned}
W &= W_{\text{unload}} + W_{\text{reload}} \\
&= -\int_{0}^{\Delta} (F_3 - k_3x) \, dx + \int_{0}^{\Delta} (F_3 - k_3\Delta + k_3x) \, dx \\
&= -F_3\Delta + \frac{k_3\Delta^2}{2} + (F_3 - k_3\Delta)\Delta + \frac{k_3\Delta^2}{2} \\
&= k_3\Delta^2
\end{aligned} 
\quad (A.3) $$

The energy dissipated due to friction during the block's movement is:

$$ 
\begin{aligned}
E_{\text{dissipated}} &= \mu \frac{1}{2}(F'_3 + F_3)\Delta_s = \mu(F_3 - \frac{1}{2}k_3\Delta) \cdot \frac{\mu k_3\Delta}{k_1 + k_2} \\
&= \frac{\mu^2 k_3 F_3 \Delta}{k_1 + k_2} - \frac{\mu^2 k_3^2 \Delta^2}{2(k_1 + k_2)}
\end{aligned} 
\quad (A.4) $$

The potential energy change accounts for all three springs after the block moves by $\Delta_s$ can be calculated accordingly:

* **Spring 1:** Initial displacement $x_1 = \frac{F_1}{k_1}$. New displacement $x_1 - \Delta_s$.

$$ 
\begin{aligned}
\Delta U_1 &= \frac{1}{2} k_1 \left( \frac{F_1}{k_1} - \Delta_s \right)^2 - \frac{1}{2} k_1 \left( \frac{F_1}{k_1} \right)^2 \\
&= -F_1 \Delta_s + \frac{1}{2} k_1 (\Delta_s)^2
\end{aligned} 
\quad (A.5) $$

* **Configurational Force** of the system:

$$ F_1^{\text{conf}} = \frac{\Delta U_1}{\Delta_s} = -F_1 + \frac{\mu k_1 k_3 \Delta}{2(k_1 + k_2)} \quad (A.6) $$

* **Spring 2:** Initial displacement $x_2 = \frac{F_2}{k_2}$. New displacement $x_2 + \Delta_s$.

$$ 
\begin{aligned}
\Delta U_2 &= \frac{1}{2} k_2 \left( \frac{F_2}{k_2} + \Delta_s \right)^2 - \frac{1}{2} k_2 \left( \frac{F_2}{k_2} \right)^2 \\
&= F_2 \Delta_s + \frac{1}{2} k_2 (\Delta_s)^2
\end{aligned} 
\quad (A.7) $$

* **Spring 3:** Energy is recovered after the reloading.

* **Configurational Energy** of the system:

$$ 
\begin{aligned}
\Delta U &= \Delta U_1 + \Delta U_2 + \Delta U_3 \\
&= -F_1 \Delta_s + \frac{1}{2} k_1 (\Delta_s)^2 + F_2 \Delta_s + \frac{1}{2} k_2 (\Delta_s)^2 \\
&= -\frac{\mu^2 k_3 F_3 \Delta}{k_1 + k_2} + \frac{\mu^2 k_3^2 \Delta^2}{2(k_1 + k_2)}
\end{aligned} 
\quad (A.8) $$

* **Configurational Force** of the system:

$$ F_{\text{conf}} = \frac{\Delta U}{\Delta_s} = -\mu F_3 + \frac{1}{2} \mu k_3 \Delta = F_2 - F_1 + \frac{1}{2} \mu k_3 \Delta \quad (A.9) $$

$\Delta U$ is the configurational energy for this system. $\Delta U_i$ is the configurational energy for elastic contact i. And $F_{\text{conf}}$ is the “configurational force” defined by the variation of the total energy of the system with respect to changes in the material configuration. It can be considered as an “internal active force” that breaks the detailed balance. After an unloading-loading cycle, the energy previously stored in Spring 1 is transmitted to Spring 2. However, the energy is not fully transmitted. The change in configurational energy of the system is related to the unloading displacement $\Delta$. When $\Delta < \Delta_{th} = \frac{2F_3}{k_3}$, the system loses some potential energy. $\Delta_{th}$ represents the unloading displacement when $\Delta U = 0$. In the meantime, the force of the system boundary also changes from $F_B = F_1 - F_2$ to $F_B = F_1 - k_1\Delta_s - (F_2 + k_2\Delta_s)$, so the change of external force of the local system of the system is $\Delta F_B = -(k_1 + k_2)\Delta_s = -\mu k_3\Delta$, which is not the same as the configurational force. When $\Delta > \Delta_{th}$, positive in configurational energy will be generated.

$\Delta F_B$ further provides a perturbation on the adjacent system with stiffness $k_4$, with a displacement $\Delta' = \frac{\Delta F_{B,1}}{k_4} = \mu \frac{k_1 k_3}{k_4(k_1 + k_2)} \Delta$, which indicates that the perturbation $\Delta$ propagates with a factor $\mu \frac{k_3}{k_4}$ with the existence of friction.