1 Introduction

Granular materials stand as the archetype of materials that offer a variety of remarkable features, despite an a priori very simplistic composition of individual particles interacting collectively at contacts. The mechanical properties of granular assemblies have attracted much interest within many scientific communities, ranging from soil mechanics to statistical physics [16, 20, 50]. As an illustration, it has been shown that, when loaded along a given mechanical loading path, granular materials can enter bifurcation regimes where the occurrence of various alternate failure modes is possible [72]. In fact, despite granular materials displaying seemingly elementary particle configurations at the mesoscopic level, they yet represent an ideal example of complex systems where the existence of several, separate scales—from the grain scale to the specimen scale through the mesoscale—gives rise to emergent, salient properties at the upper scale. On another note, the behavior of granular materials is also closely related to the second law of thermodynamics that refers to the notion of entropy with the irreversibility of processes necessitating a positive change in entropy. A thermodynamic process will always evolve so long the maximum amount of work that can be extracted from it is minimized.

To understand thermodynamics, the concept of microstates versus macrostates has been developed within a statistical framework by Boltzmann in the second part of the nineteen’s century and extended later by Gibbs through the notion of statistical canonical ensembles. We can normally measure the state variables characterizing a microstate, such as pressure as an intensive variable, whereas the microstates consisting of kinetic information (position and velocity) about all the individual molecules or atoms constituting the system are inaccessible except from a statistical viewpoint. This was the basic argument favoring the illuminating statistical approach initiated by Maxwell and Boltzmann [9, 44], then extended and formalized in a convenient statistical physics framework by Gibbs [25].

From a statistical viewpoint, a system will always tend to adopt a macrostate that corresponds to the largest number of microstates, which means that the number of microstates is maximized. This fundamental idea is embedded within the famous H-theorem demonstrated by Boltzmann in 1872, giving way to the statistical interpretation of the second law of thermodynamics, reformulated later on through the celebrated Boltzmann’s formula \(S=k \text{ln}\Omega\) that relates entropy S to the number \(\Omega\) of microstates compatible with the macroscopic state. In short, the probability of achieving a macrostate is proportional to the number of possible compatible microstates. Referring to the microcanonical ensemble statistics for isolated systems, the probability of all compatible microstates is the same. In other words, as stated in the ergodicity postulate, an isolated system at equilibrium will occupy successively all the accessible microstates for the same duration, assuming that a sufficiently large time range is considered.

Recently, this general framework of statistical physics has attracted some attention in granular solid mechanics as the system is constituted by a set of individual granules that can interact with each other. At a first glance, it could be tempting to apply the above framework to these materials by just regarding the granules as the molecules in a liquid or a gas. This pioneering idea dates to three decades ago and can be attributed to Edwards’ paper [21] where the typical case of powders was considered. As powder particles are very fine, any macroscopic specimen contains a huge number of constituents so that the classical concepts of statistical physics can be applied within a certain degree of confidence.

In the Edward’s paper, it is shown that unlike gas or liquids, the volume could play the role of kinetic energy, then proposing the measure of compactness as a convenient notion equivalent to the thermodynamic temperature. This was the first attempt paving the road to many contributions situated along this line of reasoning (see for example [2, 7, 8, 32, 36]). In particular, granular gas and fluids (very loose granular matter) were systematically described by applying concepts borrowed from statistical mechanics as these materials are first and foremost disordered ones whose internal variables such as particle locations or velocities are random fields that can be described through a suitable statistics [40, 64]. One key concept was the introduction of granular temperature, constructed in a similar way as in statistical physics from the root mean square of the difference between the mean velocity and the actual particle velocity field,see for example, a detailed review of this question in [30, 62].

When it comes to investigating the case of granular solids which are dense granular matter, several issues arise. In particular, the high compactness of these materials compromises a direct application of the above-mentioned particle velocity concepts. It turns out that the governing mechanisms within granular solids are totally different from those taking place within granular fluids and gas, mainly governed by the velocity field of particles that collide and move along a free path between successive collisions. By contrast, processes that occur on the microscopic scale within a granular solid are mainly dictated by the mechanical interaction of particles at contacts. All contacts flock together to form a fabric pattern made up of small grain clusters involving a few grains. These clusters degenerate into so-called grain loops in 2D conditions [57, 68, 70, 71]. In addition, nearly linear chains, called force chains and carrying the main fraction of the internal contact forces, develop within the assembly [51, 58]. In line with Edwards’ conjecture, it is thus believed that such materials can be couched within the above thermodynamic framework.

Restricting henceforth our analysis to two-dimensional specimens and exploring further an illuminating idea suggested by Wanjura et al. [74] and Sun et al. [66], the number of microstates in the assembly can be likened to the myriad of mesoscopic loops the collection of particles can form so that their evolution (genesis and breakage) is reminiscent of a chemical reaction. Also, the propensity or spontaneity of a change in loops topology is just like what happens in the kinetics of a chemical reaction dictated by entropy and rate of reactions, here again within the frame of the second law of thermodynamics.

The manuscript will be organized as follows. The basic thermochemistry background will be first reviewed in Section 2, and the constitutive concepts of a proper configurational granular mechanics will be developed in Section 3, leading to the introduction of a configurational entropy term. Then, Section 4 will elaborate an extension of the continuum thermodynamic framework to include the concepts of the configurational granular mechanics. Finally, a closing discussion will shed light on the potential advances that can be realized in the framework of such a proposed approach.

2 Thermochemistry background

We consider in this section a mixture of m different chemical species \({C}_{i}\) that can react with each other so that the number \({N}_{i}\) of each compound of a given species ‘i’ can evolve. Classically, a chemical reaction involves reactive compounds that can react with each other to form products. Adopting a general formalism, the chemical transformation operating within the mixture can be expressed as:

$${\nu }_{1}{C}_{1}+{\nu }_{2}{C}_{2}+\cdots +{\nu }_{m}{C}_{m}=0$$
(1)

where \({\nu }_{i}\) is the stochiometric coefficient of the species ‘i’ with \({\nu }_{i}\) being positive for products, and negative for reactive species.

The effect of a change in the number of compounds \({N}_{i}\) on the internal energy \(U\) of the system is described by Gibbs’ equation which involves the chemical potential \({\mu }_{i}={\left(\frac{\partial U}{\partial {N}_{i}}\right)}_{S,V,{N}_{j}}\) of each species as follows:

$$\dot{U}=\theta \dot{S}-P\dot{V}+\sum_{i=1}^{m}{\mu }_{i}{\dot{N}}_{i}$$
(2)

where \(S\) is the entropy of the system at a given thermodynamic temperature \(\theta\) and pressure \(P\). Equation (2) can be thus rearranged as:

$$\dot{S}=\frac{1}{\theta } \dot{U}+\frac{P}{\theta } \dot{V}-\sum_{i=1}^{m}\frac{{\mu }_{i}}{\theta } {\dot{N}}_{i}$$
(3)

where the chemical potential of the species ‘i’ now reads:

$${\mu }_{i}=-\theta {\left(\frac{\partial S}{\partial {N}_{i}}\right)}_{U,V,{N}_{j}}$$
(4)

The chemical reaction of species, as expressed in Eq. (4), undergoes the concept of chemical affinity A, defined as:

$$A=-\sum_{i=1}^{m}{\nu }_{i} {\mu }_{i}$$
(5)

By invoking the notion of extent of reaction \(\xi\), with \({\dot{N}}_{i}={\nu }_{i}\;\dot{\xi }\), the internal entropy production due to the extension of the chemical reaction under constant internal energy and volume reads:

$$\dot{S}=\frac{A\dot{\xi }}{T}=-\sum_{i=1}^{m}\frac{{\mu }_{i}}{T} {\dot{N}}_{i}$$
(6)

Equation (6) allows the entropy production to be expressed in a standard formalism as the product of a thermodynamic force (chemical affinity divided by the thermodynamic temperature) and a conjugate flux (rate of extent of reaction). The second law of thermodynamics imposes that \(\dot{\xi }\ge 0\). In other words, the system will evolve (i.e., the chemical reactions extent) until the affinity vanishes: \(A=0\). It is worth mentioning that once the affinity is zero, the system is in a stationary state where chemical reactions may continue progressing in a way that the fraction \({\alpha }_{i}=\frac{{N}_{i}}{N}\) of each species remains constant on average, with \(N=\sum_{i}^{m}{N}_{i}\) being the total number of compounds.

It is useful at this stage to draw the parallel with the topology change of loops of different orders (i.e., number of particles) that occur within a granular system [66, 74]. The topological evolution of loops can be compared to some extent with a chemical reaction in the sense that chemical reactions are governed by chemical bonding processes involving the sharing or loss of electrons in the outermost layer, while granular materials under an external mechanical loading experience a contact activity by forming or losing contacts between adjoining grains, which leads to new loop topologies.

In order to exploit the above-mentioned analogy, it is convenient to describe a two-dimensional granular assembly as a set of elementary, interconnected grain loops (Fig. 1). Grain loops are minimally closed contact chain structures with the number of grains involved in a given loop defining the loop order. Thus, a loop containing i grains will be denoted \({L}_{i}\). Referring to the fundamentals of statistical physics, a granular assembly in a given mechanical state (macroscopic state) defined by a set of macroscopic variables (such as the stress field), can therefore be associated with a set of microscopic states characterized by the topological configuration of the grains through a distribution of loops. By analogy with statistical physics, all the loop distributions compatible with the macroscopic state thus constitute a statistic ensemble denoted as a canonical ensemble.

Fig. 1
figure 1

Example of a loop tessellation in a 2D granular assembly. The tessellation is obtained by joining the centers of the particles in contact with branch lines, forming polygons. Each polygon corresponds to a specific loop, the number of sides giving the order of the loop (after [42])

Full size image

The key idea of this manuscript is to borrow the concepts of thermochemistry recalled herein to describe the evolution of the topological microstructure within a granular assembly. This is done by drawing a link between the notion of chemical reaction of species and the notion of ‘topological’ reaction within a collection of elementary loops that can reorganize themselves by gaining or losing a contact.

3 The concept of configurational granular mechanics

3.1 Preamble

We consider a system consisting of a given two-dimensional granular material, initially in a configuration \({C}_{o}\) (initial configuration) with a volume \({V}_{o}\) (with \({\Gamma }_{o}=\partial {V}_{o}\)). After a loading history, the system is in a strained configuration C and occupies a volume V (with \(\Gamma =\partial V\)), and is supposed to be in mechanical equilibrium under the current prescribed external loading. This loading is imposed through either boundary static or kinematic parameters, referred to as the control parameters, that make the entropy evolve while energy exchanges develop. The entropy production \(\dot{S}\) (rate of entropy) of the system can be split into the internal entropy production \({\dot{S}}^{\text{int}}\) (involving the entropy produced within the system) and the external entropy production \({\dot{S}}^{ext}\) (concerning the entropy exchanged through the boundary of the system with the external world), i.e.

$$\dot{S}={\dot{S}}^{\text{int}}+{\dot{S}}^{ext}$$
(7)

In soft matter physics, granular materials under standard mechanical loading are considered athermal, glassy materials [6, 49] where thermal fluxes can be omitted as a first approximation. The thermodynamic temperature is assumed constant. This assumption is reasonable in most situations but can fail when dealing with very large-scale problems such as in earthquakes where the mobilized frictional energy under very high tectonic pressures induces fast and significant temperature rises. As shown in Nicot et al. [47], the internal entropy production \({\dot{S}}^{\text{int}}\) is proportional to the plastic dissipation power \({\dot{W}}^{pl}={\int }_{V}{\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p} dv\):

$${\dot{S}}^{\text{int}}=\frac{1}{\theta }{\int }_{V}{\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p} dv$$
(8)

where \(\theta\) is the thermodynamic temperature of the system, and \(\sigma\) (resp.\({\dot{\varepsilon }}^{pl}\)) corresponds to the Cauchy stress tensor (resp. the plastic part of the strain tensor rate) field acting within the system. Basically, plastic dissipation within athermal granular systems stems from the frictional sliding or rotations at contacts between adjoining particles [46]. It is well admitted that the frictional dissipation between two solids in contact along an interface plane can be described by Coulomb’s law, which states that sliding occurs whenever the tangential contact force \({F}_{t}\) within the tangent plane equals the normal contact force \({F}_{n}\) multiplied by the friction coefficient \(\mu\). The Coulomb criterion reads therefore:

$$\begin{array}{cc}{F}_{t}<\mu {F}_{n}& \text{no sliding occurs}\end{array}$$
(9a)
$$\begin{array}{cc}{F}_{t}=\mu {F}_{n}& \text{the sliding can occur along the interface}\end{array}$$
(9b)

Thus, the system reorganizes itself through microstructural rearrangements stemming from the relative motion between the grains. An often-debated question then arises: What happens when the friction between the particles is ideally set to zero? In such a case, Eq. (9b) imposes that sliding occurs in any contact plane under zero tangential force. It turns out that the specimen mainly rearranges in an irreversible way under external loading with no frictional dissipation. Whereas the microstructure evolves irreversibility, Eq. (8), as such, leads to the counter-intuitive result that no entropy production takes place. This comes to infer that Eq. (8) might be incomplete, suggesting the existence of a missing term capable of properly accounting for microstructural evolutions alone, irrespective of any frictional dissipation contribution. It is thought that this missing term can be identified with the so-called configurational entropy.

The concept of configurational entropy introduced in the middle of the last Century [1, 24] was repeatedly proposed as a way to express the entropy of discrete, athermal media by taking inspiration from Boltzmann’s approach [54], or, equivalently, from Shannon’s information theorem [63]. It should be noted that both theories have been demonstrated to be equivalent by Jaynes, in 1957, see, for example, a complete overview in Jaynes [34, 35].

The range of eligible materials encompassing configurational entropy goes far beyond granular materials, including for example biology components such as proteins [19]. Basic approaches were essentially developed by advocating the existence of a probability density function modeling the randomness of a given topological descriptor such as the contact distribution in granular-like assemblies.

In this work, we propose an alternative approach, rooted in a different understanding of granular media. Basically, a granular solid is governed by both grain sliding and contact opening or closing. These two mechanisms can be regarded as the core ingredients that control the mechanical response of a granular assembly under a given loading. Grain sliding causes deformation of loops, and therefore (frictional) plastic dissipation related to a large fabric rearrangement. On the other hand, contact losses and gains cause a configurational change with no frictional dissipation. Both of them are obviously linked as particles sliding, when large enough, can induce contact opening or closing, which in turn leads to microstructural rearrangements. Thus, a way to account for these physical ingredients is to introduce an additional entropy term that specifically accounts for the contact opening or closing, without any plastic effect.

The contact opening and closing mechanism is associated with the concept of configurational entropy that will be detailed out in the next section, while frictional plasticity processes are associated with the classical internal entropy as given in Eq. (6). We shall emphasize that the configurational entropy developed in this work refers to an instantaneous event, consisting of contact creation or deletion, excluding any finite grain motion. By the same token, the entropy variation related to a chemical reaction refers to the creation or deletion of chemical bonds, in exclusion of any diffusion effect involving atomic or molecular motion within the chemical mixture.

The purpose of the next sections is to derive the expression of this configurational entropy by elaborating a novel framework herein denoted as the configurational granular mechanics.

3.2 Configurational framework for an elementary granular reaction

The granular system introduced in Section 3.1 can be described, from a microstructural point of view, by a microstate \({E}^{\text{mic}}\) corresponding to a distribution of loops \({L}_{i}\). It is convenient to express \({E}^{\text{mic}}\) as follows:

$${E}^{\text{mic}}=\left\{\left({L}_{3}, {N}_{3}\right),\left({L}_{4}, {N}_{4}\right),\dots ,\left({L}_{m}, {N}_{m}\right)\right\}$$
(10)

where the collection set includes all the loops and orders contained in the assembly at a given state.

The order of loops ranges from a minimal size (equal to 3) to a maximal size m [15, 57, 68, 70, 71, 76, 77]. For a given macrostate \({E}^{\text{mac}}\) controlled by the boundary conditions and described from extensive variables such as the stress state, a very large number \({n}_{E}\) of related microstates \({E}^{\text{mic}}\) exists, corresponding to all loop distributions compatible with \({E}^{{\text{ma}}{\text{c}}}\).

As previously mentioned, all these compatible microstates constitute a canonical ensemble. Basically, the order of magnitude of \({n}_{E}\) scales with \({N}^{m}\), where \(N\) is the number of particles contained in the assembly. Under an incremental change in the boundary conditions, the loop distribution will evolve from a given microstate \({E}_{1}^{\text{mic}}\) to a new microstate \({E}_{2}^{\text{mic}}\). Inspired by the pioneering ideas presented in Wanjura et al. [74] to build a rational framework for describing the detailed balance of topological changes in a 2D granular assembly, we consider that the loop distribution evolves through elementary, instantaneous loop transformations referred to as elementary configurational reactions. Ignoring as a first approximation the generation of rattlers, these configurational reactions, as exemplified in Fig. 2, can be formulated as follows in a very general way:

$$\begin{array}{cc}\left({R}_{i,j}\right)& {L}_{i}\rightleftarrows {L}_{j}+{L}_{i+2-j}\end{array}$$
(11)

for any \(i\in \left\{4,\dots ,m\right\}\), with \(j<i\)

Fig. 2
figure 2

Example of an elementary configurational reaction. A loop with 8 grains transforms reversibly into a loop with 7 grains and a loop with 3 grains by contact opening/closing between grains 1 and 2

Full size image

Turning to the mesoscopic scale, the configurational reaction \({R}_{i,j}\) transforms reversibly the meso-configuration \(C\left(i, 0\right)\) into the meso-configuration \(\left(j, i+2-j\right)\), where the arguments between brackets correspond to the order of loops composing each of the two meso-configurations. Over an infinitesimal duration \(dt\), let \(d{N}_{i,j}\) be the infinitesimal change in the amount of loops \({L}_{i}\) through the configurational reaction \({R}_{i,j}\) taking place within the assembly. We do recall that all reactions are considered instantaneous. Thus, the extent of the configurational reaction, by analogy with the thermochemistry formalism, is given by \(d{\xi }_{i,j}=-d{N}_{i,j}\). As both reactions \({R}_{i,j}\) and \({R}_{i,i+2-j}\) are identical, it is worth noting that \(d{N}_{i,j}=d{N}_{i,i+2-j}\).

It is emphasized here that the configurational approach proposed in this manuscript focuses on the contact network, and its evolution over contact gain and loss that entail closed loops evolution. This key idea constitutes the basis of the subsequent configurational granular mechanics.

Let’s consider an isolated loop \({L}_{i}\) containing i distinguishable particles and i contacts. If the chain of particles was not closed, there would be \(i!\) different possible configurations to arrange the i contacts. A configuration consists of a sequence of the i contacts (and therefore of the i grains), irrespective of the geometrical shape of the chains. For a closed loop, there will be \(\left(i-1\right)!\) different configurations, as all the \(i\) cyclic permutations should be removed.

By denoting \({S}_{i,0}\) the entropy of the configuration \(C\left(i, 0\right)\), composed of a unique loop of order i, and postulating a connection with Boltzmann’s formula, the entropy can be ideally expressed as a function of the number of compatible microstates, i.e.,

$${S}_{i,0}=k \text{ln}\left(i-1\right)!$$
(12)

where k is a Boltzmann-like constant, the magnitude of which should depend on the effective number of grains under consideration and a large enough number of configurations of grain arrangements to define an ensemble average as the state of the system.

It should be recalled that for large systems such as usual (molecular) gases and liquids, the number of particles (atoms or molecules) is typically on the order of the Avogadro’s number, i.e. \({10}^{23}\), which scales with the inverse of the standard Boltzmann constant, \({k}_{B}\approx 1.38\;{10}^{-23}\) J.K−1.

In a similar way, the configuration \(C\left(j,i+2-j\right)\) is composed of two loops \({L}_{j}\) and \({L}_{i+2-j}\), that will be assumed to be isolated from any external constraints. Each of the two loops constitutes a set of contacts (\(j\) contacts and \(i+2-j\) contacts) that will be all considered distinct from each other. As such, the entropy \({S}_{j,i+2-j}\) of the configuration \(C\left(j, i+2-j\right)\) can therefore be expressed as:

$${S}_{j,i+2-j}=k \text{ln}\left(j-1\right)!+k \text{ln}\left(i+1-j\right)!=k \text{ln}\left(\left(j-1\right)! \left(i+1-j\right)!\right)$$
(13)

By analogy with the chemical potential defined in Eq. (4), we introduce the notion of configurational potential \({\mu }_{i}\) associated with the loop \({L}_{i}\), as follows:

$${\mu }_{i}=k \theta \text{ln}\left(i-1\right)!$$
(14)

The physical meaning of this potential is that the more contacts a loop has, the higher its propensity to split into 2 smaller loops.

Again, by analogy with the chemical affinity previously defined in Eq. (5), the notion of configurational affinity \({A}_{i,j}\) associated with the reaction \({R}_{i,j}\), can readily be defined as:

$${A}_{i,j}=-\left(-{\mu }_{i}+{\mu }_{j}+{\mu }_{i+2-j}\right)=k \theta \text{ln}\left(\frac{(i-1)!}{(j-1)!(i+1-j)!}\right)$$
(15)

Straightforward algebraic manipulations make it possible to show that \({A}_{i,j}\) is a strictly positive quantity (see Appendix 1). Thus, the configurational entropy production \({\dot{S}}_{i,j}^{conf}\) related to the reactions \({R}_{i,j}\) can be written as:

$${\dot{S}}_{i,j}^{conf}=\frac{{A}_{i,j}}{\theta }{\dot{\xi }}_{i,j}=\frac{{\mu }_{i}-{\mu }_{j}-{\mu }_{i+2-j}}{\theta }{\dot{\xi }}_{i,j}$$
(16)

that is:

$${\dot{S}}_{i,j}^{conf}=-k \text{ln}\left(\frac{(i-1)!}{(j-1)!(i+1-j)!}\right){\dot{N}}_{i,j}$$
(17)

From Eq. (16), it can be inferred that whenever the reaction \({R}_{i,j}\) advances toward the right-hand side (\({\dot{\xi }}_{i,j}>0\)), forming two smaller loops \({L}_{j}\) and \({L}_{i+2-j}\), the configurational entropy production \({\dot{S}}_{i,j}^{conf}\) is strictly positive. It is negative along the reverse way, when a larger loop \({L}_{i}\) is formed from two smaller ones. The direction of the configurational reaction is therefore governed by the potentials of the different loops under consideration. If the potential of \({L}_{j}\) (left hand loop) is greater than the potential of \({L}_{j}+{L}_{i+2-j}\) (right hand loops), the reaction progresses toward the right, that is from the highest potential toward the lowest potential. This clarifies the meaning of the configurational affinity, that characterizes the propensity of the reaction to progress toward a particular side.

3.3 Ensemble of configurational granular reactions

A rational framework has been developed in the previous section to describe a single, elementary reaction \({R}_{i,j}\). Over an infinitesimal duration \(dt\), a distribution of \(d{N}_{i,j}\) reactions \({R}_{i,j}\) take place within the whole granular assembly, with \(3\le j\le \text{int}(i/2)+1\) and \(4\le i\le m\). The distribution is limited to \(j\le \text{int}(i/2)+1\), as reactions \({R}_{i,j}\) and \({R}_{i,i+2-j}\) are identical. According to the additivity property of the entropy, the configurational entropy production for the whole assembly can be expressed as:

$${\dot{S}}^{conf}=\sum_{i=4}^{m}\sum_{j=3}^{\text{int}(i/2)+1}{\dot{S}}_{i,j}^{conf}=-\frac{\sum_{i=4}^{m}\sum_{j=3}^{\text{int}(i/2)+1}{A}_{i,j}}{\theta }{\dot{N}}_{i,j}$$
(18)

with \(A=\sum_{i=4}^{m}\sum_{j=3}^{\text{int}(i/2)+1}{A}_{i,j}\) being the mutual affinity between all the configurational species composing the whole assembly.

The infinitesimal change \(d{N}_{i}\) in the number of loops \({L}_{i}\) over a duration \(dt\) can be related to the infinitesimal change \(d{N}_{i,j}\) in the amount of loops \({L}_{i}\) through the reactions \({R}_{i,j}\). Indeed, a loop \({L}_{i}\) can merge with a loop \({L}_{j+2-i}\) to form a larger loop \({L}_{j}\) (with \(i+1\le j\le m\)), or split into two smaller loops \({L}_{j}\) and \({L}_{i+2-j}\) (with \(3\le j\le \text{int}(i/2)+1\)). Thus, we have:

$$\begin{array}{cc}d{N}_{3}=-\sum_{j=4}^{m}\left(1+{\delta }_{i,j-1}\right)d{N}_{j,3}& \text{when }i=3\end{array}$$
(19a)
$$\begin{array}{cc}d{N}_{i}=\sum_{j=3}^{\text{int}(i/2)+1}d{N}_{i,j}-\sum_{j=i+1}^{m}\left(1+{\delta }_{i,j+2-i}\right)d{N}_{j,i}& \text{when }4\le i\le m-1\end{array}$$
(19b)
$$\begin{array}{cc}d{N}_{m}=\sum_{j=3}^{\text{int}(m/2)+1}d{N}_{m,j}& \text{when }i=m\end{array}$$
(19c)

where \({\delta }_{k,l}\) is the Kronecker symbol, with \({\delta }_{k,l}=1\) if \(k=l\), and \({\delta }_{k,l}=0\) otherwise. It should be noted that for the particular situation where \(j=2i-2\), the loop \({L}_{j}\) splits into two identical loops \({L}_{i}\). For example, a loop \({L}_{4}\) can split in two loops \({L}_{3}\) when two opposite grains contact.

The configurational entropy of the assembly, as given in Eq. (18), can be rewritten by virtue of Eqs. (19). The following expression can readily be obtained (see Appendix 2):

$${\dot{S}}^{conf}=-\frac{1}{\theta }\sum_{i=3}^{m}{\mu }_{i}{\dot{N}}_{i}+\frac{1}{\theta } \dot{\chi }$$
(20)

with \(\dot{\chi }=\sum_{i=4}^{m}\sum_{j=3}^{\text{int}(i/2)+1}\left({\mu }_{j}+{\mu }_{i+2-j}\right){\dot{N}}_{i,j}-\sum_{i=3}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}\;\left(1+{\delta }_{i,j+2-i}\right){\dot{N}}_{j,i}\).

Using tensorial manipulations, it can be shown that this last term is always nil with the proof of this result given in Appendix 3. This allows us to formally recover the classical Gibbs equation which was given in Eqs. (3) and (6) for the entropy of a mixture of m chemical reacting species under constant volume and internal energy. Thus, analogously we have the configurational entropy as:

$${\dot{S}}^{conf}=-\frac{1}{\theta }\sum_{i=3}^{m}{\mu }_{i} {\dot{N}}_{i}$$
(21)

At this point, it should be stressed that the instantaneous contact opening and closing event is associated with no volume nor energy changes. As such, ending up with a relation that is formally identical to the Gibbs equation should be regarded as a strong proof of consistency of the approach developed in this work.

4 Extending the continuum thermodynamics framework

4.1 Fundamentals of thermodynamics

The fundamentals of the standard continuum thermodynamics are first reviewed in this section. The internal energy of any material system is given by the first law of thermodynamics, i.e.

$$\dot{U}=\dot{Q}+\underset{V}{\int }{\sigma }_{ij} {\dot{\varepsilon }}_{ij} dv$$
(22)

where the heat power \(\dot{Q}\) reads as:

$$\dot{Q}=\underset{V}{\int }\rho \dot{r} dv-\underset{\partial V}{\int }{\dot{q}}_{i} {n}_{i} ds$$
(23)

The first integral corresponds to the radiated heat within the system body, and the second integral accounts for the boundary conduction heat transfer with a heat flux \(\dot{q}\).

Equation (22) expresses the internal energy \(U\) within the system in an integral form. Noting \(U=\underset{V}{\int }\rho e dv\), where e denotes the specific internal energy and \(\rho\) the local density of the material, Eq. (22) can be expressed following a local formulation as:

$$\rho \dot{e}=\rho \dot{r}+{\sigma }_{ij} {\dot{\varepsilon }}_{ij}-\frac{\partial {\dot{q}}_{i}}{\partial {x}_{i}}$$
(24)

As we have \(\dot{S}={\dot{S}}^{ext}+{\dot{S}}^{\text{int}}\), with

$${\dot{S}}^{ext}=\frac{\dot{Q}}{\theta }=\underset{V}{\int }\rho \frac{\dot{r}}{\theta } dv-\underset{\partial V}{\int }\frac{{\dot{q}}_{i}}{\theta } {n}_{i} ds$$
(25)

and

$${\dot{S}}^{int}=\underset{V}{\int }\frac{\Phi }{\theta } dv$$
(26)

where \(\Phi\) denotes the total dissipation rate within the system, and noting \(S=\underset{V}{\int }\rho \text{\hspace{0.33em}}\eta \text{\hspace{0.33em}}dv\), where \(\eta\) is the specific entropy, it can be shown that:

$$\Phi =\rho \theta \dot{\eta }-\rho \dot{r}+\frac{\partial {\dot{q}}_{i}}{\partial {x}_{i}}-\frac{{\dot{q}}_{i}}{\theta }\frac{\partial \theta }{\partial {x}_{i}}$$
(27)

where the term \({\Phi }^{\text{int}}=\rho \theta \dot{\eta }-\rho \dot{r}+\frac{\partial {\dot{q}}_{i}}{\partial {x}_{i}}\) stands for the intrinsic dissipation rate, whereas the term \({\Phi }^{th}=-\frac{{\dot{q}}_{i}}{\theta }\frac{\partial \theta }{\partial {x}_{i}}\) represents the conduction heat dissipation rate. Equation (27) leads to the classical Clausius–Duhem inequality.

4.2 Entropy as a combination of dissipative and configurational processes

We now introduce the effective specific entropy \({\eta }^{*}\), defined as the difference between the specific entropy \(\eta\) and the configurational part \({\eta }^{conf}\), with \({S}^{conf}=\underset{V}{\int }\rho {\eta }^{conf}\;dv\). Thus:

$${\eta }^{*}=\eta -{\eta }^{conf}$$
(28)

It should be emphasized that the configurational entropy accounts only for the topological changes induced by instantaneous contact opening and closing events in the absence of any other processes such as grain sliding and rotation, or grain deformation. Therefore. the effective entropy corresponds to the usual entropy stemming from standard dissipation processes, excluding any configurational processes considered in this work. Thus, \({\eta }^{*}\) can be related to the Helmholtz free energy through the classical Legendre transform:

$$\Psi =e-\theta {\eta }^{*}$$
(29)

Combining Eqs. (24) and (29) yields the following relation:

$$\rho \dot{\Psi }=\rho \dot{r}-\frac{\partial {\dot{q}}_{i}}{\partial {x}_{i}}-{\Phi }^{th}-\rho \dot{\theta } {\eta }^{*}-\rho \theta \left(\dot{\eta }-{\dot{\eta }}^{conf}\right)+{\sigma }_{ij} {\dot{\varepsilon }}_{ij}$$
(30)

which can be rearranged as:

$${\Phi }^{\text{int}}+{\Phi }^{th}=-\rho \left(\dot{\Psi }+\dot{\theta } {\eta }^{*}\right)+\rho \theta {\dot{\eta }}^{conf}+{\sigma }_{ij} {\dot{\varepsilon }}_{ij}$$
(31)

Restricting the analysis to athermal materials makes it possible to omit the term \(\dot{\theta }\;{\eta }^{*}\). Furthermore, the thermodynamic temperature within the material system is assumed to be constant (isothermal conditions), so that \({\Phi }^{th}=0\). Then, using the standard additive strain rate decomposition \({\dot{\varepsilon }}_{ij}={\dot{\varepsilon }}_{ij}^{e}+{\dot{\varepsilon }}_{ij}^{p}\), with \({\dot{\varepsilon }}_{ij}^{e}\) being the elastic strain part and \({\dot{\varepsilon }}_{ij}^{e}\) being the plastic strain counterpart, yields:

$${\Phi }^{\text{int}}={\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{e}-\rho \dot{\Psi }+{\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p}+\rho \theta {\dot{\eta }}^{conf}$$
(32)

Noting that the rate of the Helmholtz free energy is \(\rho\dot{\Psi }={\sigma }_{ij}\;{\dot{\varepsilon }}_{ij}^{e}\), the internal dissipation potential given in Eq. (32) reduces to:

$${\Phi }^{\text{int}}={\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p}+\rho \theta {\dot{\eta }}^{conf}$$
(33)

This corresponds to the intrinsic dissipation rate, which is always positive according to the second law of thermodynamics. It turns out that the intrinsic dissipation rate is composed of two terms. The first one, \({\Phi }^{\text{int},p}={\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p}\), corresponds to the plastic (frictional-based) processes induced by grain sliding at contacts, while the second one, \({\Phi }^{\text{int},conf}=\rho \theta {\dot{\eta }}^{conf}\), pertains to the configurational transformation related to contacts opening and closing.

As \({\dot{S}}^{int}=\underset{V}{\int }\frac{{\Phi }^{\text{int}}}{\theta } dv=\underset{V}{\int }\rho {\dot{\eta }}^{\text{int}}\;dv\), where \({\eta }^{\text{int}}\) corresponds to the specific internal entropy, it can also be written that:

$$\rho {\dot{\eta }}^{\text{int}}=\frac{1}{\theta }{\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p}+\rho {\dot{\eta }}^{conf}$$
(34)

Equation (34) constitutes a local formulation of the internal entropy production written at the material point scale. By extending the formulation to the whole granular assembly, it can eventually be obtained that:

$${\dot{S}}^{\text{int}}=\underset{V}{\int }\frac{1}{\theta } {\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p} dv-\frac{1}{\theta }\sum_{i=3}^{m}{\mu }_{i} {\dot{N}}_{i}$$
(35)

where the configurational potential \({\mu }_{i}\) of each loop \({L}_{i}\) is given by Eq. (14).

5 Closing discussion

5.1 Configurational irreversibility

Equation (34) embeds the important physical notion that the internal entropy, and thereby the dissipation potential, is composed of a plastic dissipation term due to sliding of particles and a complementary term stemming from transformations of the internal configurational topology. It should be emphasized that loops deform mainly through plastic sliding at contacts as well as rolling motion, inducing changes in the conformation of loops, i.e., their geometry and arrangement. If these changes do not give rise to contact opening and closing, the distribution of loops stays the same, and the dissipation within the system is entirely captured by the plastic term \({\dot{S}}^{\text{int},p}=\underset{V}{\int }\frac{1}{\theta } {\sigma }_{ij} {\dot{\varepsilon }}_{ij}^{p} dv\). Otherwise, when the deformation of the loops involves contact opening or closing, the mesostructures transform with potential subsequent changes in the loop distribution. The dissipation associated with this change in topology is properly described by the configurational term \({\dot{S}}^{\text{int},conf}=-\frac{1}{\theta }\sum_{i=3}^{m}{\mu }_{i} {\dot{N}}_{i}\) (for the sake of simplicity, and to stay consistent with the notations of Section 3.2, this term will be noted \({\dot{S}}^{conf}\) in the sequel).

When it comes to deal with very low friction materials, such as lubricated assemblies, the contribution of the plastic dissipation in the entropy can become negligible. In the limiting and ideal case where the friction is ideally zero, the term \({\dot{S}}^{\text{int},p}\) disappears so that the entropy production is then solely supported by the configurational changes within the specimen. This is indeed meaningful as irreversible strains are expected to occur, whereas no plastic dissipation develops, and no plastic dissipation-based entropy production takes place.

It is crucial to quantify the direction of variation of \({\dot{S}}^{conf}\), according to the changes in configuration. As mentioned in Section 3, the configurational entropy production \({\dot{S}}_{i,j}^{conf}\) is strictly positive whenever the reaction \({R}_{i,j}\) advances toward the right-hand side (\({\dot{\xi }}_{i,j}>0\)), forming smaller loops \({L}_{j}\) and \({L}_{i+2-j}\), and is negative along the reverse way, when a larger loop \({L}_{i}\) is formed. Thus, according to the second law of thermodynamics, it is expected that under a given loading, a frictionless granular specimen should evolve in a way that promotes the creation of smaller loops. The largest loops disappear to the benefit of smaller ones, making the specimen contract and its specific volume decrease.

The above-mentioned result can indeed be checked from numerical simulations based on a discrete element method [12]. The mechanical response of a two-dimensional frictionless granular specimen, axially compressed under a constant lateral pressure, is simulated. As shown in Fig. 3, the configurational entropy, normalized by the Boltzmann-like constant, increases monotonously (from an initial, arbitrary zero value) until reaching an asymptotic value on average. The fluctuations observed in Fig. 3 will be discussed in Section 5.3.

Fig. 3
figure 3

Simulation of the response of a dense, frictionless 2D specimen made up of a polydisperse assembly of 20,000 spherical particles along a biaxial loading under a constant lateral stress of 100 kPa, by using a discrete element method (YADE open-source code, [65]). The evolution of both the normalized deviatoric stress (The normalized deviatoric stress is given by \(q/{\sigma }_{2}\), where \(q={\sigma }_{1}-{\sigma }_{2}\), \({\sigma }_{1}\) is the axial stress, and \({\sigma }_{2}\) is the constant lateral stress) and the configurational entropy is given in terms of the axial strain

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5.2 Critical state regime in the light of the configurational mechanics

As another illustration of the applicability of the configurational granular mechanics framework developed in this work, we propose to address and clarify in a general manner the issue of the critical state regime that occurs in solid materials under shearing. This intriguing feature has been questioning the scientific community for many decades [3, 15, 42, 50, 60]. Granular materials experience an ultimate regime (denoted as the critical state regime), bringing the material to a unique mechanical state defined by a constant specific volume and a constant stress state, irrespective of the initial porosity of the material, while shear deformation processes are being maintained. Furthermore, for a sufficiently densely packed specimen, a phase transition is observed after a certain level of shearing coinciding with the formation of a strain localization pattern with one or multiple shear bands developing [17]. It was recently shown that this phase transition is consistent with the extremal entropy production theorems [47], leading to an optimal dissipative structure (namely, the shear band pattern) able to dissipate the most part of the external energy put into the system, and making the system able to sustain external loading without collapsing [73].

It has repeatedly been observed that dense granular specimens under shearing experience a spontaneous break in symmetry transitioning to a lower state of energy before reaching the critical state regime. This break in symmetry corresponds to a phase transition from a statistically homogeneous state to a heterogenous one, giving way to a shear band pattern fully developed once the critical state regime is reached. Thus, the specimen is composed of different spatial domains, with totally different properties. While the domain located outside the shear band pattern remains dense with nearly no dissipation processes occurring, the domain inside the shear band undergoes a marked dilatancy associated with an intense plastic dissipation.

The above-described transition toward a structured material has recently been investigated [47] by invoking the extremal entropy production theorems [5, 18, 26,27,28, 33, 43, 52, 55, 69, 75]. Such phase transitions can be encountered in various situations such as the Rayleigh-Bénard cells [4, 59], the chemical clocks [31], or many biological processes based on cellular self-organizations. All these transitions might suggest that the second principle of thermodynamics is violated, as they give rise to a gain in order. In particular, the formation of the shear band is accompanied by a dilatant behavior promoting the formation of larger loops to the detriment of smaller ones. Thus, the configurational reactions given in Eq. (10) advance toward the left side, which corresponds to a decrease in the configurational entropy.

This apparent paradox is in fact resolved by noting that the decrease in configurational entropy within the shear band pattern is mitigated by the increased plastic dissipation capacity of the shear band. As we have recently shown [47, 73], the shear band is an optimal dissipative structure that allows the specimen to resist the shear loading. By the same token, a fluid heated at the bottom should trigger a structuring mechanism through Rayleigh-Bénard convective cells, to undergo the external energy provided beyond a certain level of heating. In fact, this self-organization process is likely to concern any complex systems. This key concept that gathers self-organization, structure, and dissipation through logical links was at the core of Prigogine’s developments during the past Century [45, 55, 56]. The configurational granular mechanics proposed herein clearly provides a rational interpretation framework.

As a numerical illustration, Fig. 4 gives the evolution of the normalized configurational entropy, together with the deviatoric stress, along an axial compression under a constant lateral stress of 100 kPa. The same numerical specimen as in Fig. 3 has been used, with a non-zero intergranular friction (\({\varphi }_{g}=35\) deg) between the grains. As expected from the above theoretical framework, the configurational entropy is decreasing (from an initial, arbitrary, zero value), until reaching on average an asymptotic value consistent with the critical state regime.

Fig. 4
figure 4

Simulation of the response of a dense, frictional 2D specimen made up of a polydisperse assembly of 20,000 spherical particles along a biaxial loading under a constant lateral stress of 100 kPa, by using a discrete element method (YADE open-source code, [65]). The evolution of both the normalized deviatoric stress and the configurational entropy is given in terms of the axial strain

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5.3 Stationary regime and fluctuations

Once a shear band pattern has formed, its topological structure, basically defined through the loop distribution, stays constant on average, which then allows us to describe the critical state regime as a proper stationary regime [14, 37, 38, 67, 76]. As a matter of fact, this stationary regime corresponds to the microstate \({E}^{\text{mic}}=\left\{\left({L}_{3}, {N}_{3}\right),\left({L}_{4}, {N}_{4}\right),\dots ,\left({L}_{m}, {N}_{m}\right)\right\}\) staying the same on average with fluctuations in the loop distribution occurring continuously [15, 41, 76, 77]. In other words, dissipative rearrangements pull the system to a stationary evolution state [53] where configurational changes reach a stationary value dominated by instantaneous, reversible contact gains and loss. More specifically, loop transformations keep on developing, with no effect on the microstate. The configurational reactions \({R}_{i,j}\) are always active, transforming reversibly the meso-configurations \(C\left(i, 0\right)\) into the meso-configurations \(C\left(j, i+2-i\right)\). Locally, the system is configurationally reversible, but these reversible reactions give way to macroscopic irreversibility on the specimen scale.

These configurational reactions can be regarded as local fluctuations (or configurational fluctuations) that develop within the shear band, thereby corresponding to a region undergoing very intense configurational activity. This exhumes the famous paradoxical issue that crossed the past century without finding a clear answer, challenging the fundamental Boltzmann’s principle that results from the statistical conception of the second principle of thermodynamics. Any macroscopic system (gas, liquid) brought out of equilibrium experiences a macroscopic irreversibility, in accordance with the second principle, even though the evolution of each microscopic component, basically governed by the second Newton’s law, is theoretically reversible. It is worth noting that this paradox was clarified recently through the dissipation theorem [23]. A thorough review discussing this paradox can be found in [18].

The configurational fluctuations occurring at the critical state correlate with stress fluctuations developing within the granular assembly loaded until the critical state regime takes place. As shown in Fig. 5, where a biaxial loading path is simulated by using a discrete element method, stress fluctuations develop after the stress peak is reached, and become well marked in the critical state regime. As pointed out in Zhu et al. [77] and Clerc et al. [11], these stress fluctuations stem from continuous microstructural reorganizations occurring within the assembly. Considering the evolution of two loop categories (loops with 3 grains, and loops with 6 grains), it can be observed in Fig. 6 that the ratios of these two loop categories experience the same fluctuation regime, well correlated with the stress fluctuations displayed in Fig. 5.

Fig. 5
figure 5

Simulation of the response of a dense, frictional 2D specimen made up of a polydisperse assembly of 20,000 spherical particles along a biaxial loading under a constant lateral stress \({\sigma }_{2}\) of 100 kPa, by using a discrete element method (YADE open-source code, [65]). The evolution of the deviatoric stress \(q={\sigma }_{1}-{\sigma }_{2}\) is given in terms of the axial strain

Full size image
Fig. 6
figure 6

Evolution of the ratio of two grain loop categories (L3, left; L6, right) along a biaxial loading path under constant lateral stress. The fluctuations develop after the deviatoric stress peak is reached, simultaneously with the stress fluctuations (Fig. 3)

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The fluctuations in the two loop populations suggest that new loops are continuously created, while older ones disappear through configurational reactions \({R}_{j,3}\) and \({R}_{6,k}\) or \({R}_{l,6}\). Thus, configurational reactions \({R}_{6,k}\) keep on developing during the critical state regime, even though the loop population distribution is constant on average. Such configurational transformations, inherent in the critical state regime, can be regarded as configurational fluctuations associated with a change in configurational entropy. For example, for a fluctuation \(\Delta {N}_{i,j}\) related to the configurational reactions \({R}_{i,j}\), the corresponding change in configurational entropy \(\Delta {S}_{i,j}^{conf}\) is given by (Eqs. (15) and (16)):

$$\Delta {S}_{i,j}^{conf}=-k \text{ln}\frac{(i-1)!}{(j-1)!(i+1-j)!}\Delta {N}_{i,j}=\frac{{A}_{i,j}}{\theta }\Delta {N}_{i,j}$$
(36)

It is worth noting that \(\Delta {S}_{i,j}^{conf}\) can be positive or negative. This is in accordance with the Boltzmann’s statistical approach, as the critical state can be regarded as a stationary state (thermodynamic equilibrium). The local fluctuations \(\Delta {S}_{i,j}^{conf}\) in the configurational entropy are responsible for the global fluctuations that can be observed for the configurational entropy on the specimen scale, as depicted in Figs. 3 and 4.

Finally, by invoking the fluctuation–dissipation theorem [10, 29, 48, 61], it should be emphasized that any local increase in entropy will be compensated by a local dissipation involving other configurational transformations elsewhere in the specimen, then ensuring the whole system to be brought back to the thermodynamic equilibrium on the macroscopic scale [13, 22, 39].

5.4 Closing words

To conclude, it has been demonstrated how granular materials as complex systems are able to adapt and transform under specific loading conditions. The proposed microscopic reinterpretation of granular materials with large numbers of particles as a thermodynamic system within which a configurational entropy and an associated dissipation can be derived represents a plausible framework that certainly needs further investigation. It goes without saying that extending the framework to three-dimensional granular systems should be regarded as an urgent objective. For such materials, the fabric topology is much more difficult to describe, requiring the concept of 3D grain clusters to be defined in a clear mathematical way. Finally, marrying synergistically complementary subject areas such as statistical physics, fundamental thermodynamics and micromechanics will make it possible to shed a new light on many intriguing mechanisms occurring within complex systems such as granular materials, that have been questioning a broad scientific community since many decades.