Configurational mechanics in granular media

Abstract

Granular materials belong to the class of complex materials within which rich properties can emerge on large scales despite a simple physics operating on the microscopic scale. Most notable is the dissipative behaviour of such materials mainly through non-linear frictional interactions between the grains which go out of equilibrium. A whole variety of intriguing features thus emerges in the form of bifurcation modes in either patterning or un-jamming. This complexity of granular materials is mainly due to the geometrical disorder that exists in the granular structure. Diverse configurations of grain collections confer to the assembly the capacity to deform and adapt itself against different loading conditions. Whereas the incidence of frictional properties in the macroscopic plastic behavior has been well described for long, the role of topological reorganizations that occur remains much more elusive. This paper attempts to shed a new light on this issue by develo** ideas following the configurational entropy concept within a proper statistical framework. As such, it is shown that contact opening and closing mechanisms can give rise to a so-called configurational dissipation which can explain the irreversible topological evolutions that granular materials undergo in the absence of frictional interactions.

Graphical Abstract

Appendices

Appendix 1

Positiveness of the configurational affinity ${A}_{i,j}$

The configurational affinity ${A}_{i,j}$ is defined by the relation ${A}_{i,j}=k \theta \text{ln}\frac{(i-1)!}{(j-1)!(i+1-j)!}$. The combinatory term in the logarithm function can be transformed as follows:

$$\frac{(i-1)!}{(j-1)!(i+1-j)!}=\left(\frac{i-1}{j-1}\right)\left(\frac{i-2}{j-2}\right)\cdots \left(\frac{i-(j-2)}{2}\right)$$

(37)

As $i\ge j+1$, the quotients appearing in Eq. (37) are all strictly greater than 1. Thus:

$$\frac{(i-1)!}{(j-1)!(i+1-j)!}>1$$

(38)

which proves the strict positiveness of all terms${A}_{i,j}$, for any couple $\left(i,\;j\right)$ such that $i\in \left\{4,\dots ,m\right\}$ and $j<i$

Appendix 2

Derivation of Eq. (20)

Starting from Eq. (18), and combining with Eq. (15) gives:

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

(39)

which can be rearranged as:

$$\begin{array}{c}{\dot{S}}^{conf}=-\frac{1}{\theta }\sum_{i=4}^{m-1}\left({\mu }_{i}\left(\sum_{j=3}^{\text{int}(i/2)+1}{\dot{N}}_{i,j}-\sum_{j=i+1}^{m}\left(1+{\delta }_{i,j-i+2}\right){\dot{N}}_{j,i}\right)\right)\\ +\frac{1}{\theta }\sum_{i=4}^{m-1}\left(\left(\sum_{j=3}^{\text{int}(i/2)+1}\left({\mu }_{j}+{\mu }_{i+2-j}\right){\dot{N}}_{i,j}-\sum_{j=i+1}^{m}{\mu }_{i}\left(1+{\delta }_{i,j-i+2}\right){\dot{N}}_{j,i}\right)\right)\\ +\frac{1}{\theta }\sum_{j=3}^{\text{int}(m/2)+1}\left({\mu }_{j}+{\mu }_{m+2-j}-{\mu }_{m}\right){\dot{N}}_{m,j}\end{array}$$

(40)

Taking advantage of Eq. (19b), Eq. (40) can be rewritten as:

$$\begin{array}{c}{\dot{S}}^{conf}=-\frac{1}{\theta }\sum_{i=4}^{m-1}{\mu }_{i}{\dot{N}}_{i}\\ +\frac{1}{\theta }\sum_{i=4}^{m-1}\left(\left(\sum_{j=3}^{\text{int}(i/2)+1}\left({\mu }_{j}+{\mu }_{i+2-j}\right){\dot{N}}_{i,j}-\sum_{j=i+1}^{m}{\mu }_{i}\left(1+{\delta }_{i,j-i+2}\right){\dot{N}}_{j,i}\right)\right)\\ +\frac{1}{\theta }\sum_{j=3}^{\text{int}(m/2)+1}\left({\mu }_{j}+{\mu }_{m+2-j}-{\mu }_{m}\right){\dot{N}}_{m,j}\end{array}$$

(41)

Using Eqs. (19a) and (19c), we get:

$$\begin{array}{c}{\dot{S}}^{conf}=-\frac{1}{\theta }\sum_{i=3}^{m}{\mu }_{i} {\dot{N}}_{i}\\ +\frac{1}{\theta }\sum_{i=4}^{m-1}\left(\left(\sum_{j=3}^{\text{int}(i/2)+1}\left({\mu }_{j}+{\mu }_{i+2-j}\right){\dot{N}}_{i,j}-\sum_{j=i+1}^{m}{\mu }_{i}\left(1+{\delta }_{i,j-i+2}\right){\dot{N}}_{j,i}\right)\right)\\ +\frac{1}{\theta }\sum_{j=3}^{\text{int}(m/2)+1}\left({\mu }_{j}+{\mu }_{m+2-j}\right){\dot{N}}_{m,j}-\frac{1}{\theta }\sum_{j=4}^{m}{\mu }_{3}\left(1+{\delta }_{3,j-1}\right){\dot{N}}_{j,3}\end{array}$$

(42)

which finally gives:

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

(43)

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

Appendix 3

The objective of this appendix is to demonstrate the nullity of the term:

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

For this purpose, let us introduce the square matrix $M\in {\mathbb{R}}_{m,m}$ of general term:

$$\begin{array}{cc}{M}_{i,j}=\left({\mu }_{j}+{\mu }_{i+2-j}\right){\dot{N}}_{i,j}& {\text{if}} \, 4\le i\le m {\text{and}} \, {3}\le j\le \text{int}(i/2)+1\\ {M}_{i,j}=0& {\text{otherwise}}\end{array}$$

(44)

Thus, the first term ${\dot{\chi }}_{1}=\sum_{i=4}^{m}\sum_{j=3}^{\text{int}(i/2)+1}\left({\mu }_{j}+{\mu }_{i+2-j}\right){\dot{N}}_{i,j}$ corresponds to the sum of all the elements of the matrix: ${\dot{\chi }}_{1}=\sum_{i=1}^{m}\sum_{j=1}^{m}{M}_{i,j}$.

Furthermore, $M$ can be split in two matrixes, $M={M}^{1}+{M}^{2}$, with:

$$\begin{array}{cc}{M}_{i,j}^{1}={\mu }_{j}{\dot{N}}_{i,j}& {\text{if}} \, 4\le i\le m {\text{and}} \, {3}\le j\le \text{int}(i/2)+1\\ {M}_{i,j}^{1}=0& {\text{otherwise}}\end{array}$$

(45)

and

$$\begin{array}{cc}{M}_{i,j}^{2}={\mu }_{i+2-j}{\dot{N}}_{i,j}& {\text{if}} \, 4\le i\le m {\text{and}} \, {3}\le j\le \text{int}(i/2)+1\\ {M}_{i,j}^{2}=0& {\text{otherwise}}\end{array}$$

(46)

Both matrices ${M}^{1}$ and ${M}^{2}$ can be transformed by suitable permutation operations into matrices ${T}^{1}$ and ${T}^{2}$, as follows:

$$\begin{array}{cc}{T}_{i,j}^{1}={M}_{j,i}^{1}& {\text{if}} \, {1}\le i\le m \text{and 1}\le j\le m\end{array}$$

(47)

and

$$\begin{array}{cc}{T}_{i,j}^{2}={M}_{i+2-j,i}^{1}& {\text{if}} \, {1}\le i\le m \text{and 1}\le j\le m\end{array}$$

(48)

These permutations will induce no change in the term ${\dot{\chi }}_{1}=\sum_{i=1}^{m}\sum_{j=1}^{m}{M}_{ij}$, so that we have:

$${\dot{\chi }}_{1}=\sum_{i=1}^{m}\sum_{j=1}^{m}{T}_{i,j}^{1}+\sum_{i=1}^{m}\sum_{j=1}^{m}{T}_{i,j}^{2}$$

(49)

As an illustration, the case where $m=10$ can be considered:

$${M}^{1}=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{4,3}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{5,3}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{6,3}}& {\mu }_{4} {\dot{N}}_{\text{6,4}}& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{7,3}}& {\mu }_{4} {\dot{N}}_{\text{7,4}}& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{8,3}}& {\mu }_{4} {\dot{N}}_{\text{8,4}}& {\mu }_{5} {\dot{N}}_{\text{8,5}}& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{9,3}}& {\mu }_{4} {\dot{N}}_{\text{9,4}}& {\mu }_{5} {\dot{N}}_{\text{9,5}}& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{10,3}}& {\mu }_{4} {\dot{N}}_{\text{10,4}}& {\mu }_{5} {\dot{N}}_{\text{10,5}}& {\mu }_{6} {\dot{N}}_{\text{10,6}}& 0& 0& 0& 0\end{array}\right]$$

(50)

$${M}^{2}=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{3} {\dot{N}}_{\text{4,3}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{4} {\dot{N}}_{\text{5,3}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{5} {\dot{N}}_{\text{6,3}}& {\mu }_{4} {\dot{N}}_{\text{6,4}}& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{6} {\dot{N}}_{\text{7,3}}& {\mu }_{5} {\dot{N}}_{\text{7,4}}& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{7} {\dot{N}}_{\text{8,3}}& {\mu }_{6} {\dot{N}}_{\text{8,4}}& {\mu }_{5} {\dot{N}}_{\text{8,5}}& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{8} {\dot{N}}_{\text{9,3}}& {\mu }_{7} {\dot{N}}_{\text{9,4}}& {\mu }_{6} {\dot{N}}_{\text{9,5}}& 0& 0& 0& 0& 0\\ 0& 0& {\mu }_{9} {\dot{N}}_{\text{10,3}}& {\mu }_{8} {\dot{N}}_{\text{10,4}}& {\mu }_{7} {\dot{N}}_{\text{10,5}}& {\mu }_{6} {\dot{N}}_{\text{10,6}}& 0& 0& 0& 0\end{array}\right]$$

(51)

And after transformation, we get:

$${T}^{1}=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\mu }_{3} {\dot{N}}_{\text{4,3}}& {\mu }_{3} {\dot{N}}_{\text{5,3}}& {\mu }_{3} {\dot{N}}_{\text{6,3}}& {\mu }_{3} {\dot{N}}_{\text{7,3}}& {\mu }_{3} {\dot{N}}_{\text{8,3}}& {\mu }_{3} {\dot{N}}_{\text{9,3}}& {\mu }_{3} {\dot{N}}_{\text{10,3}}\\ 0& 0& 0& 0& 0& {\mu }_{4} {\dot{N}}_{\text{6,4}}& {\mu }_{4} {\dot{N}}_{\text{7,4}}& {\mu }_{4} {\dot{N}}_{\text{8,4}}& {\mu }_{4} {\dot{N}}_{\text{9,4}}& {\mu }_{4} {\dot{N}}_{\text{10,4}}\\ 0& 0& 0& 0& 0& 0& 0& {\mu }_{5} {\dot{N}}_{\text{8,5}}& {\mu }_{5} {\dot{N}}_{\text{9,5}}& {\mu }_{5} {\dot{N}}_{\text{10,5}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\mu }_{6} {\dot{N}}_{\text{10,6}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(52)

and

$${T}^{2}=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\mu }_{3} {\dot{N}}_{\text{4,3}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& {\mu }_{4} {\dot{N}}_{\text{5,3}}& {\mu }_{4} {\dot{N}}_{\text{6,4}}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {\mu }_{5} {\dot{N}}_{\text{6,3}}& {\mu }_{5} {\dot{N}}_{\text{7,4}}& {\mu }_{5} {\dot{N}}_{\text{8,5}}& 0& 0\\ 0& 0& 0& 0& 0& 0& {\mu }_{6} {\dot{N}}_{\text{7,3}}& {\mu }_{6} {\dot{N}}_{\text{8,4}}& {\mu }_{6} {\dot{N}}_{\text{9,5}}& {\mu }_{6} {\dot{N}}_{\text{10,6}}\\ 0& 0& 0& 0& 0& 0& 0& {\mu }_{7} {\dot{N}}_{\text{8,3}}& {\mu }_{7} {\dot{N}}_{\text{9,4}}& {\mu }_{7} {\dot{N}}_{\text{10,5}}\\ 0& 0& 0& 0& 0& 0& 0& 0& {\mu }_{8} {\dot{N}}_{\text{9,3}}& {\mu }_{8} {\dot{N}}_{\text{10,4}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\mu }_{9} {\dot{N}}_{\text{10,3}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(53)

It is worth noting that the matrix $T={T}^{1}+{T}^{2}$ takes the convenient form:

$$T=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 2{\mu }_{3} {\dot{N}}_{\text{4,3}}& {\mu }_{3} {\dot{N}}_{\text{5,3}}& {\mu }_{3} {\dot{N}}_{\text{6,3}}& {\mu }_{3} {\dot{N}}_{\text{7,3}}& {\mu }_{3} {\dot{N}}_{\text{8,3}}& {\mu }_{3} {\dot{N}}_{\text{9,3}}& {\mu }_{3} {\dot{N}}_{\text{10,3}}\\ 0& 0& 0& 0& {\mu }_{4} {\dot{N}}_{\text{5,3}}& 2{\mu }_{4} {\dot{N}}_{\text{6,4}}& {\mu }_{4} {\dot{N}}_{\text{7,4}}& {\mu }_{4} {\dot{N}}_{\text{8,4}}& {\mu }_{4} {\dot{N}}_{\text{9,4}}& {\mu }_{4} {\dot{N}}_{\text{10,4}}\\ 0& 0& 0& 0& 0& {\mu }_{5} {\dot{N}}_{\text{6,3}}& {\mu }_{5} {\dot{N}}_{\text{7,4}}& 2{\mu }_{5} {\dot{N}}_{\text{8,5}}& {\mu }_{5} {\dot{N}}_{\text{9,5}}& {\mu }_{5} {\dot{N}}_{\text{10,5}}\\ 0& 0& 0& 0& 0& 0& {\mu }_{6} {\dot{N}}_{\text{7,3}}& {\mu }_{6} {\dot{N}}_{\text{8,4}}& {\mu }_{6} {\dot{N}}_{\text{9,5}}& 2{\mu }_{6} {\dot{N}}_{\text{10,6}}\\ 0& 0& 0& 0& 0& 0& 0& {\mu }_{7}{ \dot{N}}_{\text{8,3}}& {\mu }_{7} {\dot{N}}_{\text{9,4}}& {\mu }_{7} {\dot{N}}_{\text{10,5}}\\ 0& 0& 0& 0& 0& 0& 0& 0& {\mu }_{8} {\dot{N}}_{\text{9,3}}& {\mu }_{8} {\dot{N}}_{\text{10,4}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& {\mu }_{9} {\dot{N}}_{\text{10,3}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(54)

Generalizing to any order m, it follows that the general term of the two matrices ${T}^{1}$ and ${T}^{2}$ reads:

$$\begin{array}{cc}{T}_{i,j}^{1}={\mu }_{i}{\dot{N}}_{j,i}& \text{if 3}\le i\le \text{int}(m/2)+1 {\text{and}} \, 2(i-1)\le j\le m\\ {T}_{i,j}^{1}=0& {\text{otherwise}}\end{array}$$

(55)

Likewise:

$$\begin{array}{cc}{T}_{i,j}^{2}={\mu }_{i} {\dot{N}}_{j,j+2-i}& {\text{if}} \, {3}\le i\le \text{int}(m/2)+1 {\text{and}} \, i+1\le j\le 2\left(i-1\right)\\ {T}_{i,j}^{2}={\mu }_{i} {\dot{N}}_{j,j+2-i}& {\text{if}} \, \text{int}\left(\frac{m}{2}\right)+2\le i\le m-1 {\text{and}} \, i+1\le j\le m\\ {T}_{i,j}^{2}=0& {\text{otherwise}}\end{array}$$

(56)

Thus, the particular form of the matrices ${T}^{1}$ and ${T}^{2}$ allows us to write:

$$\sum_{i=1}^{m}\sum_{j=1}^{m}{T}_{i,j}^{1}=\sum_{i=3}^{\text{int}(m/2)+1}\sum_{j=2(i-1)}^{m}{\mu }_{i}{\dot{N}}_{j,i}$$

(57)

$$\sum_{i=1}^{m}\sum_{j=1}^{m}{T}_{i,j}^{2}=\sum_{i=3}^{\text{int}(m/2)+1}\sum_{j=i+1}^{2(i-1)}{\mu }_{i}{\dot{N}}_{j,j+2-i}+\sum_{i=nt(m/2)+2}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}{\dot{N}}_{j,j+2-i}$$

(58)

Recalling now that ${\dot{N}}_{j,j+2-i}={\dot{N}}_{j,i}$, Eqs. (57) and (58) yield:

$${\dot{\chi }}_{1}=\sum_{i=3}^{\text{int}(m/2)+1}\sum_{j=2(i-1)}^{m}{\mu }_{i}{\dot{N}}_{j,i}+\sum_{i=3}^{\text{int}(m/2)+1}\sum_{j=i+1}^{2(i-1)}{\mu }_{i}{\dot{N}}_{j,i}+\sum_{i=nt(m/2)+2}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}{\dot{N}}_{j,i}$$

(59)

The summation $\sum_{i=3}^{\text{int}(m/2)+1}\sum_{j=2(i-1)}^{m}{\mu }_{i}{\dot{N}}_{j,i}+\sum_{i=3}^{\text{int}(m/2)+1}\sum_{j=i+1}^{2(i-1)}{\mu }_{i}{\dot{N}}_{j,i}$ can be merged in one single term, after noting that the index $j=2(i-1)$ is counted twice. As this corresponds to $i=j+2-i$, it is convenient to introduce the Kronecker symbol ${\delta }_{i,j+2-i}$, so that:

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

(60)

Furthermore, as $j\le m$ $j=2(i-1)$ requires that $i\le \text{int}(m/2)+1$. Thus, the third term $\sum_{i=nt(m/2)+2}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}{\dot{N}}_{j,i}$ can also be expressed as:

$$\sum_{i=nt(m/2)+2}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}{\dot{N}}_{j,i}=\sum_{i=nt(m/2)+2}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}\left(1+{\delta }_{i,j+2-i}\right){\dot{N}}_{j,i}$$

(61)

Finally, combining Eq. (59) with (60) and (61) yields:

$${\dot{\chi }}_{1}=\sum_{i=3}^{m-1}\sum_{j=i+1}^{m}{\mu }_{i}\left(1+{\delta }_{i,j+2-i}\right){\dot{N}}_{j,i}$$

(62)

which proves that $\dot{\chi }=0$