Elastic Properties of Disordered Media

Abstract

There are various physical properties that we may be interested in for a disordered material. In the previous chapter, we studied flow problems in disordered materials using the percolation system as a model disordered material. In this chapter we will address mechanical properties of the disordered material.

There are various physical properties that we may be interested in for a disordered material. In the previous chapter, we studied flow problems in disordered materials using the percolation system as a model disordered material. In this chapter we will address mechanical properties of the disordered material.

We will address the behavior of the disordered material in the limit of fractal scaling. In this limit we expect material properties such as Young’s modulus to display a non-trivial dependence on system size. That is, we will expect material properties such as Young’s modulus to have an explicit system size dependence. We will use the terminology and techniques already developed to study percolation to address the mechanical behavior of disordered systems such as the coefficients of elasticity [4, 11, 20, 28, 40]

10.1 Rigidity Percolation

What are the elastic properties of a percolation system? First, we need to decide on how to convert a percolation system into an elastic system. We will start by modeling an elastic material as a bond lattice, where each bond represents a local elastic element. The element will in general have resistance to stretching and bending. Systems with only stretching stiffness are termed central force lattices. Here, we will address systems with both stretching and bending stiffness.

Models for Stretching and Bending Stiffness

We can formulate the effect of bending and stretching through the elastic energy of the system. The energy will have terms that depend on the elongation of bonds—these will be the terms that are related to stretching resistance. In addition, there will be terms related to the bending of bonds. Here we will introduce the bending terms through the angles between bonds. For any two bonds connected to the same site, there will be an energy associated with changes in the angle of the bond. This can be expressed as

$$\displaystyle \begin{aligned} U = \sum_{ij} \frac{1}{2} k_{ij} ({\mathbf{u}}_i - {\mathbf{u}}_j)^2 + \sum_{ijk} \frac{1}{2} \kappa_{ijk} \phi_{ijk}^2 \; , \end{aligned} $$

(10.1)

where U is the total energy, the sums are over all particle pairs ij or all particle triplets ijk. The force constant is $k_{ij} = k$ for bonds in contact and zero otherwise, and $\kappa _{ijk} = \kappa$ for triplets with a common vertice, and zero otherwise. The vector ${\mathbf {u}}_i$ gives the displacement of node i from its equilibrium position. The various quantities are illustrated in Fig. 10.1

Fig. 10.1

Illustration of the initial bond lattice (dashed, gray), and the deformed bond lattice. Three nodes i, j, k are illustrated. The angle $\phi _{ijk}$ is shown. The displacements ${\mathbf {u}}_i$ and ${\mathbf {u}}_j$ are shown respectively with cyan vectors

Elastic Modulus

Let us address the effective elastic behavior of the percolation system using a material property such as Young’s modulus, E, or the shear modulus, G. Let us consider a three-dimensional sample with cross-sectional area $A = L^2$ and length L. Young’s modulus, E, relates the tensile stress, $\sigma _{zz}$, applied normal to the surface with area A to the elongation $\varDelta L$ in the z-direction.

$$\displaystyle \begin{aligned} \sigma_{zz} = \frac{F_{z}}{A} = E \frac{\varDelta L_{z}}{L} \; , \end{aligned} $$

(10.2)

We can therefore write the relation between the force $F:{z}$ and the elongation $\varDelta L_{z}$ as

$$\displaystyle \begin{aligned} F_{z} = \frac{E A}{L} \varDelta L = \frac{E L^2}{L} \varDelta L = L^{d-2} E \varDelta L \:. \end{aligned} $$

(10.3)

We recognize this as a result similar to the relation between the conductance and the conductivity of the sample, and we will call $K = L^{d-2} E$ the compliance of the system. We recognize this as being similar to the spring constant of a spring.

Elastic Properties When $p<p:c$

What happens to the compliance of the system as a function of p? When $p<p:c$ there are no connecting paths from one side to another, and the compliance will therefore be zero. It requires zero force $F:{z}$ to generate an elongation $\varDelta L_{z}$ in the system. Notice that we are only interested in the infinitesimal effect of deformation. If we compress the sample, we will of course eventually generate a contacting path, but we are only interested in the initial response of the system.

Elastic Properties When $p>p_c$

When $p \geq p_c$ there will be at least one path connecting the two edges. For a system with a bending stiffness, there will be a load-bearing path through the system, and the deformation $\varDelta L_z$ of the system requires a finite force, $F:{z}$. The compliance K will therefore be larger than zero. We have therefore established that for a system with bending stiffness, the percolation threshold for rigidity coincides with the percolation threshold for connectivity. For a central force lattice, we know that the spanning cluster at $p:c$ will contain may singly connected bonds. These bonds will be free to rotate, and as a result a central force network will have a rigidity percolation threshold which is higher than the connectivity threshold. Indeed, rigidity percolation for central force lattices will have very high percolation thresholds in three dimensions and higher. Here, we will only focus on lattices with bond bending terms.

Behavior of E Close to $p:c$

Based on our experience with percolation systems, we may hypothesize that Young’s modulus will follow a power-law in $(p-p:c)$ when p approaches $p:c$:

$$\displaystyle \begin{aligned} E \propto \left\{ \begin{array}{ll} 0 & \text{ for } p<p_c \\ (p - p_c)^{\tau} & \text{ for } p>p_c \end{array} \right. \; . \end{aligned} $$

(10.4)

where $\tau$ is an exponent describing the elastic system. We will now use our knowledge of percolation to show that this behavior is indeed expected, and to determine the value of the exponent $\tau$.

10.1.1 Developing a Theory for $E(p,L)$

Let us address the Young’s modulus $E(p,L)$ of a percolation system with occupation probability p and a system size L. We could also write E as a function of the correlation length $\xi = \xi (p)$, so that $E = E(\xi ,L)$. Young’s modulus is in general related to the compliance through $E(\xi ,L) = K(\xi ,L) L^{d-2}$. We can therefore address the compliance of the system and then calculate Young’s modulus.

Dividing the System into Boxes of Size $\xi$

We will follow an approach similar to what we used to derive the behavior of $P(p,L)$. First, we address the case when the correlation length $\xi \ll L$. In this case, we can subdivide the $L^d$ system into boxes of linear size $\xi$ as illustrated in Fig. 10.2. There will be $(L/\xi )^d$ such boxes. On this scale the system is homogeneous. Each box will have a compliance $K(\xi ,\xi )$, and the total compliance will be $K(\xi ,L)$.

Fig. 10.2

Illustration of subdivision of a system with $p = 0.60$ into regions with a size corresponding to the correlation length, $\xi$. The behavior inside each box is as for a system at $p = p_c$, whereas the behavior of the overall system is that of a homogeneous system of boxes of linear size $\xi$

Compliance of the Combined System

We know that the total compliance of n elements in series is $1/n$ times the compliance of a single element. You can easily convince yourself of this addition rule for spring constants, by addressing two springs in series. Similarly, we know that adding n elements in parallel will make the total system n times stiffer, that is, the compliance will be n times the compliance of an individual element. The total compliance $K(\xi ,L)$ of this system of $(L/\xi )^d$ boxes is therefore:

$$\displaystyle \begin{aligned} K(\xi,L) = K(\xi,\xi) ( \frac{L}{\xi} ) ^{d-2} \; . \end{aligned} $$

(10.5)

Young’s modulus can then be found as

$$\displaystyle \begin{aligned} E(\xi,L) = L^{-(d-2)} K(\xi,L) = \frac{K(\xi,\xi)}{\xi^{d-2}} \; . {} \end{aligned} $$

(10.6)

In order to progress further we need to find the compliance $K(\xi ,\xi )$. This is the compliance of the percolation system at $p = p_c$ when the system size L is equal to the correlation length $\xi$. We are therefore left with the problem of finding the compliance of the spanning cluster at $p = p_c$ as a function of system size L.

10.1.2 Compliance of the Spanning Cluster at $p = p_c$

Again, we expect from experience that the compliance will scale with the system size with a dimension $\tilde {\zeta }_{K}$:

$$\displaystyle \begin{aligned} K \propto L^{\tilde{\zeta}_{K}} \; . \end{aligned} $$

(10.7)

We will follow our now standard approach: We assume a scaling behavior, establish a set of bounds for K, which will also serve as a proof of the scaling behavior of K, and then use this result to develop a general theory for $K(p,L)$.

Energy, Force and Elongation of the System

We will use arguments based on the total energy of the system. The total energy of a system subjected to a force $F = F_z$ resulting in an elongation $\varDelta L$ is:

$$\displaystyle \begin{aligned} U = \frac{1}{2} K (\varDelta L)^2 \; , \end{aligned} $$

(10.8)

where the elongation $\varDelta L$ is related to the force F through, $\varDelta L = F/K$. Consequently,

$$\displaystyle \begin{aligned} U = \frac{1}{2} K (\frac{F}{K})^2 = \frac{1}{2} \frac{F^2}{K} \; . \end{aligned} $$

(10.9)

We can therefore relate the elastic energy of a system subjected to the force F directly to the compliance of that system.

Upper Bound for the Compliance

Our arguments will be based on the geometrical picture we have of the spanning cluster when $p = p_c$. The cluster consists of singly connected bonds, blobs, and dangling ends. The dangling ends do not influence the elastic behavior, and can be ignored in our discussion. It is only the backbone that contribute to the elastic properties of the spanning cluster. We can find an upper bound for the compliance by considering the singly connected bonds. The system consists of blobs and singly connected bonds in series. The compliance must include the effect of all the singly connected bonds in series. However, adding the blobs in series as well will only contribute to lowering the compliance. We will therefore get an upper bound on the compliance, by assuming all the blobs to be infinitely stiff, and therefore only include the effects of the singly connected bonds.

Let us therefore study the elastic energy in the singly connected bonds when the cluster is subjected to a force F. The energy, U, can be decomposed in a stretching part, $U:s$, and a bending part, $U:b$: $U = U_s + U_b$.

For a singly connected bond from site i to site j, the change in length, $\delta \ell _{ij}$, due to the applied force F is $\delta \ell _{ij} = F/k$, where k is the force constant for a single bond. The energy due to stretching, $U:s$, is therefore

$$\displaystyle \begin{aligned} U_{s} = \sum_{ij} \frac{1}{2} k \delta \ell_{ij}^2 = \sum_{ij} \frac{1}{2} k (\frac{F}{k})^2 = \frac{1}{2} \frac{M_{SC}}{k} F^2 \; , \end{aligned} $$

(10.10)

where $M:{SC}$ is the mass of the singly connected bonds.

We can find a similar expression for the bending terms. For a bond between sites i and j, the change in angular orientation, $\delta \phi _{ij}$ is due to the torque $T = r_i F$, where $r:i$ is the distance to bond i in the direction normal to the direction of the applied force F: $\delta \phi _{ij} = T/\kappa$. The contribution from bending to the elastic energy is therefore

$$\displaystyle \begin{aligned} U_{b} = \sum_{ij} \frac{1}{2} \kappa (\delta \phi_{ij})^2 = \frac{1}{2} \sum_{ij} \kappa (\frac{r_i F}{\kappa})^2 = \frac{1}{2\kappa} M_{SC} R_{SC}^2 F^2 \; , \end{aligned} $$

(10.11)

where

$$\displaystyle \begin{aligned} R_{SC}^2 = \frac{1}{M_{SC}} \sum_{ij} r_i^2 \; , \end{aligned} $$

(10.12)

where the sum is taken over all the singly connected bonds.

The elastic energy of the singly connected bonds is therefore:

$$\displaystyle \begin{aligned} U_{SC} = (\frac{1}{2k} + \frac{R_{SC}^2}{2 \kappa}) M_{SC} F^2 \; , \end{aligned} $$

(10.13)

and the compliance of the singly connected bonds is

$$\displaystyle \begin{aligned} K_{SC} = \frac{F^2}{2U} = \frac{1}{( 1/k + R_{SC}^2/\kappa) M_{SC} } \; . \end{aligned} $$

(10.14)

which is an upper bound for the compliance K of the system.

Lower Bound for the Compliance

We can make a similar argument for a lower bound for the compliance K of the system. The minimal path on the spanning cluster provides the minimal compliance. The addition of any bonds in parallel will only make the system stiffer, and therefore increase the compliance. We can determine the compliance of the minimal path by calculating the elastic energy of the minimal path. We can make an identical argument to what we did above, but we need to replace $M:{SC}$ with the mass, $M:{min}$, of the minimal path, and the radius of gyration $R:{SC}^2$ with the radius of gyration of the bonds on the minimal path $R:{min}^2$.

Kantor [20] has provided numerical evidence that both $R:{min}^2$ and $R:{SC}^2$ are proportional to $\xi ^2$. When we are studying the spanning cluster at $p = p_c$ this corresponds to $R:{min}$ and $R:{SC}$ being proportional to L. This shows that the dominating term for the energy is the bending and not the stretching energy when p is approaching $p:c$.

Bounded Expression for the Compliance K

We have therefore determined the scaling relation

$$\displaystyle \begin{aligned} K_{min} \leq K \leq K_{SC} \; , \end{aligned} $$

(10.15)

where we have found that when $L \gg 1$, $K_{min} \propto L^{-(D_{min} + 2)}$ and $K_{SC} \propto L^{-(D_{SC} + 2)}$. That is:

$$\displaystyle \begin{aligned} L^{-(D_{min} + 2)} \leq K(L) \leq L^{-(D_{SC} + 2)} \; . \end{aligned} $$

(10.16)

Because $K(L)$ is bounded by two power-laws in L (for all values of L), we have also demonstrated that $K(L)$ also is a power-law in L with an exponent $\tilde {\zeta }_K$ satisfying the relation

$$\displaystyle \begin{aligned} -(D_{min} + 2) \leq \tilde{\zeta}_K \leq - (D_{SC} + 2) \; . \end{aligned} $$

(10.17)

10.1.3 Finding Young’s Modulus $E(p,L)$

This scaling relation gives us $K(p:c,L)$. We use this expression to find $K(\xi ,\xi )$, the compliance of a system of size $\xi$ from (10.6):

$$\displaystyle \begin{aligned} E(\xi,L) = \frac{K(\xi,\xi)}{\xi^{d-2}} \propto \frac{\xi^{\tilde{\zeta}_K}}{\xi^{d-2}} \propto \xi^{\tilde{\zeta}_K - (d-2)} \: . \end{aligned} $$

(10.18)

We have therefore found a relation for the scaling exponent $\tau$:

$$\displaystyle \begin{aligned} E(p,L) = \xi^{-( d-2 - \tilde{\zeta}_K)} \propto (p - p_c)^{( d-2 - \tilde{\zeta}_K)\nu} \propto (p - p_c)^{\tau} \; . \end{aligned} $$

(10.19)

The exponent $\tau$ is therefore in the range:

$$\displaystyle \begin{aligned} (d-2 + D_{SC} + 2)\nu \leq \tau \leq (d-2 + D_{min} + 2)\nu \; , \end{aligned} $$

(10.20)

Bounds on the Exponent $\tau$

The resulting bounds on the scaling exponents are:

$$\displaystyle \begin{aligned} \left( D_{SC} + 2 \right) \nu \leq \tau \leq \left( D_{min} + 2 \right) \nu \; , \end{aligned} $$

(10.21)

For two-dimensional percolation the exponents are approximately

$$\displaystyle \begin{aligned} 3.41 \leq \tau \leq 3.77 \; , \end{aligned} $$

(10.22)

Similarity Between the Flow and the Elastic Problems

We see that the bounds are similar to the bounds we found for the exponent $\tilde {\zeta }_{R}$. This similarity lead Sahimi [29] and Roux [27] to conjecture that the elastic coefficient E and the conductivity g is related through

$$\displaystyle \begin{aligned} \frac{E}{g} \propto \xi^{-2} \; . \end{aligned} $$

(10.23)

and therefore that

$$\displaystyle \begin{aligned} \tau = \mu + 2\nu = (d + \tilde{\zeta}_R)\nu \; . \end{aligned} $$

(10.24)

which is well supported by numerical studies.

In the limit of high dimensions, $d \ge 6$, the relation $\tau = \mu + 2 \nu = 4$ becomes exact. However, we can use as a rule of thumb that the exponent $\tau \simeq 4$ in all dimensions $d \geq 2$.