Non-contact stress measurement in granular materials via neutron and X-ray diffraction: theoretical foundations

Abstract

Model validation remains a serious problem within the field of computational granular materials research. In all cases the rigor of the validation process is entirely dependent on the quality and depth of the experimental data that forms the point of comparison. Neutron and X-ray diffraction methods offer the only quantitative non-contact method for determining the spatially resolved triaxial stress field within granular materials under load. Measurements such as this can provide an unprecedented level of detail that will be invaluable in validating many models. In this paper the theoretical foundation underpinning diffraction-based strain measurements, their conversion to local stress in the particles and ultimately into the bulk stress field is developed. Effects such as elastic anisotropy within the particles of the granular material, particle plasticity and locally inhomogeneous stress distribution are shown to not offer any obstacles to the method and a detailed treatment of the calculation of the bulk stresses from the particle stresses is given.

Appendix: Bulk stress within the gauge volume—a micromechanical definition

Over recent years, micromechanical definitions of bulk stress have received a great deal of attention (e.g. [19–32]). The most common approach is to define bulk stress as the volume average of actual stress within the RVE. The problem that remains is then to define a way in which to calculate the average from a discrete set of contact forces within an assembly. Historically, the majority of approaches apply the mean stress theory which is a direct outcome of the Gauss–Ostrogradsky divergence theorem from the calculus of vector fields. The mean stress theory states that, in the absence of body stresses, the average stress tensor within a closed volume $V$ can be calculated via the following surface integral over the boundary $\partial V$;

$$\begin{aligned} {\bar{\varvec{\sigma }}}=\frac{1}{V}\int \limits _V \varvec{\sigma } dV =\frac{1}{V}\int \limits _{\partial V} {\tau \otimes xdA} , \end{aligned}$$

(12)

In this expression $\tau (x)=\varvec{\sigma }\hat{{n}}$ represents the tractive force (per unit area) experienced at the point $x\in \partial V$, which has unit normal $\hat{{n}}$. The symbol $\otimes$ denotes the dyadic (tensoral) product between vectors.

The traditional approach to calculating bulk stress considers an RVE that contains whole particles only, allowing the bulk stress to be calculated as a discrete sum of dyadic products of force and position over all of the contact on the boundary (first outlined by Love [20]). Unfortunately, this approach cannot directly be used in respect to the stress measured by diffraction methods as the boundary of the gauge volume intersects many particles. ⁸ We will develop a slightly different (but statistically equivalent) calculation of the bulk stress in an assembly in order to draw direct comparisons with the experimental measurements. We will begin by calculating the stress measured by ND as the ensemble average of the stress within the solid material inside the gauge volume.

Say that the gauge volume, V, contains N particles, some of which overlap the boundary $\partial V$ (see Fig. 8). We can consider this material from two perspectives; an assembly of grains (Fig. 8a) or a granular continuum that consists of both particles and voids (Fig. 8b).

Fig. 8

A representation of granular material within the gauge volume. a At the level of individual particles, the grains are subject to discrete contact forces. b The equivalent granular continuum is subject to distributed traction forces over the area of intersection between the boundary and the grains

From the perspective of the assembly of grains, consider a single particle, $n$, under static equilibrium, lying in a region of space at least partially within $V$ (see Fig. 9). We denote the portion of this particle that lies within $V$ as $V^{n}$ (i.e. $V\cap V^{n}=V^{n}$, and $V\cup V^{n}=V)$. This particle is subject to a number of forces located at the points of contact with other particles that lie within $V$. We will denote the force on particle $n$ originating from the contact with particle $m$ as $f^{nm}$ with a point of action of $x^{nm}\in V$. In the case where the particle overlaps the boundary we can form a set of section planes at $\partial V^{n}=\partial V\cap V^{n}$ and represent the effect of contacts external to $V$ by a distribution of traction forces $\tau ^{n}(x),\forall x\in \partial V^{n}$. Note that we can also express the tractive force per unit area of each contact in the form;

$$\begin{aligned} \tau ^{nm}\left( x \right) =f^{nm}\delta ^{2}\left( {x-x^{nm}} \right) \end{aligned}$$

(13)

Here $\delta ^{2}$ is the two dimensional Dirac delta function defined over a manifold representing the particle surface, with $x-x^{nm}$ expressed in a suitable two-dimensional curvilinear coordinate system.

Fig. 9

A single particle that lies (at least partially) within the gauge volume. This particle is subject to contact forces from within the gauge volume and tractions applied at the boundary

From Eq. 12, we can now express the average stress within $V^{n}$ as⁹;

$$\begin{aligned} {\bar{\varvec{\sigma }}^{n}}=\frac{1}{V^{n}}\left( {\sum _{m(\ne n)=1}^N {f^{nm}\otimes x^{nm}} +\int \limits _{\partial V^{n}} {\tau ^{n}\otimes xdA} } \right) \end{aligned}$$

(14)

As was demonstrated earlier, this expression represents the contribution of particle n towards the average stress measured by diffraction methods. The final measurement is equivalent to the ensemble average of this tensor over all particles within V, weighted by volume;

$$\begin{aligned} \left\langle {\bar{\varvec{\sigma }}} \right\rangle&= \frac{1}{V^{P}}\sum _{n=1}^N {V^{n}{\bar{\varvec{\sigma }}}^{n}} =\frac{1}{V^{P}}\left( \sum _{n=1}^N {\sum _{m\left( {\ne n} \right) =1}^N {f^{nm}\otimes x^{nm}}} \right. \nonumber \\&\quad \left. +\sum _{n=1}^N{\left( \int \limits _{\partial V^{n}} {\tau ^{n}\otimes xdA} \right) } \right) , \end{aligned}$$

(15)

where;

$$\begin{aligned} V^{P}=\sum _{n=1}^N {V^{n}} \end{aligned}$$

(16)

Equation 15 can be dramatically simplified by the observation that the internal contact forces come in equal and opposite pairs acting at the same location in space; i.e. $f^{nm}=-f^{mn}$ and $x^{nm}=x^{mn}$. From this perspective, the second summation in Eq. 15 vanishes and we are left with;

$$\begin{aligned} \left\langle {\bar{\varvec{\sigma }}} \right\rangle =\frac{1}{V^{P}}\sum _{n=1}^N {\left( {\int \limits _{\partial V^{n}} {\tau ^{n}\otimes xdA}} \right) } \end{aligned}$$

(17)

This expression is remarkably similar to the bulk stress within the granular continuum which is defined as follows; consider the material inside the gauge volume as being a continuous body (an element of granular continuum) subject to the set of boundary tractions $\left\{ {\tau ^{n}} \right\}$ as defined earlier. The ‘bulk’ stress in this element is simply the average of the stress tensor over all points within the volume. By applying Eq. 12 we can calculate an expression for the bulk stress as¹⁰;

$$\begin{aligned} \varvec{\sigma }_b =\frac{1}{V}\sum _{n=1}^N {\left( {\int \limits _{\partial V^{n}} {\tau ^{n}\otimes xdA} } \right) } \end{aligned}$$

(18)

As mentioned earlier, many (hopefully consistent) definitions of the bulk stress in a granular material can be found in the literature. Each contribution presents some variation to the homogenization process; such as variations on the nature of the boundary ([19, 27]), averaging over planes instead of volumes ([23, 29]), the use of branch vectors rather than absolute coordinates ([20–22]), formulating the problem in terms of graph theory ([24, 30]), the use of virtual work ([22, 23, 26]), and the use of course-graining approaches ([25, 31, 32]). In particular, there was some debate over the possibility that several of the formulations may result in an asymmetric stress tensor for a system in equilibrium [26], however this issue has largely been resolved in systems without contact moments (e.g. [27, 28]). Regardless of the controversy, in all cases an important point of commonality is that the calculation is based on a representative volume that includes both particles and voids in between (as in Eq. 18). This is in direct contrast with diffraction methods which measure the average stress within the particles alone (the voids have no crystal structure to probe).

This observation is apparent in the previous calculations; the bulk stress (Eq. 18) and the stress measured by diffraction methods (Eq. 17) are identical apart from the volume over which the ensemble average is calculated. By combining these two expressions, we find that;

$$\begin{aligned} \varvec{\sigma }_b =\frac{V^{P}}{V}\left\langle {\bar{\varvec{\sigma }}} \right\rangle \end{aligned}$$

(19)

where the ratio ${V^{P}}/V$ is recognised as the volume solid ratio within the gauge volume $e$ (as in Eq. 11).