Stress, strain, and bulk microstructure in a cohesive powder

Spin-echo small-angle neutron scattering is able to characterize powders in terms of their density-density correlation function. Here we present a microstructural study on a fine cohesive powder undergoing uniaxial compression. As a function of compression, we measure the autocorrelation function of the density distribution. From these measurements we quantify the typical sizes of the heterogeneities as well as the fractal nature of the powder packing. The fractal dimension increases with increasing stress, creating a more space-filling structure with rougher phase boundaries. The microscopic stress-strain relation showed the same nonlinear behavior as the macroscopic relation. In this way it was possible to link the macroscopic mechanical response with the evolution of microstructure inside the bulk of the cohesive powder. The total macroscopic compressive strain is in agreement with a corresponding decrease in microstructural length scales.

Figure 1

Top: Examples of reconstructed 1D density distributions $\rho \left(x\right)$ based on the correlation function Eq. (12). Bottom: Corresponding correlation functions. The characteristic size is here $a=1$.

Figure 2

Load cell used in the stress-strain measurements on the cohesive powder. The initial height of the powder packing was 6.5 mm and it was subsequently strained by nine increments of 0.25 mm. The stress was measured using Flexiforce load sensors situated at the first Al window.

Figure 3

Stress versus grain packing fraction ${\varphi }_{\text{grain}}$ relationship for the Sipernat-310 powder. The inset shows a linear relationship between the logarithm of the stress versus strain $\delta$. Note that the first two points yielded no measurable stress on the Flexiforce load sensors.

Figure 4

Polarization plotted as a function of $z$. From top to bottom these measurements correspond to a uniaxial stress (strain) of 120 kPa (35%), 7.4 kPa (23%), 1 kPa (15%), and 0 kPa (3.8%). The full curves are fitted according to Eqs. (13, 14, 8). Unless shown, the error falls within the marker symbol.

Figure 5

Initial slopes of the polarization $d\text{ln}\left(P\right)/dz$. The slopes are proportional to the grain packing fraction and the sample thickness multiplied by a constant [see Eq. (19)]. Dividing out the known thickness and the packing fraction $\left[C={\varphi }_{\text{grain}}\left(1-{\varphi }_{\text{grain}}\right)t\right]$ should yield a constant term for all measurements as seen in the lower right figure. This analysis makes it possible to determine the solid grain density, yielding about $1.1\text{g}/{\text{cm}}^3$.

Figure 6

Characteristic size $a$ and Hurst exponent $H$ obtained when applying the model Eq. (13) to the measurement. The model shows that the characteristic size as well as the Hurst exponent $H$ decrease as a function of compression (higher ${\varphi }_{\text{grain}}$). The packing fractions are here expressed in terms of the grain packing fraction ${\varphi }_{\text{grain}}$ defined in Eq. (18)

Figure 7

Logarithm of the stress plotted as a function of microstructural strains $\delta a$ and $\delta \xi$ together with the macroscopic stress-strain relationship.