Strain tensor determination of compressed individual silica sand particles using high-energy synchrotron diffraction

Abstract

The three-dimensional x-ray diffraction (3DXRD) nondestructive technique was used to measure lattice strains within individual sand particles subjected to compressive loading. Three experiments were conducted on similar single columns of silica sand particles with particle sizes between 0.595 mm and 0.841 mm. In each experiment, three sand particles were placed inside an acrylic mold with an inner diameter of 1 mm. Multiple in situ 3DXRD scans were acquired for each sand column as compressive load was increased. The volume-averaged lattice strain tensor was calculated for each sand particle. In addition, particle orientation and volumetric strain were calculated for individual sand particles. The axial normal strain  zz exhibited a linear response in the range of 0 to 10 -3 when the applied compressive axial load (F) increased from 0 to ~30 N when one particle in the sand column fractured. Stress tensor of individual particles was calculated from the acquired lattice strain measurements and elastic constants of silica sand that were reported in the literature. To the best of our knowledge, there have been no reported experimental measurements of the lattice strain tensor measurements within individual silica sand particles. The quantitative measurements reported in this paper at the particle level are very valuable for developing, validating or calibrating micromechanics-based finite element and discrete element models to predict the constitutive behavior of granular materials. 3DXRDrepresents an exciting new non-destructive technique to directly measure constitutive behavior at the scale of individual particles.

3D Experimental Measurement of Lattice Strain and Fracture Behavior of Sand Particles Using Synchrotron X-Ray Diffraction and Tomography

“…The computation of the elastic strain tensor is based on the following simplified relationship: grain-averaged lattice strain is directly proportional to the 2θ shifts of reflections recorded in unloaded and loaded conditions. In this paper, the lattice strain derivation is explained briefly and further details can be found in Margulies, et al (2002) and Alshibli, et al (2013). The lattice strain expression can be derived by differentiating Bragg's law and expressed as (Margulies et al 2002): (2) 11…”

“…Therefore, the lattice strain calculations were performed using Friedel pairs to minimize the error in the diffraction data analysis. Further information about the formulation and details of the lattice strain computation can be found in Poulsen (2004) and Alshibli et al (2013).…”

“…Recently, 3D synchrotron x-ray diffraction (3DXRD) has emerged as a novel nondestructive technique that measures the volume-averaged lattice strain of individual particles (i.e., the average strain in the crystal lattices of sand particles) within granular specimens composed of few particles (Hall et al 2011;Alshibli et al 2013;Hall and Wright 2015). Lattice strain tensors can be measured experimentally and used to compute the corresponding stress tensor assuming an elastic stress-strain relationship.…”

“…These lattice stress-strain measurements along with the 3D kinematics of particles can provide essential experimental measurements to develop and calibrate 4 advanced micro-mechanics based constitutive models using real experimental particle-scale measurements. Alshibli, et al (2013) measured the strain tensor of individual particles in a column composed of three natural silica sand particles subjected to 1D compression. Wright (2014, 2015) measured the average lattice elastic strain values of 31 synthetic single crystal silica particles in a granular assembly of 96 particles.…”

“…Wright (2014, 2015) measured the average lattice elastic strain values of 31 synthetic single crystal silica particles in a granular assembly of 96 particles. This paper builds on the work of Alshibli, et al (2013), and focuses on measuring the particle-averaged lattice strain of a larger number of natural silica sand particles in a specimen that was loaded under 1D compression. In addition, the beamline incorporated SMT capabilities that can image the same specimen.…”