Numerical study of one-dimensional compression of granular materials. I. Stress-strain behavior, microstructure, and irreversibility.

The behavior of a model granular material, made of slightly polydisperse beads with Hertz-Mindlin elastic-frictional contacts, in oedometric compression (i.e., compression along one axis, with no lateral strain) is studied by grain-level numerical simulations. We systematically investigate the influence of the (idealized) packing process on the microstructure and stresses in the initial, weakly confined equilibrium state, and prepare both isotropic and anisotropic configurations differing in solid fraction Φ and coordination number z. Φ (ranging from maximally dense to moderately loose), z (which might vary independently of Φ in dense systems), fabric and force anisotropy parameters, and the ratio K_{0} of lateral stresses σ_{2}=σ_{3} to stress σ_{1} in the compression direction are monitored in oedometric compression in which σ_{1} varies by more than three orders of magnitude. K_{0} reflects the anisotropy of the assembling process and may remain nearly constant in further loading if the material is already oedometrically compressed (as a granular gas) in the preparation stage. Otherwise, it tends to decrease steadily over the investigated stress range. It is related to force and fabric anisotropy parameters by a simple formula. Elastic moduli, separately computed with an appropriate matrix method, may express the response to very small stress increments about the transversely isotropic well-equilibrated states along the loading path, although oedometric compression proves an essentially anelastic process, mainly due to friction mobilization, with large irreversible effects apparent upon unloading. While the evolution of axial strain ε_{1} and solid fraction Φ (or of the void ratio e=-1+1/Φ) with axial stress σ_{1} is very nearly reversible, especially in dense samples, z is observed to decrease (as previously observed in isotropic compression) after a compression cycle if its initial value was high. K_{0} relates to the evolution of internal variables and may exceed 1 in unloading. The considerably greater irreversibility of oedometric compression reported in sands, compared to our model systems, should signal contact plasticity or damage.

Numerical study of one-dimensional compression of granular materials. II. Elastic moduli, stresses, and microstructure.

The elastic moduli of a transversely isotropic model granular material, made of slightly polydisperse elastic-frictional spherical beads, in equilibrium along a one-dimensional (oedometric) compression path, as described in the companion paper [M. H. Khalili et al., Phys. Rev. E 95, 032907 (2017)]10.1103/PhysRevE.95.032907, are investigated by numerical simulations. The relations of the five independent moduli to stresses, density, coordination number, fabric and force anisotropies are studied for different internal material states along the oedometric loading path. It is observed that elastic moduli, as in isotropic packs, are primarily determined by the coordination number, with anomalously small shear moduli in poorly coordinated systems, whatever their density. Such states also exhibit faster increasing moduli in compression, and larger off-diagonal moduli and Poisson ratios. Anisotropy affects the longitudinal moduli C_{11} in the axial direction and C_{22} in the transverse directions, and the shear modulus in the transverse plane C_{44}, more than the shear modulus in a plane containing the axial direction C_{55}. The results are compared to available experiments on anisotropic bead packs, revealing, despite likely differences in internal states, a very similar range of stiffness level (linked to coordination), and semiquantitative agreement as regards the influence of anisotropy. Effective medium theory (the Voigt approach) provides quite inaccurate predictions of the moduli. It also significantly underestimates ratios C_{11}/C_{22} (varying between 1 and 2.2) and C_{55}/C_{44} (varying from 1 to 1.6), which characterize elastic anisotropy, except in relatively weakly anisotropic states. The bulk modulus for isotropic compression and the compliance corresponding to stress increments proportional to the previous stress values are the only elastic coefficients to be correctly estimated by available predictive relations. We discuss the influences of fabric and force anisotropies onto elastic anisotropy, showing in particular that the former dominates in sample series that are directly assembled in anisotropic configurations and keep a roughly constant lateral to axial stress ratio under compression.

Three-dimensional full-field X-ray orientation microscopy.

A previously introduced mathematical framework for full-field X-ray orientation microscopy is for the first time applied to experimental near-field diffraction data acquired from a polycrystalline sample. Grain by grain tomographic reconstructions using convex optimization and prior knowledge are carried out in a six-dimensional representation of position-orientation space, used for modelling the inverse problem of X-ray orientation imaging. From the 6D reconstruction output we derive 3D orientation maps, which are then assembled into a common sample volume. The obtained 3D orientation map is compared to an EBSD surface map and local misorientations, as well as remaining discrepancies in grain boundary positions are quantified. The new approach replaces the single orientation reconstruction scheme behind X-ray diffraction contrast tomography and extends the applicability of this diffraction imaging technique to material micro-structures exhibiting sub-grains and/or intra-granular orientation spreads of up to a few degrees. As demonstrated on textured sub-regions of the sample, the new framework can be extended to operate on experimental raw data, thereby bypassing the concept of orientation indexation based on diffraction spot peak positions. This new method enables fast, three-dimensional characterization with isotropic spatial resolution, suitable for time-lapse observations of grain microstructures evolving as a function of applied strain or temperature.

Effect of Water on Elastic and Creep Properties of Self-Standing Clay Films.

We characterized experimentally the elastic and creep properties of thin self-standing clay films, and how their mechanical properties evolved with relative humidity and water content. The films were made of clay montmorillonite SWy-2, obtained by evaporation of a clay suspension. Three types of films were manufactured, which differed by their interlayer cation: sodium, calcium, or a mixture of sodium with calcium. The orientational order of the films was characterized by X-ray diffractometry. The films were mechanically solicited in tension, the resulting strains being measured by digital image correlation. We measured the Young's modulus and the creep over a variety of relative humidities, on a full cycle of adsorption-desorption for what concerns the Young's modulus. Increasing relative humidity made the films less stiff and made them creep more. Both the elastic and creep properties depended significantly on the interlayer cation. For the Young's modulus, this dependence must originate from a scale greater than the scale of the clay layer. Also, hysteresis disappeared when plotting the Young's modulus versus water content instead of relative humidity. Independent of interlayer cation and of relative humidity greater than 60%, after a transient period, the creep of the films was always a logarithmic function of time. The experimental data gathered on these mesoscale systems can be of value for modelers who aim at predicting the mechanical behavior of clay-based materials (e.g., shales) at the engineering macroscopic scale from the one at the atomistic scale, for them to validate the first steps of their upscaling scheme. They provide also valuable reference data for bioinspired clay-based hybrid materials.

ESEM study of the humidity-induced swelling of clay film.

We measured the humidity-induced swelling of thin self-standing films of montmorillonite clay by a combination of environmental scanning electron microscopy (ESEM) and digital image correlation (DIC). The films were about 40 µm thick. They were prepared by depositing and evaporating a suspension of clay and peeling off the highly oriented deposits. The rationale for creating such original samples was to obtain mesoscopic samples that could be used to bridge experimentally the gap between the scale of the clay layer and the engineering scale of a macroscopic clay sample. Several montmorillonite samples were used: the reference clay Swy-2, the same clay homoionized with sodium or calcium ions, and a sodium-exchanged Cloisite. The edges of the clay films were observed by ESEM at various relative humidity values between 14% and 95%. The ESEM images were then analyzed by DIC to measure the swelling or the shrinkage of the films. We also measured the adsorption/desorption isotherms by weighing the film samples in a humidity-controlled environment. In order to analyze our results, we compared our swelling/shrinkage and adsorption/desorption data with previously published data on the interlayer spacing obtained by X-ray diffraction and with numerical estimates of the interlayer water obtained by molecular dynamics simulation. The swelling and the hysteresis of this swelling were found to be comparable for the overall macroscopic films and for the interlayer space. The same correspondence between film and interlayer space was observed for the amount of adsorbed water. This suggests that, in the range of relative humidities values explored, the films behave like freely swelling oriented stacks of clay layers, without any significant contribution from the mesoporosity. The relevance of this result for the behavior of clayey sedimentary rocks and the differences with the behavior of nonoriented samples (powders or compacted powders) are briefly discussed.