Micro-macro modeling of damage and healing in rock

Our lab has developed a wide range of phenomenological and micro-mechanical models of damage and healing for rock materials, particularly for salt rock, which has the ability to self-heal under conditions of pressure and temperature that are commonly encountered in deep geological storage facilities. This open-access article gives an overview of the contributions that we made between 2010 and 2020. Since then, we have applied the homogenization theory not only to model the effects of damage and healing on mechanical properties, but also on diffusion properties. Indeed, pressure solution involves mass transfer by dissolution, diffusion, and precipitation in pores or at grain interfaces, which may result in mechanical healing. Dislocation glide is another deformation mechanism that plays a significant role in the behavior of polycrystals. We used Eshelby’s self-consistent homogenization scheme with imperfect interfaces to calculate the macroscopic mechanical and diffusive properties of an elasto-viscoplastic porous composite made of imperfectly bonded crystals. Using halite as a model material, the proposed self-consistent model was calibrated and verified against published results of experimental creep tests. Simulations highlight that healing by grain boundary precipitation (by contrast with in-pore precipitation) is a limiting factor for pressure solution, because healed interfaces have lower diffusivity than fluid-filled interfaces. The homogenization approach provides an explanatory framework for the lower creep deformation observed for larger grains, and forecasts lower diffusivity for smaller grains. Sensitivity analyses show that grain boundary healing decelerates specimen compaction, while precipitation in the pores controls the evolution of effective diffusivity.

Characterization of microstructural features that are determinant to the behavior of matrix-inclusion composites

Variational Auto-Encoder for characterization of the microstructure features that most affect the local stress field

A non-linear Variational Auto-Encoder (NLVAE) was developed to reconstruct the plane-strain stress field in a solid with embedded cracks subjected to uniaxial tension, uniaxial compression and shear loading paths. The NLVAE successfully captures stress concentrations that develop across the loading steps as a result of crack propagation. Correlations observed between the evolution of fabric descriptors and that of their significant stress latent features indicate that the NLVAE can capture important microstructure transitions during the loading process. Crack connectivity, crack eccentricity, and the distribution of zones of highly connected opened cracks vs. zones with no cracks are the fabric descriptors that best explain the sequences of latent features that are the most important for the reconstruction of the stress field. Notably, the distributional shape, tail behavior, and symmetry of microstructure descriptor distributions have more influence on the stress field than basic measures of central tendency and spread.

Correlations between microstructure disturbances and effective property disturbances

A rigorous statistical analysis further highlighted the importance of microstructure descriptors that depend on correlation functions within and among the composite phases (matrix and inclusions). The study considered an enriched set of micro-descriptors, containing both “classical” descriptors (e.g., inclusion volume fraction, size distribution, orientation distribution) as well as “non-classical” descriptors which quantify the spatial correlations of any two given micro material points inside a random heterogeneous material. We focused on 2D composites consisting of a matrix with embedded inhomogeneities (or inclusions) of random spatial arrangement. Both phases were treated as homogeneous, linearly elastic and isotropic. Starting from a rich database of reference microstructures, new datasets of perturbed microstructures were created, by inducing changes emulating the physical processes of inclusion nucleation and growth. All microstructures were characterized using the enriched set of micro-descriptors, while their apparent stiffness tensor was computed numerically with the finite element (FE) method. A sensitivity analysis between the changes of the micro-descriptors and corresponding changes of the apparent stiffness tensor revealed that the “non-classical” descriptors are consistently highly important to the macroscopic behavior. This suggests that enhanced homogenization models, made dependent on the identified pertinent “non-classical” micro descriptors, could be of higher predictive capability than existing approaches.

Limitations of the homogenization schemes that do not account for the microstructural features identified as important to the behavior of composites

Transitions of micro-structure from a state that can be modeled by classic homogenization theories to a state that cannot

A statistical analysis complemented by a machine learning model allowed detection of the limitations of the Mori-Tanaka scheme when micro-cracks coalesce in a solid. The study aimed to detect in which microstructure conditions the Mori-Tanaka scheme is inappropriate to calculate the effective stiffness of a two-phase matrix-inclusion system. We analyzed the discrepancy between numerical and Mori-Tanaka stiffness estimates in 2D solids with crack-like flat cavities. The maximum transfer entropy that occurs between a microstructure feature and a stiffness component discrepancy can not only detect the “phase change” between a Mori-Tanaka-like cracked solid to a non-Mori-Tanaka-like cracked solid, but also reveal at which load step that phase change first occurs, and which microstructure features most affect that phase change. Further analysis with a binary classifier based on a Support Vector Machine (SVM) algorithm shows that the systematic calculation of nine microstructure features based on six statistical crack network descriptors at each step of a loading path can inform on the detection of a microstructure transition. The microstructure features identified here could thus be used to trigger the transition from one homogenization scheme to another during incremental stiffness updates, for example during the simulation of a load path.

Comparison between a classic homogenization theory and a variational method that depends on correlation-functions

We investigated the validity of two different analytical homogenization methods: the Mori-Tanaka mean-field theory and Milton’s correlation function-dependent bounds. We focused on biphase linearly elastic transversely isotropic composites. The composites consisted of a matrix reinforced with long fibers of either circular or irregular cross-section shapes formed by overlapping circles, with different degrees of radius polydispersity. The Mori-Tanaka effective stiffness depends on the phase moduli, volume fractions, and on a few geometric descriptors of the fibers that can be readily evaluated. In contrast, the computation of Milton’s bounds requires finer knowledge of the microstructure, in terms of two and three-point spatial correlation functions. The effective moduli estimates of the two methods were validated against the results of numerical homogenization using the finite element method. The predictions of the Mori-Tanaka scheme generally deteriorate with an increasing fiber volume fraction. By contrast, the average of Milton’s upper and lower bounds provides a highly accurate estimate for all three independent effective moduli, without any limitation on the fiber concentration. This study highlights the indisputable effect of the spatial correlation functions on the effective properties of composites.

Towards more predictive homogenization methods enriched with higher-order microstructure descriptors

We are working on high-accuracy expressions for the phase strain localization tensors, which are the fourth-order tensors that relate the average strain inside a given phase with the average strain in the composite representative volume element (RVE), and are essential for computing the effective stiffness tensor. Starting with a direct approach to the effective stiffness, our goal is to develop expressions for phase strain localization tensors that are more accurate than those employed in Eshelby-type approaches such as the Mori-Tanaka method. Next, we plan to introduce a novel approach for capturing nonlocal interactions in brittle heterogeneous (quasi-brittle) materials. The model’s predictions of the strain localization pattern will be validated against experimental measurements of the microscale strain fields in granite rock samples subjected to unconfined uniaxial compression. A second mapping will then be created, from the strain localization pattern to the effective stiffness tensor. In contrast to existing nonlocal models, the proposed framework will not require imposing any a priori internal length scale.