On the Mori-Tanaka's method in cracked bodies

Effective elastic and conductive properties of 2-D random (“disordered”) mixtures of several types are examined by computational means. It is found that an “equivalent” material of simple microgeometry – a continuous matrix with elliptical inhomogeneities – can be identified, that matches both the elastic and the conductive properties, in the entire range of property contrast between constituents. Moreover, the ellipse eccentricities are almost the same for different types of the random mixtures in the volume fraction range (0.3 – 0.7); in this range, there is no need in specifying the type of a mixture, as far as the effective properties are concerned. It is also found that the effective properties of the considered random mixtures are well described by the Mori-Tanaka-Benveniste model (in spite of the fact that this model was not intended for them).

We also examined the effect of inhomogeneity interactions in a matrix composite on the cross-property connections between the elastic and conductive properties. Whereas the interactions generally produce strong effect on each of the properties, their effect on the connections is negligible (the latter can be taken from the non-interaction approximation).

It is argued that most findings related to 2-D random mixtures should apply to 3-D ones as well.

Rock pillars in rock engineering are affected by the stress concentration, which is uneven and can lead to potential instability after loading disturbance. In this paper, considering the heterogeneity of the rock, time-dependent velocity tomography based on the Kalman filter system is adopted to invert the velocity evolution of the rock monitored via acoustic emission (AE) under diverse stresses. Furthermore, the tomography uncertainty is investigated to validate the experimental accuracy of the loaded rock. The experimental results show that Kalman filter-based tomography can characterize the damages of the loaded rock, which is influenced by the distance from the sample’s end and heterogeneity of the rock. The rock damage and its growth rate increase when the loading increases or the cross section approaches the mid-waistline. The model uncertainty analysis demonstrates that the velocity tomography based on the Kalman filter has a high precision. Our study shows that for evaluating the stability of bearing rock pillars, we can simplify the rock pillar as a uniaxial loading rock and need to consider the rock’s heterogeneity for understanding the damage distribution and evolution of the loaded rock.

The cross-property connection between the effective elastic and conductive properties of a solid with cracks has been first given by Bristow (1960) for randomly oriented cracks, assuming that cracks have circular (penny) shapes. However, cracks in materials typically have “irregular” shapes and this factor, generally, has strong effect on the said properties (the strong stiffening effect produced by contacts between crack faces is an example). In addition, shapes of 3D cracks (for example, the statistics of the mentioned contacts) may be largely unknown. These factors make it difficult to estimate the effect of cracks on the elastic properties. We show that flat cracks of arbitrary shapes and orientation distributions produce similar effects on the elastic and conductive properties, so that changes of stiffnesses can be monitored by changes in electric conductivities. Extension to transversely isotropic materials is given. Possible extensions to non-flat crack shapes are discussed.

This paper proposed a micromechanics based constitutive model of rock considering compaction deformation, incorporating the intricate failure mechanisms of microcracks (i.e., opening, closing, friction, and slip) along with the overall deformation processes of rock (i.e., compaction, elasticity, plasticity, and damage). By combining the proposed continuum constitutive model with a discrete element numerical model featuring gapped flat joint contacts, the influence of initial cracks and crack volumes on rock compaction deformation was comprehensively analyzed from both discrete and continuous perspectives. Our findings reveal that the mechanical properties of the rock can be influenced by both the number and volume of cracks. The product of these two factors determines the overall void ratio, while the number of cracks exhibits an exponential relationship with the deformation resistance of the void under fixed void ratios. Furthermore, we observe that higher confining pressures result in reduced compaction deformation, and the compaction process is independent of force directionality. The comprehensive analysis from both continuous and discrete approaches offers a profound understanding of compaction deformation mechanisms, improves the identification efficiency of discrete compaction parameters, and enables the analysis of increased compaction deformation due to changes in crack states under external conditions (e.g., high temperature).

This paper is devoted to a novel micromechanics-based damage constitutive model for describing the mechanical behavior of quasi-brittle rocks under cyclic loading considering the effects of confining pressure. The proposed model is successfully employed to simulate cyclic loading tests under various confining pressures and upper limit stresses. Inspired by previous studies that focus on time-dependent creep deformation of quasi-brittle materials, the number of cyclic loading is regarded as equivalent to time scale. In the same spirit, the scalar-valued internal damage variable used to describe the degradation of rock materials is decomposed into two parts, namely an instantaneous component due to monotonic loading, and a fatigue-related component attributed to cyclic loading. Within the framework of irreversible thermodynamics, we present a Coulomb-type friction criterion and a strain energy release rate-based damage criterion. To describe the evolution of fatigue damage, we also propose a damage criterion based on the microstructure evolution. By the use of returning mapping procedure, the constructed model is applied to simulate two representative quasi-brittle rocks, namely sandstone and argillaceous quartz siltstone, and successfully captures all significant behavior features.

The mechanical properties of rock with pre-cracks on the free surface of roadways are anchored using a new type of constant resistance and large deformation (CRLD) cable. Taking the number of cracks as a variation, the PFC(Particle Flow Codes) software was used to assess the servo control of pallet to equivalent characteristics of CRLD cable. The results showed that owing to the constant resistance support effect of the pallet, new cracks sprout from both ends of the pallet and in the middle and deep areas and then propagated to form large through-cracks. The integrity of the rock mass near the pallet was better, and the smaller the number of pre-cracks, the larger this intact rock mass was. In addition, the Mori–Tanaka method better describes the variation in the anchored rock strength according to the number of pre-cracks by introducing parameters a and b, which represent the pallet reinforcement effect and crack correction, respectively. Finally, the energy absorption evolution of the constant resistance pallet was divided into uniform and non-uniform energy absorption stages. As the number of pre-cracks increased, the proportion of the uniform energy absorption stage increased and that of the non-uniform absorption stage decreased. As the number of pre-cracks increased, the slope S of the uniform-fitted energy absorption stage increased and the growth rate of S first increased and then decreased.