Although spectacular advances in hydraulic fracturing, also known as fracking, have taken place and many aspects are well understood by now, the topology, geometry, and evolution of the crack system remain an enigma and mechanicians wonder: Why fracking works? Fracture mechanics of individual fluid-pressurized cracks has been clarified but the vital problem of stability of interacting hydraulic cracks escaped attention. First, based on the known shale permeability, on the known percentage of gas extraction from shale stratum, and on two key features of the measured gas outflow which are (1) the time to peak flux and (2) the halftime of flux decay, it is shown that the crack spacing must be only about 0.1 m. Attainment of such a small crack spacing requires preventing localization in parallel crack systems. Therefore, attention is subsequently focused on the classical solutions of the critical states of localization instability in a system of cooling or shrinkage cracks. Formulated is a hydrothermal analogy which makes it possible to transfer these solutions to a system of hydraulic cracks. It is concluded that if the hydraulic pressure profile along the cracks can be made almost uniform, with a steep enough pressure drop at the front, the localization instability can be avoided. To achieve this kind of profile, which is essential for obtaining crack systems dense enough to allow gas escape from a significant portion of kerogen-filled nanopores, the pumping rate (corrected for the leak rate) must not be too high and must not be increased too fast. Furthermore, numerical solutions are presented to show that an idealized system of circular equidistant vertical cracks propagating from a horizontal borehole behaves similarly. It is pointed out that one useful role of the proppants, as well as the acids that promote creation of debris in the new cracks, is to partially help to limit crack closings and thus localization. To attain the crack spacing of only 0.1 m, one must imagine formation of hierarchical progressively refined crack systems. Compared to new cracks, the system of pre-existing uncemented natural cracks or joints is shown to be slightly more prone to localization and thus of little help in producing the fine crack spacing required. So, from fracture mechanics viewpoint, what makes fracking work?–the mitigation of fracture localization instabilities. This can also improve efficiency by fracturing more shale. Besides, it is environmentally beneficial, by reducing flowback per m3 of gas. So is the reduction of seismicity caused by dynamic fracture instabilities (which are more severe in underground CO2 sequestration).

A brittle aggregate composite such as portland cement concrete or mortar is modeled in two dimensions as a system of perfectly rigid particles of various sizes separated by interface layers that are characterized hy a given force-displacement relation for the normal and tangential components. The force-displacement relation exhihits for the normal component a tensile strength limit followed hy a sudden drop of force to zero. The system of rigid particles is generated randomly. The particles arc not allowed to overlap and are generally not in contact; however. when the distance hetween the particles is less than a certain limit, a deformahle interface layer is introduced. Numerical simulation of a fracture specimen with a notch shows that the fracture front consists of an irregular band of interparticle cracks the width of which is ahout three maximum particle sizes. The interparticle cracks remain continuous and do not coalesce into a continuous line fracture until the deformation is very large. The load-displacement relation exhihits gradual softening after the peak force. with a rapid force decrease followed by a long tail of slowly decreasing force. These features qualitatively agree with observations of concrete.

Summary This work presents a new original formulation of the discrete element method (DEM) with deformable cylindrical particles. Uniform stress and strain fields are assumed to be induced in the particles under the action of contact forces. Particle deformation obtained by strain integration is taken into account in the evaluation of interparticle contact forces. The deformability of a particle yields a nonlocal contact model, it leads to the formation of new contacts, it changes the distribution of contact forces in the particle assembly, and it affects the macroscopic response of the particulate material. A numerical algorithm for the deformable DEM (DDEM) has been developed and implemented in the DEM program DEMPack. The new formulation implies only small modifications of the standard DEM algorithm. The DDEM algorithm has been verified on simple examples of an unconfined uniaxial compression of a rectangular specimen discretized with regularly spaced equal bonded particles and a square specimen represented with an irregular configuration of nonuniform‐sized bonded particles. The numerical results have been verified by a comparison with equivalent finite element method results and available analytical solutions. The micro‐macro relationships for elastic parameters have been obtained. The results have proved to have enhanced the modeling capabilities of the DDEM with respect to the standard DEM.

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In thermomechanics, the Taylor–Quinney coefficient specifies fraction of plastic work converted to heat. We challenge the nearly century-long interpretation. We postulate that some fraction of energy delivered to the plastically deformed material is responsible for readjustments of deformation pathways making the plastic flow a kinematically admissible process. The rerouting triggers mesoscale dynamic excitations and activates plasticity-induced heat. Another part of the energy is stored in lattice, while the rest of it contributes to the development of dislocation structures. According to this interpretation, plastic work is not converted to heat, but increases probability of the microstructural adaptiveness and, in this manner, contributes to configurational entropy of the system.

A mechanisms-based fracture model applicable to a broad class of cemented aggregates and, among them, plastic-bonded explosive (PBX) composites, is presented. The model is calibrated for PBX 9502 using the available experimental data under uniaxial compression and tension gathered at various strain rates and temperatures. We show that the model correctly captures inelastic stress-strain responses prior to the load peak and it predicts the post-critical macro-fracture processes, which result from the growth and coalescence of micro-cracks. In our approach, the fracture zone is embedded into elastic matrix and effectively weakens the material's strength along the plane of the dominant fracture

A constitutive model that includes the effects of elasticity, crystal reorientation (twinning or detwinning), phase transformations, and plasticity is presented. The model is applied to a uranium-6wt% niobium (U-6Nb) alloy, which is known to exhibit the shape memory effect. The inelastic strains for phase transformation and reorientation are based on the material’s phase diagrams. The model captures the effects of martensitic transformation and reorientation; both of the phenomena are coupled with the alloy’s phase diagrams, which are defined in terms of temperature, hydrostatic pressure, and shear stress. Computational simulations are presented that demonstrate the ability of the model to capture the relevant physical response of the material. Also, we include experimental results that demonstrate the phenomena of phase transformation, reorientation, and plasticity for low- and high-strain rates and for variations in temperature. The model replicates the experimental data for the broad range of strain rates and temperatures. Comparisons between the model and low-strain rate data, which demonstrate the shape memory effect of U-6Nb, are also provided.