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Methods of particles are now recognized as an effective tool for numerical modeling of dynamic mechanical and coupled processes in solids and liquids. This chapter is devoted to a brief review of recent advances in the development of the popular particle-based discrete element method (DEM). DEM is conventionally considered as a highly specialized technique for modeling the flow of granular media and the fracture of brittle materials at micro- and mesoscopic scales. However, in the last decade, great progress has been made in the development of the formalism of this method. It is largely associated with the works of the scientific group of Professor S. G. Psakhie. The most important achievement of this group is a generalized formulation of the method of homogeneously deformable discrete elements. In the chapter, we describe keystones of this implementation of DEM and a universal approach that allows one to apply various rheological models of materials (including coupled models of porous fluid-saturated solids) to a discrete element. The new formalism makes possible qualitative expansion of the scope of application of the particle-based discrete element technique to materials with various rheological properties and to the range of considered scales form microscopic to macroscopic. The capabilities of this method are especially in demand in the study of the features of contact interaction of materials. To demonstrate these capabilities, we briefly review two recent applications concerning (a) the effect of adhesive interaction on the regime of wear of surface asperities under tangential contact of bodies and (b) the nonmonotonic dependence of the stress concentration in the neck of the human femur on the dynamics of hip joint contact loading.

The paper presents an approach to developing the mathematical formalism of the discrete element method to numerically study the inelastic behavior and fracture of brittle materials under dynamic loading. The approach adopts the basic principles of the kinetic theory of strength which postulate the finite time of nucleation of discontinuities and relaxation of local stresses in the material. A general methodology is proposed for constructing dynamic (kinetic) models of the mechanical behavior of a discrete element based on quasi-static models and using three dynamic material parameters (time parameters). The physical meaning of these parameters is discussed, and a method is proposed for estimating the magnitude of the parameters for a considered material using standard experimental data on its mechanical characteristics. The approach is verified by a dynamic formulation of two-parameter models of inelasticity and strength of brittle materials within the method of simply deformable discrete elements. The proposed way to the dynamic generalization of conventional quasi-static mechanical models is applicable to various Lagrangian numerical methods and makes it possible to numerically study the dynamic behavior features and to predict the mechanical characteristics of brittle materials at different strain rates (up to strain rates 103 s−1) and different types of stress state.

In the paper, we theoretically studied the dynamics of mode II crack propagation in brittle solids with taking into account the finite duration of incubation of local fracture. The study was carried out by computer simulation using the movable cellular automaton method. We analyzed the influence of the magnitude of fracture incubation time on the fulfillment of the scale-invariance condition of dynamic crack propagation. We proposed a definition of this condition in terms of fracture incubation time and introduced a new physically based criterion to classify structural levels of fracture in brittle materials.

A liquid in a pore volume influences on a strength of porous permeable media, both brittle and elastic-plastic. Despite of the dilation of the latter, a pore pressure of a liquid can remain nonzero under constrained shear loading. In the result, a liquid pressure contribute into a stress-strain state of a medium, and, correspondingly, into a strength. On the other hand, a liquid pressure depends on transportation properties of a medium namely on porosity and permeability, where the latter is determined by a characteristic diameter of filtration channels in a solid skeleton. A superposition and interplay of filtration of a liquid and its compression in pores originates a dynamical distribution of a pore pressure in a bulk of material. The non-linearity and interconnectedness of the mentioned processes clearly demonstrate the necessity of application of numerical methods for a studying of strength properties of such permeable liquid-filled media. We have studied a dependence of shear strength of a water-filled elastic-plastic sample under constrained conditions in the framework of the hybrid cellular automaton method that represents a variation of a DEM with explicit account of a liquid in pores. An elastic-plastic sample has been mounted between purely elastic permeable blocks that moved in lateral direction with a constant velocity. There were periodic boundary conditions in lateral direction. In order to create an initial hydrostatic compression in a volume, a pre-loading was performed before shearing. The elastic blocks and the sample had the same values of porosity, permeability, elastic moduli etc. We varied the values of the shear rate, the value of hydrostatic compression, width of the sample and the permeability. We have found that the strength of the sample depends on a cooperation of the following processes: 1) increase of a mean stress in a medium under shear; 2) dilation of elastic-plastic sample after reaching a yield point; 3) mass transfer of a fluid in a pore volume of a sample and elastic blocks and 4) redistribution of a fluid pressure due to its mass transfer. The observed effects together with the results of numerical simulations allowed us to suggest a binomial dependence of shear strength of an elastic-plastic sample on permeability, loading parameters and geometry of a sample. The first term of this dependence describes the exponential decrease of strength of a sample due to the decrease of effective stiffness of the elastic blocks under outflow of a liquid into an excess pore volume of a sample. Note that the latter leads to the decrease of the degree of constraint. The second term characterizes the influence of filtration on an increase of effective stiffness of the sample due to the growth of pore pressure of a liquid. This, in turn, leads to the increase of the degree of constraint of the sample. The parameters of the given dependence represent the combinations of the values of loading, width of a sample, its physical-mechanical properties, including permeability, and physical mechanical properties of a liquid.

A two-scale mechanical model of multiphase materials with the hard skeleton and different content of interstitial soft matter was developed within the framework of the formalism of movable cellular automaton method. In the paper, we numerically studied a particular example of such kind of materials, namely a heterogeneous porous composite with brittle matrix and plastic soft inclusions. The model was applied to study the fracture pattern and mechanical properties of mesoscopic samples with a linear distribution of the local porosity in the depth of the material under uniaxial compression. Simulation results showed the essentially nonlinear dependence of their elastic and strength properties on the degree of pore space filling. Depending on the sign of the gradient of porosity, the value of compression strength of partially filled samples can significantly increase or remain constant with increase in the volume fraction of filled pore space. It is shown that the combination of the parameters of pore structure, including a sign of the porosity gradient and the fraction of filled pore space, determines the shape of the main cracks and their localization.

Earlier the authors developed a discrete element based numerical model of dynamic mechanical behavior of brittle materials. The model is based on the concept of kinetic theory of strength. An important problem in the application of the proposed model to the study of features of dynamic response of real materials is lack of information about the dependence of model parameters on stress relaxation times (including time of fracture development). The present paper proposes an analytical approach to determining these dependencies for different brittle materials.

The paper is devoted to the study of dynamic propagation of mode II cracks in porous brittle materials with nanoscale pore size. We compared static (shear strength) and dynamic parameters of crack growth in dry and fluid saturated nanoporous brittle materials at different degrees of confinement. We have shown that pore fluid in nanoporous brittle materials influences mainly the dynamics of crack propagation. This leads in particular to pronounced peculiarities of the dependence of the critical value of dimensionless geometrical parameter of the initial crack (it majorizes the interval of the ratios of length to thickness for the cracks that are capable to accelerate to intersonic velocity) on applied crack normal stress. The results of the study are relevant for understanding the conditions of supershear regime of propagation of mode II cracks as well as for assessment of the ability of mode II cracks in brittle materials (including nanoporous fluid-saturated solids) to develop in supershear regime.

The paper is devoted to theoretical study of the longitudinal shear (mode II) crack unstable growth dynamics in brittle materials. We considered two main regimes of the dynamic propagation of the crack (sub-Rayleigh and supershear) and their implementation conditions. The research was carried out by computer simulation with the Movable Cellular Automaton method, using the generalized kinetic fracture model, which takes into account the finite duration of local fracture (fracture incubation time). It is shown that the fracture incubation time is a key parameter, which determines the transition conditions of the shear crack growth process from the sub-Rayleigh regime to supershear.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in AIP Conference Proceedings 1783, 020207 (2016) and may be found at https://doi.org/10.1063/1.4966501. The paper is devoted to the study of the influence of crack-normal stress on the shear strength of the brittle material with initial crack and the geometrical condition of acceleration of dynamically growing crack towards the longitudinal wave speed. We considered elastic-brittle permeable materials with nanoscale pore size. We have shown that pore fluid in nanoporous brittle materials influences mainly the condition of shear crack propagation transition from conventional sub-Rayleigh regime to supershear one. The results of the study make it possible to assess the ability of initial cracks in brittle materials to develop in supershear regime under the condition of confined longitudinal shear.

A two-scale mechanical model of brittle porous material partially filled with plastic filler (inclusions) was developed within the framework of the formalism of movable cellular automaton method. The model was applied to study the mechanical properties of mesoscopic samples with a linear distribution of the local porosity in the depth of the material. Calculation results showed essentially nonlinear dependence of their elastic and strength properties on the degree of pore space filling. It is found that depending on the sign of the gradient of porosity the value of shear strength of partially filled samples can significantly increase or remain constant with increase in the value of the degree of filling.

A method of numerical simulation of liquid-saturated porous media, that represents a combination of particle method and finite-difference method, namely, the hybrid cellular automaton method was proposed. It allows taking into account inelastic deformations, dilation and fracture of solid skeleton as well as the influence of pore pressure on the stress state of the skeleton and the redistribution of a liquid in filtration volume of porous medium. The method was applied to study the influence of viscous compressible liquid in pores of material on its strength and fracture. It has been shown that the strength of brittle liquid-saturated specimens depends on the material properties and the geometry of porosity, the physical-mechanical properties of the liquid etc. The latter shows the topicality of application of numerical methods to study and predict strength properties of fluid-saturated media under loading.

In the present paper the multiscale approach to the construction of multiscale rheological models of brittle materials with heterogeneous internal structure is implemented within the numerical method of movable cellular automata (MCA). Nikolaevsky's plasticity model with non-associated flow law was chosen as a rheological model for zirconium alumina concrete at the different structural scales. The mechanical properties (including parameters of the rheological model and strength values) of zirconium alumina concrete were obtained for both macroscopic and mesoscopic scales. The effect of strain rate on material behavior was analyzed in case of uniaxial compression and tension.