Modelling the behavior of complex media by jointly using discrete and continuum approaches

Abstract

Usually, computer simulation of the behavior of materials and complex media is based on the continuum approach, which uses highly developed mathematical apparatus of continuous functions. The capabilities of this approach are extremely wide, and the results obtained are well known. However, for of a number of very important processes, such as severe plastic deformation, mass mixing, damages initiation and development, material fragmentation, and so on, continuum methods of solid mechanics face certain hard difficulties. As a result, a great interest for the approach based on a discrete description of materials and media has been growing up in recent years. Because both continuum and discrete approaches have their own advantages and disadvantages and a great number of engineering software has been created based on continuum mechanics, the main line of discrete approach development seems to be not a substitute but a supplement to continuum methods in solving complex specific problems based on a joint using of the continuum and discrete approaches. This chapter shows an example of joining discrete element method and grid method in an effort to model mechanical behavior of complex fluid-saturated poroelastic medium. The presented model adequately accounts for the deformation, fracture, and multiscale internal structure of a porous solid skeleton. The multiscale porous structure is taken into account implicitly by assigning the porosity and permeability values for the enclosing skeleton, which determine the rate of filtration of a fluid. Macroscopic pores and voids are taken into account explicitly by specifying the computational domain geometry. The relationship between the stress-strain state of the solid skeleton and pore fluid pressure is described in the approximations of a simply deformable discrete element and Biot’s model of poroelasticity. The capabilities of the presented approach were demonstrated in the case study of the shear loading of fluid-saturated samples of brittle material. Based on simulation results, a generalized logistic dependence of uniaxial compressive strength on loading rate, mechanical properties of the fluid, and enclosing skeleton and on sample dimensions was constructed. The logistic form of the generalized dependence of the strength of fluid-saturated elasticbrittle porous materials is due to the competition of two interrelated processes, such as pore fluid pressure increase under solid skeleton compression and fluid outflow from the enclosing skeleton to the environment. Another application of the presented approach is the study of the shear strength of a water-filled sample under constrained conditions. An elastic-plastic interface was situated between purely elastic permeable blocks that were loaded in the lateral direction with a constant velocity; periodic boundary conditions were applied in the lateral direction. In order to create an initial hydrostatic compression in a volume, a pre-loading was performed before shearing. The results of simulation show that shear strength of an elastic-plastic interface depends nonlinearly on the values of permeability and loading parameters. An analytical relation that approximates the obtained results of numerical simulation was proposed.