SUMMARYThis short communication discusses an algorithm suited for the generation of periodic microstructures of granular media. Its particular features are a user-defined grain size distribution, a representative volume element which is intrinsically periodic ab initio and a user-defined termination criterion, controlled by an increase of volume fraction. For low densities our particle packings resemble fluids or gases, while we aim to reach for rather dense particle packings, similar to granular solids. The generated microstructures can thus be readily incorporated into large multiscale simulations, e.g. on the integration point level of a finite element analysis of a particular sand or concrete. The individual grain size distribution of the granular medium is incorporated through the introduction of different growth rates governing the final particle size distribution. We briefly sketch the generation of the representative volume element within a serial event-driven scheme and demonstrate how periodic boundary conditions are ensured throughout the representative volume element generation process. The potential of the suggested algorithm will be illustrated through the generation of two different periodic multi-disperse microstructures. They are based on different given grain size distributions, one for a quartz sand with a low non-uniformity index and one for concrete aggregates classified as A32 by the German standard norm DIN 1045 to have a rather large variation in grain size.
This contribution sets the focal point on the macroscopic impact of microscopic boundary conditions of discrete granular assemblies. We propose a two scale homogenization approach, containing a micro and a macro level. The microscale, describing the mechanical behavior of the single grains, is modeled by a discrete element method. On the macroscale, a continuum is assumed, discretized by a standard finite element method. Each point on the macroscale is assumed to have a corresponding micro structure, linked by the concept of a representative volume element. As a representative quantity, we focus on the Reynolds principle of dilatancy. Representative numerical examples include a slope-stability test as well as a bi-axial compression test.
We present a stabilized extended finite element formulation to simulate the hydraulic fracturing process in an elasto-plastic medium. The fracture propagation process is governed by a cohesive fracture model, where a trilinear traction-separation law is used to describe normal contact, cohesion and strength softening on the fracture face. Fluid flow inside the fracture channel is governed by the lubrication equation, and the flow rate is related to the fluid pressure gradient by the 'cubic' law. Fluid leak off happens only in the normal direction and is assumed to be governed by the Carter's leak-off model. We propose a 'local' U-P (displacement-pressure) formulation to discretize the fluid-solid coupled system, where volume shape functions are used to interpolate the fluid pressure field on the fracture face. The 'local' U-P approach is compatible with the extended finite element framework, and a separate mesh is not required to describe the fluid flow. The coupled system of equations is solved iteratively by the standard Newton-Raphson method. We identify instability issues associated with the fluid flow inside the fracture channel, and use the polynomial pressure projection method to reduce the pressure oscillations resulting from the instability. Numerical examples demonstrate that the proposed framework is effective in modeling 3D hydraulic fracture propagation.where the fluid pressure is uniformly equal to zero. Therefore, the traction on dry fracture face S d is purely governed by the traction-separation (cohesive) lawwhere t c represents the cohesive traction. We note that until now, we have not assumed any particular constitutive law for the bulk material. In other words, the previous equations are valid for both linear and nonlinear bulk material laws.
In recent years there has been an increase of interest in packed granular and discontinuous media. Due to the properties of such materials, a simple continuum approach is not appropriate to describe the material behavior. Breaking and forming of new contacts between grains, distinguishing for granular media, cannot be captured. We apply a computational homogenization method [6] which is based on a two scale concept to complete the task of modeling packed granular materials. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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