We propose a new framework for fracture mechanics, based on the idea of an approximate fracture geometry representation combined with approximate interface conditions. Our approach evolves from the shifted interface method, and introduces the concept of an approximate fracture surface composed of the full edges/faces of an underlying grid that are geometrically close to the true fracture geometry. The original interface conditions are then modified on the surrogate fracture geometry, by way of Taylor expansions. The shifted fracture method does not require cut cell computations or complex data structures, since the behavior of the true fracture is mimicked with standard integrals on the approximate fracture surface. Furthermore, the energetics of the true fracture are represented within the accuracy of the underlying polynomial finite element approximation and independently of the grid topology. The computational framework is presented here in its generality and then applied in the specific context of cohesive zone models, with an extensive set of numerical experiments in two and three dimensions.
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 this paper, a microstructural morphology based domain partitioning MDP methodology is developed for materials with non-uniform heterogeneous microstructure. The comprehensive set of methods is intended to provide a concurrent multi-scale analysis model with an initial computational domain that delineates regions of statistical homogeneity and inhomogeneity. The MDP methodology is intended to be a pre-processor to multi-scale analysis of mechanical behaviour and damage of heterogeneous materials, e.g. cast aluminium alloys. It introduces a systematic three-step process that is based on geometric features of morphology. The first step simulates high resolution microstructural information from low resolution micrographs of the material and a limited number of high resolution optical or scanning electron microscopy micrographs. The second step is quantitative characterization of the high resolution images to create effective metrics that can relate microstructural descriptors to material behaviour. The third step invokes a partitioning method to demarcate regions belonging to different length scales in a concurrent multi-scale model. Partitioning criteria for domain partitioning are defined in terms of microstructural descriptors and their functions. The effectiveness of these metrics in differentiating microstructures of a 319-type cast aluminium alloy with different secondary dendrite arm spacings SDAS is demonstrated. The MDP method establishes intrinsic material length scales for the different SDAS, namely, 23, 70 and 100 µm, and consequently subdivides the computational domain for concurrently coupling macro- and micromechanical analyses in the multi-scale model.
SUMMARYThis paper develops a microstructural morphology-based domain partitioning method (MDP) as a comprehensive pre-processor for multi-scale simulation of heterogeneous multi-phase materials. The MDP method systematically creates a multi-scale image simulation-characterization methodology to enable domain partitioning that can delineate homogenizable regions from those where explicit representation of phases is necessary for analysis. The methods are strictly based on geometric features of the material morphology and do not use any mechanical response functions. The first step in this development simulates high-resolution microstructural information from low resolution images of the domain and only a limited high resolution micrographs from optical or scanning electron microscopy. The second step uses quantitative characterization of these high resolution images with, e.g. phase distribution functions, to create effective metrics that can relate microstructural features to the material's physical behaviour. The third step invokes domain partitioning to demarcate regions corresponding to different length scales in a concurrent multi-scale model. Partitioning criteria are defined in terms of descriptors of microstructural characteristics and these are used to adaptively create multi-level domain partitions. The method developed is tested on a micrograph of a cast aluminium alloy A356.
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