A disk-shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes

A novel domain integral approach is introduced for the accurate computation of pointwise J-integral and stress intensity factors (SIFs) of 3D planar cracks using tetrahedral elements. This method is efficient and easy to implement, and does not require a structured mesh around the crack front. The method relies on the construction of virtual disk-shaped integral domains at points along the crack front, and the computation of domain integrals using a series of virtual triangular and line elements. The accuracy of the numerical results computed for through-the-thickness, penny-shaped, and elliptical crack configurations has been validated by using the available analytical formulations. The average error of computed SIFs remains below 1% for fine meshes, and 2–3% for coarse ones. The results of an extensive parametric study suggest that there exists an optimum mesh-dependent domain radius at which the computed SIFs are the most accurate. Furthermore, the results provide evidence that tetrahedral elements are efficient, reliable and robust instruments for accurate linear elastic fracture mechanics calculations

On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics

This paper discusses the reproduction of the square root singularity in quarter-point tetrahedral (QPT) finite elements. Numerical results confirm that the stress singularity is modeled accurately in a fully unstructured mesh by using QPTs. A displacement correlation (DC) scheme is proposed in combination with QPTs to compute stress intensity factors (SIF) from arbitrary meshes, yielding an average error of 2–3%. This straightforward method is computationally cheap and easy to implement. The results of an extensive parametric study also suggest the existence of an optimum mesh-dependent distance from the crack front at which the DC method computes the most accurate SIFs

A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes

AbstractThis paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented

Evolution of fracture normal stiffness due to pressure dissolution and precipitation

The normal stiffness of a fracture is a key parameter that controls, for example, rock mass deformability, the change in hydraulic transmissivity due to stress changes, and the speed and attenuation of seismic waves that travel across the fracture. Non-linearity of normal stiffness as a function of stress is often attributed to plastic yield at discrete contacts. Similar surface-altering mechanisms occur due to pressure solution and precipitation over larger timescales. These processes partition the fracture surfaces into a flattened contact region, and a rough free surface that bounds the void space. Under low loads, contact occurs exclusively over the flattened part, leading to rapid, exponential stiffening. At higher loads, contact occurs over the rough surface fraction, leading to the conventional linear increase of stiffness with stress. It follows that a relationship exists between the history of in situ temperature and stress state of a rock fracture, and its subsequent deformation behavior

Quantification of fracture interaction using stress intensity factor variation maps

Accurate and flexible models of fracture interaction are sought after in the fields of mechanics and geology. Stress intensity factors (SIFs) quantify the energy concentrated at the fracture tips and are perturbed from their isolated values when two fractures are close to one another. Using a three-dimensional finite element fracture mechanics code to simulate static fractures in tension and compression, interaction effects are examined. SIF perturbations are characterized by introducing three interaction measures: the circumferential and maximum SIF perturbation provide the “magnitude” of the effect of interaction, and the amplification to shielding ratio quantifies the balance between increased and decreased SIFs along the tip. These measures are used to demonstrate the change in interaction with fracture separation and to find the separation at which interaction becomes negligible. Interaction maps are constructed by plotting the values of the interaction measures for a static fracture as a second fracture is moved around it. These maps are presented for several common fracture orientations in tension. They explore interaction by highlighting regions in which growth is more likely to occur and where fractures will grow into nonplanar geometries. Interaction maps can be applied to fracture networks with multiple discontinuities to analyze the effect of geometric variations on fracture interaction

The effect of pore shape on the Poisson ratio of porous materials

A brief review is given of the effect of porosity on the Poisson ratio of a porous material. In contrast to elastic moduli such as K, G, or E, which always decrease with the addition of pores into a matrix, the Poisson ratio ν may increase, decrease, or remain the same, depending on the shape of the pores, and on the Poisson ratio of the matrix phase, νo. In general, for a given pore shape, there is a unique critical Poisson ratio, νc, such that the addition of pores into the matrix will cause the Poisson ratio to increase if νoνc, and remain unchanged if νo=νc. The critical Poisson ratio for spherical pores is 0.2, for prolate spheroidal pores is close to 0.2, and tends toward zero for thin cracks. For two-dimensional materials, νc=1/3 for circular pores, 0.306 for squares, 0.227 for equilateral triangles, and again approaches 0 for thin cracks. The presence of a “trapped” fluid in the pore space tends to cause νc to increase, and for the range of parameters that may occur in rocks or concrete, this increase is more pronounced for thin crack-like pores than for equi-dimensional pores. Measurements of the Poisson ratio therefore may allow insight into pore geometry and pore fluid. If the matrix phase is strongly auxetic, small amounts of porosity will generally not cause the Poisson ratio to become positive

Semi-analytical method for modeling wellbore breakout development

Borehole breakout initiation, progression, and stabilization are modeled using a semi-analytical method based on Melentiev’s graphical conformal mapping procedure, and Kolosov–Muskhelishvili complex stress potentials. The only input data required are the elastic moduli of the rock, the Coulomb strength parameters (cohesion and angle of internal friction), and the far-field stresses. The stresses around the borehole wall are computed, the region in which the rock has failed is then “removed”, creating a new borehole shape. This process is iterated until a shape is obtained for which the breakout will progress no further, and a stable state has been reached. This modeling shows that stresses around the flank of the breakout evolve so as to reduce the propensity for shear failure, which helps to explain why the breakout width remains relatively constant throughout the process, even as the breakout region deepens radially. The failed area around the borehole becomes smaller and more localized, as the breakout tip sharpens and deepens. Using the Mogi–Coulomb failure criterion, a good match is obtained between the modeled breakout geometry and the geometry observed by Herrick and Haimson in laboratory experiments on an Alabama limestone. The new method leads to a correlation between breakout geometry, rock strength properties, and in situ stress. The paper ends with a critical discussion of the possibility of inferring the in situ stress state from observed breakout geometries