Finite element modeling of frictional contact and stress intensity factors in three-dimensional fractured media using unstructured tetrahedral meshes

Abstract

This thesis introduces a three-dimensional (3D) finite element (FE) formulation to model the linear elastic deformation of fractured media under tensile and compressive loadings. The FE model is based on unstructured meshes using quadratic tetrahedral elements, and includes several novel components: (i) The singular stress field near the crack front is modeled using quarter-point tetrahedral finite elements. (ii) The frictional contact between the crack faces is modeled using isoparametric contact discretization and a gap-based augmented Lagrangian method. (iii) Accurate stress intensity factors (SIFs) of 3D cracks computed using the two novel approaches of displacement correlation and disk-shaped domain integral. The main contributions in the FE modeling of 3D cracks are: (i) It is mathematically proven that quarter-point tetrahedral finite elements (QPTs) reproduce the square root strain singularity of crack problems. (ii) A displacement correlation (DC) scheme is proposed in combination with QPTs to compute SIFs from unstructured meshes. (iii) A novel domain integral approach is introduced for the accurate computation of the pointwise $J$-integral and the SIFs using tetrahedral elements. The main contributions in the contact algorithm are: (i) A square root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. (ii) A gap-based augmented Lagrangian algorithm is introduced for updating the contact forces obtained from the penalty method to more accurate estimates. The results of contact and stress intensity factors are validated for several numerical examples of cubes containing single and multiple cracks. Finally, two applications of this numerical methodology are discussed: (i) Understanding the hysteretic behavior in rock deformation; and (ii) Simulating 3D brittle crack growth. The results in this thesis provide significant evidence that tetrahedral elements are efficient, reliable and robust instruments for accurate linear elastic fracture mechanics calculations.