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1. Numerical Simulation of Fracture Initiation in Barre Granite using an Experimentally Validated XFEM Model

   Garg, P; Hedayat, A; Griffiths, D.V. (June 2020, 54th US Rock Mechanics/Geomechanics Symposium)

   null (Ed.)

   Fracturing in brittle rocks with an existing crack results in the development of a significant nonlinear region surrounding the crack tip called the fracture process zone. Various experimental and numerical studies have shown that the crack tip parameters such as the crack tip opening displacement (CTOD) and the fracture energy are critically important in characterizing the fracture process zone. In this study, numerical simulations of rock specimens with a center notch subjected to three-point bending were conducted using the extended finite element method (XFEM) along with the cohesive zone model (CZM) to account for fracture process zone. The input parameters of CZM such as the elastic and critical crack opening displacements were first estimated based on the results of three-point bending tests on the center notched Barre granite specimens. Displacements were measured using the two dimensional digital image correlation technique and used to characterize the evolution of the fracture process zone and estimate the parameters of the cohesive zone model. The results from the numerical simulations showed that CZM provided a good agreement with experimental data as it predicted all three stages of cracking from fracture process initiation to macro-crack growth. 

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2. Cohesive Zone Interpretations of Phase-Field Fracture Models

   https://doi.org/10.1115/1.4055660  

   Tran, H; Chew, H B (December 2022, Journal of Applied Mechanics)

   Abstract Unlike micromechanics failure models that have a well-defined crack path, phase-field fracture models are capable of predicting the crack path in arbitrary geometries and dimensions by utilizing a diffuse representation of cracks. However, such models rely on the calibration of a fracture energy (Gc) and a regularization length-scale (lc) parameter, which do not have a strong micromechanical basis. Here, we construct the equivalent crack-tip cohesive zone laws representing a phase-field fracture model, to elucidate the effects of Gc and lc on the fracture resistance and crack growth mechanics under mode I K-field loading. Our results show that the cohesive zone law scales with increasing Gc while maintaining the same functional form. In contrast, increasing lc broadens the process zone and results in a flattened traction-separation profile with a decreased but sustained peak cohesive traction over longer separation distances. While Gc quantitatively captures the fracture initiation toughness, increasing Gc coupled with decreasing lc contributes to a rising fracture resistance curve and a higher steady-state toughness—both these effects cumulate in an evolving cohesive zone law with crack progression. We discuss the relationship between these phase-field parameters and process zone characteristics in the material. 

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3. A numerical study of dehydration induced fracture toughness degradation in human cortical bone

   https://doi.org/10.1016/j.jmbbm.2024.106468  

   Shin, Mihee; Martens, Penny J.; Siegmund, Thomas; Kruzic, Jamie J.; Gludovatz, Bernd (May 2024, Journal of the Mechanical Behavior of Biomedical Materials)

   A 2D plane strain extended finite element method (XFEM) model was developed to simulate three-point bending fracture toughness tests for human bone conducted in hydrated and dehydrated conditions. Bone microstructures and crack paths observed by micro-CT imaging were simulated using an XFEM damage model. Critical damage strains for the osteons, matrix, and cement lines were deduced for both hydrated and dehydrated conditions and it was found that dehydration decreases the critical damage strains by about 50%. Subsequent parametric studies using the various microstructural models were performed to understand the impact of individual critical damage strain variations on the fracture behavior. The study revealed the significant impact of the cement line critical damage strains on the crack paths and fracture toughness during the early stages of crack growth. Furthermore, a significant sensitivity of crack growth resistance and crack paths on critical strain values of the cement lines was found to exist for the hydrated environments where a small change in critical strain values of the cement lines can alter the crack path to give a significant reduction in fracture resistance. In contrast, in the dehydrated state where toughness is low, the sensitivity to changes in critical strain values of the cement lines is low. Overall, our XFEM model was able to provide new insights into how dehydration affects the micromechanisms of fracture in bone and this approach could be further extended to study the effects of aging, disease, and medical therapies on bone fracture. 

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4. Smooth Lagrangian Crack Band Model Based on Spress-Sprain Relation and Lagrange Multiplier Constraint of Displacement Gradient

   https://doi.org/10.1115/1.4063896  

   Tay_Nguyen, Anh; Xu, Houlin; Matouš, Karel; Bažant, Zdeněk P (March 2024, Journal of Applied Mechanics)

   Abstract A preceding 2023 study argued that the resistance of a heterogeneous material to the curvature of the displacement field is the most physically realistic localization limiter for softening damage. The curvature was characterized by the second gradient of the displacement vector field, which includes the material rotation gradient, and was named the “sprain” tensor, while the term “spress” is here proposed as the force variable work-conjugate to “sprain.” The partial derivatives of the associated sprain energy density yielded in the preceeding study, sets of curvature resisting self-equilibrated nodal sprain forces. However, the fact that the sprain forces had to be applied on the adjacent nodes of a finite element greatly complicated the programming and extended the simulation time in a commercial code such as abaqus by almost two orders of magnitude. In the present model, Smooth Lagrangian Crack Band Model (slCBM), these computational obstacles are here overcome by using finite elements with linear shape functions for both the displacement vector and for an approximate displacement gradient tensor. A crucial feature is that the nodal values of the approximate gradient tensor are shared by adjacent finite elements. The actual displacement gradient tensor calculated from the nodal displacement vectors is constrained to the approximate displacement gradient tensor by means of a Lagrange multiplier tensor, either one for each element or one for each node. The gradient tensor of the approximate gradient tensor then represents the approximate third-order displacement curvature tensor, or Hessian of the displacement field. Importantly, the Lagrange multiplier behaves as an externally applied generalized moment density that, similar to gravity, does not affect the total strain-plus-sprain energy density of material. The Helmholtz free energy of the finite element and its associated stiffness matrix are formulated and implemented in a user’s element of abaqus. The conditions of stationary values of the total free energy of the structure with respect to the nodal degrees-of-freedom yield the set of equilibrium equations of the structure for each loading step. One- and two-dimensional examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results of the three-point bend test are independent of the orientation of a regular square mesh, capture the width variation of the crack band, the damage strain profile across the band, and converge as the finite element mesh is refined. 

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5. Sprain energy consequences for damage localization and fracture mechanics

   https://doi.org/10.1073/pnas.2410668121  

   Xu, Houlin; Nguyen, Anh Tay; Bažant, Zdeněk P (October 2024, Proceedings of the National Academy of Sciences)

   The 2023 smooth Lagrangian Crack-Band Model (slCBM), inspired by the 2020 invention of the gap test, prevented spurious damage localization during fracture growth by introducing the second gradient of the displacement field vector, named the “sprain,” as the localization limiter. The key idea was that, in the finite element implementation, the displacement vector and its gradient should be treated as independent fields with the lowest ( $C_0$) continuity, constrained by a second-order Lagrange multiplier tensor. Coupled with a realistic constitutive law for triaxial softening damage, such as microplane model M7, the known limitations of the classical Crack Band Model were eliminated. Here, we show that the slCBM closely reproduces the size effect revealed by the gap test at various crack-parallel stresses. To describe it, we present an approximate corrective formula, although a strong loading-path dependence limits its applicability. Except for the rare case of zero crack-parallel stresses, the fracture predictions of the line crack models (linear elastic fracture mechanics, phase-field, extended finite element method (XFEM), cohesive crack models) can be as much as 100% in error. We argue that the localization limiter concept must be extended by including the resistance to material rotation gradients. We also show that, without this resistance, the existing strain-gradient damage theories may predict a wrong fracture pattern and have, for Mode II and III fractures, a load capacity error as much as 55%. Finally, we argue that the crack-parallel stress effect must occur in all materials, ranging from concrete to atomistically sharp cracks in crystals. 

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