Browsing by Subject "Phase-field modeling"
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Item Applications of phase-field modeling in hydraulic fracture(2019-12) Alotaibi, Talal Eid; Landis, Chad M.; Sharma, Mukul M; Mear, Mark E; Ravi-Chandar, Krishnaswa; Foster, John TUnderstanding the mechanisms behind the nucleation and propagation of cracks is of considerable interest in engineering application and design decisions. In many applications in the oil industry, complicated fracture geometries and propagation behaviors are encountered. As a result, the development of modeling approaches that can capture the physics of non-planar crack evolution while being computationally tractable is a critical challenge. The phase-field approach to fracture has been shown to be a powerful tool for simulating very complex fracture topologies, including the turning, splitting, and merging of cracks. In contrast to fracture models that explicitly track the crack surfaces, crack propagation and the evolution thereof arise out of the solution to a partial differential equation governing the evolution of a phase-field damage parameter. As such, the crack growth emerges naturally from solving the set of coupled differential equations linking the phase-field to other field quantities that can drive the fracture process. In the present model, the physics of flow through porous media and cracks is coupled with the mechanics of fracture. Darcy-type flow is modeled in the intact porous medium, which transitions to a Stokes-type flow regime within open cracks. This phase-field model is implemented to gain insights into the propagation behavior of fluid-injected cracks. One outstanding issue with phase-field fracture models is the decomposition of the strain energy required to ensure that compressive stress states do not cause crack propagation and damage evolution. In the present study, the proper representation of the strain energy function to reflect this fracture phenomenon is examined. The strain energy is constructed in terms of principle strains in such a way that it has two parts; the tensile and the compressive. A degradation function only applies to the tensile part enforcing that the crack is driven only by that part of the strain energy. We investigated the split operator proposed by Miehe et al. [1], and then proposed a split approach based on masonry-like material behavior [2, 3]. We have found that when using Miehe’s form for the strain energy function, cracks can propagate under compressive stresses. In contrast, the approach based on a masonry-like materials constitutive model we proposed ensures that cracks do not grow under compressive stresses. To demonstrate the capabilities of phase-field modeling for fluid-driven fractures, four general types of problems are simulated: 1) interactions of fluid-driven, natural, and proppant-filled cracks, 2) crack growth through different material layers, 3) fluid-driven crack growth under the influence of in-situ far-field stresses, and 4) crack interactions with inclusions. The simulations illustrate the capabilities of the phase-field model for capturing interesting and complex crack growth phenomena. To understand how fluid-driven cracks interact with inclusions, AlTammar et al. [4] performed experiments. Three tests with tough inclusions were performed to understand the effects of orientation angle, thickness, and material properties. Additionally, one test with a weak inclusion was performed to compare the results with those of the tough inclusion cases. The experiments show a clear tendency for the fluid-driven hydraulic fracture to cross thick natural fractures filled with materials weaker and softer than the matrix and to be diverted by thick natural fractures with tougher and stiffer filling materials. To replicate these experiments numerically and to gain a mechanistic understanding, in the present study, we ran simulations using phase-field modeling. Results from both the experiments and the simulations provide clear evidence that inclusion width, angle, material properties, and distance from the injection point affect the outcome of the crack evolution. Phase-field modeling was able to capture the trends of crack deflection/crossing in all the test cases. Finally, we extended the phase-field model has been extended to three dimensions and tested it on bench-mark problems. The first bench-mark problem is a compact test for a CT specimen. In this problem, the mechanical equations are only considered. The simulation shows that the CT specimen is split into two symmetric parts. The second bench-mark problem is a fluid-driven circular crack. The simulation for this problem shows that the crack grows in a radial direction.Item Characterization and modeling of mixed-mode I+III fracture in brittle materials(2015-12) Pham, Khai Hong; Ravi-Chandar, K.; Landis, Chad M; Liechti, Kenneth M; Mear, Mark E; Marder, Michael PMixed-mode I+III fracture in brittle materials presents spectacular, scale-independent pattern formation in nature and engineering applications; and it is one of the last remaining puzzles in linear elastic fracture mechanics. This problem has received much attention in the literature over the past few decades both from experiments and analysis, but there are still open challenges that remain. Specifically, the existence of a threshold ratio of mode III to mode I loading below which fragmentation of the crack front (formation of daughter cracks) does not occur and the length scale associated with the spacing of the fragments when they do occur are still under debate. The continued growth of cracks under remote mode I + III loading is also of interest; it is observed that in some cases the fragmented cracks coalesce, while in others they maintain their independent development. We approach this problem through carefully designed experiments to examine the physical aspects of crack initiation and growth. This is then explored further through numerical simulations of the stress state that explore the influence of perturbations on the formation of daughter cracks. We show that a parent crack subjected to combined modes I+III loading exhibits fragmentation of the crack front into daughter cracks without any threshold. The distance between the daughter cracks is dictated by the length scale corresponding to the decay of the elastic field; this decay depends on the characteristic dimension of the parent crack from which the daughter cracks are nucleated. As the daughter cracks continue growing, they coarsen in spacing also through elastic shielding. As the daughter cracks grow farther, the parent crack, pinned at the original position, experiences increased stress intensity factor and the bridging regions begin to crack and the parent crack front advances towards the daughter cracks. This establishes a steady state condition for the system of parent crack with equally spaced daughter cracks to continue growing together. Finally, direct numerical simulation of crack initiation and growth is explored using a phase-field model. The model is first validated for in-plane modes I + II through comparison to experiments, and then used to explore combined modes I + III in order to study the above mechanism of mixed-mode I + III crack growth.Item A continuum modeling approach for the deposition of enamel(2015-12) Kuang, Ye; Landis, Chad M.; Mear, Mark EIn this report continuum methods to analyze organogenesis on curved surfaces is devised. This initial study will investigate a basic system. Dental enamel is the example system used to study the simulation of organogenesis as well as pattern formation. It is observed that dental enamel is created by a number of ameloblast cells migrating generally outward from the dental enamel junction (DEJ). These cells also rearrange locally within the surface that they reside. In this report, the simulations are based on the postulate that the cell motion arises from changes in the local strain environment as the cells migrate. As opposed to a passive movement driven by external driving forces or energy gradients, this theory hypothesizes that motion can arise internally due to the migration of the individual cell influenced by the local cell density and the velocity of the cell relative to its contacting neighbors. To model this kinematically driven approach we first develop a set of continuum equations to describe the velocity of the cells. This consists of two components, one the governs the in-plane rearrangements of the cells based on local strain cues, and a second that governs the velocity of the cells normal to the DEJ, which depends upon if the cells are actively secreting or not. This second feature requires the knowledge of the location of the boundary between secretory and non-secretory cells, which we is called the commencement front. On the secretory side of the commencement front the normal velocity of the cells is a specified quantity, while on the non-secretory side the normal velocity is zero. In order to track the evolution of the commencement front a phase-field description is utilized that treats this boundary as a diffuse instead of a sharp interface. The numerical method that is used to solve the equations is described, and some initial preliminary results for simple surface geometries are presented.Item Phase-field modeling of piezoelectrics and instabilities in dielectric elastomer composites(2011-12) Li, Wenyuan, 1982-; Landis, Chad M.; Huang, Rui; Mear, Mark; Tassoulas, John L.Ferroelectric ceramics are broadly used in applications including actuators, sensors and information storage. An understanding of the microstructual evolution and domain dynamics is vital for predicting the performance and reliability of such devices. The underlying mechanism responsible for ferroelectric constitutive response is ferroelectric domain wall motion, domain switching and the interactions of domain walls with other material defects. In this work, a combined theoretical and numerical modeling framework is developed to investigate the nucleation and growth of domains in a single crystal of ferroelectric material. The phase-field approach, applying the material electrical polarization as the order parameter, is used as the theoretical modeling framework to allow for a detailed accounting of the electromechanical processes. The finite element method is used for the numerical solution technique. In order to obtain a better understanding of the energetics of fracture within the phase-field setting, the J-integral is modified to include the energies associated with the order parameter. Also, the J- integral is applied to determine the crack-tip energy release rate for common sets of electromechanical crack-face boundary conditions. The calculations confirm that only true equilibrium states exhibit path-independence of J, and that domain structures near crack tips may be responsible for allowing positive energy release rate during purely electrical loading. The small deformation assumption is prevalent in the phase-field modeling approach, and is used in the previously described calculations. The analysis of large deformations will introduce the concept of Maxwell stresses, which are assumed to be higher order effects that can be neglected in the small deformation theory. However, in order to investigate the material response of soft dielectric elastomers undergoing large mechanical deformation and electric field, which are employed in electrically driven actuator devices, manipulators and energy harvesters, a finite deformation theory is incorporated in the phase-field model. To describe the material free energy, compressible Neo-Hookean and Gent models are used. The Jaumann rate of the polarization is used as the objective polarization rate to make the description of the dissipation frame indifferent. To illustrate the theory, electromechanical instabilities in composite materials with different inclusions will be studied using the finite element methods.Item Phase-field modeling of the thermo-electro-mechanically coupled behavior of ferroelectric materials(2017-12) Woldman, Alexandra Yakovlevna; Landis, Chad M.; Mear, Mark E; Huang, Rui; Foster, John T; Ravi-Chandar, KrishnaswaFerroelectric materials are widely used in engineering and science applications due to their large nonlinear thermo-electro-mechanical coupling. Of interest recently, has been the study of the giant electrocaloric effect, a large adiabatic temperature change with the application of an electric field, due to its possible application for solid-state cooling. The electrocaloric effect is maximized near phase transitions, where entropy jumps contribute to a large nonlinear effect. This dissertation develops a continuum phase-field model for the thermo-electro-mechanically coupled behavior for ferroelectric materials. The model is derived from thermodynamic considerations and based on a phenomenological free energy function. The finite element method is applied to solve the governing equations for a selected set of boundary value problems. Mechanical displacement, electric potential, polarization and temperature are used as degrees of freedom in the formulation of the finite element implementation of the model. The a geometry for an isothermal stable two-dimensional ferroelectric to paraelectric phase boundary is developed, along with appropriate boundary conditions, and simulated using the nonlinear finite element method for a variety of ferroelectric domain widths. The dependence of the phase coexistence temperature, boundary energy, entropy jump across the boundary and closure domain shape on the ferroelectric laminate domain width is quantified. A simulation of the motion of the phase boundary through the material under entropy/heat input control is demonstrated. Next, a realistic electrocaloric cooling device based on a multilayer ferroelectric capacitor is simulated through a full thermodynamic refrigeration cycle. The model geometry and boundary conditions are chosen to match realistic device configurations. The device is driven through a cycle with two adiabatic and two constant electric field legs, and compared with the analytically computed ideal plane strain electrocaloric cooling cycle. Several inefficiencies arise in the device, including incomplete transformation, entropy loss due to phase boundary motion, and high energy zones with large stresses and closure domains at the electrode tip. Lastly, motivated by potential uses as actuators, the domain structure in three-dimensional ferroelectric nanodots is modeled by cooling from a paraelectric phase. The expected vortex domain structure forms in sufficiently small dots, but distorts upon further cooling to room temperature. The room temperature transfer of dots to a rigid substrate and actuation via an out-of-plane electric field leads to incomplete domain switching, thereby reducing actuator displacements.