scholarly journals Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation

2020 ◽  
Author(s):  
Arne Claus Hansen-Dörr ◽  
Jörg Brummund ◽  
Markus Kästner
Keyword(s):  
Crack Propagation ◽  
Phase Field ◽  
Strain Energy ◽  
Heterogeneous Media ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Jump Conditions ◽  
Modeling Framework ◽  
Material Interfaces ◽  
Mechanical Jump Conditions

Abstract In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the regularized crack. The key novelty is the combination of a strain energy split with a partial rank-I relaxation in the vicinity of the diffuse interface. The former is necessary to account for physically meaningful crack kinematics like crack closure, the latter ensures the mechanical jump conditions throughout the diffuse region. The model is verified by a convergence study, where a circular bi-material disc with and without a crack is subjected to radial loads. For the uncracked case, analytical solutions are taken as reference. In a second step, the model is applied to crack propagation, where a meaningful influence on crack branching is observed, that underlines the necessity of a reasonable homogenization scheme. The presented model is particularly relevant for the combination of any variational strain energy split in the fracture phase-field model with a diffuse modeling approach for material heterogeneities.

A hybrid ten-species phase-field model of tumor growth

2014 ◽  
Vol 24 (13) ◽  
pp. 2569-2599 ◽  
Author(s):  
E. A. B. F. Lima ◽  
J. T. Oden ◽  
R. C. Almeida
Keyword(s):  
Tumor Growth ◽  
Phase Field ◽  
Computational Models ◽  
Heterogeneous Media ◽  
Phase Field Model ◽  
Mixed Finite Elements ◽  
Mixture Theory ◽  
Diffuse Interface ◽  
Simultaneous Treatment ◽  
Moving Interfaces

The development of predictive computational models of tumor initiation, growth, and decline is faced with many formidable challenges. Phenomenological models which attempt to capture the complex interactions of multiple tissue and cellular species must cope with moving interfaces of heterogeneous media and the sprouting vascular structures due to angiogenesis and their evolution. They must be able to deliver predictions consistent with events that take place at cellular scales, and they must faithfully depict biological mechanisms and events that are known to be associated with various forms of cancer. In the present work, a ten-species vascular model for the tumor growth is presented which falls within the framework of phase-field (or diffuse-interface) models suggested by continuum mixture theory. This framework provides for the simultaneous treatment of interactions of multiple evolving species, such as tumor cells, necrotic cell cores, nutrients, and other cellular and tissue types that exist and interact in living tissue. We develop a hybrid model that couples the tumor growth with sprouting through angiogenesis. The model is able to represent the branching of new vessels through coupling a discrete model for which the angiogenesis is started upon pre-defined conditions on the nutrient deprivation in the continuum model. Such conditions are represented by hypoxic cells that release tumor growth factors that ultimately trigger vascular growth. We discuss the numerical approximation of the model using mixed finite elements. The results of numerical experiments are also presented and discussed.

Explicit Dynamics of Diffuse Interface in Phase‐Field Model

2020 ◽  
pp. 2000162
Author(s):  
Chao Yang ◽  
Houbing Huang ◽  
Wenbo Liu ◽  
Junsheng Wang ◽  
Jing Wang ◽  
...  
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Explicit Dynamics

scholarly journals Study of the Fracture Behavior of Tetragonal Zirconia Polycrystal with a Modified Phase Field Model

Materials ◽  
2020 ◽  
Vol 13 (19) ◽  
pp. 4430 ◽  
Author(s):  
Jingming Zhu ◽  
Jun Luo ◽  
Yuanzun Sun
Keyword(s):  
Phase Transformation ◽  
Crack Propagation ◽  
Phase Field ◽  
Fracture Behavior ◽  
Tetragonal Zirconia ◽  
Finite Size ◽  
Phase Field Model ◽  
Crack Tip ◽  
Propagation Mode ◽  
Tetragonal Zirconia Polycrystal

The superior fracture toughness of zirconia is closely correlated with stress-induced martensitic phase transformation around a crack tip. In this study, a modified phase field (PF) model coupling phase transformation and fracture is proposed to study the fracture behavior and toughening effect of tetragonal zirconia polycrystal (TZP). The stress-induced tetragonal to monoclinic (t–m) phase transformation around a static or propagating crack is characterized with PF simulations. It is shown that the finite size and shape of the transformation zone under different loads and ambient temperatures can be well predicted with the proposed PF model. The phase transformation may decrease the stress level around the crack tip, which implies the toughening effect. After that, crack propagation in TZP is studied. As the stress field is perturbed by the phase transformation patterns, the crack may experience deflection and branching in the propagation process. It is found that the toughness of the grain boundaries (GBs) has important influences on the crack propagation mode. For TZP with strong GBs, the crack is more likely to propagate transgranularly while, for TZP with weak GBs, intergranular crack propagation is prevalent. Besides that, the crystal orientation and the external load can also influence the topology of crack propagation.

Application of Interface-Tracking Method Based on Phase-Field Model to Numerical Analysis of Isothermal and Thermal Two-Phase Flows

2007 ◽  
Author(s):  
Naoki Takada
Keyword(s):  
Phase Field ◽  
Density Ratio ◽  
Phase Field Model ◽  
Field Method ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Navier Stokes ◽  
Interface Tracking ◽  
Two Phase Flows ◽  
Two Phase

For interface-tracking simulation of two-phase flows in various micro-fluidics devices, the applicability of two versions of Navier-Stokes phase-field method (NS-PFM) was examined, combining NS equations for a continuous fluid with a diffuse-interface model based on the van der Waals-Cahn-Hilliard free-energy theory. Through the numerical simulations, the following major findings were obtained: (1) The first version of NS-PFM gives good predictions of interfacial shapes and motions in an incompressible, isothermal two-phase fluid with high density ratio on solid surface with heterogeneous wettability. (2) The second version successfully captures liquid-vapor motions with heat and mass transfer across interfaces in phase change of a non-ideal fluid around the critical point.

scholarly journals A regularized phase-field model for faceting in a kinetically controlled crystal growth

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200227
Author(s):  
T. Philippe ◽  
H. Henry ◽  
M. Plapp
Keyword(s):  
Crystal Growth ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Interface Problem ◽  
Kinetically Controlled ◽  
Interface Theory ◽  
Coarsening Dynamics

At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen–Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t , with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.

Isogeometric Analysis of 3D Dynamic Thermo-Mechanical Phase-Field Model for Cubic-to-Tetragonal Phase Transformations in Shape Memory Alloys

2015 ◽  
Author(s):  
Rakesh Dhote ◽  
Kamran Behdinan
Keyword(s):  
Shape Memory ◽  
Phase Transformations ◽  
Phase Field ◽  
Isogeometric Analysis ◽  
Domain Walls ◽  
Landau Theory ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Free Energy Function ◽  
Martensitic Phase Transformations

In this paper, we study the dynamic thermo-mechanical behaviors of 3D shape memory alloy (SMA) nanostructures using the phase-field (PF) model. The PF model is based on the Ginzburg-Landau theory and requires a non-convex free energy function for an adequate description of the cubic-to-tetragonal martensitic phase transformations. We have developed a model that includes domain walls, treated as a diffuse interface, which leads to a fourth-order differential equation in a strain-based order parameter PF model. Arising numerical challenges have been overcome based on an isogeometric analysis (IGA) framework. Microstructure morphology evolution and consequent thermo-mechanical properties have been studied on SMA nanostructures of different geometries. The numerical results are in agreement with experimental observations. The developed coupled dynamic model has provided a better understanding of underlying microstructures and behaviors, which can be used for development of better SMA-based devices.

Phase-field elasticity model based on mechanical jump conditions

2015 ◽  
Vol 55 (5) ◽  
pp. 887-901 ◽  
Author(s):  
Daniel Schneider ◽  
Oleg Tschukin ◽  
Abhik Choudhury ◽  
Michael Selzer ◽  
Thomas Böhlke ◽  
...  
Keyword(s):  
Phase Field ◽  
Jump Conditions ◽  
Model Based ◽  
Mechanical Jump Conditions

scholarly journals A Thermodynamically­Consistent Phase Field Model for Electromechanical Diffusive Crack Propagation

PAMM ◽  
2010 ◽  
Vol 10 (1) ◽  
pp. 117-118
Author(s):  
Martina Hofacker ◽  
Christian Miehe ◽  
Fabian Welschinger
Keyword(s):  
Crack Propagation ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model

scholarly journals A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

2016 ◽  
Vol 8 ◽  
pp. 9-18
Author(s):  
Jie Liao
Keyword(s):  
Binary Alloy ◽  
Phase Field ◽  
Field Model ◽  
Contact Region ◽  
Contact Point ◽  
Boundary Surface ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
Interface Problem ◽  
Alloy Solidification

A phase field model for binary alloy solidification with boundary interface intersection is developed. In the phase field model, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The thin interface asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

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