scholarly journals A unified phase-field model of fracture in viscoelastic materials

2021 ◽  
Author(s):  
Franz Dammaß ◽  
Marreddy Ambati ◽  
Markus Kästner
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Numerical Study ◽  
Phase Field Model ◽  
Energy Functional ◽  
Internal Variables ◽  
Model Parameters ◽  
Rate Dependency ◽  
Viscoelastic Materials

AbstractThe phase-field approach has proven to be a powerful tool for the prediction of crack phenomena. When it is applied to inelastic materials, it is crucial to adequately account for the coupling between dissipative mechanisms present in the bulk and fracture. In this contribution, we propose a unified phase-field model for fracture of viscoelastic materials. The formulation is characterized by the pseudo-energy functional which consists of free energy and dissipation due to fracture. The free energy includes a contribution which is related to viscous dissipation that plays an essential role in coupling the phase-field and the viscous internal variables. The governing equations for the phase-field and the viscous internal variables are deduced in a consistent thermodynamic manner from the pseudo-energy functional. The resulting model establishes a two-way coupling between crack phase-field and relaxation mechanisms, i.e. viscous internal variables explicitly enter the evolution of phase-field and vice versa. Depending on the specific choice of the model parameters, it has flexibility in capturing the possible coupled responses, and the approaches of recently published formulations are obtained as limiting cases. By means of a numerical study of monotonically increasing load, creep and relaxation phenomena, rate-dependency of failure in viscoelastic materials is analysed and modelling assumptions of the present formulation are discussed.

scholarly journals A Numerical Study on Stratified Shear Layers With Relevance to Oil-Boom Failure

2014 ◽  
Author(s):  
David Kristiansen ◽  
Odd M. Faltinsen
Keyword(s):  
Numerical Model ◽  
Phase Field ◽  
Field Model ◽  
Numerical Study ◽  
Phase Field Model ◽  
Processing Unit ◽  
Stable Model ◽  
Interface Instability ◽  
Two Phase ◽  
Oil Boom

Interface dynamics of two-phase flow, with relevance for leakage of oil retained by mechanical oil barriers, is studied by means of a 2D lattice-Boltzmann method combined with a phase-field model for interface capturing. A Multi-Relaxation-Time (MRT) model of the collision process is used to obtain a numerically stable model at high Reynolds-number flow. In the phase-field model, the interface is given a finite but small thickness where the fluid properties vary continuosly across a thin interface layer. Surface tension is modelled as a volume force in the transition layer. The numerical model is implemented for simulations with the graphic processing unit (GPU) of a desktop PC. Verification tests of the model are presented. The model is then applied to simulate gravity currents (GC) obtained from a lock-exchange configuration, using fluid parameters relevant for those of oil and water. Interface instability phenomena are observed, and obtained numerical results are in good agreement with theory. This work demonstrates that the numerical model presented can be used as a numerical tool for studies of stratified shear flows with relevance to oil-boom failure.

On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit

2010 ◽  
Vol 140 (6) ◽  
pp. 1161-1186 ◽  
Author(s):  
Wolfgang Dreyer ◽  
Christiane Kraus
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Van Der Waals ◽  
Phase Field Model ◽  
Entropy Inequality ◽  
Sharp Interface ◽  
Energy Estimates ◽  
Sharp Interface Limit ◽  
Entropy Flux

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

Numerical study of a morphology diagram in the large undercooling limit using a phase-field model

1994 ◽  
Vol 50 (2) ◽  
pp. 1005-1008 ◽  
Author(s):  
Raz Kupferman ◽  
Ofer Shochet ◽  
Eshel Ben-Jacob
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Numerical Study ◽  
Phase Field Model ◽  
Large Undercooling

scholarly journals Phase Field Model for Solidification with Boundary Interface Interaction

2016 ◽  
Vol 9 ◽  
pp. 1-8
Author(s):  
Jie Liao
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Contact Region ◽  
Contact Point ◽  
Boundary Surface ◽  
Phase Field Model ◽  
Heat Diffusion ◽  
Diffuse Interface ◽  
Interface Problem

By incorporation the surface free energy in the free energy functional, a phase field model for solidification with boundary interface intersection is developed. In this model, the bulk equation is appropriately modified to account for the presence of heat diffusion 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 asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

A monolithic phase-field model of a fluid-driven fracture in a nonlinear poroelastic medium

2018 ◽  
Vol 24 (5) ◽  
pp. 1530-1555 ◽  
Author(s):  
CJ van Duijn ◽  
Andro Mikelić ◽  
Thomas Wick
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Galerkin Approximation ◽  
Phase Field Model ◽  
Coupled System ◽  
Energy Functional ◽  
Poroelastic Medium ◽  
Existence Of A Solution ◽  
Finite Dimensional ◽  
Biot Equations

In this paper, we present a full phase-field model for a fluid-driven fracture in a nonlinear poroelastic medium. The nonlinearity arises in the Biot equations when the permeability depends on porosity. This extends previous work (see Mikelić et al. Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 2015; 19: 1171–1195), where a fully coupled system is considered for the pressure, displacement, and phase field. For the extended system, we follow a similar approach: we introduce, for a given pressure, an energy functional, from which we derive the equations for the displacement and phase field. We establish the existence of a solution of the incremental problem through convergence of a finite-dimensional Galerkin approximation. Furthermore, we construct the corresponding Lyapunov functional, which is related to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation. Specifically, our numerical findings confirm differences with test cases using the linear Biot equations.

scholarly journals On the Diffuse Interface Models for High Codimension Dispersed Inclusions

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2206
Author(s):  
Elizaveta Zipunova ◽  
Evgeny Savenkov
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Field Model ◽  
Three Dimensional ◽  
Phase Field Model ◽  
Diffuse Interface ◽  
State Variables ◽  
Interface Models ◽  
Dispersed Inclusions ◽  
Derivatives Of

Diffuse interface models are widely used to describe the evolution of multi-phase systems of various natures. Dispersed inclusions described by these models are usually three-dimensional (3D) objects characterized by phase field distribution. When employed to describe elastic fracture evolution, the dispersed phase elements are effectively two-dimensional (2D) objects. An example of the model with effectively one-dimensional (1D) dispersed inclusions is a phase field model for electric breakdown in solids. Any diffuse interface field model is defined by an appropriate free energy functional, which depends on a phase field and its derivatives. In this work we show that codimension of the dispersed inclusions significantly restricts the functional dependency of the free energy on the derivatives of the problem state variables. It is shown that to describe codimension 2 diffuse objects, the free energy of the model necessarily depends on higher order derivatives of the phase field or needs an additional smoothness of the solution, i.e., its first derivatives should be integrable with a power greater than two. Numerical experiments are presented to support our theoretical discussion.

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