Phase-inherent linear visco-elasticity model for infinitesimal deformations in the multiphase-field context

AbstractA linear visco-elasticity ansatz for the multiphase-field method is introduced in the form of a Maxwell-Wiechert model. The implementation follows the idea of solving the mechanical jump conditions in the diffuse interface regions, hence the continuous traction condition and Hadamard’s compatibility condition, respectively. This makes strains and stresses available in their phase-inherent form (e.g. $$\varepsilon ^{\alpha }_{ij}$$ ε ij α , $$\varepsilon ^{\beta }_{ij}$$ ε ij β ), which conveniently allows to model material behaviour for each phase separately on the basis of these quantities. In the case of the Maxwell-Wiechert model this means the introduction of phase-inherent viscous strains. After giving details about the implementation, the results of the model presented are compared to a conventional Voigt/Taylor approach for the linear visco-elasticity model and both are evaluated against analytical and sharp-interface solutions in different simulation setups.

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

Archive of Applied Mechanics ◽

10.1007/s00419-020-01759-3 ◽

2020 ◽

Cited By ~ 1

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.

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The Finite Phase-Field Method - A Numerical Diffuse Interface Approach for Microstructure Simulation with Minimized Discretization Error

MRS Proceedings ◽

10.1557/opl.2012.510 ◽

2011 ◽

Vol 1369 ◽

Cited By ~ 3

Author(s):

Janin Eiken

Keyword(s):

Phase Field ◽

Finite Size ◽

Numerical Differentiation ◽

Field Method ◽

Discretization Error ◽

Sharp Interface ◽

Diffuse Interface ◽

Phase Field Method ◽

Interface Width ◽

Microstructure Simulation

ABSTRACTThe Phase-field method is recognized as the method of choice for space-resolved microstructure simulation. In theoretic phase-field approaches, the underlying diffuse interface representation is discussed in the sharp interface limit. Applied phase-field models, however, have to cope with interfaces of finite size. Numerical solution based on finite differences naturally implies a discretization error. This error may result in significant deviations from the analytical sharp-interface solution, especially in cases of interface-controlled growth. Benchmark simula-tions revealed a direct correlation between the accuracy of the finite-difference solution and the number of numerical cells used to resolve the finite-sized interface width. This poses a problem, because high numbers of interface cells are unfavorable for numerical performance. To enable efficient high-accuracy computations, a new Finite Phase-Field approach is proposed, which closely links phase-field modeling and numerical discretization. The approach is based on a parabolic potential function, corresponding to phase-field solutions with a sinusoidal interface pro-file. Consideration of this profile during numerical differentiation allows an exact quantification of the bias evoked by grid spacing and interface width, which then a priori can be compensated.

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Study of biaxial mechanical properties of the passive pig heart: material characterisation and categorisation of regional differences

International Journal of Mechanical and Materials Engineering ◽

10.1186/s40712-021-00128-4 ◽

2021 ◽

Vol 16 (1) ◽

Author(s):

Fulufhelo Nemavhola

Keyword(s):

Mechanical Properties ◽

Constitutive Model ◽

Computational Models ◽

Heart Diseases ◽

Interventricular Septum ◽

Model Material ◽

Biaxial Test ◽

Material Behaviour ◽

Material Parameters ◽

Biaxial Tests

AbstractRegional mechanics of the heart is vital in the development of accurate computational models for the pursuit of relevant therapies. Challenges related to heart dysfunctioning are the most important sources of mortality in the world. For example, myocardial infarction (MI) is the foremost killer in sub-Saharan African countries. Mechanical characterisation plays an important role in achieving accurate material behaviour. Material behaviour and constitutive modelling are essential for accurate development of computational models. The biaxial test data was utilised to generated Fung constitutive model material parameters of specific region of the pig myocardium. Also, Choi-Vito constitutive model material parameters were also determined in various myocardia regions. In most cases previously, the mechanical properties of the heart myocardium were assumed to be homogeneous. Most of the computational models developed have assumed that the all three heart regions exhibit similar mechanical properties. Hence, the main objective of this paper is to determine the mechanical material properties of healthy porcine myocardium in three regions, namely left ventricle (LV), mid-wall/interventricular septum (MDW) and right ventricle (RV). The biomechanical properties of the pig heart RV, LV and MDW were characterised using biaxial testing. The biaxial tests show the pig heart myocardium behaves non-linearly, heterogeneously and anisotropically. In this study, it was shown that RV, LV and MDW may exhibit slightly different mechanical properties. Material parameters of two selected constitutive models here may be helpful in regional tissue mechanics, especially for the understanding of various heart diseases and development of new therapies.

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A Rigorous Sharp Interface Limit of a Diffuse Interface Model Related to Tumor Growth

Journal of Nonlinear Science ◽

10.1007/s00332-016-9352-3 ◽

2016 ◽

Vol 27 (3) ◽

pp. 847-872 ◽

Cited By ~ 8

Author(s):

Elisabetta Rocca ◽

Riccardo Scala

Keyword(s):

Tumor Growth ◽

Sharp Interface ◽

Diffuse Interface ◽

Interface Model ◽

Diffuse Interface Model ◽

Sharp Interface Limit

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

Volume 2: Fora, Parts A and B ◽

10.1115/fedsm2007-37567 ◽

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.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2020.0227 ◽

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.

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Correction to: Small strain multiphase-field model accounting for configurational forces and mechanical jump conditions

Computational Mechanics ◽

10.1007/s00466-017-1485-1 ◽

2017 ◽

Vol 61 (3) ◽

pp. 297-297

Author(s):

Daniel Schneider ◽

Ephraim Schoof ◽

Oleg Tschukin ◽

Andreas Reiter ◽

Christoph Herrmann ◽

...

Keyword(s):

Field Model ◽

Small Strain ◽

Configurational Forces ◽

Jump Conditions ◽

Mechanical Jump Conditions

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THERMODYNAMICALLY CONSISTENT, FRAME INDIFFERENT DIFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE TWO-PHASE FLOWS WITH DIFFERENT DENSITIES

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511500138 ◽

2012 ◽

Vol 22 (03) ◽

pp. 1150013 ◽

Cited By ~ 176

Author(s):

HELMUT ABELS ◽

HARALD GARCKE ◽

GÜNTHER GRÜN

Keyword(s):

Surface Diffusion ◽

Sharp Interface ◽

Diffuse Interface ◽

Momentum Balance ◽

Two Phase ◽

Interface Models ◽

Natural Energy ◽

Soluble Species ◽

Sharp Interface Models ◽

Dissipation Inequalities

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation, we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

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