Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics

2019 ◽  
Vol 53 (2) ◽  
pp. 659-699 ◽  
Author(s):  
S. Almi ◽  
S. Belz ◽  
M. Negri
Keyword(s):  
Phase Field ◽  
Phase Field Model ◽  
Gradient Flow ◽  
Time Discretization ◽  
Field Variable ◽  
Small Time ◽  
One Step ◽  
The One ◽  
Discrete Finite Element ◽  
Minimization Scheme

We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovered a posteriori by a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateral L2-gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments.

scholarly journals A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

2019 ◽  
Vol 12 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Matteo Negri
Keyword(s):  
Energy Balance ◽  
Phase Field ◽  
Gradient Flow ◽  
Field Variable ◽  
Discrete Solution ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Static Limit ◽  
Viscosity Limit ◽  
Minimization Scheme

AbstractWe consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic {L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of {W^{1,2}(0,T;L^{2})}.

scholarly journals Phase-field model of precipitation processes with coherency loss

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Tian-Le Cheng ◽  
You-Hai Wen
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Ostwald Ripening ◽  
Phase Field Model ◽  
Energy Criterion ◽  
Field Variable ◽  
Ginzburg Landau ◽  
Underlying Mechanisms ◽  
Flood Fill ◽  
Coherency Loss

AbstractA phase-field model is proposed to simulate coherency loss coupled with microstructure evolution. A special field variable is employed to describe the degree of coherency loss of each particle and its evolution is governed by a Ginzburg-Landau type kinetic equation. For the sake of computational efficiency, a flood-fill algorithm is introduced that can drastically reduce the required number of field variables, which allows the model to efficiently simulate a large number of particles sufficient for characterizing their statistical features during Ostwald ripening. The model can incorporate size dependence of coherency loss, metastability of coherent particles, and effectively incorporate the underlying mechanisms of coherency loss by introducing a so-called differential energy criterion. The model is applied to simulate coarsening of Al3Sc precipitates in aluminum alloy and comprehensively compared with experiments. Our results clearly show how the particle size distribution is changed during coherency loss and affects the coarsening rate.

An Approach for Modeling Transformation Plasticity Using a Phase Field Model

2011 ◽  
Vol 320 ◽  
pp. 285-290 ◽  
Author(s):  
Takuya Uehara
Keyword(s):  
Thermal Expansion ◽  
Phase Transformation ◽  
Phase Field ◽  
Transformation Temperature ◽  
Field Model ◽  
Volume Fraction ◽  
Phase Field Model ◽  
Field Variable ◽  
Transformation Plasticity ◽  
Temperature Strain

In this paper, an approach for modeling transformation plasticity using a phase field model is presented. A conventional formula is utilized to represent the strain due to transformation plasticity as well as thermal expansion and transformation dilatation. A phase-field variable is introduced to express the state of phase in material instead of volume fraction, and numerical simulations under simplified conditions are demonstrated. As a result, the strain induced by phase transformation is suitably regenerated, and qualitatively appropriate temperature-strain curves are obtained. In addition, the effect of each parameter is investigated, and various dependencies, such as transformation temperature and stress, on the induced strain are demonstrated. It is then concluded that the results indicate the applicability of the presented model for practical use by adjusting the parameters.

Linear stability analysis and metastable solutions for a phase-field model

1999 ◽  
Vol 129 (3) ◽  
pp. 571-600 ◽  
Author(s):  
A. Jiménez-Casas ◽  
A. Rodríguez-Bernal
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Phase Field ◽  
Field Model ◽  
Equilibrium Points ◽  
Phase Field Model ◽  
One Dimensional ◽  
Stability And Instability ◽  
The One

We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.

A Phase-Field Model for the Formation of Martensite and Bainite

2014 ◽  
Vol 922 ◽  
pp. 31-36 ◽  
Author(s):  
Tansel T. Arif ◽  
Rong Shan Qin
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Materials Science ◽  
Current Model ◽  
Phase Field Model ◽  
Field Method ◽  
Field Variable ◽  
Phase Field Method ◽  
Cubic Symmetry ◽  
Chemical Free Energy

The phase field method is rapidly becoming the method of choice for simulating the evolution of solid state phase transformations in materials science. Within this area there are transformations primarily concerned with diffusion and those that have a displacive nature. There has been extensive work focussed upon applying the phase field method to diffusive transformations leaving much desired for models that can incorporate displacive transformations. Using the current model, the formation of martensite, which is formed via a displacive transformation, is simulated. The existence of a transformation matrix in the free energy expression along with cubic symmetry operations enables the reproduction of the 24 grain variants of martensite. Furthermore, upon consideration of the chemical free energy term, the model is able to utilise both the displacive and diffusive aspects of bainite formation, reproducing the autocatalytic nucleation process for multiple sheaves using a single phase field variable. Transformation matrices are available for many steels, one of which is used within the model.

scholarly journals Thermodynamics and Analysis of Predicted Responses of a Phase Field Model for Ductile Fracture

Materials ◽  
2021 ◽  
Vol 14 (19) ◽  
pp. 5842
Author(s):  
Aris Tsakmakis ◽  
Michael Vormwald
Keyword(s):  
Fracture Mechanics ◽  
Ductile Fracture ◽  
Phase Field ◽  
Damage Mechanics ◽  
Field Model ◽  
Phase Field Model ◽  
Isotropic Hardening ◽  
Linear Elastic ◽  
Isotropic Damage ◽  
The One

The fundamental idea in phase field theories is to assume the presence of an additional state variable, the so-called phase field, and its gradient in the general functional used for the description of the behaviour of materials. In linear elastic fracture mechanics the phase field is employed to capture the surface energy of the crack, while in damage mechanics it represents the variable of isotropic damage. The present paper is concerned, in the context of plasticity and ductile fracture, with a commonly used phase field model in fracture mechanics. On the one hand, an appropriate framework for thermodynamical consistency is outlined. On the other hand, an analysis of the model responses for cyclic loading conditions and pure kinematic or pure isotropic hardening are shown.

A phase field model of the premelting of grain boundaries in pure materials

2000 ◽  
Vol 652 ◽  
Author(s):  
Alexander E. Lobkovsky ◽  
James A. Warren
Keyword(s):  
Grain Boundaries ◽  
Phase Field ◽  
Field Model ◽  
Solid Phase ◽  
Phase Field Model ◽  
Field Variable ◽  
High Angle ◽  
Unified Framework ◽  
Crystalline Orientation ◽  
Solid Liquid

ABSTRACTWe present a phase field model of solidification which includes the effects of the crystalline orientation in the solid phase. This model describes grain boundaries as well as solid-liquid boundaries within a unified framework. With an appropriate choice of coupling of the phase field variable to the gradient of the crystalline orientation variable in the free energy, we find that high angle boundaries undergo a premelting transition. As the melting temperature is approached from below, low angle grain boundaries remain narrow. The width of the liquid layer at high angle grain boundaries diverges logarithmically.

scholarly journals Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms

2020 ◽  
Vol 10 (1) ◽  
pp. 24-65
Author(s):  
Matthias Ebenbeck ◽  
Kei Fong Lam
Keyword(s):  
Phase Field ◽  
Tumour Growth ◽  
Fluid Velocity ◽  
Phase Field Model ◽  
Logarithmic Potential ◽  
Stationary Solutions ◽  
Field Variable ◽  
Reaction Diffusion Equation ◽  
Singular Potentials ◽  
Source Terms

Abstract We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

Sign in / Sign up

Export Citation Format

Share Document