scholarly journals Cahn–Hilliard equations incorporating elasticity: analysis and comparison to experiments

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120342 ◽  
Author(s):  
Thomas Blesgen ◽  
Isaac Vikram Chenchiah
Keyword(s):  
Energy Density ◽  
Global Existence ◽  
Three Dimensions ◽  
Elasticity Analysis ◽  
Micro Structure ◽  
Existence Of Weak Solutions ◽  
Hilliard Equation ◽  
Elastic Energy Density ◽  
Theoretical Predictions ◽  
Cahn Hilliard Equation

We consider a generalization of the Cahn–Hilliard equation that incorporates an elastic energy density which, being quasi-convex, incorporates micro-structure formation on smaller length scales. We explore the global existence of weak solutions in two and three dimensions. We compare theoretical predictions with experimental observations of coarsening in superalloys.

scholarly journals Global existence and asymptotics of solutions of the Cahn–Hilliard equation

2007 ◽  
Vol 238 (2) ◽  
pp. 426-469 ◽  
Author(s):  
Shuangqian Liu ◽  
Fei Wang ◽  
Huijiang Zhao
Keyword(s):  
Global Existence ◽  
Asymptotics Of Solutions ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Shape Evolution and Splitting of a Single Coherent Particle

1997 ◽  
Vol 481 ◽  
Author(s):  
J. D. Zhang ◽  
D. Y. Li ◽  
L. Q. Chen
Keyword(s):  
Lattice Mismatch ◽  
Elastic Anisotropy ◽  
Equilibrium Shape ◽  
Elastic Strain Energy ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Energy Contribution ◽  
Hilliard Equation ◽  
The Matrix ◽  
Cahn Hilliard Equation

ABSTRACTThe morphology and its evolution of a single coherent precipitate was investigated using the Cahn-Hilliard equation and Khachaturyan's continuum elasticity theory for solid solutions. A cubic solid solution with negative elastic anisotropy and isotropic interfacial energy was considered. The lattice mismatch between the precipitate and the matrix was assumed to be purely dilatational and its compositional dependence obeys the Vegard's law. Both two- and three-dimensional systems were studied. The Cahn-Hilliard equation was numerically solved using a semi-implicit Fourier-spectral method. It was demonstrated that, with increasing elastic energy contribution, the equilibrium shape of a coherent particle gradually changes from a circle to a square in two dimensions, and from a sphere to a cube in three dimensions, and the composition profile becomes increasingly inhomogeneous within the precipitate with the minimum at the center of the particle, consistent with previous theoretical studies and experimental observations. It was also shown that, with sufficiently large elastic strain energy contribution, a coherent particle may split to four particles from a square, or eight particles from a sphere, during its evolution to equilibrium. For both two and three dimensions, the splitting starts by nucleating the matrix phase at the center of the particle.

Numerical Study of the Cahn-Hilliard Equation in Three Dimensions

1988 ◽  
Vol 60 (22) ◽  
pp. 2311-2314 ◽  
Author(s):  
Raúl Toral ◽  
Amitabha Chakrabarti ◽  
James D. Gunton
Keyword(s):  
Numerical Study ◽  
Three Dimensions ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CAHN–HILLIARD EQUATION WITH INERTIAL TERM

2012 ◽  
Vol 23 (09) ◽  
pp. 1250087 ◽  
Author(s):  
YIN-XIA WANG ◽  
ZHIQIANG WEI
Keyword(s):  
Asymptotic Behavior ◽  
Global Existence ◽  
Dimensional Space ◽  
Solution Space ◽  
Contraction Mapping Principle ◽  
Asymptotic Behavior Of Solutions ◽  
Inertial Term ◽  
Hilliard Equation ◽  
The Cauchy Problem ◽  
Cahn Hilliard Equation

In this paper, we investigate the Cauchy problem for Cahn–Hilliard equation with inertial term in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a solution space. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.

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