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

Blesgen, Thomas; Chenchiah, Isaac V.

doi:10.1098/rsta.2012.0342

“…Blesgen & Chenchiah in [25] consider the effects of elastic energy (i.e. stress) on the evolution of microstructure formation on smaller length scales, known as coarsening, which are modelled by time-dependent nonlinear PDEs.…”

Section: Entropy and Convexity For Nonlinear Pdes And Related Areasmentioning

confidence: 99%

“…In contrast, the equilibrium state for coarsening driven by the elastic energy alone is mathematically challenging and is a microstructure on an infinitesimal scale. In [25], the key experimental observations pertaining to coarsening in elastic solids are summarised; the Cahn-Larché model, a generalisation of the Cahn-Hillard equation that incorporates a quasiconvex elastic energy density obtained from relaxation, is analyzed; some recent developments concerning the two-scale model for elasticity-influenced coarsening are presented, including several motivations, analytical results, and comparisons of the model with experimental results; and some further mathematical reflections and questions are addressed.…”

Section: Entropy and Convexity For Nonlinear Pdes And Related Areasmentioning

confidence: 99%

Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction

Ball,

Chen

2013

Preprint

0

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Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The concept of entropy originated in thermodynamics and statistical physics during the 19 th century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the theme issue is intended to provide an introduction to entropy, convexity, and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this theme issue.Partial differential equations (PDEs) are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological, and social processes. The behaviour of every material object, with length scales ranging from sub-atomic to astronomical and timescales ranging from picoseconds to millennia, can be modelled by PDEs or by equations having similar features.The concept of entropy originated in thermodynamics and statistical physics during the 19 th century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear PDEs in recent decades. For example, for discontinuous or singular solutions to nonlinear conservation laws which may contain shock waves and concentrations, the notion of entropy solutions is based on entropy conditions involving convexity, motivated by and consistent with the Second Law of Thermodynamics. In addition, entropy methods have become one of the most efficient methods in the analysis of physically relevant discontinuous or singular solutions. The notions of convexity appropriate for multi-dimensional problems, such as polyconvexity, quasiconvexity, and rank-one convexity, are responsible for several recent major advances. In the last three decades, various nonlinear methods involving entropy and convexity have been developed to deal with discontinuous and singular solutions in different areas of PDEs, especially in nonlinear conservation laws, the calculus of variations, and gradient flows.

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“…By contrast, the equilibrium state for coarsening driven by the elastic energy alone is mathematically challenging and is a microstructure on an infinitesimal scale. In [22], the key experimental observations pertaining to coarsening in elastic solids are summarized; the Cahn-Larché model, a generalization of the Cahn-Hillard equation that incorporates a quasi-convex elastic energy density obtained from relaxation, is analysed; some recent developments concerning the two-scale model for elasticity-influenced coarsening are presented, including several motivations, analytical results and comparisons of the model with experimental results; and some further mathematical reflections and questions are addressed.…”

Section: Entropy and Convexity For Nonlinear Pdes And Related Areasmentioning

confidence: 99%

“…Ideas involving entropy have played a crucial role in developing many important mathematical approaches and methods, such as variational principles, Lyapunov functionals, relative entropy methods, monotonicity methods, weak convergence methods, divergence-measure fields and kinetic methods (cf. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). …”

Section: Entropymentioning

confidence: 99%

See 1 more Smart Citation

Entropy and convexity for nonlinear partial differential equations

Ball

¹

2013

Phil. Trans. R. Soc. A.

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Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.

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Structure‐Preserving Approximation of the Cahn‐Hilliard‐Biot System

Brunk,

Fritz

2024

Numerical Methods Partial

0

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In this work, we propose a structure‐preserving discretization for the recently studied Cahn‐Hilliard‐Biot system using conforming finite elements in space and problem‐adapted explicit‐implicit Euler time integration. We prove that the scheme is thermodynamically consistent, that is, the balance of global phase and global volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realized as a splitting method, that is, decoupling the Cahn‐Hilliard subsystem from the poro‐elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.

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

Cited by 5 publications

References 51 publications

Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction

Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction

Entropy and convexity for nonlinear partial differential equations

Structure‐Preserving Approximation of the Cahn‐Hilliard‐Biot System

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