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2014 | Book

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Phase Separation Coupled with Damage Processes

Analysis of Phase Field Models in Elastic Media

Authors: Christian Heinemann, Christiane Kraus

Publisher: Springer Fachmedien Wiesbaden

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The authors explore a unifying model which couples phase separation and damage processes in a system of partial differential equations. The model has technological applications to solder materials where interactions of both phenomena have been observed and cannot be neglected for a realistic description. The equations are derived in a thermodynamically consistent framework and suitable weak formulations for various types of this coupled system are presented. In the main part, existence of weak solutions is proven and degenerate limits are investigated.

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Table of Contents

Frontmatter

1. Introduction

Abstract

A better understanding of the mechanism and the interplay between phase separation and damage processes in elastically stressed solids is of big interest in material sciences. Various technological applications concerning the manufacturing and lifetime prediction of microelectronic devices are directly related to these phenomena. For example, solder joints in microelectronic packages connect the microchips to the circuit-boards and are, consequently, very critical components for the reliability engineering (see [LSC⁺04]). Solder materials usually consist of two or three component alloys whose aging process is influenced by temperature cycling. At high temperatures, solder alloys energetically favor one homogeneous phase consisting of a specific mixture of their chemical components. However, as soon as alloys are quenched sufficiently, phase separation or spinodal decomposition leads to fine-grained structures of different chemical compositions on a short time-scale.

Christian Heinemann, Christiane Kraus

2. Mathematical preliminaries

Abstract

The following sections acquaint the reader with some mathematical background for the analytical part of this thesis. After the basic notations are given in Section 2.1, we will present some standard and nonstandard techniques for variational problems and since the presentation can, of course, not be comprehensive, we will refer to the monographs in functional analysis for more details at the corresponding parts.

Christian Heinemann, Christiane Kraus

3. PDE modeling and thermodynamic consistency

Abstract

This chapter is devoted to phase-field models for damage processes and phase separation in elastically stressed alloys. The associated PDE systems for both models have been separately derived from balance laws and constitutive equations by M. E. Gurtin [Gur96] for phase separation as well as by M. Frémond and B. Nedjar [FN96] for damage processes, respectively.

Christian Heinemann, Christiane Kraus

4. Cahn-Hilliard systems with polynomial chemical potentials coupled with damage processes and homogeneous elasticity

Abstract

The present chapter covers certain existence results for Calm-Hilliard equations which are coupled with elasticity and partial damage processes (see Definition 3.4.1). We assume a binary mixture, a polynomial growth condition for the chemical potential, a homogeneous elastic energy density (with respect to the chemical concentration) and a p-Laplacian with p> nin the differential inclusion for the damage propagation law.

Christian Heinemann, Christiane Kraus

5. Cahn-Hilliard systems with logarithmic chemical potentials coupled with damage processes and inhomogeneous elasticity

Abstract

The existence results for weak solutions from Chapter 4 are generalized to a broader class of coupled PDE systems in this chapter. More specifically, we will be able to treat

multi-component Cahn-Hilliard systems,
inhomogeneous elastic energy densities,
chemical potentials of polynomial or logarithmic type,
quadratic gradient term of the damage variable in the energy, i.e., p= 2 in (1.3).

Additionally, we show that the results also apply to elastic Allen-Cahn systems coupled with damage processes. This case is even easier to treat.

Christian Heinemann, Christiane Kraus

6. Complete damage processes

Abstract

In the preceding chapters, Cahn-Hilliard equations have been coupled with incomplete damage processes. The uniform convexity assumptions in (4.2a) as well as in (5.2a), respectively, prevents the PDE system (3.27) from degeneration (in the elastic energy). However, for a more precise description of damage phenomena, the elastic energy should be allowed to degenerate on maximally damaged regions. Studying this case requires further mathematical tools such as Γ -convergence of regularized free energies, representation of shrinking sets with Lipschitz domains and space-time local Sobolev spaces.

Christian Heinemann, Christiane Kraus

7. Cahn-Hilliard systems coupled with complete damage processes and homogeneous elasticity

Abstract

This chapter combines the approach from Chapter 4 with the ideas from the preceding chapter. More specifically, we are going to investigate existence of weak solutions for complete damage systems which are coupled with degenerating Cahn-Hilliard equations. The diffusion mobility tensor depends on the damage variable and vanishes when the damage is maximal. Therefore, we have two degenerating terms in the resulting system: the elastic energy density and the mobility tensor.

Christian Heinemann, Christiane Kraus

8. Conclusion

Abstract

In this work, we have investigated mathematical models describing both phenomena, phase separation and damage processes, in a unifying approach. Phase separation is modeled by elastic Cahn-Hilliard equations, whereas the damage processes are described by a doubly nonlinear differential inclusion. The forces are assumed to be in a quasi-static equilibrium. We have introduced the corresponding PDE model (1.1)-(1.2) in Chapter 3 and we have shown thermodynamic consistency. The main aim has been to prove existence of weak solutions of the coupled system for various types of free energy densities of the form (1.3). The damage dissipation potential is rate-dependent and specified by (1.4).

Christian Heinemann, Christiane Kraus

Backmatter

Title: Phase Separation Coupled with Damage Processes
Authors: Christian Heinemann
Christiane Kraus
Copyright Year: 2014
Publisher: Springer Fachmedien Wiesbaden
Electronic ISBN: 978-3-658-05252-2
Print ISBN: 978-3-658-05251-5
DOI: https://doi.org/10.1007/978-3-658-05252-2

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