Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- 1 Introduction
- Part I Continuum mechanics and thermodynamics
- Part II Atomistics
- Part III Atomistic foundations of continuum concepts
- Part IV Multiscale methods
- 10 What is multiscale modeling?
- 11 Atomistic constitutive relations for multilattice crystals
- 12 Atomistic–continuum coupling: static methods
- 13 Atomistic–continuum coupling: finite temperature and dynamics
- Appendix A Mathematical representation of interatomic potentials
- References
- Index
12 - Atomistic–continuum coupling: static methods
from Part IV - Multiscale methods
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- 1 Introduction
- Part I Continuum mechanics and thermodynamics
- Part II Atomistics
- Part III Atomistic foundations of continuum concepts
- Part IV Multiscale methods
- 10 What is multiscale modeling?
- 11 Atomistic constitutive relations for multilattice crystals
- 12 Atomistic–continuum coupling: static methods
- 13 Atomistic–continuum coupling: finite temperature and dynamics
- Appendix A Mathematical representation of interatomic potentials
- References
- Index
Summary
In Chapter 2, we reviewed the essential concepts of continuum mechanics (which are covered in full detail in the companion book to this one [TME12]) and talked at length about atomistic models in Chapter 5 and static solution methods for these models in Chapter 6. The focus of the current chapter is on ways to couple the two approaches – continuum and atomistic – in search of a “best of both worlds” model that combines their strengths.
The models discussed in this chapter achieve this coupling by using a discretized approximation to continuum mechanics called the finite element method (FEM). We provide a brief review of FEM in the next section (see [TME12] for a more detailed discussion).
Finite elements and the Cauchy–Born rule
An overview of finite elements The problem we wish to solve with FEM is the static boundary value problem embodied in Fig. 12.1(a). A body B0 in the reference configuration has surface ∂B0 with surface normal N. This surface is divided into a portion (∂B0u) over which the displacements are prescribed as ū and the remainder (∂B0t) which is either free or subject to a prescribed traction. Our goal is to determine the resulting stress, strain and displacement fields throughout the body due to the applied loads.
This boundary-value problem is conveniently recast as an energy minimization problem using the principle of minimum potential energy from Section 2.6.
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- Chapter
- Information
- Modeling MaterialsContinuum, Atomistic and Multiscale Techniques, pp. 601 - 657Publisher: Cambridge University PressPrint publication year: 2011