Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Overview and background topics
- Part II Density functional theory
- Part III Important preliminaries on atoms
- Part IV Determination of electronic structure: the three basic methods
- Part V Predicting properties of matter from electronic structure – recent developments
- Appendix A Functional equations
- Appendix B LSDA and GGA functionals
- Appendix C Adiabatic approximation
- Appendix D Response functions and Green's functions
- Appendix E Dielectric functions and optical properties
- Appendix F Coulomb interactions in extended systems
- Appendix G Stress from electronic structure
- Appendix H Energy and stress densities
- Appendix I Alternative force expressions
- Appendix J Scattering and phase shifts
- Appendix K Useful relations and formulas
- Appendix L Numerical methods
- Appendix M Iterative methods in electronic structure
- Appendix N Code for empirical pseudopotential and tight-binding
- Appendix O Units and conversion factors
- References
- Index
Appendix G - Stress from electronic structure
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Overview and background topics
- Part II Density functional theory
- Part III Important preliminaries on atoms
- Part IV Determination of electronic structure: the three basic methods
- Part V Predicting properties of matter from electronic structure – recent developments
- Appendix A Functional equations
- Appendix B LSDA and GGA functionals
- Appendix C Adiabatic approximation
- Appendix D Response functions and Green's functions
- Appendix E Dielectric functions and optical properties
- Appendix F Coulomb interactions in extended systems
- Appendix G Stress from electronic structure
- Appendix H Energy and stress densities
- Appendix I Alternative force expressions
- Appendix J Scattering and phase shifts
- Appendix K Useful relations and formulas
- Appendix L Numerical methods
- Appendix M Iterative methods in electronic structure
- Appendix N Code for empirical pseudopotential and tight-binding
- Appendix O Units and conversion factors
- References
- Index
Summary
Summary
The subject of this appendix is the macroscopic stress that enters mechanical properties of matter in the form of stress–strain relations. The stress tensor is the generalization of pressure to all the independent components of dilation and shear, and the “stress theorem” is the generalization of the virial theorem for scalar pressure to all components of the stress tensor. In condensed matter, the state of the system is specified by the forces on each atom and the stress, which is an independent variable. The conditions for equilibrium are: (1) that the total force vanishes on each atom, and (2) that the macroscopic stress equals the externally applied stress.
Macroscopic stress and strain
Stress and strain are important concepts in characterizing the states of condensed matter [177, 721, 722, 890]. A body is in a state of stress if it is acted upon by external forces or if one part of the body exerts forces upon another part. If we consider two types of forces as illustrated in Fig. G.1: those acting interior to a volume element and those that act upon (or through) the surface of the element due to the surrounding material, which are shown as arrows in the figure. The latter forces (per unit area) are the stresses transmitted throughout the interior of the volume. Since these forces balance on any surface in equilibrium, the stress can be determined in terms of only the intrinsic internal forces; i.e. stress is an intrinsic property of a material in a given state.
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- Information
- Electronic StructureBasic Theory and Practical Methods, pp. 512 - 518Publisher: Cambridge University PressPrint publication year: 2004