*Adrian P. Sutton*

- Published in print:
- 2020
- Published Online:
- August 2020
- ISBN:
- 9780198860785
- eISBN:
- 9780191893001
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198860785.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials, Crystallography: Physics

In a Volterra dislocation the relative displacement by the Burgers vector appears abruptly in the dislocation core so that the core has no width. This leads to divergent stresses and strains, which ...
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In a Volterra dislocation the relative displacement by the Burgers vector appears abruptly in the dislocation core so that the core has no width. This leads to divergent stresses and strains, which are unrealistic. Hybrid models correct this failure by considering a balance of forces that results in a finite core width, and finite stresses and strains throughout. Interatomic forces tend to constrict the core and elastic forces tend to widen it. The Frenkel-Kontorova model comprises two interacting linear chains of atoms as a representation of an edge dislocation, with linear springs between adjacent atoms of each chain. The Peierls-Nabarro model assumes the core is confined to two parallel atomic planes sandwiched between elastic continua. This model enables the stress to move the dislocation to be calculated, and it leads to the concept of dislocation kinks. These models highlight the role of atomic interactions in affecting ductility.Less

In a Volterra dislocation the relative displacement by the Burgers vector appears abruptly in the dislocation core so that the core has no width. This leads to divergent stresses and strains, which are unrealistic. Hybrid models correct this failure by considering a balance of forces that results in a finite core width, and finite stresses and strains throughout. Interatomic forces tend to constrict the core and elastic forces tend to widen it. The Frenkel-Kontorova model comprises two interacting linear chains of atoms as a representation of an edge dislocation, with linear springs between adjacent atoms of each chain. The Peierls-Nabarro model assumes the core is confined to two parallel atomic planes sandwiched between elastic continua. This model enables the stress to move the dislocation to be calculated, and it leads to the concept of dislocation kinks. These models highlight the role of atomic interactions in affecting ductility.

*Alfonso Sorrentino*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0004
- Subject:
- Mathematics, Applied Mathematics

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical ...
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This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.Less

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

*Kaloshin Vadim and Zhang Ke*

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to ...
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This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.Less

This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the *t* component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.