Continuum Models for Phase Transitions and Twinning in Crystals

Pitteri, Mario; Zanzotto, Giovanni

doi:10.1201/9781420036145

Cited by 156 publications

(375 citation statements)

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“…• When the Lie group G is nilpotent (with corresponding Lie algebra with rational structure constants), Cermelli and Parry [4] have shown that the corresponding discrete subgroups give either a simple lattice or a 4-lattice (in Pitteri and Zanzotto's terminology [23]) even though the composition function in G is not additive. For such groups, Parry and Sigrist [22] construct explicitly all sets of generators of a given discrete subgroup.…”

Section: Introductionmentioning

confidence: 99%

A Classification of the Symmetries of Uniform Discrete Defective Crystals

Nicks

2014

J Elast

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Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometrical symmetries of these structures can be computed in terms of the changes of generators of the discrete subgroup which preserve the discrete set of points. Here a classification of the symmetries for the discrete subgroups of a particular class of three-dimensional solvable Lie group is presented. It is a fact that there are only three mathematically distinct types of Lie groups which model uniform defective crystals, and the calculations given here complete the discussion of the symmetries of the corresponding discrete structures. We show that those symmetries corresponding to automorphisms of the discrete subgroups extend uniquely to symmetries of the ambient Lie group and we regard these symmetries as (restrictions of) elastic deformations of the continuous defective crystal. Other symmetries of the discrete structures are classified as 'inelastic' symmetries.

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Section: Introductionmentioning

confidence: 99%

A Classification of the Symmetries of Uniform Discrete Defective Crystals

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2014

J Elast

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“…Here we shall catalogue the various multilattices that arise as discrete structures corresponding to uniform defective crystals, and for each multilattice that arises, we give a representation (1.2) which is such that the integer n is least (i.e. we give an 'essential' description of M , in the terminology of Pitteri & Zanzotto [3]). Elsewhere, we shall investigate the geometrical symmetries of the multilattices with a view to prescribing material symmetries of corresponding continuum strain energy functions.…”

Section: Introductionmentioning

confidence: 99%

The Structure of Uniform Discrete Defective Crystals

Cermelli

Parry

2006

Continuum Mech. Thermodyn.

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In the continuum context, a uniform crystal has dislocation density tensor constant in space. A simple iteration procedure generates an infinite set of points which is associated with uniform defective crystals. When certain necessary conditions are satisfied, there is a minimum (non-zero) separation of points in this set, so the set is discrete. We describe the structure of such sets explicitly, and show in particular that any such set is either a simple lattice or a 4-lattice.

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“…Since no configurations of higher complexity are considered, all our results are local, and in the configuration space the nonessential multilattices form smooth submanifolds of strictly lower dimension (see [44]); and since N is the union of disjoint SO(3)-invariant neighborhoods N + and N − of ε 0 σ and −ε 0 σ , respectively, we can assume, without loss of generality, that in N + all descriptors (e a , p r ) are essential, and s = s 0 := sgn(e 0 1 · e 0 2 × e 0 3 ). The following analogues of (13) 1 , (18) 2 hold:…”

Section: Phase Changes In a Wt-nbhdmentioning

confidence: 99%

“…I sketch here the bare essentials for the rest of the paper, addressing the reader to [44] for more information. In particular, as there, I use the summation convention and 'running indices' without specifying their range; for instance in expressions like 'the lattice basis e a ' instead of 'the lattice basis {e a , a = 1, 2, 3}', or 'the functionφ(e a , p r , θ)' instead of 'the functionφ(e 1 , e 2 , e 3 , p 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

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On Certain Weak Phase Transformations in Multilattices

Pitteri

The Rational Spirit in Modern Continuum Mechanics

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This paper is dedicated to the memory of Clifford Truesdell, to whom I acknowledge my debt and gratitude. I present some recent results of mine on thermomechanics of crystalline solids, a research on which I began working during my stay at The Johns Hopkins University, 1977University, -1979. Then, in the last section, I mention some unusual merits of Truesdell, which I experienced in my scientific career and are not widely known.

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Continuum Models for Phase Transitions and Twinning in Crystals

Cited by 156 publications

References 0 publications

A Classification of the Symmetries of Uniform Discrete Defective Crystals

A Classification of the Symmetries of Uniform Discrete Defective Crystals

The Structure of Uniform Discrete Defective Crystals

On Certain Weak Phase Transformations in Multilattices

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