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Gradient theory for plasticity via homogenization of discrete dislocations

Adriana Garroni, Giovanni Leoni, Marcello Ponsiglione (2010)

Journal of the European Mathematical Society

We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the Γ -limit of this energy (suitably rescaled),...

Ground states in complex bodies

Paolo Maria Mariano, Giuseppe Modica (2009)

ESAIM: Control, Optimisation and Calculus of Variations

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...

Ground states in complex bodies

Paolo Maria Mariano, Giuseppe Modica (2008)

ESAIM: Control, Optimisation and Calculus of Variations

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...

Internal constraints and linear constitutive relations for transversely isotropic materials

Paolo Podio-Guidugli, Maurizio Vianello (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

All internal constraints compatible with transverse isotropy are determined and representation formulae are given for the constitutive relations of arbitrarily constrained, transversely isotropic materials.

Invariant line integrals in the theory of defective crystals

Gareth P. Parry, Miroslav Šilhavý (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In a continuum theory of crystals with defects, invariant line integrals measure the line defects of the lattice structure. It is shown that the integrands of invariant line integrals can always be taken to have the transformation properties of covariant vector-valued functions.

Motion of spiral-shaped polygonal curves by nonlinear crystalline motion with a rotating tip motion

Tetsuya Ishiwata (2015)

Mathematica Bohemica

We consider a motion of spiral-shaped piecewise linear curves governed by a crystalline curvature flow with a driving force and a tip motion which is a simple model of a step motion of a crystal surface. We extend our previous result on global existence of a spiral-shaped solution to a linear crystalline motion for a power type nonlinear crystalline motion with a given rotating tip motion. We show that self-intersection of the solution curves never occurs and also show that facet extinction never...

On a variational problem arising in crystallography

Alexander J. Zaslavski (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We study a variational problem which was introduced by Hannon, Marcus and Mizel [ESAIM: COCV9 (2003) 145–149] to describe step-terraces on surfaces of so-called “unorthodox” crystals. We show that there is no nondegenerate intervals on which the absolute value of a minimizer is π / 2 identically.

Sufficient conditions for the validity of the Cauchy-Born rule close to SO ( n )

Sergio Conti, Georg Dolzmann, Bernd Kirchheim, Stefan Müller (2006)

Journal of the European Mathematical Society

The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke...

Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Mechanical twinning and growth twinning. Nota II

Giovanni Zanzotto (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note II we continue the analysis of the phenomenon of mechanical twinning that we began in a preceding Note I ( 1 ) . Furthermore, we point out some fundamental properties useful in the study of growth twins, for which a fully comprehensive thermoelastic theory is not yet available.

Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Generalities and mechanical twinning

Giovanni Zanzotto (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the present Note I and in a following Note II (Zanzotto 1988), we discuss, taking into account some available experimental data, the results of a thermoelastic theory of twinning in crystalline solids. Various noteworthy problems emerge, some of which involve the hypotheses that are at the very basis of the theory.

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