Effective Theories for Brittle Materials : a Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models.

Annotation

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Friedrich, Manuel
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Berlin : Logos Verlag Berlin, 2015.
Colección:Augsburger Schriften Zur Mathematik, Physik und Informatik Ser.
Temas:
Mathematical models.
Models, Theoretical
Modèles mathématiques.
mathematical models.
Mathematical models
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Intro; 1 The model and main results; 1.1 The discrete model; 1.2 Boundary values and scaling; 1.3 Limiting minimal energy and cleavage laws; 1.4 A specific model: The triangular lattice in two dimensions; 1.5 Limiting minimal configurations; 1.6 Limiting variational problem; 1.6.1 Convergence of the variational problems; 1.6.2 Analysis of a limiting variational problem; 1.6.3 An application: Fractured magnets in an external field; 2 Preliminaries; 2.1 Elementary properties of the cell energy; 2.2 Interpolation; 2.3 An estimate on geodesic distances; 2.4 Cell energy of the triangular lattice
  • 3 Limiting minimal energy and cleavage laws3.1 Warm up: Proof for the triangular lattice; 3.2 Estimates on a mesoscopic cell; 3.2.1 Mesoscopic localization; 3.2.2 Estimates in the elastic regime; 3.2.3 Estimates in the intermediate regime; 3.2.4 Estimates in the fracture regime; 3.2.5 Estimates in a second intermediate regime; 3.3 Proof of the cleavage law; 3.4 Examples: mass-spring models; 3.4.1 Triangular lattices with NN interaction; 3.4.2 Square lattices with NN and NNN interaction; 3.4.3 Cubic lattices with NN and NNN interaction; 4 Limiting minimal energy configurations
  • 4.1 Fine estimates on the limiting minimal energy4.2 Sharp estimates on the number of the broken triangles; 4.3 Convergence of almost minimizers; 4.3.1 The supercritical case; 4.3.2 The subcritical case; 4.3.3 Proof of the main limiting result; 5 The limiting variational problem; 5.1 Convergence of the variational problems; 5.1.1 The Gamma-lim inf-inequality; 5.1.2 Recovery sequences; 5.2 Analysis of the limiting variational problem; 6 The model and main results; 6.1 Rigidity estimates; 6.2 Compactness; 6.3 Gamma-convergence and application to cleavage laws; 6.4 Overview of the proof
  • 6.4.1 Korn-Poincaré-type inequality6.4.2 SBD-rigidity; 6.4.3 Compactness and Gamma-convergence; 7 Preliminaries; 7.1 Geometric rigidity and Korn: Dependence on the set shape; 7.2 A trace theorem in SBV2; 8 A Korn-Poincaré-type inequality; 8.1 Preparations; 8.2 Modification of sets; 8.3 Neighborhoods of boundary components; 8.3.1 Rectangular neighborhood; 8.3.2 Dodecagonal neighborhood; 8.4 Proof of the Korn-Poincaré-inequality; 8.4.1 Conditions for boundary components and trace estimate; 8.4.2 Modification algorithm; 8.4.3 Proof of the main theorem; 8.5 Trace estimates for boundary components
  • 8.5.1 Preliminary estimates8.5.2 Step 1: Small boundary components; 8.5.3 Step 2: Subset with small projection of components; 8.5.4 Step 3: Neighborhood with small projection of components; 8.5.5 Step 4: General case; 9 Quantitative SBD-rigidity; 9.1 Preparations; 9.2 A local rigidity estimate; 9.2.1 Estimates for the derivatives; 9.2.2 Estimates in terms of the H1-norm; 9.2.3 Local rigidity for an extended function; 9.3 Modification of the deformation; 9.4 SBD-rigidity up to small sets; 9.4.1 Step 1: Deformations with least crack length

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