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Mechanical Response at Very Small Scale Lecture 2: The Classical Theory of Elasticity

Mechanical Response at Very Small Scale Lecture 2: The Classical Theory of Elasticity Anne Tanguy University of Lyon (France). II. The classical Theory of continuum Elasticity. The mechanical behaviour of a classical solid can be entirely described by a single continuous field:

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Mechanical Response at Very Small Scale Lecture 2: The Classical Theory of Elasticity

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  1. Mechanical Response at Very Small Scale Lecture 2: The ClassicalTheory of Elasticity Anne Tanguy University of Lyon (France)

  2. II. The classicalTheory of continuum Elasticity. The mechanical behaviour of a classical solid can be entirely described by a single continuous field: The displacement field u(r) of the volume elements constituying the system. • 1) Whatis a « continuous » medium? • 2) The local strains. • 3) The description of local forces (stress). • 4) The Landau expansion of the MechanicalEnergy • and the ElasticModuli. • J. Salençon « Handbook of Continuum Mechanics » Springer ed. (2001) • Landau « Elasticity » Mir ed.

  3. Whatis a « continuous » medium? • Two close elementsevolve in a similarway. • In particular: conservation of proximity. • « Field » = physicalquantityaveraged • over a volume element. • = continuousfunction of space. • Hypothesis in practice, to bechecked. • Atthisscale, forces are short range (surface forces between volume elements)

  4. In general, it is valid at scales >>characteristic scale in the microstructure. Examples: crystals d >> interatomic distance (~ Å ) polycrystals d >> grain size (~nm ~mm) regular packing of grains d >> grain size (~ mm) liquids d >> mean free path disordered materials d >> ???

  5. Al polycristal (Electron Back Scattering Diffraction) Dendritic growth in Al: Cu polycristal : cold lamination (70%)/ annealing. TiO2 metallic foams, prepared with different aging, and different tensioactif agent: Si3N4 SiC dense

  6. Classical elasticity: displacement field

  7. Examples of linearizedstraintensors: Traction: Shear: Hydrostatic Pressure: Units: %. Order of magnitude:elasticity OK if e<0.1% (metal) e<1% (polymer, amorphous) L L+u L-v u

  8. Local stresses: General expression for the internal rate of work: symmetric. antisymmetric Rigid motion Rigid rotation models the internal forces (Pa)

  9. Equations of motion: internal forces external forces (volume) external forces (at the boundaries) acceleration with , for any subsystem. Equilibrium equation: Boundary conditions:

  10. Force per unit surface exerted along the x-direction, on the face normal to the direction y. Local stresses: Expression of forces: surface vector normal Units: Pa (1atm = 105 Pa) Order of magnitude: MPa =106 Pa

  11. Examples of stress tensors: Traction: Shear: Hydrostatic Pressure: F S u By definition, pressure

  12. The Landau expansion of the MechanicalEnergy and the ElasticModuli: Expression of the rate of work of internal forces: Mechanical Energy: per unit volume It means that

  13. The Landau expansion of the MechanicalEnergy and the ElasticModuli: General expansion of the Mechanical Energy, per unit volume: No dependence in (translational invariance) No dependence in (rotational invariance) Thus Hoole’s Law ut tensio sic vis 21 Elastic Moduli Cabgd in the most general 3D case.

  14. Symmetries of the tensor of ElasticModuli: General symmetries: + Specific symmetries of the crystal: Operator of symmetry Example of an isotropic and homogeneousmaterial: Units:J.m-3 , or Pa. Order of Magnitude: -1<n ≈ 0.33<0.5 and E ≈ Gpa ≈ sY/10-3

  15. Voigt notation:

  16. Examples of elasticmoduli in homogeneous and isotropicsys: Traction: Shear: Hydrostatic Pressure: F E, Young modulus n, Poisson ratio u m, shear modulus P c, compressibility.

  17. Examples of anisotropic materials (crystals) FCC 3 moduli C11 C12 C44 HCP 5 moduli C11 C12 C13 C33 C44 C66=(C11-C12)/2 Co: HC  FCC T=450°C

  18. 3 moduli (3 equivalent axis) 6 (5) moduli (rotational invariance around an axis)

  19. 6 moduli

  20. 6 moduli (2 equivalent symmetry axis)

  21. 9 moduli (2 orthogonal symmetry planes) 13 moduli (1 plane of symmetry) 21 moduli

  22. Bibliography: I. DisorderedMaterials K. Binder and W. Kob « GlassyMaterials and disorderedsolids » (WS, 2005) S. R. Elliott « Physics of amorphousmaterials » (Wiley, 1989) II. Classical continuum theory of elasticity J. Salençon « Handbook of Continuum Mechanics » (Springer, 2001) L. Landau and E. Lifchitz « Théorie de l’élasticité ». III. Microscopic basis of Elasticity S. Alexander Physics Reports 296,65 (1998) C. Goldenberg and I. Goldhirsch « Handbook of Theoretical and Computational Nanotechnology » Reithed. (American scientific, 2005) IV. Elasticity of DisorderedMaterials B.A. DiDonna and T. Lubensky « Non-affine correlations in Randomelastic Media » (2005) C. Maloney « Correlations in the ElasticResponse of Dense Random Packings » (2006) Salvatore Torquato « RandomHeterogeneousMaterials » Springer ed. (2002) V. Sound propagation Ping Sheng « Introduction to wavescattering, Localization, and Mesoscopic Phenomena » (AcademicPress 1995) V. Gurevich, D. Parshin and H. SchoberPhysicalreview B 67, 094203 (2003)

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