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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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**Mechanical Response**at Very Small Scale Lecture 2: The ClassicalTheory of Elasticity Anne Tanguy University of Lyon (France)**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.**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)**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 >> ???**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**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**Local stresses:**General expression for the internal rate of work: symmetric. antisymmetric Rigid motion Rigid rotation models the internal forces (Pa)**Equations of motion:**internal forces external forces (volume) external forces (at the boundaries) acceleration with , for any subsystem. Equilibrium equation: Boundary conditions:**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**Examples of stress tensors:**Traction: Shear: Hydrostatic Pressure: F S u By definition, pressure**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**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.**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**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.**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**3 moduli**(3 equivalent axis) 6 (5) moduli (rotational invariance around an axis)**6 moduli**(2 equivalent symmetry axis)**9 moduli**(2 orthogonal symmetry planes) 13 moduli (1 plane of symmetry) 21 moduli**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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