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: The displacement field u(r) of the volume elements constituying the system. 1) What is a « continuous » medium? 2) The local strains. 3) The description of local forces (stress). 4) The Landau expansion of the Mechanical Energy and the Elastic Moduli. J. Salençon « Handbook of Continuum Mechanics » Springer ed. (2001) Landau « Elasticity » Mir ed.

What is a « continuous » medium? Two close elements evolve in a similar way. In particular: conservation of proximity. « Field » = physical quantity averaged over a volume element. = continuous function of space. Hypothesis in practice, to be checked. At this scale, 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

Classical elasticity: displacement field

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

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

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

Order of magnitude: MPa =106 Pa Local stresses: Force per unit surface exerted along the x-direction, on the face normal to the direction y. Expression of forces: vector normal surface 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 Mechanical Energy and the Elastic Moduli: Expression of the rate of work of internal forces: Mechanical Energy: It means that per unit volume

The Landau expansion of the Mechanical Energy and the Elastic Moduli: 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 Elastic Moduli: General symmetries: + Specific symmetries of the crystal: Operator of symmetry Example of an isotropic and homogeneous material: Units: J.m-3 , or Pa. Order of Magnitude: -1<n ≈ 0.33<0.5 and E ≈ Gpa ≈ sY/10-3

Voigt notation:

Examples of elastic moduli in homogeneous and isotropic sys: 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 6 (5) moduli (3 equivalent axis) (rotational invariance around an axis)

6 moduli

6 moduli (2 equivalent symmetry axis)

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

I. Disordered Materials Bibliography: I. Disordered Materials K. Binder and W. Kob « Glassy Materials and disordered solids » (WS, 2005) S. R. Elliott « Physics of amorphous materials » (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 » Reith ed. (American scientific, 2005) IV. Elasticity of Disordered Materials B.A. DiDonna and T. Lubensky « Non-affine correlations in Random elastic Media » (2005) C. Maloney « Correlations in the Elastic Response of Dense Random Packings » (2006) Salvatore Torquato « Random Heterogeneous Materials » Springer ed. (2002) V. Sound propagation Ping Sheng « Introduction to wave scattering, Localization, and Mesoscopic Phenomena » (Academic Press 1995) V. Gurevich, D. Parshin and H. Schober Physical review B 67, 094203 (2003)