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Mechanical Response at Very Small Scale Lecture 3: The Microscopic Basis of Elasticity

Mechanical Response at Very Small Scale Lecture 3: The Microscopic Basis of Elasticity Anne Tanguy University of Lyon (France). III. Microscopic basis of Elasticity. The Cauchy-Born theory of solids (1915). General expression of microscopic and continuous energy .

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Mechanical Response at Very Small Scale Lecture 3: The Microscopic Basis of Elasticity

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  1. Mechanical Response at Very Small Scale Lecture 3: The Microscopic Basis of Elasticity Anne Tanguy University of Lyon (France)

  2. III. Microscopic basis of Elasticity. • The Cauchy-Born theory of solids (1915). • General expression of microscopic and continuousenergy. • The microscopic expression for Stresses. • The microscopic expression for ElasticModuli. • B. The coarse-grainedtheory for microscopicelasticity (2005). • Coarse-graineddisplacement and fluctuations • The microscopic expression for Stresses. • The computation of Local ElasticModuli. S. Alexander, Physics Reports 296,65 (1998) C. Goldenberg and I. Goldhirsch (2005)

  3. Microscopic expression for the local ElasticModuli: Simple example of a cubiccrystal. On each bond: strain stress elastic modulus

  4. A. The Cauchy-Born Theory of Solids (1915). Regular expression of the Many-particles Energy: j i j k N particles D dimensions N.D parameters -D(D+1)/2 rigid translations and rotations N.D –D(D+1)/2 independent distances 2-body interactions (Cauchy model) Ex. Lennard-Jones Foams BKS model for Silica 3-body inter. Ex. Silicon

  5. Expression of local forces: Internal force exerted on atom i: Force of atom j on atom i: with with Tension of the bond (i,j) in the configuration {r}. The equilibrium on each atom i writes: thus

  6. Particlesdisplacement, and strain: uijP uij uijT rij uj rijeq ui j i

  7. First orderexpansion of the energy, and local stresses: To compare with:

  8. First orderexpansion of the energy, and local stresses: To compare with: « Site stress »: Local stress:

  9. Second orderexpansion of the energy, local ElasticModuli: with rotation bound elongation Local stiffness

  10. Born-Huang approximation for local ElasticModuli: Tij=0 To compare with:

  11. Born-Huang approximation for local ElasticModuli: 2-body contribution (central forces): (i1i2)=(i3i4)  n=1/2 i 3-body contribution (angular bending): i=i1 and i=i3 or i=i4 n=2/3 i i 4-body interactions (twists): (i1i2) ≠ (i3i4)  n=2/4

  12. Number of independentElasticModuli, from the microscopic expression: Cabgd=Cbagd and Cabgd=Cabdg 36 moduli Cabgd=Cgdab 21 moduli Additional symetries , for 2-body interactions (Cauchy model): Permutations of all indices: Caabb=Cabab and Cabgg=Cagbg (Cauchy relations for 2-body interactions)  3 Caaaa + 6 Caaab + 3 Caabb + 3 Cabgg  15 moduli. Warning:CabgdMACRO ≠ < CabgdMICRO (i) > (cf. lecture 4)

  13. B. The coarse-grainedtheory for microscopicelasticity For ex. with and

  14. 1) Coarse-grained displacement: Velocity dependent

  15. Separate coarse-grained (continuous) response, and « fluctuations »: continuous Coarse-grained displacement and fluctuations: F gaussian funct. of width w C. Goldenberg et I. Goldhirsch (2004)

  16. 2) Microscopic expression for Stresses

  17. cf. Note that, at this level, there is no explicit linear relation between and !!

  18. Use of the coarse-grained (continuous) disp. field for the computation of local elastic moduli: strain F Gaussian with a width w ~ 2 stress 2D case: using 3 independent deformations for a 2D system

  19. Maps of localelastic moduli: C1 ~ 2 m1 C2 ~ 2 m2 C3 ~ 2 (l+m) 2D Jennard-Jones w=5a N = 216 225 L = 483 a

  20. Large scale convergence to homogeneous and isotropic elasticity: Elastic Moduli: 2l+2m ~ 1/w 2m Locally inhomogeneous and anisotropic. Progressive convergence to the macroscopic moduli l and m, homogeneous and isotropic. Faster convergence of compressibility. No size dependence, but no characteristic size !

  21. ? 1% Which characteristic size ? Departure from local Hooke’s law, for r < 5 a. At small scale w: ambigous definition of elastic moduli (9 uncoherent equations for 6 unknowns) Error function: Local rotations? Long-range interactions ? Role of the « fluctuations » ?

  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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