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. A.The Cauchy-Born theory of solids (1915). 1)General expression of microscopic and continuous energy. 2)The microscopic expression for Stresses. 3)The microscopic expression for Elastic Moduli. B. The coarse-grained theory for microscopic elasticity (2005). 1)Coarse-grained displacement and fluctuations 2)The microscopic expression for Stresses. 3)The computation of Local Elastic Moduli. S. Alexander, Physics Reports 296,65 (1998) C. Goldenberg and I. Goldhirsch (2005)

3. Microscopic expression for the local Elastic Moduli: Simple example of a cubic crystal. On each bond: strain stress elastic modulus

4. A. The Cauchy-Born Theory of Solids (1915). i j k j Regular expression of the Many-particles Energy: 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 Tension of the bond (i,j) in the configuration {r}. The equilibrium on each atom i writes: thus

6. Particles displacement, and strain: i j uiui ujuj r ij eq r ij u ij u ij P u ij T

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

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

9. Second order expansion of the energy, local Elastic Moduli: with Local stiffness bound elongation rotation

10. Born-Huang approximation for local Elastic Moduli: T ij =0 To compare with:

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

12. Number of independent Elastic Moduli, from the microscopic expression: Warning: C  MACRO ≠ (cf. lecture 4) C  =C  and C  =C   36 moduli C  =C   21 moduli Additional symetries, for 2-body interactions (Cauchy model): Permutations of all indices: C  =C  and C  =C  (Cauchy relations for 2-body interactions)  3 C  + 6 C  + 3 C  + 3 C   15 moduli.

13. B. The coarse-grained theory for microscopic elasticity For ex. with and

14. 1) Coarse-grained displacement: Velocity dependent

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

16. 2) Microscopic expression for Stresses

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

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

20. C 1 ~ 2  1 C 2 ~ 2  2 C 3 ~ 2 ( +  2D Jennard-Jones w=5 a N = 216 225 L = 483 a Maps of local elastic moduli:

21.   Large scale convergence to homogeneous and isotropic elasticity: Elastic Moduli: Locally inhomogeneous and anisotropic. Progressive convergence to the macroscopic moduli and  homogeneous and isotropic. Faster convergence of compressibility. No size dependence, but no characteristic size ! ~ 1/w

22. 1% Departure from local Hooke’s law, for r < 5 a. Which characteristic size ? ? 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 » ?

23. 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)