Elasticity of Disordered Materials: Equations, Bounds, and Examples

1. Mechanical Response at Very Small Scale Lecture 4: Elasticity of DisorderedMaterials Anne Tanguy University of Lyon (France)

2. IV. Elasticity of disorderedMaterials. 1) General equations of motion for a disorderedmaterial 2) Rigorousbounds for the elasticmoduli. 3) Examples. Ping Sheng « Introduction to wavescattering, Localization, and MesoscopicPhenomena » (1995) B.A. DiDonna and T. Lubensky « Non-affine correlations in Randomelastic Media » (2005) C. Maloney « Correlations in the ElasticResponse of Dense RandomPackings » (2006) Salvatore Torquato « RandomHeterogeneousMaterials » Springer ed. (2002)

3. General equations: In case of homogeneous strain: But in general

4. Inhomogeneousstrainfield:

5. Example of a lennard-Jones glass: A. Tanguy et coll. Phys. Rev. B (2002), J.P. Wittmer et coll. Europhys. Lett. (2002), A. Tanguy et coll. App. Surf. Sc. (2004) F. Léonforte et coll. Phys. Rev. B (2004), F. Léonforte et coll. Phys. Rev. B (2005), F. Léonforte et coll. Phys. Rev. Lett. (2006), A. Tanguy et coll. (2006), C. Goldenberg et coll. (2007), M. Tsamados et coll. (2007), M. Tsamasos et coll. (2009). Atomic displacements Inhomogeneous response, rotational displacements in the non-affine part. • A.Tanguy et al. • (2002,2004,2005) • A.Lemaître et C. Maloney • (2004,2006) • J.R. Williams et at. (1997) • G. Debrégeas et al. (2001) • S. Roux et al. (2002) • E. Kolb et coll. (2003) • Weeks et al. (2006)

6. foams Granular materials other examples of inhomogeneous strain emulsions, colloids, … Weeks et al. (2006) F.Radjai, S.Roux (2002) E.Kolb et al. (2003) J.R. Williams et al. (1997) G.Debrégeas, A.Kabla, J.-M. di Méglio (2001,2003)

7. Dynamical Heterogeneities [Keys, Abate, Glotzer, DJDurian (preprint, 2007)]

8. Large distribution of local ElasticModuli:

9. Cartes de modules élastiques locaux: Large distribution of Elastic Moduli: C1 ~ 2 m1 C2 ~ 2 m2 C3 ~ 2 (l+m) 2D Jennard-Jones N = 216 225 L = 483

10. Lennard-Jones glass: homogeneous and then isotropic W>20a

11. General bounds for the Effective ElasticModuli:

12. General bounds for the effective macroscopicelasticmoduli of an inhomogeneoussolid. Example of fibers in a matrix: EL,T effective Young modulus Ef Fiber’s Young modulus Em Young modulus of matrix Voigt (1889) E EL ET Vf/V Reuss (1929)

13. General bounds for the effective macroscopicelasticmoduli of an inhomogeneoussolid. Quadratic part of the local elastic energy: Effective Stiffness Tensor: with

14. Preliminary results: then

15. VoigtBound (1889) for any deformation at equilibrium, homogeneously applied at the boundaries. with equality only if

16. ReussBound (1929) for any deformation at equilibrium, homogeneously applied at the boundaries. with equality only if

17. OtherBounds: with Ex. Exact kth order perturbative solution (n=2 Hashin and Shtrikman, 1963) then

18. Examples: N. Teyssier-Doyen et al. (2007) Voigt Reuss

19. Example 2: Lennard-Jones glass 2l+2m M. Tsamados et al. (2009) ~ 1/w Local Elastic Moduli: 2m Progressive convergence to the macroscopic moduli l and m, homogeneous and isotropic medium at large scale. Faster convergence of compressibility (homogenesous density)

20. Example of an Anisotropic Material: Wood for Musical Instruments Elastic Moduli Young’s Moduli: EL>>ER ~ ET

21. Parallel to the Fibres: Perpendicular to the Fibers: Holographic Interferometry, Hutchins (1971) Simplified expresison of the Eigenmodes of an Harmonic Table: E//≈ 11,6 GPa E┴ ≈ 0,716 GPa r ≈ 0.39 t.m-3 Large variety of resonant Frequencies

22. Choice of a sandwich material, allowing for the same mass. Eigenfrequencies are comparable to those of PRFC, with the following choice for the thicknesses: d1 ≈ 0.63 d2 d2 ≈ 0.66 dwood Looking for a Material with Analogous Anisotropy: E/// E┴ ≈ 16. E//≈ rf.Vf + rm.(1-Vf) PRFC with Vf ≈ 13% E┴ ≈ 1/ (Vf/rf + (1-Vf)/rm) then E// = 53 GPa Mass Density: rPRFC = 1,25 t.m-3 Comparing the Eigenfrequencies imposes: a thickness dPRFC = 0.75 x dwood ≈ 2.52 mm Then the Total Mass of the Harmonic Table is very large MPRFC ≈ 2.69 x Mwood !!!

23. C. Besnainou (LAM, Paris) « sandwich » material Plaster Mould in a Vacuum Bag, Heated at 140°C. Heating with Silicone Rubbers. Heating Ramp < 1/2h. Wood Acrylic Foam Unidirectional Carbon Fiber glued in epoxy … convenient also for lutes: Consequences: llight, stable, humidity-resistant, less damping,

24. …cellos, and string basses « COSI » Solidity and stability, especially against humidity, With the help of composite materials with Carbon Fibers. Richness of tone?

25. End

26. 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)