Deformation of glassy systems: insights from simulation studies

Transcription

1 Deformation of glassy systems: insights from simulation studies Jean-Louis Barrat Université Joseph Fourier, Grenoble Institut Universitaire de France

2 Overview Introduction Atomic scale investigation of plastic deformation Mesoscale models Deformation of a structured material

3 Soft amorphous systems Colloidal pastes, complex fluids Foams Granular systems Structurally disordered materials Solid like behavior until yield stress/strain is reached Flow easily under moderate stresses (Pa to kpa)

4 Characterized by flow curve: (shear) stress vs strain rate in steady shear Non-linear rheology and yield stress(es): Static Yield stress log σ σ Y flow. σ γ α Newtonian fluid Dynamic Yield stress σ d log. γ

5 «Hard» amorphous systems Metallic glasses Polymer glasses Oxide glasses Stress-Strain curves rather than flow curves (tensile, compressive, uni/tri-axial..) Elastic, solid like behavior until yield stress/strain is reached Plastic flow observed before breaking

6 Si PCML université Lyon I Covalent glasses: a-si, silicates Examples of hard glasses Metallic glasses: Zr-Ti-Cu-Ni-Be, Mg-Cu-Y, Fe-B Polymeric glasses: PMMA,PC, PS SiO 2, F. Léonforte Metallic glasses, Vitreloy g(ω)/ω 2 Vibration modes, : Boson Peak. Inamura et coll. (2000) ω High yield stress:

7 Stress strain curves at imposed strain rate Bulk metallic glass (Johnson, UCLA) Polymers (VanMelick, Eindhoven) Rate and temperature effects are important

8 Ideal elastoplastic behaviour Plastic flow Elastic Unrecoverable plastic deformation To observe plastic flow, it is necessary - to avoid fracture - to avoid shear banding or strain localisation (or to have access to the strained region) This is possible in practice in: - Polymer glasses ( close to T g ) - Metallic glasses ( close to T g ) - simulation

9 Stress-strain curves can also be monitored in «soft» systems (shear deformation) => models for «hard» systems (Bragg 1930, Argon 1975) Colloids (Lequeux) Plastic response of a foam ; I. Cantat, O. Pitois, Phys. of fluids 2006 Note the jerky aspect of the response in the «athermal» system (foam) => well identified, localized plastic events

10 «T1» events in foams (Princen, 1981) (Dennin 2006) Colloidal glass, confocal microscopy Science 318 (2007) «Shear transformation zones» (Argon, Langer)

11 Why are they different from crystalline materials? Flow of crystalline solids: flow defects identified (dislocations, Volterra 1930), seen by TEM (1960) interaction and motion understood (Peierls, Nabarro, Friedel, 1950) dislocation dynamics in computer codes (1980) Flow of amorphous solids: Flow defects? Interaction and Motion?? Mesoscale codes???

12 Overview Introduction Atomic scale investigation of plastic deformation Mesoscale models Deformation of a layered polymer

13 QUASI STATIC DEFORMATION IN A 2D SYSTEM Polydisperse Lennard-Jones system, quench to T=0 from liquid state; quasistatic shear (minimization after each deformation step strain increments) Low strain rate, Low temperature stress strain curve (see also similar work by C. Maloney, A. Lemaitre)

14 Very small strain : elastic deformation (heterogeneous, non affine part is important!) Linear elastic Map of local shear modulus (Tsamados, Tanguy, JLB PRE 2009) and a low frequency vibrational mode

15 Onset of plastic deformation Localized, quadrupolar events (shear transformation zones!) Irreversible displacement Stress drop Δσ xy

16 Similar phenomenology in a 3d, polymer glass (dissipation at very small strain, anelasticity or internal friction ) Molecular plasticity of glasses in the elastic regime, Phys. Rev. E. 77, (2008)

17 L=104.0,N=10000,shear conf23-n2339 Plastic flow regime Larger stress drops This large stress drops correspond to cascades of successive quadrupolar events. Cooperativity between plastic events.

18 Overview Introduction Atomic scale investigation of plastic deformation Mean field and mesoscale models Deformation of a structured material

19 First mean field description :Eyring s model (1936) Energy Stress biased energy landscape Strain jump +/- shear rate :

20 High temperature/low shear : linear regime Viscous fluid : Low temperature/high shear : non-linear flow curve Does not account for all the complexity observed (single time scale, no yield stress) Introduce «Rate and state» ideas: additional internal variables, effective T (Rice Ruina for friction ; STZ model ; SGR model)

21 Soft Glassy Rheology (Sollich, Lequeux, Hébraud,Sollich, Fielding) distribution of energy barriers glass transition (trap model) l strain variable, increases linearly with time P(L,E,t) distribution of systems in different «traps» and at different strains is the internal variable Fixed strain rate evolution Activated escape from traps due to «mechanical noise»

22 Dynamical equation for the strain distribution function P(E,l,t) similar in spirit to the trap model of glasses (Bouchaud) Trap depths are distributed x = mechanical noise temperature Very successful model, describes many features of the flow of glassy systems a glass transition at x=x g =1 for x< x g : aging, yield stress σ Y, σ=σ Y +A γ 1-x But.. mechanical temperature x is not defined self consistently the model is still mean-field (lacks spatial information)

23 Elasto-Plastic discrete element models Argon Bulatov 1994, Picard Bocquet et al 2002 Try to capture the generic scenario Elastic resistance Localized yield event Stress redistribution but keep non-local effects to study collective behavior Picard et al. PRE (2005); earlier models by Argon Bulatov, Roux

24 Stress dynamics in the model of Picard et al Plastic activity (number of sites that reach yield point) Global elastic forcing U Plastic relaxations U -U -U

25 Stress strain and flow curves

26 Models exhibit dynamical heterogeneities at low strain rates Cumulated plastic activity (+1 if plastic) Low shear rate behavior : localized plastic bursts «always» crack in the same region «fragile» zones Heterogeneous behavior at low shear rates

27 Perspectives and challenges 3 dimensional models Input from microscopic simulations Analysis of avalanches, correlations, diffusion Role of temperature, convection Coarse graining to nonlocal constitutive equations Conditions of strain localisation

28 Overview Introduction Atomic scale investigation of plastic deformation Mesoscale models Deformation of a layered polymer material (coarse grained molecular dynamics simulations)

29 Lamellar triblock copolymer alternating rubbery and glassy lamellae (e.g. PS/PB/PS) Uniaxial deformation Deformation supported by rubbery phase Poisson stress in the transverse direction on glassy phase =>buckling

30 PS-PB-PS under 300% strain. Y. Cohen et al, Macromol. 33, 6502 (2000)

31 Elastic interpretation: The chevron folding instability in thermoplastic elastomers and other layered materials D J Read, R A Duckett, J Sweeney, T C B McLeish J. Phys. D: Appl. Phys. 32 (1999) Analysis predicts buckling instability above a critical strain, starting at wavevector k=0 (i.e. wavelength proportional to cell size). In the simulations a characteristic wavelength seems to emerge:

32 Competition between the rate of strain variation and the rate of growth of different instabilities determines the observed wavelength.

33 growth rate k 0 2k 0 Increasing deformation wavevector Amplitude of first and second mode as a function of time Failure mode depends on strain rate? To be checked experimentally

34 Collaborations Glasses: Michel Tsamados, Anne Tanguy, Lydéric Bocquet, Kirsten Martens (Université Lyon 1) Polymers: Ali Makke, Michel Perez, Olivier Lame (INSA Lyon)