Continuum modeling of ferroelectric materials and devices

Transcription

1 Continuum modeling of ferroelectric materials and devices S. Aubry, M. Fago, J. Knap, O. Schneider, A. Yavari and M. Ortiz ARO/MURI Kick-off meeting Caltech, March 30, 2001

2 Objectives To develop physics-based, predictive, engineering models of ferroelectric material behavior accounting for: Microstructure development and evolution, including: Domain structure Microstructural size Multiscale Macroscopic effective behavior, including: Hysteresis Modeling Scaling and size effect To inform multiscale continuum models with atomistic content: Elastic moduli, transformation strains, polarization Wall structure, energy Wall mobility To verify, validate and calibrate the multiscale continuum models against experimental data To develop methods of analysis enabling the use of multiscale continuum models in engineering design of devices

3 Objectives Multiscale modeling PbTiO 3 High temperature (non-polar cubic) Pb Ti O c a c =1.065 a Room temperature (<001> polarized tetragonal) (Quasicontinum) False color images of domain structure in BaTiO 3 (Burcsu, Ravichandran and Bhattacharya, 2000)

4 Objectives Multiscale modeling BaTiO 3 domain structure (BRB, 2000) Biaxial IBAD texture MEMS microactuators (Polla and Francis, 1998) Tent micropump (Cui and James, 2000)

5 Objectives Validation/Calibration σ σ 0 V V σ σ (Burcsu, Ravichandran and Bhattacharya, 2000)

6 Ferroelectric materials - Continuum theory (Shu and Bhattacharya, 2001) Total energy of a ferroelectric crystal subject to a stress, σ o, and external electric field, E o The electric potential, φ, satisfies Maxwell s Equation W = crystal free energy density F = deformation gradient P = polarization in reference configuration θ = temperature σ = stress φ = electric potential ρ f = charge

7 Ferroelectric materials Continuum modeling Phase transitions spontaneous strain and polarization BaTiO 3 Ba 2+ Ti 4+ O 2- c c a = 1.01 a Cubic (high temp) Tetragonal (room temp)

8 Martensitic materials Continuum theory Principle of Minimum Potential Energy: where: Energy lacks lower-semicontinuity, infimum not attained

9 Martensitic materials Continuum theory Observed microstructures in Cu-Al-Ni (Chu and James, 1999)

10 Martensitic materials - Relaxation Relaxed problem: where: Relaxed energy density: Minimum energy density attainable by consideration of all possible microstructures consistent with a macroscopic or average deformation. No constructive method is known for relaxing general energy densities consider special microstructures Sequential laminates Rank-1 convexification No practical algorithm is known for computing the rank-1 convexification of a general energy density exactly

11 Martensitic materials Sequential lamination Special microstructures: Sequential laminates Compatibility equations: Averaging:,

12 Martensitic materials Hierarchical relaxation Objective: To formulate a practical rank-1 convexification algorithm. Laminate energy: Configurational equilibrium: Microstructural evolution: Branching and pruning of leaves (Aubry, Fago and Ortiz, 2001)

13 Martensitic materials Hierarchical construction branching pruning Accept transition if and only if it lowers the energy of the laminate Re-equilibrate laminate after each transition

14 Martensitic materials Microstructure size Nonlocal energy: Wall energy: θ d Boundary layer: l Optimal size: l l limited by grain size, device geometry

15 Example: Cu-Al-Ni Energy vs deformation Stress vs deformation Volume fraction vs deformation Simulation of deformation path from austenite to martensitic variant (Aubry, Fago and Ortiz, 2001)

16 Example: Cu-Al-Ni Energy vs deformation Simulation of simple shear deformation path (Aubry, Fago and Ortiz, 2001)

17 Martensitic materials - Hysteresis Domain wall motion dissipates energy Kinematics of domain motion: General form of kinetic equation: Incremental energy density: Solutions are now path-dependent and dissipative

18 Example: Cu-Al-Ni (BRB, 2001) Volume fraction vs stress Simulation of deformation path from austenite to martensitic variant (Aubry, Fago and Ortiz, 2001)

19 FE Simulation of Al-Cu-Ni indentation Spherical indenter (1.5mm radius) Contact modelled by penalty method Dynamic relaxation & conjugate gradients Constitutive calculations done in parallel Quadratic tets 2 mm Quadrant of FE model

20 FE Simulation of Al-Cu-Ni indentation (indenter radius = 1.5 mm) Unrelaxed Relaxed, small grain Relaxed, large grain (Aubry, Fago and Ortiz, 2001)

21 FE simulation of Al-Cu-Ni indentation

22 Connection with atomistics: Outlook Empirical potentials Wall energies Wall mobility, kinetic equation Validation, simulation of BRB experiments Extend computational approach to fully-coupled problem Simulation of 3D effects in tent micropumps, hinge structure. Simulation of fracture and fatigue