1. Graeme Ackland March 2010 Elasticity and Plasticity Graeme Ackland University of Edinburgh

2. Graeme Ackland March 2010 ceiiinossssttuu (Hooke 1675) Ut tensio, sic vis i.e. linear response of stress to strain Tensor quantities Nine Components 3 rotations 1 volume 5 shears

3. Graeme Ackland March 2010 Strain There are two definitions of strain Engineering strain = L /Lo ; = F / Ao True strain = ln(L / Lo); = F / A Both depend on a reference state.

4. Graeme Ackland March 2010 Cubic materials - elasticity 81 ratios between 9 shears and 9 strains. Notation, e.g. xyxy = 66; xxyy = 12 C 12 = xx / yy Symmetry reduces this considerably. E.g. 2: Shear and bulk moduli (homogenous material) 3: C12, C11 C44 (cubic) 5: C11 C33 C12 C13 C44 (hexagonal)

5. Graeme Ackland March 2010 Calculation of elastic moduli Second derivative of the free energy. Apply a strain, calculate stress tensor. Apply a special strain, calculate energy. Practical concern: Even at 0K, atoms may change their positions in response to shear. Relax the atoms Elastic moduli drop significantly with temperature

6. Graeme Ackland March 2010 Actual deformation modes Not all elastic moduli correspond to appliable strains: in cubic B=(C 11 +2C 12 ) / 3 C= C 11 -C 12 / 2 C 44 These combinations must be positive: the Born Stability Criteria

7. Graeme Ackland March 2010 Harmonic Phonons: Free energy Calculate the energy as a function of small displacements (second term is zero at equilibrium). Create a matrix equation of motion The eigenvalues of this dynamical matrix are the phonon energies, eigenvectors are the phonon modes. They are independent harmonic oscillators.

8. Graeme Ackland March 2010 Harmonic Free energy Once the phonon frequencies are known, the temperature-dependent free energy is just statistical mechanics (Born & Huang etc.) (neglects phonon-phonon interactions, anharmonicity, thermal expansion) Unless … One of the eigenvalues is negative. Which it is in -Ti

9. Graeme Ackland March 2010 Dynamical stability Negative phonon eigenvalue means Energy goes down as structure distorts Crystal structure is UNSTABLE at T=0 Can define phonon by curvature of the energy, or by autocorrelation function (MD). In practice, there is always some limit to the phonon modulations.

10. Graeme Ackland March 2010 Phonon modes in Zr (similar to Ti) Phonon dispersion curve at 1400 K. Dashed lines: MD calculation Circles are neutron scattering results Solid lines quasiharmonic lattice-dynamics calculations for perfect bcc Imaginary frequencies corresponding to unstable phonons are shown as negative.

11. Graeme Ackland March 2010 Dislocations An extra half plane of atoms creating a line defect

12. Graeme Ackland March 2010 Dislocations Characterised by: Burgers vector the plane, how much slip you get Glide Plane Plane swept out by the line as it moves Screw, edge or mixed.

13. Graeme Ackland March 2010 Dislocation Mobility Small burgers vectors move more easily. Intermetallic compounds are hard. Obstacles can hold up dislocations (precipitates, impurities, other dislocations) More on dislocations tomorrow…

14. Graeme Ackland March 2010 Dislocation simulation In modelling, dislocations are treated in three ways: Molecular dynamics: explicit atom geometry Dislocation dynamics: simulation of interacting lines Finite element: unspecified source of plasticity

15. Graeme Ackland March 2010 Five slip systems: a counting exercise A unit cell is defined by a parallelipiped, three vectors, a 3x3 matrix. Nine degrees of freedom Three rotations One dilatation (volume) So to deform to a general shape requires the remaining five to be slip systems

16. Graeme Ackland March 2010 Creep and Climb (QT movies) Dislocations cannot move in the plane of the extra half plane without adding atoms. This is climb, depends on vacancy migration and stress. Very slow, but requires only low stresses to bias vacancies arriving vs leaving. Not restricted to a particular slip system.

17. Graeme Ackland March 2010 Deformation Twinning - Micrographs Fast, large shear, no long range strain : High activation energy easy to calculate twin boundary energy

18. Graeme Ackland March 2010 Martensitic Twins: Cycling Damage Accumulates, defects store memory. Austenite energy Martensite energy Martensite Structure

19. Graeme Ackland March 2010 An MD simulation of a dislocation (bcc iron) Starting configuration Periodic in xy, fixed layer of atoms top and bottom in z Dislocation in the middle Move fixed layers to apply stress Final Configuration (n.b. periodic boundary) Dislocation has passed through the material many times: discontinuity on slip plane

20. Graeme Ackland March 2010 Molecular dynamics simulation of twin and dislocation deformation