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

Graeme AcklandMarch 2010

Elasticity and Plasticity

Graeme AcklandUniversity of Edinburgh


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


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.


Cubic materials - elasticity

81 ratios between 9 shears and 9 strains.

Notation, e.g. xyxy = 66; xxyy = 12

C12 = 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)


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


Actual deformation modes Not all elastic moduli correspond

to appliable strains: in cubic B=(C11+2C12) / 3 C’= C11-C12 / 2 C44

These combinations must be positive: the Born Stability Criteria


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.


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


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.


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.


Dislocations

An extra half plane of atoms creating a line defect


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.


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…


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


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 requiresthe remaining five to be slip systems


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.


Deformation Twinning - Micrographs

Fast, large shear, no long range strain :

High activation energy

easy to calculate twin boundary energy


Martensitic Twins: Cycling

Damage Accumulates, defects store memory.

Austenite energy

Martensite energy

Martensite Structure


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


Molecular dynamics simulation of twin and dislocation deformation

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

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