Metals- Cutting and Bowing

Question 1

Coherent precipitates

Answer 1

On-to-one lattice matching across the interphase between the precipitate and the matrix. Gives rise to high strain energy and low interfacial energy

Question 2

Incoherent precipitates

Answer 2

No lattice matching across the interphase. Low strain energy, high interfacial energy.

Question 3

Semi-coherent precipitates

Answer 3

Some lattice matching, intermediate energies, very orientation dependent.

Question 4

Misfit strain formula

Answer 4

εmisfit=(a(ppt)-a(matrix))/a(matrix)
Where a sub ppt is lattice parameter for precipitate
a sub matrix is lattice parameter for matrix
Only valid for cubic crystals

Question 5

How does precipitate cutting work?

Answer 5

Small coherent precipitates may be sheared by dislocations. The dislocations can slip from the matrix into the precipitate and cause slip in the precipitate as well before transitioning back into the matrix. Part of the precipitate slides past the other by b

Question 6

Increment of strength arising from dislocations cutting through precipitates formula

Answer 6

τ=rt((3k^3ε^3G^2 f r)/2πb)
Where k is constant
ε is misfit strain
G is shear modulus
f is volume fraction of precipitates
r is precipitate radius
b is Burgers vector
Means τ proportional to rt(r)

Question 7

Graph of shear stress vs precipitate radius for cutting

Answer 7

y equals root x graph

Question 8

How to estimate volume fraction of precipitates

Answer 8

Assume precipitates form a cubic array in the material.
f=Volume of precipitates/Volume of cube.
Volume of cube is x^3 where x is interparticle spacing. Assume spherical precipitates and each one 1/8 of its volume on one cube (1 full precipitate in cube). So volume fraction is:
F=(4/3 πr0^3)/x^3 = 4/3 π(r0/x)^3
Where r0 is radius of one precipitate

Question 9

When can dislocations not pass into precipitates and what do they do instead?

Answer 9

For larger precipitates or incoherent precipitates. They must instead bow around the precipitates leaving a dislocation loop around the precipitate.

Question 10

Formula for shear stress from Orowan bowing

Answer 10

τ=(Gb/r)(3f/2π)^1/2
Where G is shear modulus
b is Burgers vector 
f is volume fraction of precipitates 
r is radius of precipitates

Question 11

What is shear strength proportional to for bowing?

Answer 11

τ proportional to 1/r

Question 12

For what type of precipitates can both Orowan bowing and cutting be operational?

Answer 12

Coherent

Question 13

Effect of precipitate radius on dislocation mechanisms

Answer 13

If they’re small they may be sheared readily providing little hardening. As radii increase, resistance resistance to cutting increases with r^1/2. Eventually becomes easier for dislocations to bow between precipitates rather than cut them. Further increase in radii results in decrease in shear stress with 1/r

Question 14

Optimum particle radius

Answer 14

An optimum particle size exists that results in the highest strength of the material.
Found by equating equations for cutting and bowing and solve for r

Question 15

Features of age hardening as precipitates grow

Answer 15

Solute hardening: short ageing times, solid solution main strengthening mechanism, decreases because atoms rejected to form precipitates.
Coherency strains: precipitates forming distort lattice, coherent particles easily sheared by dislocations.
Precipitate cutting: coherent precipitates get larger, significantly larger stresses required to shear the precipitates.
Orowan bowing: precipitates large, dislocation bypass by bowing possible/easier, easier as grow further so drop in stress (over-ageing)

Question 16

What do the age hardening factors apply to?

Answer 16

Only for coherent precipitates as can’t manipulate incoherent