lecture 3.0 structural defects mechanical properties of solids
TRANSCRIPT
Defects in Crystal StructureDefects in Crystal Structure
Vacancy, Interstitial, ImpuritySchottky DefectFrenkel DefectDislocations – edge dislocation, line,
screwGrain Boundary
Defect EquilibriumDefect EquilibriumSc= kBln gc(E)
Sb= kBln Wb EntropySs= kBln Ws
dFc = dE-TdSc-TdSs, the change in free energy
dFc ~ 6 nearest neighbour bond energies (since break on average 1/2 the bonds in the
surface)
Wb=(N+n)!/(N!n!) ~(N+n+1)/(n+1) ~(N+n)/n (If one vacancy added)
dSb=kBln((N+n)/n)
For large crystals dSs<<dSb
n ~ N exp –dFc/kBT
Mechanical Properties of SolidsMechanical Properties of Solids
Elastic deformation– reversible
• Young’s Modulus• Shear Modulus• Bulk Modulus
Plastic Deformation– irreversible
• change in shape of grains
Rupture/Fracture
Mechanical PropertiesMechanical Properties
Stress, xx= Fxx/A
Shear Stress, xy= Fxy/A
Compression
Yield Stress yield ~Y/10
yield~G/6 (theory-all
atoms to move together)
Strain, =x/xo
Shear Strain, =y/xo
Volume Strain = V/Vo
Brittle Fracture– stress leads to crack– stress concentration at crack tip
=2(l/r)– Vcrack= Vsound
Effect of Structure on Effect of Structure on Mechanical PropertiesMechanical PropertiesElasticityPlastic DeformationFracture
Strain
Str
ess Plastic
Deformation
Fracture
Elastic DeformationElastic Deformation
Pulling on a wire decreases its diameter l/lo= -l/Ro
Poisson’s Ratio, 0.5 (liquid case=0.5)
Young’s Modulus– Y(or E)= (F/A)/(l/lo)
Shear Modulus– G=/= Y/(2(1+))
Bulk Modulus• K=-P/(V/Vo)
• K=Y/(3(1-2))
Microscopic Elastic DeformationMicroscopic Elastic Deformation
Interatomic Forces FT =Tensile Force
FC=Compressive Force
Note F=-d(Energy)/dr
Forc
e
0
FC
FT
r
ao
Repulsion
Attraction
Plastic DeformationPlastic Deformation
Single Crystal– by slip on slip
planes
30/
)cos(cos
coscoscos/
cos
max
Gstressshearyield
Yielding
a
A
o
oyield
Shear Stress
Deformation of WhiskersDeformation of Whiskers
Without DefectsRupture
With Defectsgenerated by high stress
Slip Systems in MetalsSlip Systems in Metals
CrystalStructure
Slip Planes SlipDirections
Number ofSlipSystems
Examples
fcc {111} <1-10> 12 Al, Cu, Nibcc {110}
{211}{321}
<-111><-111><-111>
121224
Fe,Ta,W
hcp {0001}{10-10}{10-11}
<11-20><11-20><11-20>
336
Be, Mg,Zn,Ti, Zr, Re
Plastic DeformationPlastic Deformation
Poly Crystals– by grain boundaries
– by slip on slip planes
– Engineering Stress, Ao
– True Stress, Ai
ooii
oyield
lAlA
sizegraind
dk
Ao
Ai
Movement at Edge DislocationMovement at Edge Dislocation
Slip Plane is the plane on which the dislocation glides
Slip plane is defined by BV and I
Plastic DeformationPlastic Deformation-Polycrystalline sample-Polycrystalline sample
Many slip planes – large amount of
slip (elongation)
Strain hardening– Increased difficulty of
dislocation motion due to dislocation density
– Shear Stress to Maintain plastic flow, =o+Gb
• dislocation density,
Strain Hardening
Strain HardeningStrain Hardening/Work Hardening/Work Hardening
Dislocation Movement forms dislocation loops– New dislocations
created by dislocation movement
Critical shear stress that will activate a dislocation source
c~2Gb/l– G=Shear Modulus
– b=Burgers Vector
– l=length of dislocation segment
Burger’s Vector-Burger’s Vector-Dislocations are characterised by their Dislocations are characterised by their Burger's vectors.Burger's vectors. These These represent the 'represent the 'failure closure'failure closure' in a Burger's circuit in imperfect (top) in a Burger's circuit in imperfect (top) and perfect (bottom) crystal.and perfect (bottom) crystal.
BV Perpendicular to DislocationBV parallel to Dislocation
Solution Hardening (Alloying)Solution Hardening (Alloying)
Solid Solutions• Solute atoms segregate to dislocations =
reduces dislocation mobility• higher required to move dislocation
– Solute Properties• larger cation size=large lattice strain• large effective elastic modulus, Y
Multi-phase alloys - Volume fraction rule
Precipitation HardeningPrecipitation Hardening
Fine dispersion of heterogeneity• impede dislocation motion
c~2Gb/ is the distance between particles
– Particle Properties• very small and well dispersed• Hard particles/ soft metal matrix
Methods to Produce– Oxidation of a metal– Add Fibers - Fiber Composites
Cracking vs Plastic DeformationCracking vs Plastic Deformation
Brittle• Poor dislocation motion• stress needed to initiate
a crack is low
– Ionic Solids• disrupt charges
– Covalent Solids• disrupt bonds
– Amorphous solids• no dislocations
Ductile• good dislocation motion
• stress needed to initiate slip is low
– Metals• electrons free to move
Depends on T and P– ductile at high T (and P)