Learning Outcomes By the end of this section you should: • be familiar with some mechanical properties of solids • understand how external forces affect crystals at the Angstrom scale • be able to calculate particle size using both the Scherrer equation and stress analysis Stress, strain and more on peak broadening
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Learning Outcomes By the end of this section you should: be familiar with some mechanical properties of solids understand how external forces affect crystals.
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Learning Outcomes
By the end of this section you should:• be familiar with some mechanical properties of solids• understand how external forces affect crystals at the
Angstrom scale• be able to calculate particle size using both the
Scherrer equation and stress analysis
Stress, strain and more on peak broadening
Material Properties
What happens to solids under different forces?
The lattice is relatively rigid, but….
Note: materials properties will be considered mathematically in PX3508 – Energy and Matter
Mechanical properties of materials
Tensile strength – tensile forces acting on a cylindrical specimen act divergently along a single line.
Compressive strength – compressive forces on a cube act convergently in a single line
Mechanical properties of materials
Shear strength – shear is created by off-axis convergent forces.
Slipping of crystal planes
Stress
Stress = force/area
In simplest form:
Normal (or tensile) stress = perpendicular to materialShear stress = parallel to material
Stress () =force
Cross-sectional area
N
m2
Stress
Thus can resolve into tensile and shear components:
Tensile stress,
Shear stress,
StrainStrain – result of stress
Deformation divided by original dimension
Strain () =deformed length – original length L
Looriginal length
=
The Stress-Strain curve
Strain ()
Stress ()
Elastic region
Plastic region
Linear slope
Yield point
Ultimate stress
Structural failure point
Onset of failure
Elastic region
In the elastic region, ideally, if the stress is returned to zero then the strain returns to zero with no damage to the atomic/molecular structure, i.e. the deformation is completely reversed
Strain ()
Str
es
s (
)
Elastic region
Linear slope
Plastic region
In the plastic region, under plastic deformation, the material is permanently deformed/damaged as a result of the loading.
Strain ()
Str
es
s (
)
Elastic region
Yield point
Plastic region
The transition from the elastic region to the plastic region is called the yield point or elastic limit
In the plastic region, when the applied stress is removed, the material will not return to original shape.
Failure
At the onset of yield, the specimen experiences the onset of failure (plastic deformation), and at the termination of the range of plastic deformation, the sample experiences a structural level failure – failure point
Tensile strengthMaximum possible engineering stress in tension.
• Metals: occurs when noticeable necking starts.• Ceramics: occurs when crack propagation starts.
Modulus
The slope of the linear portion of the curve describes the modulus of the specimen.
Young’s modulus (E) – slope of stress-strain curve with sample in tension (aka Elastic modulus)
Shear modulus (G) - slope of stress-strain curve with sample in torsion or linear shear
Bulk modulus (H) – slope of stress-strain curve with sample in compression
2m
NEHooke’s law: = E
Modulus - properties
Higher values of modulus (steeper gradients of slope in stress-strain curve) relates to a more stiff/brittle material – more difficult to deform the material
Lower values of modulus (shallow gradients of slope in stress-strain curve) relates to a more ductile material.
Spider silk
e.g. (GPa) • Teflon 0.5 Bone 10-20• Concrete 30• Copper 120 • Diamond 1100
Now back to diffraction…
X-ray diffraction patterns can give us some information on strain
Remember..
BcosB
kt
Scherrer formula where k=0.9
(micro) Strain : uniform
• Uniform strain causes the lattice to expand/contract isotropically
• Thus unit cell parameters expand/contract• Peak positions shift
(micro) Strain : non-uniform
• Leads to systematic shift of atoms• Results in peak-broadening• Can arise from
– point defects (later)– poor crystallinity– plastic deformation
tan4B
Williamson-Hall plots
Take the Scherrer equation and the strain effect
cost
0.9B
cosB
0.9t C
C
tan4BStr
sin4t
0.9cosB
tan4cosθt
λ0.9B
tot
tot
So if we plot Bcos against 4sin we (should) get a straight line with gradient and intercept 0.9/t
Example
0.138 = 0.9/t
gradient
y = 0.0104x + 0.1378y = 0.0202x + 0.1383
y = 0.0303x + 0.1379
y = 0.0499x + 0.1389
y = 0.0703x + 0.1379
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.5 1 1.5 2 2.5 3 3.5
4 sin
B c
os
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
Crystallite size
Halfwidth: as before
Can give misleading results
Crystallite size
Integral breadth
Summary
External forces affect the underlying crystal structure
Strained materials show broadened diffraction peaks
Width of peaks can be resolved into components due to particle size and strain