Stress, strain and more on peak broadening

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Stress, strain and more on peak broadening

• 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

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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
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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
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Mechanical properties of materials

• Shear strength shear is created by off-axis
  convergent forces.

Slipping of crystal planes
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Stress

• Stress force/area
• In simplest form

Normal (or tensile) stress perpendicular to
material Shear stress parallel to material
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Stress

• Thus can resolve into tensile and shear
  components

Tensile stress, ? Shear stress, ?
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Strain

• Strain result of stress
• Deformation divided by original dimension

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The Stress-Strain curve
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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

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Plastic region

• In the plastic region, under plastic deformation,
  the material is permanently deformed/damaged as a
  result of the loading.

In the plastic region, when the applied stress is
removed, the material will not return to original
shape.
The transition from the elastic region to the
plastic region is called the yield point or
elastic limit
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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

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Example

• www.iop.org

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Tensile strength

• Maximum possible engineering stress in tension.

• Metals occurs when noticeable necking starts.
• Ceramics occurs when crack propagation starts.

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Modulus

• The slope of the linear portion of the curve
  describes the modulus of the specimen.
• Youngs 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

Hookes law ? E ?
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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.

• e.g. (GPa)
• Teflon 0.5 Bone 10-20
• Concrete 30
• Copper 120
• Diamond 1100

Spider silk
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Now back to diffraction

• X-ray diffraction patterns can give us some
  information on strain
• Remember..

Scherrer formula where k0.9
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(micro) Strain uniform

• Uniform strain causes the lattice to
  expand/contract isotropically
• Thus unit cell parameters expand/contract
• Peak positions shift

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(micro) Strain non-uniform

• Leads to systematic shift of atoms
• Results in peak-broadening
• Can arise from
• point defects (later)
• poor crystallinity
• plastic deformation

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Williamson-Hall plots

• Take the Scherrer equation and the strain effect

So if we plot Bcos? against 4sin ? we (should)
get a straight line with gradient ? and intercept
0.9?/t
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A Williamson-Hall plot (figure 2) indicates that
the cause of the broadening is strain, and most
of this will be the result of chemical disorder
the mean particle size is 4 pm, leading to
insignificant size broadening. . The increase in
slope with decreasing temperature clearly
indicates an increase in the rhombohedral
distortion with falling temperature.
C N W Darlington and R J Cernik J. Phys.
Condens. Matter 1 (1989) 6019-6023.
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Example

• 0.138 0.9?/t
• gradient ?

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Crystallite size
Halfwidth as before
Can give misleading results
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Crystallite size
Integral breadth
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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