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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

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stress strain and more on peak broadening
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
material properties
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
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 materials1
Mechanical properties of materials

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

Slipping of crystal planes

stress

force

N

Stress () =

Cross-sectional area

m2

Stress

Stress = force/area

In simplest form:

Normal (or tensile) stress = perpendicular to material

Shear stress = parallel to material

stress1
Stress

Thus can resolve into tensile and shear components:

Tensile stress, 

Shear stress, 

strain

L

deformed length – original length

=

Strain () =

original length

Lo

Strain

Strain – result of stress

Deformation divided by original dimension

the stress strain curve

Onset of failure

Structural failure point

Yield point

Stress ()

Ultimate stress

Linear slope

Plastic region

Elastic region

Strain ()

The Stress-Strain curve
elastic region

Linear slope

Stress ()

Elastic region

Strain ()

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

plastic region

Yield point

Stress ()

Plastic region

Elastic region

Strain ()

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

failure

Onset of failure

Structural failure point

Ultimate stress

Stress ()

Plastic region

Strain ()

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

example
Example

www.iop.org

tensile strength
Tensile strength

Maximum possible engineering stress in tension.

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

Hooke’s law:  = E 

modulus properties
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

now back to diffraction
Now back to diffraction…

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

Remember..

Scherrer formula where k=0.9

micro strain uniform
(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
(micro) Strain : non-uniform
  • Leads to systematic shift of atoms
  • Results in peak-broadening
  • Can arise from
    • point defects (later)
    • poor crystallinity
    • plastic deformation
williamson hall plots
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

example1
Example

0.138 = 0.9/t

gradient 

crystallite size
Crystallite size

Halfwidth: as before

Can give misleading results

crystallite size1
Crystallite size

Integral breadth

slide23

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