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

Strain ()

Str

es

s (

)

Plastic region

Structural failure point

Onset of failure

Ultimate stress

Example

www.iop.org

0

100

200

300

400

0.1 0.2 0.3 0.4 0.5 0.6 0.7 39.0 39.1 39.2strain /%

+

0

fracture

plastic region,extension uniformalong length

plastic region,necking hasbegun

elasticregion

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