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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: force N Stress () =**

Cross-sectional area N m2 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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**deformed length – original length**

Strain Strain – result of stress Deformation divided by original dimension Strain () = deformed length – original length L Lo original length =

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**The Stress-Strain curve**

Elastic region Plastic region Linear slope Yield point Ultimate stress Structural failure point Onset of failure

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Elastic region Strain () Stress () Elastic region Linear slope 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. Strain () Stress () Elastic region Yield point Plastic region 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 Strain () Stress () Plastic region Structural failure point Onset of failure Ultimate stress 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

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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. 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

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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 Bone 10-20 Concrete 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 k=0.9

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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)

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Example 0.138 = 0.9/t gradient

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**Can give misleading results**

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

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