2 -- annotated -- Crystals, dislocations and slip

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Yale University *

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Oct 30, 2023

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Week 2: Defects, dislocations, slip, and deformation and strengthening mechanisms
Miller indices for atomic planes Report the plane as:
Families of planes What family of planes of FCC are close-packed?
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Direction indices Planes Directions Specific instance Family
Normal stress and strain These are called the engineering or nominal stress and strain ‘True’ stress and strain (different definitions) coming later in the course
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Shear stress and strain
Stresses: tensile and shear
Hydrostatic pressure and volumetric strain
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Examples of stress states and their combinations Uniaxial (strip of material under tension; column under compression) Biaxial (balloon) Torsion (shear along the length as well as around the circumference) Bending (show that shear must be present)
Poisson’s ratio Relationships between elastic moduli depend on 𝜈𝜈 Isotropy is assumed here. Plenty of materials are not isotropic. Discuss Poisson’s ratios of metals, glassy polymers, rubbers, cork, woven fabrics…. For an isotropic material, any two of 𝐸𝐸 , 𝐺𝐺 , 𝜈𝜈 , 𝐾𝐾 are enough to define the material’s elastic behavior
Poisson’s ratio examples
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Expected Young’s modulus based on many parallel bonds Macroscopic implication of interatomic forces
Magnitudes of elastic moduli for different bond types and materials classes
‘Ideal’ rupture strength of materials Explanation: Maximum of 𝐹𝐹 − 𝑟𝑟 curve occurs at 𝑟𝑟 = 𝑟𝑟 𝐷𝐷 1.25 𝑟𝑟 0 (roughly) ( 𝑟𝑟 𝐷𝐷 is the dissociation radius; 𝑟𝑟 0 is the equilibrium atomic spacing) The stress 𝜎𝜎 (i.e., F scaled up to consider a large array of atoms) when 𝑟𝑟 = 𝑟𝑟 𝐷𝐷 is referred to as the ideal strength, �𝜎𝜎 . This is the hypothetical stress that would be needed to cause the material to disintegrate if tensile stretching of atomic bonds were the means of failure. To estimate �𝜎𝜎 , consider extrapolating the slope, E , of the stress-strain curve at 𝑟𝑟 = 𝑟𝑟 0 . This slope intersects with 𝑟𝑟 = 𝑟𝑟 𝐷𝐷 at 𝜎𝜎 = 2 �𝜎𝜎 . Therefore: 2 �𝜎𝜎 ≈ 0. 25𝐸𝐸 , so �𝜎𝜎 ≈ 𝐸𝐸 /8 More thorough analysis that considers the attractive and repulsive forces from non-nearest-neighbor atoms as well arrives at an estimate closer to �𝜎𝜎 ≈ 𝐸𝐸 / 15 (still very approximate). Very few materials approach this ‘ideal’ strength in bulk. This is because materials don’t really fail this way. Next we will discuss how materials do deform permanently and fail…
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Real strengths of materials 𝜎𝜎 𝑦𝑦 𝜎𝜎 𝑦𝑦 / E
Why are real strengths often so much lower than ‘ideal’ ones? (Hydrogel bead model demonstration in lecture) Plastic deformation is by planes of atoms sliding over each other Known as ‘slip’ Deformation is highly localized to specific bands of atomic planes Slip directions are oriented at an angle to the normal load ( 45°) Plastic deformation is incompressible (no volume change)
Comparing ideal force-displacement curve with a ductile metal Not to scale: Ductile: from Latin ducere , to lead
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Why is plastic deformation by slip? Enabled by dislocations Disruptions of the regular lattice structure of a material Make it much easier for planes to move over each other Found in metals and ceramics, but since individual bonds tend to be much stronger in ceramics dislocations are more emphasized in processing of metals
Edge dislocations: behavior Imagine cutting into the lattice, offsetting by one atomic spacing b , and re-bonding Structural effect is akin to an ‘extra half-plane of atoms’ Motion: analogy to a moving ruck in a rug b is the Burgers vector When an edge dislocation moves, it Dislocation line out of page
Motion ( glide ) of edge dislocations https://www.doitpoms.ac.uk/tlplib/dislocations/dislocation_glide.php
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The nature of an edge dislocation Figure 10.3 An edge dislocation, (a) viewed from a continuum standpoint (ignoring the atoms) and (b) showing the positions of the atoms near the dislocation. Only the atom centers are shown.
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Motion of edge dislocations Figure 10.4 How an edge dislocation moves through a crystal. (a) How the atomic bonds at the center of the dislocation break and reform to allow the dislocation to move. (b) A complete sequence for the introduction of a dislocation into a crystal from the left side, its migration through the crystal, and its expulsion on the right side; this process causes the lower half of the crystal to slip by a distance b under the upper half.
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Force needed to move a dislocation
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Wide vs narrow dislocations Ceramics – narrow; Ductile metals – wide (easier to move) Disregistry (offset from perfect lattice) is always b /2 at the center of a perfect edge dislocation https://www.doitpoms.ac.uk/tlplib/dislocations/dislocationwidth.php
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Compare force needed to move a dislocation to that needed for dislocation-free slip With dislocations (i.e. what actually happens): Without dislocations (hypothetical case to show how much higher the yield stress would be without dislocations):
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Line tension of a dislocation Clarify: force needed to move dislocation cf line tension of a dislocation
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How do we know dislocations exist? This is possibly the first transmission electron micrograph (TEM) of a dislocation At the time, resolution of TEM was 1 nm; most metal lattices ~0.2 nm. Instead, Sir James Menter used platinum phthalocyanine, which has dia 1.2 nm and a Pt atom in center to provide contrast in the microscope. Ability to etch surface pits also – etching goes deeper into dislocation lines
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How do we know dislocations exist? This is possibly the first transmission electron micrograph (TEM) of a dislocation At the time, resolution of TEM was 1 nm; most metal lattices ~0.2 nm. Instead, Sir James Menter used platinum phthalocyanine, which has dia 1.2 nm and a Pt atom in center to provide contrast in the microscope. Ability to etch surface pits also – etching goes deeper into dislocation lines
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Video of a dislocation moving through a thin film of ferritic steel At elevated temperature, taken in a TEM. https://www.doitpoms.ac.uk/vidlib/full_record.php?id=69
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Each crystal contains many dislocations Figure 10.10 An electron microscope picture of dislocation lines in stainless steel. The picture was taken by firing electrons through a very thin slice of steel about 100 nm thick. The dislocation lines here are only about 1000 atom diameters long because they have been “chopped off” where they meet the top and bottom surfaces of the thin slice. But a sugar-cube-sized piece of any engineering alloy contains about 10 5 km of dislocation line.
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Screw dislocations: behavior When a screw dislocation moves, it Before dislocation introduced: two layers of regular lattice (purple, green), one stacked directly above the other:
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Screw dislocation model
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