From ab786a21b8fbf88c22fc4702b25f1d84e9a45f8d Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Wed, 22 Dec 2021 15:43:08 +0000 Subject: [PATCH] Add notes on metals --- mechanical/mmme1029_materials.md | 163 +++++++++++++++++++++++++++++++ 1 file changed, 163 insertions(+) diff --git a/mechanical/mmme1029_materials.md b/mechanical/mmme1029_materials.md index ebdc78f..3d27982 100755 --- a/mechanical/mmme1029_materials.md +++ b/mechanical/mmme1029_materials.md @@ -902,3 +902,166 @@ is any multiple, $n$, of $\lambda$. Or: $$n\lambda = 2d\sin\theta$$ This is Bragg's Law. + +# Metals + +## Defects on the Atomic Scale + +Defects on the atomic scale have a significant effect on yield stress, ultimate tensile stress, and +ultimate fracture stress. + +The yield stress of a real metal(-alloy) is much lower than the theoretical yield stress for +the perfect metal(-alloy) crystal. +This difference is because of the defects in the metal, particularly dislocations, as the +dislocations allow the ions to slide past each other at much lower yield stresses. + +The 5 types of defects are: + +- Grain boundaries +- Vacancies (missing ion) +- Dislocations (missing row of ions) +- Impurity ions +- Crystalline includison + +![](./images/vimscrot-2021-12-22T13:02:28,180694109+00:00.png) + +### Dislocation Movement vs. Simple Sliding + +The layers of ions in a crystalline metal could simply over each other: + +![](./images/vimscrot-2021-12-22T13:16:39,506214227+00:00.png) + +However, the stress required for simple sliding is much higher than the stress required to move a +dislocation. +This is because dislocation motion is successive sliding of the partial plane of ions under applied +shear stress (black arrow). +The vacancy in the slip plane (yellow arrow) moves in steps in sequence from left to right. + +![](./images/vimscrot-2021-12-22T13:19:51,513367988+00:00.png) + +If there are no dislocations then plastic deformation is delayed to a higher applied stress, +meaning the yield stress of the metal would be much higher. + +Dislocations move more easily on specific planes and in specific directions called the +slip planes and slip directions which make up what is known as the +[slip system](#slip-systems-in-metals). + +There are a very large amount of dislocations in metals and alloys. +Dislocation density is expressed as total length of dislocations per unit volume. + +## Single Crystal Metals + +![](./images/vimscrot-2021-12-22T12:58:16,773351925+00:00.png) + +Normally when a molten metal is cooled to a solid, then lots of tiny crystals (grains) grow in +different directions until they impinge. +The grain boundaries are a source of mechanical weakness. + +A single crystal metal is one for which the casting is cooled to form just one giant crystal: + +1. The molten metal is cast into a mould +2. At the very base of the mould, the temperature is dropped and the alloy crystallises into many + little crystals +3. The crystals grow upwars through the liquid and meet a spiral tube and are constricted +4. This tube only allows one crystal to grow through the spiral and then into the main mould + +## Polycrystalline Metals + +Most normal metals you see everyday are polycrystalline. + +![](./images/vimscrot-2021-12-22T12:58:38,918714742+00:00.png) + +![Acid etched surface of a polycrystalline metal](./images/vimscrot-2021-12-22T12:59:11,940527867+00:00.png) + +## Elastic and Plastic Strain in Metals + +When you apply a tensile stress to a mteal, this will produce a shear stress in any part of the +metallic lattice that is not parallel or perpendicular to the applied stress. +Under the action of shear stress, the metallic lattice will tend to experience a combination of +elastic strain and plastic strain: + +![](./images/vimscrot-2021-12-22T13:46:45,608572706+00:00.png) + +## Raising the Yield Stress of a Metal + +There are 4 main ways to raise the yield stress of a metal: + +- Make a solid-solution---by metal alloying or atomic addition +- Precipitate crystalline inclusions---by metal alloying or atomic additions and then heat treatment +- Work-harden --- by processing and/or cold-working +- Decrease the grain size --- by processing and/or heat-treatment + +### Make a Solid-Solution + +Adding an alloying element, B, to the host, A, forms a solid-solution as the ions or atoms of B +dissolve in A. + +The impurity particles of B are a different size from the particles of A, distorting the metal +lattice. +The larger the difference in radii of the particles, the bigger the distortion. + +![Substitutional addition replaces ions in the host](./images/vimscrot-2021-12-22T14:21:25,321894455+00:00.png) + +![Interstitial addition adds particles between the ions in the host](./images/vimscrot-2021-12-22T14:21:32,009562988+00:00.png) + +The particles of B tend to diffuse to dislocations and immobilise them. +This is why alloying increases the yield stress. + +Impurity particles generate lattice strain in the structure too: + +- Smaller particles introduce a compressive strain in the surrounding lattice +- Larger particles introduce a tensile strain in the surrounding lattice + +![How Ni content in Cu affects Yield and Ultimate Tensile Stress](./images/vimscrot-2021-12-22T14:25:58,567632767+00:00.png) + +### Precipitating Crystalline Inclusions + +When adding an element, B, to a host, A, exceeds the solubility, the result is the formation of a +solid-solution with a fixed ratio of B to A, but also precipitated crystals of a different ratio of +B to A. + +![](./images/vimscrot-2021-12-22T14:30:14,141230179+00:00.png) + +Crystalline inclusions are really difficult to shear, especially if they are small, numerous, and +have high Vickers' hardness. +This slows down dislocation movement, increasing yield stress. + +### Work-Hardening and Cold Working + +We can use room temperature deformation to increase the number of dislocations present in a metal. +As the % cold-work (%CW) is increased, the number of dislocations present also increases: + +$$\% CW = \frac{A_0 - A_d}{A_0} \times 100\%$$ + +where $A_0$ is the initial cross sectional area and $A_d$ is the final cross sectional area. + +A carefully prepared sample has a dislocation density, $\rho_d$ of around $10^3$ mm mm$3$, +whereas for a heavily deformed sample it is around $10^{10}$. + +A high density of dislocations means they are more likely to get entangled with each other, +making it harder for dislocations to move. +Therefore as $\rho_d$ increases, yield stress does too. + +### Decreasing the Grain Size + +- Most metals are polycrystalline with many grains. +- Different grains will have a different crystal orientation. +- Grains impede dislocation motion + +As you decrease grain size, you get more grain boundaries which basically creates more barriers +to prevent slip. + +This is because a dislocation would have to change orientation across a grain boundary and "ionic +disorder in the grain boundary results in discontinuity of slip" (A.B Seddon University of +Nottingham 2020) (I think that's repeating it but it said it on the slideshow sooo...). + +So for any given metal, the fine grained is harder and has greater yield stress than the coarse +grained version of it. + +#### Hall Petch Equation + +$$\sigma_{yield} = \sigma_0 + k_yd^{-0.5}$$ + +where $d$ is the grain size and $\sigma_0$ and $k_y$ are material constants. + +Therefore a plot of $\sigma_{yield}$ against $d^{-0.5}$ would results in a straight line.