Material toughness

mechanical/mmme1029_materials.md

@@ -5,6 +5,8 @@ title: MMME1029 // Materials
 tags: [ uni, nottingham, mechanical, engineering, mmme1029, materials ]
 ---
 
+\tableofcontents
+
 # Lecture 1 (2021-10-04)
 
 ## 1A Reading Notes
@@ -387,3 +389,129 @@ There are two ways to make polymers:
 
 - [Addition Poymerisation](http://www.chemguide.co.uk/14to16/organic/addpolymers.html)
 - [Condensation Polymerisation](https://www.chemguide.uk/14to16/organic/condpolymers.html)
+
+# Elastic Deformaion
+
+Elastic deformation is deformation where the material will return to original shape after the
+applied stresses are removed.
+
+Elastic deformation is the first type of deformation that happens when stresses are applied to
+a material and is represented by the straight line at the beginning of a stress-strain curve.
+
+## Modulus of Resillience ($E_r$)
+
+This is the area under the elastic portion of a stress-strain graph of a material.
+
+# Plastic Deformation
+
+## Toughness (Absorbing Energy Through Plastic Deformation)
+
+- The toughness of a material is its ability to absorb energy through plastic deformation
+  without fracturing
+- The material toughness of a ductile material can be determined by finding the area under its
+  stress-strain curve (e.g. by integrating the graph)
+- Brittle materials like ceramics and glasses exhibit no material toughness
+- Ductile materials have a possibility of achieving large material toughness
+
+Ductility measures how much something deforms plastically before fracture, but just because a
+material is ductile does not make it tough.
+
+*The key to high material toughness is a good combination of large ultimate fracture stress and
+large ductility*.
+
+- The unit of toughness is energy per unit volume as toughness can be mathematically expressed as:
+
+  $$toughness = \int^{\varepsilon_f}_0\! \sigma \,\mathrm{d}\varepsilon 
+    = \frac{\text{Energy}}{\text{Volume}} $$
+
+- A metal may have satisfactory toughness under static loads but fail under dynamic loads or impact
+
+  This may be caused by the fact that ductility and toughness usually decrease as rate of loading
+  increases.
+- Ductility and toughness decreasee with temperature
+- Notches in the material affect the distribution of stress in the material, potentially changing
+  it from a uniaxial stress to multiaxial stress
+
+### Charpy Impact Test
+
+Measures material toughness by determining the amount of energy absorbed during fracture.
+
+It works by essentially dropping a hammer into a sample whose dimensions are standardized
+(usually either by BSI or ISO) and measuring how high the hammer goes up on the other side,
+after it breaks the material
+
+The height of the hammer after impact will tell you how much enery is left in it, and therefore
+how much has been aborbed by the now broken sample.
+
+Under a microscope, more ductile fractures appear fibrous or dull, whereas less ductile surfaces
+have granular or shiny surface texture.A
+
+The charpy test has a couple issues:
+
+- Results are prone to scatter as it is difficult to achieve a perfectly shaped notch
+- Temperature has to be strictly controlled since it affects a material's ductility
+
+#### The setup of a charpy impact test
+
+1. Sample is made to standardized dimensions, with a notch
+2. Sample is placed on support
+3. A very heavy hammer pendulum of mass $m$ is dropped from rest at $h_0$ to swing about a pivot,
+   reaching $E_{kmax}$ vertically below the pivot.
+4. 
+
+   a. If no sample is in place then the hammer will swing back up on the other side to a height of
+      $h_h$ where theoretically $h_h = h_0$
+   b. With a sample placed vertically below, some of the $E_k$ is transferred to the sample to bend
+      and (usually) break the sample.
+
+      If breaks the sample, it will swing up to the other side, where its max height, $h_f$ can be
+      used to calculate how much energy was used to break the sample:
+
+      $$E = mg(h_h-h_f)$$
+
+      Where $g$ is acceleration due to gravity.
+
+# Ductility
+
+Ductility is the plastic deformation a material withstands before fracture.
+
+# Griffith Surface Flaws
+
+These flaws vary in size and shape.
+They limit the ability of any material, brittle or ductile, to withstand tensile stresses as they
+concentrate the tensile forces applied to a smaller area.
+
+The stress at the tip of the flaw:
+
+$$\sigma_{actual} = 2\sigma\sqrt{\frac a r}$$
+
+For deep ($a$ is large) or thin ($r$ is small) the stress is magnified and, if it exceeds the UFS
+in a brittle material, the flaw will grow into a crack, resulting in the brittle material
+fracturing.
+
+However in a ductile material, the tip of the flaw can heal, reducing $a$ and increasing $r$.
+This is due to the chemical structure of ductile materials like metals.
+
+![](./images/vimscrot-2021-11-08T13:51:17,152036728+00:00.png)
+
+## Stress Intensity Factor
+
+Stress Intesity Factor, $K$:
+
+$$K = f\sigma\sqrt{\pi a}$$
+
+where:
+
+- $f$ is the geometry factor (1 would represent an infinite width sample, and 0 a 0 width sample)
+- $\sigma$ is applied tensile strength
+- $a$ is flaw depth
+
+## Fracture Toughness
+
+![An example sample for testing fracture toughenss. From: <https://www.researchgate.net/figure/Compact-tension-sample-geometry-used-for-fracture-toughness-measurement_fig2_340037774> [accessed 8 Nov, 2021]](./images/Compact-tension-sample-geometry-used-for-fracture-toughness-measurement.png)
+
+The value of $K$ that causes the notch to grow and cause fractures.
+This is value is known as the fracture toughness, $K_c$.
+
+At low thicknesses fracture toughness depends on thickness but as thickness increases, $K_c$
+decreases to the constant value, the plane strain fracture toughness, $K_{1c}$.