mmme2053 notes on fatigue and fracture

uni/mmme/2xxx/2053_mechanics_of_solids/fracture.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2053 // Fracture
+tags: [ uni, mmme2053, fracture, materials, engineering ]
+uuid: 17315e63-3870-428b-b65d-a5d249768c05
+---
+
+# Fracture
+
+- Consider the stress concentration factor (SCF) for an elliptical hole in a large, linear-elastic
+  plate subjected to a remote, uniaxial stress
+
+  ![](./images/vimscrot-2022-11-03T16:16:29,022777996+00:00.png)
+
+- It can be shown that SCF can be expressed as:
+
+  $$K_t = \frac{\sigma_\text{max}^\text{el}}{\sigma_\text{nom}} = 1 + 2\frac{a}{b}$$
+
+- Therefore as $b \rightarrow 0$, the hole degenerates to a crack and $\frac ab \rightarrow \infty$
+  $\therefore K_t \rightarrow \infty$, provided the material behaviour remains linear elastic
+
+# Basis of the Energy Approach to Fracture Mechanics
+
+There are three modes of loading cases: $K_\text{I}$, $K_\text{II}$, $K_\text{III}$.
+
+- Generally, for geometries with finite boundaries, $K_\text{I}$ is used:
+
+  $$K_\text{I} = Y\sigma\sqrt{a\pi}$$
+
+  where $Y$ is a function of the crack and $a$ is never mentioned in
+  [the slides](./lecture_slides/fatigue_and_failure_2.pdf) (slide 6)
+
+  > TODO: find out what $Y$ and $a$ are
+
+- $K_\text{I}$ is the Mode-1 stress-intensity factor which defined the magnitude of the elastic stress
+  field in the vicinity of the crack tip
+- $K_\text{II}$ and $K_\text{III}$ are similar
+- The energy release release rate under mixed loading is given by
+
+  $$K_\text{total} = K_\text{I} + K_\text{II} + K_\text{III}$$
+
+![](./images/stress-intensity-factors.png)
+
+## Typical Fracture Toughness Values
+
+Material                      | $\sigma_y$ / Nm$^{-2}$ | $K_\text{Ic}$ / Nm$^{-1.5}$
+----------------------------- | ---------------------- | ---------------------------
+Mild steel                    | 220                    | 140 to 200
+Pressure vessel steel (HY130) | 1700                   | 170
+Aluminium Alloys              | 100 to 600             | 45 to 23
+Cast Iron                     | 200 to 1000            | 20 to 6
+
+# Fatigue Crack Growth
+
+- for a wide range of conditions, there is a logarithmic linear between crack growth rate and
+  intensity factor range during cyclic loading of cracked components
+- it allows crack growth to be modelled and estimated based on
+
+  - knowledge of crack and component geometry
+  - loading conditions
+  - empirical crack growth data
+
+Considering a load cycle as shown in figure \ref{fig:p-vs-t} which gives rise to a load acting on
+a cracked body
+
+![A graph of P vs t \label{fig:p-vs-t}](./images/P_vs_t.png)
+
+- The load range and crack geometry gives rise to a cyclic variation in stress intensity factor:
+
+  $$\Delta K = K_\text{max} - K_\text{min}$$
+
+- Paris showed that subsequent crack growth can be modelled by following equation
+
+  $$\frac{\mathrm{d}a}{\mathrm{d}N} = C\Delta K^m$$
+
+  where $C$ and $m$ are empirically determined material constants.
+
+
+- Fatigue crack growth data is often as $\log \frac{\mathrm{d}a}{\mathrm{d}N}$ against $\log{\Delta K}$
+
+  ![](./images/fatigue_and_failure_2_-019.png)
+
+- Below $K_\text{th}$, no observable crack growth occurs
+- Region II shows a near linear relationship---this is the region which fail by fatigue failure spend
+  most of their life
+- In region III rapid crack grown occurs and little life is involved
+
+- the fatigue crack growth life of the component can be obtained by integrating the Paris equation
+  between the limits of the initial crack size and final crack size, given that you know the
+  stress intensity factor
+
+
+## Typical Values for $\Delta K_\text{th}$, $m$, and $\Delta K$
+
+Material        | $\Delta K_\text{th}$ | $m$ | $\Delta K$ for $\frac{\mathrm{d}a}{\mathrm{d}N} = 10^{-6}$
+--------------- | -------------------- | --- | -------------
+Mild Steel      | 4 to 7               | 3.3 | 6.2
+Stainless Steel | 4 to 6               | 3.1 | 6.3
+Aluminium       | 1 to 2               | 2.9 | 2.9
+Copper          | 1 to 3               | 3.9 | 4.3
+Brass           | 2 to 4               | 4.0 | 4.3 to 66.3
+Nickel          | 4 to 8               | 4.0 | 8.8