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