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+ + + Derivation + + + + $$F = \frac{OB}{OA}$$ + + from similar triangles we get + + $$\frac{S_a}{\frac{S_u}{F} - S_m} = \frac{S_e}{K_fS_u}$$ + +
+ + +# Failure Examples + +## Bicycle Crank Arm + +![](./images/vimscrot-2022-11-03T14:37:49,949154012+00:00.png) + +![](./images/vimscrot-2022-11-03T14:37:57,733079705+00:00.png) + +## D.H.-106 Comet Failure + +- 1st production jet liner (debut in 1952) +- several crashed in 1954 led to an inquiry +- water tank testing and examination of a recovered fuselage showed that failure originated at a + square corner window +- future designs used oval windows + + +![](./images/vimscrot-2022-11-03T14:39:35,250528225+00:00.png) + +![](./images/vimscrot-2022-11-03T14:39:41,710507511+00:00.png) + +# Glossary (of Symbols) + +- notch stress concentration factor, $K_f$ +- stress concentration factor, $K_t$ +- alternating stress, $S_a$ +- fatigue strength, $S_e$ --- hypothetical value of stress range at failure for exactly $N$ cycles +- mean stress, $S_m$ +- ultimate tensile stress, $S_u$ +- yield strength, $S_y$ diff --git a/uni/mmme/2xxx/2053_mechanics_of_solids/fracture.md b/uni/mmme/2xxx/2053_mechanics_of_solids/fracture.md new file mode 100755 index 0000000..bf82acb --- /dev/null +++ b/uni/mmme/2xxx/2053_mechanics_of_solids/fracture.md @@ -0,0 +1,103 @@ +--- +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 | 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z_O=*_3Lc`L$HmBy{wOq@e;5rR;@Z+F5RHb@s0bP73l&9C_Wr13ynj@TjPH*|#mxbY zNWrnuNMMH9cxhxR38zsJd|Mh=MfSEd45!hF`1Pd|>A3#rBoe#N2$4(%G4QALK!_M_ z-GGLhKZHnwXms41A_$7}4?#$D_BlWhjKsbk2ttL}(NJ7ll!)NkqC^^w4JA?8*8@BW zxHU&m+K&ML^SqEyB5n*i8DqciQ4A%rw?#1u!tM(RqvCu9-ci{5BZEhVofmjaXCH$? zrI6X}FfvBOYgF95i-8=&-XE1hr{UuUkwV4C7YYrc(eWBaglGsvLm?UoqLCpQ1)@P> zA_n2WARJ(Qaq$$~LwF6sL4|ODXCL1d!a;>_P$3*t2nQ9yL4|ODIEnLx2H~JVIA{!a>J#ARt$; zkAV>J90-U*cso1?i0%+OJO_yG5IZ~vi1-jY2nWd1IR8L?!)p)@5bN-E5DpNp@OBUm zkbCfU5Do;wfkHS?2nWb{xG_ME!fOx?P>12|ARH)!1BGy)5DpTAg9PCK`4{I43Bm#D z2)rGHg9PCqK{!Yd4iba|)M2=B$q)`QgahP8d|L#t)AG8t@TiM-sqcWWFy#LNBfvkaj8PrM`OGa2que>wH%DWD10^8$) + +# Effect of Mean Stress + +- mean stress has a significant effect on fatigue behaviour in cyclically loaded components +- in figure \ref{fig:effect-of-mean-stress} you can see tensile mean stresses reduce fatigue life +- compressive stresses increase fatigue life + +![\label{fig:effect-of-mean-stress}](./images/vimscrot-2022-11-03T14:31:30,946882039+00:00.png) + +- effect of mean stress commonly represented as a plot of $S_a$ against $S_m$ for a given fatigue + life +- attempts have been made to generalise the relationship, as shown in figure \ref{fig:s_a-s_m} + +![\label{fig:s_a-s_m}](./images/vimscrot-2022-11-03T14:45:42,986596633+00:00.png) + +- modified Goodman line assumes linear relationship, where gradient and intercept are defined by + fatigue life, $S_e$, and material UTS, $S_u$, respectively +- Gerber parabola employs same intercepts but relationship is a parabola +- Soderberg line assumes linear relationship but the x intercept (mean axis end point) is taken + as yield stress, $S_y$ + +- these curves can be extended into the compressive mean stress region to give increasing allowable + alternating stress with increasing compressive mean stress +- this is normally taken to be horizontal for design purposes and conservatism + + +# Effect of Stress Concentrations + +- fatigue failure is most commonly associated with notch-type features +- stress concentrations associated with notch-type features typically leads to local plastic strain + and eventually fatigue cracking +- the estimation of stress concentration factors (SCFs) are typically expressed in terms of an + elastic SCF, $K_t$: + + $$K_t = \frac{\sigma^{\text{el}}_{\text{max}}}{\sigma_{\text{nom}}}$$ + +- the fatigue strength of a notched component can be predicted with the fatigue notch factor, $K_f$, + which is defined as the ratio of the fatigue strengths: + + $$K_f= \frac{S_a^{\text{smooth}}}{S_a^{\text{notch}}}$$ + + > i thought $S_a$ is alternating stress and $S_e$ is fatigue strength but the + > [uni slides](./lecture_slides/fatigue_and_failure_1.pdf) (slide 18) say otherwise :sob: + > TODO: find out what it's meant to be for sure + + - however $K_f$ is found to vary with both alternating stress level and mean stress level and thus + number of cycles + +- figure \ref{fig:effect-of-notch} shows the effect of a notch, with $K_t = 3.4$, on the fatigue + behaviour of wrought aluminium alloy + + + ![\label{fig:effect-of-notch}](./images/vimscrot-2022-11-03T15:22:29,993209954+00:00.png) + +# S-N Design Procedure for Fatigue + +- constant life diagrams plotted as $S_a$ against $S_m$ (also known as Goodman diagrams) + (figure \ref{fig:goodman-diagram}) can be used in design to give safe estimates of fatigue life + and loads + +![\ref{fig:goodman-diagram}](./images/vimscrot-2022-11-03T15:40:37,514148113+00:00.png) + +- the fatigue strength for zero mean stress is is reduced by the fatigue notch factor, $K_f$ +- $K_t$ is used if $K_f$ is not known +- for static loading of a ductile component with a stress concentration, failure still occurs + when mean stress, $S_m$, is equal to UTS +- failure at intermediate values of mean stress is assumed to be given by the dotted line + +- in order to avoid yield of whole cross-section of component, maximum nominal stress must be less + than the yield stress, $S_y$: + + $$S_m + S_a < S_y$$ + +## Safety Factor, $F$ + +- determined from the position of the point relative to the safe/fail boundary: + + $$\frac1F = \frac{S_aK_f}{S_e} + \frac{S_m}{S_u}$$ + +