210 lines
7.6 KiB
Markdown
Executable File
210 lines
7.6 KiB
Markdown
Executable File
---
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author: Akbar Rahman
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date: \today
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title: MMME2053 // Fatigue
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tags: [ mmme2053, fatigue, materials, uni, engineering ]
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uuid: 23852418-9fbb-44b8-a697-3c8b566e5143
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---
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# Stages of Fatigue
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## 1: Crack Initiation
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- happens on a micro-structural level
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- causes the start fatigue cracks
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- persistent slip bands form at the surface
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- they are the result of dislocations moving along crystallographic planes
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- leads to slip band intrusions and extrusions on the surface
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- act as stress concentrations, **leading to crack initiation**A
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- crystallographic slip is controlled by shear stresses rather than normal stresses
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- therefore cracks tend to initially grow in a plane of maximum shear stress range
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- this leads to short cracks, usually on the order of a few grains
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## 2: Crack Propagation
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- the fatigue cracks tend to join together with more cycles
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- they grow along planes of maximum tensile stress
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## 3: Final Fracture
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- occurs when crack reaches critical length
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- results in either
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- ductile tearing (plastic collapse)
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- cleavage (brittle fracture)
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# Total Life Approach (Estimating Lifetime of a Part)
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- based on lab tests
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- carried out under controlled loading conditions
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- either stress or strain controlled loading conditions
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- performed on idealised specimens
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- specimens usually have finely polished defects (minimises surface roughness effects, affecting
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stage 1 crack initiation)
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- tests give number of loading cycles to the initiation of a measurable crack as a function of
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applied stress or strain parameters
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- measurability is dictated by the accuracy of the crack detection method used
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- this is typically between 0.75 mm to 1.00 mm
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- the challenge of fatigue design is to then relate the tests to actual component lives under
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real loading conditions
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- traditionally, most fatigue testing was based stress controlled conditions with mean stress,
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$S_m = 0$, which is known as a fully reversed load
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- the data was presented in the form of S-N curves (either semi-log or log-log plots) of alternating
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stress, $S_a$, against cycles to failure, $N$ (failure defined as fracture)
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- figure \ref{fig:typical-s-n} contains schematic representations of two typical S-N curves
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- part _(a)_ shows a continuously sloping curve
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- part _(b)_ shows a discontinuity ("knee") in the curve---this is only found in a few materials
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(notably low strength steels) between $10^6$ and $10^7$ cycles under non-corrosive conditions
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- the curves are normally drawn through the median life value
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- therefore represent 50 percent expected failure
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- fatigue strength, $S_e$, is a hypothetical value of stress range at failure for exactly $N$ cycles
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as obtained from an S-N curve
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- fatigue limit (or endurance limit) is the limiting value of the median fatigue strength as $N$
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becomes very large ($>10^8$)
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# Effect of Mean Stress
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- mean stress has a significant effect on fatigue behaviour in cyclically loaded components
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- in figure \ref{fig:effect-of-mean-stress} you can see tensile mean stresses reduce fatigue life
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- compressive stresses increase fatigue life
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- effect of mean stress commonly represented as a plot of $S_a$ against $S_m$ for a given fatigue
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life
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- attempts have been made to generalise the relationship, as shown in figure \ref{fig:s_a-s_m}
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- modified Goodman line assumes linear relationship, where gradient and intercept are defined by
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fatigue life, $S_e$, and material UTS, $S_u$, respectively
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- Gerber parabola employs same intercepts but relationship is a parabola
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- Soderberg line assumes linear relationship but the x intercept (mean axis end point) is taken
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as yield stress, $S_y$
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- these curves can be extended into the compressive mean stress region to give increasing allowable
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alternating stress with increasing compressive mean stress
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- this is normally taken to be horizontal for design purposes and conservatism
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# Effect of Stress Concentrations
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- fatigue failure is most commonly associated with notch-type features
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- stress concentrations associated with notch-type features typically leads to local plastic strain
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and eventually fatigue cracking
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- the estimation of stress concentration factors (SCFs) are typically expressed in terms of an
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elastic SCF, $K_t$:
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$$K_t = \frac{\sigma^{\text{el}}_{\text{max}}}{\sigma_{\text{nom}}}$$
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- the fatigue strength of a notched component can be predicted with the fatigue notch factor, $K_f$,
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which is defined as the ratio of the fatigue strengths:
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$$K_f= \frac{S_a^{\text{smooth}}}{S_a^{\text{notch}}}$$
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> i thought $S_a$ is alternating stress and $S_e$ is fatigue strength but the
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> [uni slides](./lecture_slides/fatigue_and_failure_1.pdf) (slide 18) say otherwise :sob:
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> TODO: find out what it's meant to be for sure
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- however $K_f$ is found to vary with both alternating stress level and mean stress level and thus
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number of cycles
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- figure \ref{fig:effect-of-notch} shows the effect of a notch, with $K_t = 3.4$, on the fatigue
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behaviour of wrought aluminium alloy
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# S-N Design Procedure for Fatigue
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- constant life diagrams plotted as $S_a$ against $S_m$ (also known as Goodman diagrams)
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(figure \ref{fig:goodman-diagram}) can be used in design to give safe estimates of fatigue life
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and loads
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- the fatigue strength for zero mean stress is is reduced by the fatigue notch factor, $K_f$
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- $K_t$ is used if $K_f$ is not known
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- for static loading of a ductile component with a stress concentration, failure still occurs
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when mean stress, $S_m$, is equal to UTS
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- failure at intermediate values of mean stress is assumed to be given by the dotted line
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- in order to avoid yield of whole cross-section of component, maximum nominal stress must be less
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than the yield stress, $S_y$:
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$$S_m + S_a < S_y$$
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## Safety Factor, $F$
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- determined from the position of the point relative to the safe/fail boundary:
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$$\frac1F = \frac{S_aK_f}{S_e} + \frac{S_m}{S_u}$$
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<details>
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<summary>
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Derivation
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<summary>
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$$F = \frac{OB}{OA}$$
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from similar triangles we get
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$$\frac{S_a}{\frac{S_u}{F} - S_m} = \frac{S_e}{K_fS_u}$$
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</details>
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# Failure Examples
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## Bicycle Crank Arm
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## D.H.-106 Comet Failure
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- 1st production jet liner (debut in 1952)
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- several crashed in 1954 led to an inquiry
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- water tank testing and examination of a recovered fuselage showed that failure originated at a
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square corner window
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- future designs used oval windows
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# Glossary (of Symbols)
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- notch stress concentration factor, $K_f$
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- stress concentration factor, $K_t$
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- alternating stress, $S_a$
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- fatigue strength, $S_e$ --- hypothetical value of stress range at failure for exactly $N$ cycles
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- mean stress, $S_m$
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- ultimate tensile stress, $S_u$
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- yield strength, $S_y$
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