--- author: Akbar Rahman date: \today title: MMME2053 // Fatigue tags: [ mmme2053, fatigue, materials, uni, engineering ] uuid: 23852418-9fbb-44b8-a697-3c8b566e5143 --- # Stages of Fatigue ## 1: Crack Initiation - happens on a micro-structural level - causes the start fatigue cracks - persistent slip bands form at the surface - they are the result of dislocations moving along crystallographic planes - leads to slip band intrusions and extrusions on the surface - act as stress concentrations, **leading to crack initiation**A ![](./images/vimscrot-2022-11-03T14:11:47,770744805+00:00.png) - crystallographic slip is controlled by shear stresses rather than normal stresses - therefore cracks tend to initially grow in a plane of maximum shear stress range - this leads to short cracks, usually on the order of a few grains ![The loading in this figure is horizontal tension](./images/vimscrot-2022-11-03T14:13:13,910050859+00:00.png) ## 2: Crack Propagation - the fatigue cracks tend to join together with more cycles - they grow along planes of maximum tensile stress ![](./images/vimscrot-2022-11-03T14:15:30,939765129+00:00.png) ## 3: Final Fracture - occurs when crack reaches critical length - results in either - ductile tearing (plastic collapse) - cleavage (brittle fracture) # Total Life Approach (Estimating Lifetime of a Part) - based on lab tests - carried out under controlled loading conditions - either stress or strain controlled loading conditions - performed on idealised specimens - specimens usually have finely polished defects (minimises surface roughness effects, affecting stage 1 crack initiation) - tests give number of loading cycles to the initiation of a measurable crack as a function of applied stress or strain parameters - measurability is dictated by the accuracy of the crack detection method used - this is typically between 0.75 mm to 1.00 mm - the challenge of fatigue design is to then relate the tests to actual component lives under real loading conditions - traditionally, most fatigue testing was based stress controlled conditions with mean stress, $S_m = 0$, which is known as a fully reversed load - the data was presented in the form of S-N curves (either semi-log or log-log plots) of alternating stress, $S_a$, against cycles to failure, $N$ (failure defined as fracture) ![](./images/vimscrot-2022-11-03T14:31:30,946882039+00:00.png) - figure \ref{fig:typical-s-n} contains schematic representations of two typical S-N curves - part _(a)_ shows a continuously sloping curve - part _(b)_ shows a discontinuity ("knee") in the curve---this is only found in a few materials (notably low strength steels) between $10^6$ and $10^7$ cycles under non-corrosive conditions ![\label{fig:typical-s-n}](./images/vimscrot-2022-11-03T14:34:36,090286120+00:00.png) - the curves are normally drawn through the median life value - therefore represent 50 percent expected failure - fatigue strength, $S_e$, is a hypothetical value of stress range at failure for exactly $N$ cycles as obtained from an S-N curve - fatigue limit (or endurance limit) is the limiting value of the median fatigue strength as $N$ becomes very large ($>10^8$) # 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: $$\frac1F = \frac{S_aK_f}{S_e} + \frac{S_m}{S_u}$$
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, $S_y$