notes/uni/mmme/2xxx/2053_mechanics_of_solids/fatigue.md

author	date	title	tags	uuid
Akbar Rahman	\today	MMME2053 // Fatigue	mmme2053 fatigue materials uni engineering	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 initiationA

• 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

2: Crack Propagation

• the fatigue cracks tend to join together with more cycles
• they grow along planes of maximum tensile stress

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)

• 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

• 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

• 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}

• 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 (slide 18) say otherwise 😭 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

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

• 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

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

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