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

---
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:

  $$\frac1F = \frac{S_aK_f}{S_e} + \frac{S_m}{S_u}$$

  <details>
  <summary>

  Derivation

  <summary>

  $$F = \frac{OB}{OA}$$

  from similar triangles we get

  $$\frac{S_a}{\frac{S_u}{F} - S_m} = \frac{S_e}{K_fS_u}$$

  </details>


# 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, $S_y$