1. STRESS CONCENTRATION AT NOTCHES
One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”) is the consideration of stress concentration
Stress concentration at geometrical notches are always present in a real structure
Notches introduce inhomogeneous stress distribution with a stress concentration at the root of the notch
Stress concentration factor:
Kt is referred as the theoretical stress concentration factor: it is based in the assumption of linear elastic material behavior
Kt describes the severity of the notch and depends on the geometry of the notch configuration (shape factor of the notch)

2. Common examples of stress concentration
Gear teeth
Shaft keyway
Bolt threads
Shaft shoulder
Riveted or bolted joint
Welded joint
all these components might be subjected to cyclic loads !

3. usual alternative DEFINITIONS: For the previous example: therefore:
following the definitions of R.E. Peterson in Stress Concentration Factors, John Wiley & Sons, New York (1974)
In general Kt is the preferred factor to indicate stress concentration

4. THE “MODEL” STRESS CONCENTRATION CASE: the circular hole in an infinite sheetq r0
Along the edge of the circular hole:
compression at q = 0
q = 30° or
j = p /2 – q = 60° from sqq max
q for sqq = 0 ?

5. Stress Profiles along the normal to the edge of the circular hole:
We are interested in evaluating:
Situation for compressive remote stress (–S): presence of tensile stress ?  YES
Gradient of Stress in the direction normal to the edge of the hole at the location of speak: strong gradient
Gradient of Stress along the edge of the hole at the location of speak : slow decrease of stress along the edge of the notch
Volume of material subjected to high Stress around the root of the notch: larger for larger notches! (significant to understand notch size effects on fatigue)

6. sC Presence of Local Tensile Stress upon Remote Compression Stress
* Fatigue Crack Growth under Compressive Stress?
* Effect on Brittle Materials: Failure Criteria Based on Maximum Normal Stress for Brittle Materials sC
How would a cylinder of brittle material with a distribution of defects fail under compression?
Spherical voids and sharp like cracks are often produced during processing of brittle materials:
Spherical voids  sintering of ceramic powders (remnants of initial porosity)
Microcraks  thermal expansion mismatch

7. b) Gradient of Stress in the direction NORMAL to the edge of the hole at the location of speak
* Although the peak stress is of great importance, it is also interesting to know how fast the stress decreases away from the root of the notch  Stress Gradient
for the circular hole:
The Stress Gradient at the root of a notch should give an indication of the volume of material under high stresses  estimate the distance d along the normal to the root for a drop from speak to 0.9 speak (10 % decrease) for a circular hole with r0 = 2.5 mm:
d = 0.1 mm= 100 mm
if grain size 50 mm  the depth d corresponds only to few grains
The grains at the noth root surface are subjected to high loads and this is very important for fatigue

8. c) y d) Gradient of Stress ALONG the edge of the hole at the location of speak
As fatigue crack nucleation is a surface phenomenon it is of interest to know how fast the tangential stress along the edge of the notch is decreasing S
* Slow decrease of the stress along the edge compared with the decrease from the edge at the location of speak
* Larger notches have a larger material surface along the root of the notch  very important to understand notch size effects on fatigue
Note: even when a particular case was analyzed here (circular hole in an infinite plate) the conclusions are of general validity  similar peak stresses and notch root radii give comparable stress distribution around the root of an arbitrary notch

9. Reason of the existence of Notch Size Effects in Fatigue
Effect of notch geometry on Kt
Geometrically similar specimens have the same Kt (Kt is dimensionless) …
… but different stress gradients (stress gradient is not dimensionless)
Larger specimens have larger volumes and larger notch surface areas of highly stressed material
Reason of the existence of Notch Size Effects in Fatigue
Also: Importance of surface quality (method of production / fabrication)  surface defects due to manufacturing in a highly stressed region along the wall of a hole
Further examples / further aspects of stress concentrators 

10. The elliptical hole in an infinite sheet
a / b
1/3 1 3
r/ a 9
1/9 Kt
1.67 7
use large radii on surface parallel to applied stress to reduce stress concentration !

11. Stress Concentration for an elliptical hole under biaxial loading:
b: biaxiality ratio S
* For the case of a thin walled pressure vessel under pressure b = 0.5 and for the case of a circular hole (a = b):
lower than 3 S for uniaxial loading
* Same case but elliptical hole with b/a = 2:
lower than 3 S for uniaxial loading (actually, sqq = 1.5 S along the edge of the hole)
compare with the square hole (dashed line) with rounded corners with r 10% of hole width: Kt = 4.04 for b = 0.5

12. Stress Concentration for a circular hole in a plate under pure shear:
Fatigue cracks growing from holes in a shaft subjected to cyclic torsion !

13. Comparison of Kt values for a lug and an open hole
Pin - loaded hole:
Comparison of Kt values for a lug and an open hole
Conection between a lug and a clevis:
* Lugs are fatigue critical parts (also prone to fretting corrosion)  values of d/W below 1/3 are usually avoided to keep Kt below 3.5

14. If a relativelly small notch is added to the root of the main notch
Superposition of notches:
If a relativelly small notch is added to the root of the main notch
 Effect of superposition of notches:
This overestimates Kt because the small notch is not completely embedded in an homogeneous stress field of magnitude Kt1

15. (see Kt for Edge Notches two transparencies later)
Technique for estimating conservative limiting value for Kt for superposed notches:
“Fill” the notch (cross hatched area) leaving a single deep narrow notch
The theoretical stress concentration factor for the single deep narrow notch will always be greater than the Kt for the multiple notch
(see Kt for Edge Notches two transparencies later)

16. Examples of Superposition of notches:
Lug with small lubrication hole to the lug hole
Cross section of a fatigue crack at a sharp corner

17. Edge notches and Corrosion Pits
Corrosion pits at the material surface of an Al-alloy. Pit depth = 0.15 mm. Equivalent shape gives very high Kt values

18. Stress concentration factors for a shaft with a grove subjected to:
Axial Load
Torsion
Bending
Further information of the type that can be found in
the clasical handbook of R.E. Peterson, Stress Concentration Factors, 1974

19. … but this is not the case !
THE FATIGUE STRENGHT OF NOTCHED SPECIMENS
* STRESS – LIFE APPROACH: Notch Effects on the Fatigue Limit (Sm = 0)
Sa = Se
SeK
Similarity Principle: if Sa = Se is the fatigue limit of the smooth specimen, then Speak should give the fatigue limit SeK of the notched specimen:
meaning that:
… but this is not the case !
 The Fatigue Strength Reduction Factor or Fatigue Notch Factor Kf is introduced:
In general: Kf < Kt  fatigue limit of different materials are less notch sensitive to fatigue than predicted by Kt

20. Examples:
Effect of a Notch on S - N behavior (Tryon and Day, 2003)

21. Mechanical Behaviour of Materials (Dowling, 1999)
The examples illustrates a general observation for different materials: the finite life region is also less notch sensitive to fatigue than predicted by Kt

22. and Kt depends on geometry and mode of loading In general Kf < Kt
Kf also depends on material and notch size
and
In general Kf < Kt
Blunting effects in soft materials: Yielding at the notch root reduces peak stress from the values predicted by Kt
Fatigue strength of a notched component depends on the volume of highly stressed material near the notch  also effect of stress gradient on crack growth.

23. Empirical equations for q were proposed by different authors:
For engineering applications, the fatigue strength reduction fator Kf can be empirically related to the elastic stress concentration facto Kt by a Notch Sensitivity Factor defined as:
q = 1  material fully notch sensitive: Kf = Kt
q = 0  material not notch sensitive: Kf = 1
Empirical equations for q were proposed by different authors:
* Peterson (1959)
* Neuber (1946)
* Siebel and Stiele (1955)

24. Effect of notch root radius on Kf
* Peterson assumed that fatigue damage occurs when a the stress at a point located at a critical distance ap away from the notch root is equal to the fatigue strength of a smooth specimen and obtained the following empirical equation:
r is the notch root radius
- aP is a material constant related with material strength and ductility.
* for high strenght steels with SU > 560 MPa:
Effect of notch root radius on Kf

25. Also, q can be obtained in graphical form:
Peterson ´s notch sensitivity for steels
Also, q can be obtained in graphical form:

26. Neuber´s Notch Sensitivity curves for Al alloys
* Neuber assumed that fatigue failure occurs if the average stress over a length aN from the notch root is equal to the fatigue limit of a smooth specimen and proposed the following empirical equation:
r is the notch root radius
aN is the Neuber´s material constant related to the grain size
Neuber´s Notch Sensitivity curves for Al alloys
Relation with Hall-Petch?

27. No significant effect of Kt on  
* Siebel and Stiele (1955) introduced the Relative Stress Gradient (RSG)  to characterize the effects of fatigue strength reduction (instead of using the notch radius!)
where:
for the circular hole:
No significant effect of Kt on  
Similar dependencies are found for other geometries and for typical Kt values: 2 < Kt <5

28. Equation for the Curves:
Testing the Fatigue Strength of smooth and notched specimens they generated empirical curves relating Kf / Kt vs. 
Equation for the
Curves:
where Css is a material constant related with Sy

29. Examples:
RSG () values calculated from Siebel and Stiele by the theory of elasticity for different notched members

30. * STRESS – LIFE APPROACH: Notch Effect on Finite Life (S–N curve)
In the Finite Life region the notch may be less sensitive to what is predicted by Kf for the Fatigue Limit
 to much conservative !
Use Kf for the whole S-N curve? :
define a Fatigue Sensitivity Factor at a particular Life (and interpolate / extrapolate)
analogously to what was made for estimating S – N curves for smooth specimens  Kf for N1000 : K’f

31. An Empirical Fatigue Notch Sensitivity Factor (q’1000) can be defined at 1000 cycles:

32. Fatigue Notch Sensitivity Factor Kf depends on Cycles to Failure Nf
* Estimate of Fatigue Life for Notched Component: different approaches
Examples: A)
“Juvinall”
approach
Fatigue Notch Sensitivity Factor Kf depends on Cycles to Failure Nf
LCF REGION! B)

33. Example:
Juvinall
for the following loading cases:
Reversed bending loading
Reversed axial loading (negligible bending)
Reverse torsional loading