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:
K t describes the severity of the notch and depends on the geometry
of the notch configuration (shape factor of the notch) K t is
referred as the theoretical stress concentration factor: it is
based in the assumption of linear elastic material behavior
Slide 2
Common examples of stress concentration (a)Gear teeth (b)Shaft
keyway (c)Bolt threads (d)Shaft shoulder (e)Riveted or bolted joint
(f)Welded joint all these components might be subjected to cyclic
loads !
Slide 3
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 K t is the preferred
factor to indicate stress concentration usualalternative
Slide 4
THE MODEL STRESS CONCENTRATION CASE: the circular hole in an
infinite sheet S S r r0r0 for = 0 ? Along the edge of the circular
hole: compression at = 0 = 30 or = /2 from max
Slide 5
Stress Profiles along the normal to the edge of the circular
hole: We are interested in evaluating: a) Situation for compressive
remote stress (S): presence of tensile stress ? YES b) Gradient of
Stress in the direction normal to the edge of the hole at the
location of s peak : strong gradient c) Gradient of Stress along
the edge of the hole at the location of s peak : slow decrease of
stress along the edge of the notch d) 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)
Slide 6
a)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 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 CC CC
Slide 7
b) Gradient of Stress in the direction NORMAL to the edge of
the hole at the location of peak * 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 along the normal to the root for a
drop from peak to 0.9 peak (10 % decrease) for a circular hole with
r 0 = 2.5 mm: = 0.1 mm= 100 m if grain size 50 m the depth
corresponds only to few grains The grains at the noth root surface
are subjected to high loads and this is very important for
fatigue
Slide 8
S S c) y d) Gradient of Stress ALONG the edge of the hole at
the location of peak 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 * Slow decrease of the
stress along the edge compared with the decrease from the edge at
the location of peak * 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
Slide 9
Geometrically similar specimens have the same K t (K t is
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 but different stress
gradients (stress gradient is not dimensionless) 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 Effect of notch geometry on K t Further examples /
further aspects of stress concentrators
Slide 10
The elliptical hole in an infinite sheet a / b1/313 / a 911/9
KtKt 1.6737 S S use large radii on surface parallel to applied
stress to reduce stress concentration !
Slide 11
Stress Concentration for an elliptical hole under biaxial
loading: * For the case of a thin walled pressure vessel under
pressure = 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, = 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: K t = 4.04 for
= 0.5 : biaxiality ratio S
Slide 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 !
Slide 13
Pin - loaded hole: Comparison of K t values for a lug and an
open hole * Lugs are fatigue critical parts (also prone to fretting
corrosion) values of d/W below 1/3 are usually avoided to keep K t
below 3.5 Conection between a lug and a clevis:
Slide 14
Superposition of notches: If a relativelly small notch is added
to the root of the main notch Effect of superposition of notches:
This overestimates K t because the small notch is not completely
embedded in an homogeneous stress field of magnitude K t1
Slide 15
Technique for estimating conservative limiting value for K t
for superposed notches: The theoretical stress concentration factor
for the single deep narrow notch will always be greater than the K
t for the multiple notch (see K t for Edge Notches two
transparencies later) Fill the notch (cross hatched area) leaving a
single deep narrow notch
Slide 16
Examples of Superposition of notches: Cross section of a
fatigue crack at a sharp corner Lug with small lubrication hole to
the lug hole
Slide 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 K t values
Slide 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
Slide 19
S a = S e S eK THE FATIGUE STRENGHT OF NOTCHED SPECIMENS *
STRESS LIFE APPROACH: Notch Effects on the Fatigue Limit (S m = 0)
Similarity Principle: if S a = S e is the fatigue limit of the
smooth specimen, then S peak should give the fatigue limit S eK of
the notched specimen: meaning that: but this is not the case ! The
Fatigue Strength Reduction Factor or Fatigue Notch Factor K f is
introduced: In general: K f < K t fatigue limit of different
materials are less notch sensitive to fatigue than predicted by K
t
Slide 20
Effect of a Notch on S - N behavior (Tryon and Day, 2003)
Examples:
Slide 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 K t
Slide 22
In general K f < K t Blunting effects in soft materials:
Yielding at the notch root reduces peak stress from the values
predicted by K t 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. K t depends on geometry and
mode of loading K f also depends on material and notch size
and
Slide 23
For engineering applications, the fatigue strength reduction
fator K f can be empirically related to the elastic stress
concentration facto K t by a Notch Sensitivity Factor defined as: q
= 1 material fully notch sensitive: K f = K t q = 0 material not
notch sensitive: K f = 1 Empirical equations for q were proposed by
different authors: * Peterson (1959) * Neuber (1946) * Siebel and
Stiele (1955)
Slide 24
* Peterson assumed that fatigue damage occurs when a the stress
at a point located at a critical distance a p 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 - a P is a material constant related with material strength
and ductility. Effect of notch root radius on K f * for high
strenght steels with S U > 560 MPa:
Slide 25
Peterson s notch sensitivity for steels Also, q can be obtained
in graphical form:
Slide 26
* Neuber assumed that fatigue failure occurs if the average
stress over a length a N 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 - a N is the
Neubers material constant related to the grain size Neubers Notch
Sensitivity curves for Al alloys Relation with Hall-Petch?
Slide 27
where: for the circular hole: * Siebel and Stiele (1955)
introduced the Relative Stress Gradient (RSG) to characterize the
effects of fatigue strength reduction (instead of using the notch
radius!) No significant effect of K t on Similar dependencies are
found for other geometries and for typical K t values: 2 < K
t