STRESS INTENSITY FACTORS FOR CRACKS

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• STRESS INTENSITY FACTORS FOR CRACKS
• STARTING FROM RIVET HOLES

Stefan D. Pastrama, Paulo M. S. T. de
Castro University Politehnica of Bucharest,
Romania IDMEC, Universidade do Porto, Portugal
EU project GRD1-2000-25069 contract
G4RD-CT-2000-0396 - ADMIRE
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• Objective of the paper

- To calculate the stress intensity factor in
finite width strips with cracked holes, using the
compounding method and the weight function
technique - The compounding method is used to
determine the stress intensity factor for remote
uniform tension - The obtained results are used
as reference case in the weight function
method, for the case of point load applied on the
hole surface
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Geometry and loadings I

• A strip of finite width, having a central hole,
  with
• a. two symmetrical and equal cracks emerging
  from the hole
• b. a single crack starting from the hole.
• Studied loadings
•  
• a. remote uniform tension
• b. point load on the hole surface, that can be
  an approximation for the action of the rivet.

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Geometry and loadings II
2R/W 0.1, 0.16 and 0.2 a/R 1.01, 1.02, 1.04,
1.06, 1.08, 1.1, 1.15, 1.2, 1.25, 1.3, 1.4 and 1.5
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The compounding method
The structure containing n boundaries is
separated into ancillary configurations,
containing only one boundary which interacts
with the crack, and with known solutions
Kn the stress intensity factor for the cracked
structure having the n-th boundary, K0 the
stress intensity factor in the absence of all
boundaries. Ki interaction term (in most cases
negligible)
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The effect of the hole
The hole has a significant effect on the
displacement at the crack tip A crack having an
equivalent length 2a has to be introduced in the
ancillary configurations to replace the cracked
hole The value of the equivalent length is
obtained from the condition that such a crack in
an infinite plate should produce the same stress
intensity factor K0 as the crack of length 2a
emanating from the hole
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The general equation of the compounding method
Kn Q0Kn is the stress intensity factor for
the ancillary configuration with the n-th
boundary and containing the equivalent crack of
length 2a?. The general nondimensional form is
obtained by dividing the terms with ?(?a)1/2
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The ancillary configurations
a the studied structure (K12) b the
configuration without boundaries (K0) c, d
configurations with one boundary (free edge) and
the equivalent crack (K1 and K2)
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Effective calculation with the compounding method
The nondimensional compounding equation for n 2

• Q0 , Q1 and Q2 taken from handbooks
• Qi calculated using the stress concentration
  factor concept (the method is valid for small
  cracks)

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The interaction term Qi

• Obtained from a supplementary loading for the
  configuration without boundaries uniform
  pressure p acting on the hole perimeter
• the pressure p is derived from the equation

Kt1N stress concentration factor at the edge
of a hole in a strip subjected to uniform tension
(from handbooks) Ktn the stress
concentration factor at the prospective crack tip
due to the boundary n (for straight edges 1)
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Results for a single crack

• accurate results
• - differences less than 3 from results of Wu
  Carlsson
• small influence of the interaction term (less
  than 4.2 )

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Results for two symmetric cracks

• very accurate results
• - differences less than 0.8 from results of
  Kitagawa
• very small influence of the interaction term
  (less than 1)

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The weight function method
A method for calculating K in a cracked
structure, if one already knows KIr(a) stress
intensity factor for a reference loading
case uIr(x,a) half of the crack face
displacement field in the reference case
h(x,a) the weight function independent of the
loading ?(x) crack line stress in the
un-cracked structure in the case for which K is
calculated
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Application for the point load on the hole
?(x) is determined using finite element analyses
and curve fitting techniques The reference stress
intensity factor is taken from the previous case
(uniform traction) The reference crack face
displacement is calculated using the approximate
expression of Petroski and Achenbach
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Petroski Achenbach approximation
?0 characteristic stress for the reference case
(remote stress) F(a/L) correction factor in the
expression of KIr G(a/L) function obtained from
the equation of self consistency K Kir L
characteristic dimension L W (width of strip)
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The function G(a/L)
Can be obtained from the weight function equation
written for the reference case K KIr
?r(x) reference stress variation on the crack
line (FEM) The only unknown is G(a/L)
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Results for a single crack

• a a R ?0 P/2R
• - No results are available in the literature
• - Comparison is made with the case of an infinite
  plate
• - The free edge of the strip is far enough from
  the crack to make possible this comparison
• - Results are slightly greater due to the
  boundary influence
• - The accuracy can be considered as very good

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Conclusions
- stress intensity factor for a finite width
strip with a central cracked hole was determined
using a combination of the compound and weight
function techniques - two loading cases, (uniform
remote tensile stress and point load on the hole
surface) were considered - results were compared
with those from the literature, where available -
accuracy was excellent, proving that these two
methods can be utilized for calculating K for
other loading cases, - important information for
subsequent studies, especially for fatigue loads
can be obtained by using these methods.