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On classical statistical fitting models (CaprinoD’amore, Sendeckyj)
Caprino-d’Amore (CD in the following) start from the wear-out model (WangSuemasu-Chim, or Buggy-Dillon)
dσr
= −an−b
dn
(1)
To explicitly account for the well known dependence of the strength decay
on the stress ratio R, it was further assumed that the constant at is linearly
dependent on stress range, hence the equation really is
dσr
= −a0 σn−b
dn
(2)
CD’s assumed form of residual strength therefore is obtained as
σr (n) = σf s − A (1 − R) σ nB − 1
(3)
where A,B are two fitting constants1 . WARNING: Caprino-d’Amore and Kassapoglou both use constants named A,B!
–
CD do not use the residual strength equation to report SN data to the
equivalent static ones, like Sendeckyi. They instead write the SN curve from
their residual strength curve, obtaining (the relationship between A and a, B
and b is given in the original papers)
σ A N B − 1 (1 − R) + 1 = σf s
(4)
Comparing to Sendeckyj, at fatigue failure, the SN curve is very similar
σ [1 − (Nf − 1) C]G = σf s
(5)
whereas finally K does not obtain a SN curve from his residual strength model.
Sendeckyj generally maps residual stregth data to equivalent static ones,
based on the eqt.
G
σ fs 1/G
σr = σ
+ (N − 1) C
(6)
σ
1 Checking
the derivative
dσ r
= −A (1 − R) σ nB−1
dn
it is clear that it is different from the differential equation of K wear-out model
dσ r
= A′ σ r + B ′
dn
in that there is no constant B ′ and hence the residual strength is initially varying very slowly.
B
r = −A (1 − R) σ(n −1+1) = (σ r −σ f s )−A(1−R)σ ,
The closest we can get to compare is dσ
dn
n
n
which shows that both constants depend on n, and they also depend on stress level.
1
which gives incidentally gives
dσr
dn
= CG σ
1
G
σ fs
σ
G−1
+ (N − 1) C
= CG
= CG σ1/G σ 1−1/G
r
σ
σr
(7)
1/G
σr
(8)
which seems therefore vaguely similar and may have inspired K’s choice of wear
out of the class
dσ r
= A′ σr + B ′
(9)
dn
where A′ , B ′ are later taken in a peculiar form, so that they evetually lead
to violation of the equal rank assumption. We shall return to a more accurate
comparison of Caprino-D’Amore, Kassapoglou, and Sendeckly wear out models.
–
Now, what CD do in practise is much simpler than K. They substitute
directly from the SN curve equation, the distribution of static strength on the
RHS of
1 + A N B − 1 (1 − R) σ = σf s
(10)
and hence obtain the distribution of the LHS is the same Weibull distribution
with parameters δ, γ of static strength. using the stress-life equal rank assumption and assuming a two-parameter Weibull distribution of the monotonic tensile
strength, CD obtain the probability P(N*) to find an N value lower than N* as


δ
B
1+A N −1

P (N ∗ ) = 1 − exp − σ
(11)
γ
which is Weibull in terms of applied load, but not exactly Weibull in life –
perhaps the closest is a Weibull with exponent
αL = Bδ
In other words, CD obtain the scatter of fatigue life in terms of the product of
the Weibul parameter of static strength (of the order of 30-50) times the SN
inverse curve slope (1/10).
In that case, the K relationship that the inverse SN slope is exactly the
Weibull shape parameter of static strength seems to be a special case, when
αL = 1 so that
1 = Bδ
but this special case is obtained by coincidence, not via a true model. Notice on
the contrary that Navy database has αL = 1.25 as a mode value, ie. the most
occurring in the distribution of "scatter of scatter" of lives, but there is a full
distribution, not a single delta function like Kassapoglou. The good Caprinod’Amore and Sendeckjy models, instead, since they don’t make the wearout
fitting constants artificially disappear, can predict a distribution of "scatter of
scatter", because of the product in αL = Bδ contains both constants and slope
of SN curve which vary.
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References
Sendeckyj G.P. Fitting Models to Composite Materials Fatigue Data. Test
Methods and Design Allowables for Fibrous Composites, ASTM STP 734, Chamis,
C.C. editor, ASTM, Philadelphia, 1981. 245-260
C Kassapoglou, 2012. Predicting the Structural Performance of Composite Structures Under Cyclic Loading, PhD thesis, Delft Univ of Technology.
http://repository.tudelft.nl/assets/uuid:73a4025d-c519-4e3a-b1cd-c1c8aa0fdfeb/full_document_v3.pdf
C Kassapoglou, Fatigue Model for Composites Based on the Cycle-by-cycle
Probability of Failure: Implications and Applications, Journal of Composite
Materials February 1, 2011 45: 261-277.
Christos Kassapoglou, 2011. Fatigue Model for Composites Based on the
Cycle-by-cycle Probability of Failure: Implications and Applications. Journal
of Composite Materials 45 (3) 261-277
Christos Kassapoglou, Myriam Kaminski, (2011) Modeling damage and load
redistribution in composites under tension—tension fatigue loading, Composites:
Part A 42 1783—1792
Christos Kassapoglou, (2007) Fatigue Life Prediction of Composite Structures Under Constant Amplitude Loading
Journal of Composite Materials 2007 41: 2737. DOI: 10.1177/0021998307078735
[11] AD’Amore, G Caprino, P Stupak, J Zhou, L Nicolais (1996) Effect of
stress ratio on the
flexural fatigue behaviour of continuous strand mat reinforced plastics. Science and
Engineering of Composite Materials 5(1):1—8.
[12] G Caprino, A D’Amore. Flexural fatigue behaviour of random continuous fibre
reinforced thermoplastic composites. (1998) Composite Science and Technology Volume 58,
Issue 6, Pages 957-965
[13] A D’Amore, G Caprino, L. Nicolais, Marino G. (1999) Long-term behaviour of PEI and
PEI-based composites subjected to physical aging, Composites Science and
Technology,
Volume 59, Issue 13, 1993-2003
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