FATIGUE LIFE PREDICTION FOR MARINE STRUCTURES
J.H. Vink*, M. Mukhopadhyay** and B. Boon**** Delft University of Technology (DUT), Faculty of Design, Construction and Production, Ship Structures
Laboratory, P.O. Box 5035, 2600 GA Delft, The Netherlands, E-mail: j.h.vink()wbmt.tudelft.n1
** Visiting professor, from Indian Institute of Technology (ITT), Kharagpur, Department of Ocean Engineering
and Naval Architecture, Kharagpur 721302, India, E-mail: jit@naval.iitkgp.ernet.in
Professor, Delft University of Technology (DUT), Faculty of Design, Construction and Production, Ship
Structures Laboratory, P.O. Box 5035, 2600 GA Delft, The Netherlands, E-mail: b.boon@wbmt.tudelft.nJ
AB S' I RACT
The fracture rnechanica approach is indispensible in view of predicting the remaining fatigue life after cracks
have been detected. Especially in case of mixed mode crack growth, the non singular T-term has to be considered in view of the crack path.
The paper deals with an improved method to evaluate the elastic T-terra and its application to mixed mode crack growth simulation, using various criteria for crack growth direction. Numerical values for the T-term have been compared with results from literature, and the robustness of the method has been demonstrated by considering a
wide variation of the parameters used in the finite element model. Comparison of numerical predictions for the crack
paths with experimental results have been made for uniaxially loded plates with inclined cracks as well as for
biaxially loaded plates with a central crack.
NOTATION
I and II of circular mesh in view
Deift University of Technology
Ship Hydromechanics Laboratory
Library
Mekelweg 2, 2628 CD Deift
The Netherlands
Phone: +31 15 2786873 - Fax: ±31 15 2781836
B stress biaxiality ratio: B =
C material constant of Paris law for crack growth rate E modulus of elasticity
G strain energy release rate
K, stress intensity factors: i = mode I or mode II
'ç
stress intensity factor values: j = max. rain, cl. or eff.NR number of elements in radial direction in circular cractip mesh
N number of elements over 180 degr. in circular cractip mesh
RA radius of triangular cracktip elements
RB outer radius of circular cracktip mesh
s
ratio of stress intensity factor mode II over mode ISd strain energy density
Ti
value for T-terra according to following variants: i = s, in, u or avgw (halt) width of plate, as applicable
a half length of initial crack
a1 coefficients of strain energy density function da incremental crack extension length
incremental number of cycles for crack extension da 'np material constant of Paris law for crack growth rate
r radius measured to cracktip
u radial displacement
X crack extension coordinate, measured along initial crack direction, see fig. 3
c2' effective modulus of elasticity, depending upon stress state
e angular position measured from direction of crack advance, see fig. i stress intensity factors i = I or LI of forced displacement field applied to copy of corrected T-terra
2 load biaxiality ratio:
2 transverse deviation of crack extension, see fig. 3
V poisson ratio
INTRODUCTION
Life extension, planning of inspection periods and prediction of the remaining fatigue life make that there is a growing interest in fatigue predictions based upon crack growth rates for ship and offshore structures. As cracks may originate in arbitrary directions or may grow into areas with different orientation of the principal stress direction, the crack growth process is a mixed mode problem. While there is a lot of literature on this topic, the problem is by far not
solved. For example, several criteria for the crack growth direction are in use, and the way it is influenced by the T-term.
The paper deals with numerical and experimental investigation of mixed mode crack
propa-gation in thin walled structures. Using stress intensities as calculated with the finite element method and Paris law for subcritical crack growth, the fracture mechanics approach enables predictions of crack growth rate and crack growth path for any structural configuration in an iterative process. An inherent consequence of this approach is that it requires a considerable computing effort to accurately evaluate the relevant parameters for each interval of the crack growth process. However, before being applied as an engineering tool to predict the remaining fatigue life of marine structures after a crack has been detected, a proper evaluation of the alternatives for the above mentioned aspects with help of experimental results is of paramount inportance. For this purpose, several alternatives have been programmed in the finite element package ANSYS, and tests on uniaxially loaded plates with inclined cracks have been carried
out
The main objectives of the present investigation are:
Implemention of an automated crack propagation process in ANSYS where the finite
element model is repeatedly updated as the crack propagates, including a purpose written
macro for a circular mesh around the cracktip.
- Implementation of a routine to accurately evaluate the T-term stress near the crack tip.
- An evaluation of the validity of different alternatives for the crack direction criteria with
help of experimental results on plates with inclined cracks as earned out and by comparison
with other data from literature.
FORMULATION OF MIXED MODE CRACK PROPAGATION
For cracks propagating in thin plates which are loaded in their plane, the stress field near the
crack tip is characterized with three parameters: the stress intensity factors K and K11 for the singular stress fields due to the opening and the sliding mode respectively, and the T-term,
which represents the constant stress parallel to the crack flanks.
In case of arbitrarily oriented cracks, there are two major aspects to be dealt with in view of crack growth simulation: the crack growth rate and the crack growth direction. As both of them are dependent upon the stress field near the crack tip, a proper evaluation of the above
mentioned parameters is of paramount importance. 2.1. Evaluation of stress intensity factors
Several methods are described in literature for calculating the stress intensity factors using
the results of a finite element analysis of the cracked configuration:
* Displacement extrapolation method [1, 2], being one of the earliest approaches. Based upon
this method, mixed mode stress intensity factors can be calculated. In literature it is suggested
crack tip, combined with quarterpoint locations for the midside nodes located at element sides which are attached to the cracktip node in order that the fr displacement distribution can be
better approximated.
* Stiffness derivative method. In its initia! formulation [3] the energy release rate is evaluated,
which means that the individual stress intensity factors ca not be determined. Later it has been extended to the mixed mode problem [4]. The stiffness derivative method is said to be rather
accurate while it does not require a very dense mesh.
* Modifed crack closure integral [5]. Mixed mode stress intensity factors can be calculated with this metod, and it is rather simple to implement provided that the forces in the first nodes ahead of the existing crack can be extracted from the finite element results. It does not require
a very dense element mesh while its accuracy is to be evaluated.
* J-integral method [6, 7]. In its initial formulation it is suitable for the evaluation of the
energy release rate only, but later it was also extended to mixed mode situations [8, 9]. A
drawback of the J-integral method is that its results are slightly path dependent.
* Mode enrichment method [10]. This method is suitable for mixed mode stress intensities, and a rather coarse element mesh suffices. Its implementation in a finite element code is rather
complicated as a correction to the stiffness matrices is required for the elements attached to the
ctrackti p.
The first two mentioned methods have been used in previous research [11]:
- the dispacement extrapolation method is an ANSYS facility that can be activated by issuing the relevant commands,
- th stiffness derivative method, which was progammed in ANSYS with macros, see [11].
From comparison with theoretical results it was concluded that both methods are almost
equivalent and sufficiently accurate, while the latter requires a more dense mesh then the first. As the displacement extrapolation method is much more simple to apply, and requires less
computer time, this will be used in present work. 2.2. Evaluation of the T-term
Three methods to evaluate the T-term were found in literature, and it was decided in [12] to
implerment the Stress Extrapolation Method without any corrections as a first attempt to evalute the T-term. because of its inherent simplicety. Three variants of this method were
programmed:
- T_s: T-term based upon an extrapolation of the FEM radial stresses along crack flanks,
which are averages of all elements attached to the node, and includes the effect of tangential
and shear strains in the elements
- T_m: T-term based upon an extrapolation of the local radial strains, Edu/dr, excluding all effects of tangential and shear strains
- T_u: T-term based upon an extrapolation of the radial distribution of the average radial strain, Eu/r.
In order to evaluate the T-term, the averages of the top and bottom flank values for a number
of radial positions of these three values is used. The relevant value at R=0 is determined with a linear fit through the data points to find the corresponding value of the T-term. Finally also the
average value, T_avg, of the above three variants T_s, T_m and T_u is calculated.
After variation of a number of parameters, it appeared that the required robustness of results could not yet be obtained. As a consequence. it was concluded in [12] that a more rigorous
approach is required. Following options were proposed:
- correction of the stresses and displacements due to the singular fields near the crackitip.
2.3. Prediction of mixed mode crack path
Two aspects are important in view of path
pre-diction under mixed mode crack growth, viz:
initi-al crack growth direction for very sminiti-all
incre-ments, and crack path stability criteria, which ac-count for deviations from the initial direction for
finite crack growth increments.
2.3.1. Criteria for initial crack direction
Several criteria are in use for the prediction of the initial direction of crack growth:
- direction with vanishing shearing mode sress intensity factor K11 as applied by Cotterell & Rice [13] and by Sumi et. al. [14], see fig. 1:
K
where
S=!1
K1
As the initial crack growth direction is determined by the ratio of the stress intensity factors only, this criterion is rather simple to implement. It is understood that this formula is accurate
for small angles only.
direction with maximum tangential stress, see Erdogan and Sih [15]:
I -31
SLflOT = S
19S2
j
Comparisons of predictions using this formula with experimental results for uniaxially loaded
plates having inclined central cracks was reasonable [11].
- direction with minimum strain energy density, see Sih [16], Tanaka [17] and Mi et.al. [18]:
16rr K1_=a110--4a120S+a,20S=0 (3)
Where the coefficients a0 are:
a11 0=2sin0(-1 +2v--cos0)
a1,0=2cos0-cosO(1 -2v)-1 (4)
a22 0=2sin0(1 -2v-3cosø)
- direction with maximum energy release rate, see Wu [19]. A disadvantage of this criterion is
that the direction has to be determined by calculating the energy release rate for a number of different crack extension angles. which in priciple requires as many separate analyses as the number of directions required to rind a good maximum. In this sense it is a rather indirect
method which requires much additional computational effort as compared to the previous
ones, where the ratio of the stress intensity factors, S, is the only parameter.
- direction with maximum opening mode stress intensity factor K1, see [20]. Again, this
= -25
crack
X
de criterion for small values of S and O, Fig. 2: Comparison of initial direction criteria
= Cotterell and Rice [13] proved that the
maximum strain energy release rate criterion is equivalent with the vanishing shearing mode
criterion too.
= Furthermore, the energy release rate in mixed mode crack growth is:
I
G = a(K
- K)
(5)with a E/(1-v2) for plane strain and a = E for plane stress. Using the criterion of maximum strain energy release rate, the kink angle is such that:
'oe
"oe
ÔK ôK11
i- K (6)
Combining this relationship with the condition of vanishing mode H, it can be concluded that the direction of mixed mode crack growth will also satisfy the condition of maximum opening
mode stress intensity, as K11=O and K1O results in:
ÖK
(7)
Ramulu and Kobayashi [21] compared crack extension angles predicted by four of these
methods for a uniaxially loaded plate with inclined cracks. From fig. 2, where S is the only free
parameter, it follows that the methods are almost equivalent for S<O.25. Furthermore it is also clear that the initial angle according to the vanishing Kfl criterion as used by Cotterell &Rice
and by Sumi deviates rather much from the other criteria, and especially from the maximum
tangential stress criterion by Erdogan and Sih, in case of S>1.
According to all of these 4 criteria a crack will extend with a kink in the path direction if
KO
at the crack tip, and if a crack follows a path which continuously turns, this will implythat K11=O during the extension.
2.3.2. Crack path stability criteria
In addition to the initial crack growth direction, as discussed above, there is evidence that
the crack path may deviate from the intial crack direction as the crack progresses in cases
where an additional tensile stress field works paralleli to the crack flanks, i.e. a positive T-term. criterion requires additional numerical 90
effort to find the relevant direction.
While no uniform opinion exists which of so
these criteria is the best one to be used, there is evidence that that the first 4 of
them are equivalent for small angles:
= from the formulae given it is rather simple to prove, that the maximum
tang-ential stress criterion and the minimum
strain engergy density criterion give the
The crack path formulation of Cottereli and Rice [13] see fig. 3, as based upon a first order
perturbation procedure for a kinked and slightly curved crack, reads:
4T
3K1
\
2x T2
4x
Kwhere 00 is as indicated in§ 2.3.1
Using this result, it can be concluded that a straight crack under mode I loading, after being
inluenced by a local disturbance, will return to its original path direction in case T < O, and for T> O the tangent angle of the path increases from 00 as the crack extends, see fig. 3.
The work of Cotterell and Rice was extended by Sumi et. al. [22, 23, 14]. They included also a 'Ir term in the Williams series expansion for the stress field around the crack tip, and
arrived at a slightly different result:
The additional corrections in y represent the effect of stress redistribution due to the crack
growth in a finite body. This correction term will be small for relatively short cracks.
It is admitted in [14] that, since the perturbation solution is effective only for slightly kinked
cracks, in case of large kink angles the above formula for 00 is replaced by the results of
Amestoy and Leblond [241.
2.4. Crack growth rate
In order to reflect the repeated strain cycles at a crack tip for crack growth due to fatigue loading, the effective stress intensity range is important instead of the maximum value of the
stress intensity factor in view of the crack growth law as formulated by Paris:
=
C((
eff)mPwhere C and m are material related constants, and:
IÇff = - Kd is the stress intensity range, where Kd = stress intensity value at which crack closure takes place.
Normally, IÇ is taken to be O or 1Ç , whichever is greater. However, in case that the actual
crack closure takes place at a level higher than both these criteria, due to plastic deformation of
the crack flanks because of a previous overload or residual compressive stresses due to
wel-ding, the actual value at crack closure has to be used.
Under mixed mode loading the effec-tive stress intensity range has to account for the effective mode II stress intensity
range in a realistic way too. As a crack
under mixed mode loading
normallytends to grow in a vanishing mode II
di-rection it can be argued that the crack .
Fig. 3: Crack increment coordinates 1>0 (8)
(b)
X =0
(x) y 4L_IÇ
8T1
3K1\rma&lirional
2x+yx
corrections where: (9)growth is to be based upon the mode I stress intensity range only [25, 26]
Other models do account for the mixed mode situation, by using an equivalent range of the stress intensity factor. Using the maximum tangential stress criterion. Erdogan and Sih [15]
concluded that the equivalent stress intensity factor is:
00 0 0
K = Kcos3 -
eq '3K1cos2sín
(11)2 ¡ 2 2
Tanaka [17] proposed following formulation for the equivalent effective range of the stress intensity factor, which is based upon the minimum strain energy density criterion and using a
theory for fatigue crack propagation by Weertman (27]:
= (K7 -- 8Kí).25
(12)This formulation by Tanaka is well supported by tests on aluminium.
Some indications were found in literature {28 that the crack growth rate increases in case of a negative T-term, due to an increase of the plastic zone. This aspect will not yet be
inclu-ded in present work.
3. IMPLEMENTATION
3.1. Evaluation of the T-term
Based upon the observations in [12] a regular mesh around the cracktip was programmed in ANSYS as a first step for improvement of the T-term. This new mesh covers a circular region
with outer radius equal to the crack growth increment: RB = da, and includes a number of rings
of elements, see fig. 4. Following parameters can be given default values or may be specified by the user: NT, NR, RA, and distribution of radial lengths of elements. Outside this circular
region a mesh will be generated by ANSYS.
Using this cracktip mesh the results for the T-term were already much more consistent, but not yet sufficiently robuts and reliable.
Because of these observations it was finally decided to implement also the correction of th '' stresses and displacements due to the singular field near the cracktip. For this purpose it is to be realised that theoretically the radial stress at the crack flanks are zero, except of the T-term stress. However, the displacement fields used in the elements are approximations of the real situation, with as a result that the calculated radial stresses at the crack flanks deviate from ihe T-term stress distribution. Especially in the triangular
cracktip elements, the calculated stress and displacement
field will be be erroneous, due to the complicated stress distribution near the crack tip. In order to improve the
numerical stresses and displacements at the crack flanks in a realistic way, it is essential that the correction
inclu-des the same errors that were introduced in the original
model. For this purpose, the circular finite element mesh
as used around the crack tip was copied twice. The first copy is subjected to forced displacements (at the nodes of its outer circle only) which are equivalent with a mo-de I displacement field of fixed value K1 = , and the
second copy is subjected to a mode [I displacement field equivalent with K11 = K. As these rings will be analysed .
Fig. 4: Regular circular mesh
together with the model of the cracked plate. the
tions for the stresses and displacements at the crack flanks of
the plate can be calculated from the the results of the copies of
the circular mesh using the ratios of actual over enforced K
values: K1 a/Kl and K11 a/1I
3.2. Finite element model for crack growth simulation A number of new macros have been written which create the finite element model of the cracked plate by iterating on
the crack growth as calculated from the previous cracked
situation. The main program reads the data in an interactive session, it controls the iteration process of crack extension,
and calls the relevant other macros. Options are available for:
- Selection of initial crack growth direction criterion, mo - Selection of crack path stability criterion,
- Choice whether or not to apply correction of the T-term, Fig. 5: Test specimen - Choice of T-term variant: T_s, T_m. T_u or T_avg
- Parameters of circular mesh,
- Choice of a straight line or a sp line representation for the crack flanks.
EXPERIMENTS
Four rectangular plates having a central inclined crack with angles 15, 30, 45 and 60 deg., see fig. 5, were tested under uniaxial cyclic loading. The experimental set up as used has been
reported already in [11] including the results for two plates, viz. 30 and 45 deg.
RESULTS
5.1. Comparison of calculated T-term with theory
5.1.1. Center cracked plate loaded in tension
The stress biaxiality ratio B has been calculated for a rectangular plate with a horizontal crack, H/W=2 see fig. 6.a, and uniaxially loaded in tension,?. = o/o, =0. The calculated results
for B are compared in fig. 7 with the results of Leevers and Radon [29], line A (H/W=??), and Bilbey et. al. [30], line B (H!W=2). The calculated results of B are for T_avg values, and one
set are corrected values while the other
set are not corrected. Both sets comprise W W
8 different lines for combinations of para- t t t t t t t t t
meters: RB = 10.0, 5.0 and 2.5, N1 =12, 9
H i
and 6 and NR=12, 10, 8 and 6. The
cor-rected T_avg values are all within a very
-i
-
-t
narrow band, whereas the not correctedresults show more scatter
and have-j
slightly higher values. Especially the line
I I
for NT=6 is rather extreme. The individu- i i i t i i i i i I
o.
al lines for T s, T m and T u for both
Ysets show the same tendency but have a) CCT b) SECT
slightly more scatter, especially for the
Fig. 6: Specimen geometries
¡liii
¡ ¡ î0000 000 00 ol
00000
-.Jo
-:5
DO
Fig. 7: Results of B for center cracked plate
not corrected set. The calculated results compare well with the results of Bilbey et. al., but are
approximately 1% lower. The linear line of Leevers and Radon seems to be not very accurate.
5.1.2. Side edge cracked plate loaded in tension
The calculated values of B for a side edge cracked plate, HJW=6 see fig. 6.b, are presented
in fig. 8 together with data of Sham [311, line C. The results agree very well. 5.2. Predictions for plates with slanted cracks
Crack predictions have been made for rectangular plates with a central inclined crack with 4 different angles, see fig. 5, for which experimental results were available.
5.2.1. Crack path predictions
Five methods were used for the crack path predictions, as follows:
1.0 crack direction using the maximum tangential stress criterion by Erdogan and Sih [15]. 2.0 crack growth prediction according to Cotterell and Rice [13] including the T-term and
with the initial crack direction according to the vanishing K criterion.
2.1 as 2.0 but using the maximum tangential stress criterion for the initial crack direction.
3.0 crack growth prediction according S.E.C.T.: T avg corrected to Sumi et. al. [14] including the T-
-
-term and with initial crack direction
according to the vanishing K11 crite- 240
non. -
---I
3.1 as 3.0 but using the maximum tang- 220 RefÇ
b
ential stress criterion for the initial
3 30 -
_aç_corcrack direction.
Using a crack growth increment RB=2,
the five methods predict almost equal
crack growth paths, see fig. 9 for strip C, 30 deg. initial angle. From these results
for strips B, C, D. E (15,30,45 60deg.)
-.2 20
4C
-:
-25
08
T_avgcorrecled T avg not correctei
I,., wr, of pronfe.-.
32 03 34 35 36 07 08
a/W
method 3.1 had on average slightly better results, except for strip E (60 deg.). A plot of
predicted paths for these 4 strips using method 3.1 is presented in fig. 10, from which it can be concluded that the actual crack is in general more horizontal than the prediction and shows
more fluctuations. The prediction for strip B (15 deg.) and E (60 deg.) is rather good.
As proof of the quality of these methods, the crack path simulation was repeated with a crack
increment which was increased by a factor 4: da=RB=8. From these results it appears that: methods 2.1 and 3.1 behave in general better than methods 2.0 and 3.0.
- method 3.1 is by far the best for strip C, see fig. 11.
- for strip D methods 1.0, 2.1 and 3.1 are almost equivalent and rather good. - for strip E method 3.1 is more in error than methods 1.0 and 2.1, see fig. 12.
From these results there is no strong evidence for a better overall behaviour of one of the
methods. However there is a slight preference for method 11 because it does not show strong fluctuations in the first increments, see fig. 11, and it is on average not more in error than the
other methods.
The simulation results differed only marginally for a straight line instead of a spline representa-tion of the crack path between the end of the original crack and the regular crack tip mesh. 5.2.2. Crack growth predictions
Comparison of crack growth predictions with experimental results are presented in fig. 13.
:0
Fig. 11: Crack paths strip C for da=8 Fig. 12: Crack paths strip E for da=8
40 80
racK exter,!on X4,D (nfl,
58 54 38 .830
---5Z6 8 .4 -O 4 >-200j
C3 200 300 400 500 X mmiFig. 15: Predicted crack paths for biaxially loaded plate compared with measurements
30 :00 300 400 500 0C
:yc/' 000
Fig. 13: Crack growth predictions for strips Fig. 14: Comparison of crack growth
parameters
The criterion of Tanaka for K0q was used for these predictions, but due to a relatively large scatter in crack initiation time, the number of cycles were measured starting from the point where the crack had grown 2 mm. In fig. 14 the Paris crack growth curves da/dN as deducted from experimental results are compared with the average material parameters C and m from
Dijkstra et. al. [32] which were used for the crack growth simulations.
5.3. Predictions for biaxially loaded plate with horizontal central crack
Crack path predictions have been made for a biaxially loaded plate, H=W=500 mm, with a horizontal center crack. a=75 mm, and two values of the loading ratio, X=2 and X=3. The simulated crack paths for da = 3.125 mm are shown in fig. 15 together with measurements as reported by Leevers et. al. [33]. Also indicated are the endpoints of the simulated crack for different values of the crack increment length. The predicted crack paths compare very well
with the experimental results of Leevers and Radon, notwithstanding that a full convergence of
the predicted path was not yet achieved for R8=3.l25 mm. In this case the strongly curved
500
crack path was represented by a spline curve between the end of the original crack and the
regular circular mesh around the crack tip.
6. CONCLUSIONS
An improved method to evaluate the T-terni was implemented, for which a regular circular mesh around the crack tip was programmed together with a correction for the radial stresses along the crack flanks using the singular displacement field which was applied to the outer
boundary of a copy of this regular circular mesh.
The corrected T-term as calculated with this method is very accurate and appears to be
rather insensitive for variations in the meshing-parameters.
The crack path predictions including the T-term show an improvement of the prediction
capability, especially when using an increased crack increment. There is a slight preference for
the metod of Sumi et. al. [14], combined with an initial crack direction according to the
maximum tangential stress criterion of Erdogan and Sih [151.
The experimental crack growth speeds da/dN tallied very well with the mean line as used for
the simulations.
Crack path predictions for a biaxially loaded plate show good correlation with measured
crack paths. A rather small crack increment is required in this case for a full convergence. REFERENCES
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