PANM 17
Karel Mikeš
Comparison of crack propagation criteria in linear elastic fracture mechanics
In: Jan Chleboun and Petr Přikryl and Karel Segeth and Jakub Šístek and Tomáš Vejchodský (eds.): Programs and
Algorithms of Numerical Mathematics, Proceedings of Seminar. Dolní Maxov, June 8-13, 2014. Institute of
Mathematics AS CR, Prague, 2015. pp. 142--149.
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Programs and Algorithms of Numerical Matematics 17
J. Chleboun, P. Přikryl, K. Segeth, J. Šı́stek, T. Vejchodský (Eds.)
Institute of Mathematics AS CR, Prague 2015
COMPARISON OF CRACK PROPAGATION CRITERIA IN LINEAR
ELASTIC FRACTURE MECHANICS
Karel Mikeš
Faculty of Civil Engineering, Czech Technical University in Prague
Thákurova 7, 166 29 Prague 6, Czech Republic
karel.mikes.1@fsv.cvut.cz
Abstract
In linear fracture mechanics, it is common to use the local Irwin criterion or the
equivalent global Griffith criterion for decision whether the crack is propagating or
not. In both cases, a quantity called the stress intensity factor can be used. In this
paper, four methods are compared to calculate the stress intensity factor numerically;
namely by using the stress values, the shape of a crack, nodal reactions and the global
energetic method. The most accurate global energetic method is used to simulate the
crack propagation in opening mode. In mixed mode, this method is compared with
the frequently used maximum circumferential stress criterion.
1. Introduction
The description of crack propagation is one of the most important ingredients of
linear elastic fracture mechanics (LEFM). The main questions are: At which loading
level will the crack propagation begin and in which direction will the crack propagate?
The aim of this paper is to compare numerical implementations of most frequently
used crack propagation criteria for opening mode and mixed mode in 2D.
2. Stress intensity factor concept
The stress intensity factor is a quantity used in LEFM to describe the asymptotic
singular stress field near the crack tip. The stress in the vicinity of the crack tip is
unbounded and grows in inverse proportion to the square root of distance from the
tip. Under plane stress, the asymptotic stress field is described by
KI
θ
θ
3θ
KII
θ
θ
3θ
cos
1 − sin sin
−√
sin
2 − cos cos
,(1)
σx (r, θ) = √
2
2
2
2
2
2
2πr
2πr
KI
θ
θ
3θ
KII
θ
θ
3θ
σy (r, θ) = √
cos
sin cos cos ,
1 + sin sin
+√
(2)
2
2
2
2
2
2
2πr
2πr
θ
θ
3θ
KII
θ
θ
3θ
KI
τxy (r, θ) = √
sin cos cos
−√
cos
1 − sin sin
,
(3)
2
2
2
2
2
2
2πr
2πr
142
where KI and KII are the stress intensity factors for modes I and II which represent
the loading and geometry conditions, r is the distance from the crack tip and θ is
the polar angle; see e.g. [7].
3. Crack propagation in mode I (opening mode)
In mode I, the crack is opening without sliding. Therefore, we can assume that
the crack will propagate in the original direction and we have to decide when the
propagation starts.
3.1. Local Irwin criterion
This concept was introduced by Irwin [4] in 1957. The stress intensity factor
is used to decide about the crack propagation. The propagation will start when
the value of the stress intensity factor KI reaches its critical value so-called fracture
toughness denoted by Kc .
3.2. Global Griffith criterion
This criterion was introduced by Griffith [3] in 1920. The crack will grow if
a sufficient amount of energy is released by its propagation. The criterion is based
on the strain energy release rate, defined as
G(u, a) = −
1 ∂We (u, a)
,
t
∂a
(4)
where We (u, a) is the elastic strain energy considered as a function of the imposed
displacement u and the crack length a. The beam thickness is denoted by t; see
Figure 1.
Under plane stress and in mode I, both criteria are equivalent [2]. We can write
G(u, a) =
KI 2
.
E
(5)
The rules for crack propagation according to the local Irwin and global Griffith
criteria are summarized in Table 1, where Kc resp. Gf are material properties called
fracture toughness [Nm−3/2 ] and fracture energy [Nm−1 ], resp.
Local criterion
KI < Kc
KI = Kc
KI > Kc
Global criterion
G < Gf
G = Gf
G > Gf
Crack behaviour
⇒ no crack propagation
⇒ crack propagation
⇒ inadmissible (in statics)
Table 1: Crack propagation rules according to the local and global criteria
143
3 Výpočetní modely
Výpočty pro jednotlivé modely byly prováděny pomocí metody konečných
prvků
za rozměr
předpokladu
rovinné
výpočtům
byljedná
použit
program
fyzikální
jednotek,
protonapjatosti.
není nutné K
uvádět,
zda se
o milimetry
„OOFEM“.
Jedná ale
se oveličiny
objektově
orientovaný,
volně
software určený
nebo
centimetry,
můžeme
považovat
za šiřitelný
bezrozměrné.
3.3. Simulation
in opening
mode
pro řešení multifyzikálních problémů metodou konečných prvků, který je
Four methods have been used to calculate the stress intensity factor or the strain
vyvíjen na katedře mechaniky Stavební fakulty ČVUT [4]. Pomocí tohoto
energy release rate in opening mode;
𝐹/2 namely by using (i) the stress values, (ii) the
programu
byly(iii)
pronodal
řešený
model
hodnoty
posunů
a uzlových
reakcí
crack opening,
forces
and získány
(iv) the release
of strain
energy.
The first
three
pro
jednotlivé
uzly
a
pro
jednotlivé
prvky
byly
vypočteny
hodnoty
deformací
methods have a local character and deal with the values near the crack tip to calcualate
napětí.
Pro intensity
následnéfactor.
zpracování
těchto
hodnot
a výpočty
faktoru
the stress
The fourth
method
evaluates
the change
of intenzity
the energy
of the whole
beam whenvlastní
the crack
is extended.
Three
different
types
of triangular
napětí
byly vytvořeny
programy
v jazyce
Delphi
Object
Pascal.
ℎ
finite elements have been used for each method; namely (i) three-node element with
𝑎approximation
linearNosník
of displacement, (ii) six-node element with quadratic approx3.1
s vrubem
imation and (iii) six-node quadratic element with singular shape functions on the
Prvním
zkoumaným
modelem
bylsee
prostý
nosník
se svislou
trhlinou (vrubem)
𝐿/2
edges starting
from the
crack tip;
Figure
2. Numerical
simulations
have been
na
spodním
okraji
uprostřed
rozpětí,
namáhaný
tříbodovým
ohybem,
viz Obperformed using the open-source finite element code OOFEM [6]. The relative
error
‫ܮ‬
(a)
(b)
of all methods
in a three-point
test pro
withtento
geometry
according
Figure 1
rázek
3.1. Poměr
byl zvolenbending
4, protože
poměr
jsou v to
literatuře
Obrázek
3.2: (a)݄ geometrie
modelované
poloviny
(b) intensity
použitá síť
have
been evaluated
by comparing
the computed
value nosníku,
of the stress
factor
dostupné přibližné vzorce, které bude možné využít pro ověření správnosti
with the ,,exact” values obtained by using approximate analytic formulas
available
ܽ
,
která
přednumericky
získaných
výsledků.
Dále
zavedeme
veličinu
ߙ
ൌ
Pro
výpočet
metodou
konečných
prvků
byla
použita
relativně
dosti
jemná
in [2], [5] and [7]. The relative errors of all methods for all types݄of elements
are
trojúhelníková
síť,
která
byla
v
okolí
kořene
trhliny
ještě
výrazně
zjemněna,
shown
in
Table
2.
stavuje poměr délky trhliny ku výšce nosníku.
viz Obrázek 3.2 (b). Celkem byly vytvořeny tři sítě lišící se typem použitých
prvků. Nejprve síť obsahující nejjednodušší trojúhelníkové tříuzlové prvky,
‫ ܨ‬lineárních funkcí. Poté lepší šestiuzlokteré aproximují pole posunů pomocí
vé trojúhelníkové prvky, které k aproximaci pole posunů využívají kvadratické funkce. A na závěr upravené kvadratické prvky, u kterých byl na
úsečkách obsahující
kořen trhliny posunut mezilehlý uzel z jedné poloviny
݄
do jedné čtvrtiny
ܽ blíže ke kořeni trhliny, viz Obrázek 3.3. Tento posun vynutí u bázových funkcí nenulových v kořeni trhliny singularitu typu 𝑟 −1/2 . Te‫ܮ‬
‫ ݐ‬kořene trhdy stejnou, jaká se vyskytuje v asymptotickém poli napětí v okolí
liny. Předpokládáme, že tato úprava umožní ještě lépe vystihnout hodnoty
3.1: Geometrie
a zatížení
modelu
nosníku
vrubempoužitou
napětí vObrázek
okolí Figure
kořene
Tento
umožnil
pro stest
každou
1:trhliny.
Geometry
of thezpůsob
three-point
bending
metodu porovnat vliv typu prvků sítě. Ke generování sítí byl použit generáProtože nosník je osově symetrický, bylo pro zjednodušení výpočtu možné
tor T3D [5].
modelovat pouze jednu symetrickou polovinu nosníku, viz Obrázek 3.2 (a).
Tím se snížil počet neznámých na polovinu a výpočet probíhal výrazně rychleji. Dalším zjednodušením bylo, že nosník nebyl zatěžován silou, ale byl mu
v místě působení síly předepsán svislý konstantní posun. Velikost zatěžující
síly ‫ܨ‬, která by odpovídala předepsanému posunu, byla získána jako svislá
reakce zatěžovaného uzlu. Geometrie nosníku byla zvolena následovně: výška
݄ ൌObrázek
ͶͲ,Used
délka
‫ܮ‬ൌ
ͳ͸Ͳ, tloušťkaൌ
délka
trhliny(middle),
ܽkvadratický
ൌ ͳͲ, velikost
základ3.3:
Lineární,
kvadratický
aquadratic
upravený
prvek
Figure
2:
finite
elements:
linear ͳ,
(left),
modified
singular
ního
trojúhelníkového
prvku
ൌ
ʹ,
ͳͲ,
velikost
trojúhelníkového
prvku
v
okolí
quadratic (right)
3.1.1
Přibližné
vzorce
pro
výpočet
faktoru
intenzity
napětí
kořene trhliny ൌ Ͳǡͳ. Při výpočtu metodou konečných prvků není podstatný
𝐿
Pro nosník s vrubem s poměrem délky ku výšce = 4 je v díle [3] k dispozici
ℎ
následující vzorec:
144
8
9
Values of stress
Shape of a crack
Node reactions
method
Nano and Energetic
Macro Mechanics
2013
Linear
Quadratic Singular quadratic
> 30 %
10 - 30 %
10 - 30 %
< 10 %
2%
1%
5 - 10 %
<5%
<5%
5 %Faculty of Civil
2%
0.5in%Prague, 2013, 19th September
Engineering, CTU
Table
The
relative
of and
thefracture
computed
stress𝐾𝑐intensity
factor
different
𝐸 = 20 2:
GPa,
Poisson
ratioerrors
𝜈 = 0,2
toughness
= 4 MNm−3/2
, weK
obtain
ForceI for the
methods
anddiagram,
different
finite
elements
displacement
see in
Fig. 2.
𝑎0 = 50 mm
𝐹 [kN]
𝑎 = 100 mm
𝑎 = 150 mm
𝑎 = 200 mm
𝑎 = 250 mm
𝑎 = 300 mm
𝑎 = 350 mm
𝑎 = 400 mm
𝑎 = 450 mm
𝑎 = 480 mm
𝑢 [mm]
Fig. 2 Force-displacement diagram of the three-point bending test
Figure 3: Force-displacement diagram of the three-point bending test
𝐹 [kN]
MCSC
MSERR
The global energy method using the modified quadratic elements is the most accurate but also the most time-consuming one. Figure 3 shows the load-displacement
diagram obtained with this method for a beam of the following geometry: height
h = 0.5 m, length L = 2 m, thickness t = 0.2 m, initial crack length a0 = 0.05 m,
elastic modulus E = 20 GPa, Poisson ratio ν = 0.2 and fracture toughness Kc =
4 MNm−3/2 .
4. Crack propagation in mixed mode
The mixed mode represents a combination of tensile opening and in-plane shear.
The direction of crack propagation is not known in advance and has to be determined
by a suitable criterion.
145
4.
𝑢 [m]
CRACK PROPAGATION IN MIXED MODE
In opening mode in a plane there is a combination of an opening and a shear. The direction of the
crack propagation has to be determined by another criterion.
4.1. Maximum circumferential stress criterion (MCSC)
This criterion determines the crack propagation direction based on the maximal
circumferential stress σθ , which is defined as
σθ (r, θ) = σy cos2 θ + σx sin2 θ − 2τxy sin θ cos θ.
(6)
Substituting from (1)–(3), we obtain
KI
θ
KII
θ
θ
σθ (r, θ) = √
cos3 − 3 √
cos2 sin .
2
2
2
2πr
2πr
(7)
The angle θ with maximal circumferential stress (MCSC1) is obtained by solving the
equation
KII
sin θ +
(3 cos θ − 1) = 0
(8)
KI
with the conditions
θ ∈ (−π, π);
KI > 0;
KII sin
θ
< 0,
2
(9)
where the ratio KII /KI is obtained by fitting (1)–(3) to the values of the stress field
in a number of Gauss points near the crack tip.
Another approach (referred to as MCSC2) is based on substituting the values of
the stress field at Gauss points into the original definition of circumferential stress (5).
After smooth of these data by a polynomial function, the angle θ that maximizes
this function can be found.
Both approaches give almost the same results; see Figure 4. The first method
(MCSC1) turned out to be numerically preferable and therefore is used in the following examples.
4.2. Maximum strain energy release rate criterion (MSERRC)
This criterion determines the crack propagation in the direction that leads to the
maximum strain energy release rate defined in (4). Numerical realization consists
in simulation of a number of sufficiently small crack extensions in several different
directions. For each direction, the strain energy release rate is evaluated by subtracting the final strain energy from the original one and dividing by the increment
of the crack area. The obtained values are smoothed using a polynomial function,
for which the maximum is then found.
This criterion predicts, in most cases, similar crack trajectories to the MCSC.
However, application to the three-point bending test with an eccentric initial crack
leads to a different crack path; see Figure 4.
146
Nano and Macro Mechanics 2013
Faculty of Civil Engineering, CTU in Prague, 2013, 19th September
MCSC1
MCSC2
MSERRC
Nano and Macro Mechanics 2013
Faculty of Civil Engineering, CTU in Prague, 2013, 19th September
Fig. 3 Comparing of the different criterion in the three-point bending test
Figure 4: Comparison of crack paths according to different criteria for the three-point
bending
test withexample
an eccentric initial crack
4.3. Comparative
Another example is taken
over from [4]. It is a rectangular panel with two holes and two initial
MCSC1
MCSC2
4.3.
Comparative
example
cracks
submitted to a vertical
tensile. In this example both criterions lead to almost the same crack
paths
andlast
results
are similar
totaken
the results
4 and 5. panel with two holes and
The
example
isMSERRC
fromfrom
[1].[4],Itsee
is ina Fig.
rectangular
two initial cracks subjected to tension in the vertical direction. In this example,
both criteria lead to almost the same crack paths and the results are similar to those
from [1]; see Figure 5. The load-displacement diagrams are depicted in Figure 6.
Both criteria predict the same behaviour but the curve obtained with MCSC is not
smooth. This means that MCSC is less accurate when used to decide whether the
Fig. 3 Comparing
crack is propagating
or not.of the different criterion in the three-point bending test
MSERRC
4.3.
Comparative
MCSC example
5. Conclusion
Another
example ismode,
taken over
from results
[4]. It is are
a rectangular
withglobal
two holes
and two
initial
In the opening
the best
obtainedpanel
by the
energy
method.
cracks
submitted
to
a that
tensile.
In different
this
example
both
lead
totest
almost the same
crack
Fig.
4vertical
Comparing
of
criterion
incriterions
a vertical
tensile
In
the mixed
mode
arises
inthe
the
three-point
bending
test,
the
MSERRC
criterion
paths and results are similar to the results from [4], see in Fig. 4 and 5.
Fig. 5 Comparing of the different criterion in a vertical tensile test from [4]
5.
CONCLUSION
With using the global energetic method, we obtained very good results in mode I. In the three-point
bending test in mixed mode, criterion based on this method leads to the different crack path then the
criterions using
the circumferential stress. But both criterions give the similar results in other examples
MSERRC
MCSC
in mixed mode.
REFERENCES
Fig. 4 Comparing of the different criterion in a vertical tensile test
[1]
JIRÁSEK,
M.
and ZEMAN,
J. Přetváření
porušování to
materiálů.
Praha:
ČVUT, in
2008.
ISBN
Figure 5: Comparison
of crack
paths aaccording
different
criteria
a vertical
978-80-01-03555-9
tensile
test. Simulation in OOFEM (left) against results taken from [1] (right).
Fig. 5 Comparing of the different criterion in a vertical tensile test from [4]
5.
CONCLUSION
147
With using the global energetic method, we obtained very good results in mode I. In the three-point
bending test in mixed mode, criterion based on this method leads to the different crack path then the
criterions using the circumferential stress. But both criterions give the similar results in other examples
in mixed mode.
𝑎0 = 300 mm
𝑎0 = 350 mm
𝑎0 = 400 mm
𝑎0 = 450 mm
𝑎0 = 480 mm
𝑢 [mm]
Fig. 2 Force-displacement diagram of the three-point bending test
𝐹 [kN]
MCSC
MSERR
𝑢 [m]
4.
CRACK
PROPAGATION
INload-displacement
MIXED MODE
Figure 6:
Comparison of
diagrams for different criteria
In opening mode in a plane there is a combination of an opening and a shear. The direction of the
crack propagation has to be determined by another criterion.
based on this method leads to a different crack path than the MCSC criterion using
the circumferential stress. But both MSERRC and MCSC give similar results in
other examples in mixed mode. Both criteria seem to be accurate in prediction of
the propagation angle, but to determine whether the crack propagates it is more
appropriate to use MSERRC.
Acknowledgements
This work was supported by the Grant Agency of the Czech Technical University
in Prague, grant No. SGS14/029/OHK1/1T/11.
References
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[2] Broek, D.: Elementary engineering fracture mechanics. Martinus Nijhoff Publishers, Hague, 1984.
[3] Griffith, A. A.: The phenomena of rupture and flow in solids. Philos. Trans. R.
Soc. Lond. 221 (1921), 163–198.
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[4] Irwin, G. R.: Analysis of stresses and strains near the end of a crack traversing a
plate. J. Appl. Mech. 24 (1957), 361–364.
[5] Jirásek, M. and Zeman, J.: Přetvářenı́ a porušovánı́ materiálů. ČVUT, Praha,
2008.
[6] Patzák, B.: OOFEM – an object-oriented simulation tool for advanced modeling
of materials and structures. Acta Polytechnica 52 (2012), 59–66.
[7] Wang, C. H.: Introduction to fracture mechanics. DSTO Aeronautical and Maritime Research Laboratory, Melbourne, 1996.
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