- Email: [email protected]
Underlying fracture trends on Mode-I crack multiple-branching
Journal Pre-proofs Underlying fracture trends on Mode-I crack multiple-branching H. Yuan, Y.J. Xie, W. Wang PII: DOI: Reference:
S0013-7944(19)30847-1 https://doi.org/10.1016/j.engfracmech.2019.106835 EFM 106835
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
7 July 2019 15 December 2019 19 December 2019
Please cite this article as: Yuan, H., Xie, Y.J., Wang, W., Underlying fracture trends on Mode-I crack multiplebranching, Engineering Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.engfracmech.2019.106835
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
Underlying fracture trends on Mode-I crack multiple-branching H. Yuana, b, Y. J. Xie a 1, W. Wang a a Department b
of Mechanical Engineering, Liaoning Shihua University, Fushun, 113001, LN, P. R. China
College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao, 266000, SD, P. R. China Abstract: An energy-based modelling has been proposed to capture the physical process of multiple cracks
initiation from a Mode-I crack tip. The underlying fracture trends and instability on crack tri-branching, symmetric branching and side-branching induced by dominant Mode-I loading are formulized. The solutions of the energy release rate, crack initiation angles and K-based fracture criterion for multiple cracks initiation have been investigated. The upper limit on the number of possible crack initiation is given for a Mode-I crack. The discussions in present article indicate that some underlying fracture configurations for Mode-I crack can be triggered by using impulse loading. Many fracture behaviours for a Mode-I crack can be better explained and understanding based on the present theoretical modelling. The fracture toughness predicted by the present modelling and Griffith’s criterion agree well with experimental results. Keywords: Multiple-cracks initiation; Mode-I crack; Crack-branching. Nomenclature
angle of the boundary translation
c
critical wing crack initiation angle for a Mode-I crack
i
intersections of the θ-axis and integrand curve of normalized J2-integral
n
normalized phase angle
ij
stress components
Poisson’s ratio
ε
disturbance parameter of weak Mode-II loading to Mode-I loading
ei
boundary shifting tensor
E
elastic module
F, β
interim parameters
G
energy release rate for boundary cracking
GC
energy-based fracture toughness
Gmax
energy-based driving force
G tri b
energy release rate for crack triple-branching
GCtrib
energy-based fracture toughness for crack triple-branching
trib Gmax
energy-based driving force for crack triple-branching
sideb Gmax
energy-based driving force for crack side-branching
1
Corresponding authors. Y.J. Xie, E-mail address: [email protected]. 1
si wing si ext Gmax ， Gmax energy-based driving forces for wing crack and crack extension initiation from boundary si
respectively Ji
conservation integrals
K I , KII
stress intensity factors for the Mode-I and Mode-II crack
KIC
K-based Mode-I fracture toughness
tri b K ΙC
fracture toughness for Mode-I crack triple-branching
sideb K ΙC
fracture toughness for Mode-I crack side-branching
si wing s -ext KΙC , KΙCi K-based fracture toughness for wing crack and crack extension initiation from boundary si
respectively ni
the unit inward normal on boundary s
P
load
s, si
boundaries or integration paths around the crack tip
(sin)l
integral path within solids
Ti
stress vector acting on the integration path
u
i
displacement components
vi
directions of local boundary si shifting
w
strain energy density
1. Introduction Due the crack-branching behavior occurs, the original crack tip will split into two or more crack tips, and a distinctive impact of this behavior is the stress intensity factor of original crack tip decrease rapidly [1]. A good understanding of a crack branching conditions, including the multiple cracks initiation from a crack tip, is important and essential for the fundamental study of complex failure mechanism of brittle materials, such as glass, rocks and rock-like materials. Therefore, the mechanism of crack-branching plays an important role in crack growth retardation or arrest and other applications such as the construction of the fracture network in hydraulic fracturing. In early investigations, Yoffe [2] found the analytical solution for stress field near the head of rapidly propagating crack, and observed that when crack velocity larger than critical velocity, the crack would propagate in a direction normal to the maximum tensile stress. However, crack-branching critical speeds predicted by theoretical analysis are larger than experimental ones [3]. Clark and Irwin [4], Schardin [5] suggested that crack-branching should be investigated by critical K or G, rather than the variation of stress distribution caused by crack high velocity growth. Congleton and Petch [6] pointed out that crack-branching not always occur at a critical crack velocity but at a constant stress intensity factor. Doll [7] proved that branching depends on not only the critical velocity of crack but also the critical strain energy release rate. Besides above favorable conditions of crack-branching, Murphy et al. [8] investigated crack-branching in PMMA and reported that the crack occur successful branching in SENT specimens with shorter notches and larger thickness. In summary, the factors associated with crack-branching involving crack velocity, stress intensity factor, and energy release rate, however, the physical mechanism of crack-branching remains ambiguous to some extent. 2
Fracture criterion is the important content of fracture mechanics, which can be used to determine the conditions of quasi-brittle material fracture under static or dynamic loading and predict the trends in mixed-Mode-I/II crack branching angle, kinking angle and the crack propagation path. There are some theories so far for determining the direction of crack initiation and fracture toughness for mixed Mode-I/II crack problems [9]. The maximum tangential stress (MTS) criterion [10] (Erdogan and Sih), the minimum strain energy density (SED) criterion [11] (Sih) and the maximum energy release rate (G) criterion [12] (Hussain et al), including their modified forms based on T-stress [13]-[17], are some of the well-known mixed mode fracture criteria. In 1989, He and Hutchinson developed a crack kink criterion based on the ratio of energy release rates [18]. Subsequently, thorough discussions have been made by Hutchinson and Suo [19]. Additionally, there are some other methods such as [18] can be used also to predict the crack kinking angle. However, all quasi-static fracture criteria mentioned above work well only for predicting a new born crack initiation from a crack tip, which cannot be used for analysis of the multiple cracks initiation from a crack tip, such as the case of crack-branching under quasi-static loading. In present article, based on the fracture modelling for multiple cracks initiation, Mode-I crack tri-branching, symmetric-branching, side-branching, kinking and the instability problem are investigated in detail. The underlying and possible crack-branching angles and associated critical energy release rates are found. The concept and definition of the underlying fracture configurations, concerned K-based fracture toughness are proposed. Many underlying fracture behaviors are theoretically revealed based on the present energy-based modelling, which will be helpful for better understanding on some typical experimental results under static or dynamic loading. 2. Modelling of multiple-cracks initiation and energy release rate From the geometrical point of view, multiple cracks initiation from a boundary with singular stress, regardless whether it occurs at the crack tip or general boundary, can be understood and regarded as the multiple boundaries shifting in different directions. Then a geometrical model for multiple fractures initiation in m directions from a boundary s with s s 1 s 2 s m 0 of the special case of
is proposed, and Fig. 1 is the simplified schematic description
m 3 , i.e., three fractures initiation in 3 different directions from boundary
s s1 s2 s3 0 . The energy release rate for multiple cracks initiation can be given by [21]-[31]
G lim
s0
lim we n ds i i sl 0 , l 1 s l m
we i ni ds
s
(1)
where w is the strain energy density, ni is the unit inward normal on boundary s, e1 cos l and e 2 sin l for the two-dimensional solids, the l is the angle between shifting direction v l
of boundary
s
l
and x
1
,
which represents the direction of the boundary shifting as shown in Fig. 1 and 8. From the conservation law [21][23], Eq. (1) can be written as m
G
J s | 1
l 1
l
sl 0
cos l J 2 sl |sl 0 sin l ,
(2)
where
3
J j sl |s 0 lim wn j ds l
sl 0
sl
wn
sin l
j
Ti ui , j ds , j=1, 2,
(3)
here, s in l , (l=1, 2…m), is an integration path within the area closed by the solids boundary, and s l s in l , (l=1, 2…m), forms a closed path. The J j sl |s 0 in Eq. (3) denotes the energy release rate of the boundary sl shifting l
in x j -direction or the driving force for crack initiation from the boundary sl in xj-direction when the limits taken exist.
(a) Multiple-notches model for multiple boundaries si shifting.
(b) Model for multiple cracks initiation.
Fig. 1. Model of multiple cracks initiation from a boundary with singular stress (in case of m=3) when s 0 for two-dimensional solids. The asymptotic singular stress fields exist not only in crack tip region, but also in the edge of the Mode-I indentation [34]-[35]. Above analysis defines actually a universal modelling used to analyse such as crack tip fracture or the boundary fracture of Mode-I indentation. It is more interesting and powerful to formulize the model of multiple cracks initiation. It should be pointed out that the boundary shift or fracture must be inward to the solids, i.e., J j s l |s 0 must be positive along the direction of boundary shift or fracture, which means the energy releases l
otherwise means the energy absorbs [24][36]. The boundary fracture inducing energy absorbs is impossible for an actual state of the solids under loading. Eq. (2) must include all the contributions of the positive energy release for each sin l , (l=1, 2…m), shifting.
4
x2 2
r n
θ 1
sl
3
ro
4
θo
x1
o o
a
Crack tip
Fig. 2. The local boundary of the crack tip and ro→0+. 3. Ji-integral for mixed Mode-I/II crack Mode-I crack is the special case of mixed Mode-I/II crack when KI KII . It is helpful to further investigate the underlying fracture behaviours of Mode-I crack from the standpoint of a crack under Mode-I/II loading. 3.1 Basic characteristics on the J i -integral around mixed-Mode-I/II crack tip An enlarged and U-turn-shaped crack tip with the limit ro 0 and a circular arc path s 12 are shown in Fig. 2.
sin l s12 s23 s41
Let sl s34 in Eq. (3), a portion of the crack tip boundary, and
within the K-dominant
region. From Eq. (3), the energy-based driving forces in x j -directions can be given by J j sl | s 0 lim wn j ds l
sl 0
sl
wn
j
r ro
Ti ui , j ds
s12 s 23 s 41
wn
j
Ti ui , j ds
(4)
s12
For any integral path s l = {s ro , θ ∈ [θ 0 , θ ]} as shown in Fig. 2, substituting the stress and displacement of mixed-Mode-I/II crack within the K-dominant region into Eq. 4, the Ji -integral over the path sl can be found as J1 sl
K 1 cos 2 K 1 cos 2 2 K K
1 2 2E
(5)
sin 2 K II2 sin 2 2 K I K II 1 cos 2 d .
(6)
2 I
0
2 II
I
II
sin 2 d
and
J 2 sl
K
KI , K II
1 2 2E By introducing 2
n arctan
0
2 I
(7)
where n is a normalized phase angle and also a mixity parameter varying from 0.0 (for pure Mode-II) to 1.0 (for pure Mode-I), which depicts the relative strengths of K I and KII similar to the parameter M e proposed by Shih in [37], then the Eqs. (5) and (6) can be rewritten as J 1 sl
and J 2 sl
1 μ K 2
2 I
E
1 K 2
K II2
2
0
2 I
K II2
cos ( 2
n
)d
(8)
cos sin( n )d . E where, E is Young’s module and is Poisson’s ratio.
(9)
0
5
1.0
Inte gra
Mode-II φn=0.0 0.1 0.2 0.3
0.8
nd of J1π
0.6
0.4 0.5 0.6
E/( 1-μ 2
0.4
0.7 0.8 0.9 φn=1.0 Mode-I
)(K 2
I
+
2 0.2
KII )
0.0
-180.0
-150.0
-120.0
-90.0
-60.0
-30.0
0.0
30.0
60.0
Polar coordinate θ deg.
90.0
120.0
150.0
180.0
Fig. 3 Distribution of the integrand of normalized J1 -integral along the circle path. It should be pointed out that the process of boundary shifting or cracking must accompany the energy releasing, i.e., the boundary of fracture must be inward to the solids. The integrand of the normalized J 1 -integral, i.e., cos2 - n / 2 0 on the interval of - , indicates that J 1 -integral is always positive as shown in Fig. 3,
which means the energy releases when any part of boundary sl around crack tip shift in x1 -direction, that is to say the J 1 -integral always has a substantial contribution to the energy release rate for multiple-cracks initiation from a crack tip. However, unlike the J 1 -integral over any part of the crack tip boundary, the integrand of the normalized J 2 -integral shows a different and significant behaviour as shown Fig. 4. 0.6
Int egr an d of J2π E/( 1-μ 2 )( 2 KI +KII 2
)
Mode-I φn=1.0
0.3
Mode-II φn=0.0
0.9
0.1
0.8
0.2 0.3 0.4 0.5
0.7 0.6 0.5
0.0
-0.3
-0.6
-0.9
-1.2 -180.0
-150.0
-120.0
-90.0
-60.0
-30.0
0.0
30.0
60.0
90.0
120.0
Polar coordinate θ deg.
Fig. 4 Distribution of the integrand of normalized J2 -integral along the circle path. 6
150.0
180.0
Table 1. Normalized Ji -integrals over the paths s1 , s2 , s3 and s4 . s1 s ro , ∈ 0, 2
{
}
s3 = s ro , θ ∈ [- π , - π 2]
s2 s ro , ∈ 2 ,
s4 s ro , ∈ - 2 ,0
2J1 si E /(1 2 ) K I2
π/2
π/2
2J 2 si E /(1 2 ) K I2
-1
1
3.2 J 2 -integral around pure Mode-I crack tip When n 1.0 , which defined actually a special case for pure Mode-I deformation as shown in Fig. 4, the integrand of the normalized J 2 -integral cos sin 0 on the intervals of 0 2 and - - 2 ;
cos sin 0 on the intervals of 2
and - 2 0 . These characters determine that the whole
boundary of a crack tip should be divided into four parts, i.e., s1 s ro , ∈ 0, 2 , s2 s ro , ∈ 2 , , s3 s ro , ∈ , 2 and s4 s ro , ∈ 2 ,0 as shown in Fig. 5(a). The normalized Ji -integrals over the
paths s1，s2，s3 and s 4 are given in Table 1 [24], which imply different physical meanings. For the upper half crack tip boundaries s1 and s2 , J 2 s1 0 and J 2 s2 0 indicate theoretically that if the boundary s1 moves in x2-direction, the process of boundary cracking would accompany energy absorbing or increase strain energy, which is actually impossible to occur naturally; if the s2 moves in the x2-direction, the cracked solids will lead to release strain energy. So, the J 2 s2 has a substantial contribution to the energy release rate for multiple-cracks initiation from a Mode-I crack tip, and the contribution of J 2 s1 for Mode-I crack should be excluded in the analysis of the crack tip fracture. Similarly, for the lower half crack tip boundaries s3 and s4 , when s3 moves in the opposite x2-direction, i.e., fracture is inward to the solids, the energy release rate - J 2 s3 0 , which means that the fracture of boundary s3 inward to the solids is possible; when s4 moves in the opposite x2-direction, - J 2 s4 0 implies that the fracture will lead to energy-absorbing and is not going to happen. Then the underlying
moving trends of boundaries s1 , s2 , s3 and s4 can be schematically shown by the red arrows in Fig. 5(a). For mixed Mode-I/II crack, Eq. (9) indicates that the four θ-axis intersection points can be found from integrand cos sin( - n ) 0 for a given n , i.e.,
，2
， 3，4 k n ，k=0, -1.
(10)
These intersection points would divide the lower crack surface interval [-π, 0] or upper crack surface interval [0, π] into three subintervals as shown in Fig. 5(b). Generally, over the three subintervals, the J 2 -integrals are either positive or negative as indicated in Fig. 4, which can be used to identify how many underlying cracks should initiate from a mixed Mode-I/II crack tip and decides also the instability and trends of crack tip fracture. The potential moving trends of all parts of crack tip boundary are shown by the red arrows in Fig. 5(b).
7
x2
s2
π
x2
s2
s5
s1
πφn
x1
x1
s4
s4
s3
s1
s3
s6
(a) Pure Mode-I loading (φn=1.0).
(b) Mixed Mode-I/II loading.
Fig. 5. The interval partitioning around crack tip and the underlying moving trends. The red arrows denote the potential moving trends of local boundaries. In fact, it is very difficult to maintain an accurate experimental condition of pure Mode-I deformation. The Mode-I fracture in vast majority of experiments is actually a special mixed case of dominant Mode-I and weak Mode-II loading, which means a slight perturbation from the weak Mode-II loading to Mode-I loading [33, 34]. The relationship of n - 1.0 → ε (ε is a small quantity), which may stand for a weak Mode-II loading disturbance to Mode-I loading during experimental process, depicts actually the Mode-mixity of a dominant Mode-I loading and weak Mode-II loading. When n 1 ε , the subinterval divisions can be schematically shown in Fig. 6.
s5
s2
x2
x2
s2
s1
s1 s5 πε
x1
-πε s6 s3
x1
s4
s4
s6
s3
(a) φn→1.0-ε. (b) φn→1.0+ε, the symmetrical case of φn→1.0-ε. Fig. 6. The interval divisions around crack tip and the potential moving trends. The red arrows denote the potential moving trends of local boundaries. 4. Multiple-cracks initiation from a Mode-I crack tip As discussed in section 3, the diversity of potential moving or fracture trends for all local boundaries around a crack tip constitutes multiple complicated fracture configurations, which is interesting and quite helpful for understanding the complicated fracture behaviours in engineering practice. Actually, the crack extension, crack 8
kinking and branching can be considered as that the fractures initiation from one or some parts of the crack tip boundary are triggered. For the pure Mode-I crack, there are three typical crack multiple-branching fracture configurations, i.e., the crack tri-branching ( -shaped fracture shown in Fig. 7(a)), crack symmetric-branching ( -shaped symmetrical fracture shown in Fig. 7(b)) and crack side-branching ( -shaped fracture shown in Fig. 7(c)), which had been investigated in detail in [24]. This section will further investigate the Mode-I crack tri-branching, symmetric-branching and side-branching with the possible effect of weak Mode-II loading.
(a) Crack tri-branching.
(b) Crack symmetric-branching.
(c) Crack side-branching.
Fig. 7. Illustrations for enlarged multiple cracks initiation from a pure Mode-I crack tip.
Fig. 8. Enlarged geometrical modelling for three cracks initiation from a pure Mode-I crack tip. 4.1. Energy-based driving force and fracture angles for pure Mode-I crack tri-branching As pointed out in section 3.2, the possible fractures initiation from a Mode-I crack tip may be formed when boundaries s2 and s3 move inward to solids as two wing cracks and s1 and s4 move together in x1-direction as a form of crack extension as shown in Figs. 5(a), 7(a) and 8 in order to achieve maximum energy release rate. In this case, by using the symmetry of the tri-fractures configuration under pure Mode-I loading the energy release rate induced by crack tri-branching as shown in Fig. 8 can be found as [24] G tri-b 2 J 1 s1 cos 1 J 1 s 2 cos 2 J 2 s 2 sin 2
(11)
from Eq. (2). Substitute the results in Table 1 into Eq. (11), the following results can be found. G tri -b
1 - K 2
2E
2 I
cos 1
1 4 / 2 cos 2 - ,
(12)
where 9
tan
-1
2 / 32 .48
deg.
(13)
The energy release rate is now maximised with respect to i by setting
G tri b i
0 , i=1 and 2, i.e.,
sin 1 c 0 and sin 2c - 0 , which gives the critical crack initiation angles
1c 4 c 0 deg.
(14)
2c 3c 32.48 deg.
(15)
and
where 1c and 2c are the theoretical expected and possible crack initiation angles at which G tri b exhibits a maximum value, and explicitly this is given by tri b Gmax
1 K 2
2 I
E
Ftri b ,
(16)
where Ftri b
1 1 4 / 2 1.0927 , 2
(17)
which is the energy-based driving force for Mode-I crack tri-branching as shown in Figs. 7(a) and 8. It indicates that when tri b G max G Ctri b ,
(18)
the Mode-I crack will fracture theoretically in the form of crack tri-branching according to the Griffith’s criterion [38]. 4.2. On K-based fracture criterion for Mode-I crack tri-branching For a homogenous and isotropic brittle solid, Griffith’s criterion [38] states that a crack will propagate when the critical value Gc of the solid is equal to the driving force or applied Gmax [38][39] Gmax Gc .
(19)
For Mode-I crack extension, when Gmax Gc and Gmax Gc , we have Gc
1 K 2
E
2 IC
.
(20)
When fracture configuration of Mode-I crack tri-branching is triggered, as discussed in section 4.1, the energy-based driving forces for a wing crack initiation from boundary s2 or s3 can be found as s wing s wing Gmax Gmax 2
3
1 - K 2
E
2 I
1 4 / 2 . 4
(21)
The energy-based driving forces for cracks initiation from s1 and s4 in a form of crack extension can be expressed by s1 s 4 ext Gmax
1 - K 2
E
2 I
1 . 2
(22)
From Eqs. (19)-(22), it is not difficult to find the K-based fracture toughness s 2 wing K IC 1.84 K IC
(23) 10
and s s ext K IC 1.41K IC 1
(24)
4
for a wing crack initiation and main crack extension respectively. Then from Eqs. (16) and (18), the critical energy release rate can be found as
1- K
tri-b 2 IC
2
trib C
G
Ftri-b ,
E
(25)
where
s wing s s ext tri -b K IC max K IC , K IC 1.84 K IC . 2
1
4
(26)
s wing s s ext tri b The relationship between Gmax , Gmax and Gmax under pure Mode-I loading can be schematically shown 1
2
4
in Fig. 9. The K-based criterion for Mode-I crack tri-branching can be written as
Normalized energy-based drive forces GmaxE/(1-μ2)KIC2
trib K I K IC .
(27)
2 GCtrib E / 1 - 2 KIC
trib 2 Gmax E / 1 - 2 K IC
s s ext Gmax E / - K IC
Energy release rate for creating a new surface
s wing Gmax E / - K IC
2 G C E / 1 - 2 K IC
KI KIC 1.41
K I K IC 1.84
Stress intensity factor K I K IC s wing s s Fig. 9. The relationship between Gmax , Gmax 2
1
4 ext
tri b and Gmax under pure Mode-I loading.
4.3 Effect of weak Mode-II loading on Mode-I crack tri-branching Because of the underlying shifting trend inward to solids, the attempted shifting of boundary s6 as shown in Fig.6 (a) should stop s4 from moving in x1-direction. Similarly, the attempted shifting of the boundary s 5 inward to solids, as shown in Fig.6 (b), should stop also boundary s 1 from moving in x1-direction. However, the energy release rate for the boundary s 6 or s 5 attempted shifting is a small value, from which a wing crack initiation is difficult. So, another form of Mode-I crack tri-branching should be formed from boundaries s1 moving in x1-direction as a form of crack extension and s2 and s3 moving inward to solids as two wing cracks as shown in Fig.10. In this case, the concerned energy release rate for crack tri-branching can be written as G tri b J 1 s1 cos 1 2 J 1 s 2 cos 2 J 2 s 2 sin 2 .
Then it is not difficult to get
11
(28)
tri b Gmax
1 K 2
E
2 I
2 1 1 4 / 4 2
(29)
and the theoretical expected crack initiation angles
1c 0 deg. and 2c 3c 32.48 deg.
(30)
The energy-based driving force acting on s1 , which will induce to boundary s1 to fracture in form of crack extension, can be given by s1 ext Gmax J1 s1
1 - K 2
2 I
4E
.
(31)
From Eqs. (19), (20) and (31), the K-based fracture toughness for s1 can be found s ext K IC 2 K IC .
(32)
1
Similar to the discussion in section 4.2, the critical energy release rate for the crack tri-branching as shown in Fig.10 can be found as
1 - K
tri -b 2 IC
2
tri b C
G
E
2 1 1 4 , 4 2
(33)
where the K-based fracture toughness for Mode-I crack tri-branching becomes
s -wing s ext s ext tri-b K IC , K IC max K IC K IC 2K IC 2
1
(34)
1
tri b 2 K IC , the Mode-I crack tri-branching may be Based on K-based fracture criterion, when applied K I K IC
triggered. s 2 wing s1 ext The relationship between Gmax , Gmax
trib and Gmax for Mode-I loading with a possible weak disturbance
of Mode-II loading can be schematically shown in Fig. 11.
x2 s2 Original crack tip α2c
ro
s4 α3c
s1
x1
s6
s3
Fig. 10. Effect of weak Mode-II loading on Mode-I crack tri-branching.
12
Normalized energy-based drive forces GmaxE/(1-μ2)KIC2
GCtrib E / 1- 2
tri b 2 Gmax E / 1 - 2 K IC
2 KIC
s wing Gmax E / - K IC
Energy release rate for creating a new surface
2 GC E / 1 - 2 K IC
s ext Gmax E / - K IC
KI KIC 2 K I K IC 1.84
Stress intensity factor K I K IC s wing s ext tri b Fig. 11. The relationship between Gmax , Gmax and Gmax under Mode-I loading with a possible weak disturbance of Mode-II loading. 4.4 Effect of weak Mode-II loading on Mode-I crack side-branching
Three forms of underlying crack side-branching are shown in Fig.12. The first one is the idealized crack side-branching based on that all the boundary fracture accompanied by energy release are triggered. The other two forms are composed of a wing crack initiated from s2 or s3 and crack extension from local boundaries around crack tip, which is effected by weak Mode-II loading. Similar to the discussions in sections 4.1-4.3, for the crack side-branching configuration as shown in Fig.12(a), the concerned energy-based driving force can be written as side b Gmax
1 K 2
2 I
E
3 1 4 / 2 4
(35)
and the K-based fracture toughness for crack extension becomes s s s ext K IC 1
3
4
2 3
K IC 1.15K IC .
(36)
In this case, the K-based fracture toughness for Mode-I crack side-branching can be found as
s2 -wing s1 s3 s4 ext s2 wing side-b K IC max K IC , K IC K IC 1.84K IC
(37)
For the crack side-branching configuration as shown in Fig.12(b), the concerned energy-based driving force can be found as side b Gmax
1 K 2
E
2 I
1 1 4 / 2 4
(38)
and the K-based fracture toughness for crack side-branching
s 2 - wing s1 ext s1 ext side - b K IC max K IC , K IC K IC 2 K IC .
(39)
For the crack side-branching configuration as shown in Fig.12(c), the concerned energy-based driving force and K-based fracture toughness can be found respectively as
13
side b Gmax
1 K 2
E
2 I
2 2 1 4 / 4
(40)
and
s -wing s s ext s wing side-b , K IC K IC max K IC K IC 1.84K IC 2
1
2
(41)
2
The Eqs. (37), (39) and (41) show that the upper limit of K-based fracture toughness for Mode-I crack side-branching is 2KIC and the lower limit 1.84 KIC. 4.5 On Mode-I crack kinking and symmetric-branching As above analysis, when 1.84KIC < applied KI <2KIC and affected by a weak Mode-II loading, if a wing crack initiating from s2 or s3 is triggered, the crack kinking should be formed. If two wing cracks initiating respectively from s2 and s3 are triggered, the crack branching should occur as shown in Fig.13.
(a) Idealized crack side-branching under pure Mode-I loading.
(b) Effect of weak Mode-II loading on Mode-I crack side-branching.
14
(c) Another form of Mode-I crack side-branching effected by weak Mode-II loading. Fig.12. Possible Mode-I crack side-branching.
Fig.13. Crack symmetric-branching. Nor mal ize d str ess inte nsit y fact or KI/ KIC
2.5
2.0
Upper limit
tri -b K IC K IC 2
tri - b K IC K IC 1.84
1.5
Lower limit
1.41
1.0
0.5
0.0 0.0
t0 Time t Fig. 14. Impulse pure Mode-I loading.
5. Triggering for Model-I crack multiple-branching 15
As discussed in above sections, the Mode-I crack multiple-branching exists theoretically based on the energy-based fracture analysis. Taking the crack tri-branching for example, because of the quantitative difference s wing s ext between Gmax and Gmax , only one crack will initiate from crack tip as a form of crack kinking or main crack 2
i
extension under quasi-static Mode-I loading. Therefore, the Mode-I crack multiple-branching is rare in the quasi-static fracture tests. However, if increasing loading rate or even impulse Mode-I loading, as shown in Fig. s wing s ext 14, the Gmax and Gmax driving the boundaries s2, s3 and si to fracture respectively may reach their critical 2
i
values simultaneously or in a enough short time period. The possible Mode-I crack tri-branching, symmetric-branching or side-branching may be triggered. 6. Typical experimental fracture behaviours on Mode-I crack multiple-branching 6.1. Crack multiple-branching behaviours for GPPS specimens Mode-I crack multiple-branching behaviours of cracked specimens made of general-purpose Polystyrenes (GPPS) are investigated in this section. The GPPS is homogenous, isotropic, quasi-brittle at room temperature and exhibits linear elastic behaviour. Additionally, its optical transparency and being glassy facilitate direct observation of the fracture phenomenon during experiments. Overall geometry dimensions of the prepared CTS specimens and CTS fixture are shown in Fig. 15. The relative advantage of CTS specimen geometry is that the sufficient size to enable fully main and branched cracks to develop entirely. CTS specimens can be used also for mixed Mode I/II fracture experiments by adjusting the position of fixture. Fig. 15(c) shows the loading condition of the CTS specimens used in experiments. A CNC engraving machine was utilized to cut the overall specimens from GPPS sheet with 7.5 mm thickness and all specimens were processed from the same sheet to eliminate batch variations. A diamond wire saw with thickness of 0.42 mm was used for introducing a narrow notch with the initial depth of slightly less than 45 mm in the specimens. Then, a natural sharp crack tip was generated by sawing a fresh razor blade carefully to make the final length of each crack 45 mm. For manufactured specimens, the overall dimensions and details such as length (L) and width (W) and crack length (a) were considered the same for the sake of comparing the test results. The image of the sharpened notch tip magnified under an optical microscope is shown in Fig. 16. The set of fixture and specimen was mounted on a uniaxial tension compression INSTRON-8872 servo hydraulic test machine. In the fracture toughness tests of similar quasi-brittle materials, the loading rates are in the rage of 0.08 to 0.4 mm/min [40]-[46], especially the loading rate of 0.3 mm/min had been frequently used [40] [45]. The fracture toughness tests in present research were carried out under displacement control at loading rate of 0.3 mm/min. The low loading rate was employed to guarantee the establishment of quasi-static loading conditions and to provide sufficient time for transferring the applied load to the tested specimen uniformly. According to the ASTM-D5045 standard [47], fracture toughness tests should be replicated at least three times. It is mentioning that to enhance preciseness of the fracture test results, four specimens were provided for determinations of fracture toughness. Subsequently, increasing loading rate to capture the phenomenon of crack multiple-branching. The test data were recorded by a computer and all the experiments were carried out at room temperature.
16
For brittle materials, there is no obvious subcritical crack growth, so the measured peak fracture load can be considered as the critical fracture load, which can be used to calculate crack initiation fracture toughness. To evaluate the stress intensity factor for CTS specimens, the following expression can be used [48]. a 0.26 2.65( ) F cos wa K a , a a a 2 WB (1 ) 1 0.55( ) 0.08( ) w wa wa
(42)
where F is the applied force, w is the width of the specimen, B is the thickness of the specimen, a is the crack length and α is the angle of loading direction with respect to the crack plane. In order to ensure the condition for plane strain, ASTM-D5045 standard [47] requires the specimen thickness to exceed 2.5(KIC/σy)2, where KIC is the fracture toughness and σy is the yield stress. The test data obtained from the fracture toughness experiments have been listed in Tables 2 and 3. According to ASTM-D5045 standard, if yielding does not occur and brittle fracture is observed, the stress at fracture shall be used as yield strength in criteria to give a conservative size value. As shown in Table 2, the ultimate tensile strength at room temperature is 39.8 MPa, and take the maximum value 1.26 MPam1/2 from Table 3, i.e., plain strain conditions should certainly be achieved for thicknesses exceeding 2.51 mm. And based on it, the specimen thickness in present tests satisfied plane-strain condition.
17
(a)
(b)
(c) Fig. 15. Geometry and loading condition of CTS fixture and specimen (dimensions in mm). Table 2. Some mechanical properties of the tested GPPS. Material Property
unit
value
Elastic module
GPa
3
Poisson’s ratio
0.34
Density
g/cm3
1.044
Ultimate bending strength
MPa
59.0
Ultimate tensile strength
MPa
39.8
Tensile strain at break
1.4% 18
Fig. 16. The sharped notch tip of a test specimen shown under an optical microscope.
Fig. 17. Experimental setup for the testing of GPPS material using the CTS specimen. Table 3. Fracture toughness KIC for GPPS specimens Replicate No.
a(mm)
Fcr (N)
KIC (Mpa·m1/2)
1 2 3 4 5 average
45 45 45 45 45 45
729.46 837.72 695.75 700.27 844.48 761.54
1.08 1.25 1.03 1.04 1.26 1.13
19
Specimen a (a)
Specimen b (b)
Specimen c (c)
20
Specimen d (d)
Specimen e (e) Fig. 18. Macroscopic and microcosmic branching patterns on fracture surfaces in specimens: (a) to (c) Crack tri-branching. (d) Crack symmetric-branching. (e) Crack side-branching.
Fig. 19. A side view of fracture surface tested specimen (b). All the test samples fractured suddenly from the crack tip at a critical peak load, which confirms brittleness of the tested material. Twenty specimens were carried out for Mode-I crack multiple-branching tests. There is a 21
statistical variation in the branching patterns, however, some typical general observations can be picked out. Distinct crack tri-branching was observed in three out of twenty specimens, as shown in Figs. 18 (a)-(c). Meanwhile, crack quasi-symmetric branching and side-branching were observed and shown in Figs. 18 (d) and (e) respectively in the above specimens. For crack tri-branching configuration, it is worth mentioning that the two wing cracks in crack tri-branching are difficult to fully develop as shown in Fig. 18 (a)-(c) because the stress intensity factors for the new-born wing cracks should be much smaller than that for extended main crack.
1/2
stress intensity factor (MPa m )
3.0
2.4
2.26
1.8
KI for specimen a KI for specimen b KI for specimen c KI for specimen d KI for specimen e
1.2
0.6
0.0 0.0
0.1
0.2 0.3 Time(s)
0.4
0.5
Fig. 20. SIFs - time curves obtained from crack multiple branching experimental tests. From the Eq. (34), the fracture toughness for crack tri-branching (the upper limit) predicted by present method tri -b tri -b is K IC =2KIC, i.e., K IC =2×1.13=2.26 MPa m1/2. According to the theoretical predictions in section 4, when
the applied KI
tri -b ≥ K IC
the underlying Mode-I crack tri-branching, side-branching or quasi-symmetrical
branching should be triggered. The corresponding SIFs-time curves from experiments as shown in Fig. 20, and the tri -b theoretical value of K IC was highlighted with a red dotted line. As shown in Fig. 20, the K I for specimens a,
b, c, d and e exceeded the red dotted line, which indicates that the experimental results agree with the theoretical prediction based on present modelling. 6.2. Crack multiple-branching behaviours for PMMA specimens An investigation of the branching characteristics of small PMMA single edge notched tensile (SENT) specimens had been conducted by Murphy et al [8]. Some typical images of Mode-I crack branching are given in Fig. 21. Similar to the GPPS, stable crack branching behaviors have been observed for PMMA, which may be predicted and better understand based on the present modelling.
22
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 21. Branching patterns in specimens containing the 0.1 mm notches. Successful branching occurred in all of the 8 mm thick specimens tested. The size of the images is 25 by 20 mm in each case [8]. 7. Conclusions By further exploring the Ji-integral over the local boundary of a Mode-I crack tip, an energy-based fracture modelling for multiple fractures initiation from a crack tip has been investigated and formulized. Unlike the traditional applications of the J-integral around a crack-tip, a new partial integral path and the associated calculation method on the energy release rate for multiple cracks initiation had been defined. The present article further validates theoretically the underlying fracture trends for the Mode-I crack multiple-branching with a possible effect of weak Mode-II loading. Several possible fracture configurations on multiple-cracks initiation from a crack tip are proposed for a quasi-Mode-I crack, such as crack tri-branching, symmetric-branching and side-branching based on the present energy-based modelling. The corresponding energy-based driving forces, K-based fracture toughness 23
and fracture criteria for above crack configurations have been formalized and found. And according to the prediction based on present method, tri-branching is the upper limit on the number of possible multiple crack initiation for a Mode-I crack with a weak disturbance of Mode-II loading. The fracture mechanism for a Mode-I crack under quasi-static and dynamic loading can be better explained based on the present theoretical modelling, which is of great significance to understand the complex fracture phenomena of brittle materials and engineering applications, such as hydraulic fracturing with multiple cracks and crack growth retardation or arrest. The fracture toughness predicted by the present modelling for Mode-I crack tri-branching, side-branching and symmetrical-branching with a possible effect of weak Mode-II loading agree well with the experimental results. The proposed method provides also a potential way to predict the crack multiple-branching behaviours under the mixed Mode-I/II loading. The fracture nature of the experimental results can be easily recognized by using present analysis. By comparing the experimental fracture configurations and the theoretical one, which form of the fracture, such as crack tri-branching, side-branching and symmetric-branching, and the mechanism behind experimental phenomena can be identified. Then the corresponding energy-based driving force, fracture toughness and K-based criterion can be clearly known. Additionally, the present analysis implies also that the underlying fracture trends of multiple cracks initiation under static condition can be triggered by using impulse loading, which may be one of the effective methods to induce the Mode-I crack multiple-branching. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.: 50771052, 50971068 and 11272141) and Natural Science Foundation of Liaoning (Grant Nos.: LS2010100 and 20102129). References [1]. Meggiolaro MA, Miranda ACO. Stress intensity factor equations for branched crack growth. Engng Fract Mech 2005; 72: 2647-71. [2]. Yoffe EH. The moving Griffith crack. Philos Mag 1951; 42: 739-50. [3]. Adda-Bedia M. Brittle fracture dynamics with arbitrary paths Ⅲ. The branching instability under general loading. J Mech Phys Solids 2005; 53: 227-48. [4]. Clark ABJ, Irwin GR. Crack-propagation behaviours. Exp Mech 1966; 6: 321-30． [5]. Schardin H. Velocity effects in fracture. In Fracture : proceeding of an international conference on the atomic mechanism of fracture help in Swampscott, Massachusetts, H. Schardin, ed., 1959, pp. 297–330. [6]. Congleton J, Petch NJ. Crack branching. Philos Mag 1967; 16: 749-60. [7]. Doll W. Investigation of the crack branching energy. Int J Fract 1975; 11: 184-86. [8]. Murphy N, Ali M, Ivankovic A. Dynamic crack bifurcation in PMMA. Engng Fract Mech 2006; 73: 2569-87. [9]. Aliha MRM, Ayatollahi MR. Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int J Solids Struct 2012; 49: 1877-83. [10].Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Engng Trans ASME 1963; 85: 519-25. [11].Sih GC. Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 1974; 10: 305-21. 24
[12].Hussain MA, Pu SL, Underwood J. Strain energy release rate for a crack under combined mode I and Mode II. Fracture Analysis, 1974. ASTM STP 560. American Society for Testing and Materials, Philadelphia, pp. 2–28. [13].Akbardoost J, Ayatollahi MR, Aliha MRM, Pavier MJ, Smith DJ. Size-dependent fracture behavior of Guiting limestone under mixed mode loading. Int J Rock Mech Min Sci 2014; 71: 369-80. [14].Aliha MRM, Hosseinpour GhR, Ayatollahi MR. Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials. Rock Mech Rock Engng 2013; 46: 1023–34. [15].Mirsayar MM, Razmi A, Aliha MRM, Berto F. EMTSN criterion for evaluating mixed mode I/II crack propagation in rock materials. Engng Fract Mech 2018; 190: 186-97. [16].Aliha MRM, Berto F, Mousavi A, Razavi SMJ. On the applicability of ASED criterion for predicting mixed mode I+II fracture toughness results of a rock material. Theor Appl Fract Mech 2017; 92: 198-204. [17].Razavi SMJ, Aliha MRM, Berto F. Application of an average strain energy density criterion to obtain the mixed mode fracture load of granite rock tested with the cracked asymmetric four-point bend specimens. Theor Appl Fract Mech 2018; 97: 419-25. [18].He M, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials. Int J Solids Struct 1989; 25: 1053-1067. [19].Hutchinson JW, Suo Z. Mixed mode cracking in layered materials. Adv Appl Mech 1992; 29: 63-191. [20].Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fract 1980; 16: 155-69. [21].Eshelby JD. The Force on an Elastic Singularity. Phil Trans Roy Soc London Ser A 1951; 244: 87-112. [22].Sih GC. Dynamic Aspects of Crack Aspects of Crack Propagation. In Sih G C ed. Inelastic Behavior of Solids. New York, Mc-Graw-Hill Book, Co.; 1969: 607-39. [23].Budiansky B and Rice JR. Conservation Laws and Energy-Release Rates. ASME J Appl Mech 1973; 40: 201-3. [24].Xie YJ, Li J, Hu XZ, Wang XH, Cai YM, Wang W. Modelling of multiple crack-branching from Mode-I crack-tip in isotropic solids. Eng Fract Mech 2013; 109: 105-16. [25].Xie YJ, Wang XH and Wang YY. Stress intensity factors for cracked homogeneous and composite multi-channel beams. Int J Solids Struct 2007; 44: 4830-44. [26].Xie YJ, Xu H, and Li PN. Crack Mouth Widening Energy-Release Rate and Its Applications. Theor Appl Fract Mech 1998; 29: 195-203. [27].Xie YJ. A. Theory on Cracked Pipe. Int J Pres Ves Pip 1998; 75: 865-69. [28].Xie YJ, Zhang X, Wang XH. An exact method on penny-shaped cracked homogeneous and composite cylinders. Int J Solids Struct 2001; 38: 6953-63. [29].Xie YJ. An analytical method on circumferential periodic cracked pipes and shells. Int J Solids Struct 2000; 37: 5189-201. [30].Giannakopoulos AE, Lindley TC, Suresh S. Aspects of connections and life-prediction methodology for fretting-fatigue. Acta Mater 1998; 46: 2955-68. [31].Giannakopoulos AE, Venkatesh TA, Lindley TC, Suresh S. The role of adhesion in contact fatigue. Acta Mater 1999; 47: 4653-64. [32].Hutchinson JW. Singular behavior at the end of a tensile crack in a hardening material. J Mech Phys Solids 25
1968; 16: 13. [33].Sundaram BM. Why do growing cracks branch? An optical investigation of brittle monolithic and bilayer transparencies under stress wave loading. Doctoral dissertation, Auburn University, May 6, 2017. [34].Sundaram BM, Tippur HV. Dynamic fracture of soda-lime glass: A full-field optical investigation of crack initiation, propagation and branching. J Mech Phys Solids 2018; 120:132-53. [35].Xie YJ and Hills DA. Quasi brittle fracture beneath a flat bearing surface. Eng Fract Mech 2008; 75: 1223-30. [36].Li XH, Zheng XY, Yuan WJ, Cui XW, Xie YJ, Wang Y. Instability of cracks initiation from a mixed-mode crack tip with iso-stress intensity factors KI and KII. Theor Appl Fract Mech 2018; 96: 262-71. [37].Shih CF. Small-scale yielding analysis of mixed mode plane-strain crack problems. Fracture Analysis, ASTM STP 560, American Society for Testing and Materials, 1974, pp. 187-210. [38].Griffith AA. The Phenomena of Rupture and Flow in Solids. Phil Trans Roy Soc Lon Ser A 1921; 221: 163-98. [39].Cherepanov GP. Mechanics of Brittle Fracture, Moscow, Publish House “Nuaka,” 1974, English translation published by McGraw-Hill International Book Co., New York; 1979: 266-69. [40].Safaei S, Ayatollahi MR, Saboori B. Fracture behavior of GPPS brittle polymer under mixed mode I/III loading. Theor Appl Fract Mech 2017; 91: 103–15. [41].Zhou Jun, Wang Yang, Xia Yuanming. Mode-I fracture toughness measurement of PMMA with the Brazilian disk test. J Mater Sci 2006; 41: 5778-81. [42].Loya JA, Villa EI, Fernández-Sáez J. Crack-front propagation during three point-bending tests of polymethyl-methacrylate beams. Polym Test 2010; 29: 113-18. [43].Kailash CJ, Hareesh VT. Quasi-static and dynamic fracture behavior of particulate polymer composites: A study of nano-vs. micro-size filler and loading-rate effects. Compos Part B-E 2012; 43: 3467–81. [44].Zhang H, Zhang Z, Friedrich K, Eger C. Property improvements of in situ epoxy nanocomposites with reduced interparticle distance at high nanosilica content. Acta Mater 2006; 54: 1833-42. [45].Weerasooriya T, Moy P, Casem D, Chen M. Fracture toughness of PMMA as a function of loading rate, in: Proceedings of the 2006 SEM Annual Conference on Experimental Mechanics, 2006, pp. 5-7. [46].Ayatollahi MR, Saboori B. A new fixture for fracture tests under mixed mode I/III loading. Eur J of Mech A/Solids 2015; 51: 67-76. [47].Annual book of standards. Standard E399-90: standard test method for plane strain fracture toughness of metallic materials. Philadelphia, PA: American Society for Testing and Materials; 1997. [48].Richard HA. Bruchvorhersagen bei überlagerter normal-und schubbean-spruchung von risen VDI Forschungsheft 631. Düsseldorf: VDI-Verlag; 1985；p. 1-60.
26
Highlights • An energy-based modelling for multiple cracks initiation from a tip is proposed. • Underlying fracture trends of cracks initiation induced by dominant Mode-I loading with a weak disturbance of
Mode-II loading is formulized. • The proposed modelling can capture nearly all of crack branching patterns observed in test.
27
S0013-7944(19)30847-1 https://doi.org/10.1016/j.engfracmech.2019.106835 EFM 106835
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
7 July 2019 15 December 2019 19 December 2019
Please cite this article as: Yuan, H., Xie, Y.J., Wang, W., Underlying fracture trends on Mode-I crack multiplebranching, Engineering Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.engfracmech.2019.106835
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
Underlying fracture trends on Mode-I crack multiple-branching H. Yuana, b, Y. J. Xie a 1, W. Wang a a Department b
of Mechanical Engineering, Liaoning Shihua University, Fushun, 113001, LN, P. R. China
College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao, 266000, SD, P. R. China Abstract: An energy-based modelling has been proposed to capture the physical process of multiple cracks
initiation from a Mode-I crack tip. The underlying fracture trends and instability on crack tri-branching, symmetric branching and side-branching induced by dominant Mode-I loading are formulized. The solutions of the energy release rate, crack initiation angles and K-based fracture criterion for multiple cracks initiation have been investigated. The upper limit on the number of possible crack initiation is given for a Mode-I crack. The discussions in present article indicate that some underlying fracture configurations for Mode-I crack can be triggered by using impulse loading. Many fracture behaviours for a Mode-I crack can be better explained and understanding based on the present theoretical modelling. The fracture toughness predicted by the present modelling and Griffith’s criterion agree well with experimental results. Keywords: Multiple-cracks initiation; Mode-I crack; Crack-branching. Nomenclature
angle of the boundary translation
c
critical wing crack initiation angle for a Mode-I crack
i
intersections of the θ-axis and integrand curve of normalized J2-integral
n
normalized phase angle
ij
stress components
Poisson’s ratio
ε
disturbance parameter of weak Mode-II loading to Mode-I loading
ei
boundary shifting tensor
E
elastic module
F, β
interim parameters
G
energy release rate for boundary cracking
GC
energy-based fracture toughness
Gmax
energy-based driving force
G tri b
energy release rate for crack triple-branching
GCtrib
energy-based fracture toughness for crack triple-branching
trib Gmax
energy-based driving force for crack triple-branching
sideb Gmax
energy-based driving force for crack side-branching
1
Corresponding authors. Y.J. Xie, E-mail address: [email protected]. 1
si wing si ext Gmax ， Gmax energy-based driving forces for wing crack and crack extension initiation from boundary si
respectively Ji
conservation integrals
K I , KII
stress intensity factors for the Mode-I and Mode-II crack
KIC
K-based Mode-I fracture toughness
tri b K ΙC
fracture toughness for Mode-I crack triple-branching
sideb K ΙC
fracture toughness for Mode-I crack side-branching
si wing s -ext KΙC , KΙCi K-based fracture toughness for wing crack and crack extension initiation from boundary si
respectively ni
the unit inward normal on boundary s
P
load
s, si
boundaries or integration paths around the crack tip
(sin)l
integral path within solids
Ti
stress vector acting on the integration path
u
i
displacement components
vi
directions of local boundary si shifting
w
strain energy density
1. Introduction Due the crack-branching behavior occurs, the original crack tip will split into two or more crack tips, and a distinctive impact of this behavior is the stress intensity factor of original crack tip decrease rapidly [1]. A good understanding of a crack branching conditions, including the multiple cracks initiation from a crack tip, is important and essential for the fundamental study of complex failure mechanism of brittle materials, such as glass, rocks and rock-like materials. Therefore, the mechanism of crack-branching plays an important role in crack growth retardation or arrest and other applications such as the construction of the fracture network in hydraulic fracturing. In early investigations, Yoffe [2] found the analytical solution for stress field near the head of rapidly propagating crack, and observed that when crack velocity larger than critical velocity, the crack would propagate in a direction normal to the maximum tensile stress. However, crack-branching critical speeds predicted by theoretical analysis are larger than experimental ones [3]. Clark and Irwin [4], Schardin [5] suggested that crack-branching should be investigated by critical K or G, rather than the variation of stress distribution caused by crack high velocity growth. Congleton and Petch [6] pointed out that crack-branching not always occur at a critical crack velocity but at a constant stress intensity factor. Doll [7] proved that branching depends on not only the critical velocity of crack but also the critical strain energy release rate. Besides above favorable conditions of crack-branching, Murphy et al. [8] investigated crack-branching in PMMA and reported that the crack occur successful branching in SENT specimens with shorter notches and larger thickness. In summary, the factors associated with crack-branching involving crack velocity, stress intensity factor, and energy release rate, however, the physical mechanism of crack-branching remains ambiguous to some extent. 2
Fracture criterion is the important content of fracture mechanics, which can be used to determine the conditions of quasi-brittle material fracture under static or dynamic loading and predict the trends in mixed-Mode-I/II crack branching angle, kinking angle and the crack propagation path. There are some theories so far for determining the direction of crack initiation and fracture toughness for mixed Mode-I/II crack problems [9]. The maximum tangential stress (MTS) criterion [10] (Erdogan and Sih), the minimum strain energy density (SED) criterion [11] (Sih) and the maximum energy release rate (G) criterion [12] (Hussain et al), including their modified forms based on T-stress [13]-[17], are some of the well-known mixed mode fracture criteria. In 1989, He and Hutchinson developed a crack kink criterion based on the ratio of energy release rates [18]. Subsequently, thorough discussions have been made by Hutchinson and Suo [19]. Additionally, there are some other methods such as [18] can be used also to predict the crack kinking angle. However, all quasi-static fracture criteria mentioned above work well only for predicting a new born crack initiation from a crack tip, which cannot be used for analysis of the multiple cracks initiation from a crack tip, such as the case of crack-branching under quasi-static loading. In present article, based on the fracture modelling for multiple cracks initiation, Mode-I crack tri-branching, symmetric-branching, side-branching, kinking and the instability problem are investigated in detail. The underlying and possible crack-branching angles and associated critical energy release rates are found. The concept and definition of the underlying fracture configurations, concerned K-based fracture toughness are proposed. Many underlying fracture behaviors are theoretically revealed based on the present energy-based modelling, which will be helpful for better understanding on some typical experimental results under static or dynamic loading. 2. Modelling of multiple-cracks initiation and energy release rate From the geometrical point of view, multiple cracks initiation from a boundary with singular stress, regardless whether it occurs at the crack tip or general boundary, can be understood and regarded as the multiple boundaries shifting in different directions. Then a geometrical model for multiple fractures initiation in m directions from a boundary s with s s 1 s 2 s m 0 of the special case of
is proposed, and Fig. 1 is the simplified schematic description
m 3 , i.e., three fractures initiation in 3 different directions from boundary
s s1 s2 s3 0 . The energy release rate for multiple cracks initiation can be given by [21]-[31]
G lim
s0
lim we n ds i i sl 0 , l 1 s l m
we i ni ds
s
(1)
where w is the strain energy density, ni is the unit inward normal on boundary s, e1 cos l and e 2 sin l for the two-dimensional solids, the l is the angle between shifting direction v l
of boundary
s
l
and x
1
,
which represents the direction of the boundary shifting as shown in Fig. 1 and 8. From the conservation law [21][23], Eq. (1) can be written as m
G
J s | 1
l 1
l
sl 0
cos l J 2 sl |sl 0 sin l ,
(2)
where
3
J j sl |s 0 lim wn j ds l
sl 0
sl
wn
sin l
j
Ti ui , j ds , j=1, 2,
(3)
here, s in l , (l=1, 2…m), is an integration path within the area closed by the solids boundary, and s l s in l , (l=1, 2…m), forms a closed path. The J j sl |s 0 in Eq. (3) denotes the energy release rate of the boundary sl shifting l
in x j -direction or the driving force for crack initiation from the boundary sl in xj-direction when the limits taken exist.
(a) Multiple-notches model for multiple boundaries si shifting.
(b) Model for multiple cracks initiation.
Fig. 1. Model of multiple cracks initiation from a boundary with singular stress (in case of m=3) when s 0 for two-dimensional solids. The asymptotic singular stress fields exist not only in crack tip region, but also in the edge of the Mode-I indentation [34]-[35]. Above analysis defines actually a universal modelling used to analyse such as crack tip fracture or the boundary fracture of Mode-I indentation. It is more interesting and powerful to formulize the model of multiple cracks initiation. It should be pointed out that the boundary shift or fracture must be inward to the solids, i.e., J j s l |s 0 must be positive along the direction of boundary shift or fracture, which means the energy releases l
otherwise means the energy absorbs [24][36]. The boundary fracture inducing energy absorbs is impossible for an actual state of the solids under loading. Eq. (2) must include all the contributions of the positive energy release for each sin l , (l=1, 2…m), shifting.
4
x2 2
r n
θ 1
sl
3
ro
4
θo
x1
o o
a
Crack tip
Fig. 2. The local boundary of the crack tip and ro→0+. 3. Ji-integral for mixed Mode-I/II crack Mode-I crack is the special case of mixed Mode-I/II crack when KI KII . It is helpful to further investigate the underlying fracture behaviours of Mode-I crack from the standpoint of a crack under Mode-I/II loading. 3.1 Basic characteristics on the J i -integral around mixed-Mode-I/II crack tip An enlarged and U-turn-shaped crack tip with the limit ro 0 and a circular arc path s 12 are shown in Fig. 2.
sin l s12 s23 s41
Let sl s34 in Eq. (3), a portion of the crack tip boundary, and
within the K-dominant
region. From Eq. (3), the energy-based driving forces in x j -directions can be given by J j sl | s 0 lim wn j ds l
sl 0
sl
wn
j
r ro
Ti ui , j ds
s12 s 23 s 41
wn
j
Ti ui , j ds
(4)
s12
For any integral path s l = {s ro , θ ∈ [θ 0 , θ ]} as shown in Fig. 2, substituting the stress and displacement of mixed-Mode-I/II crack within the K-dominant region into Eq. 4, the Ji -integral over the path sl can be found as J1 sl
K 1 cos 2 K 1 cos 2 2 K K
1 2 2E
(5)
sin 2 K II2 sin 2 2 K I K II 1 cos 2 d .
(6)
2 I
0
2 II
I
II
sin 2 d
and
J 2 sl
K
KI , K II
1 2 2E By introducing 2
n arctan
0
2 I
(7)
where n is a normalized phase angle and also a mixity parameter varying from 0.0 (for pure Mode-II) to 1.0 (for pure Mode-I), which depicts the relative strengths of K I and KII similar to the parameter M e proposed by Shih in [37], then the Eqs. (5) and (6) can be rewritten as J 1 sl
and J 2 sl
1 μ K 2
2 I
E
1 K 2
K II2
2
0
2 I
K II2
cos ( 2
n
)d
(8)
cos sin( n )d . E where, E is Young’s module and is Poisson’s ratio.
(9)
0
5
1.0
Inte gra
Mode-II φn=0.0 0.1 0.2 0.3
0.8
nd of J1π
0.6
0.4 0.5 0.6
E/( 1-μ 2
0.4
0.7 0.8 0.9 φn=1.0 Mode-I
)(K 2
I
+
2 0.2
KII )
0.0
-180.0
-150.0
-120.0
-90.0
-60.0
-30.0
0.0
30.0
60.0
Polar coordinate θ deg.
90.0
120.0
150.0
180.0
Fig. 3 Distribution of the integrand of normalized J1 -integral along the circle path. It should be pointed out that the process of boundary shifting or cracking must accompany the energy releasing, i.e., the boundary of fracture must be inward to the solids. The integrand of the normalized J 1 -integral, i.e., cos2 - n / 2 0 on the interval of - , indicates that J 1 -integral is always positive as shown in Fig. 3,
which means the energy releases when any part of boundary sl around crack tip shift in x1 -direction, that is to say the J 1 -integral always has a substantial contribution to the energy release rate for multiple-cracks initiation from a crack tip. However, unlike the J 1 -integral over any part of the crack tip boundary, the integrand of the normalized J 2 -integral shows a different and significant behaviour as shown Fig. 4. 0.6
Int egr an d of J2π E/( 1-μ 2 )( 2 KI +KII 2
)
Mode-I φn=1.0
0.3
Mode-II φn=0.0
0.9
0.1
0.8
0.2 0.3 0.4 0.5
0.7 0.6 0.5
0.0
-0.3
-0.6
-0.9
-1.2 -180.0
-150.0
-120.0
-90.0
-60.0
-30.0
0.0
30.0
60.0
90.0
120.0
Polar coordinate θ deg.
Fig. 4 Distribution of the integrand of normalized J2 -integral along the circle path. 6
150.0
180.0
Table 1. Normalized Ji -integrals over the paths s1 , s2 , s3 and s4 . s1 s ro , ∈ 0, 2
{
}
s3 = s ro , θ ∈ [- π , - π 2]
s2 s ro , ∈ 2 ,
s4 s ro , ∈ - 2 ,0
2J1 si E /(1 2 ) K I2
π/2
π/2
2J 2 si E /(1 2 ) K I2
-1
1
3.2 J 2 -integral around pure Mode-I crack tip When n 1.0 , which defined actually a special case for pure Mode-I deformation as shown in Fig. 4, the integrand of the normalized J 2 -integral cos sin 0 on the intervals of 0 2 and - - 2 ;
cos sin 0 on the intervals of 2
and - 2 0 . These characters determine that the whole
boundary of a crack tip should be divided into four parts, i.e., s1 s ro , ∈ 0, 2 , s2 s ro , ∈ 2 , , s3 s ro , ∈ , 2 and s4 s ro , ∈ 2 ,0 as shown in Fig. 5(a). The normalized Ji -integrals over the
paths s1，s2，s3 and s 4 are given in Table 1 [24], which imply different physical meanings. For the upper half crack tip boundaries s1 and s2 , J 2 s1 0 and J 2 s2 0 indicate theoretically that if the boundary s1 moves in x2-direction, the process of boundary cracking would accompany energy absorbing or increase strain energy, which is actually impossible to occur naturally; if the s2 moves in the x2-direction, the cracked solids will lead to release strain energy. So, the J 2 s2 has a substantial contribution to the energy release rate for multiple-cracks initiation from a Mode-I crack tip, and the contribution of J 2 s1 for Mode-I crack should be excluded in the analysis of the crack tip fracture. Similarly, for the lower half crack tip boundaries s3 and s4 , when s3 moves in the opposite x2-direction, i.e., fracture is inward to the solids, the energy release rate - J 2 s3 0 , which means that the fracture of boundary s3 inward to the solids is possible; when s4 moves in the opposite x2-direction, - J 2 s4 0 implies that the fracture will lead to energy-absorbing and is not going to happen. Then the underlying
moving trends of boundaries s1 , s2 , s3 and s4 can be schematically shown by the red arrows in Fig. 5(a). For mixed Mode-I/II crack, Eq. (9) indicates that the four θ-axis intersection points can be found from integrand cos sin( - n ) 0 for a given n , i.e.,
，2
， 3，4 k n ，k=0, -1.
(10)
These intersection points would divide the lower crack surface interval [-π, 0] or upper crack surface interval [0, π] into three subintervals as shown in Fig. 5(b). Generally, over the three subintervals, the J 2 -integrals are either positive or negative as indicated in Fig. 4, which can be used to identify how many underlying cracks should initiate from a mixed Mode-I/II crack tip and decides also the instability and trends of crack tip fracture. The potential moving trends of all parts of crack tip boundary are shown by the red arrows in Fig. 5(b).
7
x2
s2
π
x2
s2
s5
s1
πφn
x1
x1
s4
s4
s3
s1
s3
s6
(a) Pure Mode-I loading (φn=1.0).
(b) Mixed Mode-I/II loading.
Fig. 5. The interval partitioning around crack tip and the underlying moving trends. The red arrows denote the potential moving trends of local boundaries. In fact, it is very difficult to maintain an accurate experimental condition of pure Mode-I deformation. The Mode-I fracture in vast majority of experiments is actually a special mixed case of dominant Mode-I and weak Mode-II loading, which means a slight perturbation from the weak Mode-II loading to Mode-I loading [33, 34]. The relationship of n - 1.0 → ε (ε is a small quantity), which may stand for a weak Mode-II loading disturbance to Mode-I loading during experimental process, depicts actually the Mode-mixity of a dominant Mode-I loading and weak Mode-II loading. When n 1 ε , the subinterval divisions can be schematically shown in Fig. 6.
s5
s2
x2
x2
s2
s1
s1 s5 πε
x1
-πε s6 s3
x1
s4
s4
s6
s3
(a) φn→1.0-ε. (b) φn→1.0+ε, the symmetrical case of φn→1.0-ε. Fig. 6. The interval divisions around crack tip and the potential moving trends. The red arrows denote the potential moving trends of local boundaries. 4. Multiple-cracks initiation from a Mode-I crack tip As discussed in section 3, the diversity of potential moving or fracture trends for all local boundaries around a crack tip constitutes multiple complicated fracture configurations, which is interesting and quite helpful for understanding the complicated fracture behaviours in engineering practice. Actually, the crack extension, crack 8
kinking and branching can be considered as that the fractures initiation from one or some parts of the crack tip boundary are triggered. For the pure Mode-I crack, there are three typical crack multiple-branching fracture configurations, i.e., the crack tri-branching ( -shaped fracture shown in Fig. 7(a)), crack symmetric-branching ( -shaped symmetrical fracture shown in Fig. 7(b)) and crack side-branching ( -shaped fracture shown in Fig. 7(c)), which had been investigated in detail in [24]. This section will further investigate the Mode-I crack tri-branching, symmetric-branching and side-branching with the possible effect of weak Mode-II loading.
(a) Crack tri-branching.
(b) Crack symmetric-branching.
(c) Crack side-branching.
Fig. 7. Illustrations for enlarged multiple cracks initiation from a pure Mode-I crack tip.
Fig. 8. Enlarged geometrical modelling for three cracks initiation from a pure Mode-I crack tip. 4.1. Energy-based driving force and fracture angles for pure Mode-I crack tri-branching As pointed out in section 3.2, the possible fractures initiation from a Mode-I crack tip may be formed when boundaries s2 and s3 move inward to solids as two wing cracks and s1 and s4 move together in x1-direction as a form of crack extension as shown in Figs. 5(a), 7(a) and 8 in order to achieve maximum energy release rate. In this case, by using the symmetry of the tri-fractures configuration under pure Mode-I loading the energy release rate induced by crack tri-branching as shown in Fig. 8 can be found as [24] G tri-b 2 J 1 s1 cos 1 J 1 s 2 cos 2 J 2 s 2 sin 2
(11)
from Eq. (2). Substitute the results in Table 1 into Eq. (11), the following results can be found. G tri -b
1 - K 2
2E
2 I
cos 1
1 4 / 2 cos 2 - ,
(12)
where 9
tan
-1
2 / 32 .48
deg.
(13)
The energy release rate is now maximised with respect to i by setting
G tri b i
0 , i=1 and 2, i.e.,
sin 1 c 0 and sin 2c - 0 , which gives the critical crack initiation angles
1c 4 c 0 deg.
(14)
2c 3c 32.48 deg.
(15)
and
where 1c and 2c are the theoretical expected and possible crack initiation angles at which G tri b exhibits a maximum value, and explicitly this is given by tri b Gmax
1 K 2
2 I
E
Ftri b ,
(16)
where Ftri b
1 1 4 / 2 1.0927 , 2
(17)
which is the energy-based driving force for Mode-I crack tri-branching as shown in Figs. 7(a) and 8. It indicates that when tri b G max G Ctri b ,
(18)
the Mode-I crack will fracture theoretically in the form of crack tri-branching according to the Griffith’s criterion [38]. 4.2. On K-based fracture criterion for Mode-I crack tri-branching For a homogenous and isotropic brittle solid, Griffith’s criterion [38] states that a crack will propagate when the critical value Gc of the solid is equal to the driving force or applied Gmax [38][39] Gmax Gc .
(19)
For Mode-I crack extension, when Gmax Gc and Gmax Gc , we have Gc
1 K 2
E
2 IC
.
(20)
When fracture configuration of Mode-I crack tri-branching is triggered, as discussed in section 4.1, the energy-based driving forces for a wing crack initiation from boundary s2 or s3 can be found as s wing s wing Gmax Gmax 2
3
1 - K 2
E
2 I
1 4 / 2 . 4
(21)
The energy-based driving forces for cracks initiation from s1 and s4 in a form of crack extension can be expressed by s1 s 4 ext Gmax
1 - K 2
E
2 I
1 . 2
(22)
From Eqs. (19)-(22), it is not difficult to find the K-based fracture toughness s 2 wing K IC 1.84 K IC
(23) 10
and s s ext K IC 1.41K IC 1
(24)
4
for a wing crack initiation and main crack extension respectively. Then from Eqs. (16) and (18), the critical energy release rate can be found as
1- K
tri-b 2 IC
2
trib C
G
Ftri-b ,
E
(25)
where
s wing s s ext tri -b K IC max K IC , K IC 1.84 K IC . 2
1
4
(26)
s wing s s ext tri b The relationship between Gmax , Gmax and Gmax under pure Mode-I loading can be schematically shown 1
2
4
in Fig. 9. The K-based criterion for Mode-I crack tri-branching can be written as
Normalized energy-based drive forces GmaxE/(1-μ2)KIC2
trib K I K IC .
(27)
2 GCtrib E / 1 - 2 KIC
trib 2 Gmax E / 1 - 2 K IC
s s ext Gmax E / - K IC
Energy release rate for creating a new surface
s wing Gmax E / - K IC
2 G C E / 1 - 2 K IC
KI KIC 1.41
K I K IC 1.84
Stress intensity factor K I K IC s wing s s Fig. 9. The relationship between Gmax , Gmax 2
1
4 ext
tri b and Gmax under pure Mode-I loading.
4.3 Effect of weak Mode-II loading on Mode-I crack tri-branching Because of the underlying shifting trend inward to solids, the attempted shifting of boundary s6 as shown in Fig.6 (a) should stop s4 from moving in x1-direction. Similarly, the attempted shifting of the boundary s 5 inward to solids, as shown in Fig.6 (b), should stop also boundary s 1 from moving in x1-direction. However, the energy release rate for the boundary s 6 or s 5 attempted shifting is a small value, from which a wing crack initiation is difficult. So, another form of Mode-I crack tri-branching should be formed from boundaries s1 moving in x1-direction as a form of crack extension and s2 and s3 moving inward to solids as two wing cracks as shown in Fig.10. In this case, the concerned energy release rate for crack tri-branching can be written as G tri b J 1 s1 cos 1 2 J 1 s 2 cos 2 J 2 s 2 sin 2 .
Then it is not difficult to get
11
(28)
tri b Gmax
1 K 2
E
2 I
2 1 1 4 / 4 2
(29)
and the theoretical expected crack initiation angles
1c 0 deg. and 2c 3c 32.48 deg.
(30)
The energy-based driving force acting on s1 , which will induce to boundary s1 to fracture in form of crack extension, can be given by s1 ext Gmax J1 s1
1 - K 2
2 I
4E
.
(31)
From Eqs. (19), (20) and (31), the K-based fracture toughness for s1 can be found s ext K IC 2 K IC .
(32)
1
Similar to the discussion in section 4.2, the critical energy release rate for the crack tri-branching as shown in Fig.10 can be found as
1 - K
tri -b 2 IC
2
tri b C
G
E
2 1 1 4 , 4 2
(33)
where the K-based fracture toughness for Mode-I crack tri-branching becomes
s -wing s ext s ext tri-b K IC , K IC max K IC K IC 2K IC 2
1
(34)
1
tri b 2 K IC , the Mode-I crack tri-branching may be Based on K-based fracture criterion, when applied K I K IC
triggered. s 2 wing s1 ext The relationship between Gmax , Gmax
trib and Gmax for Mode-I loading with a possible weak disturbance
of Mode-II loading can be schematically shown in Fig. 11.
x2 s2 Original crack tip α2c
ro
s4 α3c
s1
x1
s6
s3
Fig. 10. Effect of weak Mode-II loading on Mode-I crack tri-branching.
12
Normalized energy-based drive forces GmaxE/(1-μ2)KIC2
GCtrib E / 1- 2
tri b 2 Gmax E / 1 - 2 K IC
2 KIC
s wing Gmax E / - K IC
Energy release rate for creating a new surface
2 GC E / 1 - 2 K IC
s ext Gmax E / - K IC
KI KIC 2 K I K IC 1.84
Stress intensity factor K I K IC s wing s ext tri b Fig. 11. The relationship between Gmax , Gmax and Gmax under Mode-I loading with a possible weak disturbance of Mode-II loading. 4.4 Effect of weak Mode-II loading on Mode-I crack side-branching
Three forms of underlying crack side-branching are shown in Fig.12. The first one is the idealized crack side-branching based on that all the boundary fracture accompanied by energy release are triggered. The other two forms are composed of a wing crack initiated from s2 or s3 and crack extension from local boundaries around crack tip, which is effected by weak Mode-II loading. Similar to the discussions in sections 4.1-4.3, for the crack side-branching configuration as shown in Fig.12(a), the concerned energy-based driving force can be written as side b Gmax
1 K 2
2 I
E
3 1 4 / 2 4
(35)
and the K-based fracture toughness for crack extension becomes s s s ext K IC 1
3
4
2 3
K IC 1.15K IC .
(36)
In this case, the K-based fracture toughness for Mode-I crack side-branching can be found as
s2 -wing s1 s3 s4 ext s2 wing side-b K IC max K IC , K IC K IC 1.84K IC
(37)
For the crack side-branching configuration as shown in Fig.12(b), the concerned energy-based driving force can be found as side b Gmax
1 K 2
E
2 I
1 1 4 / 2 4
(38)
and the K-based fracture toughness for crack side-branching
s 2 - wing s1 ext s1 ext side - b K IC max K IC , K IC K IC 2 K IC .
(39)
For the crack side-branching configuration as shown in Fig.12(c), the concerned energy-based driving force and K-based fracture toughness can be found respectively as
13
side b Gmax
1 K 2
E
2 I
2 2 1 4 / 4
(40)
and
s -wing s s ext s wing side-b , K IC K IC max K IC K IC 1.84K IC 2
1
2
(41)
2
The Eqs. (37), (39) and (41) show that the upper limit of K-based fracture toughness for Mode-I crack side-branching is 2KIC and the lower limit 1.84 KIC. 4.5 On Mode-I crack kinking and symmetric-branching As above analysis, when 1.84KIC < applied KI <2KIC and affected by a weak Mode-II loading, if a wing crack initiating from s2 or s3 is triggered, the crack kinking should be formed. If two wing cracks initiating respectively from s2 and s3 are triggered, the crack branching should occur as shown in Fig.13.
(a) Idealized crack side-branching under pure Mode-I loading.
(b) Effect of weak Mode-II loading on Mode-I crack side-branching.
14
(c) Another form of Mode-I crack side-branching effected by weak Mode-II loading. Fig.12. Possible Mode-I crack side-branching.
Fig.13. Crack symmetric-branching. Nor mal ize d str ess inte nsit y fact or KI/ KIC
2.5
2.0
Upper limit
tri -b K IC K IC 2
tri - b K IC K IC 1.84
1.5
Lower limit
1.41
1.0
0.5
0.0 0.0
t0 Time t Fig. 14. Impulse pure Mode-I loading.
5. Triggering for Model-I crack multiple-branching 15
As discussed in above sections, the Mode-I crack multiple-branching exists theoretically based on the energy-based fracture analysis. Taking the crack tri-branching for example, because of the quantitative difference s wing s ext between Gmax and Gmax , only one crack will initiate from crack tip as a form of crack kinking or main crack 2
i
extension under quasi-static Mode-I loading. Therefore, the Mode-I crack multiple-branching is rare in the quasi-static fracture tests. However, if increasing loading rate or even impulse Mode-I loading, as shown in Fig. s wing s ext 14, the Gmax and Gmax driving the boundaries s2, s3 and si to fracture respectively may reach their critical 2
i
values simultaneously or in a enough short time period. The possible Mode-I crack tri-branching, symmetric-branching or side-branching may be triggered. 6. Typical experimental fracture behaviours on Mode-I crack multiple-branching 6.1. Crack multiple-branching behaviours for GPPS specimens Mode-I crack multiple-branching behaviours of cracked specimens made of general-purpose Polystyrenes (GPPS) are investigated in this section. The GPPS is homogenous, isotropic, quasi-brittle at room temperature and exhibits linear elastic behaviour. Additionally, its optical transparency and being glassy facilitate direct observation of the fracture phenomenon during experiments. Overall geometry dimensions of the prepared CTS specimens and CTS fixture are shown in Fig. 15. The relative advantage of CTS specimen geometry is that the sufficient size to enable fully main and branched cracks to develop entirely. CTS specimens can be used also for mixed Mode I/II fracture experiments by adjusting the position of fixture. Fig. 15(c) shows the loading condition of the CTS specimens used in experiments. A CNC engraving machine was utilized to cut the overall specimens from GPPS sheet with 7.5 mm thickness and all specimens were processed from the same sheet to eliminate batch variations. A diamond wire saw with thickness of 0.42 mm was used for introducing a narrow notch with the initial depth of slightly less than 45 mm in the specimens. Then, a natural sharp crack tip was generated by sawing a fresh razor blade carefully to make the final length of each crack 45 mm. For manufactured specimens, the overall dimensions and details such as length (L) and width (W) and crack length (a) were considered the same for the sake of comparing the test results. The image of the sharpened notch tip magnified under an optical microscope is shown in Fig. 16. The set of fixture and specimen was mounted on a uniaxial tension compression INSTRON-8872 servo hydraulic test machine. In the fracture toughness tests of similar quasi-brittle materials, the loading rates are in the rage of 0.08 to 0.4 mm/min [40]-[46], especially the loading rate of 0.3 mm/min had been frequently used [40] [45]. The fracture toughness tests in present research were carried out under displacement control at loading rate of 0.3 mm/min. The low loading rate was employed to guarantee the establishment of quasi-static loading conditions and to provide sufficient time for transferring the applied load to the tested specimen uniformly. According to the ASTM-D5045 standard [47], fracture toughness tests should be replicated at least three times. It is mentioning that to enhance preciseness of the fracture test results, four specimens were provided for determinations of fracture toughness. Subsequently, increasing loading rate to capture the phenomenon of crack multiple-branching. The test data were recorded by a computer and all the experiments were carried out at room temperature.
16
For brittle materials, there is no obvious subcritical crack growth, so the measured peak fracture load can be considered as the critical fracture load, which can be used to calculate crack initiation fracture toughness. To evaluate the stress intensity factor for CTS specimens, the following expression can be used [48]. a 0.26 2.65( ) F cos wa K a , a a a 2 WB (1 ) 1 0.55( ) 0.08( ) w wa wa
(42)
where F is the applied force, w is the width of the specimen, B is the thickness of the specimen, a is the crack length and α is the angle of loading direction with respect to the crack plane. In order to ensure the condition for plane strain, ASTM-D5045 standard [47] requires the specimen thickness to exceed 2.5(KIC/σy)2, where KIC is the fracture toughness and σy is the yield stress. The test data obtained from the fracture toughness experiments have been listed in Tables 2 and 3. According to ASTM-D5045 standard, if yielding does not occur and brittle fracture is observed, the stress at fracture shall be used as yield strength in criteria to give a conservative size value. As shown in Table 2, the ultimate tensile strength at room temperature is 39.8 MPa, and take the maximum value 1.26 MPam1/2 from Table 3, i.e., plain strain conditions should certainly be achieved for thicknesses exceeding 2.51 mm. And based on it, the specimen thickness in present tests satisfied plane-strain condition.
17
(a)
(b)
(c) Fig. 15. Geometry and loading condition of CTS fixture and specimen (dimensions in mm). Table 2. Some mechanical properties of the tested GPPS. Material Property
unit
value
Elastic module
GPa
3
Poisson’s ratio
0.34
Density
g/cm3
1.044
Ultimate bending strength
MPa
59.0
Ultimate tensile strength
MPa
39.8
Tensile strain at break
1.4% 18
Fig. 16. The sharped notch tip of a test specimen shown under an optical microscope.
Fig. 17. Experimental setup for the testing of GPPS material using the CTS specimen. Table 3. Fracture toughness KIC for GPPS specimens Replicate No.
a(mm)
Fcr (N)
KIC (Mpa·m1/2)
1 2 3 4 5 average
45 45 45 45 45 45
729.46 837.72 695.75 700.27 844.48 761.54
1.08 1.25 1.03 1.04 1.26 1.13
19
Specimen a (a)
Specimen b (b)
Specimen c (c)
20
Specimen d (d)
Specimen e (e) Fig. 18. Macroscopic and microcosmic branching patterns on fracture surfaces in specimens: (a) to (c) Crack tri-branching. (d) Crack symmetric-branching. (e) Crack side-branching.
Fig. 19. A side view of fracture surface tested specimen (b). All the test samples fractured suddenly from the crack tip at a critical peak load, which confirms brittleness of the tested material. Twenty specimens were carried out for Mode-I crack multiple-branching tests. There is a 21
statistical variation in the branching patterns, however, some typical general observations can be picked out. Distinct crack tri-branching was observed in three out of twenty specimens, as shown in Figs. 18 (a)-(c). Meanwhile, crack quasi-symmetric branching and side-branching were observed and shown in Figs. 18 (d) and (e) respectively in the above specimens. For crack tri-branching configuration, it is worth mentioning that the two wing cracks in crack tri-branching are difficult to fully develop as shown in Fig. 18 (a)-(c) because the stress intensity factors for the new-born wing cracks should be much smaller than that for extended main crack.
1/2
stress intensity factor (MPa m )
3.0
2.4
2.26
1.8
KI for specimen a KI for specimen b KI for specimen c KI for specimen d KI for specimen e
1.2
0.6
0.0 0.0
0.1
0.2 0.3 Time(s)
0.4
0.5
Fig. 20. SIFs - time curves obtained from crack multiple branching experimental tests. From the Eq. (34), the fracture toughness for crack tri-branching (the upper limit) predicted by present method tri -b tri -b is K IC =2KIC, i.e., K IC =2×1.13=2.26 MPa m1/2. According to the theoretical predictions in section 4, when
the applied KI
tri -b ≥ K IC
the underlying Mode-I crack tri-branching, side-branching or quasi-symmetrical
branching should be triggered. The corresponding SIFs-time curves from experiments as shown in Fig. 20, and the tri -b theoretical value of K IC was highlighted with a red dotted line. As shown in Fig. 20, the K I for specimens a,
b, c, d and e exceeded the red dotted line, which indicates that the experimental results agree with the theoretical prediction based on present modelling. 6.2. Crack multiple-branching behaviours for PMMA specimens An investigation of the branching characteristics of small PMMA single edge notched tensile (SENT) specimens had been conducted by Murphy et al [8]. Some typical images of Mode-I crack branching are given in Fig. 21. Similar to the GPPS, stable crack branching behaviors have been observed for PMMA, which may be predicted and better understand based on the present modelling.
22
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 21. Branching patterns in specimens containing the 0.1 mm notches. Successful branching occurred in all of the 8 mm thick specimens tested. The size of the images is 25 by 20 mm in each case [8]. 7. Conclusions By further exploring the Ji-integral over the local boundary of a Mode-I crack tip, an energy-based fracture modelling for multiple fractures initiation from a crack tip has been investigated and formulized. Unlike the traditional applications of the J-integral around a crack-tip, a new partial integral path and the associated calculation method on the energy release rate for multiple cracks initiation had been defined. The present article further validates theoretically the underlying fracture trends for the Mode-I crack multiple-branching with a possible effect of weak Mode-II loading. Several possible fracture configurations on multiple-cracks initiation from a crack tip are proposed for a quasi-Mode-I crack, such as crack tri-branching, symmetric-branching and side-branching based on the present energy-based modelling. The corresponding energy-based driving forces, K-based fracture toughness 23
and fracture criteria for above crack configurations have been formalized and found. And according to the prediction based on present method, tri-branching is the upper limit on the number of possible multiple crack initiation for a Mode-I crack with a weak disturbance of Mode-II loading. The fracture mechanism for a Mode-I crack under quasi-static and dynamic loading can be better explained based on the present theoretical modelling, which is of great significance to understand the complex fracture phenomena of brittle materials and engineering applications, such as hydraulic fracturing with multiple cracks and crack growth retardation or arrest. The fracture toughness predicted by the present modelling for Mode-I crack tri-branching, side-branching and symmetrical-branching with a possible effect of weak Mode-II loading agree well with the experimental results. The proposed method provides also a potential way to predict the crack multiple-branching behaviours under the mixed Mode-I/II loading. The fracture nature of the experimental results can be easily recognized by using present analysis. By comparing the experimental fracture configurations and the theoretical one, which form of the fracture, such as crack tri-branching, side-branching and symmetric-branching, and the mechanism behind experimental phenomena can be identified. Then the corresponding energy-based driving force, fracture toughness and K-based criterion can be clearly known. Additionally, the present analysis implies also that the underlying fracture trends of multiple cracks initiation under static condition can be triggered by using impulse loading, which may be one of the effective methods to induce the Mode-I crack multiple-branching. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.: 50771052, 50971068 and 11272141) and Natural Science Foundation of Liaoning (Grant Nos.: LS2010100 and 20102129). References [1]. Meggiolaro MA, Miranda ACO. Stress intensity factor equations for branched crack growth. Engng Fract Mech 2005; 72: 2647-71. [2]. Yoffe EH. The moving Griffith crack. Philos Mag 1951; 42: 739-50. [3]. Adda-Bedia M. Brittle fracture dynamics with arbitrary paths Ⅲ. The branching instability under general loading. J Mech Phys Solids 2005; 53: 227-48. [4]. Clark ABJ, Irwin GR. Crack-propagation behaviours. Exp Mech 1966; 6: 321-30． [5]. Schardin H. Velocity effects in fracture. In Fracture : proceeding of an international conference on the atomic mechanism of fracture help in Swampscott, Massachusetts, H. Schardin, ed., 1959, pp. 297–330. [6]. Congleton J, Petch NJ. Crack branching. Philos Mag 1967; 16: 749-60. [7]. Doll W. Investigation of the crack branching energy. Int J Fract 1975; 11: 184-86. [8]. Murphy N, Ali M, Ivankovic A. Dynamic crack bifurcation in PMMA. Engng Fract Mech 2006; 73: 2569-87. [9]. Aliha MRM, Ayatollahi MR. Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int J Solids Struct 2012; 49: 1877-83. [10].Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Engng Trans ASME 1963; 85: 519-25. [11].Sih GC. Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 1974; 10: 305-21. 24
[12].Hussain MA, Pu SL, Underwood J. Strain energy release rate for a crack under combined mode I and Mode II. Fracture Analysis, 1974. ASTM STP 560. American Society for Testing and Materials, Philadelphia, pp. 2–28. [13].Akbardoost J, Ayatollahi MR, Aliha MRM, Pavier MJ, Smith DJ. Size-dependent fracture behavior of Guiting limestone under mixed mode loading. Int J Rock Mech Min Sci 2014; 71: 369-80. [14].Aliha MRM, Hosseinpour GhR, Ayatollahi MR. Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials. Rock Mech Rock Engng 2013; 46: 1023–34. [15].Mirsayar MM, Razmi A, Aliha MRM, Berto F. EMTSN criterion for evaluating mixed mode I/II crack propagation in rock materials. Engng Fract Mech 2018; 190: 186-97. [16].Aliha MRM, Berto F, Mousavi A, Razavi SMJ. On the applicability of ASED criterion for predicting mixed mode I+II fracture toughness results of a rock material. Theor Appl Fract Mech 2017; 92: 198-204. [17].Razavi SMJ, Aliha MRM, Berto F. Application of an average strain energy density criterion to obtain the mixed mode fracture load of granite rock tested with the cracked asymmetric four-point bend specimens. Theor Appl Fract Mech 2018; 97: 419-25. [18].He M, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials. Int J Solids Struct 1989; 25: 1053-1067. [19].Hutchinson JW, Suo Z. Mixed mode cracking in layered materials. Adv Appl Mech 1992; 29: 63-191. [20].Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fract 1980; 16: 155-69. [21].Eshelby JD. The Force on an Elastic Singularity. Phil Trans Roy Soc London Ser A 1951; 244: 87-112. [22].Sih GC. Dynamic Aspects of Crack Aspects of Crack Propagation. In Sih G C ed. Inelastic Behavior of Solids. New York, Mc-Graw-Hill Book, Co.; 1969: 607-39. [23].Budiansky B and Rice JR. Conservation Laws and Energy-Release Rates. ASME J Appl Mech 1973; 40: 201-3. [24].Xie YJ, Li J, Hu XZ, Wang XH, Cai YM, Wang W. Modelling of multiple crack-branching from Mode-I crack-tip in isotropic solids. Eng Fract Mech 2013; 109: 105-16. [25].Xie YJ, Wang XH and Wang YY. Stress intensity factors for cracked homogeneous and composite multi-channel beams. Int J Solids Struct 2007; 44: 4830-44. [26].Xie YJ, Xu H, and Li PN. Crack Mouth Widening Energy-Release Rate and Its Applications. Theor Appl Fract Mech 1998; 29: 195-203. [27].Xie YJ. A. Theory on Cracked Pipe. Int J Pres Ves Pip 1998; 75: 865-69. [28].Xie YJ, Zhang X, Wang XH. An exact method on penny-shaped cracked homogeneous and composite cylinders. Int J Solids Struct 2001; 38: 6953-63. [29].Xie YJ. An analytical method on circumferential periodic cracked pipes and shells. Int J Solids Struct 2000; 37: 5189-201. [30].Giannakopoulos AE, Lindley TC, Suresh S. Aspects of connections and life-prediction methodology for fretting-fatigue. Acta Mater 1998; 46: 2955-68. [31].Giannakopoulos AE, Venkatesh TA, Lindley TC, Suresh S. The role of adhesion in contact fatigue. Acta Mater 1999; 47: 4653-64. [32].Hutchinson JW. Singular behavior at the end of a tensile crack in a hardening material. J Mech Phys Solids 25
1968; 16: 13. [33].Sundaram BM. Why do growing cracks branch? An optical investigation of brittle monolithic and bilayer transparencies under stress wave loading. Doctoral dissertation, Auburn University, May 6, 2017. [34].Sundaram BM, Tippur HV. Dynamic fracture of soda-lime glass: A full-field optical investigation of crack initiation, propagation and branching. J Mech Phys Solids 2018; 120:132-53. [35].Xie YJ and Hills DA. Quasi brittle fracture beneath a flat bearing surface. Eng Fract Mech 2008; 75: 1223-30. [36].Li XH, Zheng XY, Yuan WJ, Cui XW, Xie YJ, Wang Y. Instability of cracks initiation from a mixed-mode crack tip with iso-stress intensity factors KI and KII. Theor Appl Fract Mech 2018; 96: 262-71. [37].Shih CF. Small-scale yielding analysis of mixed mode plane-strain crack problems. Fracture Analysis, ASTM STP 560, American Society for Testing and Materials, 1974, pp. 187-210. [38].Griffith AA. The Phenomena of Rupture and Flow in Solids. Phil Trans Roy Soc Lon Ser A 1921; 221: 163-98. [39].Cherepanov GP. Mechanics of Brittle Fracture, Moscow, Publish House “Nuaka,” 1974, English translation published by McGraw-Hill International Book Co., New York; 1979: 266-69. [40].Safaei S, Ayatollahi MR, Saboori B. Fracture behavior of GPPS brittle polymer under mixed mode I/III loading. Theor Appl Fract Mech 2017; 91: 103–15. [41].Zhou Jun, Wang Yang, Xia Yuanming. Mode-I fracture toughness measurement of PMMA with the Brazilian disk test. J Mater Sci 2006; 41: 5778-81. [42].Loya JA, Villa EI, Fernández-Sáez J. Crack-front propagation during three point-bending tests of polymethyl-methacrylate beams. Polym Test 2010; 29: 113-18. [43].Kailash CJ, Hareesh VT. Quasi-static and dynamic fracture behavior of particulate polymer composites: A study of nano-vs. micro-size filler and loading-rate effects. Compos Part B-E 2012; 43: 3467–81. [44].Zhang H, Zhang Z, Friedrich K, Eger C. Property improvements of in situ epoxy nanocomposites with reduced interparticle distance at high nanosilica content. Acta Mater 2006; 54: 1833-42. [45].Weerasooriya T, Moy P, Casem D, Chen M. Fracture toughness of PMMA as a function of loading rate, in: Proceedings of the 2006 SEM Annual Conference on Experimental Mechanics, 2006, pp. 5-7. [46].Ayatollahi MR, Saboori B. A new fixture for fracture tests under mixed mode I/III loading. Eur J of Mech A/Solids 2015; 51: 67-76. [47].Annual book of standards. Standard E399-90: standard test method for plane strain fracture toughness of metallic materials. Philadelphia, PA: American Society for Testing and Materials; 1997. [48].Richard HA. Bruchvorhersagen bei überlagerter normal-und schubbean-spruchung von risen VDI Forschungsheft 631. Düsseldorf: VDI-Verlag; 1985；p. 1-60.
26
Highlights • An energy-based modelling for multiple cracks initiation from a tip is proposed. • Underlying fracture trends of cracks initiation induced by dominant Mode-I loading with a weak disturbance of
Mode-II loading is formulized. • The proposed modelling can capture nearly all of crack branching patterns observed in test.
27