V-NOTCHED SPECIMEN UNDER MIXED-MODE FRACTURE
N. Recho1, J. Li2, D.Leguillon3
LMM, University of Paris VI, France and LaMI, University Blaise Pascal of Clermont-II,France
2
LPMTM, CNRS UPR 9001, University of Paris XIII, 99 Av. J.B. Clément, 93430 Villetaneuse, France
3
LMM, University of Paris VI, France
1
ABSTRACT
A general methodology to study the crack propagation and the crack extension initiated from a V-notch was established under
mixed-mode static loading. This methodology was established on the basis of a set of criteria developed in recent studies
carried out by Leguillon [1,2]. A new criterion, by combining experimentation and numerical calculation, was developed in this
work in order to predict the beginning of the crack initiation and the crack growth path. A series of mixed-mode experiments of
a V-notch under mixed-mode loading are performed with CTS (Compact-Tension-Shear) PMMA specimens associated to a
mixed mode loading device. The effect of loading angle on fracture criterion is analyzed.
Introduction
Used in classical fracture mechanics analysis, the Williams expansion [3] allows to determine the stress field close to the notch
tip region by introducing the boundary conditions of free surface at the notch lips. Nevertheless, in this case, the signification of
the Stress Intensity Factor (SIF) is not valid. As consequence, the stress field mixity has to be defined by another way than
that based on KI/KII ratio. The work done by Seweryn et al [4] and by Leguillon et al [2] and Yosibash et al. [5] gives responses
on these two questions. The validation of such studies has not been established experimentally particularly when the notch
angle is big enough (>90°) and when the stress field mixity is dominated by the Mode II loading. This paper presents an
appropriate experimental procedure applied to specimen made from brittle material (PMMA) containing very sharp notches.
The experimental results, associated to stress field analysis, allow us to show how the fracture criterion interacts with the local
near notch tip analysis. A further work is planed to determine the crack growth path initiated at the notch tip.
Modelling of the singular stress field at a notch tip
The existing notch in a structure increases the stress concentration in the vicinity of the notch tip, thus the maximum load,
which the structure supports, can considerably be reduced. In this part the aim is to determine the stress field in the case of a
V-notch, having various notch angles, ω, see Figure 1.
As boundary conditions one has:
•
On the upper notch side: θ = π − ω = α ,
σ θθ = σ rθ = 0
2
•
And on the lower notch side: θ = −π + ω = −α,
σ θθ = σ rθ = 0
2
Let consider the stress, strains and displacements fields in the vicinity of a notch in homogeneous, isotropic, linear elastic
medium [3]. The resolution of 2D linear elastic problems consists in searching a function
2
2
the following compatibility equation:∇ (∇ Ψ)=0.
Ψ (x,y) or Ψ (r,θ), which satisfy
Figure 1.V-Notch
From this function, one deduces the stress field which has to satisfy the boundary conditions. It exists several approaches,
making it possible to obtain a singular stress field at the notch tip.
As conclusion, the stress and displacement fields have the following schematic shape:
σ ij =
K Iλ
K IIλ
c
(
θ
)
+
d ij (θ )
ij
( 2π r ) 1 − λ I
( 2π r ) 1− λ II
qi =
⎤
r ⎡ K Iλ
K IIλ
a i (θ ) +
bi (θ ) ⎥
1− λ
⎢
2 µ ⎣ ( 2π r )
( 2π r ) 1 − λ
⎦
I
(1)
(2)
II
where cij(θ),dij(θ),ai(θ),bi(θ) are trigonometric functions.
mode II.
λI and λII are respectively the singularity degrees for mode I and
Figure 2 shows the variation, determined by Seweryn et al. [4], of the exponents
ω = 360° - 2α .
λI and λII as function of the notch angle
λΙΙ(ω)
ω
λΙ(ω)
Notch angle ω (deg.)
Figure 2. λI and λII as function of the notch angle ω (Seweryn et al. [4]).
Onset criterion in case of mixed mode fracture [6]
When one uses the traditional tools for the study of the singularity in the vicinity of a notch tip, one arrives at a paradox related
to an inconsistency between the Stress criterion and the Energetic criterion.
If an Energetic criterion and a Stress criterion are considered, the first is not sufficient to explain the failure mechanism
because the energy release rate cancels itself and it cannot thus reach a critical value GC. In the stress criterion, the notch tip
presents a singularity and the stress field tends to infinity and the stresses are thus higher than their critical values
σc and τC.
In mixed mode situation the displacement field is written as follows:
U ( x1, x2 ) = U (O) + k1r λ1 u1(θ ) + k2 r λ2 u 2(θ ) + ... ,
where 1/ 2 ≤ λ1 ≤ λ2 ≤ 1 (note that in any case the two modes u1(θ ) and u 2(θ ) are different).
(3)
It’s to be noted that the two modes do not split into a symmetric and an antisymmetric parts. In order to point that, the
expression (3) rewrites:
U ( x1 , x2 ) = U (O ) + k1r λ1 ( u 1(θ ) + mu 2(θ ) ) + ...withm =
k2 λ2 − λ1
r
,
k1
k1 = 0 , (3) expands with a single mode)
The parameter m expresses the mode mixity and depends on the distance r
(4)
(note that if
of the notch tip. The energy criterion reads
now as follows: ( θ 0 is the direction of the crack extension)
−δ Wp = k12l 2λ1 ( K1 (θ 0 ) + 2mK12 (θ 0 ) + m2 K 2 (θ0 ) ) d ≥ Gc ld ,
Indeed, (5) no longer provide explicit relations for the lower bound of
identical for the upper bound derived from the stress criterion
l
since
m
(5)
depends on
l.
The situation is strictly
σ (θ0 ) = k1lλ −1 ( s1 (θ0 ) + ms2 (θ 0 ) ) ≥ σ c ,
(6)
1
Finally the non linear equation to solve numerically is:
2
l=
⎛ s1 (θ 0 ) + ms2 (θ 0 ) ⎞
Gc
⎜
⎟ ,
K1 (θ 0 ) + 2mK12 (θ 0 ) + m 2 K 2 (θ 0 ) ⎝
σc
⎠
Following, the fracture length l being determined, the mode mixity parameter
derives straightforwardly from this relation:
1− λ1
⎛
⎞
Gc
k1 ≥ kc = ⎜
⎟
2
⎝ K1 (θ 0 ) + 2mK12 (θ 0 ) + m K 2 (θ 0 ) ⎠
τ
and its critical value
τc
is also known and the initiation criterion
⎛
⎞
σc
⎜
⎟
⎝ s1 (θ 0 ) + ms2 (θ 0 ) ⎠
In this section, the stress criterion has been written using the tension component
the shear component
m
(7)
2 λ1 −1
,
(8)
σ , but it can be also carried out also with
for an interface crack [2] for instance. Moreover, since mode mixity is
invoked both must be considered and the one giving the higher upper bound has to be taken as consistent.
Once the crack length
l
is determined by equation (7) as function of
θ0 , it is replaced in equation (8) in order to determine
kc (θ 0 ) . For several values of θ0 we choose the angle (direction) giving the minimum value of kc (θ 0 ) . This direction gives
the crack extension criterion. This procedure to determine the crack extension is indeed different from the crack extension
given by a classical crack criterion such as the maximum of the circumferential stress. In this paper we present only the results
in term of resistance to failure. Nevertheless, on the basis of these experimental results, a future work is done in order to
establish a crack extension criterion [7].
Experimental
The experiments are performed by using the CTS specimens with the mixed mode loading device which was developed by
Richard [8]. All specimens are made from PMMA.Figure 3 shows the geometry of the specimens and of the mixed-mode
loading device. The specimens are of10mm thickness. Five V-notch angles (30°, 60°, 90°, 120°, and 160°) are introduced in
the specimens. There are seven holes in the loading device shown in Figure 3(b). A pure mode I loading is performed using
the No.1 and 1’ holes in the loading device and a pure mode II loading is realized using the No.7 and 7’ holes. When the other
holes (n-n’) are used, different mixed mode loading can be obtained.
The tests are conducted on a machine MTS-810 (Material Test System) at room temperature. The CTS specimens are tested
with four loading angles γ= 90°, 60°, 30°and 0° with respect to the initial crack axis as shown in Figure 4. The case of 90°
loading corresponds to the pure mode I test. Two or three specimens are used for each loading condition.
P
90
27
P
27
3
4
5
2
6
1
7
148
54
45
ω
notch
7'
54
Vnotch
6' 5'
4'
3'
2'
1'
P
(a) PMMA specimen
(b) Loading device
Figure 3. Specimens and loading device
P
P
60°
90°
P
30°
P
P
P
Figure 4. Loading angles
One can see on figure 5 the area A where the specimens are fixed and the area B where one applies the loading with the
angle γ (the angle between two holes is of 15°). Screws were used in order to install the specimen on the testing machine
MTS.
First of all, we carried out simple tensile tests on specimens without notch. The goal is to find the mechanical characteristics of
material, in particular to determine the stress to rupture σ c and the Young’s modulus E. The mean values of 10 tensile test
specimens are: σ c =75 MPa and E= 3300 MPa.
Each test is carried out under monotonic quasi-static loading (2mm/minute) until a first crack appears. Subsequently, a total
unloading occurs. Re-loading is carried out then on the specimen containing the crack until the brittle fracture. Two surfaces of
crack appear. First step corresponding to the creation of a crack and the second corresponding to brittle fracture. The second
step allows in fact to calculate the critical energy release rate from the knowledge of the critical load to rupture during this
second step and from a calculation by finite elements of the specimen containing the crack.
B
A
Figure 5. Test device
Let choose the specimens which reveal clearly the two steps on the cracking surface. One measures then the lengths L (mm)
of the cracks of the first step. From there, the test can be regarded as a tensile test on a specimen containing a crack having
length L. The second step then makes it possible to find the maximum load to rupture F (N). Going from this value, a finite
elements calculation allows us to determine Gc.
Figure 6 shows two examples of specimens after the crack extension. That enabled us to determine the crack extension angle
according to the notch angle and to the loading angle. The cracks show well that when the angle β increases, the maximum
load value is higher. That comes owing to the fact that when the angle β increases, mode II (λ II ) is increasingly important and
thus the load to rupture increases.
γ
ω
θ
Figure 6. Examples of crack propagation
Results
The various geometries and the various loadings were tested. The following figure 7 shows the maximum loads to rupture.
ω=90°
a=90
ω=30°
a=30°
3500
12000
3007
2746
2623
2577
2425
2411
force (N)
2500
2000
1500
1485
1498
1289
40
60
31
Figure
1000 1010
a=30°
1606
1580
1441
11145
10360
9086
10000
8000
force(N)
3000
B=90
6000
4000
2000
500
0
0
20
40
60
80
0
100
0
π/2−γ°
20
angle B
40
60
80
100
angle B
π/2−γ°
a=120°
ω=120°
ω=a=160
160°
6000
12000
5000
8000
6860
6000
4000
2577
2090
1794
2000
2404
2390
2105
4945
4208
3543
a=120°
force (N)
9944
10000
force (N)
2587
2549
2368
1829
1590
1485
1501
1527
1248
3000
50
a=160
2000
1000
0
0
4891
4411
4345
4333
4232
4039
4000
0
100
0
angle B
π/2−γ°
10
20
30
40
π/2−γ°
angle B
Figure 7. Loads to failure as function of loading angle and notch angle
Analysis of the Results
The structure is modelled by finite elements as shown in figure 8. A specific finite element programme integrating the local
asymptotic analysis is used in order to determine the singularity degrees (λ1 and λ2 ) of the structure as function of the notch
angle . The following table 1 gives the values of λ1 and λ2:
ω=0°
λ1=0,5 λ2=0,5
ω=30°
λ1=0,502 λ2=0,598
ω=60°
λ1=0,512 λ2=0,73
ω=90°
λ1=0,545 λ2=0,906
Table 1. Singularity degrees λ1 and λ2 as function of notch angle ω
Figure 8. Finite Elements mesh
ω=120°
λ1=0,616 λ2=1,149
ω=160°
λ1=0,819 λ2=1,628
The numerical results are put in concordance with the experimental results in term of maximum load to rupture at mode I
fracture. This loading level is considered as equal to 1. The Loading ratio, R is then defined as : the load to rupture of the
specimen under mixed Mode / the load to rupture of the specimen under Mode I. The following figure 9 shows some examples
of the comparison between the experimental results and the numerical analysis based on our criterion.
The theoretical analysis is in good agreement with experimental results. It’s to be noted that the singularity vanishes when
ω=120° with a pure Mode II dominating loading (see Table 1). Figure 10 shows one of experimental test specimens in this
condition. Failure no longer occurs at the notch root.
4,5
4
R
3,5
3
2,5
Sˇrie1
Sˇrie2
2
1,5
1
ω
0,5
0
0
20
40
60
80
100
120
140
160
180
R versus ω ( γ is equal to 90°- Pure Mode I)
5
4,5
R
4
3,5
γ
3
Sˇrie1
Sˇrie2
2,5
ω
2
θ
1,5
1
ω
0,5
0
0
20
40
60
80
100
120
140
R versus ω ( γ is equal to 30°)
N.B. Series 1 corresponds to experimental results and series 2 to numerical simulations.
Figure 9. Comparison with experimental results in term of loading ratio R
160
180
Figure 10. Case of vanishing singularity for ω=120°associated to mode II dominating (γ = 30°)
Conclusions
The V-Notch criterion is extended to Mixed Mode loading. It is shown that the knowledge of the singularity degrees in Mode I
and Mode II are necessary in establishing criteria of the failure prediction. A series of tests agree with the disappearing of any
singularity in case of pure mode II when the notch angle becomes large. The agreement between predictions and experiments
is good in term of critical load to rupture. Further work is in preparation in order to establish a specific crack extension criterion
on the basis of these experimental results.
References
1.
2.
3.
4.
5.
6.
7.
8.
Leguillon D. “Strength or toughness? A criterion for crack onset at a notch” European J. Mech. A/Solids, Vol.21 pp.61-72,
2002
Leguillon D. Siruguet K., Finite fracture mechanics – Application to the onset of a crack at a bimaterial corner, IUTAM
symposium on Analytical and computational fracture mechanics of non-homogeneous materials, Cardiff, 18-22 june 2001,
Wales, published in IUTAM Symposium on Analytical and Computational Fracture Mechanics of Non-Homogeneous
Materials, B.L. Karihaloo ed., Solid Mechanics and its Applications vol.97, Kluwer Academic Publishers, Dordrecht, 2002,
11-18.
Recho N.« Rupture par fissuration des structures »,Hermes, (1995).
Seweryn A., Krzysztof M. “Elastic stress singularities and corresponding generalized stress intensity factors for angular
corner under various boundary conditions”, Eng Fracture Mech, Vol.55, No 4, pp.529-556, 1996.
Yosibash Z., Priel E., Leguillon D., A failure criterion for brittle elastic materials under mixed-mode loading, Int. J. Fract.,
2006, 141(1), 289-310.
Leguillon D., Yosibash Z. “Crack onset at a v-notch. Influence of the notch tip radius” Int. J. of Fracture, 122(1-2), 1-21.
(2003)
Recho N., Leguillon D. “Propagation of Short Cracks near a V-Notch Tip in Brittle Material under Mixed-Mode Loading”
CFRAC – Nantes France - june 2007
Richard H.A. and Benitz K. “A loading device for the creation of mixed mode in fracture mechanics”. Int. J. Fracture, 1983;
22: R55.