EXPERIMENTAL STUDY OF THE OUT-OF-PLANE DISPLACEMENT FIELDS
FOR DIFFERENT CRACK PROPAGATION VELOVITIES
S. Hédan, V. Valle and M. Cottron
Laboratoire de Mécanique des Solides, UMR 6610
Université de Poitiers
Poitiers, FRANCE
ABSTRACT
The fundamental aim of this study is to determine the area of the 3D effects zone and the transient one at the vicinity of the
crack tip during a crack propagation in brittle materials (PMMA) using an optical method. For the experimental data, we
measure the out-of-plane displacements field by using the interferometry on SEN (Single Edge Notch) specimens loaded in
mode I with a constant loading σ. To obtain the experimental data of out-of-plane displacement fields, we used an optical
method composed by a laser, a Michelson interferometer and a high speed CCD camera. The experimental interferograms are
analysed using the MPC method (Modulated Phase Correlation), which allows extracting the phase fields from only one fringe
pattern. By the use of optical relations, we can determine the relief around the crack from the phase field. We compare the
experimental data of the out-of-plane displacements with a 2D theoretical solution and a 3D formulation. The 2D solution
characterizes out-of-plane displacements in plane stress hypothesis. The proposed 3D expression is based on works of the
literature relating to the presence of the 3D effects for stationary cracks. In our case, during a crack propagation, we modified
the Stress Intensity Factor term (SIF [10]) by introducing the crack propagation velocity and the velocities of longitudinal and
shear waves. The crack propagation velocity introduces a transient term in more at the 3D effects term present for the
stationary cracks. The presence of the 3D and transient effects results in a progressive gap between the 2D solution and the
3D formulation when we approach the crack tip. So, by a study of the detachment between the both expressions, we can
determine the area of the 3D and transient effects zone according to the crack propagation velocity. Results are shown for one
static test and two dynamic tests. The analysis of the results shows that the detachment zone of the two expressions is large
and proportional to the crack propagation velocity.
Introduction
Many works relate rupture of materials, the first having made it possible to characterize the stress field near the crack tip. The
out-of-plane displacement field could be formulated by making the assumption of the stress plane [1, 2, 3]. At the middle of the
years 1980 [4], using the method of the caustics, these authors highlighted that a difference existed between calculated out-ofplane displacement and the 2D theory for a stationary crack. During the years 1990, using the interferometric methods, works
which related to the study of out-of-plane displacement allowed to determine the out-of-plane displacement formulations for
stationary cracks [5, 6]. The study is to determine the out-of-plane displacement fields uz(R,θ) near the crack tip during a crack
propagation by using interferometric method. The confrontation of these results with theoretical formulations (eq.7 and eq.8)
makes it possible to quantify the influence of the crack propagation velocity on the 3D and transient effects zone. The first
theoretical expression (eq. 7) is the 2D solution proportional to the first stress invariant in-plane stress [1]. The second (eq.8)
based on the Humbert's work [6] permits to seem the presence of 3D effects near the crack tip (R<0,5) for stationary cracks. In
static, the 3D effects are related at the Poisson's ratio. In dynamics, the effects related at the crack propagation velocity cannot
be neglected , for that we will speak about 3D and transient effect.
Experimental setup and optical technique
For these experiments, we used the PMMA specimens with following geometric dimensions: Length L = 290mm, Width W =
190mm, thickness h = 6mm and initial crack length ap = 1mm. According to the Rotinat work [7], the mechanic characteristics
are obtain experimentally : Young’s modulus E = 3000 MPa, poisson’s coefficient ν = 0.39. The specimen are loaded in mode I
with a applied constant stress σ. The interferograms are obtain with a Michelson interferometer who is composed of a
monochromatic source divided by a beamsplitter in two beam at right angles (α
α = π/2). They are respectively reflected by the
reference mirror and the specimen surface and are recombined in the beamsplitter. Lastly, their interferences are recorded by
a high speed CCD camera.
σ
Crack
y
r
σ
LASER beam
λ = 514.5 nm
θ
x
SEN
specimen
Beamsplitter
photomultiplier
Reference
Mirror (M)
High speed
CCD camera
Figure 1. Experimental setup to measure out-of-plane displacement: Michelson interferometer
The crack propagation is realized by impacting a cutter in an initial crack. For a good optical reflective, the specimen surface is
covered with a thin aluminum layer (≈ 50 nm). To take into account the experimental conditions, we record optical data using a
high speed CCD camera (6 Mfps, time exposure less than 170 ns) which is well adapted to the dynamic experiments (the
crack propagation velocity can reach 700 m/s in PMMA, i.e. PMMA rayleigh speed). For dynamic investigation, it is necessary
to synchronize the camera with the beginning of the crack propagation using of a simple electronic device.
Experimental results
The figure 2a is an interferogram sequence obtained during a crack propagation. The determination of out-of-plane
displacement is realized with different procedures. The first consists to demodulate the image fringes for to extract the phase
ϕ(x,y). For that, we used the MPC method developed in our laboratory which permits to exploit the fringes to start only one
image [8].
Figure 2. a) Succession of interferograms stored during a crack propagation. b) Associated phase fields cartography in the
vicinity of the crack tip (MPC).
In this figure, the presence non calculated zone is due to a too important density of fringes near the crack tip. From these fields
of phase (figure 2b), we can determine the relief uexp(x,y) at the vicinity the crack tip by using an optico-geometrical relation.
u exp ( x, y ) =
ϕ (x, y )
λ
2π
2 sin α
(1)
with wave length λ =514,5 nm and incidence angle α = π/2.
After this demodulation and this relation, we don't directly the out-of-plane displacement fields near the crack tip. The use of
Michelson interferometer imposes that the surface specimen is perfectly plane. As that is never the case, we withdraw a
reference relief uref(R,θ) (i.e. the relief when the crack did not pass yet) [9]. Finally to determine out-of-plane displacement at
the vicinity of crack tip, we should know the position of the reference mirror. This one being in experiments difficult to
determine, we preferred to add an additional constant u0 (eq.2). While placing a photomultiplier in the optical field (figure 1) we
can temporally count the fringes in a point and thus obtain the value u0. We can then write the relation of out-of-plane
displacement in cylindrical coordinates centered in the crack tip.
u z ( R, θ ) = u exp ( R, θ ) − u ref ( R, θ ) + u 0
(2)
2D and 3D theoretical formulations
In crack stationary, several experimental works show a divergence between the relief experimental and the 2D solution. This
divergence translates the presence of 3D effects near the crack tip. From experimental results obtained by using an
interferometric technique, it appears a divergence due to the presence of Poisson’s effect (i.e. 3D effects) near the crack tip
(R<0.5). Different formulations have been proposed for to express the out-of-plane displacement field [5, 6]. In our paper, we
used two theoretical formulations then we compare with the experimental data. The first solution (eq.7) is deduced on the
mechanic with the plane stress hypothesis and crack stationary, the out-of-plane displacement u2D(R,θ) (eq.3) of the surface
free of a plate of thickness h can write the following relation:
u 2 D (R , θ , z = h / 2 ) = −
with
σ xx =
KI
σ yy =
KI
(
ν
h/2
E
0
)
∫ σ xx + σ yy dz
 θ  1 − sin θ sin 3θ  − (1 − k )σ


2
2 
 2 
cos
2πr
2πr
(3)
 θ  1 + sin θ sin 3θ 


2
2 
 2 
cos
where σxx, σyy are respectively the stresses in cylindrical coordinates (R, θ) centered in the crack tip in the direction x and y. KI
is SIF in loading mode I.
We can see for k=1, one finds the Westergaard's solution [1]. We impose SIF by using the expression [11]:
a

W 
K I = σ a .f 
(4)
2
with:
3
a
a
a
a
a
f   = 1.99 − 0.41 + 18.70  − 38.48  + 53.85 
W
W 
W 
W 
W 
4
(5)
a
 is function to the crack length a and width W. In the literature [12, 13], the crack propagation velocity require to modify
W 
f
the SIF by introducing a corrective term f (V ) . Now, the SIF is noted KId and can be writing with the following formulation [14]:
K Id = K I .f ( V )
( )2
f(V) =
(1+β22 )(. β12 −β22 )
4β1β 2 − 1+β 22
with:
1/2
2


and β i =  1− V  
v
  i 
i = 1, 2
(6)
where V is the crack propagation velocity and v1, v2 are respectively the velocities of longitudinal and shear waves. In our
-1
-1
study we use a specimen in PMMA with v1 and v2 equal to 2080 ms and 1000 ms .The 2D solution is centered in crack tip:
U 2 D (R , θ ) = −
ν K Id h
E
θ  1
+ cte

 2  2 πR
cos
with cte=0
(7)
with R= r/h
The second formulation is obtained for crack stationary which has the advantage of being based on the 2D solution. That
results in the addition a term at the 2D solution which cancelled when R increase. Near the crack tip, this expression takes into
account the 3D and transient effects induced respectively by the Poisson effect and the crack propagation velocity. The 3D
formulation can be writing by the following formulation :
U 3 D (R , θ ) = −
ν K Id
h
E
 c 2 e −c1R
θ  1 
−c R
+ (1 − e 3 ) cos 


 2  2 πR 
1 + c1 R
(8)
where c1, c2 and c3 are unknown constants.
We have experimental data and we compare these results with two formulations. We choose different angles θ
(0°, ±45°, ±90°, ±135°) to characterize the experimental field as wel l as possible. The determination of c1, c2 and c3 is realized
by a minimization process between this expression and the experimental results.
Comparison between the theoretical solutions and the experimental results
We have realized few tests for different crack propagation velocity: one in static (test 1) and two in dynamics (test 2 and 3). To
estimate the influence of the crack propagation velocity of the out-of plane displacement, it is necessary to compare the
experimental data with the 2D solution and the 3D formulation. To obtain these experimental results, the different data are
plotted in table 1.
-1
Test
σ [MPa]
KId [MPa√m]
V [m.s ]
a [mm]
u0 [mm]
c1
c2
c3
1
0.75
0.699
0
60
/
2
3
5.49
7.85
2.42
3.62
540
690
38.8
47.6
10
1.4
9
4.63x10
-3
1.4
0.38
0.5
4.63x10
-3
1.7
0.4
0.4
Table 1. Summary table of the experimental conditions for the three tests
In this table, for the test 1 we don’t have the constant u0 by using the photomultiplier because we use an other method [6]. The
experimental and theoretical results are traced according to R and a length of crack a. We present curves for values of definite
angles θ for an acquisition frequency f and an integration time of camera ∆t.
Test
f [kHz]
∆t [ns]
1
/
/
2
250
500
3
125
200
Table 2. Summary table of the camera data for the three tests
From the data, we can obtain the following curves for the test 2 and 3.
[mm]
0,030
0,030
Uz_exp(0°)
Uz_2D(0°)
0,025
Uz_exp(45°)
Uz_exp(-45°)
0,025
Uz_3D(0°)
Uo
0,020
0,020
0,015
0,015
Uz_2D(45°)
Uz_3D(45°)
0,010
0,010
0,005
0,005
0,000
0,000
0
0,030
[mm]
1
2
3
R=r/h
[mm]
0,030
Uz_exp(90°)
Uz_exp(-90°)
0,025
1
2
Uz_3D(90°)
0,015
0,010
0,010
0,005
0,005
0,000
R=r/h
4
Uz_exp(135°)
Uz_exp(-135°)
Uz_2D(135°)
Uz_3D(135°)
0,020
0,015
3
[mm]
0,025
Uz_2D(90°)
0,020
0
4
0,000
0
1
2
3
R=r/h
4
0
1
2
3
R=r/h
Figure 3. Field of theoretical and experimental displacements for test N°2
4
0,030
0,030
Uz_exp(0°)
[mm]
Uz_2D(0°)
0,025
Uz_exp(45°)
[mm]
Uz_exp(-45°)
0,025
Uz_3D(0°)
0,020
0,020
0,015
0,015
0,010
0,010
0,005
0,005
Uz_2D(45°)
Uz_3D(45°)
0,000
0,000
0
1
2
3
0,030
4
R=r/h
2
3
R=r/h5
Uz_2D(135°)
0,025
Uz_2D(90°)
4
Uz_exp(135°)
[mm]
Uz_exp(-90°)
0,025
1
0,030
Uz_exp(90°)
[mm]
0
5
Uz_3D(135°)
Uz_3D(90°)
0,020
0,020
Uo
0,015
0,015
0,010
0,010
0,005
0,005
0,000
0,000
0
1
2
3
4 R=r/h
5
0
1
2
3
4 R=r/h
5
Figure 4. Field of theoretical and experimental displacements for test N°3
On these figure, we see appearing differences between the 2D solution and the other data. We can see that the 3D formulation
is superimposed with the experimental data and pass by a constant u0. The zone of 3D and transient effects is more important
during a crack propagation (3<R<6 for the two tests) that the 3D effect zone in plane stress hypothesis (R<0.5) for a stationary
crack. The advantage used a photomultiplier is that the 3D formulation must pass by this point u0 and the experimental data
must be confused. The constants c1, c2 and c3 must minimize the difference between these experimental data and the 3D
formulation.
Influence of the crack propagation velocity for the out-of-plane displacement fields
The analysis of the differences between the curves resulting from the 3D formulation and the 2D solution enables us to
evaluate the area of the zone of the 3D and transients effects according to the crack propagation velocity. For that we consider
that two curves (U2D(R,θ) and U3D(R,θ)) separate when the difference between the two becomes higher than an arbitrarily
value than we define equalizes with 0.5µm.
U 2 D (R , θ ) − U 3 D (R , θ ) < 0.5 µm
(9)
For various angles θ, figure 5 shows this divergence according to the crack propagation velocity V. On this figure, we cannot
exceed the Rayleigh speed Vmax of the PMMA which we in experiments measured with 700 m/s. Higher up this value, there is
branching of material and the crack propagation velocity no increase. An analysis of this figure seems to show that the zone of
the 3D and transients effects believes in a linear way according to the crack propagation velocity.
7
theta = 0°
theta = 45°
theta = 90°
theta = 135°
R = r/h
6
5
4
3
2
Increasing θ
1
Vmax
0
0
200
600 V[m/s] 800
400
Figure 5. Area of the 3D and transient effects zone in function of the speed V for θ = 0°, 45°, 90° and 135°.
On the following figure, we traced in coordinates (R, θ) the position of the 3D and transient effects zones on the surface free of
the specimen. We can notice a homothetic increase in the curve of detachment according to the crack propagation velocity V
and this up to large value R.
Test 1
Test 2
Test 3
V = 690 m/s
V = 540 m/s
θ
V = 0 m/s
R
-4
-3
-2
-1
0
1
2
3
4
5
6
7
Figure 6. Area of the 3D and transient effects zone in function of (R, θ) for V = 0, 540 and 690 m/s.
With the figure 6, near the Rayleigh speed (≈ 700 m/s), we can see that the area of the 3D and transient zone reaches large
size (R ≈ 6). In the direction of crack propagation to be in the stress plane conditions, it is necessary to place at 36 mm.
Conclusions and prospects
The proposed experimental method is very efficient for studies of crack propagation in dynamic conditions. The proposed 3D
expression represents correctly the out-of-plane displacement fields’ evolution and takes into account the induced variations
by the 3D and transient effects. The use of a photomultiplier obliges that the experimental data passes in this point u0 and that
enables us from the relief to obtain the out-of-plane displacement near the crack tip. The comparison between the theoretical
2D and 3D formulations makes it possible to highlight the presence of 3D and transient effects to quantify its area. The
dimension of this area seems increase, for the different angles θ, when the crack propagation velocity boost and that in a
linear way. There is necessary however to remain cautious on this last point and to carry out investigations for low crack
propagation velocities in order to have more information. The 3D formulation must be used for other materials in order to
extend its validity field. A study of displacement in the plan (Ux, Uy) must also be undertaken in order to show the influence of
the crack propagation velocity on the calculation of KId in dynamics by methods based Ux, Uy. It would be interesting to couple
this work with a study on the crack roughness (fracture appearance).
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