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Statistical approach to evaluating active reduction of crack propagation in aluminum panels
with piezoelectric actuator patches
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2011 Smart Mater. Struct. 20 085009
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IOP PUBLISHING
SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 20 (2011) 085009 (11pp)
doi:10.1088/0964-1726/20/8/085009
Statistical approach to evaluating active
reduction of crack propagation in
aluminum panels with piezoelectric
actuator patches
R Platz1 , C Stapp2 and H Hanselka1,2
1
Fraunhofer Institute for Structural Durability and System Reliability LBF,
Bartningstrasse 53, 64289 Darmstadt, Germany
2
Technische Universitaet Darmstadt, System Reliability and Machine Acoustics,
Magdalenenstrasse 4, 64289, Darmstadt, Germany
E-mail: [email protected]
Received 8 April 2011, in final form 8 June 2011
Published 6 July 2011
Online at stacks.iop.org/SMS/20/085009
Abstract
Fatigue cracks in light-weight shell or panel structures may lead to major failures when used for
sealing or load-carrying purposes. This paper describes investigations into the potential of
piezoelectric actuator patches that are applied to the surface of an already cracked thin
aluminum panel to actively reduce the propagation of fatigue cracks. With active reduction of
fatigue crack propagation, uncertainties in the cracked structure’s strength, which always
remain present even when the structure is used under damage tolerance conditions, e.g. airplane
fuselages, could be lowered. The main idea is to lower the cyclic stress intensity factor near the
crack tip with actively induced mechanical compression forces using thin low voltage
piezoelectric actuator patches applied to the panel’s surface. With lowering of the cyclic stress
intensity, the rate of crack propagation in an already cracked thin aluminum panel will be
reduced significantly. First, this paper discusses the proper placement and alignment of thin
piezoelectric actuator patches near the crack tip to induce the mechanical compression forces
necessary for reduction of crack propagation by numerical simulations. Second, the potential
for crack propagation reduction will be investigated statistically by an experimental sample test
examining three cases: a cracked aluminum host structure (i) without, (ii) with but passive, and
(iii) with activated piezoelectric actuator patches. It will be seen that activated piezoelectric
actuator patches lead to a significant reduction in crack propagation.
(Some figures in this article are in colour only in the electronic version)
riveted to the host structure, leading to contact problems like
shear or additional notching effects. If a crack has already
propagated, the notching effect at the crack tip for further
crack growth could be lowered by additional drilling of a bore
hole into the crack tip [6]. Of course, for drilling holes, the
crack tip must be identified correctly in advance and has to be
sufficiently accessible for the drilling process.
As a passive and integrated alternative to reducing crack
propagation in planar structures, the potential of self-healing
concepts like adhesive capsules has been examined and
described in [10] and [12]. For that, a plastic panel host
1. Introduction
To reduce crack propagation in plane panel metal or fiber
composite structures, various different investigations have
been conducted so far, e.g. by passive or active applied or
integrated structural measures. A simple passive applied
method would be, for example, the application of an additional
and sufficiently thin plate directly on the surface of a cracked
host panel for local reinforcement [3, 4, 8, 14, 16]. However,
additional masses conflict with light-weight design constraints.
Furthermore, additional panels must be bonded, bolted or
0964-1726/11/085009+11$33.00
1
© 2011 IOP Publishing Ltd Printed in the UK & the USA
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
structure is filled with capsules that contain adhesives. If a
crack in the host panel hits and opens one of the capsules, the
adhesive exits the capsule and fills the crevasse of the crack
due to capillary action. With integrated hardeners in the host
structure, the adhesive bonds and closes the crack. This method
is fully autonomous, yet it is not reversible and the capsules
of course weaken the otherwise solid host structure. The
persistence of crack growth through sequenced and varying
loading conditions on the cracked host structure could be
reduced. Cold work or strain hardening occurs due to well and
gently applied tensile stress near the crack tip. This hardening
effect could be achieved by additional but short term single and
harmonic mechanical overloading of the host structure with
constant load amplitudes [5, 21, 22].
Piezoelectric actuators could be used for active applied
crack propagation. For example, in [24] one piezoelectric
patch is applied on the bottom surface of a three-point-bent
beam that has been cracked on the opposite or, respectively,
top surface. It has been shown via numerical simulations that
the tensile stress concentration at the crack tip could be lowered
through activation of the piezoelectric patch working against a
bending moment effected by an external loading of the beam.
Eventually, cantilevered vibrating fiber-glass reinforced
panels may enhance their durability by actively influencing the
local path of forces [20]. With applied piezoelectric patches,
propagation of delamination in the specimen due to impact
damage or notches could be prevented. It was the aim to reduce
bending stresses at the notch in the fiber-glass reinforced
host structure but, unfortunately, high scatter in the dynamic
behavior of the specimen [21], as well as the inhomogeneous
material properties of fiber-glass reinforced components are
superimposed on the effect of enhancing the durability with
applied piezoelectric patch actuators.
In this paper, the effect of piezoelectric actuator patches on
reducing damage propagation in thin homogeneous aluminum
panels will be investigated numerically and experimentally.
The basic idea is to induce mechanical compression forces near
the crack tip area of an already cracked aluminum panel to
reduce further crack propagation. First, the principal governing
relations in fracture mechanics will be introduced. Second, the
potential of activated piezoelectric actuator patches to induce
mechanical compression forces into a thin cracked aluminum
panel is summarized from [19]. Third, experimental case-bycase sample studies for crack propagation in cracked aluminum
panels with
Figure 1. Specimen for reduction of active crack propagation.
near the crack tip to nearly close it. Generally, in linear fracture
mechanics, crack propagation can be described by the well
known Paris relation
da
= CK m
dN
(1)
with the crack length a , number of load cycles N , the
cyclic stress intensity factor K at the crack tip area and
the experimental determined constants C and m for crack
mode I [17]. In (1) linear elastic material behavior is assumed
for the cracked structure. If crack propagation is to be reduced,
the cyclic stress intensity factor K should be lowered. The
cyclic stress intensity factor can be written as
K = K max − K min
(2)
with the stress intensities K max and K min depending on the
highest and lowest stresses σmax and σmin acting in the cracked
structure, leading to the stress range
σ = σmax − σmin .
(3)
This leads to another mathematical expression of the
cyclic stress intensity factor
√
K = σ πaY
(4)
with the geometry factor Y depending on the crack length a
and height h of specimen [17].
In this work the cyclic stress intensity factor K of
a cracked thin panel structure shall be reduced by active
induction of compression stresses into the crack tip area with
piezoelectric actuator patches applied to the panel’s surface
near the crack tip. For that, figure 1 shows an adequate threepoint-bent cracked specimen to verify the proposed concept
numerically and, eventually, experimentally. A thin aluminum
plate with height h and thickness t is fixed by left and right
bearing forces FBL and FBR at a distance of lB and loaded by an
external harmonic excitation force Fe . For the numerical and
experimental investigations, the tunable Fe initiates predefined
fatigue crack growth starting from a notch in the middle of the
specimen at x = l/2.
Stable and well-defined crack growth according to (1) will
be achieved if the governing stress at the crack tip
(i) no piezoelectric actuator patch applied,
(ii) a non-activated (passive) piezoelectric actuator patch
applied and
(iii) an activated piezoelectric actuator patch applied
clarify the potential of active crack propagation reduction.
2. Basic principles for crack propagation assessment
and reduction
A successful reduction of any active fatigue crack propagation
in solid continuous panel structures would depend on the
ability to induce mechanical compression forces into the area
KI
σ =√ ,
πa
2
(5)
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Figure 2. Finite element of cracked specimen.
Figure 3. Finite element of piezoelectric actuator patch and
connection (radiating lines) to the host structure.
with the stress intensity factor K I for crack mode I and crack
length a , is applied to the structure. The minimum governing
bending moment
σ Iz
(6)
M=
h/2
piezoelectric patch actuators have been modeled in finite
elements. The finite element software ANSYS, Release 11, was
used [1].
with the geometrical moment of inertia Iz and half the height
h/2 will be needed to load the specimen for crack propagation
that is necessary for crack propagation reduction tests in the
following investigations.
3.1.1. Cracked host structure. Figure 2 shows the finite
element model of the cracked host structure without any
piezoelectric actuator patch applied to it.
In ANSYS, the structure element SOLID95 is used to model
the host structure (element form: tetrahedron) and the crack
tip (prism). Fine resolution is used to model the crack tip
area and the contact point of the external harmonic excitation
force Fe from figure 1. The material parameters for the
aluminum specimen are: density ρ = 2700 kg m−3 , Young’s
modulus E = 70 kN mm−2 and Poisson ratio ν = 0.33. The
geometrical data for the whole structure according to figure 1
are: length lB = 525 mm, height h = 50 mm and thickness
t = 1.5 mm. For the numerical calculation of the cyclic stress
intensity factor K , Murti, Valiappen and Lee proposed a socalled QUARTER-POINT arrangement of knots for the structure
element SOLID95, which is also used in ANSYS, see [15] for
details.
3. Numerical investigation of active reduction of
crack propagation
As seen in (1) and (4), crack growth propagation could be
lowered by reducing the cyclic stress intensity factor K or,
respectively, the stress range σ . So the prime goal of this
work is to lower σ by inducing counteracting mechanical
stresses into the crack tip area via activated piezoelectric
actuator patches applied near the cracked surface of the cracked
panel shown in figure 1. For measuring this σ reduction, the
cyclic stress intensity factor K at the crack tip in the host
structure is calculated and compared numerically within three
cases:
X1 host structure without additional applied actuators near the
crack tip,
X2 host structure with additional applied but non-activated
(passive) actuators near the crack tip and
X3 host structure with additional applied and activated
(active) actuators near the crack tip.
3.1.2. Piezoelectric actuator patch. With the structure
element SOLID226, the software ANSYS models properties of
a piezoelectric actuator patch following the d31 -effect and
allows its transfer or, respectively, connection into the host
structure modeled by the SOLID95 elements (section 3.1.1). In
addition to material and geometrical data, electrical data like
the piezoelectric constant d31 , coupling factor k31 and voltage
U are taken into account. Figure 3 shows the finite element
model of the actuator patch with the already connected knots
(linked lines) to the host structure.
For the material type PIC151 from PICERAMIC [18], some of
the important material and electrical data for the piezoelectric
patch are: density ρp = 7800 kg m−3 , elastic compliance
constant S11 = 15 × 10−12 m2 N−1 , piezoelectric constant
d31 = −210 × 10−12 C N−1 and coupling factor k31 = 0.38.
The most suitable position and alignment of the activated
actuator patch near the crack tip that enforces the highest
decrease of the cyclic stress intensity factor K compared
to passive and non-applied patches was selected by numerical
investigations by the authors in [19]. In the following, a brief
summary of the investigations will be given.
3.1. Finite element simulation model
For simulating the proposed crack propagation reduction
method numerically, the cracked host structure and the applied
3
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Table 1. Cyclic stress intensity factors K Xi for crack length
a1 = 13 mm and a2 = 17 mm and relative change of K X3 against
K X2 .
√
a1,2 (mm) Case Xi K Xi (MPa m)
K reduction
1 − (K X3 /K X2 ) (%)
13
—
17
The geometrical data for the patch are: thickness tp = 0.3 mm
in z -direction, length lp = 20 mm in y -direction and height
h p = 10 mm in the x -direction (figure 3). In [19], the finite
element model of the piezoelectric actuator patch has been
verified by comparing analytical with numerical calculations
of the change of length
—
−0.75
This investigation evaluates statistically the influence of
controlled reinforcement of a crack tip area by actively
inducing mechanical compression forces near the crack tip
area in a cracked and loaded aluminum specimen for crack
propagation reduction. Inducing mechanical stresses will be
realized with activated piezoelectric patch actuators on both
sides of the cracked specimen (figure 4). In total, 45 similar
cracked sample specimens were tested to investigate reduction
of crack propagation within the three different cases described
in section 3:
(7)
due to an electrical field strength
U
,
tp
−0.68
4. Experimental investigation
with the strain—neglecting any external mechanical loading—
in the y -direction
Sp = d31 E p
(8)
Ep =
8.94
8.84
8.78
11.16
10.82
10.74
forces via activated applied piezoelectric actuator patches
located according to figure 4 were conducted for two different
crack lengths a1 = 13 mm and a2 = 17 mm. Table 1 shows the
stress intensities K Xi for each case X1 (none), X2 (applied
but passive patch) and X3 (applied and active patch).
Compared to K X1 of the host structure without any
patches, K X2 was lowered because of passive strengthening
of the host structure when applying additional passive patches
in case X2. The relative effect of reduction of crack
propagation by lowering K X3 with an activated patch
compared to K X2 with passive patch is 0.68–0.75 %, which
is rather small but still significant (table 1). The following
experimental investigations show that lowering the crack
propagation rate is possible—even when changes in stress
intensity by activated patches seem apparently small.
Figure 4. Best position to apply an active piezoelectric actuator
patch on the cracked host structure for lowering stress intensity K
near the crack tip by induced mechanical compression forces.
lp = Splp
X1
X2
X3
X1
X2
X3
(9)
with the electrical voltage U . With U = 100 V, the
analytically calculated change of length according to (7)
becomes lp,analyt = −1.40 μm. Maximal change in the
numerically determined length lp,numeric = −1.43 μm at the
free end due to numerical finite element simulation matches
almost completely the analytical calculations.
15 specimens without (case X1),
15 specimens with passive (case X2) and
15 specimens with active (case X3)
piezoelectric patch actuators. All host structures have the
same material and geometrical properties, the same loading
conditions and the same initial crack lengths as a starting
point for the crack growth investigation. However, crack
propagation will differ in each specimen in each case Xi . This
is due to the fact that maintaining identical load conditions and
preparation of identical host structures, identical application
of piezoelectric patches on individual specimens with identical
material properties in host structures and piezoelectric ceramic
patches is impossible. Therefore, mean values of crack
propagation versus number of load cycles will be derived
numerically and analytically on the basis of experimental data
for each of the three cases X1, X2 and X3.
3.2. Positioning of the actuator patch for best crack
propagation reduction results
It was shown by case-by-case studies in [19] that single
activated actuator patches applied alongside the crack growth
direction on each side of the cracked area of the host structure
near the crack tip lead to the highest mechanical compression
forces that reduce the stress intensity factor K , figure 4.
Numerical simulation of lowering the cyclic stress
intensity factor K by inducing mechanical compression
4
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Figure 6. Cracked specimen, host structure with propagated fatigue
crack.
Figure 5. Cracked specimen, host structure.
4.1. Specimen and test rig
For all 45 aluminum and initially cracked specimens, the host
structures with and without passive or, respectively, activated
piezoelectric actuator patches are similar, figure 5.
The numeric dimensions in figure 5 refer to the
dimensions in figure 1. A thin surrounding area with length
lm = 80 mm was milled to a thickness of tm = 1.5 mm
for fast fatigue crack propagation and sufficient reduction of
crack propagation by thin actuator patches. The thin area
with tm is a compromise between loading the specimen for
crack growth without buckling in the cracked area but allowing
thin piezoelectric patch actuators to have a noteworthy effect
on crack propagation reduction. Therefore for the purposes
of the first limited examinations based on simple samples,
the geometry of the specimen differs from the international
standards like ISO 12135 [11], ESIS-P2 in Europe [9] or
ASTM 1820 in the USA [2]. These are normally strongly
recommended for confident tests [13]. The actual propagationrecording during experimental crack growth of the fatigue
crack starts, therefore, at the end of a mechanically sawn
initial notch with length anotch = 12 mm for each of the 45
specimens to ensure reproducibility, since no defined crack
tip could be realized by a sawn notch this time. Figure 6
illustrates the propagated fatigue crack length in one specimen,
subdivided into the initial notch length anotch , an initial crack
length ainitial = 1 mm and the propagated crack length aXi, j for
j = 1, 2, . . . , J = 15 comparative experimental comparisons
in each case Xi .
Due to alternating three-point-bending described in
figure 1 and experimentally conducted in section 4.1, first an
initial crack length ainitial propagates, starting from the sawn
notch tip at anotch (figure 6). To ensure reproducibility and
comparability of fatigue crack propagation in all 45 specimens,
recording of stable crack growth in all specimens starts from
the initial crack length ainitial = 1 mm. Figure 7 shows the
milled crack area and a bonded piezoelectric patch actuator
Figure 7. Cracked specimen, host structure with piezoelectric patch
actuator.
with dimensions and properties illustrated in section 3.1.2 on
one side of the specimen. The distance between the top edge
of the specimen and the top edge of the patch is lp = 20 mm.
A second patch with same properties and position is bonded on
the other side of the host structure.
The whole test rig for investigating the reduction of crack
propagation in a bent aluminum specimen via piezoelectric
actuator patches is shown in figure 8.
Figure 9 shows details of the assembly of the test rig as
well as the measuring chain to load the aluminum specimen
with or without piezoelectric patch actuators.
For controlled crack growth, a shaker excites the specimen
with the excitation force Fe against two counteracting bearing
forces FBL and FBR at two limit stops (figure 9). A punctual
point of contact at the left and right bearing is realized by
a rectangular shaped metallic prism (see cutout at the right
bearing). In the measuring chain in figure 9, the shaker
{1} loads the specimen {2} for three-point-bending with a
preloaded harmonic force amplitude F̂e , measured by a load
cell {3}. The measured force signal is digitally processed {4}
5
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Table 2. Properties of specimen for cases X1, X2 and X3.
Number of
Case specimens
Xi
J
Patch Patch Fe,pre Fe,min Fe,max f
U
applied active (N) (N) (N) (Hz) (V)
1
2
3
No
Yes
Yes
15
15
15
—
No
Yes
150
150
150
50
50
50
250
250
250
10
10
10
—
—
100
piezoelectric patches are synchronized with pulsating tensile
stresses induced by the shaker. Recording of stable crack
growth starts at the sawn notch tip plus the initial fatigue crack
length ainitial = 1 mm, figure 6.
4.2. Crack propagation without and with bonded
passive/active piezoelectric patch actuators
Figure 8. Test rig for investigating crack propagation.
The properties of the specimen for each case Xi without
patches applied (X1), with passive patches applied (X2) and
with active patches applied (X3) are summarized in table 2.
As examples, for two out of J = 15 specimens for each
case Xi , crack lengths aXi, j within 15 × 104 load cycles N
for specimen j = 1, 2 are shown in figure 10 to evaluate both
experimental and analytical propagation trends.
Experimental crack length propagation data were retrieved
by visual checkpoints of aXi, j versus load cycles N . Now, to
estimate the mean values of crack propagation for each case
Xi , the Paris relation (1) will be adapted to the experimental
data seen in figure 10 to determine the analytical crack
propagation curve. Analytical crack length propagation can
be adapted in two ways. First, the constants C and m in the
Paris relation (1) can be approximated by regression analysis
between experimental crack propagation rate data for da/d N
and the stress intensity factor K . Second, crack propagation
could be estimated by direct fit of the Paris relation to measured
propagation data, which was the case in figure 10.
and visually checked {5} with an oscilloscope. The crack
length aXi, j for each case Xi and specimen j will be also
visually checked with a microscope {6}. If piezoelectric
actuator patches {7} are applied and activated on the specimen,
a pulsating control voltage U = 100 V at a frequency f =
10 Hz is applied and will be checked and compared visually
together with the force signal.
To initiate and propagate stable crack growth according
to (1), the shaker loads the specimen with the pulsating
excitation force amplitude F̂e = 100 N at f = 10 Hz
with the preload Fe,pre = 150 N, leading to a minimum
of Fe,min = 50 N and a maximum Fe,max = 250 N or a
stress ratio R = 0.2. Phase shift between the pulsating
control voltage U and the pulsating shaker force Fe is zero.
When the shaker force reaches its maximum Fe,max , the
control voltage U reaches 100 V. When the shaker force
reaches its minimum Fe,min , the control voltage U is zero.
So, pulsating compression forces actively induced by the
Figure 9. Schematic test rig for investigating crack propagation. Left: CAD sketch. Right: measuring chain.
6
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Figure 10. Experimental and analytical crack propagation for two specimens j = 1, 2 for each case Xi : X1 without and with bonded
passive/active piezoelectric patch actuators (top), X2 with applied but passive patch (middle) and X3 with applied and active patch (bottom).
4.2.1. Regression analysis. In a first option, constants C and
m could be estimated by a linear regression analysis. For that,
a scatter plot of the approximated crack growth rate
da
a
ak+1 − ak
≈
=
dN
N
Nk+1 − Nk
Table 3. Approximated constants C Xi and m Xi by regression
analysis.
Constants case X1
CX1 × 10
(10)
0.09
at adjacent experimental checkpoints k and k + 1, with k =
1, 2, . . . , K checkpoints are plotted in figure 11.
Then, the stress intensity factor K (4) with the stress
range
(Fe,max − Fe,min )lB h
(11)
σ =
4 Iz
2
Constants case X2
m X1
CX2 × 10
3.67
9.9
−7
Constants case X3
m X2
CX3 × 10−7
m X3
1.68
0.71
2.74
crack propagation according to the Paris relation (1) in
logarithmic scale for each case Xi with
daXi
(13)
log
= log(CXi ) + m Xi log(K Xi ).
dN
and the geometry factor [23]
a π 4
0.924 + 0.199 1 − sin X2i,hj
a π Y (aXi, j , h) =
cos X2i,hj
a π 2h
Xi, j
,
tan
×
aXi, j π
2h
−7
The constants CXi and m Xi for each case Xi can be
approximated in (13). As a result, it can already be seen
qualitatively that if no piezoelectric patch is applied in case
X1, the overall level and the gradient of propagation rate
daX1 /d N is higher than the level and gradient of propagation
rates in cases X2 and X3. Quantitatively, the constants CXi and
m X1 , with m X1 representing the gradient of the approximated
regression line, are summarized for each case Xi in table 3.
By comparing cases X2 and X3 for applied passive and
applied active piezoelectric patches, it can be seen qualitatively
(12)
are adapted with varying C and m for best fit of a linear
regression line through the experimental scatter plot in
figure 11. The regression line then represents the averaged
7
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Figure 12. Numerical and analytical estimated stress intensity factor
K Xi versus predetermined crack length a for no applied patch for
case X1 (top), applied passive patch for case X2 (middle) and applied
active patch for case X3 (bottom).
Figure 11. Approximated regression line of crack propagation rate
versus stress intensity factor for no applied patch, case X1 (top),
applied passive patch, case X2 (middle) and applied active patch,
case X3 (bottom).
analysis. Therefore, another approach to determine mean crack
propagation rates will be followed by direct fit of the Paris
relation in section 4.2.2.
and quantitatively with m X2 < m X3 in table 3 that the gradient
of the approximated regression for case X3 is higher than
the gradient for case X2. This would lead to the assumption
that the crack propagation rate with an activated patch is
higher then the crack propagation rate with a passive patch.
However, constant CX2 CX3 represents a relatively lower
level of crack propagation daX3 /d N . High scatter of the
relatively low number of experimental check points, though,
makes it difficult to have high confidence in the regression
4.2.2. Direct fit of the Paris relation. The Paris relation (1)
can be fitted to the experimental checkpoints in figure 10
directly by varying the constants C and m [7]. This procedure
was conducted for calculating the analytical curve in figure 10.
For that, K Xi for each case Xi can be calculated by finite
element simulation according to section 3. In figure 12,
8
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Table 4. Fitted constants CXi, j and m Xi, j for every specimen j = 1, 2, . . . , 15 in each case Xi and averaged constants C̃Xi and m̃ Xi by direct
fit of the Paris relation.
Constants case X1
No. of specimen j
CX1, j × 10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
3.66
0.004
2.95
0.25
7.50
7.60
8.80
0.30
0.24
8.65
0.23
0.39
1.32
5.90
0.29
−7
Constants case X2
m X1, j
CX2, j × 10
1.85
5.00
2.27
3.50
2.00
2.00
1.50
3.00
3.20
1.82
3.50
3.25
2.76
1.70
3.00
1.25
3.83
6.65
3.08
19.50
17.50
0.06
1.22
4.43
3.85
0.14
8.00
9.65
1.98
3.53
Constant case X1
Averaged see (15)
and (16)
C̃X1 × 10
Mean (˜)
0.96
−7
m̃ X1
C̃X2 × 10
2.69
2.73
−7
Constants case X3
m X2, j
CX3, j × 10−7
m X3, j
2.70
2.00
1.84
2.27
1.40
1.40
3.90
2.57
2.19
2.24
3.60
1.70
1.72
2.30
2.30
8.10
1.21
4.10
0.09
16.50
0.29
9.90
0.004
3.33
2.46
0.09
14.50
1.40
3.90
1.20
1.64
2.70
2.00
3.60
1.50
3.10
1.80
5.00
2.00
2.50
3.60
1.60
2.40
2.00
2.37
Constant case X2
eight checkpoints of numerically calculated K Xi versus
predetermined discrete a -value relations are plotted. However,
K (4) can be also estimated analytically with the given stress
range σ (11) and geometry factor Y (12). For that, Y will
be varied iteratively until (4) fits best a curve through the
numerical discrete checkpoints in figure 12. Both numerical
checkpoints and analytical fitting results for predetermined
crack lengths a between 0 and 7 mm, starting from ainitial =
1 mm, are shown in figure 12 for cases X1 (no patch), X2
(passive patch) and X3 (activated patch).
With the now given relation between crack propagation
rate daXi, j /d N and estimated stress intensity factor K Xi ,
the crack propagation chart describing crack growth aXi, j
versus load cycles N for each case Xi will be calculated by
mathematical integration of the Paris relation (1). Inserting (4)
into (1) and with separation of da and d N , the Paris relation
then becomes
Ñ
ã
−(1/2)m
−m
a
{Y (a)} da =
C{σ }m π (1/2)m d N. (14)
ã0
−7
Constant case X3
m̃ X2
C̃X3 × 10−7
m̃ X3
2.28
1.29
2.52
comparison between cases X1, X2 and X3 regarding crack
propagation reduction. On a logarithmic scale, the Paris
relation (13) for each case Xi can be easily averaged with
da
1 J
=
log(CXi, j )
log
d N Xi
J j =1
J
1
+
m Xi, j log(K Xi )
(15)
J j =1
to determine mean values C̃Xi and m̃ Xi for the constants CXi, j
and m Xi, j for each case Xi out of j = 1, 2, . . . , J = 15. These
values are summarized in table 4 at the bottom. Again—and
according to table 3 —the relation for constant m is m̃ X3 >
m̃ X2 , meaning that the gradient of da/d N for case X3 is higher
than it is for case X2, or, more precisely, 11% higher. However,
the averaged constant C̃X3 is about 50% of the value of C̃X2 ,
which leads, eventually and according to (13), to a smaller
crack propagation rate as seen later in figures 13 and 14.
The mean deviation
da
S
d N C̃,m̃,Xi,ã
2
J
1
da
da
=
−
(16)
J (J − 1) j =1
d N Xi, j,ã
d N C̃,m̃,Xi,ã
0
However, the constants C and m are still unknown. They
are determined by iterative numerical fitting of the integral (14)
to the experimental measured crack length and load cycle
relation plotted in figure 10. The fitted crack length a versus
load cycle N relations then correspond very well to the
experimental relations as seen in figure 10. Table 4 summarizes
the fitted constants CXi and m Xi for each of the 15 cracked
specimens due to the estimated K Xi –a relation shown in
figure 12 for each case Xi .
Eventually and with the known constants CXi, j and m Xi, j ,
the Paris relation (1) is evaluated for every specimen j =
1, . . . , 15 and then averaged for each case Xi (section 4.2.3).
describes the scatter of the mean Paris relation estimated
in (15) for an averaged crack length ã = 4 mm, starting from
ainitial = 1 mm. In (16), the left term of the square root
represents the crack propagation rate for specimen j in the
specific case Xi at the average crack length ã . The right term
of the square root represents the geometric mean value of the
crack propagation rate (15) of all 15 specimens in case Xi with
averaged constants C̃Xi and m̃ Xi at ã . For estimating upper
and lower bounds for mean deviation via the Paris relation, the
4.2.3. Averaging the Paris relation for each case Xi. Finally,
the Paris relation is averaged for each case Xi to allow
9
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
Figure 14. Summarized mean (——) and mean deviation (– – –)
curves of crack length propagation aXi versus load cycles N
according to no applied patch for case X1 (top), applied passive
patch for case X2 (middle) and applied active patch for case X = 3
(bottom).
activating an applied active piezoelectric actuator patch near
the crack tip with respect to an applied but passive actuator
on a similar specimen under similar boundary conditions. It
needs to be pointed out that even with the averaged constant
m̃ X2 < m̃ X3 , but with the averaged constant C̃X2 C̃X3 ,
averaged crack growth reduction in case X3 can be observed.
The statistically assessed reduction rate of crack propagation
through reinforcement by an applied but passive piezoelectric
actuator patch adds up to 12%. Both crack propagation
reduction rates for cases X2 and X3 as well as the propagation
rate when no piezoelectric actuator patch is applied, case X1,
deviate about 10% when testing 15 specimens for each case as
described in section 4.1.
The crack propagation reduction rate
γa max,i+1 =
Na max,i+1 − Na max,i
Na max,i
(17)
is determined by comparing the averaged number of load
cycles Na max,i in figures 13 and 14 for case Xi and X(i +
1) necessary for a maximum crack length amax = 7 mm.
Comparing the averaged number of load cycles Na max,2 =
138 × 103 with applied but passive piezoelectric patches for X2
with the averaged number of load cycles Na max,1 = 123 × 103
without applied piezoelectric patches for case X1 to reach
amax , according to (17) the crack propagation reduction rate
becomes γa max,2 = 12.2%. In the same way, comparing the
averaged number of load cycles Na max,3 = 165 × 103 with
applied and activated piezoelectric patches for case X3 with
the averaged number of load cycles Na max,2 with applied but
passive piezoelectric patches for case X2 to reach amax , the
crack propagation reduction rate becomes γa max,3 = 19.6%.
Figure 13. Mean, mean deviation, measured and envelope curves of
crack length propagation aXi versus load cycles N according to no
applied patch for case X1 (top), applied passive patch for case X2
(middle) and applied active patch for case X = 3 (bottom).
constant C only is varied in (15) until the mean deviation value
according to (16) is reached. In the same way, the upper and
lower envelope of measured crack propagation curves can be
estimated visually by varying the constant C in (13) until the
best fit to the steepest and flat measured curve is achieved.
Figure 13 shows the mean value curve, mean deviation
curve, all measured curves and the envelope curves of crack
length propagation aXi versus load cycles N for case X1
without a patch, X2 with a passive patch and X3 with
an activated patch. Mean and mean deviation curves are
summarized in only one graph for better distinction in
figure 14.
Figures 13 and 14 show a significant statistically assessed
reduction rate of about 20% in crack propagation when
5. Conclusion
It was shown with numerical finite element simulations
that the stress intensity factor in a thin cracked aluminum
plate can be reduced by up to nearly 1% with applied and
10
Smart Mater. Struct. 20 (2011) 085009
R Platz et al
active piezoelectric actuator patches—compared to the crack
propagation reduction by applied but not activated actuators. A
lowered stress intensity factor should also lower the effective
crack propagation rate. Experimental statistical examination
based on 45 sample specimens only seems to roughly verify
the relatively small reduction of the stress intensity and shows
a significant reduction of crack growth propagation rate even
with great scatter. However, and according to the well known
Paris relation, the crack propagation rate also depends on the
chosen constant factors, which were averaged via experimental
validation. The experimental investigations considered three
different cases, each with sets of 15 similar cracked host
structure specimens (i) without, (ii) with passive and (iii) with
activated piezoelectric actuator patches.
It could be shown statistically based on sample
examination that by activating a low voltage piezoelectric
patch actuator with 100 V, averaged active crack propagation
reduction of about 20% relative to applied but non-activated
patches is possible. An activated low voltage piezoelectric
patch actuator near the crack tip of a fatigue cracked aluminum
host structure induces mechanical compression forces in
the crack tip area sufficient to reduce crack propagation
significantly. For the 15 specimens examined in each case,
the crack propagation rate reduction varies with approximately
10% deviation.
As an outlook, further investigations to optimize the
height and control of the applied voltage as well as the
number, size, location and alignment of the piezoelectric patch
actuators may clarify the actual limits of an active crack
propagation reduction technology. For more confidence in
the statistical results for crack propagation reduction, more
specimens and a specimen geometry closer to a standard
specimen geometry like ASTM E399 could be used. Also,
more precise location of where the crack begins to grow needs
to be assured, for example making a finer notch by spark
erosion. Additionally, questions like uncertainty, reliability
and cost-effective handling of such active methods to reduce
the danger of fatigue cracks in thin load-carrying panels need
to be pursued.
[6]
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[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Acknowledgment
[20]
The authors would like to thank the Deutsche Forschungsgemeinschaft DFG for supporting this project within the Collaborative Research Centre SFB 805.
[21]
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