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Home Search Collections Journals About Contact us My IOPscience Statistical approach to evaluating active reduction of crack propagation in aluminum panels with piezoelectric actuator patches This content has been downloaded from IOPscience. Please scroll down to see the full text. 2011 Smart Mater. Struct. 20 085009 (http://iopscience.iop.org/0964-1726/20/8/085009) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 14.139.194.12 This content was downloaded on 24/02/2016 at 14:09 Please note that terms and conditions apply. 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. 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