INTEGRITY OF PIEZO-COMPOSITE BEAMS UNDER HIGH CYCLIC ELECTRO-MECHANICAL LOADS - EXPERIMENTAL RESULTS Lucy Edery-Azulay and Haim Abramovich Faculty of Aerospace Engineering, Technion, I.I.T., Haifa 32000, Israel [email protected]; [email protected]; ABSTRACT One key issue in the study of the structural integrity of smart structures is the research of high cyclic electro-mechanical (E/M) loading. To truly understand the integrity of smart structures, investigation must include active sensors/actuators embedded or bonded as a part of a smart structure, and to be loaded in a combined electro-mechanical cyclic loads as the way they will be used in real life applications. A better understanding of the effects of cyclic E/M loading is necessary to characterize the lifetime behavior of active ceramic components. The present research is a part of a comprehensive1 experimental study dealing with the structural integrity of smart structures. A laminate composite made of graphite-epoxy with a quasi-isotropic lay-up was used as a host structure for all the specimens. Two commercially available piezoelectric patches (PZT-5H, lead zirconate titanate) were used: ACX patches from QP15N Cymer Inc. U.S.A. and PIC255 from PI Ceramics, Germany. The first part of the present study investigates the behaviour of piezolaminated beams with embedded or surface bonded patches, subjected to axial tension/compression loads. Seven different specimens (numbered as AX-EM) with embedded piezoceramic patches and another two specimens (numbered as AX-SM) with four piezoceramic patches bonded to the surface of each host structure (one pair of ACX and one pair of PIC-255 patches) were tested. The second part of the present study investigates the behaviour of seven specimens (numbered as BEN1-7) with four piezoelectric patches, one pair of ACX and one pair of PIC-255, bonded on each side of the host structure. These specimens were subjected to bending cyclic loads using four-point bending test setup. The degradation in the sensing abilities of the piezoelectric patches for an increasing number of E/M load cycles was monitored and recorded. The aim of the present research is to examine the changes in the piezoceramic sensing capabilities as a piezo-laminated structure is undergoing an increasing number of electro-mechanical load cycles. These two types of integrated loads, called Electro-Mechanical (E/M) loads can be applied in-phase (both loads cause either tension or compression) or out-of-phase (one load causes tension while the other causes compression and vice versa). It is expected that the main impact of the present research will be its capability to predict the structural integrity of a given smart structure as a function of its use, yielding a balanced design with an enhanced survivability and a higher confidence in the usage of piezoelectric patches. 1 Yet, in general, structural life prediction involves analyzing data gathered from a large number of tests. The present review does not pretend to cover the whole spectrum of experiment that have to be done. INTRODUCTION The key issue in the structural integrity of smart structures used for aerospace industry is its applied cyclic loading. Therefore, to truly understanding the integrity of smart structures, investigation must include active sensors/actuators embedded or bonded as a part of a smart structure, and to be loaded in a combined electro-mechanical cyclic loads as the way they will be used in real-life applications. Surveying the published experimental studies one can found only a few relevant experimental reports where the life span of the piezoceramic patches is studied in connection to two types of stress conditions: tension/compression due to axial loads and bending stresses. Bronowicki et al. [3] performed one of the first studies on the structural integrity of a smart structure with embedded active sensor/actuator. Graphite/epoxy laminates with embedded active PZT were loaded with various external tensile mechanical loads, and then completely unloaded. PZT’s changes and degradation in its performance were monitored. Paget et al. [4] evaluated the performance of embedded piezoceramic transducers used as Lamb-wave2 generators. Composite specimens were subjected to static tensile and compressive loads as well as fatigue loading. Mall and Coleman [5] investigated the effect of embedded active PZT sensor/actuator on tensile strength and fatigue behavior of quasi-isotropic graphite/epoxy laminates. A commercially PZT (ACX QP15N)3 was inserted in the two middle 90° plies. One of the first studies that tested the influence of combined electric and mechanical loads, (E/M fatigue) was presented by Mall and Hsu [6]. The behavior of quasi-isotropic graphite/epoxy laminates with embedded PZT was studied. Only tensile mechanical loads were applied while the embedded PZT was subjected to positive or negative AC voltage, giving either an in-phase or out-of-phase. Yocum and Abramovich [2] presented another pioneering study, in which they investigated the functionality of an embedded piezoceramic actuator/sensor undergoing fully reversed tension compression E/M fatigue. The tested specimens were similar to those used in Ref. [6] and in the present study. To date, only a few studies dealt with the life span of piezoelectric sensor/actuators under bending stress conditions, although the integrity of smart structures must refer also to its behavior in the presence of bending stress conditions. Mitrovic et al [7], investigated the behavior of piezoelectric materials under combined electro-mechanical loading. Experimental results indicate strong dependence of both elastic and piezoelectric properties on the operating conditions (mechanical and electrical). Yet, only static loads were applied. Thielicke et al [8] investigated the reliability of the piezoelectric patches themselves and their adhesion on the substrate. PZT patches were subjected to cyclic mechanical loads in a four point bending system. Jones et al [9] characterized the strain accumulation of a ferroelectric/ferroelastic ceramic PZT patch under mechanical cycling using four-point bending bar geometry. The present research is a part of a comprehensive1 experimental study dealing with the structural integrity of smart structures. The first part of the present research is a direct continuance of Ref [2]. It extends the investigation of piezo-laminated beams under uniaxial electro-mechanical cyclic loads. The surface mounted piezo-composite specimens were loaded with asymmetric E/M loads profile ranged from 100% to 150% limit stresses in in-phase conditions. The behavior of two commercial PZT patches, ACX and PIC-255, was studied. Degradation in the sensing abilities of the piezoelectric patches was monitored for an increasing number of load cycles. It was found that an out-of-phase tensioncompression (T-C) E/M fatigue caused larger damage to the PZT than in-phase T-C E/M fatigue for a given applied maximum stress. Yet, far less damage was experienced by the PZT in the T-C tests than in T-T tests (as compared with Ref. [6]). A stable output was found for the surface mounted PZT patches loaded by in-phase asymmetric T-C, even when the applied loads were 150% of the limit stress. The second part of the present research investigates the behavior of a lamina with a surface mounted PZT subjected to bending cyclic loads. The specimens were mounted in a four point bending fixture. Each specimen contained two types of commercial piezoelectric patches, ACX and PIC255. The specimens were exposed to different load types: combined symmetric electro-mechanical cyclic loads, 2 Lamb wave generation is a way of determining damage in a structure. The system of embedded PZT's, as a Lamb wave generator, would be part of a built-in structural health monitoring system. 3 Similar patches were used in the present research in- or out-of-phase, only electric loads and only mechanical loads. Experimental results are presented and discussed with respect to the results of the first part of the present research, the uniaxial tension/compression loads, Ref [2, 6]. A good correlation was found. In addition, the application of only tensile electric load cycles was found to have a significant destructive effect on the patch life span. It is expected that the main impact of the present research will be its capability to predict the structural integrity of a given smart structure as a function of its use, yielding a balanced design with an enhanced survivability and a higher confidence in the usage of piezoelectric patches. Experimental system and equipment A 50 tons MTS servo-hydraulic test machine was used to apply a monotonic tensile/compression cyclic load. A KEPCO 500 power supply/ amplifier together with an IEC F31 Function generator (to alter the frequency) were used to supply an AC electric load to the specimens. A computer code uploaded on a VISHAY '6000 system' machine was used to store strains, applied loads and output sensing voltages. The applied load and the PZT sensing voltage are read in parallel on two oscilloscopes: an AGILENT S4624A oscilloscope and a TEKTRONIX 2430A oscilloscope. A hand held DVM was used to measure the capacity and resistance of the patches. All test instructions are input through a special program into a computer which then controls the MTS machine. The test setup is presented in Fig. 1a, while the tested specimens can be either axially loaded (Fig. 1b) or under four point bending loads (Fig. 1c). (b) (c) (a) Figure 1 – (a) The test setup (b) A specimen with buckling guide (c) Four point bending test setup The specimens A laminated composite beam made of graphite-epoxy with a quasi-isotropic lay-up4 was used as a host structure for all the specimens. Two commercially available piezoelectric patches (PZT-5H, lead zirconate titanate) were used: ACX patches from QP15N Cymer Inc. U.S.A. and PIC255 from PI Ceramics, Germany. The ACX patches are pre-packaged with an outer protective polyimide5 layer and extended lead wires. The PIC-255 patches have exposed silver electrodes and the wires (the terminals) have to be soldered manually by the user. Table I summarizes some of the main material properties of these two piezoceramic materials. The embedded specimens group was characterized by a sole ACX piezoceramic patch being inserted in the middle plane of the laminated composite beam (see Fig.2a), while the surface mounted specimens had four piezoelectric patches bonded externally: one pair of ACX patches and one pair of PIC-255 ones. Each pair was glued on both sides of the laminated composite beam. Back-to-back strain gauges were placed at the mid-span of all the specimens, and were referred as S.G. 11-12. Some additional strain gauges were used depending on the case load (see Fig.2b). The axially loaded specimens were clamped from both sides. As our tests were to be tensioncompression tests, we used a buckling guide in order to prevent the possibility of buckling6, (see Fig.1b). For the bending experiments, the specimens were mounted in a four-point bending fixture with inner and outer spans of 170 and 200 mm. This arrangement yields a constant tension/compression strain region located between the inner two reactions (see Fig.1c and Fig. 2d). The PZT patches (one pair or more) were glued on each side of the host structure in this constant strain region. The specimen lay in a stationary system, and was subjected to compressive mechanical loads applied by the MTS machine. Figs. 2(a-d) illustrate schematically both the embedded and surface bonded specimens and the four-point bending test setup. Electro-mechanical cyclic loads Changes in piezoceramic sensing capabilities were examined while a piezo-laminated structure was undergoing an increasing number of electro-mechanical load cycles. E/M loads can be applied in two different combinations, an in-phase or an out-of-phase condition. In-phase conditions refer to the case where both the mechanical and the electric loads would induce on the patch and the specimen a strain with equal tendency (either tension or compression). The out-of-phase condition refers to the case were the two types of loads would induce on the PZT patch and the specimen7 opposite strain tendencies. Both the electric and the mechanical loads were applied at a frequency of 10 Hz. Using a free strain test analysis with frequencies between 1-10 Hz, the 10 Hz frequency was found to have the least hysteresis for both piezoceramic actuators, ACX and PIC-255. 4 A composite laminate made of graphite-epoxy with [0/±45/90]S and [0/±45/90]3S was used as a host structure for axial and bending test, respectively. 5 The protective polyimide improves the handing of the brittle PZT material while providing mechanical stabilization and electrical insulation. 6 For more details about the complicated process of manufacturing these specimens and the design of the buckling guide see Ref. [1]. 7 The application of a negative electric field on a piezoceramic material whose piezoelectric coefficient d31 has a negative value, would induce a tension strain, ε = d 31 E , where ε is the induced strain and E is the applied electric field. Table I – Piezoelectric patches material properties Unit ACX T ε11 /ε0 Relative dielectric permittivity Density Piezoelectric charge coefficient 1800 7700 3 kg/m d33 d31 Elastic modulus E11 E33 Maximum suffer volt Size (l ⋅b ⋅ t) 8 Calculated Capacity m/V or C/N N/m 304.8 mm 228.6 mm Embedded ACX PZT 400⋅10- 12 12 -180⋅10- 12 10 6.2⋅10 10 10 4.8⋅10 ±400 46 x 26 x 0.2 75 6.9⋅10 10 5.5⋅10 ±190 46 x 26 x 0.13 116 S.G.-14 S.G. -11-12 12 -179⋅10- Volt mm ηF 1.03mm 1650 7800 350⋅10- 2 PIC-255 ACX-1 S.G.-11 ACX-2 S.G.-12 PI-3 S.G.-13 PI-4 Tab 50.8 mm 50.8 mm 50.8 PZT wire leads mm (a) (b) Front side Back side L=200 mm ACX-1 P(t)/2 P(t)/2 ACX-2 S.G- 12 S.G.- 11 PI-3 PI-4 15 mm P/2 PZT___ patches 15 mm P/2 Graphite-epoxy lamina (c) (d) Figure 2 Axially loaded specimens: (a). Schematic embedded piezoelectric patches, (b). Schematic surface bonded piezoelectric patches Bending experiments: (c) A typical schematic bending specimen (d) A schematic four-point bending setup test 8 The patch capacity was calculated as follows: C = k ⋅ ε 0 ⋅ S = k ⋅ ε 0 ⋅ l ⋅ b , h t ε 0 = 8.85 ⋅ 10 −12 Test procedure At the onset of each test (axial or bending) the polarity of the PZT patch must be known. Once the polarity is determined, in-phase or out-of-phase conditions can be imposed. In addition, the capacity of each piezoelectric patch at the beginning of the test and from time to time during the test is measured using the DVM device9. The loading conditions were specifically determined for each specimen. As a reference for the mechanical induced strains and the electrical load (voltage) we have used the limit values suggested by the ACX company, 1000µε and 100 Volt, respectively10. Prior starting the test, the initial output voltage for each PZT, Vout1, was measured. This voltage, Vout1, became an important value to be monitored and recorded as it served as an index for the piezoceramic patch health. The sensing voltage values were recorded peak-to-peak. During the E/M cyclic load tests the cycling was stopped (at least once every half decade) to determine the leftover PZT’s sensing capabilities, at the same loading conditions as Vout1 was found. If the measured voltage was greater than 50% of Vout1, then the test was continued, otherwise, the PZT patch was short-circuited and repoled. The repoling process was performed by applying a voltage of 250 DC [V] to the patch for 7-15 minutes, for the embedded and surface mounted specimens, respectively. After repoling, the output voltage is checked again. If the new output voltage is at least 70% of Vout1 then the test is continued, otherwise the PZT patch is considered to have failed. Tests results and discussion A test matrix of the completed tensile/compression and bending E/M tests is shown in Table II; Seven different specimens with one embedded ACX piezoceramic patches (See Fig. 2(a)), designated as AXEM#11, were investigated under various tensile/compression symmetric E/M loads combinations. Two specimens with two pairs of ACX and PI-255 piezoceramic patches bonded to the surface of each host structure, (See Fig. 2(b)), nick-named as AX-SM#12, were subjected to a series of tests in asymmetric tensile/compression. Seven different specimens with two pairs of piezoceramic patches (See Fig. 2(c)), designated as BEN#13, were loaded in bending. In general, during the bending tests, the PI patches would yield almost twice the initial sensing voltage, as compared to the ACX patches, namely 110.0 and 60.0 volts, respectively. This can be explained by the different piezoceramic material properties and thicknesses of the two patches. Yet, this difference became much smaller during the axial tests, which yielded significantly lower sensing values (only a few tens of volts) as compared to those monitored in the bending test. This is due to the different contributions of axial and bending deformations in the sensing models, Ref [10]. According to the detailed test procedure, from time to time repoling process was necessary. In some cases repoling enabled the patch to regain back almost all of its initial sensing voltage. Yet, during or even from the beginning of some experiments, contact and/or short circuit problems caused the "loosing" of different patches. 9 It was found that the measured value for the patch capacity, not necessary indicates the patch condition during the test. No specific limit values were provided by the PI company 11 AX-EM#, AX-axial, EM-embedded patch, # - specimen number 12 AX-SM#, AX-axial, SM-surface mounted patch, # - specimen number 13 BEN#- BEN-Bending, #- specimen number 10 Bending experiments Axially loaded experiments Specimens AX-EM 1 AX-EM 2 AX-EM 3 AX-EM 4 AX-EM 5 AX-EM 6 AX-EM 7 ASY#1 ASY#2 ASY#3 AX-SM 1 SY#4 SY#5 SY#6 ASY#1 AX-SM 2 SY#2 ASY#3 BEN1 BEN2 BEN3 BEN4 BEN5 BEN6 BEN7 Table II – Experimental test matrix Mechanical strain [µε] Test type Recorded on S.G. 11-12 ±500 In-phase ±1000 In-phase ±1500 In-phase ±1750 In-phase ±500 Out-of-phase ±1000 Out-of-phase ±1750 Out-of-phase +500/-1000 +625/-1250 +750/-1500 In-phase +1000/-1000 +1250/-1250 +1500/-1500 +500/-1000 +1000/-1000 In-phase +1500/-1000 ±1000 In-phase ±1000 In-phase Out-of-phase ±1000 Out-of-phase ±1000 Electrical load ±1000 only Electrical load ±1000 only Mechanical ±1000 Load only Electric load [Volt] ±100 ±100 ±100 ±100 ±100 ±100 ±100 +50/-100 +62.5/-125 +75/-150 +100/-100 +125/-125 +150/-150 +50/-100 +50/-100 +50/-100 ±100 ±100 ±100 ±100 ±100 Ranged or Total number of cycles 6 0-1⋅10 6 0-1⋅10 0-1⋅106 6 0-1⋅10 6 0-1⋅10 6 0-1⋅10 6 0-1⋅10 6 0-2⋅10 6 6 2⋅10 -2.5⋅10 6 6 2.5⋅10 -3.5⋅10 6 6 3.5⋅10 -4.1⋅10 6 6 4.1⋅10 -4.3⋅10 6 6 4.3⋅10 -4.7⋅10 6 0-1.5⋅10 6 6 1.5⋅10 -2.0⋅10 6 2.0⋅10 -2.1⋅106 6 0-2.65⋅10 6 0-2.65⋅10 6 0-2 ⋅10 6 0-1.65 ⋅10 6 0-2 ⋅10 6 ±100 0-0.5 ⋅10 ±100 0-2.7 ⋅10 6 Figs. 3(a-b) shows typical normalized14 sensing output voltage of the piezoceramic patches as function of the bending load cycles for specimens subjected to in-phase and out-of-phase conditions, respectively. Referring to the in-phase results, an almost stable sensing response during more then two million load cycles was encountered for the two commercial patches. On the other hand, much less stable behaviour was monitored for the patches that were tested in out-of-phase conditions. It is assumed that the in-phase condition provides a “stress relieving” effect which lowered the local stresses on the PZT patch and thus preventing its damage. The out-of-phase condition increased the local stresses on the patch, yielding a larger and faster accumulated damage, (more details in Ref. [1]). A good correlation was found with the test results obtained for specimens subjected to tensile/compression in phase E/M loads. Figs. 4(a-b) show the comparisons between the in-phase and out-of-phase tension-compression symmetric E/M load cycles tests at applied stresses of 100% and 175% of the manufacture's suggested limit, respectively. Fig. 4(b) also includes the results of one test that was performed under mechanical loads only, at an applied maximum stress of 175% of the manufacturer’s recommended strain limit of the PZT. It is clear, that the manufacturer’s recommended strain limit of ±1000ηε together with ±100 Volt for axial and bending functioning, is a good limit for both patches, ACX and PIC-255. Concerning the bending tests, the application of compression loads yielded a special case were the two patches (ACX1 & PI3) bonded on the upper surface of the beam were subjected to tension mechanical loads only while the two lower patches (ACX2 & PI4) were subjected to compression mechanical loads only. Figs. 5(a-b) present the lifetime behaviour of typical upper (ACX1) and lower (ACX2) patch, 14 All data was normalized by the initial sensing voltage, Vout1. respectively15. It is clear from both graphs that application of only tensile electric load cycles has a significant destructive effect on the patch health. In this case, the patch is loaded with only negative (against the patch poling direction) electric loads and never gets a repoling regime from the applied external voltage. 1.20 1.00 0.80 0.60 0.40 0.20 ACX1-BEN1 PI3-BEN1 ACX1-BEN2 ACX2-BEN2 PI4-BEN1 PI4-BEN2 Load cycles 0.00 0 50000 100000 150000 200000 Normalized sensing voltage Normalized sensing voltage 1.40 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 ACX2-BEN4 PI3-BEN4 PI4-BEN4 ACX1-BEN3 PI4-BEN3 Load cycles 0 250000 ACX1-BEN4 500000 1000000 1500000 2000000 (a) (b) Figure 3 Lifetime behaviour of typical patches in different E/M bending loads; (a) In-phase (b) Out of phase Normalized sensing voltage Normalized sensing voltage 1 0.9 In phase 0.8 Out of phase Load Cycles 0.7 1 10 100 1000 0.9 0.7 In phase Out of phase Mechanical 0.5 Load Cycles 0.3 10000 100000 1E+06 1 10 100 1000 10000 100000 1E+06 (a) (b) Figure 4 Symmetric Tension-Compression E/M cyclic load, in-phase and out-of-phase results at manufacturer's suggested limit loads (a) 100% (b) 175% ACX1-BEN1 ACX1-BEN7 ACX1-BEN6 1.40 1.20 ACX1-BEN2 ACX1-BEN5 ACX1-BEN4 1.00 0.80 0.60 0.40 0.20 Load cycles 0.00 0 50000 100000 150000 ACX2-BEN2 ACX2-BEN5 ACX2-BEN3 1.40 Normalized sensing voltage Normalized sensing voltage 1.60 1.20 ACX2-BEN7 ACX2-BEN6 1.00 0.80 0.60 0.40 0.20 Load cycles 0.00 200000 250000 0 50000 100000 150000 200000 (a) (b) Figure 5 The lifetime behaviour of typical patches in different E/M bending loads (a) Patch bonded to the upper surface of the beam (b) Patch bonded to the lower surface of the beam 15 We choose to present results for patches from the same company, ACX. 250000 CONCLUSIONS The present study investigated the functionality and fatigue response of a piezoceramic actuator/sensor (PZT) embedded or bonded on the surface of a laminated composite beam. Two commercially available PZT's were used: ACX patches from QP15N Cymer Inc. and PIC255 from PI Company. A graphite/epoxy laminate with a quasi-isotropic lay-up was used as a host structure. Different combinations of electromechanical (E/M) loading cycles profiles were applied on each specimen, supplying either in or out of phase conditions. Combined tension-compression E/M loading cycles with symmetric and asymmetric profiles were applied on the specimens according to the limit stresses (strains) suggested by the manufacturers, (1000µε together with 100 Volts). In addition, tests with bending E/M cyclic loads using a four-point bending setup were performed. The degradation in the sensing abilities of the piezoelectric patches for an increasing number of E/M load cycles was monitored and recorded. The effect of different E/M loads combinations on the sensing patch health was investigated. All results were analyzed and a comprehensive comparison was performed between the present results (Ref. [1]) and the results published in literature [6]. It was found that for a given electrical load, the increasing of the mechanical load will raise the sensing degradation. Under equal direction electro-mechanical loads, namely the in-phase condition, leads to only a minor degradation of the PZT patch while for a out-of-phase condition the damage is higher and the degradation appears earlier. For a given mechanical load applying a positive electric load will improve the patch’s sensing capabilities as compared to a negative applied one. The application of only an electrical load, which would induce tensile stresses on the PZT patch, has a significant destructive effect on the patch health. It was also found that asymmetric E/M loads, where the compression load is larger then the tension one, would improve the piezoceramic patch health. In contrast, application of asymmetric load where the tension load is larger then the compression one would lead to a rapid degradation of the PZT Patch. ACKNOWLEDGEMENTS The authors would like to acknowledge the outstanding experimental work performed by Mr. A. Grunwald and his exceptional assistance in setting the tests and dedicated assistance in performing them, and the assistance of Mrs. R. Yaffe, both from the Aerospace Structures Laboratory, Faculty of Aerospace Engineering, Technion, Haifa, Israel REFERENCES 1. Edery-Azulay, L. and Abramovich, "Piezo-Composite Beams under High Cyclic Electro-Mechanical Loads - Experimental Results", TAE Report No.971 Haifa, Israel: Faculty of Aerospace Engineering, Technion, IIT; January 2007. 2. Yocum, M., Abramovich, H., Grunwald, A. and Mall, S., “Fully Reversed Electromechanical Fatigue Behavior of Composite Laminate with Embedded Piezoelectric Actuator/sensor”, Smart Materials and Structures, 12 (2003), 1-9. 3. Bronowicki, A. J., McIntyre, L. J., Betros, R. S., and Dvorksy, G. R., “Mechanical Validation of Smart Structures” , Smart Materials & Structures, Vol. 5, 1996, 129-139. 4. Paget, C. A. and Levin, K., “Structural Integrity of Composites with Embedded Piezoelectric Ceramic Transducer” , SPIE Conference on Smart Structures and Integrated Systems (Newport Beach, CA, March), SPIE Vol. 3668, 1999, 306-313. 5. Mall, S. and Coleman, J. M., “Monotonic and Fatigue Loading Behavior of Quasi-Isotropic Graphite/Epoxy Laminate Embedded with Piezoelectric Sensor”, Smart Materials & Structures, Vol. 7, 1998, 822-832. 6. Mall, S. and Hsu, T. L., “Electromechanial Fatigue Behavior of Graphite/Epoxy Laminate Embedded with Piezoelectric Actuator”, Smart Materials & Structures, Vol. 9, 2000, 78-84. 7. Mitrovic, M., Carman, G. P. and Straub, F. K., "Response of piezoelectric stack actuators under combined electromechanical loading", International Journal of Solids and Structures, Volume 38, Issues 24-25, Pages 4357-4374. 8. Thielicke, B., Gesang, T., and Wierach, P., "Reliability of piezoceramic patch sensors under cyclic mechanical loading", Smart Mater. Struct. 12 993–996, 2003. 9. Jones, J. L., Salz, C. R. J. and Hoffman, M., " Ferroelastic Fatigue of a Soft PZT Ceramic", Journal of the American Ceramic Society, 88 [10] 2788–2792, 2005 10. Edery-Azulay, L. and Abramovich, H., Actuation and sensing of shear type piezoelectric patches-closed form solutions, Composite Structures, 64, pp. 443-453, 2004.
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