Northwestern University, C. Dagdeviren An Analytic Model .

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Y. ShiState Key Laboratory of Mechanicsand Control of Mechanical Structures,Nanjing University of Aeronauticsand Astronautics,Nanjing 210016, China;Department of Civiland Environmental Engineering,Department of Mechanical Engineering,Center for Engineering and Health,Skin Disease Research Center,Northwestern University,Evanston, IL 60208C. DagdevirenDepartment of Materials Scienceand Engineering,Frederick Seitz Materials Research Laboratory,University of Illinois at Urbana–Champaign,Urbana, IL 61801J. A. RogersDepartment of Materials Scienceand Engineering,Frederick Seitz Materials Research Laboratory,University of Illinois at Urbana–Champaign,Urbana, IL 61801C. F. GaoState Key Laboratory of Mechanicsand Control of Mechanical Structures,Nanjing University of Aeronauticsand Astronautics,Nanjing 210016, ChinaAn Analytic Model for SkinModulus Measurement ViaConformal Piezoelectric SystemsThe Young’s modulus of human skin is of great interests to dermatology, cutaneouspathology, and cosmetic industry. A wearable, ultrathin, and stretchable device providesa noninvasive approach to measure the Young’s modulus of human skin at any location,and in a way that is mechanically invisible to the subject. A mechanics model is developed in this paper to establish the relation between the sensor voltage and the skin modulus, which, together with the experiments, provides a robust way to determine theYoung’s modulus of the human skin. [DOI: 10.1115/1.4030820]Keywords: skin modulus, piezoelectric actuators and sensors, analytic modelY. Huang1Department of Civiland Environmental Engineering,Department of Mechanical Engineering,Center for Engineering and Health,Skin Disease Research Center,Northwestern University,Evanston, IL 60208e-mail: y-huang@northwestern.edu1IntroductionHuman skin plays an essential role in protecting the organismfrom the environment. Change in its mechanical propertiesreflects the tissue modifications caused by aging, disease, or stimulation of the environments [1]. The mechanical properties ofhuman skin, such as the Young’s modulus, are of great interests todermatology, cutaneous pathology, and cosmetic industry [2].The Young’s modulus of human skin has been obtained fromthe linear elastic part of the stress–strain curve [3]. Thestress–strain curve is measured by one of the following three techniques: (1) twist of the skin [3], (2) indentation [2], and (3)method of suction [4]. These tests, however, all involve relativelycomplex setup in the laboratory, which prevent simple and1Corresponding author.Contributed by the Applied Mechanics Division of ASME for publication in theJOURNAL OF APPLIED MECHANICS. Manuscript received April 4, 2015; final manuscriptreceived April 13, 2015; published online June 25, 2015. Editor: Arun Shukla.Journal of Applied Mechanicsportable applications outside the lab. In addition, such testsinvolve relatively large deformation of the skin, and therefore aredifficult to repeat quickly because it takes hours for the skin distortion to disappear upon unloading [3].Dagdeviren et al. [5] developed a wearable, ultrathin, andstretchable modulus measurement device that is much more robustthan the existing techniques. The device consists of a series ofmicroflexible lead zirconate titanate (PZT) [6] actuators and sensors laminated on a thin elastomeric matrix. It provides a noninvasive approach to measure the Young’s modulus of human skin atany location, normal conditions and upon administration of pharmacological and moisturizing agents, and in a way that is mechanically invisible to the subject. As to be shown in the analyticmodel in Sec. 2, the Young’s modulus of human skin is linearlyproportional to the sensor’s output voltage (for each given actuating voltage). Therefore, the measured output voltage, togetherwith the analytic model, determines the Young’s modulus ofhuman skin.C 2015 by ASMECopyright VSEPTEMBER 2015, Vol. 82 / 091007-1Downloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Fig. 1 An array of PZT ribbons sandwiched by the super- and substrates. (a) A schematic diagram and (b) thesuper- and substrates adhere together to form air-gaps at each end of the PZT ribbon.The objective of this paper is to establish an analytic model inSec. 2, particularly the linear relation between the sensor’s outputvoltage and the Young’s modulus. The analytic model is validatedin Sec. 3, and the application of this model to human skin is discussed in Sec. 4.2An Analytic ModelFigure 1(a) shows an array of n PZT ribbons sandwiched bytwo semi-infinite media (the super- and substrates are muchthicker than the PZT ribbons). The thickness h of PZT ribbons ismuch smaller than their width 2b and spacing 2w such that thesubstrate and superstrate adhere over the parts without the PZTribbon, as illustrated in Fig. 1(b). This leaves an air-gap at eachterminal of the PZT ribbon, where the length 2c of air-gap is to bedetermined by the work of adhesion between two media [7], asgiven at the end of this section. Each PZT ribbon, together withthe two air-gaps around its ends, can be modeled as an interfacialcrack with the length 2b þ 4c, with the center part (length 2b) incontact with the PZT ribbon.One PZT ribbon is subjected to an actuating voltage UA. Without the surrounding media, it would expand freely by (see theAppendix)Du ¼c11 e33 c13 e31UAc11 c33 c213(1)due to piezoelectricity of PZT, where cij and eij are the elastic andpiezoelectric constants of PZT, respectively, given in the Appendix. The expansion of this PZT ribbon with the actuating voltagecauses deformation in the surrounding media, of which the elasticmoduli are several orders of magnitude smaller than that of PZT.As a result, the contraction of this PZT ribbon with the actuatingvoltage due to deformation in the surrounding media is negligibleas compared to Du, as shown in the Appendix. Therefore, theboundary conditions for this crack, due to actuating voltage UA,are the constant opening displacement Du in the center part (oflength 2b) and traction free around the two ends (each of length2c), as illustrated in Fig. 2(a).8 The deformation in the surrounding media, in turn, causes theother PZT ribbons (without the actuating voltage) to expand. LetUS,i denote the sensing voltage in the ith PZT ribbon, which isseveral orders of magnitude smaller than UA, as shown in theAppendix, such that the deformation induced by US,i is also negligible. Therefore, the boundary conditions for these cracks, due toactuating voltage UA, are the vanishing opening displacement inthe center part (of length 2b) and traction free around the two ends(each of length 2c), as illustrated in Fig. 2(a). In fact, such a crack(of length 2b þ 4c) can be considered as two smaller cracks (eachof length 2c) since the center part (of length 2b) does not open up.The above problem (illustrated in Fig. 2(a)) can be decomposedto the following two subproblems:(1) A single crack (of length 2b þ 4c) subjected to the openingdisplacement in Eq. (1) over the center part (of length 2b),which is modeled as a crack with a rigid wedge (Fig.2(b)).(2) The collinear cracks (each of length 2c) subjected crackface pressure to negate the normal tractions on the crackface induced by 1), as illustrated in Fig. 2(c), such as theair-gap remain traction free.For the first subproblem illustrated in Fig. 2(b), the Westergaardfunction is given by [8]Z Du ðzÞ ¼E0 ðb þ ��ffiffiffiffiffiffiffiffiffiffiffiffiffi# ��ffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2)2 z2 b22b2z ðb þ 2cÞ4K1 ðb þ 2cÞ2where E0 is the plane-strain, effective modulus of the media and isto be discussed in detail at the end of this section, 1 2Ð p 2 1 k2 sin2 udu is the complete elliptic integralK ðkÞ ¼ 0pffiffiffiffiffiffiffiof first kind, z ¼ x þ iy, with i ¼ 1 and (x, y) is the local coordinate with the origin at the center of the crack. The normal stressalong the crack line (y ¼ 0) is given byE0 ðb þ ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;hib22 ðb þ 2cÞ2 ðx2 b2 Þ4K1 xðxÞ¼rDuy ðb þ 2cÞ2 :0; b j xj b þ 2c091007-2 / Vol. 82, SEPTEMBER 2015j xj b þ 2c(3)Transactions of the ASMEDownloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Fig. 2 A schematic illustration of the mechanics model. (a) PZT ribbons with air-gaps, withone of them (actuator) subjected to the actuating voltage, which results expansion in the actuator and therefore opening displacement at the interface between the adhered super- and substrates. (b) Subproblem 1, a single crack, representing the actuator and its two air-gaps,subjected to the opening displacement. (c) Subproblem 2, negation of the normal tractionsinduced by subproblem 1 at all air-gaps.where 0 in the second line represents the vanishing normal stress traction over two air-gaps around the PZT subjected to the actuatingvoltage UA.For the second subproblem illustrated in Fig. 2(c), the collinear cracks have the length 2c and center-to-center spacing alternating betweend1 ¼ 2b þ 2c and d2 ¼ 2w 2c. As shown at the end of this section, the air-gap is much shorter than the spacing d1 and d2 such that the normal stress to be negated on the jth crack can be approximated by the corresponding value at the center (x ¼ xj and y ¼ 0) of the crack, i.e.,8 E0 ðb þ 2cÞDu ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; xj b þ 2c hi 2bpj ¼(4)4K1 x2j ðb þ 2cÞ2 x2j b22 ðbþ2cÞ :0; b xj b þ 2cwhere 1 j 2n (n is the number of PZT ribbons), the negative sign in the first line of Eq. (4) denotes the negation. The Westergaardfunction for the second subproblem is then given by [9,10]322n Q22 1 2 ðn xk Þ cj 1 ð x þc2n 67X176ð 1Þ pj j k¼1(5)dn þ aj 1 zj 1 7ZðzÞ ¼ 2n h6i1 254Qpz nxj c2j¼12ðz xk Þ ck¼1Journal of Applied MechanicsSEPTEMBER 2015, Vol. 82 / 091007-3Downloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

where 2n is the total number of air-gaps around n PZT ribbons,and the coefficients aj for the polynomial are determined by thesingle valued condition of displacement for each crackþIm½Z ð xÞ dx ¼ 0(6)where E0super and E0sub are the plane-strain moduli of the super- andsubstrates, respectively. The length of air-gap is governed by thecompetition between the deformation energy (due to adherence ofthe super- and substrates) and adhesion energy and is given analytically by [7]CjHere, Cj (1 j 2n) is the closed loop for the jth crack. The sumof integrals for 1 j 2n is equivalent to a2n 1 ¼ 0 [9].The normal stress along the line y ¼ 0 is given by [8]ry ð xÞ ¼ Re½Z ð xÞ It gives the average pressure qi at the ith sensor asði1 x2i c hry ð xÞ þ rDuqi ¼y ð xÞ dx2b x2i 1 þc(8)3(10)11þE0super E0sub2Z3 ðzÞ ¼(11)(7)where k33 is the dielectric constant of PZT given in the Appendix.For the incompressible super- and substrates that sandwich thePZT ribbons, the plane-strain, effective modulus E0 of the mediais given by [7]2E0 h28pcwhere c is the work of adhesion between the super and substrates.For the super- and substrates moduli around 0.1 MPa, and a representative work of adhesion 50.6 mJ/m2 [7], the length of air-gap ison the order of 1 lm, which is much smaller than the width of thesensor (200 lm).The corresponding output voltage US,i at the ith sensor is thenobtained as (see the Appendix for details) qi (9)US;i ¼ h 2 e þ c33 e31 c13 e33 e þ c11 c33 c13 k 333133 c11 e33 c13 e31c11 e33 c13 e31E0 ¼2c ¼Special Case: One Actuator and One SensorThe special case of two PZT ribbons, with the left and rightones serving as the actuator and sensor, respectively, is consideredin this section to further illustrate the analysis. There are four collinear cracks (two air-gaps for each PZT ribbon), and their centershave the coordinates (with the origin at the center of the actuator)x1 ¼ b c; x2 ¼ b þ c;x4 ¼ 3b þ 2w þ c(12)Equation (4) then becomesp1 ¼ p2 ¼ 0;p3 E0 bDu;8pwðb þ wÞp4 ¼ E0 bDu8pðb þ wÞð2b þ wÞ(13)for small air-gap length c b; w, i.e., only the nonzero pressurep3 and p4 need to be considered. Accordingly, the Westergaardfunction in Eq. (5) can be written as the sum of those for p3 andp4, ZðzÞ ¼ Z3 ðzÞ þ Z4 ðzÞ, where 1 24 Q ðn xk Þ2 c2 6p ð x3 þc16 3k¼16hi41 2 4 pQx3 cðz xk Þ2 c2x3 ¼ b þ 2w c;z n37ð3Þð3Þð3Þ 7dn þ a0 þ a1 z þ a2 z2 75k¼12Z4 ðzÞ ¼14 hQðz xk Þ2 c 1 24 Q ðn xk Þ2 c2 6 p ð x4 þc6 4k¼1 i1 2 64px4 c2z n3(14)7ð4Þð4Þð4Þ 7dn þ a0 þ a1 z þ a2 z2 75k¼1where the vanishing coefficient a3 ¼ 0 for z3 has been used. For c b; w, Eq. (14) can be simplified to32 1 2 22 ðn xðÞ c376 E0 b2 Du x3 þc 16ð3 Þð3 Þð3 Þ 7Z3 ðzÞ ¼ 4 hdn þ a0 þ a1 z þ a2 z2 76 i21 254Qpz nx3 cðz xk Þ2 c2k¼1Z4 ðzÞ ¼4Q2h1ðz xk Þ2 c26E0 b2 Du6i1 2 64 p2 1 2 ð x4 þc ðn x4 Þ2 c2 x4 cz n37ð4Þð4Þð4Þ 7dn þ a0 þ a1 z þ a2 z2 75(15)k¼1ð3ÞHere, the coefficients aj (j ¼ 0, 1, and 2) are determinedby the single valued condition in Eq. (5) around thecracks 1, 2, and 4 but not crack 3 (to avoid a singular,091007-4 / Vol. 82, SEPTEMBER 2015Cauchy-principle integral) because the equivalent conditiona3 ¼ 0 for z 3 has already been imposed in Eq. (15). ThisgivesTransactions of the ASMEDownloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

E0 b2 Du 2c4pðb þ wÞE0 b2 Du 2ð3 Þð3 Þð3Þ(16)ca0 þ a1 x2 þ a2 x22 ¼ 4pwE0 bDu 2ð3 Þð3 Þð3Þca0 þ a1 x4 þ a2 x24 ¼4pð4ÞSimilarly, the coefficients aj (j ¼ 0, 1, and 2) are determinedby the single valued condition in Eq. (5) around the cracks 1,ð3 Þð3 Þð3Þa0 þ a1 x1 þ a2 x21 ¼ E0 b2 Dup2ð3Þð4Þ2, and 3. The resulting aj and aj are linear proportional toc2 Du.The average pressure on the sensor is then obtained from Eqs.(7) and (8) asðo1 x4 c nRe½Z3 ð xÞ þ Z4 ð xÞ þ rDuðxÞdx(17)q2 ¼y2b x3 þcwhere 1 2 ð x3 þc ðn x3 Þ2 c2 ð3Þð3Þð3Þdn a0 a1 x a2 x2x nx3 cZ3 ð xÞ ¼hi1 2 hi1 2ðx x1 Þðx x2 Þ ðx x3 Þ2 c2ðx4 xÞ2 c2 1 2 22 ððn x4 Þ c E0 b2 Du x4 þc ð4Þð4Þð4Þ dn a0 a1 x a2 x2p2x nx4 cZ4 ð xÞ ¼hi1 2 hi1 2ðx x1 Þðx x2 Þ ðx x3 Þ2 c2ðx4 xÞ2 c2(18)which are much smaller than rDuy ð xÞ for c b; w such that Eq. (17) can be approximated by"#E0 Dub2ln 1 q2 8pbðb þ wÞ2(19)The sensor voltage in Eq. (9) then becomesE0 hU S ¼ UA4p 2bc11 e33 c13 e31c11 c33 c2131lnc33 e31 c13 e33c11 c33 c213b2e33 þe31 þk33 1 c11 e33 c13 e31c11 e33 c13 e31ðb þ wÞ2It is linearly proportional to the actuator voltage UA, the effectivemodulus E0 , and the thickness to width ratio h/(2b) of PZT. It alsodepends on the material properties of PZT, and the spacing towidth ratio w/b through ln½1 b2 ðb þ wÞ2 1 . Figure 3 showsthis function versus (b þ w)/b, which decreases rapidly as the spacing increases. Here, (b þ w)/b is the ratio of center-to-center distance between the actuator to sensor to the sensor width. It is notedthat Eq. (20) is independent of the air-gap length 2c for c b; w.The ratio of sensor to actuator voltage US/UA, together with thematerial parameters of PZT and thickness to width ratio h/(2b)and spacing to width ratio w/b, gives the effective modulus E0 ,and therefore the substrate modulus (if the superstrate modulus isknown).4 Approximate Solution for Multiple Actuators andSensorsThe analysis in Sec. 3 clearly suggests that, for c b; w,Z3 ð xÞ þ Z4 ð xÞ resulting from the second subproblem in Sec.2 is negligible as compared to rDuy ð xÞ resulting from thefirst subproblem. This observation also holds for multiplesensors such that the average pressure on the ith sensor inEq. (8) becomesJournal of Applied Mechanics1qi 2bð x2i cx2i 1 þcrDuy ð xÞdx"#E0 Dub2ln 1 8bpm2 ðb þ wÞ2(20)(21)where m is the number of sensor away from the actuator, andm ¼ 1 degenerates to Sec. 3. The voltage in Eq. (9) for all sensorsthen becomesc11 e33 c13 e31E0 hc11 c33 c213U S ¼ UA4p 2bc33 e31 c13 e33c11 c33 c213e33 þe31 þk33c11 e33 c13 e31c11 e33 c13 e311ln(22)b21 m2 ðb þ wÞ2It has the same relation with UA, E0 , h/(2b), and the material properties of PZT as Eq. (20), but now depends on m (the number ofsensor away from the actuator) through the ratio m(b þ w)/b,which is the ratio of center-to-center distance between the actuatorto sensor to the sensor width. As shown in Fig. 3,no 1[versus m(b þ w)/b] decreasesD ¼ ln 1 ½mðb þ wÞ b 2SEPTEMBER 2015, Vol. 82 / 091007-5Downloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

5ConclusionsAnalytic expressions of the sensor voltage are obtained for anarray of piezoelectric actuators and sensors between super- and substrates. Together with experimentally measured sensor voltage [5]and the material properties and geometric parameters of piezoelectric actuators and sensors, these expressions provide a simple way todetermine the Young’s modulus of the substrate if that of the superstrate is known. This is particularly useful to determine the Young’smodulus of the skin, as demonstrated in recent experiments [5].AcknowledgmentY.S. was partially supported by the National Basic ResearchProgram of China (No. 2015CB351900) and the National NaturalScience Foundation (NNSF) of China (No. 11320101001). C.D.and J.A.R. acknowledge the support from the U.S. DOE, Office ofBasic Energy Sciences, Division of Materials Sciences and Engineering (No. DE-FG02-07ER46471), through the Frederick SeitzMaterials Research Laboratory at the University of Illinois atUrbana-Champaign. C.F.G. acknowledges the support from NNSF(Nos. 11472130 and 11232007). Y.H. acknowledges the supportfrom U.S. National Science Foundation (No. CMMI-1400169).Fig. 3 The function lnf1 ½mðb þ wÞ b 2 g 1 , which the sensorvoltage is linearly proportional to, versus its variable m(b 1 w)/brapidly as the number of sensor away from the actuator mincreases. Similar to Eq. (20), Eq. (20) is also independent of theair-gap length 2c for c b; w.89 8r11 c11 cr22 12 c r 3313¼ 0r 23 r 0 31 ::; 0r12c12c11c13c130000c13000c330000c440000c4408 9 8 D1 0 0D2 ¼ : :; D3e3100e3100e330e150091007-6 / Vol. 82, SEPTEMBER 2015The constitutive model of piezoelectric material gives [11]989 8e11 0 e22 0 0 0e33 02e23 0 0 e15 2e31 ::;;0ðc11 c12 Þ 22e1200e1500where rij , eij , Ei , and Di represent the stress, strain, electrical field,and electrical displacement, respectively, cij , eij , and kij are theelastic, piezoelectric, and dielectric parameters of the material,and the subscript “3” denotes the polarization (vertical, Fig. 2(a))direction of the PZT layer.The electric field in polarization direction is Ea3 ¼ UA h whenthe PZT ribbon is subjected to the actuating voltage UA , wherethe superscript a denotes the actuator. Under plane-strain deformation ea22 ¼ ea12 ¼ ea23 ¼ 0, electric field boundary conditionEa1 ¼ Ea2 ¼ 0 and the approximate traction free condition ra33 ¼ 0(by neglecting the traction from the soft super- and substrates),Eq. (A1) gives8 c11 c33 c213 ea11 þ ðc13 e33 c33 e31 ÞEa3 ra11 ¼c33aa eE ce 3313311 ea ¼:33c33Appendix(A3)89e11 e 89 22 k110 e 3300þ :; 2e23 0 0 2e 31 :;2e120k220000e15009e31 8 9 e31 E1 e33 E2 0 : ; E3 0 ;098 90 E1 0E2 : ; ;E3k33(A1)(A2)ÐThe vanishing membrane force, h ra11 dz ¼ 0, gives ea33 . Its integration then leads to the expansion of the actuator Du in Eq. (1).For the pressure qi on the ith sensor and the vanishing stressand electric displacement fields ri33 ¼ qi and Di3 ¼ 0 (where thesuperscript i denotes the ith sensor), Eqs.Ð (A1) and (A2), togetherwith the vanishing of membrane force, h ri11 dz ¼ 0, give the following equations to determine ei11 , ei33 , and Ei3 :8 c11 c33 c213 ei11 þ ðc13 e33 c33 e31 ÞEi3 þ c13 qi 0¼ c33 iiqi þ e33 E3 c13 e11 ei33 ¼ c33 :iie31 e11 þ e33 e33 þ k33 Ei3 ¼ 0(A4) This gives the output voltage US;i ¼ Ei3 h in Eq. (9). Theexpansion Dui ¼ ei33 h of the ith sensor isTransactions of the ASMEDownloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

c11 k33 þ e231 qi hDui ¼c11 e233 þ c11 c33 k33 þ c33 e231 2c13 e31 e33 c213 k33(A5)Its ratio to the expansion of the actuator given in Eq. (1) is thengiven by326Dui E0 hc11 k33 þ e2316ln¼6Du 8bp c11 e233 þ c11 c33 k33 þ c33 e231 2c13 e31 e33 c213 k33 41 which is extremely small as illustrated in main text.References[1] Escoffier, C., Derigal, J., Rochefort, A., Vasselet, R., Leveque, J. L., andAgache, P. G., 1989, “Age-Related Mechanical Properties of Human Skin: AnIn Vivo Study,” J. Invest. Dermatol., 93(3), pp. 353–357.[2] Pailler-Mattei, C., Bec, S., and Zahouani, H., 2008, “In Vivo Measurements ofthe Elastic Mechanical Properties of Human Skin by Indentation Tests,” Med.Eng. Phys., 30(5), pp. 599–606.[3] Agache, P. G., Monneur, C., Leveque, J. L., and De Rigal, J., 1980,“Mechanical Properties and Young’s Modulus of Human Skin In Vivo,” Arch.Dermatol. Res., 269(3), pp. 221–232.[4] Diridollou, S., Patat, F., Gens, F., Vaillant, L., Black, D., Lagarde, J. M., Gall,Y., and Berson, M., 2000, “In Vivo Model of the Mechanical Properties of theHuman Skin Under Suction,” Skin Res. Technol., 6(4), pp. 214–221.[5] Dagdeviren, C., Shi, Y., Joe, P., Ghaffari, R., Balooch, G., Usgaonkar, K., Gur,O., Tran, P. L., Crosby, J. R., Meyer, M., Su, Y., Webb, R. C., Tedesco, A. S.,Journal of Applied Mechanics[6][7][8][9][10][11]1b2i2 ðb þ wÞ27775(A6)Slepian, M. J., Huang, Y., and Rogers, J. A., 2015, “Conformal PiezoelectricSystems for Clinical and Experimental Characterization of Soft Tissue Biomechanics,” Nat. Mater. (in press).Dagdeviren, C., Yang, B. D., Su, Y., Tran, P. L., Joe, P., Anderson, E., Xia, J.,Doraiswamy, V., Dehdashti, B., Feng, X., Lu, B., Poston, R., Khalpey, Z.,Ghaffari, R., Huang, Y., Slepian, M. J., and Rogers, J. A., 2014, “ConformalPiezoelectric Energy Harvesting and Storage From Motions of the Heart, Lung,and Diaphragm,” Proc. Natl. Acad. Sci. USA, 111(5), pp. 1927–1932.Huang, Y., Zhou, W. X., Hsia, K. J., Menard, E., Park, J. U., Rogers, J. A., andAlleyne, A. G., 2005, “Stamp Collapse in Soft Lithography,” Langmuir, 21(17),pp. 8058–8068.Tada, H., Paris, P. C., and Irwin, G. R., 2000, The Stress Analysis of CracksHandbook, 3rd ed., ASME Press, New York.Muskhelishvili, N. I., 1953, Some Basic Problems of the Mathematical Theoryof Elasticity, P. Noordhoff Ltd., Amsterdam.Wang, L., and Lin, X., 1985, “Problems on Infinite Plate With Arbitrary Collinear Cracks,” Acta Mech. Sin., 17(3), pp. 243–252.Fang, D., and Liu, J., 2014, Fracture Mechanics of Piezoelectric and Ferroelectric Solids, Tsinghua University Press/Springer, Beijing.SEPTEMBER 2015, Vol. 82 / 091007-7Downloaded From: .org/ on 07/08/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, IL 61801 J. A. Rogers Department of Materials Science and Engineering, Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, IL 61801 C. F. Gao State Key Laboratory of Mechanics

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