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Research Papers

An Analytic Model for Skin Modulus Measurement Via Conformal Piezoelectric Systems OPEN ACCESS

[+] Author and Article Information
Y. Shi

State Key Laboratory of Mechanics
and Control of Mechanical Structures,
Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China
Department of Civil
and Environmental Engineering,
Department of Mechanical Engineering,
Center for Engineering and Health,
Skin Disease Research Center,
Northwestern University,
Evanston, IL 60208

C. Dagdeviren, 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
and Control of Mechanical Structures,
Nanjing University of Aeronautics and Astronautics,
Nanjing 210016, China

Y. Huang

Department of Civil
and Environmental Engineering,
Department of Mechanical Engineering,
Center for Engineering and Health,
Skin Disease Research Center,
Northwestern University,
Evanston, IL 60208
e-mail: y-huang@northwestern.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 4, 2015; final manuscript received April 13, 2015; published online June 25, 2015. Editor: Arun Shukla.

J. Appl. Mech 82(9), 091007 (Sep 01, 2015) (7 pages) Paper No: JAM-15-1185; doi: 10.1115/1.4030820 History: Received April 04, 2015; Revised April 13, 2015; Online June 25, 2015

The Young's modulus of human skin is of great interests to dermatology, cutaneous pathology, and cosmetic industry. A wearable, ultrathin, and stretchable device provides a 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 the Young's modulus of the human skin.

FIGURES IN THIS ARTICLE
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Human skin plays an essential role in protecting the organism from the environment. Change in its mechanical properties reflects the tissue modifications caused by aging, disease, or stimulation of the environments [1]. The mechanical properties of human skin, such as the Young's modulus, are of great interests to dermatology, cutaneous pathology, and cosmetic industry [2].

The Young's modulus of human skin has been obtained from the linear elastic part of the stress–strain curve [3]. The stress–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 relatively complex setup in the laboratory, which prevent simple and portable applications outside the lab. In addition, such tests involve relatively large deformation of the skin, and therefore are difficult to repeat quickly because it takes hours for the skin distortion to disappear upon unloading [3].

Dagdeviren et al. [5] developed a wearable, ultrathin, and stretchable modulus measurement device that is much more robust than the existing techniques. The device consists of a series of microflexible 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 at any 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 analytic model in Sec. 2, the Young's modulus of human skin is linearly proportional to the sensor's output voltage (for each given actuating voltage). Therefore, the measured output voltage, together with the analytic model, determines the Young's modulus of human skin.

The objective of this paper is to establish an analytic model in Sec. 2, particularly the linear relation between the sensor's output voltage and the Young's modulus. The analytic model is validated in Sec. 3, and the application of this model to human skin is discussed in Sec. 4.

Figure 1(a) shows an array of n PZT ribbons sandwiched by two semi-infinite media (the super- and substrates are much thicker than the PZT ribbons). The thickness h of PZT ribbons is much smaller than their width 2b and spacing 2w such that the substrate and superstrate adhere over the parts without the PZT ribbon, as illustrated in Fig. 1(b). This leaves an air-gap at each terminal of the PZT ribbon, where the length 2c of air-gap is to be determined by the work of adhesion between two media [7], as given at the end of this section. Each PZT ribbon, together with the two air-gaps around its ends, can be modeled as an interfacial crack with the length 2b + 4c, with the center part (length 2b) in contact with the PZT ribbon.

One PZT ribbon is subjected to an actuating voltage UA. Without the surrounding media, it would expand freely by (see the Appendix)Display Formula

(1)Δu=c11e33-c13e31c11c33-c132UA

due to piezoelectricity of PZT, where cij and eij are the elastic and piezoelectric constants of PZT, respectively, given in the Appendix. The expansion of this PZT ribbon with the actuating voltage causes deformation in the surrounding media, of which the elastic moduli are several orders of magnitude smaller than that of PZT. As a result, the contraction of this PZT ribbon with the actuating voltage due to deformation in the surrounding media is negligible as compared to Δu, as shown in the Appendix. Therefore, the boundary conditions for this crack, due to actuating voltage UA, are the constant opening displacement Δu in the center part (of length 2b) and traction free around the two ends (each of length 2c), as illustrated in Fig. 2(a).

The deformation in the surrounding media, in turn, causes the other PZT ribbons (without the actuating voltage) to expand. Let US,i denote the sensing voltage in the ith PZT ribbon, which is several orders of magnitude smaller than UA, as shown in the Appendix, such that the deformation induced by US,i is also negligible. Therefore, the boundary conditions for these cracks, due to actuating voltage UA, are the vanishing opening displacement in the 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 + 4c) can be considered as two smaller cracks (each of length 2c) since the center part (of length 2b) does not open up.

The above problem (illustrated in Fig. 2(a)) can be decomposed to the following two subproblems:

  1. (1)A single crack (of length 2b + 4c) subjected to the opening displacement in Eq. (1) over the center part (of length 2b), which is modeled as a crack with a rigid wedge (Fig. 2(b)).
  2. (2)The collinear cracks (each of length 2c) subjected crack-face pressure to negate the normal tractions on the crack face induced by 1), as illustrated in Fig. 2(c), such as the air-gap remain traction free.

For the first subproblem illustrated in Fig. 2(b), the Westergaard function is given by [8]Display Formula

(2)ZΔu(z)=E'(b+2c)Δu4K[1-b2(b+2c)2]1z2-(b+2c)21z2-b2

where E' is the plane-strain, effective modulus of the media and is to be discussed in detail at the end of this section, K(k)=0π/2(1-k2sin2ϕ)-1/2dϕ is the complete elliptic integral of first kind, z = x + iy, with i=-1 and (x, y) is the local coordinate with the origin at the center of the crack. The normal stress along the crack line (y = 0) is given byDisplay Formula

(3)σyΔu(x)={E'(b+2c)Δu4K[1-b2(b+2c)2][x2-(b+2c)2](x2-b2),|x|>b+2c0,b<|x|<b+2c

where 0 in the second line represents the vanishing normal stress traction over two air-gaps around the PZT subjected to the actuating voltage UA.

For the second subproblem illustrated in Fig. 2(c), the collinear cracks have the length 2c and center-to-center spacing alternating between d1 = 2b + 2c and d2 = 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 = xj and y = 0) of the crack, i.e.,Display Formula

(4)pj={-E'(b+2c)Δu4K[1-b2(b+2c)2][xj2-(b+2c)2](xj2-b2),|xj|>b+2c0,b<|xj|<b+2c

where 1j2n (n is the number of PZT ribbons), the negative sign in the first line of Eq. (4) denotes the negation. The Westergaard function for the second subproblem is then given by [9,10]Display Formula

(5)Z(z)=1Πk=12n[(z-xk)2-c2]1/2j=12n[(-1)j-1pjπxj-cxj+cΠk=12n|(ξ-xk)2-c2|1/2z-ξdξ+aj-1zj-1]

where 2n is the total number of air-gaps around n PZT ribbons, and the coefficients aj for the polynomial are determined by the single valued condition of displacement for each crackDisplay Formula

(6)ΓjIm[Z(x)]dx=0

Here, Γj (1j2n) is the closed loop for the jth crack. The sum of integrals for 1j2n is equivalent to a2n−1 = 0 [9].

The normal stress along the line y = 0 is given by [8]Display Formula

(7)σy(x)=Re[Z(x)]

It gives the average pressure qi at the ith sensor asDisplay Formula

(8)qi=12bx2i-1+cx2i-c[σy(x)+σyΔu(x)]dx

The corresponding output voltage US,i at the ith sensor is then obtained as (see the Appendix for details)Display Formula

(9)US,i=h|qie33+c33e31-c13e33c11e33-c13e31e31+c11c33-c132c11e33-c13e31k33|

where k33 is the dielectric constant of PZT given in the Appendix.

For the incompressible super- and substrates that sandwich the PZT ribbons, the plane-strain, effective modulus E' of the media is given by [7]Display Formula

(10)E'=21E'super+1E'sub

where E'super and E'sub are the plane-strain moduli of the super- and substrates, respectively. The length of air-gap is governed by the competition between the deformation energy (due to adherence of the super- and substrates) and adhesion energy and is given analytically by [7]Display Formula

(11)2c=E'h28πγ

where γ 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 is on the order of 1 μm, which is much smaller than the width of the sensor (200 μm).

The special case of two PZT ribbons, with the left and right ones serving as the actuator and sensor, respectively, is considered in this section to further illustrate the analysis. There are four collinear cracks (two air-gaps for each PZT ribbon), and their centers have the coordinates (with the origin at the center of the actuator)Display Formula

(12)x1=-b-c,x2=b+c,x3=b+2w-c,x4=3b+2w+c

Equation (4) then becomesDisplay Formula

(13)p1=p2=0,p3-E'bΔu8πw(b+w),p4=-E'bΔu8π(b+w)(2b+w)

for small air-gap length cb,w, i.e., only the nonzero pressure p3 and p4 need to be considered. Accordingly, the Westergaard function in Eq. (5) can be written as the sum of those for p3 and p4, Z(z)=Z3(z)+Z4(z), whereDisplay Formula

(14)Z3(z)=1Πk=14[(z-xk)2-c2]1/2[p3πx3-cx3+cΠk=14|(ξ-xk)2-c2|1/2z-ξdξ+a0(3)+a1(3)z+a2(3)z2]Z4(z)=1Πk=14[(z-xk)2-c2]1/2[-p4πx4-cx4+cΠk=14|(ξ-xk)2-c2|1/2z-ξdξ+a0(4)+a1(4)z+a2(4)z2]

where the vanishing coefficient a3 = 0 for z3 has been used. For cb,w, Eq. (14) can be simplified toDisplay Formula

(15)Z3(z)=1Πk=14[(z-xk)2-c2]1/2[-E'b2Δuπ2x3-cx3+c|(ξ-x3)2-c2|1/2z-ξdξ+a0(3)+a1(3)z+a2(3)z2]Z4(z)=1Πk=14[(z-xk)2-c2]1/2[E'b2Δuπ2x4-cx4+c|(ξ-x4)2-c2|1/2z-ξdξ+a0(4)+a1(4)z+a2(4)z2]

Here, the coefficients aj(3) (j = 0, 1, and 2) are determined by the single valued condition in Eq. (5) around the cracks 1, 2, and 4 but not crack 3 (to avoid a singular, Cauchy-principle integral) because the equivalent condition a3 = 0 for z3 has already been imposed in Eq. (15). This givesDisplay Formula

(16)a0(3)+a1(3)x1+a2(3)x12=-E'b2Δu4π(b+w)c2a0(3)+a1(3)x2+a2(3)x22=-E'b2Δu4πwc2a0(3)+a1(3)x4+a2(3)x42=E'bΔu4πc2

Similarly, the coefficients aj(4) (j = 0, 1, and 2) are determined by the single valued condition in Eq. (5) around the cracks 1, 2, and 3. The resulting aj(3) and aj(4) are linear proportional to c2Δu.

The average pressure on the sensor is then obtained from Eqs. (7) and (8) asDisplay Formula

(17)q2=12bx3+cx4-c{Re[Z3(x)+Z4(x)]+σyΔu(x)}dx

whereDisplay Formula

(18)Z3(x)=E'b2Δuπ2x3-cx3+c|(ξ-x3)2-c2|1/2x-ξdξ-a0(3)-a1(3)x-a2(3)x2(x-x1)(x-x2)[(x-x3)2-c2]1/2[(x4-x)2-c2]1/2Z4(x)=-E'b2Δuπ2x4-cx4+c|(ξ-x4)2-c2|1/2x-ξdξ-a0(4)-a1(4)x-a2(4)x2(x-x1)(x-x2)[(x-x3)2-c2]1/2[(x4-x)2-c2]1/2

which are much smaller than σyΔu(x) for cb,w such that Eq. (17) can be approximated byDisplay Formula

(19)q2-E'Δu8πbln[1-b2(b+w)2]

The sensor voltage in Eq. (9) then becomesDisplay Formula

(20)US=UAE'4πh2bc11e33-c13e31c11c33-c132e33+c33e31-c13e33c11e33-c13e31e31+c11c33-c132c11e33-c13e31k33ln11-b2(b+w)2

It is linearly proportional to the actuator voltage UA, the effective modulus E', and the thickness to width ratio h/(2b) of PZT. It also depends on the material properties of PZT, and the spacing to width ratio w/b through ln[1-b2/(b+w)2]-1. Figure 3 shows this function versus (b + w)/b, which decreases rapidly as the spacing increases. Here, (b + w)/b is the ratio of center-to-center distance between the actuator to sensor to the sensor width. It is noted that Eq. (20) is independent of the air-gap length 2c for cb,w.

The ratio of sensor to actuator voltage US/UA, together with the material parameters of PZT and thickness to width ratio h/(2b) and spacing to width ratio w/b, gives the effective modulus E', and therefore the substrate modulus (if the superstrate modulus is known).

The analysis in Sec. 3 clearly suggests that, for cb,w, Z3(x)+Z4(x) resulting from the second subproblem in Sec. 2 is negligible as compared to σyΔu(x) resulting from the first subproblem. This observation also holds for multiple sensors such that the average pressure on the ith sensor in Eq. (8) becomesDisplay Formula

(21)qi12bx2i-1+cx2i-cσyΔu(x)dx-E'Δu8bπln[1-b2m2(b+w)2]

where m is the number of sensor away from the actuator, and m = 1 degenerates to Sec. 3. The voltage in Eq. (9) for all sensors then becomesDisplay Formula

(22)US=UAE'4πh2bc11e33-c13e31c11c33-c132e33+c33e31-c13e33c11e33-c13e31e31+c11c33-c132c11e33-c13e31k33ln11-b2m2(b+w)2

It has the same relation with UA, E', h/(2b), and the material properties of PZT as Eq. (20), but now depends on m (the number of sensor away from the actuator) through the ratio m(b + w)/b, which is the ratio of center-to-center distance between the actuator to sensor to the sensor width. As shown in Fig. 3, D=ln{1-[m(b+w)/b]-2}-1 [versus m(b + w)/b] decreases rapidly as the number of sensor away from the actuator m increases. Similar to Eq. (20), Eq. (20) is also independent of the air-gap length 2c for cb,w.

Analytic expressions of the sensor voltage are obtained for an array 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 to determine the Young's modulus of the substrate if that of the superstrate is known. This is particularly useful to determine the Young's modulus of the skin, as demonstrated in recent experiments [5].

Y.S. was partially supported by the National Basic Research Program of China (No. 2015CB351900) and the National Natural Science Foundation (NNSF) of China (No. 11320101001). C.D. and J.A.R. acknowledge the support from the U.S. DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (No. DE-FG02-07ER46471), through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign. C.F.G. acknowledges the support from NNSF (Nos. 11472130 and 11232007). Y.H. acknowledges the support from U.S. National Science Foundation (No. CMMI-1400169).

Appendix

The constitutive model of piezoelectric material gives [11]Display Formula

(A1){σ11σ22σ33σ23σ31σ12}={c11c12c13000c12c11c13000c13c13c33000000c44000000c44000000(c11-c12)/2}{ɛ11ɛ22ɛ332ɛ232ɛ312ɛ12}-{00e3100e3100e330e150e1500000}{E1E2E3}
Display Formula
(A2){D1D2D3}={0000e150000e1500e31e31e33000}{ɛ11ɛ22ɛ332ɛ232ɛ312ɛ12}+{k11000k22000k33}{E1E2E3}

where σij, ɛij, Ei, and Di represent the stress, strain, electrical field, and electrical displacement, respectively, cij, eij, and kij are the elastic, 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 E3a=UA/h when the PZT ribbon is subjected to the actuating voltage UA, where the superscript a denotes the actuator. Under plane-strain deformation ɛ22a=ɛ12a=ɛ23a=0, electric field boundary condition E1a=E2a=0 and the approximate traction free condition σ33a=0 (by neglecting the traction from the soft super- and substrates), Eq. (A1) givesDisplay Formula

(A3){σ11a=(c11c33-c132)ɛ11a+(c13e33-c33e31)E3ac33ɛ33a=e33E3a-c13ɛ11ac33

The vanishing membrane force, hσ11adz=0, gives ɛ33a. Its integration then leads to the expansion of the actuator Δu in Eq. (1).

For the pressure qi on the ith sensor and the vanishing stress and electric displacement fields σ33i=qi and D3i=0 (where the superscript i denotes the ith sensor), Eqs. (A1) and (A2), together with the vanishing of membrane force, hσ11idz=0, give the following equations to determine ɛ11i, ɛ33i, and E3i :Display Formula

(A4){0=(c11c33-c132)ɛ11i+(c13e33-c33e31)E3i+c13qic33ɛ33i=qi+e33E3i-c13ɛ11ic33e31ɛ11i+e33ɛ33i+k33E3i=0

This gives the output voltage US,i=|E3i|·h in Eq. (9). The expansion Δui=ɛ33i·h of the ith sensor isDisplay Formula

(A5)Δui=(c11k33+e312)qihc11e332+c11c33k33+c33e312-2c13e31e33-c132k33

Its ratio to the expansion of the actuator given in Eq. (1) is then given byDisplay Formula

(A6)ΔuiΔu=E'h8bπc11k33+e312c11e332+c11c33k33+c33e312-2c13e31e33-c132k33ln[11-b2i2(b+w)2]

which is extremely small as illustrated in main text.

Escoffier, C., Derigal, J., Rochefort, A., Vasselet, R., Leveque, J. L., and Agache, P. G., 1989, “Age-Related Mechanical Properties of Human Skin: An In Vivo Study,” J. Invest. Dermatol., 93(3), pp. 353–357. [CrossRef] [PubMed]
Pailler-Mattei, C., Bec, S., and Zahouani, H., 2008, “In Vivo Measurements of the Elastic Mechanical Properties of Human Skin by Indentation Tests,” Med. Eng. Phys., 30(5), pp. 599–606. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
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 the Human Skin Under Suction,” Skin Res. Technol., 6(4), pp. 214–221. [CrossRef] [PubMed]
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., Slepian, M. J., Huang, Y., and Rogers, J. A., 2015, “Conformal Piezoelectric Systems for Clinical and Experimental Characterization of Soft Tissue Biomechanics,” Nat. Mater. (in press). [CrossRef]
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, “Conformal Piezoelectric Energy Harvesting and Storage From Motions of the Heart, Lung, and Diaphragm,” Proc. Natl. Acad. Sci. USA, 111(5), pp. 1927–1932. [CrossRef]
Huang, Y., Zhou, W. X., Hsia, K. J., Menard, E., Park, J. U., Rogers, J. A., and Alleyne, A. G., 2005, “Stamp Collapse in Soft Lithography,” Langmuir, 21(17), pp. 8058–8068. [CrossRef] [PubMed]
Tada, H., Paris, P. C., and Irwin, G. R., 2000, The Stress Analysis of Cracks Handbook, 3rd ed., ASME Press, New York.
Muskhelishvili, N. I., 1953, Some Basic Problems of the Mathematical Theory of 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.
Copyright © 2015 by ASME
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References

Escoffier, C., Derigal, J., Rochefort, A., Vasselet, R., Leveque, J. L., and Agache, P. G., 1989, “Age-Related Mechanical Properties of Human Skin: An In Vivo Study,” J. Invest. Dermatol., 93(3), pp. 353–357. [CrossRef] [PubMed]
Pailler-Mattei, C., Bec, S., and Zahouani, H., 2008, “In Vivo Measurements of the Elastic Mechanical Properties of Human Skin by Indentation Tests,” Med. Eng. Phys., 30(5), pp. 599–606. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
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 the Human Skin Under Suction,” Skin Res. Technol., 6(4), pp. 214–221. [CrossRef] [PubMed]
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., Slepian, M. J., Huang, Y., and Rogers, J. A., 2015, “Conformal Piezoelectric Systems for Clinical and Experimental Characterization of Soft Tissue Biomechanics,” Nat. Mater. (in press). [CrossRef]
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, “Conformal Piezoelectric Energy Harvesting and Storage From Motions of the Heart, Lung, and Diaphragm,” Proc. Natl. Acad. Sci. USA, 111(5), pp. 1927–1932. [CrossRef]
Huang, Y., Zhou, W. X., Hsia, K. J., Menard, E., Park, J. U., Rogers, J. A., and Alleyne, A. G., 2005, “Stamp Collapse in Soft Lithography,” Langmuir, 21(17), pp. 8058–8068. [CrossRef] [PubMed]
Tada, H., Paris, P. C., and Irwin, G. R., 2000, The Stress Analysis of Cracks Handbook, 3rd ed., ASME Press, New York.
Muskhelishvili, N. I., 1953, Some Basic Problems of the Mathematical Theory of 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.

Figures

Grahic Jump Location
Fig. 2

A schematic illustration of the mechanics model. (a) PZT ribbons with air-gaps, with one 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 tractions induced by subproblem 1 at all air-gaps.

Grahic Jump Location
Fig. 1

An array of PZT ribbons sandwiched by the super- and substrates. (a) A schematic diagram and (b) the super- and substrates adhere together to form air-gaps at each end of the PZT ribbon.

Grahic Jump Location
Fig. 3

The function ln{1−[m(b+w)/b]−2}−1, which the sensor voltage is linearly proportional to, versus its variable m(b + w)/b

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