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.

## Introduction

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.

## An Analytic Model

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 2*b* and spacing 2*w* 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 2*c* 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 2*b* + 4*c*, with the center part (length 2*b*) in contact with the PZT ribbon.

*U*

_{A}. Without the surrounding media, it would expand freely by (see the Appendix)

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 $\Delta u$, as shown in the Appendix. Therefore, the boundary conditions for this crack, due to actuating voltage *U*_{A}, are the constant opening displacement $\Delta u$ in the center part (of length 2*b*) and traction free around the two ends (each of length 2*c*), 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 *U*_{S,}* _{i}* denote the sensing voltage in the

*i*th PZT ribbon, which is several orders of magnitude smaller than

*U*

_{A}, as shown in the Appendix, such that the deformation induced by

*U*

_{S,}

*is also negligible. Therefore, the boundary conditions for these cracks, due to actuating voltage*

_{i}*U*

_{A}, are the vanishing opening displacement in the center part (of length 2

*b*) and traction free around the two ends (each of length 2

*c*), as illustrated in Fig. 2(a). In fact, such a crack (of length 2

*b*+ 4

*c*) can be considered as two smaller cracks (each of length 2

*c*) since the center part (of length 2

*b*) does not open up.

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

- (1)
A single crack (of length 2

*b*+ 4*c*) subjected to the opening displacement in Eq. (1) over the center part (of length 2*b*), which is modeled as a crack with a rigid wedge (Fig. 2(b)). - (2)
The collinear cracks (each of length 2

*c*) 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.

*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 by

where 0 in the second line represents the vanishing normal stress traction over two air-gaps around the PZT subjected to the actuating voltage *U*_{A}.

*c*and center-to-center spacing alternating between

*d*

_{1}= 2

*b*+ 2

*c*and

*d*

_{2}= 2

*w*− 2

*c*. As shown at the end of this section, the air-gap is much shorter than the spacing

*d*

_{1}and

*d*

_{2}such that the normal stress to be negated on the

*j*th crack can be approximated by the corresponding value at the center (

*x*=

*x*and

_{j}*y*= 0) of the crack, i.e.,

*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]

*n*is the total number of air-gaps around

*n*PZT ribbons, and the coefficients

*a*for the polynomial are determined by the single valued condition of displacement for each crack

_{j}Here, *Γ _{j}* ($1\u2264j\u22642n$) is the closed loop for the

*j*th crack. The sum of integrals for $1\u2264j\u22642n$ is equivalent to

*a*

_{2}

_{n}_{−1}= 0 [9].

*q*at the

_{i}*i*th sensor as

*U*

_{S}

*at the*

_{,}_{i}*i*th sensor is then obtained as (see the Appendix for details)

where *k*_{33} is the dielectric constant of PZT given in the Appendix.

where $\gamma $ 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/m^{2} [7], the length of air-gap is on the order of 1 *μ*m, which is much smaller than the width of the sensor (200 *μ*m).

## Special Case: One Actuator and One Sensor

*p*

_{3}and

*p*

_{4}need to be considered. Accordingly, the Westergaard function in Eq. (5) can be written as the sum of those for

*p*

_{3}and

*p*

_{4}, $Z(z)=Z3(z)+Z4(z)$, where

*a*

_{3}= 0 for

*z*

^{3}has been used. For $c\u226ab,w$, Eq. (14) can be simplified to

*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

*a*

_{3}= 0 for

*z*

^{3}has already been imposed in Eq. (15). This gives

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 $c2\Delta u$.

It is linearly proportional to the actuator voltage *U*_{A}, the effective modulus $E'$, and the thickness to width ratio *h*/(2*b*) 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 2*c* for $c\u226ab,w$.

The ratio of sensor to actuator voltage *U*_{S}/*U*_{A}, together with the material parameters of PZT and thickness to width ratio *h*/(2*b*) and spacing to width ratio *w*/*b*, gives the effective modulus $E'$, and therefore the substrate modulus (if the superstrate modulus is known).

## Approximate Solution for Multiple Actuators and Sensors

*i*th sensor in Eq. (8) becomes

It has the same relation with *U*_{A}, $E'$, *h*/(2*b*), 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 2*c* for $c\u226ab,w$.

## Conclusions

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].

## Acknowledgment

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

where $\sigma ij$, $\u025bij$, $Ei$, 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.

*a*denotes the actuator. Under plane-strain deformation $\u025b22a=\u025b12a=\u025b23a=0$, electric field boundary condition $E1a=E2a=0$ and the approximate traction free condition $\sigma 33a=0$ (by neglecting the traction from the soft super- and substrates), Eq. (A1) gives

The vanishing membrane force, $\u222bh\sigma 11adz=0$, gives $\u025b33a$. Its integration then leads to the expansion of the actuator Δ*u* in Eq. (1).

*q*on the

_{i}*i*th sensor and the vanishing stress and electric displacement fields $\sigma 33i=qi$ and $D3i=0$ (where the superscript

*i*denotes the

*i*th sensor), Eqs. (A1) and (A2), together with the vanishing of membrane force, $\u222bh\sigma 11idz=0$, give the following equations to determine $\u025b11i$, $\u025b33i$, and $E3i$ :

*i*th sensor is

which is extremely small as illustrated in main text.