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Research Papers

# Thermomechanical Analysis of Epidermal Electronic Devices Integrated With Human SkinOPEN ACCESS

[+] Author and Article Information
Yuhang Li

Institute of Solid Mechanics,
Beihang University (BUAA),
Beijing 100191, China;
Key Laboratory of Soft Machines and Smart
Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China;
State Key Laboratory of Digital Manufacturing
Equipment and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China

Jianpeng Zhang, Yufeng Xing

Institute of Solid Mechanics,
Beihang University (BUAA),
Beijing 100191, China

Jizhou Song

Key Laboratory of Soft Machines and Smart
Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China;
Department of Engineering Mechanics and Soft
Matter Research Center,
Zhejiang University,
Hangzhou 310027, China
e-mail: jzsong@zju.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 9, 2017; final manuscript received August 18, 2017; published online September 12, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(11), 111004 (Sep 12, 2017) (7 pages) Paper No: JAM-17-1435; doi: 10.1115/1.4037704 History: Received August 09, 2017; Revised August 18, 2017

## Abstract

Epidermal electronic devices (EEDs) are very attractive in applications of monitoring human vital signs for diagnostic, therapeutic, or surgical functions due to their ability for integration with human skin. Thermomechanical analysis is critical for EEDs in these applications since excessive heating-induced temperature increase and stress may cause discomfort. An axisymmetric analytical thermomechanical model based on the transfer matrix method, accounting for the coupling between the Fourier heat conduction in the EED and the bio-heat transfer in human skin, the multilayer feature of human skin and the size effect of the heating component in EEDs, is established to study the thermomechanical behavior of the EED/skin system. The predictions of the temperature increase and principle stress from the analytical model agree well with those from finite element analysis (FEA). The influences of various geometric parameters and material properties of the substrate on the maximum principle stress are fully investigated to provide design guidelines for avoiding the adverse thermal effects. The thermal and mechanical comfort analyses are then performed based on the analytical model. These results establish the theoretical foundation for thermomechanical analysis of the EED/skin system.

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## Introduction

Unlike conventional electronic devices, epidermal electronic devices (EEDs) are made of laminated layers with effective elastic modulus, bending stiffness, and mass density close to those of human skin [1]. With these unique properties, EEDs can be mounted onto the skin and retain conformal contact with the skin even under compression, tension and twist, as shown in Fig. 1(a). Their advent makes the real-time monitoring of human vital signs (e.g., body temperature [24], blood flow [5], skin hydration [6], blood glucose [7,8], blood oxygen [9], skin pressure [10,11], etc.) and mechanical properties (e.g., modulus) of human skin [12,13] possible for diagnostic, therapeutic or surgical functions, and has attracted much attention recently. For example, Webb et al. [2] developed ultrathin conformal devices that can measure the skin temperature continuously and precisely. Gao et al. [6] developed an ultrathin wireless epidermal photonic device for the measurement of the blood flow and skin hydration to establish their relevance to cardiovascular health and skin care. Lee et al. [7] fabricated a wearable sweat-based glucose monitoring device without invasion into the human body. Gao et al. [8] developed a fully integrated wearable sensor arrays to measure the sweat metabolites, skin temperature and electrolytes. Li et al. [9] developed a new strategy to design an epidermal inorganic optoelectronic device for the blood oxygen measurement.

Based on whether EEDs have thermal effects, EEDs can be classified into two categories. The first category is EEDs consisting of components without thermal effects or with negligible thermal effects such as temperature sensors [2], chemical sensors [7,8], and pressure sensors [10,11]. For this type of EEDs, the main critical issue is to remain perfect conglutination between EEDs and human skin, which has been investigated by many researchers [1417]. For example, Cheng and Wang [14] developed an analytical mechanics model to establish a criterion for conformal contact between EEDs and human skin when subjected to an external loading. Liu and Lu [15] investigated the conformal contact between a thin film and a slightly wavy surface. Lu et al. [16] studied the fracture at the interface between EEDs and a compliant substrate. Huang et al. [17] studied the interfacial delamination of inorganic films on viscoelastic substrates. The second kind is EEDs consisting of components with thermal effects such as heating sensors to measure the blood flow [6] and light-emitting diodes for blood oxygen measurement [9]. For this type of EEDs, in spite of the perfect conglutination between EEDs and human skin, knowledge on thermomechanical properties of the EED/skin system due to the physical coupling of EEDs to the skin is very critical since excessive heating-induced temperature increase or stress in the skin may cause discomfort.

In order to accurately predict the thermomechanical properties of the EED/skin system and prevent the heating-induced discomfort, the points below should be considered: (1) the coupling of Fourier heat conduction in EEDs and bio-heat conduction in human skin; (2) the multilayer feature of human skin, which consists of three layers (epidermis, dermis, and fat from top to bottom) with much different mechanical properties (Young’s modulus may vary from 0.01 MPa for fat to 100 MPa for epidermis) [18,19]; (3) the size effect of the heating components in EEDs since their in-plane size (∼100μm) may be much smaller than the thickness of human skin (∼5 mm). Although there exist some studies on thermal behaviors of EED/skin system [2022], the above three factors complicate the study on the thermo-mechanical behaviors of the EED/skin system with very seldom reports, which motivates us to perform thermomechanical analysis of the EED/skin system.

This paper aims to establish an analytical thermomechanical model based on the transfer matrix method to study the thermomechanical behaviors of the EED/skin system. The model is further validated by finite element analysis (FEA). Section 2 describes the mathematic modeling of the EED/skin system. The results and discussion are given in Sec. 3. The main conclusion is summarized in Sec. 4.

## Mathematic Modeling of the System

A typical EED made of microscale inorganic light emitting diodes (μ-ILEDs) integrated with human skin is taken as an example to illustrate our analytical model. Figure 1(b) schematically shows the cross-sectional structure of the EED/skin system. The μ-ILED, located on top of the polydimethylsiloxane (PDMS) substrate with the thickness denoted by $hP$, serves as the heating component and is modeled as a circular heat source with the radius $r0$ and the heat flux $Q0$. The PDMS substrate is attached to the human skin, which consists of three layers from top to bottom: epidermis, dermis and fat. Let $hE$, $hD$, and $hF$, denote the thicknesses of epidermis, dermis, and fat, respectively. The in-plane dimensions of the PDMS substrate and the human skin are on the order of ∼centimeters, which is much larger than the system thickness (∼millimeters). Therefore, it is reasonable to assume the in-plane dimensions of the PDMS substrate and the human skin to be infinite. For simplicity, an axisymmetric analytical thermo-mechanical model is established for the EED/skin system. It should be noted that the results are also applicable for a square μ-ILED [23,24]. At the bottom surface of the fat layer, the temperature is set as the body temperature and the displacements are set to be zero. At the top surface of the PDMS, the heat flux $Q0$ is applied within the region of the μ-ILED while the heat flux is assumed to be zero outside the μ-ILED, which is reasonable to approximate the natural convection boundary [23,24] in practical applications. Furthermore, the top surface is stress free.

The steady Pennes bio-heat transfer equation [18,19] takes the form of $λ∇2T−ωρc(T−Tb)+qmet=0$, which accounts for the effects of blood perfusion and metabolic heat generation, and suffices for modeling the heat transfer in the skin although the bio-heat transfer may be modeled by the dual-phase-lag model or thermal wave model to further account for the lengthy thermal relaxation time in biological tissue [25]. Here $∇2$ is the Laplace operator, $λ$ is the thermal conductivity of the skin, $ω$ is the blood perfusion rate, $ρ$ is the blood density, $c$ is the heat capacity of the blood, $qmet$ is the metabolic heat generation, and $Tb$ is the blood temperature, which is usually taken as the body temperature. The effect of the metabolic heat generation in the skin on the temperature increase is negligible [18] such that the steady Pennes bio-heat transfer equation can be written as $λ∇2T−ωρc(T−Tb)=0$. If the effect of blood perfusion is also not included, the Pennes bio-heat transfer equation degenerates to the Fourier heat conduction equation, i.e., $λ∇2T=0$. The EED/skin system is a multilayer structure consisting of PDMS, epidermis, dermis and fat as shown in Fig. 1(b). Since the blood perfusion only exists in the dermis layer [18,19], we will apply the Pennes bio-heat transfer equation in the dermis layer while the Fourier heat conduction equation in other layers including PDMS, epidermis, and fat.

For the dermis layer with the blood perfusion, the equilibrium equations and the heat transfer equation in the cylindrical coordinate system with the origin located at the bottom surface of the fat layer (as shown in Fig. 1(b)) are given by Display Formula

(1)$11−2μD∂e∂r+∇2u−ur2=2αD(1+μD)1−2μD∂θ∂r11−2μD∂e∂z+∇2w=2αD(1+μD)1−2μD∂θ∂zλD∇2θ=ωDρDcDθ$

where u and w are the displacements along the radial and axial directions, respectively, $e=∂u/∂r+u/r+∂w/∂z$, $μD$ and $αD$ are the Poisson’s ratio and the thermal expansion coefficient of dermis, respectively, $θ=T−Tb$ is the temperature increase from the body temperature, $ωD$ is the blood perfusion rate in the dermis layer, $ρD$ and $cD$ are the density and heat capacity of the blood in the dermis layer, and $λD$ is the thermal conductivity of dermis. The constitutive relations for the dermis are given by Display Formula

(2)$σr=2GD[μD1−2μDe+∂u∂r]−αDEDθ1−2μDσθ=2GD[μD1−2μDe+ur]−αDEDθ1−2μDσz=2GD[μD1−2μDe+∂w∂z]−αDEDθ1−2μDτzr=GD[∂u∂z+∂w∂r]$

where $ED$ is the Young’s modulus of dermis and $GD=ED/[2(1+μD)]$ is the shear modulus of dermis. The summation of applying the operator $∂/∂r+1/r$ on the first equation in Eq. (1) and partial derivative with respect to z on the second equation in Eq. (2) yields Display Formula

(3)$∇2e=αD(1+μD)1−μD∇2θ$

The first-order Hankel transform $f̃(ξ,z)=∫0∞f(r,z)J1(ξr)rdr$ [26] of the first equation in Eq. (1) and the zero-order Hankel transform $f̃(ξ,z)=∫0∞f(r,z)J0(ξr)rdr$ [26] of the second and third equations in Eqs. (1) and (3) give the following ordinary differential equations: Display Formula

(4)$d2ũdz2−ξ2ũ−ξẽ1−2μD+2αD(1+μD)ξθ̃1−2μD=0d2w̃dz2−ξ2w̃+11−2μDdẽdz−2αD(1+μD)1−2μDdθ̃dz=0d2ẽdz2−ξ2ẽ=αD(1+μD)1−μDv2θ̃d2θ̃dz2−β2θ̃=0$
with $J0$ and $J1$ as the zeroth-order and first-order Bessel function of the first kind, $v2=ωDρDcD/λD$, $β2=v2+ξ2$, $ũ=∫0∞uJ1(ξr)rdr$, $w̃=∫0∞wJ0(ξr)rdr$, $θ̃=∫0∞θJ0(ξr)rdr$ and $ẽ=∫0∞eJ0(ξr)rdr$. Equation (4) has the solutions Display Formula
(5)$θ̃=A1eβz+B1e−βzẽ=A2eξz+B2e−ξz+αD(1+μD)1−μD(A1eβz+B1e−βz)ũ=A3eξz+B3e−ξz−αD(1+μD)ξ1−μD(A1eβz+B1e−βz)+z(A2eξz−B2e−ξz)2−4μDw̃=A4eξz+B4e−ξz+αD(1+μD)β1−μD(A1eβz−B1e−βz)−z(A2eξz+B2e−ξz)2−4μD$

where the coefficients $Ai$ and $Bi(i=1,2,3,4)$ are to be determined by the boundary and continuity conditions. The Hankel transform of $e=∂u/∂r+u/r+∂w/∂z$ gives $ẽ=ξũ+∂w̃/∂z$, which yields $A2=2ξ(2μD−1)(A3+A4)/(4μD−3)$ and $B2=2ξ(2μD−1)(B3−B4)/(4μD−3)$. Combined with Eq. (5), the zero-order Hankel transform of $σz$ and $τrz$ in Eq. (2) and the heat flux equation $Q̃=λD∂θ̃/∂z$ give the following equation: Display Formula

(6)${σ̃z(ξ,z)τ̃zr(ξ,z)ũ(ξ,z)w̃(ξ,z)θ̃(ξ,z)Q̃(ξ,z)}=[M(ξ,z)]{A1B1A3B3A4B4}$

where $[M(ξ,z)]$ is a 6 × 6 matrix, which is given in the Appendix. The coefficients $Ai$ and $Bi(i=1,3,4)$ on the right hand side of Eq. (6) can be obtained by applying z to be $hF$ such that Eq. (6) becomes Display Formula

(7)${σ̃z(ξ,z)τ̃zr(ξ,z)ũ(ξ,z)w̃(ξ,z)θ̃(ξ,z)Q̃(ξ,z)}=[ΦD(ξ,z)]{σ̃z(ξ,hF)τ̃zr(ξ,hF)ũ(ξ,hF)w̃(ξ,hF)θ̃(ξ,hF)Q̃(ξ,hF)}$

where $[ΦD(ξ,z)]$ is the transfer matrix given by Display Formula

(8)$[ΦD(ξ,z)]=[M(ξ,z)][M(ξ,hF)]−1$

For the other layers without the blood perfusion including the PDMS, epidermis and fat layers, we can follow the same procedure for the dermis layer but replace the bio-heat transfer equation (the third equation in Eq. (1)) by the Fourier heat conduction equation $λ∇2θ=0$. The constitutive relations for the PDMS, epidermis and fat layers remain same as Eq. (2) except to replace the subscript D for the dermis layer by F, E, and P for the fat, epidermis, and PDMS layer, respectively. The transfer matrix can then be obtained as Display Formula

(9)$[ΦF(ξ,z)]=[NF(ξ,z)][NF(ξ,0)]−1[ΦE(ξ,z)]=[NE(ξ,z)][NE(ξ,hF+hD)]−1[ΦP(ξ,z)]=[NP(ξ,z)][NP(ξ,hF+hD+hE)]−1$
for the fat, epidermis, and substrate layer, respectively. Here, $[N(ξ,z)]$ is given in the Appendix with the subscripts F, E, and P for the fat, epidermis, and substrate layer, respectively. At the bottom surface of the fat layer ($z=0$), the temperature increase from the body temperature is zero, which gives $θ̃(ξ,0)=0$. The fixed boundary at the bottom surface gives $ũ(ξ,0)=0$ and $w̃(ξ,0)=0$. The heat flux condition at the top surface ($z=hP+hE+hD+hF$) with $Q0$ within the heating region ($r≤r0$) and zero outside the heating region ($r>r0$) gives $Q̃(ξ,hP+hE+hD+hF)=Q0r0J1(ξr0)/ξ$. The stress free boundary at the top surface gives $σ̃z(ξ,hP+hE+hD+hF)=0$ and $τ̃rz(ξ,hP+hE+hD+hF)=0$. The interfaces between different layers satisfy the continuity condition, i.e., the temperature increase, heat flux, displacements, and stresses are continuous across the interfaces. Based on the transfer matrix method, we have Display Formula
(10)${σ̃z(ξ,hP+hE+hD+hF)τ̃zr(ξ,hP+hE+hD+hF)ũ(ξ,hP+hE+hD+hF)w̃(ξ,hP+hE+hD+hF)θ̃(ξ,hP+hE+hD+hF)Q̃(ξ,hP+hE+hD+hF)}=[Λ]{σ̃z(ξ,0)τ̃zr(ξ,0)ũ(ξ,0)w̃(ξ,0)θ̃(ξ,0)Q̃(ξ,0)}$

where Display Formula

(11)$[Λ]=[ΦP(ξ,hP+hE+hD+hF)][ΦE(ξ,hE+hD+hF)][ΦD(ξ,hD+hF)][ΦF(ξ,hF)]$

By applying the boundary conditions at the top and bottom surfaces, the stresses and the heat flux at the bottom surface can be obtained by Display Formula

(12)${σ̃z(ξ,0)τ̃zr(ξ,0)Q̃(ξ,0)}=[Λ11Λ12Λ16Λ21Λ22Λ26Λ61Λ62Λ66]−1{00Q0r0J1(ξr0)/ξ}$

where $Λij$ is the component at the ith row and jth column of the matrix $[Λ]$. The stresses, displacements, and temperature increase in the epidermis layer ($hF+hD≤z≤hF+hD+hE$) can be obtained by the transfer matrix method as Display Formula

(13)${σ̃z(ξ,z)τ̃zr(ξ,z)ũ(ξ,z)w̃(ξ,z)θ̃(ξ,z)Q̃(ξ,z)}=[Ω]{σ̃z(ξ,0)τ̃zr(ξ,0)000Q̃(ξ,0)}$
followed by the inverse Hankel transform where: Display Formula
(14)$[Ω]=[ΦE(ξ,z)][ΦD(ξ,hD+hF)][ΦF(ξ,hF)]$

For example, the temperature increase and displacements in the epidermis layer are given by Display Formula

(15)$θ(r,z)=∫0∞Ω56Q̃(ξ,0)J0(ξr)ξdξ$
Display Formula
(16)$u(r,z)=∫0∞[Ω31σ̃z(ξ,0)+Ω32τ̃zr(ξ,0)+Ω36Q̃(ξ,0)]J1(ξr)ξdξ$
Display Formula
(17)$w(r,z)=∫0∞[Ω41σ̃z(ξ,0)+Ω42τ̃zr(ξ,0)+Ω36Q̃(ξ,0)]J0(ξr)ξdξ$

The substitution of the temperature increase and displacements in Eqs. (15)(17) into Eq. (2) by replacing the subscript D for the dermis layer by E for the epidermis layer and then setting z by $hF+hD+hE$ yields stresses $σr$, $σz$ and $τrz$ at the EED/skin interface, which give the maximum principle stress determining whether thermal stress-induced discomfort occurs.

## Results and Discussion

The material properties and geometric parameters in the calculations below are listed in Table 1. The density and heat capacity of the blood are 1060 kg/m3 and 3770 J/kg/K, respectively [18]. The radius $r0$ of the heating region, corresponding to the size of heating component, at the top surface is set as 2 mm with the heat flux $Q0$ as 2.3 mW/mm2. An axisymmetric thermomechanical finite element model is also developed with abaqus to investigate the temperature increase and stress distribution in the EED/skin system. The CAX4T element is used to discretize the geometry with the uniform size of 0.03 mm, which ensures the results to be convergent. The in-plane dimension (i.e., the radius) of the substrate and skin is taken as 10 cm, which is much larger than the system thickness to eliminate the effect of the in-plane size. The top surface has a heat flux boundary $Q0$ within the region of $r≤r0$ and zero flux boundary outside the region of $r>r0$. The bottom surface’s displacements and temperature increase are set to be zero.

Figure 2(a) shows the distribution of the temperature increase at the EED/skin interface along the radial direction. The solid line denotes the result from the analytical prediction while the dot is from finite element analysis. The good agreement between the analytical prediction and FEA validates the accuracy of the analytical model. The maximum temperature increase at the EED/skin interface occurs at the center ($r=0$) and then decreases as $r$ increases. When $r$ increases from 0 to 4 mm, the temperature increase decays quickly from 3.2 °C to 0.4 °C, and then drops slowly to 0.01 °C as $r$ further increases to 9 mm. Figure 2(b) shows the distribution of the temperature increase along the thickness direction with $r=0$. The maximum temperature occurs at the center location of heat source ($z=0$) and then decreases as the distance to the center location of heat source increases. The temperature increase in the skin ($z≤6 mm$) remains small since the PDMS substrate with a small thermal conductivity prevents the heat dissipation into the skin.

Figure 3 shows the distribution of the maximum principle stress $σmax(r,z=hE+hD+hF)$ at the EED/skin interface along the radial direction. The analytical prediction (solid line) agrees well with FEA (dot), which further validates the accuracy the analytical thermomechanical model. The maximum principle stress reaches the maximum and remains almost unchanged within the heating region ($r≤2 mm$) and decreases as $r$ increases. The maximum principle stress $σmax=σmax(r=0,z=hE+hD+hF)$ at the EED/skin interface is very important since it may determine whether stress-induced discomfort occurs [27]. Therefore, we further studied the influences of geometric parameters and material properties of the substrate on $σmax$ to obtain design guidelines for EEDs to avoid discomfort.

The influences of the thickness and the thermal conductivity of the substrate on $σmax$ are shown in Fig. 4(a). As the thermal conductivity of the substrate increases, $σmax$ decreases because a large thermal conductivity may lead to a low temperature increase at the EED/skin interface, which reduces the thermal mismatch at the EED/skin interface. The increase of the substrate thickness first reduces the compressive stress and then increases the tensile stress to be maximum at some critical thickness. When the thickness further increases, the tensile stress $σmax$ decreases to zero. This can be understood by the following. When the substrate thickness is small, the thermal expansion of the skin within the heating region is mainly constrained by its surrounding skin, which yields a compressive stress. When the substrate thickness increases, the larger thermal expansion of the substrate comparing to that of skin stretches the skin to reduce the compressive stress and increase the tensile stress. When the substrate thickness is large such that the temperature increase at the skin surface can be approximated to be zero, the tensile stress then decreases from the maximum value to zero. It is interesting to observe that $σmax$ may equal to zero at some critical thicknesses such that the EED becomes completely “mechanical invisible” to the skin. For example, the critical thicknesses are 0.75 mm and 1.5 mm for the cases with thermal conductivity of 0.15 W/m/K and 0.45 W/m/K, respectively. Figure 4(b) shows the influences of the Young’s modulus and the coefficient of thermal expansion of the substrate on $σmax$. For the same temperature increase, a larger Young’s modulus or thermal expansion coefficient of the substrate induces a larger thermal mismatch at EED/skin interface to yield a larger thermal stress.

The analytical model here is critical to design EEDs to avoid the adverse thermal effects and perform comfort analysis. Xu et al. [28] pointed out that the critical temperature increase to induce thermal discomfort is about 6 °C, which is adopted here to define the thermal comfort region. In addition, the thermal-induced stress may cause mechanical discomfort. We adopt the minimum value 20 kPa of the maximum principle stress that skin can feel [27] to define the mechanical comfort region of skin. The size of the heating component $r0$ and the substrate thickness $hP$ are taken as design parameters to obtain the thermal and mechanical comfort region as shown in Fig. 5 for the heat generation of 45 mW. Figure 5(a) shows the maximum temperature increase at the EED/skin interface varies with $r0$ and $hP$. The black line in Fig. 5(a) determines the critical design parameters, which gives the maximum temperature increase at the EED/skin interface to be equal to 6 °C. The region above the line corresponds to the thermal comfortable region. It is shown that a larger size of heating component or substrate thickness is helpful to reduce the maximum temperature increase at the EED/skin interface. Figure 5(b) shows the maximum principle stress $σmax$ at the EED/skin interface varies with $r0$ and $hP$. The white and black lines in Fig. 5(b) determine the critical design parameters, which give $σmax$ to be equal to −20 kPa and 20 kPa, respectively. The regions above the white and black lines correspond to the mechanical comfortable region.

## Conclusions

An axisymmetric analytical thermomechanical model based on the transfer matrix method is established to predict the temperature increase and mechanical behaviors of the EED integrated with human skin. The analytical model accounts for the coupling between the Fourier heat conduction in the EED and bio-heat transfer in human skin, the multilayer feature of human skin and the size effect of the heating component in EEDs. The temperature increase and the principle stress predicted from the analytical model agree very well with FEA. The influences of various geometric parameters and material properties of the substrate on the maximum principle stress are investigated. The thermal and mechanical comfort analyses are then performed based on the analytical model. These results pave the theoretical foundation for thermomechanical analysis of the EED/skin system.

## Acknowledgements

The authors acknowledge the support from the National Basic Research Program of China (Grant No. 2015CB351901), the Research Fund for the Doctoral Program of Higher Education of China (20131102110039), and the Fundamental Research Funds for the Central Universities.

## Funding Data

• National Natural Science Foundation of China (Grant Nos. 11502009, 11772030, 11172028, 11372021, and 11621062).

• National Research Foundation of Korea (Grant No. NRF-2017M3A7B404946).

## Appendices

###### Appendix: $[M(ξ,z)]and$$[N(ξ,z)]$

The matrix $[M(ξ,z)]$ is given by Display Formula

(A1)$[M(ξ,z)]=[M11M12M13M14M15M16M21M22M23M24M25M26M31M32M33M34M35M36M41M42M43M44M45M46M51M52M53M54M55M56M61M62M63M64M65M66]$
with

$M11=2GDαD(1+μD)ξ2eβz(1−μD)ν2,M12=2GDαD(1+μD)ξ2e−βz(1−μD)ν2,M13=(−2ξz+4μD−2)GDξeξz3−4μD,M14=(2ξz+4μD−2)GDξe−ξz3−4μD,M15=(2ξz+4μD−4)GDξeξz4μD−3,M16=(−2ξz+4μD−4)GDξe−ξz3−4μD,M21=2GDαD(1+μD)ξeβzβ(−1+μD)ν2,M22=2GDαD(1+μD)ξe−βzβ(1−μD)ν2,M23=(−2ξz+4μD−4)GDξeξz4μD−3,M24=(2ξz+4μD−4)GDξe−ξz3−4μD,M25=(2ξz+4μD−2)GDξeξz3−4μD,M26=(−2ξz+4μD−2)GDξe−ξz3−4μD,M31=αD(1+μD)ξeβz(−1+μD)ν2,M32=αD(1+μD)ξe−βz(−1+μD)ν2,M33=(−ξz+4μD−3)eξz4μD−3,M34=(ξz+4μD−3)e−ξz4μD−3,M35=−ξzeξz4μD−3,M36=−ξze−ξz4μD−3,M41=−αD(1+μD)βeβz(−1+μD)ν2,M42=αD(1+μD)βe−βz(−1+μD)ν2,M43=ξzeξz4μD−3,M44=ξze−ξz4μD−3,M45=(ξz+4μD−3)eξz4μD−3,M46=(−ξz+4μD−3)e−ξz4μD−3,M51=eβz,M52=e−βz,M53=M54=M55=M56=0,M61=λDβeβz,M62=−λDβe−βz,M63=M64=M65=M66=0$
The matrix $[N(ξ,z)]$ is given by Display Formula
(A2)$[N(ξ,z)]=[N11N12N13N14N15N16N21N22N23N24N25N26N31N32N33N34N35N36N41N42N43N44N45N46N51N52N53N54N55N56N61N62N63N64N65N66]$
with

$N11=αG(1+μ)(4ξz−2)eξz3−4μ,N12=αG(1+μ)(4ξz+2)e−ξz3−4μ,N13=(−2ξz+4μ−2)Gξeξz3−4μ,N14=−(2ξz+4μ−2)Gξe−ξz4μ−3,N15=(2ξz+4μ−4)Gξeξz4μ−3,N16=−(−2ξz+4μ−4)Gξe−ξz4μ−3,N21=αG(1+μ)(4ξz+2)eξz4μ−3,N22=αG(1+μ)(4ξz−2)e−ξz4μ−3,N23=(−2ξz+4μ−4)Gξeξz4μ−3,N24=−(2ξz+4μ−4)Gξe−ξz4μ−3,N25=−(2ξz+4μ−2)Gξeξz4μ−3,N26=−(−2ξz+4μ−2)Gξe−ξz4μ−3,N31=2αz(1+μ)eξz4μ−3,N32=−2αz(1+μ)e−ξz4μ−3,N33=(−ξz+4μ−3)eξz4μ−3,N34=(ξz+4μ−3)e−ξz4μ−3,N35=−ξzeξz4μ−3,N36=−ξze−ξz4μ−3,N41=−2αz(1+μ)eξz4μ−3,N42=−2αz(1+μ)e−ξz4μ−3,N43=ξzeξz4μ−3,N44=ξze−ξz4μ−3,N45=(ξz+4μ−3)eξz4μ−3,N46=(−ξz+4μ−3)e−ξz4μ−3,N51=eξz,N52=e−ξz,N53=N54=N55=N56=0,N61=λξeξz,N62=−λξe−ξz,N63=N64=N65=N66=0$

## References

Kim, D.-H. , Lu, N. S. , Ma, R. , Kim, Y.-S. , Kim, R.-H. , Wang, S. D. , Wu, J. , Won, S. M. , Tao, H. , Islam, A. , Yu, K. J. , Kim, T. I. , Chowdhury, R. , Ying, M. , Xu, L. Z. , Li, M. , Chung, H. J. , Keum, H. , McCormick, M. , Liu, P. , Zhang, Y. W. , Omenetto, F. G. , Huang, Y. G. , Coleman, T. , and Rogers, J. A. , 2011, “ Epidermal Electronics,” Science, 333(6044), pp. 838–843. [PubMed]
Webb, R. C. , Bonifas, A. P. , Behnaz, A. , Zhang, Y. H. , Yu, K. J. , Cheng, H. Y. , Shi, M. , Bian, Z. , Liu, Z. , Kim, Y.-S. , Yeo, W.-H. , Park, J. S. , Song, J. , Li, Y. , Huang, Y. , Gorbach, A. M. , and Rogers, J. A. , 2013, “ Ultrathin Conformal Devices for Precise and Continuous Thermal Characterization of Human Skin,” Nat. Mater., 12(10), pp. 938–944. [PubMed]
Lee, J. W. , Xu, R. X. , Lee, S. , Jang, K. I. , Yang, Y. C. , Banks, A. , Yu, K. J. , Kim, J. , Xu, S. , Ma, S. Y. , Jang, S. W. , Won, P. , Li, Y. H. , Kim, B. H. , Choe, J. Y. , Huh, S. , Kwon, Y. H. , Huang, Y. G. , Paik, U. , and Rogers, J. A. , 2016, “ Soft, Thin Skin-Mounted Power Management Systems and Their Use in Wireless Thermography,” Proc. Natl. Acad. Sci. U. S. A., 113(22), pp. 6131–6136. [PubMed]
Zhang, Y. H. , Webb, R. C. , Luo, H. Y. , Xue, Y. G. , Kurniawan, J. , Cho, N. H. , Krishnan, S. , Li, Y. H. , Huang, Y. G. , and Rogers, J. A. , 2016, “ Theoretical and Experimental Studies of Epidermal Heat Flux Sensors for Measurements of Core Body Temperature,” Adv. Healthcare Mater., 5(1), pp. 119–127.
Webb, R. C. , Ma, Y. , Krishnan, S. , Li, Y. , Yoon, S. , Guo, X. , Feng, X. , Shi, Y. , Seidel, M. , Cho, N. H. , Kurniawan, J. , Ahad, J. , Sheth, N. , Kim, J. , Taylor, J. G. , Darlington, T. , Chang, K. , Huang, W. , Ayers, J. , Gruebele, A. , Pielak, R. M. , Slepian, M. J. , Huang, Y. , Gorbach, A. M. , and Rogers, J. A. , 2015, “ Epidermal Devices for Noninvasive, Precise, and Continuous Mapping of Macrovascular and Microvascular Blood Flow,” Sci. Adv., 1(9), p. e1500701. [PubMed]
Gao, L. , Zhang, Y. H. , Malyarchuk, V. , Jia, L. , Jang, K. I. , Webb, R. C. , Fu, H. R. , Shi, Y. , Zhou, G. Y. , Shi, L. K. , Shah, D. , Huang, X. , Xu, B. X. , Yu, C. J. , Huang, Y. G. , and Rogers, J. A. , 2014, “ Epidermal Photonic Devices for Quantitative Imaging of Temperature and Thermal Transport Characteristics of the Skin,” Nat. Commun., 5, p. 4938. [PubMed]
Lee, H. , Choi, T. K. , Lee, Y. B. , Cho, H. R. , Ghaffari, R. , Wang, L. , Choi, H. J. , Chung, T. D. , Lu, N. S. , Hyeon, T. , Choi, S. H. , and Kim, D. H. , 2016, “ A Graphene-Based Electrochemical Device With Thermoresponsive Microneedles for Diabetes Monitoring and Therapy,” Nat. Nanotechnol., 11(6), pp. 566–572.
Gao, W. , Emaminejad, S. , Nyein, H. Y. Y. , Challa, S. , Chen, K. V. , Peck, A. , Fahad, H. M. , Ota, H. , Shiraki, H. , Kiriya, D. , Lien, D. H. , Brooks, G. A. , Davis, R. W. , and Javey, A. , 2016, “ Fully Integrated Wearable Sensor Arrays for Multiplexed In Situ Perspiration Analysis,” Nature, 529(7587), p. 509. [PubMed]
Li, H. , Xu, Y. , Li, X. , Chen, Y. , Jiang, Y. , Zhang, C. , Lu, B. , Wang, J. , Ma, Y. , Chen, Y. , Huang, Y. , Ding, M. , Su, H. , Song, G. , Luo, Y. , and Feng, X. , 2017, “ Epidermal Inorganic Optoelectronics for Blood Oxygen Measurement,” Adv. Healthcare Mater., 6(9), p. 1601013.
Dagdeviren, C. , Su, Y. W. , Joe, P. , Yona, R. , Liu, Y. H. , Kim, Y. S. , Huang, Y. A. , Damadoran, A. R. , Xia, J. , Martin, L. W. , Huang, Y. G. , and Rogers, J. A. , 2014, “ Conformable Amplified Lead Zirconate Titanate Sensors With Enhanced Piezoelectric Response for Cutaneous Pressure Monitoring,” Nat. Commun., 5, p. 4496. [PubMed]
Wang, C. , Hwang, D. , Yu, Z. B. , Takei, K. , Park, J. , Chen, T. , Ma, B. W. , and Javey, A. , 2013, “ User-Interactive Electronic Skin for Instantaneous Pressure Visualization,” Nat. Mater., 12(10), pp. 899–904. [PubMed]
Shi, Y. , Dagdeviren, C. , Rogers, J. A. , Gao, C. F. , and Huang, Y. , 2015, “ An Analytical Model for Skin Modulus Measurement Via Conformal Piezoelectric Systems,” ASME J. Appl. Mech., 82(9), p. 091007.
Yuan, J. H. , Shi, Y. , Pharr, M. , Feng, X. , Rogers, J. A. , and Huang, Y. , 2016, “ A Mechanics Model for Sensors Imperfectly Bonded to the Skin for Determination of the Young’s Moduli of Epidermis and Dermis,” ASME J. Appl. Mech., 83(8), p. 084501.
Cheng, H. Y. , and Wang, S. D. , 2014, “ Mechanics of Interfacial Delamination in Epidermal Electronics Systems,” ASME J. Appl. Mech., 81(4), p. 044501.
Liu, W. , and Lu, N. S. , 2016, “ Conformability of a Thin Elastic Membrane Laminated on a Soft Substrate With Slightly Wavy Surface,” ASME J. Appl. Mech., 83(4), p. 041007.
Lu, N. S. , Zhang, Z. , Yoon, J. , and Suo, Z. G. , 2012, “ Singular Stress Fields at Corners in Flip-Hip Packages,” Eng. Fract. Mech., 86, pp. 38–47.
Huang, Y. , Yuan, J. , Zhang, Y. , and Feng, X. , 2016, “ Interfacial Delamination of Inorganic Films on Viscoelastic Substrates,” ASME J. Appl. Mech., 83(10), p. 101005.
Xu, F. , Lu, T. J. , and Steffen, K. A. , 2008, “ Biothermomechanics of Skin Tissues,” J. Mech. Phys. Solids, 56(5), pp. 1852–1884.
Xu, F. , Lu, T. J. , Steffen, K. A. , and Ng, E. Y. K. , 2009, “ Mathematical Modeling of Skin Bioheat Transfer,” Appl. Mech. Rev., 62(5), p. 050801.
Song, J. , Feng, X. , and Huang, Y. , 2016, “ Mechanics and Thermal Management of Stretchable Inorganic Electronics,” Natl. Sci. Rev., 3(1), p. 128. [PubMed]
Cui, Y. , Li, Y. H. , Xing, Y. F. , Yang, T. Z. , and Song, J. Z. , 2016, “ One-Dimensional Thermal Analysis of the Flexible Electronic Devices Integrated With Human Skin,” Micromachines, 7(11), p. 210.
Cui, Y. , Li, Y. H. , Xing, Y. F. , Ji, Q. G. , and Song, J. Z. , 2017, “ Thermal Design of Rectangular Microscale Inorganic Light-Emitting Diodes,” Appl. Therm. Eng., 122, pp. 653–660.
Lü, C. , Li, Y. , Song, J. , Kim, H. S. , Brueckner, E. , Fang, B. , Hwang, K. C. , Huang, Y. , Nuzzo, R. G. , and Rogers, J. A. , 2012, “ A Thermal Analysis of the Operation of Microscale, Inorganic Light-Emitting Diodes,” Proc. R. Soc. London A, 468(2146), pp. 3215–3223.
Li, Y. , Shi, Y. , Song, J. , Lü, C. , Kim, T. , Rogers, J. A. , and Huang, Y. , 2013, “ Thermal Properties of Microscale Inorganic Light-Emitting Diodes in a Pulsed Operation,” J. Appl. Phys., 113(14), p. 144505.
Xu, F. , Seffen, K. A. , and Lu, T. J. , 2008, “ Non-Fourier Analysis of Skin Biothermomechanics,” Int. J. Heat Mass Transfer, 51(9–10), pp. 2237–3704.
Carslaw, H. S. , and Jaeger, J. C. , 1959, Conduction of Heat in Solids, 2nd ed., Carendon Press, Oxford, UK.
Lee, C. H. , Ma, Y. , Jang, K. I. , Banks, A. , Pan, T. , Feng, X. , Kim, J. S. , Kang, D. , Raj, M. S. , McGrane, B. L. , Morey, B. , Wang, X. , Ghaffari, R. , Huang, Y. , and Rogers, J. A. , 2015, “ Soft Core/Shell Packages for Stretchable Electronics,” Adv. Funct. Mater., 25(24), pp. 3698–3704.
Xu, F. , Wen, T. , Seffen, K. , and Lu, T. , 2008, “ Modeling of Skin Thermal Pain: A Preliminary Study,” Appl. Math. Comput., 205(1), pp. 37–46.
View article in PDF format.

## References

Kim, D.-H. , Lu, N. S. , Ma, R. , Kim, Y.-S. , Kim, R.-H. , Wang, S. D. , Wu, J. , Won, S. M. , Tao, H. , Islam, A. , Yu, K. J. , Kim, T. I. , Chowdhury, R. , Ying, M. , Xu, L. Z. , Li, M. , Chung, H. J. , Keum, H. , McCormick, M. , Liu, P. , Zhang, Y. W. , Omenetto, F. G. , Huang, Y. G. , Coleman, T. , and Rogers, J. A. , 2011, “ Epidermal Electronics,” Science, 333(6044), pp. 838–843. [PubMed]
Webb, R. C. , Bonifas, A. P. , Behnaz, A. , Zhang, Y. H. , Yu, K. J. , Cheng, H. Y. , Shi, M. , Bian, Z. , Liu, Z. , Kim, Y.-S. , Yeo, W.-H. , Park, J. S. , Song, J. , Li, Y. , Huang, Y. , Gorbach, A. M. , and Rogers, J. A. , 2013, “ Ultrathin Conformal Devices for Precise and Continuous Thermal Characterization of Human Skin,” Nat. Mater., 12(10), pp. 938–944. [PubMed]
Lee, J. W. , Xu, R. X. , Lee, S. , Jang, K. I. , Yang, Y. C. , Banks, A. , Yu, K. J. , Kim, J. , Xu, S. , Ma, S. Y. , Jang, S. W. , Won, P. , Li, Y. H. , Kim, B. H. , Choe, J. Y. , Huh, S. , Kwon, Y. H. , Huang, Y. G. , Paik, U. , and Rogers, J. A. , 2016, “ Soft, Thin Skin-Mounted Power Management Systems and Their Use in Wireless Thermography,” Proc. Natl. Acad. Sci. U. S. A., 113(22), pp. 6131–6136. [PubMed]
Zhang, Y. H. , Webb, R. C. , Luo, H. Y. , Xue, Y. G. , Kurniawan, J. , Cho, N. H. , Krishnan, S. , Li, Y. H. , Huang, Y. G. , and Rogers, J. A. , 2016, “ Theoretical and Experimental Studies of Epidermal Heat Flux Sensors for Measurements of Core Body Temperature,” Adv. Healthcare Mater., 5(1), pp. 119–127.
Webb, R. C. , Ma, Y. , Krishnan, S. , Li, Y. , Yoon, S. , Guo, X. , Feng, X. , Shi, Y. , Seidel, M. , Cho, N. H. , Kurniawan, J. , Ahad, J. , Sheth, N. , Kim, J. , Taylor, J. G. , Darlington, T. , Chang, K. , Huang, W. , Ayers, J. , Gruebele, A. , Pielak, R. M. , Slepian, M. J. , Huang, Y. , Gorbach, A. M. , and Rogers, J. A. , 2015, “ Epidermal Devices for Noninvasive, Precise, and Continuous Mapping of Macrovascular and Microvascular Blood Flow,” Sci. Adv., 1(9), p. e1500701. [PubMed]
Gao, L. , Zhang, Y. H. , Malyarchuk, V. , Jia, L. , Jang, K. I. , Webb, R. C. , Fu, H. R. , Shi, Y. , Zhou, G. Y. , Shi, L. K. , Shah, D. , Huang, X. , Xu, B. X. , Yu, C. J. , Huang, Y. G. , and Rogers, J. A. , 2014, “ Epidermal Photonic Devices for Quantitative Imaging of Temperature and Thermal Transport Characteristics of the Skin,” Nat. Commun., 5, p. 4938. [PubMed]
Lee, H. , Choi, T. K. , Lee, Y. B. , Cho, H. R. , Ghaffari, R. , Wang, L. , Choi, H. J. , Chung, T. D. , Lu, N. S. , Hyeon, T. , Choi, S. H. , and Kim, D. H. , 2016, “ A Graphene-Based Electrochemical Device With Thermoresponsive Microneedles for Diabetes Monitoring and Therapy,” Nat. Nanotechnol., 11(6), pp. 566–572.
Gao, W. , Emaminejad, S. , Nyein, H. Y. Y. , Challa, S. , Chen, K. V. , Peck, A. , Fahad, H. M. , Ota, H. , Shiraki, H. , Kiriya, D. , Lien, D. H. , Brooks, G. A. , Davis, R. W. , and Javey, A. , 2016, “ Fully Integrated Wearable Sensor Arrays for Multiplexed In Situ Perspiration Analysis,” Nature, 529(7587), p. 509. [PubMed]
Li, H. , Xu, Y. , Li, X. , Chen, Y. , Jiang, Y. , Zhang, C. , Lu, B. , Wang, J. , Ma, Y. , Chen, Y. , Huang, Y. , Ding, M. , Su, H. , Song, G. , Luo, Y. , and Feng, X. , 2017, “ Epidermal Inorganic Optoelectronics for Blood Oxygen Measurement,” Adv. Healthcare Mater., 6(9), p. 1601013.
Dagdeviren, C. , Su, Y. W. , Joe, P. , Yona, R. , Liu, Y. H. , Kim, Y. S. , Huang, Y. A. , Damadoran, A. R. , Xia, J. , Martin, L. W. , Huang, Y. G. , and Rogers, J. A. , 2014, “ Conformable Amplified Lead Zirconate Titanate Sensors With Enhanced Piezoelectric Response for Cutaneous Pressure Monitoring,” Nat. Commun., 5, p. 4496. [PubMed]
Wang, C. , Hwang, D. , Yu, Z. B. , Takei, K. , Park, J. , Chen, T. , Ma, B. W. , and Javey, A. , 2013, “ User-Interactive Electronic Skin for Instantaneous Pressure Visualization,” Nat. Mater., 12(10), pp. 899–904. [PubMed]
Shi, Y. , Dagdeviren, C. , Rogers, J. A. , Gao, C. F. , and Huang, Y. , 2015, “ An Analytical Model for Skin Modulus Measurement Via Conformal Piezoelectric Systems,” ASME J. Appl. Mech., 82(9), p. 091007.
Yuan, J. H. , Shi, Y. , Pharr, M. , Feng, X. , Rogers, J. A. , and Huang, Y. , 2016, “ A Mechanics Model for Sensors Imperfectly Bonded to the Skin for Determination of the Young’s Moduli of Epidermis and Dermis,” ASME J. Appl. Mech., 83(8), p. 084501.
Cheng, H. Y. , and Wang, S. D. , 2014, “ Mechanics of Interfacial Delamination in Epidermal Electronics Systems,” ASME J. Appl. Mech., 81(4), p. 044501.
Liu, W. , and Lu, N. S. , 2016, “ Conformability of a Thin Elastic Membrane Laminated on a Soft Substrate With Slightly Wavy Surface,” ASME J. Appl. Mech., 83(4), p. 041007.
Lu, N. S. , Zhang, Z. , Yoon, J. , and Suo, Z. G. , 2012, “ Singular Stress Fields at Corners in Flip-Hip Packages,” Eng. Fract. Mech., 86, pp. 38–47.
Huang, Y. , Yuan, J. , Zhang, Y. , and Feng, X. , 2016, “ Interfacial Delamination of Inorganic Films on Viscoelastic Substrates,” ASME J. Appl. Mech., 83(10), p. 101005.
Xu, F. , Lu, T. J. , and Steffen, K. A. , 2008, “ Biothermomechanics of Skin Tissues,” J. Mech. Phys. Solids, 56(5), pp. 1852–1884.
Xu, F. , Lu, T. J. , Steffen, K. A. , and Ng, E. Y. K. , 2009, “ Mathematical Modeling of Skin Bioheat Transfer,” Appl. Mech. Rev., 62(5), p. 050801.
Song, J. , Feng, X. , and Huang, Y. , 2016, “ Mechanics and Thermal Management of Stretchable Inorganic Electronics,” Natl. Sci. Rev., 3(1), p. 128. [PubMed]
Cui, Y. , Li, Y. H. , Xing, Y. F. , Yang, T. Z. , and Song, J. Z. , 2016, “ One-Dimensional Thermal Analysis of the Flexible Electronic Devices Integrated With Human Skin,” Micromachines, 7(11), p. 210.
Cui, Y. , Li, Y. H. , Xing, Y. F. , Ji, Q. G. , and Song, J. Z. , 2017, “ Thermal Design of Rectangular Microscale Inorganic Light-Emitting Diodes,” Appl. Therm. Eng., 122, pp. 653–660.
Lü, C. , Li, Y. , Song, J. , Kim, H. S. , Brueckner, E. , Fang, B. , Hwang, K. C. , Huang, Y. , Nuzzo, R. G. , and Rogers, J. A. , 2012, “ A Thermal Analysis of the Operation of Microscale, Inorganic Light-Emitting Diodes,” Proc. R. Soc. London A, 468(2146), pp. 3215–3223.
Li, Y. , Shi, Y. , Song, J. , Lü, C. , Kim, T. , Rogers, J. A. , and Huang, Y. , 2013, “ Thermal Properties of Microscale Inorganic Light-Emitting Diodes in a Pulsed Operation,” J. Appl. Phys., 113(14), p. 144505.
Xu, F. , Seffen, K. A. , and Lu, T. J. , 2008, “ Non-Fourier Analysis of Skin Biothermomechanics,” Int. J. Heat Mass Transfer, 51(9–10), pp. 2237–3704.
Carslaw, H. S. , and Jaeger, J. C. , 1959, Conduction of Heat in Solids, 2nd ed., Carendon Press, Oxford, UK.
Lee, C. H. , Ma, Y. , Jang, K. I. , Banks, A. , Pan, T. , Feng, X. , Kim, J. S. , Kang, D. , Raj, M. S. , McGrane, B. L. , Morey, B. , Wang, X. , Ghaffari, R. , Huang, Y. , and Rogers, J. A. , 2015, “ Soft Core/Shell Packages for Stretchable Electronics,” Adv. Funct. Mater., 25(24), pp. 3698–3704.
Xu, F. , Wen, T. , Seffen, K. , and Lu, T. , 2008, “ Modeling of Skin Thermal Pain: A Preliminary Study,” Appl. Math. Comput., 205(1), pp. 37–46.

## Figures

Fig. 1

(a) An EED consisting of temperature sensors and heaters on a skin in a twisting motion (Reproduced with permission from Webb et al. [2]. Copyright 2013 by Nature Publishing Group.) and (b) schematic diagram of the cross-sectional structure for the EED/skin system.

Fig. 2

(a) The distribution of the temperature increase along the radial direction at the EED/skin interface and (b) the distribution of the temperature increase along the thickness direction with r = 0

Fig. 3

The distribution of the maximum principle stress at the EED/skin interface along the radial direction

Fig. 4

(a) The influences of the thermal conductivity and thickness of the substrate on the maximum principle stress σmax at the EED/skin interface and (b) the influences of the Young’s modulus and thermal expansion coefficient of the substrate on the maximum principle stress σmax at the EED/skin interface

Fig. 5

(a) The maximum temperature increase and (b) the maximum principle stress σmax at the EED/skin interface varies with the size of heating component and the substrate thickness

## Tables

Table 1 Parameters of human skin [18] and PDMS substrate [22]

## Discussions

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