## Abstract

Silicone-based biofidelic surrogates are used in many biomedical applications. Apart from mimicking the mechanical behavior of bodily tissues, there is an increasing requirement for these materials to be electrically conductive and piezoresistive to facilitate direct instrumentation. Carbon nanotubes (CNTs) have been extensively investigated as fillers to impart electrical conductivity and piezoresistivity to polymeric materials including silicone. In this paper, we fabricate, test, and characterize a two-part silicone/CNT sheet sandwich composites that exhibit conductivity, piezoresistivity, and biofidelic with mechanical properties corresponding to that of the white matter of human brain tissue. The electromechanical performance of the sandwich composite improves in subsequent loading after the core fracture during initial loading. Analytical models developed for discontinuous core sandwich structures are used to analyze and explain the experimental results. The results indicate the potential for using this discontinuous core biofidelic-piezoresistive sandwich nanocomposite for biomedical applications without deploying external deformation sensors.

## 1 Introduction

Biofidelic materials are a class of materials which can mimic the mechanical properties of a biological system or a tissue. The most common application of biofidelic materials is as surrogates for animal tissues where they can provide the same mechanical response to static or impact load in research and testing. The use of surrogate materials for testing addresses the inadequate availability of biological tissues and inaccessibility, especially for human tissues due to ethical considerations [1]. Even if a limited amount of the human tissue can sometimes be successfully obtained from a cadaver, inherent biological variability and mechanical behavior changes due to tissue dehydration, degradation, and death [2,3] make synthetic biofidelic materials more desirable and economical in many situations.

When used as surrogates for human tissues, biofidelic materials can help evaluate the effectiveness of personal protective equipment [4], test firearms, and bullets [5] and assess the body injury mechanisms like traumatic brain injury [68]. They can also be used to practice and model surgery both by surgeons and robotic equipment [9]. In most current test setups using biofidelic material systems, strain and deformation sensors are required to be embedded or attached to human tissue surrogates to measure the mechanical response during the mechanical testing. For example, in the testing of physical human surrogate torso model by Roberts et al., piezoresistive pressure sensors and accelerometers were embedded at multiple locations to sense the mechanical response of surrogate tissues [10]. Biofidelic materials with electrical conductivity and/or inherent sensing capability will reduce the need for external sensors and expand the capabilities of tissue surrogates significantly. Moreover, biocompatible and biofidelic material systems with high sensitivity can potentially detect low biomechanical strains [11] such as those generated by arterial activity for body health monitoring. Thus, piezoresistivity and electrical conductivity are highly desired for biofidelic materials to simplify the test procedures and extend the field of applications.

Electrically conductive nanofillers such as carbon nanotubes (CNTs) and graphene nanoplatelets have been used to impart electrical conductivity and piezoresistivity to insulating composites. While CNTs exhibit electrical conductivity and piezoresistivity intrinsically [12], tunneling mechanism between fillers, resulting in the formation of a percolated network is considered to be the principal mechanism responsible for conductivity in CNT/polymer composites [1321]. The change in the conductive network structure due to applied strain is responsible for the piezoresistivity of nanocomposites. CNTs are often used in the form of CNT sheet or buckypaper, which is essentially a thin macroscopic membrane containing entangled CNT networks loosely bonded through van der Waals interactions [22,23]. CNT sheets exhibit relatively high electrical and thermal conductivity due to their continuous CNT network. In addition, they are mechanically stable, flexible, and possess high-surface area because of their porous structure. CNTs and CNT sheets have been used as fillers to fabricate multifunctional composites with several polymeric matrix materials, e.g., polyethylene [24], polycarbonate [17,25], polypropylene [26], polyamides [27,28], and epoxies [21]. Among the polymers that have been used as matrix materials, silicone or polydimethylsiloxane (PDMS) is notable for its biomedical applications due to its biocompatibility and biofidelity [11]. Several researchers showed that CNT/PDMS composites exhibit high levels of piezoresistivity and strain sensitivity in addition to high failure strain [11,2931].

Silicone is a popular biofidelic material and has been used to mimic several biological systems such as muscles, skin, and brain tissues [8,10,3235]. The mechanical properties of silicone can be varied to match those of many biological tissues, for example, Chanda et al. developed proprietary blends of two-part silicone to mimic the nonlinear mechanical behaviors of the human skin [35], as well as human brain tissues [8] by changing the mix ratios of the components and modeled the experimental stress–strain curves using hyperelastic material models.

Incorporating CNT tunneling network into this biofidelic silicone matrix will enable electrical conductivity and will result in a biofidelic piezoresistive multifunctional composite. In particular, silicone-CNT nanocomposites have been proposed for several biomedical applications such as artificial nerve guidance [36], neurite interfacing [37], and robotic arm sensors [38]. Additionally, a sandwich structure with an infiltrated CNT sheet core layer can have high-dielectric permittivity in the core layer and low-dielectric loss provided by insulating facesheets [39]. A piezoresistive biofidelic sandwich composite will eliminate the need for external sensor deployment and will be of great value in applications like studying injury mechanisms and protective equipment design.

The objective of this study is to fabricate, characterize, and analyze a biofidelic piezoresistive sandwich composite with properties of human brain tissue. First, sandwich nanocomposites with CNT sheet as core and the two-part silicone developed by Chanda et al. [8] as the matrix are fabricated and characterized. Then, the composition of silicone mixtures is varied to retain biofidelity and piezoresistivity, while accounting for stiffness increase due to the addition of CNT sheet core. The sandwich composite exhibits increased piezoresistivity after core fracture during initial loading. Finally, we use the variational methods developed for cracked laminates and discontinuous core sandwich structures [40,41] to analyze and discuss the mechanical and piezoresistive behavior of the nanocomposite sandwich structures.

## 2 Experimental

### 2.1 Materials.

Multiwall carbon nanotubes sheets consisting of 100% free-standing nanotubes were procured from NanoTechLabs, Inc. The manufacturer reports an areal density of 21.7 g/m2 and surface electrical resistivity 1.5 Ω/m2. For the silicone matrix material, we used a recently developed two-part silicone mix that mimics the nonlinear tensile properties of human brain tissue [8]. Experiments indicate that human brain tissues are very soft materials with low stiffness [42,43]. Correspondingly, the silicone biofidelic surrogate is a low stiffness two-part silicone fabricated using a platinum catalyst with a Shore hardness of 10, procured from Smooth-On, Inc. The part A of the silicone material is the hardener, which polymerizes only when added to part B, which activates a silicone functional group to crosslink. An electrically conductive silicone adhesive procured from Silicon Solutions was used to connect copper electrodes and CNT sheet.

### 2.2 Sandwich Composites Fabrication.

The fabrication method of silicone/CNT sheet sandwich composites was the stacking up process using a prefabricated mold. In the stacking up process, a sandwich composite is formed with top and bottom silicone skins and a conductive silicone-infiltrated CNT sheet core layer. First, copper electrodes (10 × 10 mm2) were attached to the CNT sheet of 50 × 10 mm2 dimensions to facilitate stable measurement of electrical resistance. Electrically conductive silicone adhesive was applied on one side of the copper electrodes to bond with the CNT sheet on both ends.

An ABS mold with four grooves each measuring 50 mm × 10 mm × 3 mm (see Fig. 1) was 3D printed using Flashforge 3D printer Creator Pro. Parts A and B of brain tissue simulant were weighted separately and were thoroughly mixed using a wooden stick in a portion control cup in different ratios from 50%A–50%B to 20%A–80%B. The mold was then placed on the scale, and the mixture was filled into one of the grooves until the tare weight reading equaled half the calculated weight for the corresponding mold volume. After 2 min, the liquid level of the mixture in the groove flattened to form the bottom face sheet of the sandwich composite. The CNT sheet with attached electrodes was then carefully placed on the liquid surface of the bottom face sheet as shown in Fig. 1(a). More mixture was cast into the groove until the liquid level was flush to the edge of the mold. Any excess material was wiped off with a smooth blade. Through this approach, the CNT sheet could be located at the central surface of the brain tissue simulant matrix. The fabricated specimens after curing for 3 h at the room temperature are shown in Fig. 1(c).

Fig. 1
Fig. 1
Close modal

### 2.3 Testing and Characterization.

The mechanical and electromechanical properties were measured by a tensile test using a Chatillon CS225 force tester at room temperature. Load and displacement data directly collected by the embedded computer on the force tester was converted into engineering stress–strain curves. The electrical resistance of specimens was measured based on the four-terminal sensing approach, which was combined with the displacement data from the tensile testing machine to plot the resistance-strain curves. The tensile test setup is shown in Fig. 2.

Fig. 2
Fig. 2
Close modal

All of the specimens were tested at a low displacement rate which is 2.5 mm/s to compare with prior results of Chanda et al. [8] and human brain tissue mechanical tests [39]. The maximum displacement was set to 30 mm to obtain the data in the strain range 0–1. Twelve specimens divided into three sets were prepared for each test. The silicone specimen were fabricated with different compositions including 50%A–50%B, 40%A–60%B, 30%A–70%B, and 20%A–80%B. A low current of 0.01 A was applied to the specimen to obtain a wide measuring range for resistivity and to prevent ohmic heating from affecting test results. Unless otherwise specified, the mechanical and electromechanical test results shown in Sec. 3 correspond to an average of four specimens with error bars given by the standard deviation. Scanning electron microscopy (SEM) micrographs were captured using FEI Quanta FEG 650 for cross-sectional measurements and analysis of fracture surfaces. Specimens for the cross-section surface observation needed to be gold coated before imaging.

## 3 Results and Discussion

### 3.1 Mechanical Behavior.

As a baseline measurement, we first tested 50%A–50%B silicone and compared with past results on human brain white matter [42] and brain tissue simulant results with the same silicone composition from Chanda et al. [8]. There is a good match between the stress–strain plots for the three cases as shown in Fig. 3, which verifies our result in comparison with the past result and demonstrates the biofidelity of the neat silicone for this composition. Further to investigate the effect of load history, a second-time loading was applied on the specimens which had been tested once. The curves for first loading and second loading are very similar as shown in Fig. 3. This indicates that the loading history of silicone in the strain range of 0–0.9 does not significantly affect its mechanical behavior.

Fig. 3
Fig. 3
Close modal

Figures 4(a) and 4(b) show the mechanical behavior and load history of the silicone (50%A, 50%B)/CNT sheet sandwich. Figure 4(a) shows the stress–strain plot for the sandwich for its first loading and when the specimen is reloaded the second time; also shown is a plot corresponding to the neat silicone matrix. We calculate the tensile modulus of neat silicone in the linear strain range of 0–0.4 to be 26.40 ± 5.6 kPa, while the corresponding modulus for the sandwich is 196.35 ± 15.3 kPa for the first loading of the sandwich structure. In the second loading, the tensile stiffness drops to 49.58 ± 6.00 kPa indicating an intrinsic structural change.

Fig. 4
Fig. 4
Close modal

Note that the core of the sandwich structure is not CNT sheet alone, but it is CNT sheet infiltrated with silicone which is stiffer than both matrix and neat CNT sheet (see schematic in Fig. 1); therefore, there is a significant stiffening of the sandwich during the first loading when this core is the primary load-bearing component. Visual examination indicated that the infiltrated CNT sheet core layer is not fractured up to a strain of 0.4. Cracks start to appear at a tensile strain of around 0.4 and at a strain of 0.6, the cracks became wider gaps, which divided the infiltrated CNT sheet core into several sections. The configuration of a sample in these three phases is shown in Figs. 5(a)5(c).

Fig. 5
Fig. 5
Close modal

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

### 3.2 Electromechanical Behavior.

We now look into the electromechanical behavior of the sandwich structures. During the tensile test, the electrical resistance of samples was also recorded using a four-point probe measurement system, and the resistivity–strain curves were recorded. Resistivity–strain curve of one silicone (50%A–50%B)/CNT sheet composite coupon is plotted in Fig. 8. The initial resistivity of the sample at zero strain is 169.92 × 10−5 Ω m. Though there is a small increase, the resistivity change in the strain range of 0–0.6, it is insignificant when compared with the dramatic increase beyond the strain of 0.6. Note that the fracture of the infiltrated CNT sheet core layer starts around the strain of 0.4; however, there probably is a continuity of conductive layer which is broken at a higher strain around 0.6, which explains the sudden increase in resistance. The resistivity becomes steady at strains above 0.9 because of the limitation of the maximum measuring range of data acquisition system around 2000 × 10−5 ohm m.

Fig. 8
Fig. 8
Close modal

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

### 3.3 Analytical Model for Discontinuous Core Sandwich Structure.

The sandwich structure investigated here after the core fracture in the first loading can be considered to be a discontinuous core sandwich. Other similar discontinuous core structures in the literature include discontinuous tile ceramics used in applications like armor [41,46] and masonry structures [47]. The formulations used for analysis of such structures consider the discontinuous core as a continuous core with periodic cracks, similar to that of transverse cracks in a cross-ply composite laminate [40,4850]. Interfacial adhesion between the core and the facesheet layers is particularly important here to facilitate the load transfer between the continuous facesheet and the discontinuous core. The analysis methods for these problems in composites and ceramic sandwich structures include strength approaches [5153] based on the shear-lag model and energy-based approaches [40,4850,54,55]. The strength approaches have been effective in matching experimental results; however, these methods use shear load transfer approximation and require a clearly defined long shear-lag region to avoid interactions between cracks. Hashin [40] pioneered a variational approach for this problem using complementary energy to determine stress state and effective modulus. This approach does not make any assumptions regarding shear load transfer and only makes an assumption regarding constant normal stress through thickness and is in very good agreement with experimental results. This model has been extended to axial and shear loading of discontinuous core ceramic tile composites by Huang et al. [41] and to thermal stresses by Gwandi et al. [46]. We add this formulation to model and analyze our experimental results. The silicone matrix used in the study is generally considered to be a nonlinear elastic material; however, we limit our analysis to low strains (less than 10%) where the matrix and sandwich can both be considered linear and elastic.

The axial stresses in the continuous core sandwich which corresponds to the first loading (see Fig. 6) are given by
$σxx0(c)=εxxE(c)σxx0(f)=εxxE(f)σxx0(ad)=εxxE(ad)$
(1)
Here, the superscript (c) denotes the core layer, (f) denotes the facesheet layer, and (ad) is for the adhesive layer. The load transfer to core is facilitated by an adhesive layer, which becomes more crucial for discontinuous cores but is mentioned here for completeness. Note that the core for the current problem is silicone-infiltrated CNT sheet while the face sheet is silicone.

We apply this isostrain model to the low-strain (<than 10%) elastic part of the first-loading response for the two compositions (Fig. 6). The continuous core sandwich elastic modulus (E0(s)) and facesheet elastic moduli (E(f)) are known from the stress–strain plots. We use the model to calculate the elastic modulus of the core layer (E(c)). Table 1 lists the parameters and the calculated core modulus for the two matrix compositions considered here.

Table 1

Geometric and elastic parameters for the analytical model

Facesheettf (mm)tc (mm)tad (mm)a (mm)E(f)–Expt. (kPa)E0(s)–Expt. (kPa)E0(c)–calc. (kPa)
50%A–50%B Silicone1.40.180.031026.4 ± 5.6196.35 ± 15.32858.9
20%A–80%B Silicone1.40.180.031013.5 ± 1.0181.57 ± 7.22814.7
Facesheettf (mm)tc (mm)tad (mm)a (mm)E(f)–Expt. (kPa)E0(s)–Expt. (kPa)E0(c)–calc. (kPa)
50%A–50%B Silicone1.40.180.031026.4 ± 5.6196.35 ± 15.32858.9
20%A–80%B Silicone1.40.180.031013.5 ± 1.0181.57 ± 7.22814.7

Note: The elastic modulus of continuous core sandwich (E0(s)) and the facesheet modulus (E(f)) are obtained experimentally.

Now, consider the post-core fracture scenario which corresponds to subsequent loading after the first loading. Figure 11 explains the geometric parameters and coordinate system for this situation. The structure can be treated as a discontinuous core sandwich, and the stresses in the core, facesheet, and adhesive regions can be obtained using the variational analysis developed by Huang et al. [41]. The axial stresses for this case are dependent on a stress function ψ as
$σxx(c)(x)=σxx0(c)(x)−ψ(x)σxx(f)(x)=σxx0(f)(x)+AcAfψ(x)σxx(ad)(x)=σxx0(ad)(x)$
(2)
Ac and Af are the core and facesheet cross-sectional areas. Further, the following boundary conditions are used: (1) normal and shear stresses at core crack edge and facesheet free surface are zero and (2) normal and shear stresses are continuous at interfaces between facesheet core and adhesive regions. Huang et al. [41] have obtained the stress function ψ using a total energy definition which is dependent on elastic energies in individual components obtained from the laminate theory. They solve for the stress function ψ that minimizes total energy from Euler–Lagrange relation as given below to obtain an analytical expression for ψ [41].
$d4ψdx4+pd2ψdx2+qψ=0$
(3)
Here, p and q are functions of elastic constants and geometric parameters. Once ψ is known as a function of x coordinate, elastic properties of core, facesheet, and interfacial adhesive regions, and the geometric parameters (Fig. 11), the stress distributions in the core and facesheet are completely defined. While this approach has been used for axial, normal, shear, and thermal stresses, for the current problem, we focus on axial stresses only. Following Gwandi et al. [46], the average strain in the facesheet $(e¯xf)$ and the effective modulus of the discontinuous core sandwich can be evaluated using equations below:
$E¯=σ¯xe¯xfwheree¯xf=1a∫exfdx$
(4)
Fig. 11
Fig. 11
Close modal

Previous investigations have validated this model by comparing with finite element simulations [41,46]. Here, we use the model to obtain stresses in individual components of the discontinuous core sandwich and effective modulus of the sandwich structure and are compared with the experimental data. The model is populated with parameters in Table 1. The average length of the cracked regions is considered to be the core length. The core thickness is computed using SEM pictures (see Fig. 11(b)). The elastic moduli for the facesheet are obtained from experimental results and the core elastic modulus from continuous core calculation in Table 1.

Adhesion between the discontinuous core and the sandwich layers and the core length are the two primary parameters determining the load transfer to the core and the mechanical behavior of the discontinuous core sandwich. Figure 12 shows the variation of core stresses $((σxx(C)(x))/(σxxo(C)))$ along the core length (a = 10 mm) for different adhesive region stiffness for 20%A–80%B silicone/CNT sheet core sandwich structure. The stress is normalized by the axial stress in the continuous core computed using the isostrain model. The interface stiffness in the range of 10% of matrix stiffness is considered to be a strong interface [46,56]. When we vary the adhesive region stiffness, the load transferred to the core varies significantly and alters the effective elastic modulus of the sandwich structure. Figure 13 shows the parametric variation of the effective elastic modulus of the discontinuous core sandwich (20%A–80%B silicone matrix) for different core lengths and adhesive stiffness. For small core length (e.g., 2 mm), the load transfer to core does not occur irrespective of the interface effectiveness. As the core length increases, the interface stiffness has a greater impact on the effective modulus of the discontinuous core sandwich. The experimental sandwich stiffness from the second loading with the corresponding core length is also shown in Fig. 13. This indicates a correlation between experiment and model for specific interface properties. Note that this structure is biofidelic and has the same elastic modulus as human brain white matter. This approach can be used to design a discontinuous core sandwich stiffness by adjusting the core length and interface adhesion.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

### 3.4 Discussion.

The biofidelic behavior demonstrated in Fig. 7, when combined with the piezoresistive property of the sandwich structures, potentially enables biomedical applications without external instrumentation. The extent of piezoresistivity is commonly measured using gauge factor (G) as
$G=ΔRR0ε$
(5)
where ɛ is the applied strain, R0 is the initial unstrained resistance, and ΔR is the change in resistance after deformation. The gauge factor of metal foil sensors is typically around 1.5–2, and it can be higher in semiconductors [57]. Individual single-wall nanotubes have been reported to have an outstanding intrinsic piezoresistive response, with measured gauge factors ranging from 400 to 2900 [12,58]. This behavior does not necessarily translate to nanocomposites. The piezoresistivity and the electrical conductivity of CNT nanocomposites are governed by the tunneling mechanism between fillers, resulting in the formation of a percolated network. The changes in the conductive network structure due to the applied strain are considered to be responsible for piezoresistivity [1321]. Reports suggest that gauge factors of CNT nanocomposites are commonly in the range of 0.5–39.3 [21,2931,59], and for certain combinations of fillers and matrix materials, they reach even higher values [60].

Table 2

Gauge factors for biofidelic sandwich nanocomposites

Several experimental studies [43,44,60] and numerical simulations [20] indicate that the piezoresistivity and the gauge factor of CNT composites is highest around percolation and reduces if additional CNTs are added. The resistance change in nanocomposites with percolation network is associated with the changes in the number of percolation paths under applied strain. Many percolation paths are available in the high CNT content composites. While some of the paths are disrupted due to the application of strain, availability of many other percolation paths limits the extent of the resistance change. In contrast, composites with lower CNT content near percolation threshold have fewer percolation paths, and when these paths are disrupted through deformation, there is a bigger change in resistance which results in higher gauge factor. This mechanism applied to the discontinuous core sandwich structure is explained in Fig. 14.

Fig. 14
Fig. 14
Close modal
The discontinuous core sandwich structure can be considered to be composed of resistors in series alternating between the silicone-infiltrated CNT sheet core layer and cracked regions (see Fig. 14). Therefore, the resistivity of the unit cell at a given strain can be expressed as
$Rs(e¯xf)=Rc(ex=0c)+Rf(ex=af)$
(6)

It is apparent that CNT sheet core layer has high CNT content; however, the conductivity of the sandwich structure after core fracture in the first loading indicates that the matrix in cracked regions also contains a network of remnant CNTs. This is confirmed by the micrographs in Fig. 9. The piezoresistivity of this low CNT content cracked region is likely higher compared with that of the core layer. This can be observed in the first loading plots in Figs. 8 and 10, where the slope of the resistivity–strain curves increases significantly after the core fractures because this corresponds to piezoresistivity in the cracked regions. Given the large difference in stiffness between the silicone matrix and the core layer, the deformation is expected to be concentrated in the cracked region. This is further affected by the adhesion between core and facesheet and the length of the discontinuous core as evidenced in Fig. 12. Calculations indicate that the strain of facesheet at the crack edge can be as high as 15% when the overall sandwich strain is 2%. The higher strain in the region with higher gauge factor is responsible for the sharp increase in piezoresistivity in the second loading and subsequently as observed in Fig. 10 resulting in the increase in gauge factor from 1.14 to 24.93. As a part of future research, the discontinuities in the sandwich structures could be designed by notching the CNT sheet at the desired locations. The ability of engineering the desired biofidelic mechanical behavior in silicone—CNT sheet discontinuous core sandwich structures, combined with their characteristic of repeatable resistance change when subject to deformation indicates potential use in many biomechanical applications.

## 4 Conclusions

The primary scientific conclusions of the study are as follows:

• Silicone-based CNT sandwich composites can be designed and fabricated to exhibit the same mechanical properties as human brain tissue while exhibiting intrinsic strain sensing and electrical conductivity.

• The matrix composition (20%A–80%B) and the load history that lead to both biofidelic and piezoresistive characteristics in the sandwich composites are identified.

• The load history of the sandwich composites has a key role in increasing the piezoresistivity. Core fracture during the first loading creates a series of alternating high CNT density and low CNT density areas. The low CNT density areas result in an increase in the gauge factor from 1.14 in initial loading up to 24.93 in subsequent loading for the biofidelic sandwich structure.

• A variational model developed for discontinuous core sandwich structures used earlier for ceramic composites is adapted to model these sandwich structures. The results show good agreement between the model and the experiment and provide an analytical method for designing the effective stiffness of the sandwich structure by varying the core length and the adhesive layer stiffness.

## Acknowledgment

SN and NJ acknowledge support from ERAU (Funder ID: 10.13039/100008218) internal grants and NASA (Funder ID: 10.13039/100000104) SBIR. VU would like to acknowledge the support of NASA EPSCoR SID grants.

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