The Role of Fracture Patterns on Crack-Based Strain Sensors

Strain sensors, which can transduce mechanical stimuli to electrical signals, play an important role in various applications, such as artificial skins [1–3], wearable health monitoring devices [4–6], and soft robots [7,8]. Recently, crack-based strain sensors (CBS) have attracted much attention due to their high sensitivity and flexibility [9–17]. This technology is inspired by the mechanism that spiders use to detect ambient vibrations. As shown in Fig. 1(a), spiders have an organ called “slit sensilla” near their leg joints [10]. This organ has multiple slits with different lengths embedded in its exoskeleton as illustrated in Fig. 1(b) [18]. This specific geometry allows tiny vibrations to be captured via opening and closing of the slits. Nerve cells that are connected to the slits can transmit movement information to the spiders’ brains in real time. To mimic this sensing mechanism, researchers have developed CBS that can change the resistance of an electrical circuit due to the crack opening and closure. Typically, CBS consists of two layers: a nanoscale conductive layer that is usually made of metals or carbon-based materials, and a polymer microscale substrate layer as shown in Fig. 1(c). The slit-like cracks are created by propagating surface cracks on the conductive layer to the polymer substrate through roll-to-roll pressing, bending, or pre-stretching [10,11,16]. These pre-cracks work as resistors in a circuit. When the pre-cracks start to open, resistance of the resistors tends to increase due to the decreased conductive surface area that remains in contact. Once the conductive surfaces are fully disconnected, the sensor becomes nonfunctional as the resistance immediately reaches infinity. The corresponding applied strain can be considered as the sensing range.

Sensitivity of strain sensors is evaluated by the gauge factor (GF), which is the relative change in resistance over strain. Therefore, high sensitivity requires a large change in resistance under the same amount of applied strain. It has been reported that crack geometry plays an important role in sensitivity. Kang et al. [10] initially developed CBS without controlling the crack pattern. They found that the GF can reach 2000 at 2% strain. On the basis of the same design, Park et al. [11] performed additional bending and stretching to further increase the pre-crack depth to 96 nm. They found that the GF can be increased from 2000 to 16,000 at 2% strain. In this approach, improvement of sensitivity was at the expense of sacrificing its sensing range as deeper pre-cracks correspond to larger crack opening under the same amount of stretching. As a result, the conductive surfaces were easier to disconnect. Since reversible sensor design requires no pre-crack propagation during stretching, there exists a crack depth limit. Choi et al. [16] created a star-shaped crack pattern on the conductive surface. They found that this design not only improves the GF to over $2×106$ but also extends the sensing range to 10% applied strain. In recent years, techniques such as laser engraving [19], laser direct writing [20], and additive manufacturing [21] have allowed crack patterns to be better controlled and precisely resolved in CBS. A computational model that can predict sensing performance as the function of crack information will significantly save time and cost for future CBS design and manufacturing.

In this article, a coupled mechanical–electrical finite element model is developed to systematically evaluate the role of crack patterns on sensitivity and sensing range of CBS. This model can simultaneously capture the deformation history and resistance change under different designs of pre-crack orientation, density, and distribution. Simulations in this work concern CBS with a platinum (Pt) conductive layer bonded to a polydimethylsiloxane (PDMS) substrate layer. The model developed here can be applied to any other material compositions through different constitutive modeling.

In this work, the coupled mechanical–electrical finite element model is developed in abaqus using a 3D square-shaped specimen with a side length of 30 μm, tied between positive and negative gold electrodes, as illustrated in Fig. 2(a). The specimen consists of a top Pt conductive layer with a thickness of 0.3 μm and a bottom PDMS substrate with a thickness of 5 μm. In this example, six through cracks with a depth of 0.5 μm are illustrated. Each crack includes two artificially created sawtooth groups that resemble the surface roughness of preprocessed cracks, resulting seven Pt pieces with surface pre-cracks. Each Pt piece is generated using solidworks and tied to the PDMS substrate. The friction between saw teeth is ignored, and the electrical conductivity is assumed to be a large number (10¹⁰ Ω⁻¹ m⁻¹). This specimen is subjected to mode I loading with 5% tensile strain (equivalent to 1.5 μm displacement) applied to each electrode. 0 V and 1 V are applied to the outer surface of the negative and the positive electrode, respectively. Resistance between the electrodes and the conductive layer is ignored in this study. The bottom surface of the PDMS substrate is constrained to move solely in the transverse direction (x-z direction). Refined meshes shown in Fig. 2(c) are applied to each crack-tip region. The maximum mesh size in the Pt layer and the PDMS substrate is 0.4 μm and 1 μm, respectively, in consideration of both computational accuracy and efficiency according to Fig. 2(b). The mesh type for the Pt layer and the substrate is Q3D8 (eight-node trilinear displacement, electric potential, and temperature) and C3D8R (8-node linear), respectively.

In this work, the constitutive relationship of all the materials is linear elastic, and the detailed material properties are shown in Table 1.

The finite element model developed here can keep track of the evolution of resistance and CMOD during the course of stretching. When the conductive area decreased to zero, resistance reached infinity, and further stretch would not yield any resistance change. The applied strain that corresponded to this critical moment is extracted as the sensing range. The gauge factors are calculated by the relative change of resistance ΔR/R₀ divided by the applied strain.

Three sets of crack patterns are generated to study the effect of crack orientation/distribution, crack density, and crack spacing on sensitivity and sensing range. The first set of designs, as illustrated in Fig. 4(a), consider a single through crack of 0 deg, 10 deg, and 20 deg tilting angle with respect to the x-axis, as well as ten 0 deg nonthrough discrete cracks with the same total crack length. The second set of designs considers 1, 3, 6, and 10 parallel through cracks with 0 deg tilting angle as shown in Fig. 4(b). These cracks are evenly distributed in the z-axis so that the spacing between two adjacent cracks (e.g., l₂) equates to the margin distance from the top (bottom) crack to the sample edge (e.g., l₁ and l₇), as illustrated in Fig. 4(c). In the third set of design, we create four distribution scenarios for six parallel through cracks with 0 deg tilting angle. In scenario 1, cracks share the same interspace as l₂ = l₃ = · · · = l₆. In scenario 2, cracks follow an alternating spacing with l₂/l₃ = l₅/l₄. Scenario 3 concerns an increasing crack spacing gradient as l_i+1 = 1.3 × l_i when 1 ≤ i ≤ 5. Scenario 4 studies random crack spacing. In all the aforementioned cases, the margin distance is calculated as l₁ = l₇ = L − (l₂ + l₃ + · · · + l₆). All the crack patterns generated earlier are implemented to the coupled mechanical–electrical simulations as discussed in Sec. 3.

In this section, we consider surface crack patterns as summarized in Fig. 4. Crack depth a = 0.5 μm is employed for all the cases. Other specimen dimensions follow the descriptions shown in Fig. 2.

We start the analysis with one single through crack with 0 deg, 10 deg, and 20 deg tilting angle as illustrated in Fig. 4(a). Figure 5(a) shows the evolution of relative change in resistance ΔR/R₀ during the stretching process. An abrupt increase in ΔR/R₀ signify the diminishing conductive surface between the crack teeth. Once the output current reduces to zero, conductive surfaces are fully disconnected. The corresponding applied strain is extracted as the sensing range. Sensitivity is evaluated by the maximum GF, which is the slope of ΔR/R₀ versus applied strain curve right before the disconnecting point, as illustrated in Fig. 5(a). According to Figs. 5(c) and 5(d), a larger tilting angle corresponds to the lower sensing range and sensitivity. This trend was observed in the previous experimental work by Choi et al. [16].

Figure 6 compares the electric current distribution in specimens with 0 deg and 10 deg tilting angle. When the applied strain increases from 0.75% to 0.83%, most of the teeth remain connected in the 0 deg crack as indicated in Fig. 6(a). Under the same amount of deformation, all the teeth are disconnected when the crack is tilted by 10 deg. This is because the local crack teeth can experience combined mode I and mode II opening under the uniaxial stretch. As illustrated in Fig. 6(b), mode II displacement intensifies crack teeth disconnection, leading to CBS failure at lower applied strain. If the 0 deg through crack is divided into ten equal nonthrough cracks that are randomly distributed on the conductive layer, sensitivity is significantly decreased according to Figs. 5(b) and 5(c). This observation also aligns with the experimental finding from Zhu et al. [12].

Compared with the single through crack design, CBS with discrete nonthrough cracks exhibits significantly lower sensitivity. This is because discrete cracks can effectively mitigate the deformation so that the variation in each conductive area is reduced. According to Fig. 7, the relative change in conductive area in a through crack specimen and a discrete crack specimen is 76.68% and 0.898%, respectively, when the applied strain increases from 0.62% to 0.92%, as shown in Fig. 7. Although the discrete crack design exhibits low sensitivity, its sensing range is expected to outperform the single through crack design as a much larger applied strain is required to fully disconnect all the discrete crack teeth. This example also addresses the fundamental challenge of improving sensitivity and sensing range at the same time.

It can be concluded from the earlier discussion that if the crack density is a constant, a specimen with mode I through crack can exhibit better sensing performance than a specimen with a tilted crack or with discrete random nonthrough cracks. In this study, we keep the single mode I through crack specimen as the reference case and create three additional specimens with 3, 6, and 10 parallel through cracks, as illustrated in Fig. 4(b). Here, crack spacing is a constant in each specimen. According to the results in Fig. 8(a), increasing the number of cracks has a negligible effect on sensitivity. However, it significantly improves the sensing range. Specifically, the sensing range in specimens with 3, 6, and 10 parallel through cracks is increased by 289.38%, 572.57%, and 991.15%, respectively, compared with the reference case, according to Fig. 8(b). This is due to the fact that the crack in the center of the specimen has the largest crack opening under mode I loading. Once more cracks are introduced to the conductive layer, crack opening of individual crack is reduced under the same load. Therefore, higher amount of strain is required to fully disconnect the conductive layer, leading to improved sensing range. For example, at 3.33% applied strain, crack opening in the reference case is 0.9995 μm. It reduces to 0.1014 μm when 10 cracks are present. Since the shape of sawtooth remains unchanged, the rate of change in conductive area is very similar in different cracks. Therefore, no substantial change in resistance is observed.

In this set of study, we employ specimens with six parallel mode I through cracks. Four types of crack distributions are considered as summarized in Fig. 4(c). It is found that crack spacing, although plays a negligible role on sensitivity, can largely affect the sensing range of CBS. For example, in a specimen with Gradient 1.3 × crack patterns, the largest opening distance of 0.32 μm is observed in crack 6 when the specimen is subjected to 3.33% applied strain. This is almost five times larger than the smallest opening distance of 0.0633 μm in crack 1. The sensing performance of other designs can be found from both Fig. 9 and Table 2. This result indicates that the sensing range is determined by the first group of cracks that becomes disconnected. This is similar to the weakest link model in fracture mechanics [22–24]. It can be inferred that the fundamental avenue to improve the sensing range under this crack pattern is to reduce the maximum crack opening distance by optimizing the crack spacing. According to our analysis, the specimen with 4.3 μm equal spacing has the lowest maximum crack opening distance. Other configurations such as alter 1:8 and alter 2:6, exhibit similar sensing ranges. This is because the specimens with alternating spacing (Fig. 4(c)) are created based on the “3-crack equal-spacing” case in Fig. 2(b). Here, each original crack is replaced by two identical cracks positioned above and below it, with a predefined distance between them. When the predefined distance is small, the crack mouth opening that was present in each original single crack is almost equally distributed between the two new cracks and almost doubled compared to the three-crack equal-spacing scenario (sensing range is 3.27% as shown in Fig. 8(b)). Once the predefined distance increases, the interactions between different new crack groups become more prominent. As a result, the two new cracks cannot equally accommodate the crack mouth opening that was present in each original single crack. This disparity in crack behavior leads to different levels of crack disconnection as samples with cracks that disconnect earlier leading to lower sensing range. This is the primary reason that the alter 3.5:7 specimen tends to have lower sensing range.

It is noted from Fig. 10(a) that CMOD values predicted from the finite element model are very close to the analytical solution. For simplicity, we extract GF at a thickness of 0.5 μm as the baseline case. As shown in Fig. 10(b), the normalized GF with respect to a crack depth of 0.1 μm, 0.3 μm, and 0.7 μm is approximately 0.64, 0.70, and 1.09, respectively. It can be inferred that there is a linear increase in sensitivity when the crack depth increases. Additionally, a more pronounced stress concentration is observed at the crack tip when a deeper crack is present according to Fig. 11. When the specimens are subjected to 2% strain, the corresponding J values of CBS with 0.1 μm, 0.3 μm, 0.5 μm, and 0.7 μm crack depth are 4.444 × 10⁻⁴ J/m², 1.324 × 10⁻³ J/m², 2.208 × 10⁻³ J/m², and 3.052 × 10⁻³ J/m², respectively. It should be noted that the critical J for crack propagation in PDMS is approximately 360 J/m² [26], which is significantly higher than the predicted J values. This indicated that crack propagation in the depth direction is not a concern for sensing range. Improvement of sensing range will require lower CMOD upon the same applied strain. It is found that increasing the number of surface cracks can mitigate the deformation, leading to decreased CMOD and higher sensing range as indicated in Fig. 12.

A coupled electrical–mechanical finite element model is developed to evaluate the effects of fracture patterns on sensing performance of CBS. This model can keep track of the evolution of resistance R and CMOD during the course of stretching. Calculations in this work concern CBS with a platinum (Pt) conductive layer and a PDMS substrate layer. Three sets of crack patterns are generated to study the effect of crack orientation/distribution, crack density, and crack spacing on sensitivity and sensing range. Based on each crack pattern, sensitivity is evaluated by the slope of ΔR/R₀ versus strain curve right before the conductive surfaces are fully disconnected. Applied strain at this moment corresponds to the sensing range. It is found that mode I through cracks can yield better sensing performance than nonthrough cracks in the same orientation or through cracks with other orientations. Either increasing crack density or creating even crack spacing can lead to improved sensing range. However, these approaches have a negligible effect on sensitivity. Increasing crack depth can improve sensitivity at the expense of reducing sensing range. The forgoing analyses point out that the fundamental avenue to have a balanced sensitivity and sensing range is to create high-density mode I through cracks with even spacing.

As an initial work, we simplify the model to assume idealized crack patterns and linear-elastic constitutive relationship. This simplification does not account for factors such as friction and temperature-dependent nonlinearities. This computational framework can be extended to resolve more realistic crack morphologies and material response. In addition to the quasi-static loading conditions, our future work will address the effects of vibration on CBS performance as vibration at certain frequencies can cause the CBS to resonate, leading to changes in their mechanical properties. This resonance can result in altered crack opening or closure behavior, affecting the sensor's output signal. Vibration can also induce additional stress on pre-cracks, potentially accelerating crack propagation and dynamic deformations in the sensor structure.

• National Aeronautics & Space Administration (NASA) (Award No. 80NSSC22M0175).

• The start-up funds from Thayer School of Engineering at Dartmouth College.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

a =

crack depth

t =

conductive layer thickness

A =

instantaneous conductive area

E =

Young’s modulus

I =

instantaneous current

J =

strain energy release rate

L =

sample length

N =

total number of saw teeth

T =

substrate thickness

U =

applied voltage

W =

total energy

d₀ =

initial contact length

l_i =

spacing between crack to sample edge or between two adjacent cracks

u₀ =

strain energy density

W₁ =

mechanical strain energy

W₂ =

electrical potential energy

d⁽ⁱ⁾ =

instantaneous contact length of ith saw teeth pair

J^E =

current density

CMOD =

instantaneous crack mouth opening distance

GF =

gauge factor

ɛ =

applied strain

ɛ_ij =

strain tensor

ν =

Poisson’s ratio

ρ_q =

electric charge density

σ^E =

electrical conductivity

σ_ij =

normal stress tensor

τ_ij =

shear stress tensor

φφ =

electrical potential