Fiber-reinforced soft composites (FrSCs) are seeing increasing use in applications involving soft actuators, four-dimensional printing, biomimetic composites, and embedded sensing. The three-dimensional (3D) printing of FrSCs is a layer-by-layer material deposition process that alternates between inkjet deposition of an ultraviolet (UV) curable polymer layer and the stamping of electrospun fibers onto the layer, to build the final part. While this process has been proven for complex 3D geometries, it suffers from poor fiber transfer efficiencies (FTEs) that affect the eventual fiber content in the printed part. In order to address this issue, it is critical to first understand the mechanics of the fiber transfer process. To this end, the objective of this paper is to develop a cohesive zone-based finite element model that captures the competition between the “fiber–carrier substrate” adhesion and the “fiber–polymer matrix” adhesion, encountered during the stamping process used for 3D printing FrSCs. The cohesive zone model (CZM) parameters are first calibrated using independent microscale fiber peeling experiments involving both the thin-film aluminum carrier substrate and the UV curable polymer matrix. The predictions of the calibrated model are then validated using fiber transfer experiments. The model parametric studies suggest the use of a roller-based stamping unit design to improve the FTE of the FrSC 3D printing process. Preliminary experiments confirm that for a 0.5 in diameter roller, this new design can increase the FTE to ∼97%, which is a substantial increase from the 55% efficiency value seen for the original flat-plate stamping platen design. The model has broader applications for the transfer-printing of soft material constructs at the submicron scale.

## Introduction and Motivation

Fiber-reinforced soft composites (FrSCs) are an emerging class of materials that are “made up of polymeric fibers with specific material properties and hierarchical length scales, embedded within another soft-polymer matrix” [1]. Recently, Spackman et al. [1,2] demonstrated a novel three-dimensional (3D) printing process for manufacturing FrSCs relevant to applications such as soft actuators, four-dimensional printing, biomimetic composites, and embedded sensing [35]. While their process has been proven for complex 3D geometries, it suffers from poor fiber transfer efficiencies that affect the resulting fiber content in the printed part [1].

Figure 1 outlines the FrSC 3D printing process developed by Spackman et al. [1]. As seen in Fig. 1, the process first involves the direct-writing of polymer fiber coupons on a thin-film aluminum carrier substrate, using the near-field electrospinning process (step 1). This step creates aligned polymeric fibers that are then transferred along with the aluminum carrier substrate onto a five-axis motion platform capable of printing ultraviolet (UV) curable inks using an inkjet nozzle [2]. Once a polymer layer has been printed (step 2 in Fig. 1), a stamping operation is then conducted to transfer the fibers from the aluminum carrier substrate onto the printed polymer layer (steps 3 and 4 in Fig. 1). As these steps are repeated in a layer-by-layer fashion, it is possible to manufacture aligned FrSCs with varying fiber concentrations.

The process study investigations of Spackman et al. [1] revealed that the fiber transfer efficiency (FTE) of the stamping process (step 4 in Fig. 1) is a function of both the effective area coverage of the fiber coupon (defined as the percentage area of the aluminum carrier substrate that is covered by the fibers) and the surface energy of the carrier substrate. It was found that by increasing the effective area coverage above 60% and using a polytetrafluoroethylene (PTFE)-coated aluminum substrate, the FTEs could be increased to >95%. However, this meant that mats with lower fiber concentrations could not be effectively transferred using the current stamping protocol involving the vertical motion of a flat-plate stamping platen (Fig. 1) [1,2].

In order to increase the FTEs encountered during the 3D printing of FrSCs, it is critical to first understand the mechanics of the fiber transfer process using a suitable stamping process model. To this end, the objective of this paper is to develop a cohesive zone-based finite element model [6] that captures the competition between the fiber–carrier substrate adhesion and the fiber–polymer matrix adhesion encountered during the stamping process. The cohesive zone model (CZM) parameters are first calibrated using independent microscale fiber peeling experiments involving the aluminum substrate and the polymer matrix. The predictions of the calibrated model are then validated using fiber transfer experiments. The validated model is then used to perform parametric studies that provide design insights into improving the FTEs for the FrSC 3D printer.

The remainder of this paper is organized as follows: Section 2 provides an overview of the modeling approach and the use of well-designed experiments to estimate the model parameters. Section 3 provides the model implementation details including the model calibration and validation. Section 4 provides the results from the model parametric studies and the corresponding experimental trials. Finally, Sec. 5 presents the specific conclusions that can be obtained from this work.

## Modeling Approach and Materials

### Model Overview.

Figure 2 outlines the overall modeling strategy to develop a cohesive zone-based finite element model for the fiber transfer process encountered during the 3D printing of FrSCs. As seen in Fig. 2, the overall strategy comprises two key stages, viz., Stage 1: estimation of the cohesive zone parameters (for modeling the interfaces) and Stage 2: development of the 3D fiber transfer model. abaqus ExplicitTM is used as the finite element solver for both stages [68]. Both modeling stages use the fiber diameter distribution and spacing parameters identified from the microscopy images as their key input (refer top gray colored box in Fig. 2).

The first stage in the model development involves the estimation of the cohesive zone parameters that capture the behavior of the two interfaces, viz. (1) the fiber–aluminum carrier substrate interface and (2) the fiber–polymer matrix interface, under conditions encountered during the stamping operation. This stage involves the following three steps:

• Step 1: Estimating the area of the two contact interfaces using a two-dimensional (2D) finite element simulation that models the compression encountered during the stamping operation;

• Step 2: Performing two independent calibration experiments that measure the steady-state peel force encountered during the failure of the two different interfaces (see Fig. 2, lower gray box); and

• Step 3: Combining steps 1 and 2 with a 3D fiber-peeling finite element simulation to estimate the cohesive zone parameters for the two interfaces.

It should be noted that at the end of step 3, the cohesive parameters of both the interfaces would have been estimated independent of each other.

The second stage in the modeling process is to combine the CZM parameters with a 3D fiber transfer model that explicitly simulates the three-material system involving the aluminum carrier substrate, fibers and the polymer matrix (Fig. 2). The behavior of the two interfaces encountered in this system is captured using the cohesive zone parameters estimated in Stage 1 (see step 3). In addition, the input to this stage also includes the properties (stress–strain curves, density) of the three materials and the necessary boundary conditions that replicate the fiber transfer process. The finite element solver is then used to predict the steady-state peel force. This value of the peel force can then be validated using fiber transfer experiments that are similar to the Stage 1 calibration tests but this time using the three-material system (i.e., aluminum carrier substrate, fibers, and the polymer matrix). Once the model has been validated, it can be used to perform process planning experiments to improve the fiber transfer efficiency of the stamping process.

### Materials.

The materials selected for this modeling study were identical to ones chosen for the Spackman et al. [1] experimental study, viz., Nylon-6 for the fibers, TangoBlackTM (a commonly used UV curable polymer) as the polymer matrix, and PTFE-coated aluminum foil as the fiber–carrier substrate, respectively. Table 1 summarizes the key material properties relevant to these three materials. In order to capture both the elastic and (possibly) plastic deformation of the fibers and the polymer matrix, the tensile stress–strain curves reported in Ref. [1] were also used as input to the model.

The scope of the modeling work was only limited to a fiber mat with a 45% effective area coverage. The microscopy experiments reported by Spackman et al. [1] indicate that for such a fiber mat, the average center-to-center distance of the fibers is 2.6 μm, and the fiber diameters follow a log-normal distribution with an mean of 1.34 μm and a standard deviation of 0.403 μm. These experimentally measured values were used to simulate the fiber diameters and spacing in the model.

## Model Implementation

This section will describe the model implementation details involved in the two-stage model.

### Stage 1—Cohesive Zone Parameter Estimation.

The overall model accuracy is critically hinged on capturing the adhesion behavior of the fiber–aluminum carrier substrate interface and the fiber–polymer matrix interface. CZMs have been proven in literature to capture the debonding behavior of interfaces [6,11]. CZMs are dependent on two factors, viz., the effective area of the contact interface and the traction separation law assigned to the interfacial cohesive elements. The effective area of the contact interface governs the number and the size of the cohesive elements connecting the two surfaces, whereas the traction–separation law governs the opening of the interface leading to debonding.

Stage 1 of the model is aimed at estimating the cohesive zone parameters. As outlined in Sec. 2, this stage comprises of three steps, viz. (1) estimating the area of contact (AOC) interfaces; (2) performing calibration experiments to obtain the interface-specific force–displacement curves; and (3) estimating the cohesive zone parameters using a 3D fiber-peeling simulation. The details of each of these three steps are outlined in the remainder of this section.

#### Step 1: Estimating the AOC for the Interfaces.

In order to capture the fiber peeling force accurately, it is critical to estimate the area of the two contact interfaces. The interface between the fiber and the aluminum carrier substrate is the result of the near-field electrospinning process that deposits “wet” fibers onto the substrate [1215], whereas the interface between the fiber and the polymer matrix is a result of the stamping operation that ensures contact between the fibers and the UV curable polymer surface.

The AOC between the fiber and the aluminum carrier substrate is a function of the near-field electrospinning process parameters [1215]. It should be noted that due to the presence of the carrier solvent, the fibers are wet when deposited on the aluminum carrier substrate [1215]. This results in a surface contact between the fiber and the carrier substrate as opposed to the line contact that one would expect if the fibers were “dry” before landing on the substrate. Experimental studies in the literature including the microscopy studies conducted by Spackman et al. [1] have shown that the width of the contact between the fiber and the substrate is ∼25% to 35% of the fiber diameter. Therefore, the interface between the fiber and the aluminum substrate is approximated as a rectangular area whose width is 20% of the diameter of the fiber while its length extends along the entire fiber. Cohesive zone elements (CZEs) will be placed along this rectangular area while modeling the interface between the fiber and the aluminum carrier substrate.

Estimating the AOC between the fiber and the polymer matrix substrate is nontrivial since this is an interface that is formed as a result of the stamping operation. Given the small diameters of the fibers (∼1 μm), it is difficult to accurately estimate this interface area using experiments. Therefore, this interfacial area is estimated using a 2D simulation that models the stamping engagement between the fiber-carrying aluminum substrate and the polymer matrix.

Figure 3 outlines the details of the 2D simulation. As shown in Fig. 3(a), the 2D model assumes perfect contact between the aluminum carrier substrate and the fibers (based on the AOC estimated previously). While the substrate remains stationary (i.e., fixed boundary condition), a line load, P, equivalent to 50 N/m is applied to the block to ensure contact between the carrier substrate and the surface of the polymer matrix (Fig. 3(a)). The value of the line load is based on the magnitude of the load applied during the stamping operation. As stated in Sec. 2.2, the fiber diameter variation and the spacing in this simulation are based on experimental measurements. The number of fibers in the model was limited to six to ensure reasonable computational time.

Figure 3(b) shows the steady-state result of this 2D simulation. As can be seen, the application of the stamping load allows for deformation of the softer polymer substrate around the stiffer fiber. As a result, a contact arc region is observed between the fiber and polymer matrix substrate (arc S in Fig. 3(c)). Ideally, cohesive elements should be introduced along this arc that covers around 10–25% of the circumference. However, given the limitation of the cohesive elements available in abaqusTM, it is challenging to do so at the end of the stamping simulation. Therefore, based on the results of this 2D simulation, this arc length is approximated as a straight edge that is 30–75% of the fiber diameter (li in Fig. 3(c)). Fibers with larger diameters fall on the higher end of this range as they are the first to contact the 3D-printed substrate and therefore compress the substrate more than the smaller diameter fibers (Fig. 3(b)). This approximation means that the contact area between the fiber and the substrate will be modeled as a rectangular area whose width is 30–75% of the fiber diameter while its length extends along the entire fiber. Cohesive zone elements will be placed on this rectangular area while modeling the interface between the fiber and the polymer matrix substrate.

#### Step 2: Calibration Experiments.

These experiments are designed to extract the specific load–displacement curves for the fiber–aluminum carrier substrate interface and the fiber–polymer matrix interface. While 90-deg peel experiments have been used in literature to extract CZM parameters for pliable macroscale interfaces [15], the experiments here are challenging because the forces developed during the peeling of the microscale electrospun fibers are too low to be detected using conventional load cells. Therefore, a microscale tensile/compression system MicroSquisher (CellScale, Waterloo, ON, Canada) was used. Figure 4(a) shows the general layout of the system that uses a digital camera and pixel-tracking software to calculate the force exerted during the test. This force measurement is based on the movement of the piezoelectric stages (input) and the relative movement of the tip of a tungsten microbeam. The software calculates the force magnitude by solving a basic beam-deflection equation. Data are output at 5 Hz, with a resolution of ±0.1 μN.

The MicroSquisher system is typically used to measure the stiffness and surface tension of microscale soft materials and biomaterials. Therefore, its standard sample mounting protocols were modified for the purposes of these calibration experiments. Two sets of samples were manufactured for these calibration tests, viz., samples where the fibers were attached to the aluminum carrier substrate (product of the near field electrospinning) and samples where the fibers were stamped onto the UV curable polymer matrix substrate. These samples were created such that the length of the fibers extended beyond the substrates. For each of the tests, a bundle of ∼4 to 8 free fibers ends was bonded to the microbeam of the instrument, using an adhesive. The adhesive was dried for 30 min before conducting the peel test by moving the piezoelectric stage vertically along the positive Y-direction (shown in Fig. 4(a)). The peel rate, the peel angle, and the diameter of the microbeam were selected to be 0.005 mm/s, ∼90 deg, and 0.30 mm, respectively. The inset in Fig. 4(a) shows the view from the camera during a peel test used to extract parameters for fiber–polymer substrate interface. Figure 4(b) shows a representative force–displacement curve from each of two sets of samples. As expected, the data clearly show that the bond between the fiber and the polymer matrix is significantly stronger than that between the fiber and the aluminum carrier substrate. It should be noted here that in these curves the region before the steady-state peel region (Fig. 4(b)) is a function of the slack in the fibers that needs to be taken up by the system. A total of three replicates were performed for each of the two sets of measurements denoted in Fig. 4(b).

#### Step 3: Estimating the Cohesive Zone Parameters Using a 3D Fiber-Peeling Simulation.

Figures 5(a) and 5(b) shows the 3D finite element framework used to model the fiber-peeling experiments that were conducted in step 2. As seen, this model has 3D fibers that have modified contact areas as estimated in step 1. Two separate simulations were conducted for the two tests represented in Fig. 4(b). For the peeling simulation involving the polymer substrate and the fiber, cohesive zone elements were only attached along their interfacial area of contact (as shown in Fig. 5(a)). This allows the estimation of the CZM parameters specific to the fiber–polymer matrix interface by tuning the CZM parameters to match the steady-state peel force from the calibration experiments. The same process was repeated but this time for the aluminum carrier substrate and the fiber combination where the CZEs were only attached along their interfacial area of contact (Fig. 5(b)). For both these simulations, a velocity boundary condition (0.005 mm/s) was applied to the free fiber ends to mimic the conditions of the calibration experiment. The substrate in both cases was maintained stationary.

The simulation snapshots in Figs. 5(a) and 5(b) show clearly that the cohesive zone elements allow the fiber–substrate interface to open and then subsequently fail as the peeling process proceeds. This behavior of the CZEs is controlled by the interface-specific traction separation curves that have a direct bearing on the steady-state peel force predicted by the model.

Figure 5(c) shows a schematic of the triangular traction separation curve used for the interfaces modeled in this study. The key parameters to be estimated for both interfaces include the damage initiation stress (tult), the damage initiation opening (δc), and the displacement to failure (δf–δc). For the purposes of this model, δc was assumed to be negligible, i.e., the damage initiation is assumed to be instantaneous. In order to estimate the damage-initiation stress and the displacement-to-failure values for both the interfaces, some initial values for these parameters were first assumed. These values were then updated sequentially using an error minimization approach aimed at minimizing the difference between simulated and experimental steady-state peel force values [16].

Table 2 also provides the fracture energy values, i.e., the area under the curve in Fig. 5(c), for each of the two interfaces. The cohesive zone models in literature involving the peeling of adhesive tape on a glass substrate have reported a fracture energy value of 0.08 N/mm [15,17]. The values obtained for the two interfaces encountered during the 3D printing of FrSCs indicate a weaker bond compared to those studies.

### Stage 2—3D Fiber Transfer Model.

Now that the CZM parameters have been calibrated, they can be used as input to the 3D fiber transfer model to predict the steady-state peel-force for the three-material system involving the aluminum carrier substrate, nylon fiber, and the polymer matrix. This 3D model will essentially capture the competition between the two interfaces that eventually results in the transfer of fibers from the aluminum carrier substrate onto the polymer matrix surface. Once the predictions made by this 3D fiber transfer model have been validated using comparable experimental data, the model can be used to design fiber stamping units that ensure higher fiber transfer efficiencies.

#### Model Prediction.

This Stage-2 3D fiber transfer model makes the following assumptions:

Assumption 1: The stamping operation has been completed and fiber transfer process is being modeled at the onset of the subsequent peeling operation. This assumption implies that the CZEs at both the fiber–aluminum carrier substrate interface and the fiber–polymer matrix interface are engaged as in Stage 1 simulations.

Assumption 2: The fiber diameter and spacing distribution per unit width of the fiber mat remain constant. This allows the comparison of the model predictions with the experimental data irrespective of the width of the sample.

Assumption 3: The stamping pressure remains the same as in Stage 1. This assumption implies that the contact area of the two interfaces is maintained as in Stage 1. Furthermore, except for the area of contact of the interfaces, all other geometrical features of the model are nonconsequential to the behavior of the interface that dictates the steady.

In light of the aforementioned assumptions, the fibers and aluminum carrier substrate are modeled similar to that in Stage 1, with CZEs connecting the two. Given that the model explicitly assumes contact between the fiber and the polymer surface, the surface of the substrate can no longer be modeled to be geometrically flat (as in Stage 1). Rather it has to be modeled to have undulations that ensure contact between the fibers (with varying diameter) and the polymer substrate (Fig. 7(a)). However, these undulations are purely geometric in nature to ensure contact. It has no bearing on the steady-state peel load since the area of contact between the fiber and the polymer substrate is the same as in Stage 1 (as per Assumption 3 mentioned previously).

Figure 7(a) shows the overall configuration of the Stage 2 fiber-transfer model, whereas Fig. 7(b) shows two distinct time-stamps of the fiber-transfer simulation. The overall dimensions of the carrier substrate simulated in Fig. 7 are 0.05 mm (length) × 0.02 mm (width) × 0.016 mm (thickness). The boundary condition is that of a vertical velocity of 0.005 m/s applied to the aluminum carrier substrate, while the polymer substrate is kept stationary. In Fig. 7(b), after 0.16 s into the simulation, the CZEs are seen to fail at different rates based on the diameter of the fibers. After 1.89 s, the simulation results show the effective transfer of the fiber from the aluminum carrier substrate and onto the polymer substrate.

#### Model Validation.

In order to validate the Stage 2 model, the setup described in Fig. 4(a) was used again to perform peeling experiments involving the three-material system modeled in Fig. 7. The validation experiment was set up by first placing a fiber-carrying aluminum substrate, 20 mm (length) × 1 mm (width) × 0.016 mm (thick), on the 3D-printed polymer substrate. This substrate size was limited by the smallest size that could be physically handled to run the peeling experiment. A weight of 5 g was then placed on the top surface of the fiber-carrying aluminum carrier substrate to simulate the stamping process. This is equivalent to the line load of 50 N/m used in Stage 1 simulations (Fig. 3). With the weight positioned on top of the aluminum carrier substrate, its extremity was attached to the microbeam (0.4064 mm diameter) using an adhesive (Fig. 8). After drying the adhesive for 30 min, the peel test was initiated and the peeling force versus displacement data were recorded as the microbeam was raised vertically at a rate of 0.005 mm/s. Based on multiple experiments, the steady-state peel force values were measured to be in the 1500–2500 μN range. It should be noted that this value encompasses both the competition between the two interfaces and the force needed to move the aluminum carrier substrate.

The width of the experimental sample was 50 times the width simulated in the model. Assuming that fiber diameter and spacing distribution per unit width of the fiber mat remains constant (Assumption 2, Sec. 3.2.1), this means that in order to effectively compare the simulation data with the experimental value, it has to be scaled by a factor of 50. Upon scaling the forces, the model is seen to predict an average steady-state peel force of 2350 μN, which implies that the model predictions lie toward the higher end of the experimentally measured range of 1500–2500 μN. This trend can be attributed to the following plausible reasons:

1. (1)

The difference in the scale of the experiment and the simulation introduces errors, the key one being due to the fiber diameter/spacing distribution assumption. Given that the experimental sample is 50 times wider than the simulated region, it is likely that the tested fiber mats have different diameter and spacing distributions from that assumed in the model;

2. (2)

The contact at the fiber interfaces may be intermittent as opposed to the continuous contact assumed by the model; and

3. (3)

Unlike the simulated fibers, the electrospun fibers are likely to have defects that act as stress concentrators (see Fig. 10(a) ahead).

These findings imply that the predictions of the model need to be interpreted conservatively.

## Model-Enabled Design Improvements for the Fiber Stamping Unit

The fiber transfer model developed in Sec. 3 can be used to improve the design of the fiber stamping unit, which is a critical part of the 3D printer configuration. The original fiber stamping unit designed by Spackman et al. [1] consisted of a flat plate onto which the fiber-carrying aluminum carrier substrate was mounted (Fig. 1). After the stamping operation was completed, the peeling of the fibers from the aluminum substrate (and its subsequent deposition on the polymer matrix substrate) was achieved by a vertical motion of the flat-plate. This flat-plate design and the corresponding vertical peeling velocity boundary condition was seen to result in low fiber transfer efficiencies (∼50%) for fiber mats that have an effective area coverage <50% [1].

In order to improve the overall fiber transfer efficiency of the 3D printer, a new roller-based design is explored in this paper. It is hypothesized that such a design is more suitable for the transfer of fibers, for two main reasons. First, given that the region of contact will be localized around the point of tangency of the roller and the substrate, the curvature of the roller can be expected to provide a more localized and uniformly distributed stamping load than the flat-plate design. Irrespective of the fiber number density of the mat, this can be expected to result in a good contact length at the fiber–polymer matrix interface that promotes adhesion to the polymer substrate. Second, the natural peeling action imposed by the rotation of the roller is expected to induce lower stress-levels in the fibers than those encountered with the abrupt vertical motion of the flat-plate design. These lower stress levels will translate to higher FTEs since the fibers can be expected to transfer to the polymer substrate without tears propagating across the fiber mat.

In order to confirm this hypothesis, first the 3D fiber transfer model predictions are used to evaluate the design parameter space for the roller-based configuration. The peak stress levels in the fibers are then used as a measure to evaluate the efficacy of the design. Model-educated experiments are then run to confirm the fiber transfer efficiencies obtained by the new design. The remainder of this section presents the results of these model-enabled design efforts.

### Boundary Conditions for a Roller-Based Stamping Unit Design.

The experiments used to calibrate and validate the 3D model predictions were performed using the microtensile platform described in Fig. 4(a). However, the implementation of the stamping operation on the 3D printer involves not only significantly larger length scales but also entirely different boundary conditions than those encountered during the peeling operation using the microtensile platform. For the model to be useful for realistic process planning, the boundary conditions in the model must be mapped to that expected during the stamping process encountered on the 3D printer.

Figure 9(a) shows the schematic of the new roller design envisioned for the fiber stamping and transfer process. As shown in Fig. 9(a), for this roller design, the two process parameters that can be varied during the stamping process include the radius of the roller, R, and the angular velocity of the roller, ω. The product of these two values is the tangential velocity of the roller, VT, that has a magnitude of . In this configuration, since the fibers are mounted on the roller, the substrate has to be translated horizontally with a linear velocity to ensure a no-slip boundary condition at the bottom-dead-center (BDC) position of the roller.

In the roller design, the stamping force is maximum at the BDC of the roller and the peeling of the fibers from the aluminum carrier substrate only kicks in at a small angular distance Θ1 from the BDC position (Fig. 9(a)). For the purposes of this design evaluation, Θ1 is assumed to be a small value of 1 deg. The peeling action now needs to be translated into appropriate velocity boundary conditions for the 3D fiber transfer model presented in Sec. 3.2. In order to do this, the segment to be simulated in the model is assumed to be the region that is bounded between angles Θ1 and Θ2. The arc length S, given by the below equation, is the length of the sample being simulated by the model
(1)

Based on the simulation results in Sec. 3.2, it can be safely assumed that the steady-state peel force is reached over an arc length of 50 μm, which is the length of the sample simulated in Sec. 3.2. Therefore, for any given radius of the roller R, the angle Θ2 is selected based on Eq. (1) as the value that would give an arc length, S, equal to 50 μm, which is the length of the model used of the simulation.

Given that the velocity boundary conditions in the 3D model are applied only to the aluminum carrier substrate, for the case depicted in Fig. 9, the relative velocity vector between the aluminum carrier substrate and the polymer substrate has to be used. This relative velocity vector $VT′$ can be written as
$VT′(Θ, R,ω)=[ωR cos (Θ)−ωR]î+ωR sin (Θ)ĵ+0k̂$
(2)

For a given R and ω combination, the velocity boundary conditions to be applied at the two ends of the aluminum carrier substrate in the model are shown in Fig. 9(b). These boundary conditions are given by the vector expressions $VT′(Θ1, R,ω)$ and , respectively.

### Model Results for Design Evaluation.

Table 3 shows the design space of the parameters chosen to be evaluated using the model. The lowest value of the roller radius (R) in the table corresponds to the smallest fiber carrier substrate coupon that could be tested on the printing platform, whereas the largest value of the roller radius was limited by the working volume of the motion platform. The tangential velocities (VT) of the roller were chosen based on the limitations of the hardware. For each of the R and VT combination outlined in Table 3, Eqs. (1) and (2) were first used to arrive at the corresponding boundary conditions to be applied to the 3D model. The model was then used to predict the peak stress values seen in the fibers.

The stresses encountered by the fiber during the transfer process have a correlation with the FTEs. As the peak stress increases in the fibers, they are more likely to fail and tear away from the mat without being transferred onto the printed substrate, thereby lowering the corresponding FTE value. This phenomenon is further aided by the fact that the electrospun fibers have various defects, such as beads and tears [18], that significantly reduce their cross section at different regions of the mat (Fig. 10(a)) [19]. These fiber defects are not modeled in the 3D finite element model that assumes a uniform cross-sectional area for the fibers. Therefore, the yield stress of the fiber, which is a lower threshold value than the ultimate tensile strength, is used as the criterion for evaluating the design performance.

The model predictions in Fig. 10(b) show a correlation between the peak stresses induced in the fibers (during the fiber transfer process) and the tangential velocity of the roller. It should be noted that the horizontal line in Fig. 10(b) represents the yield stress for the Nylon-6 fiber. For each of the radius tested, the lowest peak stress value corresponds to the lowest tangential velocity condition. This implies that a lower roller velocity may be preferred to ensure higher FTEs. The radius of the roller is also seen to affect the stress level in the fibers. The stresses generated by the smaller roller with a radius of 0.25 in do not exceed the yield stress for Nylon-6 fibers for any of the speeds tested. However, for the roller with a radius of 1.75 in, the higher speed of 10 mm/s is clearly resulting in stress levels higher than the yield stress. This trend predicted by the simulations suggests that the smaller roller will have higher FTEs at all tangential velocities, whereas the larger roller may have a comparatively poor FTE, especially at a tangential velocity of 10 mm/s.

### Experimental Confirmation of Model Results.

In order to confirm the model results, a follow-on experimental study was conducted to evaluate the FTEs encountered when the roller-based stamp design is implemented “at-scale” on the 3D printing platform. Figure 11 shows the details of this experiment that was conducted on a three-axis computer controlled motion platform with a ±1 μm positional accuracy. As seen in Fig. 11, a rotary drum replicating the roller-based design was attached to the horizontal X–Y stage of the motion platform. This polyvinyl chloride rotary drum was designed with a region of foam padding (Fig. 11(b)) onto which the fiber-laden aluminum carrier substrate was mounted. The foam pad was added to ensure a uniform stamping force on the fibers during the rotation of the roller. The fiber mat used in this experiment had a 45% effective area coverage value and was obtained from the same processing batch that was reported in Ref. [1]. The inkjet system reported by Spackman et al. [1] was then used to print a 40 μm think layer of the UV curable polymer (TangoblackTM) on an aluminum substrate. This substrate was mounted on the vertical (Z) axis of the motion platform with its polymer coated surface facing the roller. The vertical stage was then brought down (in the Z-axis) until there was contact with the foam region of the rotary drum. Once contact was made with the rotary drum, the stage holding the aluminum substrate was translated over the drum at the desired tangential velocity (VT in Fig. 11). After this single pass, the fiber carrier substrate was imaged using a microscope and the FTE was quantified using the process outlined in Ref. [1].

Figure 12 shows the fiber transfer efficiencies measured from the experimental study. The benchmark value to evaluate the performance of the roller-based design is the FTE value reported for the same mat while using a flat-plate design. This value was reported by Spackman et al. [1] and is denoted by the dotted line in Fig. 12. The experimental data show that the smallest roller (radius of 0.25 in) significantly improved the fiber transfer efficiencies over those seen for the flat-plate design. The data also show that an increase in the tangential velocity does not negatively impact the FTE for this roller. Both these findings for this roller are backed up by the model predictions in Fig. 10(b) that show that at all values of the velocities tested, the stress levels in the fiber are the lowest for this roller size. The experimental data also show that the drum with the largest diameter (radius of 1.75 in) was more sensitive to the tangential velocity. While the lowest velocity of 0.1 mm/s was seen to improve the FTE over that of the flat plate design, this advantage was lost at higher speeds of 1 and 10 mm/s. This experimental finding is also backed up by the model predictions for this roller (Fig. 10(b)). At 10 mm/s, the stress levels predicted in Fig. 10(b) are seen to be higher than the yield stress, which correlates with a drop in the FTE in Fig. 12. It should be noted here that the model predictions in Fig. 10(b) assume that the cohesive zone parameters are independent of the tangential velocity. However, even under this assumption, the use of the model-predicted peak stress in the fibers as a measure of the fiber transfer efficiency appears to be valid (Fig. 12).

### Possible Implementation Strategy on 3D Printing Platform.

Figure 13 shows one possible translation of the findings from Secs. 4.2 and 4.3 into a fiber stamping unit design concept for the FrSC 3D printer [1,2]. The realization of this concept would involve assembling the carrier substrates onto a feed reel. The carrier substrates would then travel over a stamping roller (0.25 in radius) as the 3D-printed part translates at a speed equal to the tangential velocity of the roller. A take-up reel would then collect the empty carrier substrates. Such an implementation is expected to improve the fiber transfer efficiency encountered during the stamping step for the FrSC 3D printer [1,2].

## Conclusions

The objective of this paper is to develop a cohesive zone-based finite element model that captures the competition between the fiber–carrier substrate adhesion and the fiber–polymer matrix adhesion, encountered during the stamping process used for 3D printing FrSCs. In order to achieve this objective, the standard sample mounting protocols of the MicroSquisher microtensile unit were successfully modified for the purposes of the model calibration and validation experiments. These modifications allowed for peeling experiments that were sensitive to the force variations predicted by the model. The following specific conclusions can be drawn from this study:

1. (1)

The effective area of contact between the soft fibers and the aluminum carrier substrate was estimated to be 20% (of the initial fiber diameter) for the fiber–carrier substrate interface and to be 30–75% (of the initial fiber diameter) for the fiber–polymer matrix interface.

2. (2)

The cohesive zone parameters for the fiber–carrier substrate interface and the fiber–polymer matrix interface were estimated using an error minimization approach aimed at minimizing the difference between simulated and experimental steady-state peel force values. These parameters resulted in a predicted steady-state peel forces that were within 5.2% of the average experimental values.

3. (3)

The 3D fiber transfer model for the three-material system involving the aluminum carrier substrate, nylon fiber, and the polymer matrix yields steady-state peel force predictions that lie toward the higher end of the experimentally measured range of 1500–2500 μN. This is attributed to factors such as defects in the fibers, poor interfacial contact, and a possible violation of the model assumption that the fiber diameter and spacing distribution per unit width of the sample remain constant.

4. (4)

The model-based parametric study validates the hypothesis that roller-based designs are superior to the original flat-plate stamping platen design of Spackman et al. [1,2]. Smaller diameter rollers (radius ∼0.25 in) are preferred since their tangential velocities do not detrimentally affect the fiber transfer efficiencies. However, if the design requires a larger diameter (1.75 in), then it is advantageous to run it at the lowest speed to ensure higher fiber transfer efficiencies.

The aforementioned findings are expected to educate critical design improvements that are needed to further the nascent field of 3D printing FrSCs. Furthermore, the modeling approach and the experimental validation techniques proposed in this paper also have implications for the transfer-printing of soft material constructs at the submicron scale [20].

## Acknowledgment

The authors acknowledge support from the US National Science Foundation, Manufacturing Machines and Equipment program and internal funds from the Rensselaer Polytechnic Institute, NY, USA. J.F. Nowak acknowledges support by the National Science Foundation Graduate Research Fellowship Program (NSF GRFP).

## Funding Data

• Division of Civil, Mechanical and Manufacturing Innovation (Award No. CMMI 14-62648).

• Division of Graduate Education (Grant No. DGE12-47271).

## References

References
1.
Spackman
,
C. C.
,
Frank
,
C. R.
,
Picha
,
K. C.
, and
Samuel
,
J.
,
2016
, “
3D Printing of Fiber-Reinforced Soft Composites: Process Study and Material Characterization
,”
J. Manuf. Processes
,
23
, pp.
296
305
.
2.
Spackman
,
C. C.
,
Picha
,
K. C.
,
Gross
,
G. J.
,
Nowak
,
J. F.
,
Smith
,
P. J.
,
Zheng
,
J.
,
Samuel
,
J.
, and
Mishra
,
S.
,
2015
, “
A Novel Multimaterial Additive Manufacturing Technique for Fabricating Laminated Polymer Nanocomposite Structures
,”
J. Micro Nano-Manuf.
,
3
(
1
), p.
11008
.
3.
Tibbits
,
S.
,
2014
, “
4D Printing: Multi-Material Shape Change
,”
Archit. Des.
,
84
(
1
), pp.
116
121
.
4.
,
A.
,
Matsumoto
,
E. A.
,
Nuzzo
,
R. G.
,
,
L.
, and
Lewis
,
J. A.
,
2016
, “
Biomimetic 4D Printing
,”
Nat. Mater.
,
15
(
4
), pp.
413
418
.
5.
Muth
,
J. T.
,
Vogt
,
D. M.
,
Truby
,
R. L.
,
Mengüç
,
Y.
,
Kolesky
,
D. B.
,
Wood
,
R. J.
, and
Lewis
,
J. A.
,
2014
, “
Embedded 3D Printing of Strain Sensors Within Highly Stretchable Elastomers
,”
,
26
(
36
), pp.
6307
6312
.
6.
Elices
,
M.
,
Guinea
,
G. V.
,
Gómez
,
J.
, and
Planas
,
J.
,
2002
, “
The Cohesive Zone Model: Advantages, Limitations and Challenges
,”
Eng. Fract. Mech.
,
69
(
2
), pp.
137
163
.
7.
Spring
,
D. W.
, and
Paulino
,
G. H.
,
2014
, “
A Growing Library of Three-Dimensional Cohesive Elements for Use in ABAQUS
,”
Eng. Fract. Mech.
,
126
, pp.
190
216
.
8.
Peng
,
R. D.
,
Zhou
,
H. W.
,
Wang
,
H. W.
, and
Mishnaevsky
,
L.
,
2012
, “
Modeling of Nano-Reinforced Polymer Composites: Microstructure Effect on Young's Modulus
,”
Comput. Mater. Sci.
,
60
, pp.
19
31
.
9.
Collins
,
J. A.
,
Busby
,
H.
, and
Staab
,
G.
,
2010
,
Mechanical Design of Machine Elements and Machines: A Failure Prevention Perspective
,
2nd ed.
,
Wiley
,
New York
.
10.
Stratasys
, 2017, “
PolyJet Technology
,” Stratasys Ltd., Eden Prairie, MN, accessed Dec. 1, 2016, http://www.stratasys.com/3d-printers/technologies/polyjet-technology
11.
Rabinovitch
,
O.
,
2008
, “
Debonding Analysis of Fiber-Reinforced-Polymer Strengthened Beams: Cohesive Zone Modeling Versus a Linear Elastic Fracture Mechanics Approach
,”
Eng. Fract. Mech.
,
75
(
10
), pp.
2842
2859
.
12.
Liu
,
Z. H.
,
Pan
,
C. T.
,
Lin
,
L. W.
,
Huang
,
J. C.
, and
Ou
,
Z. Y.
,
2013
, “
Direct-Write PVDF Nonwoven Fiber Fabric Energy Harvesters Via the Hollow Cylindrical Near-Field Electrospinning Process
,”
Smart Mater. Struct.
,
23
(
2
), p.
25003
.
13.
Sun
,
D.
,
Chang
,
C.
,
Li
,
S.
, and
Lin
,
L.
,
2006
, “
Near-Field Electrospinning
,”
Nano Lett.
,
6
(
4
), pp.
839
842
.
14.
Zheng
,
G.
,
Li
,
W.
,
Wang
,
X.
,
Wu
,
D.
,
Sun
,
D.
, and
Lin
,
L.
,
2010
, “
Precision Deposition of a Nanofibre by Near-Field Electrospinning
,”
J. Phys. D. Appl. Phys.
,
43
(
41
), p.
415501
.
15.
Kovalchick
,
C.
, and
Molinari
,
A.
,
2009
, “
Mechanics of Peeling for Extensible Elastic Adhesive Tapes
,” 11th Pan-American Congress of Applied Mechanics (
PACAM XI
), Foz do Iguaçu, Brazil, Jan. 4–8, pp.
4
9
.http://www.set.eesc.usp.br/proceedings/papers/area1-solidMechanics/PAC0349-KovalchickC.pdf
16.
Thouless
,
M. D.
, and
Yang
,
Q. D.
,
2008
, “
A Parametric Study of the Peel Test
,”
,
28
(
4–5
), pp.
176
184
.
17.
Sauer
,
R. A.
,
2011
, “
Enriched Contact Finite Elements for Stable Peeling Computations
,”
Int. J. Numer. Methods Eng.
,
87
(
6
), pp.
593
616
.
18.
Deitzel
,
J. M.
,
Kleinmeyer
,
J.
,
Harris
,
D.
, and
Beck Tan
,
N. C.
,
2001
, “
The Effect of Processing Variables on the Morphology of Electrospun Nanofibers and Textiles
,”
Polymer
,
42
(
1
), pp.
261
272
.
19.
Megelski
,
S.
,
Stephens
,
J. S.
,
Bruce Chase
,
D.
, and
Rabolt
,
J. F.
,
2002
, “
Micro- and Nanostructured Surface Morphology on Electrospun Polymer Fibers
,”
Macromolecules
,
35
(
22
), pp.
8456
8466
.
20.
Chen
,
Y.
,
Shi
,
J.
, and
Wang
,
L.
,
2009
, “
Microcontact Printing and Lithographic Patterning of Electrospun Nanofibers
,”
Langmuir
,
25
(
11
), pp.
6015
6018
.