A mechanical relation model for biaxial tension of nanofibrous membrane

Electrospinning is an economical and straightforward technique that is widely used in the fabrication of continuous ultrafine fibers with a high surface-to-volume ratio, high porosity, and variable pore size distribution. Such features highlight the importance of electrospinning for biomedical applications.^1–3 One of the most commonly used electrospun biomaterials is poly(ε-caprolactone) (PCL). PCL is degraded by the hydrolysis of its ester linkages under physiological conditions (such as in the human body) and has received a great deal of attention for use as a biomaterial, including for skin,⁴ bones,⁵ heart,⁶ vessels,⁷ and drug delivery,⁸ owing to its satisfactory mechanical performance.

Understanding nanofibrous membranes’ mechanics is essential when evaluating their mechanical properties at various structural levels, both during processing and final applications. Computer modeling of electrospun nanofiber membranes is a difficult task because of the randomness of the mesostructure of the fiber web and its sophisticated mechanical response. These responses often involve large deformations and rotations, bonding and fiber fracture, fiber slippage, and the continuous recombination of the fiber network topology. The literature shows that the mechanical properties of an electrospun mat are tunable by changing the characteristics of the polymer,⁹ the diameter of the constituent fibers,¹⁰ fiber alignment,¹¹ porosity,¹² and bonding at the cross-points of the fibers.¹³ Wei et al.¹⁴ showed that the fusion between fibers affects the tensile strength of fiber membranes. More fusion points can increase the strength of the membrane, while over-fusing may decrease the fracture energy. However, this study was only at the nanoscale, not in a multi-stage medium. It could not explain why excessive fusion of fiber bonding points did not significantly affect the tensile strength of the membranes. The study of Chavoshnejad et al.¹⁵ shows that a large percentage of interfiber bonding at a predefined porosity of a mat does not increase the elastic modulus of the mat, nor does it have considerable effects on the failure behavior. Moreover, the effect of interfiber bonding increases with a mat’s porosity. Even after considering those parameters, it remains challenging to link the overall bulk material behavior to the small-scale constituent elements.

In recent years, many researchers have investigated the uniaxial mechanics of polymer nanofibers and membranes.^16–21 However, the force of nanofibrous membranes in practical applications is mainly a two-dimensional plane force. Uniaxial stress analysis cannot entirely reflect the actual stress state of the material. Therefore, double-axis mechanical tensile analysis must be conducted on the electrospun nanofibrous mats. Yin and Xiong²² established the biaxial tensile relationship between the fiber and fiber membrane according to the microstructure characteristics of the electrospinning nanofiber membrane, which was verified by the biaxial tensile experiment of the silk fibroin/polycaprolactone composite nanofiber membrane. The results of the study confirmed that the model developed for the biaxial tensile relationship could better explain the elastic–plastic tensile mechanical behavior of the polymer. Chavoshnejad and Razavi²³ studied the tensile mechanical behavior of the membranes by considering the microstructure of the electrostatically spun fiber membranes and the percentage of intersections that were bonded, and they developed three finite element models with no interactions between fibers, half of the intersections acting, and all of the intersections acting, respectively, assuming that the distribution of fibers in the fiber membranes was uniform. The biaxial tensile mechanical properties of the fiber membranes showed that at a fiber membrane porosity of 0.25, inter-fiber fusion could increase the nanofiber membranes’ stiffness, independent of the tensile loading mode and the mechanical properties of the individual fibers. However, as the porosity of the fiber film increases, the inter-fiber bonding will be weakened. These studies, in some aspects, elucidated the relationship between the composition, structure, and properties of the electrostatically spun nanofiber membranes under external tension. However, these studies are more or less complicated with derivation calculations and high operating costs, which are not conducive to the microstructural design and practical application of electrostatically spun nanofiber membranes.

In this study, an analytical model was developed to predict the biaxial behaviors of randomly oriented nanofibrous membranes. Stress–strain data from a single nanofiber were tested for use in the model. The model starts from the mechanical properties of single fibers and the membrane structure and studies the biaxial tensile mechanical properties of the membrane in tension by calculating the energy change in the membrane during deformation. Four control groups with different ratios of stretching speed during biaxial tension were investigated. The applicability of the analytical model was examined by a comparison between modeling prediction and experimental data. This model has the advantages of a simple calculation process and low calculation cost, and its derivation results have sure accuracy, which is very convenient for the design of the biaxial tensile mechanical properties of the electrospinning nanofiber membrane. It is envisaged that this work will provide insights to understand the biaxial mechanical characteristics of nanofibrous membranes in applications.

By referring to Fig. 1, the following assumptions can be made before the derivation of the constitutive relationship.

First, nanofibers in the electrospun membrane were simplified as continuous and straight filaments. The membrane consisted of layers of randomly oriented nanofibers in the in-plane direction.

Second, there was no interlayer effect. Nanofibers were deposited and randomly overlapped ideally in sequence during the entire electrospinning process.

Third, time-dependent properties were not considered. Because tensile speed was controlled to ensure that the specimens were under quasi-static stretching, neither the strain rate effect nor stress relaxation was considered.

Suppose that there is a nanofibrous membrane with the volume V, the porosity ratio p, and the average distance r₀ among nodes. The biaxial deformation gradient F in the two-dimensional membrane is described as

where x and X are the displacement coordinates before and after the deformation, respectively. Both F₁₂ and F₂₁ equal zero because no shear deformation is considered during biaxial stretching. According to the geometric relationship, the strain of single fibers ɛ_f in the membrane is described as

where θ is the orientation angle of the nanofiber in the membrane.

During stretching, the translation energy component U_λ contributed by the axial stress and strain can be calculated by

where r′ is the length of the nanofiber at present stretching, which can be expressed as r′ = r₀(ɛ_f + 1); d is the diameter of the nanofiber; and σ_f is the stress along the nanofiber’s orientation. However, since the diameters of the nanofibers are not uniform, stress–strain data obtained from the single-nanofiber tests should not be used directly. Therefore, mathematical fitting, as shown in Fig. 2 and Table I, was applied to obtain stable results from the test data. Then, the harmonic stress $σ f ̄$ is based on the variation of the diameters of the nanofibers in the membrane as follows:

where a, b, and c are the fitting parameters and D is the harmonic average of the diameters of the nanofibers.

During stretching, the rotation energy component U_θ contributed by the changing of the nanofiber’s orientation can be calculated by

where E is Young’s modulus, I is the cross-sectional moment of inertia, θ is the original orientation angle of the nanofiber, and θ′ is the angle at present stretching. θ′ is obtained by

The total energy U consists of the translation energy component U_λ and the rotation energy component U_θ. Therefore, U is described as

where δ_λ and δ_θ are the energy density distribution functions, which can be obtained by

The statistics of nanofibers in harmonic diameter D, the average distance r₀, and porosity ratio p was obtained by Image-Pro Plus^® in the SEM processing program.

The PCL solution prepared for electrospinning was dissolved by 98% formic acid (FA) with 18% (w/v) concentration at room temperature for 2 h. The FA solvent (purity 98.0%) was obtained from Shanghai Lingfeng Chemical Reagent Co., Ltd. (China) and used as received without further purification. After that, the solution was kept under uninterrupted stirring for 4 h.

The PCL solution was transferred to a 10 ml syringe with a stainless-steel syringe needle (22 G) and loaded into a syringe pump. The feeding rate was 0.6 ml/h. A potential of 15 kV from a high-voltage power supply was applied between the metal needle tip and aluminum sheet collector mounted on the surface of the vertical metal mesh. The distance between the needle and the collector was fixed as 12 cm. The relative humidity of the laboratory during electrospinning was kept below 60%.

The morphology of the electrospun PCL nanofibers was observed using a Carl Zeiss (Oberkochen, Germany) field-emission scanning electron microscope (SEM) operated at an acceleration voltage of 3 kV. Samples were sputter-coated for 50 s with gold to increase conductivity. To assess the average diameter, over 100 individual fibers were measured by using the Image-Pro Plus 6.2 software (ICube, Crofton, MD, USA) using SEM images. The PCL nanofibers in the membrane are presented as the SEM images in Fig. 3, and the statistics are presented in Table II.

Stress along the axial direction of a single nanofiber was tested with the Agilent T150 system (Agilent UTM T150, Santa Clara, CA, USA). The Agilent T150 provides superior nano-mechanical characterization via a unique actuating transducer that produces a tensile force (load on nanofibers) via electromagnetic actuation combined with a precise capacitive gauge. The length of single-fiber specimens was 6 mm. The drawing speed applied was 0.006 mm/s.

Biaxial stress–strain curves of the nanofibrous membranes were tested using the KSM-BX545 tensile system (Kato-Tech Company, Kyoto, Japan), as shown in Fig. 4. The normal size of the specimens was 60 mm (length) × 60 mm (width). Specimens were fixed by biaxial clamps along mutually orthogonal directions. The effective size of the specimens was 50 mm (length) × 50 mm (width). Four control groups with different ratios of stretching speed were investigated. The drawing speeds along the x- and y-axis were set as 0.01 mm/s vs 0.01 mm/s, 0.02 mm/s vs 0.01 mm/s, 0.05 mm/s vs 0.01 mm/s, and 0.08 mm/s vs 0.01 mm/s, respectively.

After fiber membrane preparation and single fiber tensile experiments, the tensile mechanical properties of electrospun PCL fibers can be obtained, taking three tensile curves to explain their properties. Three curves are the best mechanical properties, the worst mechanical properties, and one tensile curve in between, as shown in Fig. 5. The PCL fiber tensile process is first subjected to linear elastic stretching with a modulus of about 0.5 GPa, followed by plastic stretching with elongation at break between 40% and 70% and a strong force at break between 8 MPa and 18 MPa. As mentioned earlier, given the small diameter of the PCL electrostatic spun fibers, the polymeric materials can have scale effects at small scales due to molecular orientation, etc. The single-fiber data obtained by direct stretching need to be transformed.

Figure 6 demonstrates the typical deformative characteristics of the PCL nanofibrous membrane under biaxial tension. Membranes were stretched until fracture occurred. It was observed that the square specimen was forced into a rectangular shape, and it catastrophically failed, with two main cracks from the edge of the y-axis because of the relatively high stretching speed along the x-axis.

Figure 7 shows the biaxial stress–strain curves of the PCL membranes under different stretching speeds. Post-fracture behaviors were not retained in all curves. If the ratio of the stretching speed was 1:1, there was no difference in the stress–strain curves along the x- and y-axis, respectively. Stress along the x-axis had advantages during biaxial deformation with the increase in the ratio from 2:1 to 8:1. This indicates that biaxial stress–strain behaviors were sensitive to the ratios of the applied stretching speed.

In Fig. 6, we can see that the destruction of the fiber membranes starts near the clamping edge, which implies the presence of stress concentrations at the clamping boundary. This setup can significantly underestimate the failure strength of the membrane. However, Fig. 7 shows that the experimental and theoretical values near the fiber membranes’ failure point are more consistent with each other. After analysis, we believe that this is because as the tensile process proceeds, the fibers are arranged closer together and the friction energy between fibers of the same layer increases. Due to the synergistic effect of these several factors, we observed this interesting phenomenon.

By comparing the modeling prediction and experimental data, it was found that the applicability of the analytical model was roughly desirable. The proposed modeling prediction had a similar tendency as the testing results. It demonstrated that the mechanical behaviors of membranes could be predicted based on the proposed method. However, the experimental stress was lower than the predicted stress until large plastic deformation occurred. Possible reasons for this based on the morphology of the membranes are as follows: (1) the nanofibrous membranes were not ideally uniform, which may have led to the existence of some weak regions; (2) the nanofibers were not ideally straight, which may have led to the slow increase in stress at the initial stage by the adjustment of the nanofibers themselves; (3) some nanofibers in the membranes were not effectively loaded during the tensile process; and (4) there might have been a counteraction effect by the prestress in the nanofibers, which was preserved with the quick evaporation of the solvent and the possible shrinkage of the nanofibers during the depositional process of electrospinning.

It can also be observed that the agreement between modeling prediction and experimental data along the x-axis was better than that along the y-axis with the increase in the stretching speed ratio from 2:1 to 8:1. On the one hand, this was caused by the relative shrinking tendency along the y-axis due to the slower stretching speed. Such a relative shrinking tendency along the y-axis resulted in lower stress responses in the experimental tests because external y-axial loads were partially used to counteract the shrinking tendency. On the other hand, the higher stretching speed caused more nanofibers to be oriented along the x-axis. Such stretch-induced orientation in membranes brings better mechanical properties along the oriented direction.

Four groups with different ratios of stretching speed were investigated under biaxial tension. The conclusions drawn from the experimental tests and modeling prediction are as follows:

• Biaxial stress–strain behaviors were sensitive to the ratios of stretching speed applied. Stress along the x-axis had advantages during biaxial deformation with the increase in the ratio of the stretching speed.

• The modeling analysis was proposed based on the tensile properties of single fibers and the structural characteristics of nanofibrous membranes. The applicability of the analytical model was verified by the comparison between modeling prediction and experimental data.

• Experimental stress was lower than predicted stress until large plastic deformation occurred because of structural imperfections, prestress, and the stretch-induced orientation in the membranes.

Y.Y. and H.Y. obtained the research funding; Y.Y. conceived the study, made charts, and drafted this manuscript; and H.L. and J.Z. performed the experiments and made suggestions on the revision of this manuscript.

This research was funded by the Program for Interdisciplinary Direction Team in the Zhongyuan University of Technology (Grant No. 32500032) and the Research Foundation for Young Doctor Teachers of the Zhongyuan University of Technology (Grant No. 34110557).

The authors declare no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.