Abstract
The mechanical properties of regenerated silk fibroin-based fibers have attracted much attention, but the related theoretical research and the mechanical model are insufficient. Thus, recycled regenerated silk fibroin (RRSF)/polyvinyl alcohol (PVA) blended as-spun fibers were taken as an example. Transformations of the apparent morphology, microstructure, and two-phase distribution of the as-spun fibers were explored at different tensile stages. The appropriate mechanical models were established with model formulas. The results showed that the RRSF component in the fiber gradually presented an axial fibril-like dispersion and split fragmentary areas under uniform stretching. Then, the PVA component undertook the external force, and ductile fracture finally occurred. At the micro level, the β-sheet content of the RRSF component reached a maximum of 54.7% at 50% elongation. The value of I⊥/I∥ reached the highest point (1.25) when stretched to 100%. The tensile process was divided into low-, negative-, and high-viscoelasticity deformation stages. Nonlinear spring or sticky pot elements were introduced to establish a two-element parallel mechanical model, and the fitting degree of the model formula was higher than 0.993.
Keywords
Silk fibers obtained from Bombyx mori silkworm cocoons possess outstanding mechanical properties, with an excellent equilibrium between strength and toughness 1 . Many researchers are still exploring biomimetic spinning methods and expect to obtain regenerated silk fibroin (RSF) fibers. However, humans have encountered tremendous difficulties in artificially producing 2 RSF fibers with excellent mechanical properties. The mechanical properties of fibers have always been the focus of researchers because the mechanical defects of RSF fibers impact the practicality of materials and limit their application and the feasibility of industrial production.
In 1933, Esselen et al. 3 were inspired by cellulose fiber spinning and successfully prepared RSF fibers by wet spinning. However, it was not until 1960 that the mechanical properties of the fibers were first recorded. Yazawa 4 dissolved natural silk in magnesium nitrate and produced RSF fibers with a strength of 2.205 cN/dtex and an elongation of 20–25%.However, at that time, the diameters of RSF fibers were generally thick (>100 µm), five times that of natural silk fibers.5,6 Subsequently, Wei7,8 used the dry spinning method to improve the mechanical properties of fibers by increasing the concentration of the spinning solution (20–50 wt.%).The prepared RSF fibers had diameters as low as 2 µm, their strength was close to that of natural silk, and their toughness was twice that of natural silk. To further increase the added value of RSF fibers, researchers are also adding other materials to blend them with RSF fibers.9–12 To date, silk fibroin has been combined with various biomass polymers,13,14 synthetic polymers,15,16 and nanoparticles17–19 to produce fiber materials. Adding these mixed materials improved the crystallinity of the blended fibers to a certain extent, enhanced the mechanical properties, and ensured a particular function of the fibers, such as tunable mechanical properties, anticoagulant activity, osteogenic induction, and self-warming properties. According to current studies, the mechanical properties of RSF fibers can be close2,20–24 or even superior 25 to those of natural silks in certain aspects when the degradation of the silk fibroin is not excessive during dissolution and when postdrawing treatment16,26,27 is sufficient. Although there have been various studies designed to improve the mechanical properties of RSF fibers,28,29 there are still few related theoretical discussions and deep theoretical investigations of RSF blended fibers. A two-component distribution, also called phase behavior, affects morphology and performance and significantly influences geometric structures. 30 However, effort has not been devoted to research on this aspect.
The considerable waste of silk resources has gained increasing attention. Recycling and utilization of precious silk fibroin resources from wasted silk can fully realize the high-value reuse of waste silk. However, reuse of waste silk fibroin is still limited to dense films,31,32 hydrogels, 33 nanofiber membranes, 34 and other forms.35,36 Such products have seen limited application due to their weak mechanical properties and can only be produced on a small scale, which is not beneficial for recycling waste silk resources. Recycled regenerated silk fibroin (RRSF) fiber is expected to achieve mass production and is of great significance in improving the value of recycled silk fibroin, reducing waste silk resources, and relieving environmental pressure. However, due to the high technical requirements for production and because preparation of RRSF fibers and their comprehensive mechanical properties still have certain gaps compared with those of natural silk fibers, there have been few reports on RRSF fibers, 27 improvement of their mechanical properties, or theoretical research.
Hence, the main aim of this paper is to demonstrate the importance of recycling waste silk resources and provide a theoretical study on the mechanical properties of RRSF blended fibers. We used waste silk fibers as a raw material with which to prepare RRSF/polyvinyl alcohol (PVA) blended as-spun fibers. Then, we investigated the tensile deformation mechanism and established a theoretical foundation. The main purpose was to scientifically formulate a postprocessing technology for RRSF/PVA blended fibers according to the mechanical requirements of various application scenarios. Further, we selected fibers at different tensile stages as samples for characterization of the apparent morphology and structure. Moreover, a change in the two-phase distribution of the fibers was also observed during tensile testing. Based on the above experiments, we divided the tensile curve into three stages according to the fiber deformation characteristics and constructed tensile mechanics models. Then, the model formula was used for the simulation to verify the theoretical model.
Materials and methods
Materials
Discarded silk quilts used for over 10 years were recycled as a raw material from which to extract silk fibroin. PVA (95% hydrolyzation) with a viscosity of 50–60 mPa · s was obtained from J&K Scientific Co., Ltd (Beijing, China). Ethanol was sold by Chinasun Specialty Products Co., Ltd (Jiangsu, China). The other chemical reagents, such as LiBr, Na2CO3, and polyethylene glycol 20000 (PEG20000), were of analytical grade and used without further purification.
Preparation of RRSF/PVA blended fibers
Figure 1 presents the preparation process of the RRSF/PVA spinning solution. The waste silk fibers from the discarded quilts were degummed by boiling them in a 0.2 wt.% Na2CO3 aqueous solution for 25 min. After being washed several times with deionized water, the waste silk fibroin fibers were air-dried overnight at room temperature. A 10% (w/v) solution of waste silk fibroin fibers in a 9.3 mol/L LiBr aqueous solution was prepared by continuous stirring for 4 h at 60°C. After dialysis against flowing deionized water by a cellulose semipermeable membrane (MWCO: 3.5 kDa) at 4°C for 3 days to remove salts, the RRSF aqueous solution was centrifuged at 6000 r/min for 10 min to filter impurities. A RRSF aqueous solution with a 20 wt.% concentration was condensed by dialysis against a 2 L PEG20000 solution with a concentration of 15% (w/v). PVA (20 wt.%) was dissolved in deionized water at 90°C and then allowed to stand overnight to defoam. The 20 wt.% blended aqueous solutions with a RRSF/PVA ratio (w/w) of 4/6 were obtained as a spinning dope.

Schematic illustration of the preparation procedure for the recycled regenerated silk fibroin (RRSF)/polyvinyl alcohol (PVA) blended solution.
The RRSF/PVA blended fibers were spun through homemade dry–wet spinning equipment, as shown in Figure 2. The blended solutions were extruded through needle tubing at a constant rate of 2 mL/h (16.0 mm/s) using a syringe pump and a 0.21-mm-diameter spinneret. Through the 20-mm air gap, the solution was placed into a 100% ethanol coagulation bath. The solution promptly solidified into a nearly cylindrical shape as it passed through the coagulation bath. Then, the blended fiber was successively collected on a winding device with a guide roller and reciprocating cross-transfer device. The length of the coagulation bath was 1150 mm, the winding speed of the collecting device was 14.7 rpm (19.2 mm/s), and the outer diameter of the winding drum was 25 mm. The process used for optimization of the spinning parameters is detailed in another paper. 37

Schematic illustration of dry–wet spinning equipment.
Characterization and evaluation
The stress–strain curves of the RRSF/PVA blended fibers were measured using a tensile tester (Instron 3365, Norwood, MA). The crosshead speed and the initial tensions were 20 mm/min and 0.1 cN, respectively. The mechanical indices were calculated by averaging 20 samples (length = 20 mm) randomly selected from among the tested fibers, as shown in Figure 3(a). The breaking fibers were retained to observe the fracture morphology. While observing the real-time stress–strain curves displayed on the data screen, we stopped the drafting procedure when the fiber was stretched to the yield point or when the elongation was 50%, 100%, 150%, or 200%. The fibers were removed from the testing machine and fixed on cardboard by applying double-sided tape at both ends to prevent shrinkage, as shown in Figure 3(b). Actual fiber stretching is shown in Figure 3(c). The structures and properties of the fibers obtained at different elongation stages were tested to fully explore the structural transformation.

(a) As-spun recycled regenerated silk fibroin/polyvinyl alcohol blended fibers. (b) Schematic diagram for fiber collection at constant elongation and (c) Stretched fibers.
Fourier transform infrared (FTIR) spectra were obtained with a Nicolet iS5 spectrophotometer (Thermal Fisher Scientific, USA) over the wavenumber range of 800–4000 cm−1. The resolution was 4 cm−1, with 16 scans obtained for each spectrum. The smoothed absorption FTIR spectra were corrected by background calibration and normalization.
A Raman spectrometer (LabRAM XploRA, HORIBA Scientific, France) with a wavelength of 532 nm and a He-Ne laser was used to obtain Raman spectra for the blended fibers aligned parallel or vertical to the long axis. All of the spectra were normalized with respect to the peak intensity at 1450 cm−1. The wavenumber range of 800–4000 cm−1 was measured six times to calculate the average.
A laser scanning confocal microscopy (LSCM) system (TCS SP5, Leica, Germany) was used to observe the intrinsic fluorescence of silk fibroin in the RRSF/PVA blended fibers. The excitation wavelength was 488 nm and the emission wavelength range was 500–550 nm. The objective lens magnification was adjusted, the scanning mode was switched, the scanning depth and scanning times were selected according to the sample image, layer scanning was performed, and a clear image was finally synthesized.
The surface images of RRSF/PVA blended fibers with different elongations, which were prepared by sputtering with a thin layer of platinum, were taken with a tabletop scanning electron microscope (TM3030, Hitachi, Tokyo, Japan) at a voltage of 15 k. The cross-sectional morphology was determined by immersing the fiber in liquid nitrogen and fragmenting it. The surface C, O, and N elemental distributions and their relative contents in the samples were analyzed by X-ray energy dispersion spectrometry (EDX).
Results and discussion
Tensile deformation mechanism of RRSF/PVA blended fibers
Static mechanical properties of RRSF/PVA blended fibers
The as-spun RRSF/PVA blended fibers prepared in this paper were taken as the research object. Figure 4 shows the stress–strain curve of the RRSF/PVA blended fiber. In the beginning, the curve exhibited a linear upward trend, which was in accordance with Hooke's law. The initial modulus of the fiber measured at this stage was 12.22 ± 2.55 cN/dtex. Then, the stress increased slowly and the curve entered the yield zone. After reaching the yield point, a considerable downward trend occurred, followed by consistent growth with slight fluctuations. Eventually, the fibers fractured with a breaking stress of 0.29 ± 0.04 cN/dtex and a breaking strain of 439 ± 129%.

Tensile curve for recycled regenerated silk fibroin/polyvinyl alcohol blended fibers (the red dots in the figure indicate where the samples were selected). (Color online only.)
To further explore the tensile deformation mechanism of the RRSF/PVA fibers, the fibers were stretched to 50%, 100%, 150%, and 200% for morphological and microstructural characterization. At the same time, fibers stretched to the yield point (strain of 3%) were also selected for testing so that the selected strain points could be spread throughout different tensile deformation characteristics.
Morphology of RRSF/PVA blended fibers at different stretching stages
To analyze the microstructural transformation of the fibers during tensile deformation, we selected the samples mentioned above for comparison with the as-spun fibers. Figure 5 displays the longitudinal and cross-sectional morphologies of fibers at different stretching stages. In the electron microscopy image of the longitudinal fiber, the surface grooves gradually became less obvious when the as-spun fiber was stretched to the yield point. Then, the fiber surface gradually became smooth, and the diameter of the fibers decreased until axial vertical cracks gradually appeared on the surface and were gradually aggravated with further tensile action. When stretched to 200%, the fibers exhibited a striking dehiscence phenomenon in the skin layer and revealed a relatively independent internal layer. The cross-sectional morphology of RRSF/PVA was nearly round, and there was a hole structure in the core layer. Overall, the cross-section of the blended fibers did not present great differences or obvious variations in characteristics under tensile loading.

Longitudinal apparent morphology of (a) as-spun fibers and fibers (b) at the yield point, with (c) 50%, (d) 100%, (e) 150%, and (f) 200% elongation (the upper right corners show the cross-sectional morphologies). The scale bars are 40 μm.
Fracture morphologies of RRSF/PVA blended fibers
The rupture morphologies of RRSF/PVA blended fibers can be observed in Figure 6(b), and they reveal the typical features of ductile fracture. The break started at the fiber surface and was followed by extended tearing perpendicular to the fiber axis. Ductile tearing continued across the rest of the link section of the fiber, and the break moved from the surface to the inner layer. The fracture morphology was not regular ladder-like. There were localized fibrillar subunits in which the internal voids separated into independent entities. These fibrillars experienced ductile extension and broke at different weak points. Overall, fiber failure occurred at approximately the symmetric portion of the initial fracture with fast crack growth.

X-ray energy dispersion spectrometry spectra of a fracture in the recycled regenerated silk fibroin/polyvinyl alcohol blended fibers; (a) skin part, (b) fracture morphology and (c) core part, where the inset tables show the corresponding atomic proportions and the weight proportions of the detected elements.
In detecting the surface elements in different parts of the fiber fracture, Figures 6(a) and (c) show that the content of N was higher in the outer part of the fiber, while there was almost no N in the inner part. N is the main component of RRSF and is nonexistent in PVA. N was be detected in the surface layer of the fiber but not in the core layer area. This indicated that the RRSF component could not withstand the external tension and was pulled off. After that, the PVA in the inner part was the main component bearing the external force.
Two-phase distribution of RRSF/PVA blended fibers
To fully explain the distribution of the two-phase structure during the tensile process, LSCM was performed. LSCM images showed that the blended fibers of RRSF exhibited green fluorescence upon excitation at specific wavelengths, while PVA remained black due to a lack of fluorescence. 9 Based on this special effect, the morphological distributions of RRSF and PVA in the blended fiber can be observed.
As shown in Figure 7, green fibril fluorescence was observed along with the longitudinal distribution in the formed blended fiber. As the tensile force was further strengthened, the boundary between the green and black bands became more obvious. With the strengthening of the tensile effect, the proportion of black areas increased gradually. In contrast, the green areas gradually broke up into discrete chunks. Accordingly, the RRSF distributed in the surfaces of the fibers could not bear a high external tensile force, and the PVA component played a main role in the late tensile stage. This result was consistent with the elemental analysis at the fracture morphology of the fibers in the Fracture morphologies of RRSF/PVA blended fibers section.

Confocal laser images of blended fibers: (a) as-spun fiber, fibers at (b) the yield point and at (c) 50%, (d) 100%, (e) 150%, and (f) 200% elongation.
Molecular structures of RRSF/PVA blended fibers at different stretching stages
The molecular structure is the focus when exploring fiber properties. Figure 8(a) shows FTIR spectra of the fibers at six tensile stages. The characteristic peaks of silk fibroin (red area) and PVA (blue area) are highlighted. There was no noteworthy discrepancy in the six spectral curves. Using the FTIR data, the secondary structure of the silk fibroin can be analyzed quantitatively. The β-sheet structure was the dominant crystalline structure in the fiber, which affected the strength of the fiber, while α-helices and random coils were important components of the amorphous region and affected the toughness of the fibers. Quantitative calculation of the secondary structure content can provide a deep understanding of the transformations occurring in the fiber internal structure at different tensile stages. The specific calculation method can be briefly summarized as follows: the amide I region (1600–1700 cm−1) was selected, smoothing was performed to remove the influence of the signal-to-noise ratio, and deconvolution was performed to weaken the overlaps between subpeaks. The second derivative was taken to determine the peak position corresponding to each secondary structure. Finally, fitting software was used for peak fitting to obtain the specific subpeak spectrum and peak area. The final secondary structure content was obtained from the area proportions. The specific method was as described previously in the literature. 38

Microstructural transformations of blended fibers at different tensile stages: (a) infrared spectra; (b) secondary structure content of the recycled regenerated silk fibroin; (c) Raman spectra generated with light polarized in different directions and (d) ratio of amide I peak intensities for vertically polarized light and parallel polarized light. (Color online only.)
Figure 8(b) shows the secondary structure content of RRSF at different tensile stages. The β-sheet content of the as-spun fibers reached 48.9%, followed by an abrupt drop as stretching was continued. Then, from 43.5% in the yield zone, the β-sheet content reached the highest value of 54.7% when stretched by 50%. A distinct drop occurred as the elongation continued to increase. There was a marginal difference in β-sheet content, 45.2% and 42.8%, respectively, when the elongation was 100% and 150%. However, when the elongation reached 200%, the β-sheet content reached a low value of 33.0%.
In addition, Raman spectroscopy is a common method used in molecular structure research. The different directions of polarized light during the test lead to different information on the vibration and rotation of molecules. Because the β-sheet structure is revealed by a stretching vibration of the C=O bond, the vibration direction is perpendicular to the axis of the molecular chain. Therefore, the change in molecular chain orientation can be determined by measuring the spectrum of the fiber with vertical and parallel polarized light and understood by comparing intensities of the amide I peak for β-sheets. 24 As shown in Figures 8(c) and (d), with stretching of the fiber, the extent of molecular chain orientation first fell, then rose, and eventually declined. Starting at 1.26, the value of I⊥/I∥ immediately decreased to 0.90. Then, the ratio reached a peak of 1.25 at 100% elongation, followed by 0.91 at 150%. Then, the degree of orientation remained low at approximately 0.9.
Overall, the as-spun RRSF/PVA blended fibers yielded under the minimal deformation condition. Although the apparent morphology did not change significantly, the crystallinity and molecular chain orientation of the RRSF component decreased obviously due to relaxation of the molecular chains in the yield region. After the yield point, the microstructure of the RRSF was improved by tensile action; hence, the crystallinity and molecular chain orientation were significantly enhanced. However, according to electron microscopy and confocal images of the blended fibers, the RRSF component could not withstand high external tension, and when stretched to approximately 50%, cracks began to appear on the fiber surface. Then, the RRSF molecules and aggregates were destroyed, and the relevant value decreased gradually. Subsequently, the fiber stretching curve rose gradually after passing through a plateau region, where the main chains of the PVA macromolecule were subjected to stress. Therefore, the postdraft ratios of the as-spun fibers can be controlled within this strengthening region to improve the mechanical properties.
Tensile mechanical model of RRSF/PVA fibers
The mechanical model for the tensile curve was constructed by the subsection method. Figure 9(a) shows the stretching curve of the RRSF/PVA blended fibers and enlarged views of the different stages. The tensile process was divided into three stages according to the characteristics of the tensile curves. These are as follows: (1) the low-viscoelasticity deformation stage: at the initial stage of the tensile curve, the stress grew rapidly with increasing strain, as shown in Figure 9(b); (2) the negative-viscoelasticity deformation stage: the curve showed that the stress decreased with increasing strain, as shown in Figure 9(c); (3) the high-viscoelasticity deformation stage: the curve showed a gentle and prolonged upward trend after the strain reached approximately 70%, as shown in Figure 9(d). The tensile mechanics model of the RRSF/PVA blended fibers was therefore established.

(a). Tensile curve for recycled regenerated silk fibroin/polyvinyl alcohol blended fibers. (b) Low-viscoelasticity deformation stage. (c) Negative-viscoelasticity deformation stage and (d) High-viscoelasticity deformation stage.
Tensile mechanical model in the low-viscoelasticity deformation stage
Low-viscoelasticity deformation is the stage before the yield strain (approximately 3%). The curve gradually changed from an elastic deformation region with a constant slope to a flat area with a decreasing slope until the slope was almost zero. As shown in Figure 10, the tensile curve was composed of a straight rise and an arc rise, which represented the elastic deformation stage and yield stage of the blended fibers, respectively. Spring and sticky pot elements can be used in parallel to construct the tensile mechanical model. According to the literature on pure RSF fibers and pure PVA fibers, RSF fibers
39
usually have a higher initial modulus. After a short strain, the fiber undergoes yield deformation. PVA fibers with high strength and high modulus often need to undergo posttensile treatment.40,41 Therefore, as-spun PVA fibers or fibers with a low draft usually have lower molecular orientation and crystallinity, leading to a lower initial modulus and longer breaking elongation. Therefore, the tensile curve for the blended fibers at this stage mainly represented the characteristic effect of the RRSF. The RRSF component rapidly shifted from the elastic region to the yield region, while the PVA component was still in the elastic deformation stage. Hence, a nonlinear spring element was introduced in this stage to represent the modulus change in this region under the action of the two components. In the low-viscoelasticity deformation stage, the modulus of spring E decreased with increasing strain. The relationship between the two is assumed to be as follows

Fitting effect of the low-viscoelasticity deformation stage.
Because the drawing was obtained under uniform conditions
Substituting Equations (1) and (3) into Equation (2), we obtain the following
After verification, when n0 = 2, the fitting effect was best, and the fitting results are shown in Table 1.
Fitting parameters for the low-viscoelasticity deformation stage
Tensile mechanical model for the negative-viscoelasticity deformation stage
Negative-viscoelasticity deformation is the stage between the yield point and the point at which the stress value stops dropping, as shown in Figure 11(a). The tensile curve first exhibited a striking drop and then a gently falling trend until the slope of the curve was zero again. According to the literature, there is no phenomenon for which the stress value decreases with increasing strain in the tensile curve of degummed silk after the yield stage,42,43 while regenerated silk fiber has this obvious feature.
44
However, the strain range (approximately 2%) and stress drop area (approximately 6%) experienced in this stage were small, unlike the negative-viscoelasticity deformation stage of the blended fibers in this paper. Based on the study and analysis of the two-phase morphology distribution for the blended fibers mentioned above, the main occurrence at this stage was continuous deformation of RRSF fibrils independently distributed in the PVA component. Based on the microstructural analysis of the blended fibers, the random coil structure in RRSF was stretched and oriented at this stage, and α-helices began to transform into β-sheets, which led to a gradual increase in β-sheet content and orientation structure. The external force was transferred from the amorphous region to the crystallized region, and the RRSF gradually entered the strengthening region, as shown in Figure 11(b). As the breaking elongation of RSF fibers is usually relatively low
45
(approximately 10%–50%), continuous fracture of RRSF fibrils may occur in the negative-viscoelasticity deformation stage, as shown in Figure 11(c), which results in a significant stress reduction. Due to the large elongation characteristics of PVA fibers, this component may still experience inelastic deformation or be in the yield state during this stage. Thus, the parallel model with the nonlinear dashpot and linear spring elements was introduced. In the stage of negative-viscoelasticity deformation, the modulus for the coefficient of viscosity η decreases with increasing strain. The relationship between the two is assumed to be as follows

(a) Fitting effect of the negative-viscoelasticity deformation stage. (b) Schematic diagram for recycled regenerated silk fibroin (RRSF) microstructural deformation and (c) Schematic diagram for the fracture of the fibrillary RRSF component.
Substituting Equations (5) and (3) into Equation (6), we obtain the following
The fitting results are shown in Table 2.
Fitting parameters for the negative-viscoelasticity deformation stage
Tensile mechanical model for the high-viscoelasticity deformation stage
High-viscoelasticity deformation is the stage in which the stress shows a proportional trend with strain again, as shown in Figure 12. The stretching curve rises almost linearly. With continuous stretching, the RRSF in the blended fibers first broke completely and formed blocky or fragmentary shapes, which resulted in the chapped surfaces of the blended fibers. Therefore, the β-sheet contents and the extent of molecular orientation decreased obviously. The PVA component assumed the external tensile action burden and displayed high elasticity at this stage. The Kelvin model was used for characterization by considering a spring element connected with a dashpot element, for which the elastic modulus and viscosity coefficient were constant

Fitting effect for the high-viscoelasticity deformation stage.
Substituting Equation (3) into Equation (8), we obtain the following
The fitting results are shown in Table 3.
Fitting parameters for the high-viscoelasticity deformation stage
Conclusion
The mechanical behavior of as-spun RRSF/PVA blended fibers was studied in this paper. We analyzed the deformation mechanisms of fibers by exploring the apparent morphologies, microstructures, and two-phase distributions at different tensile stages and discovering the fracture morphology characteristics and elemental distribution. The mechanical models were established based on comprehensive mechanical property analyses, and stress–strain relationship equations were established for different deformation stages. The following conclusions can be drawn.
Under uniform stretching, the RRSF component, which presented axial fibril-like dispersion, gradually split into fragmentary areas. After elongation by approximately 50%, the PVA component began to undertake the external force and finally underwent ductile fracture. At the micro level, the β-sheet content reached the highest value of 54.7% when stretched by 50% and the value of I⊥/I∥ achieved a peak of 1.25 at 100% elongation. The stretching process was divided into low-, negative-, and high-viscoelasticity deformation stages. A nonlinear spring or dashpot element was introduced. The modulus or viscosity coefficient of the element was adjusted according to the different microstructures of the fibers. A mathematical model representing parallel connection was established. The equations for the stress–strain curves of the different tensile processes were simulated as follows: σ(ε) = –k0
Footnotes
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Third-Priority Academic Program Development of Jiangsu Higher Education Institutions, Primary Research & Developement Plan of Jiangsu Province (grant number BE2019045), the Science and Technology Guidance Project of the China National Textile and Apparel Council (grant number 2020102), and the Qing Lan Project.
