Finite element simulations to evaluate deformation of polyester tubular braided structures

Abstract

The use of narrow tubular braided structures for biological tissue support has made it possible to produce highly flexible and robust soft tissue reinforcement structures. These attributes make the braids ideal in supporting ruptured and broken tissues during healing and regeneration. There have been continued efforts to improve the design in order to reinforce tissues while still maintaining their flexibility; this has been undertaken by exploring the deformation behavior of these structures. Mechanical modeling, which provides an in-depth understanding of the deformation mechanism of structures, plays an important role in designing structural changes in tubular braids. This paper reports the results of numerical and experimental investigations into the radial contraction and deformation mode of two types of tubular braided fabrics—single and double braided—subjected to uniaxial tensile loading under quasi-static conditions. Realistic geometrical structures were developed for mechanical modeling of tubular braids in terms of tensile loads, elongation, radial contraction and braid angle. The results indicated that there was a good match between experimental and simulated tensile behavior of the braided structures. It was established that the amount of braided yarns within the structure had the likelihood of influencing the radial contraction and braid angle in the braided structure under uniaxial tensile deformation. The results portrayed that braided structures would undergo large deformations at low loads. It was also established that there would be more structural stability as the yarns increased, evidenced by more loads in the double-braided structure as compared to the single-braided tubular structure.

Keywords

braided structures orthopedic stents quasi-static tubular braid finite element modeling

Tubular braided fabrics are circular structures formed by interlacing two sets of helical yarns in clockwise and anticlockwise directions. The helical assembly of tubular braids provides structural flexibility and mechanical integrity which makes them suitable for a wide array of applications. Subsequently, tubular braided structures have found applications in areas including medicine, aerospace, automobiles, train components, and reinforced hoses.¹ The unique net-like shape of tubular structures make it possible to design very specialized products for various applications.²

In the medical industry, tubular braided structures are used for their compressive properties as stents or for their tensile attributes in orthopedics,³ most notably in the repair of fractured bones. The main reason for the wide range of applications of tubular braids in the medical field is their unique structural response: under compressive loading they often “balloon”, while when subjected to tensile loading they undergo radial contraction.^4,5 This has been attributed to the fact that braids are structural assemblies that exhibit a special mode of deformation when axially extended or compressed. The extended structure is capable of substantial accommodation of displacement, since the initially inclined yarns are free to pivot to a position more parallel to the direction of the stress. As the tubular braid extends under uniaxial tensile loading, its diameter decreases until the structure reaches a point of maximum packing density, called the tensile or extensive jamming point. Equally, a braid forced to contract in length will have its structural yarns re-aligning more perpendicular to the direction of compressive stress, increasing its diameter up to a compressive jam point. The extensive and compressive jamming are dependent on the deformation of the structure due to axial loading. Numerical simulation of tensile loading of braided structures can be a valuable tool in analyzing the deformation behavior of the structure. The aim is to obtain a realistic link between the deformation of the structure and the axial contraction behavior of the braid in terms of the construction parameters of the braid: angle and crimp; and in terms of the physical properties of the used material: elasticity, bending stiffness, and friction. Hence, there has been increased interest in investigating the mechanical behavior of braids using realistic three-dimensional modeling and analysis techniques. Recently, there have been studies on tubular braided structures that have mainly focused on three-dimensional modeling of the braid structure,⁶ while the dynamics of the structural deformations have focused on the analysis of the jamming of the yarns due to tensile or compressive loads applied to the structure.^7,8 The use of finite element modeling (FEM) has been adopted in studies on tubular braids. Most researchers have concentrated on studies focusing on self-expanding stents. While concentrating on mechanical aspects of braid deployment using FEM simulations, it was established that stents can be comfortably implanted while minimizing the over pressure on to the vessel walls, due to the thermo-responsive shape memory behavior of the material used, for example, the shape memory metal alloy, nitinol.^9,10 There have also been studies on braids focusing on the effects of compressive loading and the structural deformation in catheter stents.^11–13 However, the investigation of the mechanical deformations in a double-braided structure alongside a single-braided structure using finite element analysis (FEA) have been missing in these studies, especially considering the fact that more braided yarns within the structure affect the mechanical behavior of the structure. In this study, therefore, mechanical modeling for tubular braids is performed to investigate the tensile deformation in terms of radial contraction and changes in braid angle between the constituent yarns. In this context, the objective of this study was thus to evaluate the results of FEM simulations applied to a tubular braided structure proposed for ligament tissue engineering. This was undertaken by comparing the evolution of the radial contraction of braid geometries and braid angles within the structure, computed from FEM simulations and obtained from experimental data acquired during a tensile test. In this regard, this study presents realistic geometrical models and simulation of tubular braided structure deformation under uniaxial quasi-static loading with the aim of studying the influence of the type of braided structure on radial contraction and braid angle. The structures investigated were categorized into two, namely single-braided and double-braided structures. Tensile loading was applied to the structure, which translates into the structure undergoing radial contraction, as illustrated in Figure 1. The figure shows clockwise and anticlockwise braided yarns before and after tensile strain with the displacement of the end nodes constrained, resulting in radial contraction behavior of the structure.

Quasi-static tensile testing

Materials

The tubular braids that were used as samples for the uniaxial tensile tests were fabricated from polyester monofilaments using a 20-head braiding machine with one filament yarn and two filament yarns, using 0.22 mm diameter polyester monofilaments on an 8-mm mandrel. The experimental tests were based on five samples made from single-braided yarns and five samples made from double-braided yarns, whose mean results were computed and used to validate the finite element models. The two categories of samples were made up of 20 and 40 yarns, respectively, while the braid angle between the filament yarns was 45°. The material characterization was done by determining the mechanical properties of the monofilaments on a STATIMAT® tensile tester using a load cell of 1 kN at a test speed of 100 mm/min. The Young's modulus was computed from tests on five polyester monofilament yarns (the yarns tested were the same as those used for the tubular braid samples); the mean for the Young's was established to be 3500 ± 32.79 MPa. The monofilament yarns had a linear density of 10 tex. The density and Poisson's ratio for polyester were quoted from literature, where the values of the poisson's ratio for the polyester monofilament is recorded to be with the range of 0.2–0.3.¹⁴

Methods

The quasi-static tensile test was performed on an Instron® tensile machine equipped with a 500-N load cell at a test speed of 5 mm/min. The tubular braid samples were placed in the tensile machine, secured at both ends, and stretched in the uniaxial direction. The load and elongation were measured and recorded by the tensile machine. Figure 2 shows the test setup for the tensile testing of the tubular braided samples. The samples were tested within a gauge length of 20 mm; this was also considered as the original length for the test samples. The samples were induced with a uniaxial tensile loading up to a maximum extension of 4 mm, which translated into an axial strain limit of 0.2. This was established as the elastic limit of the narrow braided structures considered in this paper.

Finite element modeling

The tubular braids used in this work are modeled as helical braid structures with mutually interlacing yarns in clockwise and anticlockwise directions. A boundary condition was assigned to the nodes at the opposite ends of the structure to constrain radial movement of the end nodes. In simulating uniaxial tensile loading and radial contraction mechanisms under quasi-static conditions, the FEA software ABAQUS® was selected. The ABAQUS/Explicit approach was used for the nonlinear FEM analysis, as it is most suited for problems that involve complex contacts between finite elements. The material model considered for FEM was an isotropic linear elastic material with the experimentally determined properties for the polyester monofilament. The creation of the geometrical structures used in the simulation of the mechanical behavior of tubular braided structures was done with a python programming interface—PyFormex®. The python scripts developed allowed the construction of a realistic braided structure from a planar base module of two crossing braided filament yarns. The resulting geometrical structures also accounted for the crimp between the interlacing yarns at the crossing points. The geometrical structures were imported into ABAQUS/Explicit in order to implement the material properties and compute their mechanical response to tensile loading conditions. In the discretization of the filament yarns in the braid model, beam elements (B31) were selected as the most suitable type of element. A mesh size of 0.25 mm was used. These elements offer flexibility associated with transverse shear deformation between the beam's axis and its cross-section directions.

Contact formulation

The general contact procedure of the ABAQUS/Explicit solver was used to define contacts in the assembly. The modeling of friction between the braided yarns was based on the fact that for two dry solid surfaces sliding against one another, the magnitude of the kinetic friction exerted through the surface is independent of the speed of the slipping of the surfaces against each other. Therefore, the force required to initiate sliding is proportional to the normal force acting on the plane of contact.¹⁵ The coefficient of friction between polyester filaments yarns was found to be 0.45.¹⁶

FEM simulation of quasi-static tensile loading

In the numerical simulation of quasi-static tensile loading of the tubular braided structure, the uniaxial tensile loading of the structure was modeled. This was done by applying a uniform displacement on selected end nodes of each braided yarn with constrained radial and longitudinal displacements. The tensile loading was applied using a smooth step amplitude in ABAQUS/Explicit to emulate the experimental tensile tests. The nodes at the opposite end of the structure were constrained in tangential, axial and radial displacements: fixed (X, Y, Z) coordinates. The action of the uniform displacement in the braided structure translates into load, uniaxial strain, and radial contraction of the structure and change in braid angle within the structure. Simulation results were computed from the sum of the reaction forces, elongation, radial displacement, and braid angle. The simulation results were compared to experimental test data.

Comparison of experimental and numerical results

The results established from the experimental tests and from the two types of simulations were first compared in terms of tensile deformation, by evaluating their load–elongation behavior; this was followed by a comparison of braid angle change with elongation, and finally radial contraction was also plotted as a function of the elongation. The radial contraction was formulated by considering mesh nodes n_i, where i = 1, … , N; the geometrical structure was deformed as follows

n_{i} : [x_{i, A}, y_{i, A}, z_{i, A}] \to [x_{i, B}, y_{i, B}, z_{i, B}]

where [x_i,A, y_i,A, z_i,A] are the initial global coordinates of the structure and [x_i,B, y_i,B, z_i,B] are the global coordinates after the structure has been deformed. The maximum contraction ΔD can then be estimated from the coordinates of each node before and after deformation as

Δ D = 2 [{max}_{\forall i} (\sqrt{(x_{i, A}^{2} + y_{i, A}^{2}) - (x_{i, B}^{2} + y_{i, B}^{2})})]

Results and discussion

Figure 3 shows the deformed tubular braids during the experiment and the simulation. They indicate that the overall deformed shape of the geometric model and test sample was quite similar.

Experimental and simulated tensile responses from the two categories of braided structures in terms of load and elongation are represented in Figure 4. The figure shows that the experimental and FEM data are in good agreement until the point where the elongation of the structure is 1.17 of the original length. There is a closer fit between the experimental and FEM data for the single-braided structure. Deviations between the curves start at the point where the experimental tests samples indicated signs of yielding, which is around an elongation of 1.18 of the original length.

The FEM structure did not show the yielding behavior because the structures are modeled using isotropic elastic material properties, hence the polyester filaments within the structure show behavior similar to the viscoelasticity of polyester in which they undergo elongation recovery after a constant rate of extension.¹⁷ In the case of the single-braided structure, the loads increased with an increase in elongation, showing close correspondence up to an elongation of 1.18 of the original length, after which a maximum value of 11 N is reached at an elongation of 1.2 of the original length for the experimental data and 12 N for the simulation data. The double-braided structure showed that the load increased with the increase in elongation. Good agreement is obtained up to an elongation of 1.18 of the original length after which there is a maximum load value of 27 N for the experimental data and 54 N for the simulation results. The experimental results established that the double-braided structure had the ability to withstand more load (22 N) before showing signs of yielding compared to the single-braided structure (8 N). As the model did not include plasticity in the material model, the simulations predicted realistic tensile properties up to an elongation of 1.17 of the original length. This was illustrated by the deformed images in Figure 3 and the close convergence of the data shown in Figure 4. These results are consistent with findings in literature,^18–20 which suggest that narrow tubular braided structures and their constituent braided yarns often undergo substantial deformations at low loads due to their high sensitivity to tensile tension. This is also confirmed by the trend of the curves in Figure 4, where the structural loads were quite low at the initial stages of the tensile tensions but produced considerably high elongations for both types of braids. This explains the profile of the curves for the elongation below 1.1 of the original length for both results, after which there was a nonlinear increase between the load and elongation. This is due to the fact that the braided yarns within the structure at the beginning of the tensile loading process were simply pivoting on each other within the structure as they are displaced in longitudinal direction while the structure extends. When the loads in the structures are increased further the yarns reached a point where they started touching each other, which translated to a sliding behavior. This leads to an emergence of significant sliding friction within the structure due to more yarns rubbing against each other, which translates into a higher rate of increase of the load with increasing elongation; hence, the nonlinear profile in the curves after the flat regions. This is in agreement with Jedwarb and Clerc,⁷ who established that when a single-braided yarn is isolated, it would exhibit a slightly different geometry from that which it exhibits when in the braided structure. That is, the yarns are not totally independent, but exert a frictional force on each other that contributes to the load in the structure. This was also supported by Stefano et al.,²¹ who established that braided structures undergo jamming while under tensile loading and that rubbing of the jammed yarns would give extra strength to the braided structure. They also established that braids with smaller braid angles had a stiffer response after jamming occurred with respect to those having larger braid angles. This can be confirmed by the deformation images in Figure 3 and the results plotted in Figure 4 in which tensile jamming within the structure could be attributed to increased friction due to rubbing yarns and tensile loads in the structures. This explains the movement of yarns close to each other in Figure 3 and also the increase in load with elongation being higher after an elongation of 1.1 of the original length in Figure 4. The structural deformation of the braid due to the tensile loading indicated by the results in Figure 4 led to an evolution of changes in the braid angle with the elongation of the braided structure, which translated into radial contraction of the braids as a result of the yarns being displaced towards each other as the spaces between them decreased. This is illustrated by Figures 5 and 6.

Braid angle with elongation

The braid angle is the angle at which the clockwise set of yarns crosses the anticlockwise yarns. It was established that with increasing elongation, the angle between the yarns decreased, as shown in Figure 5. The correlation between the rate of change of the braid angle with elongation for experimental and simulation results is computed with the gradient of the regression line at a 95% confidence level. It was also established that in the single braid the experimental results underwent more change in braid angle (slope = –138.62, R² = 0.969, p = 0.05) compared to the simulation results (slope = –117.72, R² = 0.983, p = 0.05). The experimental and simulation results for the double braid established that the rate at which the braid angle changes was greater in the simulation results (slope = –119.18, R² = 0.983, p = 0.05) as compared to the experimental results (slope = –117.25, R² = 0.987, p = 0.05).

The curves in Figure 5 show that the braid angle decreases with increasing elongation of the braided structures in a linear trend. Good agreement between the simulation and the experiment was obtained. It was observed that the single-braided structure had a larger change in braid angle than the double-braided structure. This is attributed to the fact that during elongation the single-braided structure has more space between the yarns, and thus, less pivoting and sliding frictional resistance. Hence, the yarns will be displaced more quickly towards each other, which will reduce the angle between them. In the case of the double-braided structure, the increase in the number of yarns per unit area implies that the spaces between them will be reduced and there will be more resistance to movement due to interactions between the yarns. Hence, the rate at which they move towards each other becomes less as compared to the single braid yarns.

Radial contraction with elongation

The evolution of the structural radial contraction with the elongation for the tensile test and FEM simulations for the two types of geometries is represented in Figure 6. The figure shows the percentage contraction in the braids as a function of elongation for the experimental and FEM data. The trend is linear and quite similar for the two structures. The rate at which the braided structure underwent radial contraction with elongation for experimental and simulation results was evaluated from the gradient of the regression line at a 95% confidence level.

For the single-braided structure, the experimental specimen was undergoing fast contraction during elongation (slope = 311.82, R² = 0.940, p = 0.05) compared to the simulation model (slope = 234.58, R² =0.997, p = 0.05). In the double braid the experimental sample showed faster change in the braid angle (slope = 245.88, R² = 0.955, p = 0.05) as compared to the simulation model (slope = 232.22, R² = 0.995, p = 0.05). Furthermore, the single-braided structure was contracting faster than the double-braided structure. This is attributed to the sensitivity of the narrow braids causing deformation at low tensile loading.

Quasi-static sensitivity analysis

The extent to which the explicit simulation follows quasi-static conditions was investigated by considering the internal energy and kinetic energy. As shown in Figures 7 and 8, the energy results indicate reliable simulation energy values for the models of both braided structures, as illustrated by the low kinetic energy as compared to the internal energy for most parts of the simulations. It was established that for both simulations the kinetic energy was lower for most parts of the simulations than the internal energy within the acceptable stable level of below 5% of the internal energy, as recommend by ABAQUS FEA.²²

Conclusion

Single- and double-braided structures were fabricated for potential medical applications. The structures were analyzed both experimentally and numerically in order to investigate the relationship among the polymeric braided structural parameters that are critical in end-use applications. In this regard, finite element models were developed using python programming language, which has the ability to compute realistic tubular braided structures. The computation of the parameters of the braided structures was done in terms of the evolution of radial contraction of the structures and also in terms of angular changes between the braid yarns under tensile loading at the different levels of elongation. It was established that the simulation of the tensile loads in the braided structure due to the elongation had close convergence with the experimental results, which suggested the finite element models could be reliable in predicting the deformation of a tubular braided structure under uniaxial tension. The results from the experimental tests and simulations both accounted for the evolution of the resistance between yarns at crossing points in the single-braided and double-braided structures established by the load–elongation profiles of the tubular braids. The experimental results and simulation models also enabled prediction of the trends for contraction and braid angle when moving from single to double braid; it was established that there was a deviation of the rate at which the structural parameters change for the two types of braids in terms of radial contraction and braid angle. It was also possible to ascertain that the explicit dynamic simulations implemented in this study followed quasi-static conditions, which is a requirement for credible FEM results; this suggests that FEM analysis using ABAQUS/Explicit is able to reproduce satisfactorily the quasi-static load–deformation of tubular braided structures. It was evident that there was a significant accuracy of the FEM simulations to reproduce results on tensile loading, elongation, radial contraction, and changes in braid angle of the braided structures during the tensile test. It could be suggested that given the complexity of these structures, there was approximate agreement between these quantities extracted from experimental test data and simulation results and their evolution during tensile tests. However, the predictability of the simulation models could be improved by accounting for the plasticity of the braided structures in the finite element models.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the funding from the VLIR-UOS textile project.

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