Mechanical analysis of fiber-reinforced composite running-specific prosthesis using a matrix of TPU or PLA

Abstract

Running-specific prostheses (RSPs) allow amputees to run competitively and recreationally, offering stability, flexibility, and energy return. Despite advancements, RSP can still improve in materials and manufacturing. This study investigated the mechanical behavior of running-specific prosthesis (RSP) manufactured with a composite of carbon fiber and matrices of thermoplastic polyurethane (TPU) or polylactic acid (PLA). The research objectives were to experimentally determine the most suitable matrix (TPU or PLA) according to mechanical behavior and for the selected matrix to compute the mechanical response for using 8, 10, and 12 carbon fiber layers and a 6, 8, and 10 mm matrix thickness. The matrices were 3D printed and the carbon fiber layers were incorporated by a hand lay-up process. The applied load corresponded to a 60 kg user. Experimental measurements identified that the TPU matrix presents low stiffness compared to the PLA matrix. Numerical analysis indicated a delamination failure in the RSP with 8 layers, whereas with 10 and 12 layers, no failure was observed. No evidence of failure was exhibited for the TPU matrix. Because of its compression capacity, load resistance, and the absence of damage, the TPU is suitable for its use in the manufacturing of RSP.

Keywords

Finite Element Method polylactic acid running prosthesis thermoplastic polyurethane

Introduction

People who have had amputations of lower limbs are limited to their physical activities. This sedentary lifestyle can contribute to weight gain and obesity, which can lead to other health problems like diabetes and high blood pressure, among others. The disuse of certain muscle groups, due to the lack of physical activity, can lead to muscle atrophy, reducing human body mobility. Due to these factors, amputees may face mental health issues such as depression and anxiety.^1,2 An option to replace the loss of lower limbs is running-specific prosthesis (RSP), which can reintegrate amputees into physical activities.

Prosthetic sports feet serve for different sports activities, and according to Poonsiri et al.,³ the performance of prosthetic sports feet in any sport is the most important characteristic for the users. In running sports, an RSP contributes to improving the running performance through enhancing step frequency, ground-contact and kinetics. An RSP functions like a spring, in which the compression depends on the level of the stiffness. The RSP is compressed during both the initial contact between the heel and the ground and the mid-stance stages, and then uncompressed during takeoff.^4–6 The energy stored during compression is released, converting into forward propulsion for the runner. As a result, the amputee requires less effort to achieve a faster running gait, almost identical to a biological leg.⁷ According to^8,9 the magnitude of both stored and returned mechanical energy is inversely proportional to the prosthesis stiffness and depends on the magnitude and orientation of the force applied to the leg, therefore, a RSP with lower stiffness will store more energy. Considering both the stiffness and the resistance to compression,¹⁰ it is assumed that the greater the compression, the greater the forward propulsion of the user.

Several investigations have been carried out to understand the behavior of RSP stiffness. Beck et al.¹¹ conducted research on the evaluation of the performance of commercial RSP. They tested four different models and evaluated their mechanical properties. They found that for the case of C-shaped RSP, when the user is running at 3 and 6 m/s, the peak of the reaction forces occurs at angles of 15.1 and 10°, respectively, and the load is 2.5 and 2.7 times the body weight of the RSP user. Also, the study concluded that when RSP users change prosthetics, adjustments in height and sagittal plane alignment cause changes in stiffness, which impact the performance of the RSP. Beck et al. investigated the effects of stiffness and height of different RSP models on the metabolic cost of running.¹² They conducted tests on five individuals with bilateral transtibial amputations. They found that stiffness and the RSP shape or model influence the metabolic cost of running, but not the height. The effect of the angle of alignment of a J-shaped RSP was studied by Groothuis and Houdijk to identify its effect on the stiffness and the gait pattern during running.¹³ They conducted the tests under step frequency imposed condition and free condition at alignment angles of 0, 5, 10, and 15°. For both step frequency imposed and free condition, their results show that when the alignment angle increases, the RSP stiffness decreases. The RSP users require gait adaptation because the actual RSP stiffness is dependent on the alignment angle during running. Beck et al.¹⁴ studied the effect of running speed, height, and prosthetic stiffness on the biomechanics of athletes with bilateral transtibial amputations. In a set of trials, they tested running speed ranging from 3 to 9 m/s, with increments of 1 m/s. Their conclusion suggests that stiffness, rather than height, has a greater impact on distance running performance compared to sprinting performance. Prosthetic stiffness had a mitigating effect on biomechanics during faster running, but height had no impact.

Composite materials are widely used in the fabrication of RSP, with carbon fiber being the most used material. The selection of carbon fiber is primarily due to its potential to store and release energy during a running sprint.^15–17 Additionally, the use of carbon fiber increases flexural strength, which leads to a stiffness reduction.¹⁸ Ismail et al.¹⁵ evaluated the mechanical properties of carbon matrix composites for a RSP. They tested carbon fiber composites on matrices of four resins: epoxy bakelite, casting resin, PMMA (polymethyl methacrylate) Orthocryl, and polyester. Their findings showed that carbon fiber composites with Orthocryl PMMA matrix exhibited superior mechanical properties for RSP, including high bending stress, the highest stress, and the lowest impact energy. Sehar et al.¹⁹ investigated the effect of the number of layers on the mechanical strength of carbon fiber composites for prosthetic applications. The authors carried out tensile tests to evaluate the effect of the number of layers on the carbon fiber specimens according to the ASTM D3039 standard. They applied the hand lay-up method to manufacture the specimens. These authors concluded that the manufacturing process is effective in the construction of the prostheses and also an increase of the mechanical strength was observed when more carbon fiber layers were used. In Ref. 20,²⁰ a new design of laminated-type prosthetic foot was proposed. They used rubber as the matrix material to improve the bending of the prosthesis. The authors concluded that the design performance is like the commercial LP Vari-flex model. Besides,²⁰ other authors have tested different types of polymer matrix materials in laminated composites as well as various manufacturing methods.^20–23 These modifications had been made to design prostheses that are durable, functional and lightweight.^24–26 The thermoplastic polyurethane (TPU) is a polymer whose properties include high mechanical resistance and rubber-like flexibility, making it ideal for RSP designs.²⁷

Another challenge to solve is reducing the cost of commercial prostheses because the high cost makes them unaffordable for most potential users. To address this issue, it is necessary to investigate and evaluate new composite alternatives and propose more cost-effective manufacturing processes.^28,29

In this work, a simple manufacturing process is proposed to construct a carbon fiber polymer matrix composite RSP. Two different polymers were compared experimentally to determine the best RSP performance based on stiffness and mechanical strength: polylactic acid (PLA) and (TPU). Matrices were manufactured by fused deposition modeling and the composite structure was by hand lay-up process. The materials were chosen because they can be processed using a standard 3D printer. The properties of both materials are different, with TPU being more flexible than PLA. However, results show how these properties were modified due to the addition of carbon fiber. After identifying the best material, the effect of increasing both the number of layers and the matrix thickness was analyzed using the finite element method. The calculated responses from the numerical study include the Tsai-Wu failure index, $F I$ , vertical compression, $C_{v}$ , and matrix stresses, $σ_{M}$ . For finite element simulations, the TPU is considered hyperelastic; therefore, experimental tests were carried out to define the model that numerically better fit the experimental results. The RSP was designed specifically for a particular age group, such as adolescents.

Experimental and numerical methodology

This work was carried out in four stages: prosthesis construction, mechanical tests, hyperelastic modeling, and finite element computations. During the first stage, using the hand lay-up method, two RSP laminated composites were manufactured. The layers of carbon fiber and matrices of thermoplastic polyurethane (TPU) or polylactic acid (PLA) made up the composite material. In the second stage, measurements were performed to evaluate $C_{v}$ and the ultimate strength of both RSP models. The hyperelastic model used in the FEM computations was validated in the third stage. Finally, in the fourth stage, the entire FEM model was validated using experimental measurements, then two design parameters of the RSP, number of carbon fiber layers, and TPU matrix thickness were analyzed. The workflow is presented in Figure 1. The methodology applied in these stages is described in the following sections.

Figure 1.

Workflow stages of experimental and numerical studies.

Prosthesis construction

The shape of the RSP was based on the Össur Flex-Run Junior Model. To approximate the curved contour, a side view image was imported into CAD software using the prosthesis height as a reference length. The contour was traced over the image in the sketch to replicate its shape and dimensions. The height of the matrix, $h_{M} = 220$ mm was determined to fit its size to the 3D printer, whereas the width, $w$ , was 50 mm and its thickness, $t$ , was 8 mm (Figure 2).

Figure 2.

RSP matrix dimensions with values: $h_{M}$ = 200 mm, $w$ = 50 mm, and $t$ = 8 mm.

The process continued with printing the RSP matrices made of PLA and TPU. The 3D printer used was a Creality Ender 3 V2 with print dimensions of 220 × 220 × 250 mm and a nozzle diameter of 0.4 mm. The materials were PLA and Ninjaflex^® TPU, both filaments have a diameter of 1.75 mm. The print settings for PLA are well known since it is one of the most used materials, and its properties facilitate printing. In the case of TPU material, even though the supplier settings were applied, two issues were encountered during the print process: cracking and warping. For such reasons, several tests were carried out to identify the proper settings and to avoid the occurrence of such defects. In Table 1, the settings used to print the defect-free RSP matrices are shown, whereas in Figure 3, the printed RSP matrices are presented.

Table 1.

Settings used for 3D printing.

	PLA	TPU
Print temperature (°C)	215	240
Bed temperature (°C)	80	60
Print speed (mm/s)	30	40
Infill density (%)	100	100
Layer height (mm)	0.2	0.2
Travel speed (mm/s)	60	80

Figure 3.

3D printed RSP matrices. (a) PLA filament and (b) TPU filament.

The next step was the composite manufacturing using the hand lay-up, involving manual application of fiber layers to form the composite structure.³⁰^,³¹ The fiber used was a bi-directional carbon fiber fabric with a 2 × 2 twill patterns, 6k tow with a thickness of 0.43 mm from Composite Envisions™. A resin mixture was prepared as a blend of Thin Epoxy Resin 1159A™ and Epoxy Hardener Slow Cure 1160B™ at a 2:1 ratio. The surface of the RSP matrix was smoothly sanded and then cleaned to ensure a proper adhesion. Using a brush, the whole internal RSP surface was coated with the resin mixture after which the first layer of carbon fiber was laid on one side, and this process was done in alternating steps, applying the resin mixture followed by a layer of carbon fiber, continuing this sequence until 10 layers of carbon fiber were completed. The curing time was 36 hours at room temperature. The same process was repeated on the external side of the RSP. After the curing time of the remaining side, the excess fiber was trimmed. By following this procedure, the two composite RSPs (PLA or TPU matrix) were manufactured. The total height $h$ of the RSP changes due to the addition of layers, being $h = 228.6$ mm, which resulted from the sum of the $h_{M} + 0.43 N$ where N is the number of total layers and 0.43 mm is the thickness of carbon fiber. Both constructed prostheses are presented in Figure 4.

Figure 4.

RSP manufactured using the hand lay-up method. (a) PLA and (b) TPU.

Mechanical test

The mechanical tests were conducted to determine which of the two materials exhibits better performance, in terms of $C_{v}$ and mechanical strength, that contributes to the development of improved RSP. Additionally, their ultimate strength and maximum vertical compression were measured. The tests were conducted using a Gunt WP 310 unit with a maximum force of 50 kN. Figure 5 illustrates the placement of the RSP models in the test machine. The weight in standing conditions was set as 60 kg, considering that the potential users will be young individuals with an age of up to 18 years old. For dynamic or running conditions, this magnitude increases by 2.5 times the standing weight.¹¹ Considering this fact, the maximum force applied to characterize dynamic conditions was set at $F_{D} = 1500 N$ . This force was applied to compare $C_{v}$ between both RSP models. The ultimate strength and maximum vertical compression were identified in another test in which the force was applied until the failure occurred. Due to limitations of experimental facilities, the load was applied only at 0°.

Figure 5.

Compression testing of the RSP model.

Hyperelastic modeling

Because TPU is considered a hyperelastic material,^32,33 before conducting FEM simulations, it was necessary to model its elastic response to ensure an accurate simulation capable of predicting the behavior of TPU. For this purpose, in the third stage, uniaxial tests were carried out to obtain stress-strain data, which were subsequently used to define the hyperelastic model that best approximates the experimental results. The specimen dimensions,³³ shown in Figure 6, were defined based on the DIN EN ISO 527-1 type 1BA standard.

Figure 6.

Specimen dimensions: $l_{1} =$ 30 mm, $l_{2} =$ 58 mm, $l_{3} =$ 75 mm, $b_{1} = 5$ 0 mm, $b_{2} =$ 10 mm, $L_{0} =$ 25 mm, and $L = l_{2} + 2$ mm.

The five specimens, shown in Figure 7(a), were printed using the same settings employed in the printing of the TPU matrix (see Table 1). The measurements were conducted in accordance with the DIN EN ISO 527-1 standard. The testing machine was a Sintec 20/G with a load capacity of 100 kN. The specimen mounted during the test can be observed in Figure 7(b).

Figure 7.

Uniaxial tests based on DIN EN ISO 527-1 standard. (a) Specimens and (b) specimen during testing.

Finite element calculations

The effects of the number of carbon fiber layers and matrix thickness were investigated in the fourth stage using FEM. Three different matrix thicknesses were analyzed (8, 10, and 12 mm) and for each thickness, the number of layers per side varied from 8, 10, and 12 layers. Three domains were modeled, two of them correspond to the external and internal fiber layers, and the third domain corresponds to the hyperelastic material matrix (Figure 8). The TPU has a density of 1040 kg/m³ and the rest of its mechanical properties were defined based on the selection of the hyperelastic model, whereas the mechanical properties of the composite are presented in Table 2.

Figure 8.

Geometrical model of RSP.

Table 2.

Mechanical properties of composite.

Property	Units	Value
$E_{x}$	MPa	59160
$E_{y}$	MPa	59160
$E_{z}$	MPa	7500
$υ_{x y}$	------	0.04
$υ_{y z}$	------	0.3
$υ_{x z}$	------	0.3
$ρ$	kg/m³	1451
$X_{1 T}$	MPa	750
$Y_{1 c}$	MPa	650
$X_{2 T}$	MPa	60
$Y_{2 C}$	MPa	185
$S_{12}$	MPa	100
$S_{23}$	MPa	65
$S_{13}$	MPa	65

The load was applied at 600 and 1500 N to address standing and dynamic conditions, respectively, considering that the weight of prospective users is 60 kg.³⁴ The load was applied at two angles (Figure 9). According to Beck,¹¹ 15° and 10° correspond to a running speed of 3 and 6 m/s, respectively, for a C-shape RSP design. The domain was fixed at the lower surface of the base, which was in contact with the ground.

Figure 9.

Force and constraints applied to the RSP.

To determine the effects of the number of layers and matrix thickness, three major results were obtained from FEM calculations: the first one was the stresses in the TPU matrix, $σ_{M}$ ; the second one was the vertical compression of the entire RSP, $C_{v}$ ; and the third one was to evaluate the structural integrity of the composite material by the Tsai-Wu failure index, $F I$ . This³⁵ criterion was employed in other studies focused on lower limb prosthetics and laminates.^36,37 This $F I$ establishes that a composite will fail according to the following expression:

F I = F_{1} σ_{1} + F_{11} σ_{1}^{2} + F_{2} σ_{2} + F_{22} σ_{2}^{2} + {2 F}_{12} σ_{1} σ_{2} + F_{66} τ_{12}^{2}; F I < 1

(1)

where

σ_{1}

σ_{2}

, and

τ_{12}^{2}

are the normal stresses,

S

is the in-plane shear strength, and the tensile and compressive strength parameters were calculated using³⁸:

F_{1} = \frac{1}{X_{1 T}} - \frac{1}{Y_{1 c}}

(2)

F_{2} = \frac{1}{X_{2 T}} - \frac{1}{Y_{2 c}}

(3)

F_{11} = \frac{1}{X_{1 T} Y_{1 c}}

(4)

F_{22} = \frac{1}{X_{2 T} Y_{2 c}}

(5)

F_{66} = \frac{1}{S_{12}^{2}}

(6)

F_{12} = - \frac{1}{2} \sqrt{F_{11} F_{22}}

(7)

The finite element model was constructed using elements of a size of 4 mm based on a convergence study where $F I$ was evaluated. The compared element sizes were 4, 5, and 6 mm, considering that a smaller element size results in a larger mesh. The percentage difference between 4 and 5 mm for $F I$ was less than 1%. According to these results, the element size selected for this study was 4 mm. The meshed model and the results of mesh convergence are presented in Figure 10.

Figure 10.

Meshed model and mesh convergence study.

In addition to the validation of the hyperelastic material behavior, simulations were performed to validate the full model, ranging from loads lower than standing conditions to dynamic conditions. The load was applied at 0°, and the rest of the FEM settings were the same as previously described.

Results

Experimental results

The two manufactured RSPs were tested to determine whether the material, PLA or TPU, presents a better trade-off between higher compression and its mechanical resistance. In Figure 11, the $C_{v}$ of both RSP designs is compared. These results are presented in nondimensional form; in the case of $C_{v}$ was normalized concerning the height, $h$ , of the RSP, whereas the applied force was normalized concerning the design force, $F_{D}$ . The results demonstrated that the prosthesis with a TPU matrix compressed approximately 27% of the total height, whereas the prosthesis with a PLA matrix compressed only 6%. The TPU prosthesis achieved this displacement with a lower load than PLA, around 1.8 times $F_{D}$ . PLA needed higher loads to increase its displacements. From this measurement, it can be concluded that prostheses constructed with TPU exhibited higher $C_{v}$ due to lower stiffness and can support 1.8 times the weight of the user. Consequently, this material was more suitable for RSP designs.

Figure 11.

Comparison of TPU and PLA stiffness.

To determine the strength of the RSP, the force was increased until failure occurred in both models. The TPU model failed under a load of around 2.5 kN and its displacement was 61 mm, whereas the PLA model was subjected to a load of 7 kN and achieved a displacement of 20 mm. A key finding from the experimental measurements was that the PLA material can withstand a higher load; however, the TPU displacement was three times greater. In both models, the crack took place in the internal layers of the prosthesis, as can be observed in Figure 12. The failure occurred due to delamination, extending along the entire side of the prosthesis in the case of PLA. The delamination in the case of the TPU matrix occurred in a portion of the cross-sectional area of the internal side. The failure mechanics for this delamination was interlaminar shear failure because as FEM computations indicated, there was high stress in such areas. In both cases, the matrix did not suffer any damage.

Figure 12.

Failure of RSP designs. (a) PLA and (b) TPU.

Numerical results and discussion

The effects of the number of layers and the thickness of the matrix were evaluated using finite element calculations. Considering the properties of TPU, the first step was to select the hyperelastic model that best approximates its behavior. The comparison of the models and the result from the uniaxial experimental test is shown in Figure 13. During the experimental test, the material TPU exhibited a strain greater than 400% and stresses lower than 30 MPa. The hyperelastic model that best approximates the TPU behavior is the Mooney-Rivlin 5 parameters model because, as can be observed in Figure 13, it presented good agreement with experimental data. Considering that the prostheses deformation magnitudes are lower, this model is the best option to compute $F I$ , $σ_{M}$ , and $\frac{C_{V}}{h}$ using FEM. The resultant strain energy density function for the 5 parameters Mooney-Rivlin model is:

W = C_{10} (I_{1} - 3) + C_{01} (I_{2} - 3) + C_{11} (I_{1} - 3) (I_{2} - 3) + C_{20} {(I_{1} - 3)}^{2} + C_{02} {(I_{2} - 3)}^{2}

(8)

where

I_{1}

and

I_{2}

are the first and second strain invariants and

C_{10}

C_{01}

C_{11}

C_{20}

, and

C_{02}

are the material constants, which are presented in Table 3.

Figure 13.

Comparison between uniaxial test results and hyperelastic models.

Table 3.

Constants of the Money-Rivlin model.

Material	C₁₀ (MPa)	C₀₁ (MPa)	C₁₁ (MPa)	C₂₀ (MPa)	C₀₂ (MPa)
TPU	−8.49	15.23	−0.044	0.379	1.93

For the composite RSP with TPU matrix, the comparison of experimental and numerical vertical compression due to the force applied is presented in Figure 14 using the normalized vertical compression and the normalized applied force. The composite RSP is intended to be used between the range of standing conditions, $\frac{F}{F_{d}} = 0.4$ , to dynamic running conditions, $\frac{F}{F_{d}} = 1$ . For this range, the numerical predictions align closely to the experimental results and the average difference is 4.32% and the simulated results will be reliable. For values below standing conditions, $\frac{F}{F_{d}} < 0.4$ , the difference between both results is higher; however, the FEM computations varying thickness and number of layers in RSP were carried out for the range of standing conditions, $\frac{F}{F_{d}} = 0.4$ , to dynamic running conditions, $\frac{F}{F_{d}} = 1$ , as shown below.

Figure 14.

Comparison of experimental and finite element results.

The results of $σ_{M}$ , $\frac{C_{V}}{h}$ and the $F I$ at standing and running conditions are presented in Figure 15. At standing conditions, the load was minimal and did not affect the mechanical resistance of the RSP, and this caused low magnitudes of $σ_{M}$ , $\frac{C_{V}}{h}$ , and the $F I$ . The $\frac{C_{V}}{h}$ is around 10.5% of the initial length of the RSP. It was observed that the difference in magnitudes between 6 and 8 mm of thickness was greater than between 8 and 10 mm. This trend was repeated in the rest of the cases where the number of layers varied. As was expected, at 3 and 6 m/s, $σ_{M}$ , $\frac{C_{V}}{h}$ , and the $F I$ undergo a significant increase in their magnitudes due to the application of 2.5 times the weight of RSP user. With 8 layers the value of $F I$ exceeded or was so close to the limit of 1, which indicated the failure of the RSP. In contrast, with 10 and 12 layers, the $F I$ decreased around 20 and 30 %, respectively. These results showed that 8 layers were not appropriate for designs of RSP for users weighing 60 kg or above. Meanwhile, the increase in the number of layers reduced the $F I$ . As can be observed, in all cases, the increase in the matrix thickness was negligible in changing the $F I$ ; however, analyzing cases with 10 and 12 layers, speed did influence it. At 3 and 6 m/s, the $F I$ was 0.8525 and 0.89, respectively, which corresponds to a 4.4% increase despite the applied load being the same in both cases. The height of the RSP depended on the matrix thickness and the number of layers, therefore the value of $h$ in $\frac{C_{V}}{h}$ varied for all cases. The results showed that unlike the $F I$ , the $\frac{C_{V}}{h}$ was affected by the matrix thickness since the compression of the RSP decreases with ticker matrices. The RSP designs that achieved the maximum compression are those with 8 layers, between 0.3 and 0.27. However, these models presented an unacceptable $F I$ . With 10 and 12 layers, the $\frac{C_{V}}{h}$ diminished. As noticed in standing conditions, the difference in compression was greater between 8 and 10 layers than between 10 and 12 layers. This indicates that with increasing layers of the RSP, the variation of $\frac{C_{V}}{h}$ is less significant. Then it is expected that stiffness decreases as the number of layers increases. Another fact to consider is that there was a slight decrease in compression because of changes in speed. For example, the design with 10 layers and 8 mm of matrix thickness compresses 9.2% more for running speeds of 3 m/s than for 6 m/s; also, the design with 12 layers and 10 mm of matrix thickness compresses 9.5% more for running speeds of 3 m/s. The reason for this behavior is due to the variation in the direction of the force components which were applied at different angles for both running speed conditions. At 3 m/s, the forces in $x$ and $y$ directions were 388 and 1449 N, respectively, whereas at 6 m/s were 260 and 1477 N, respectively. Even though the force in $y$ direction was higher by around 2 % at 6 m/s, the force in $x$ direction at 3 m/s increases by an additional 128 N, around 50 %. Then, it was possible to assume that the force in $x$ direction affects the magnitude of $\frac{C_{V}}{h}$ . The matrix stresses increased almost 3 times from standing to dynamic conditions. Here, the stresses of matrix thickness were not affected by the running speed and the differences between both conditions are negligible. With 8 layers and 6 mm of thickness, stresses exceeded 10 MPa and decreased to 7.4 and 5.2 MPa for 10 and 12 layers, respectively. According to FEM computations when the matrix thickness was increased, the stresses decreased for all cases. For 12 layers and 10 mm of thickness, a minimum stress of 4 MPa was reached, for both 3 and 6 m/s. Comparing the magnitude of the computed stresses, there is no evidence that the TPU matrix would fail at these conditions, which agreed with the experimental results where a delamination failure in the carbon fiber was observed.

Figure 15.

Comparison of stress, C_v/h, and FI varying thickness and number of layers. Top row: standing conditions; middle row: 3 m/s; bottom row: 6 m/s.

The RSP design with 10 layers and 8 mm of thickness was selected to present the contours of FI, $σ$ , and $\frac{C_{v}}{h}$ in Figure 16. For the case of $F I$ only four layers, 1 5, 6, and 10, per side were chosen to be displayed. The value of FI decreases from layer 1, which is attached to the matrix, to layer 5, where FI is lower. However, from layer 6 to the last layer, layer 10, it increases, reaching its maximum value. The same trend was observed on both sides of the RSP; however, the values of $F I$ were highest on the internal side. The elongation was more constrained due to the number of adjacent layers, and this was the reason why at the center of the carbon fibers the $F I$ increased. Based on the FEM calculations, it was anticipated that as the load increases, the area most susceptible to failure will be the curved region on the internal side of the RSP, and it will initiate in the outermost layers. This was validated qualitatively with the experimental results (Figure 12) where the location of the crack coincided with the numerical results. It was notable that there was a difference in the location of the critical zone between the internal and external last layers. The stresses in the matrix also appeared in the same region as in the last layer of the internal side; however, a concentration zone was observed at the central lateral position. The contact between RSP and ground generated an increment of the stresses in the matrix, being around 75% of the maximum. It is recommended to expand the research to evaluate the ground reaction forces to understand their effects on the stress in the matrix. The RSP compression varies longitudinally, being higher at the top, then compression decreased being almost zero at the bottom. This difference was a result of the spring-like mechanism of the prosthesis.

Figure 16.

Finite element contours of: (a) $F I$ , (b) $σ_{M}$ , and (c) $\frac{C_{v}}{h}$ in the RBP design with 10 layers and matrix thickness of 8 mm.

Conclusions

In this paper, RSPs were compared experimentally to evaluate their mechanical behavior. The RSPs were made with composite materials including a matrix of PLA and TPU. The best RSP design was analyzed using FEM calculations to identify the influence of matrix thickness and the number of carbon fiber layers. The carbon fiber TPU matrix composite RSP demonstrated a good performance due to its low stiffness, while also withstanding 1.8 times the force before the failure occurred. The failure mechanism occurred due to delamination on the internal side of the prosthesis. The Mooney-Rivlin 5 parameters model was the best to fit the TPU behavior. Increasing the carbon fiber layers will reduce the FI, but also increase the stiffness of the RSP. Designs with more than 8 layers are highly recommended for adolescents weighing 60 kg. No signs of matrix damage were observed in any of the studied cases. Both $C_{v}$ and $F I$ were more sensitive to the layers increment than the matrix thickness increment. The direction and magnitude of force components change according to speed, mainly in $x$ direction. It was concluded that these changes affect the $C_{v}$ , but do not the $F I$ . The manufacturing of the RSP by the suggested procedure was successful; the hand lay-up process simplifies manufacturing and can be used in low volume production. The authors also suggest evaluating the use of other thermoplastics, like TPE, to determine their functionality and performance in the design of RSP.

Footnotes

Author contributions

Omar Dávalos: conceptualization, investigation, methodology, formal analysis, validation, and writing—original draft. J.C. Garcia: methodology, data curation, formal analysis, validation, and writing—review and editing. Clemente Mirafuentes, C. M.: investigation, formal analysis, validation, and writing—original draft.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Jose Davalos

Juan Carlos Garcia

Christian Clemente

Data availability statement

Data will be made available on request.*

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