Predicting Compression Pressure of Knitted Fabric Using a Modified

materials

Article Predicting Compression Pressure of Knitted Fabric Using a Modiﬁed Laplace’s Law

Yetanawork Teyeme 1,2,* , Benny Malengier 1 , Tamrat Tesfaye 2 , Simona Vasile 3 , Wolelaw Endalew 2 and Lieva Van Langenhove 1

1 Centre for Textile Science and Engineering, Department of Materials, Textiles and Chemical Engineering, Ghent University, 9052 Ghent, Belgium; [email protected] (B.M.); [email protected] (L.V.L.) 2 Ethiopian Institute of Textile and Fashion Technology, Bahir Dar University, Bahir Dar 6000, Ethiopia; [email protected] (T.T.); [email protected] (W.E.) 3 Fashion and Textiles Innovation Lab (FTILab+), HOGENT University of Applied Sciences and Arts, 9052 Ghent, Belgium; [email protected] * Correspondence: [email protected]

Abstract: The aim of this study is to develop a mathematical model for the prediction of compression pressure based on fabric parameters, such as engineering stress, engineering strain and engineering modulus of elasticity. Four knitted compression fabrics with different fibrous compositions and knit structures were used. Rectangular-cut strips were employed for the force–elongation characterization of the fabrics. The experimental pressure values between the fabric and rigid cylinder were assessed using a Picopress pressure measuring device. The mechanical and physical parameters of the fabric that influence the interface pressure, such as strain, elasticity modulus/stress and thickness, were determined and integrated into Laplace’s law. A good correlation was observed between the experimental and calculated pressure values for all combinations of fabrics, mounted with variable Citation: Teyeme, Y.; Malengier, B.; tension on the cylinder. Over the considered range of pressures, the difference between the two Tesfaye, T.; Vasile, S.; Endalew, W.; datasets was generally less than 0.5 mmHg. The effect of washing after five, ten and fifteen washing Van Langenhove, L. Predicting cycles on the fabric–cylinder interface pressure was found to be significant. Compression Pressure of Knitted Fabric Using a Modified Laplace’s Law. Materials 2021, 14, 4461. Keywords: stretch fabric; interface pressure; Laplace’s law; compression garment https://doi.org/10.3390/ma14164461

Academic Editor: Philippe Boisse 1. Background Received: 30 May 2021 Graduated compression has been used in diverse medical applications for several Accepted: 4 August 2021 years [1–3]. Compression garments are believed to assist in relieving ailments by provid- Published: 9 August 2021 ing evidence for their use in the sporting business [4,5]. However, the interface pressure applied by garments has not been accurately measured in the majority of research papers Publisher’s Note: MDPI stays neutral in this area [6,7]. Studies that have reported interface pressure measurements have com- with regard to jurisdictional claims in monly evaluated only the calf and the mid-thigh, which frequently reflect the exhibition of published maps and institutional affil- graduated compression [8–10]. iations. Different postures and positions during the wear of compression garments induce variations in the garment–body interface pressure. The interface pressure exerted by garments of different sizes is an area that should still be investigated further. Studies have suggested that undersized garments exert more pressure than recommended size Copyright: © 2021 by the authors. garments [11]. To quantify differences between the garments assessed, a complete pressure Licensee MDPI, Basel, Switzerland. profile should be established. A lot has been done to quantify the pressure exerted by This article is an open access article compression garments on lower body parts, however to the authors’ knowledge, there distributed under the terms and is no study that examines the pressure exerted by compression garments on the upper conditions of the Creative Commons body of cyclists. This study aims to investigate the interface pressure between various Attribution (CC BY) license (https:// sportswear compression fabrics and a rigid cylinder of size comparable to that of a human creativecommons.org/licenses/by/ arm. In addition to this, it also looks at the relationship between the experimental values, 4.0/).

Materials 2021, 14, 4461. https://doi.org/10.3390/ma14164461 https://www.mdpi.com/journal/materials Materials 2021, 14, 4461 2 of 13

measured by a Picopress instrument [12], and the predicted values obtained using the modiﬁed Laplace law that uses easy to measure fabric mechanical and physical parameters as input [13].

2. Materials and Methods 2.1. Materials Four commercial knitted fabrics (Liebaert Marcel Nv, Deinze, Belgium) with 25–42% elastane, suitable for sports compression wear applications, were selected for this study. Table1 lists the ﬁber composition, fabric structural parameters and means values (STDEV) of area density, thickness [14] and air permeability [15]. Area density of the fabrics varied from 150 g/m2 (fabric LB) to 310 g/m2 (fabric TL), they had a thickness between 0.29 mm (fabric LB) and 0.58 mm (fabric SW) and exhibited an air permeability with values between 74.93 mm/s (fabric DB) and 394.5 mm/s (fabric LB).

Table 1. Fabric information and properties.

Air Nylon/Elastane Fabric Area Density Thickness Sample Code 2 Permeability Wales/cm Courses/cm Composition (%) Construction (g/m ) (mm/s) (mm) Warp knit— 394.5 0.29 LB 74/26 150 31 37 Tricot single face (11.59) (0.008) Warp knit— 74.93 0.39 DB 58/42 200 29 39 Tricot double face (6.79) (0.00) Warp knit—Tricot 183.25 0.55 TL 65/35 310 22 26 with pillar stitch (6.94) (0.01) Warp knit— 314.0 0.58 SW 75/25 247 22 38 Tricot (1 × 1) (27.18) (0.01)

Figure1 shows the technical face (a) and back face (b) surfaces of the compression fabrics (LB, DB, TL and SW). Usually, the fabric has raised wales on the face side of the fabric, whereas the inner side is smooth. A plain fabric back surface provides a comfortable smooth surface to the skin and also decreases stress concentration during compression. Moreover, the compression force was mostly developed by one-dimensional fabric stretching. This means that for producing effective pressure in compression garment design, the wale direction with Spandex or Lycra® (Du Pont, Wilmington, DE, USA) is normally used [16,17].

2.2. Methods Hereafter, all test methods are briefly described. First, a method is presented for the experimental assessment of the pressure (Section 2.2.1) exerted by the four fabrics on a rigid cylinder, approaching a human body part such as an arm. The test methods for the assessment of fabric modulus of elasticity (Section 2.2.3)[18] and theoretical prediction of interface pressure of the fabric–rigid cylinder are further presented (Section 2.2.4). Moreover, the effects of domestic washing after five, ten and fifteen wash cycles on the fabric–cylinder interface pressure were investigated (Section 2.2.2). Prior to testing, all samples were placed for at least 24 h in a conditioning chamber (Memmert, France), controlled at 21 ± 2 ◦C and a relative humidity of 65% ± 4% [19].

Figure 1. Structure of sample compression fabrics (LB, DB, TL and SW): technical face (a) and back face (b). Figure 1. Structure of sample compression fabrics (LB, DB, TL and SW): technical face (a) and back face (b).

2.2. MethodsClothing pressure can be obtained by many methods, including theoretical calcula- tionsHereafter, [22], simulations all test methods [23,24], direct are briefly tests, descri or indirectbed. First, tests [a25 method,26]. The is garmentpresented pressure for the experimental(mmHg) of the assessment stretchable of knitted the pressure fabrics was(Secti assessedon 2.2.1) using exerted a PicoPress by the four® pressure fabrics sensor on a rigid(Microlab cylinder, Elettronica approaching SAS, Padua, a human Italy) body [12 part], which such is as shown an arm. in FigureThe test2. Thismethods instrument for the assessmentconsists of aof manometer fabric modulus connected of elasticity to a probe, (Section namely 2.2.3) a[18] ﬂexible, and theoretical circular plastic prediction bladder of interfacewith a diameter pressure of of 5 the cm. fabric–rigid This device cylinder is accurate, are further reliable presented and measures (Section the 2.2.4). actual More- pres- over,sure experiencedthe effects of bydomestic the wearer washing [27]. after A rigid five, PVC ten (Polyvinyland fifteen chloride) wash cycles cylinder on the of fabric– 11 cm cylinderdiameter interface (34.56 cm pressure circumference), were investigated approximating (Section the 2.2.2). circumference Prior to testing, of an adult all samples human upper arm, was used in this protocol. Fabric samples were cut and sewn into circular tubes, Materials 2021, 14, x FOR PEER REVIEW 4 of 14

were placed for at least 24 h in a conditioning chamber (Memmert, France), controlled at 21 ± 2 °C and a relative humidity of 65% ± 4% [19].

2.2.1. Interface Pressure Evaluation Adequate and stable pressure is crucial for compression garments. Extensibility and elastic recovery are important characteristics, due to which compression garments can exert continuous, constant pressure on the human body. The mechanical properties of the fabric and the curvature radius of the respective body parts are of great importance when constructing compression garments. An improper compression garment would influence the energy, work efficiency and health of the wearer. Insufficient pressure will limit effi- cacy, while too high a pressure will make people feel uncomfortable, cause breathing dif- ficulty, and can also cause serious damage to health [20,21]. Clothing pressure can be obtained by many methods, including theoretical calculations [22], simulations [23,24], direct tests, or indirect tests [25,26]. The garment pressure (mmHg) of the stretchable knitted fabrics was assessed using a PicoPress®® pressure sensor (Microlab Elettronica SAS, Padua, Italy) [12], which is shown in Figure 2. This instrument consists of a manometer connected to a probe, namely a flexible, circular plastic bladder with a diameter of 5 cm. This device is accurate, reliable and measures the actual pressure experienced by the wearer [27]. A rigid PVC (Polyvinyl chloride) cylinder of 11 Materials 2021, 14, 4461 4 of 13 cm diameter (34.56 cm circumference), approximating the circumference of an adult human upper arm, was used in this protocol. Fabric samples were cut and sewn into circular tubes, whose circumferences were 10%, 20% and 30% less than the circumference of the whoserigid PVC circumferences cylinder [28]. were The 10%, fabric 20% tube and wa 30%s mounted less than on the the circumference cylinder, the of bladder the rigid (pres- PVC surecylinder sensor) [28]. was The placed fabric tubebetween was the mounted cylinder on and the cylinder,the fabric the (Figure bladder 2) and (pressure the interface sensor) waspressure placed of betweenthe cylinder–fabric the cylinder was and measured. the fabric (FigureFor each2) sample, and the interfacethe static pressure interface of pres- the surecylinder–fabric was recorded was at measured. five different For each locations sample, and the the static mean interface values pressure (mmHg) was were recorded calculated.at ﬁve different locations and the mean values (mmHg) were calculated.

®® Figure 2. PicoPress® pressure pressuresensor sensor device device ( left(left)) used used to to measure measure fabric–rigid fabric–rigid cylinder cylinder interface interface pressure pressure in in vitro vitro ( right). (right). 2.2.2. Effects of Washing on Pressure 2.2.2. Effects of Washing on Pressure The effect of domestic washing after five, ten and fifteen wash cycles on the fabric– cylinderThe interfaceeffect of domestic pressure washing was investigated after five, [ 29ten]. and The fifteen pressure wash exerted cycles by on the the circular fabric– cylindersample oninterface the rigid pressure cylinder was was investigated also tested [2 after9]. The 5, 10 pressure and 15 domesticexerted by washes the circular of the samplefabric tube, on the performed rigid cylinder according was to also ISO tested 6330:2012(E) after 5, [1029 ].and After 15 eachdomestic washing washes cycle of (of the a fabricduration tube, of performed 30 min), the according fabrics were to ISO flat-dried 6330:2012(E) at chamber [29]. After temperature each washing and the cycle interface (of a pressure of the dry fabric tubes was subsequently measured.

2.2.3. Fabric Elasticity and Flexural Modulus

Measurements of fabric elasticity and the flexural modulus of elasticity were carried out using an Instron 3334 Tensiometer (Instron, Boechout, Belgium) (Figure3). Assessment of the stretch recovery was determined according to EN 14704-1:2005 [18]. The fabric strip of 50 mm width was tested under a gauge length of 100 mm and at a constant test speed of 500 mm/min. The load (N) was preset at a value determined as a function of the fabric’s elastane content and the value of strain (%) was determined for five cycles. The strain value of the first cycle was used in further calculations. For each fabric, five specimens were tested, and the mean values were calculated [30].

2.2.4. Theoretical Method for Prediction of Compression To calculate the pressure applied by the compression garments, a method for vali- dating the reliability of Laplace’s law was investigated (Section 3.4.1). Furthermore, the physical and mechanical properties (such as fabric elasticity, weight and thickness) of nylon/elastane knitted fabrics were studied and compared to conﬁrm their capability to prove satisfactory compression. Such a study is necessary to understand whether a fabric is suitable for developing a compression garment for a determined application and for estimating the required compression force needed when designing a custom-made compression garment with sports compression fabrics. Materials 2021, 14, x FOR PEER REVIEW 5 of 14

duration of 30 min), the fabrics were flat-dried at chamber temperature and the interface pressure of the dry fabric tubes was subsequently measured.

2.2.3. Fabric Elasticity and Flexural Modulus Measurements of fabric elasticity and the flexural modulus of elasticity were carried out using an Instron 3334 Tensiometer (Instron, Boechout, Belgium) (Figure 3). Assess- ment of the stretch recovery was determined according to EN 14704-1:2005 [18]. The fabric strip of 50 mm width was tested under a gauge length of 100 mm and at a constant test speed of 500 mm/min. The load (N) was preset at a value determined as a function of the Materials 2021, 14, 4461 fabric’s elastane content and the value of strain (%) was determined for five cycles.5 ofThe 13 strain value of the first cycle was used in further calculations. For each fabric, five specimens were tested, and the mean values were calculated [30].

FigureFigure 3. 3. StripStrip tests tests for for stretch-recovery stretch-recovery measurem measurementent using an Instron 3334 Tensiometer.

2.2.4.3. Results Theoretical and Discussion Method for Prediction of Compression 3.1. InterfaceTo calculate Pressure the Evaluationpressure applied by the compression garments, a method for vali- datingThe the results reliability showed of Laplace’s that the tricot law was with investigated pillar stitch (TL)(Section exerted 3.4.1). the Furthermore, highest pressure the physicalon the cylinder, and mechanical with average properties values (such between as fabric 5.7 mmHg elasticity, (at 10%weight stretch) and thickness) and 16.6 mmHg of nylon/elastane(at 30% stretch). knitted The fabrics maximum were attained studied pressure and compared in this study to confirm falls under their the capability moderator to provecompression satisfactory category compression. descriptor Such and a canstudy help is necessary to reduce to Deep understand Vein Thrombosis whether a (DVT)fabric isin suitable normal-risk for developing patients, especially a compression during ga periodsrment for of longa determined travel. Previous application studies and also for estimatingshowed that the patients required with compression symptoms force of mild needed venous when insufficiency designing benefita custom-made from wearing com- pressioncompression garment garments with providingsports compression a pressure fabrics. of 10–20 mmHg [31]. However, to the authors’ knowledge, there is no known study available which quantifies the pressure produced by 3.compression Results and garments Discussion and the interactions between the garment and the upper body parts 3.1.of a Interface cyclist. The Pressure maximum Evaluation pressure attained in this study could reduce the range of motion during cycling, improve body stability, support the muscles of cyclists and improve the The results showed that the tricot with pillar stitch (TL) exerted the highest pressure blood flow. The extent of pressure depends on the garment fit and construction, the size on the cylinder, with average values between 5.7 mmHg (at 10% stretch) and 16.6 mmHg and shape of the part of the body to which it is applied, structure and physical properties (at 30% stretch). The maximum attained pressure in this study falls under the moderator of the fabric and the type of sport activity. The future study will entail an interaction of size compression category descriptor and can help to reduce Deep Vein Thrombosis (DVT) in and shape of body parts of cyclists on pressure applied and the use of this compression normal-risk patients, especially during periods of long travel. Previous studies also garment for treating different venous diseases. showed that patients with symptoms of mild venous insufficiency benefit from wearing 3.2. Effects of Washing on Pressure Figure4 shows the pressure in mmHg exerted by the knitted fabrics on the cylinder at zero washing cycles (noted as 0×). Among the tested fabrics, the tricot with pillar stitch (TL) mounted on the cylinder at 30% stretch exhibited the highest pressure (c) before washing (0×). This fabric had a greater compact structure and higher areal density, and consequently led to more pressure than other knit structures. The influence of various numbers of washing cycles (5×, 10×, 15×) on the pressure exerted by fabrics mounted at (a) 10%, (b) 20% and (c) 30% on the cylinder can be also seen in Figure4. Materials 2021, 14, x FOR PEER REVIEW 6 of 14

compression garments providing a pressure of 10–20 mmHg [31]. However, to the authors’ knowledge, there is no known study available which quantifies the pressure produced by compression garments and the interactions between the garment and the upper body parts of a cyclist. The maximum pressure attained in this study could reduce the range of motion during cycling, improve body stability, support the muscles of cyclists and improve the blood flow. The extent of pressure depends on the garment fit and construction, the size and shape of the part of the body to which it is applied, structure and physical properties of the fabric and the type of sport activity. The future study will entail an interaction of size and shape of body parts of cyclists on pressure applied and the use of this compression garment for treating different venous diseases.

3.2. Effects of Washing on Pressure Figure 4 shows the pressure in mmHg exerted by the knitted fabrics on the cylinder at zero washing cycles (noted as 0×). Among the tested fabrics, the tricot with pillar stitch (TL) mounted on the cylinder at 30% stretch exhibited the highest pressure (c) before washing (0×). This fabric had a greater compact structure and higher areal density, and Materials 2021, 14, 4461 consequently led to more pressure than other knit structures. The influence of various6 of 13 numbers of washing cycles (5×, 10×, 15×) on the pressure exerted by fabrics mounted at (a) 10%, (b) 20% and (c) 30% on the cylinder can be also seen in Figure 4.

FigureFigure 4.4. PressurePressure exertedexerted byby thethe 10%10% stretchedstretched fabricsfabrics ((aa),), 20%20% stretchedstretched fabricsfabrics ((bb)) andand 30%30% stretchedstretched fabricsfabrics ((cc)) onon thethe tubetube afterafter 5,5, 1010 andand 1515 washingwashing cycles.cycles.

and tricot 1 × 1 SW, respectively. In general, tricot DB and tricot TL retained better pressure after 5, 10 and 15 washes than tricot LB and tricot SW, but not tricot 1 × 1 SW, which had a better pressure retained than DF at 20% stretch. In general, there was not much difference in terms of pressure loss between the four knitted structures after 5 washes, but this difference was quite signiﬁcant after 10 and 15 washes, which suggests that the knitted stretchable fabrics tend to lose their compression ability after multiple washes. Moreover, a t-test (α = 0.05) showed signiﬁcant differences between the pressure exerted by the fabric on the cylinder due to levels of stretch (p < 0.05).

3.3. Fabric Elasticity Double knits (DB) are in general more stable and heavier than single knits due to the opposite loop orientation of the double guide bar patterns. Single knit structures are dimensionally unstable and split easily when damaged. It is also quite evident in the strain of fabrics (Figure5) that the tricot single face (LB) has the highest strain (190 mm) at maximum load of cycle 1, due to its looser structure (Figure1) that makes it stretch quite easily. Similarly, tricot double face (DB) with the most compact structure exhibited the lowest strain (116 mm). Materials 2021, 14, x FOR PEER REVIEW 7 of 14

Table 2 shows the pressure loss (in %) after 5, 10 and 15 washing cycles. For instance, at 30% stretch and after 15 washes, the pressure of all four knitted structures was reduced by 0.4, 0.2, 0.4 and 0.2 mmHg. After 5 washes, a pressure loss of 0.2, 0.0, 0.0 and 0.2 mmHg was noticed for the tricot single face LB, tricot double face DF, tricot with pillar stitch TL and tricot 1 × 1 SW, respectively. In general, tricot DB and tricot TL retained better pressure after 5, 10 and 15 washes than tricot LB and tricot SW, but not tricot 1 × 1 SW, which had a better pressure retained than DF at 20% stretch. In general, there was not much difference in terms of pressure loss between the four knitted structures after 5 washes, but this difference was quite significant after 10 and 15 washes, which suggests that the knitted stretchable fabrics tend to lose their compression ability after multiple washes. More- over, a t-test (α = 0.05) showed significant differences between the pressure exerted by the fabric on the cylinder due to levels of stretch (p < 0.05).

Table 2. Pressure loss after 5, 10 and 15 washing cycles of the fabrics LB, DB, TL and SW mounted at 10%, 20% and 30% stretch on the rigid cylinder.

Pressure Losing Percentage (%) Washes 5× Washes 10× Washes 15× Fabric Name 10% 20% 30% 10% 20% 30% 10% 20% 30% Materials 2021, 14, 4461 Stretch Stretch Stretch Stretch Stretch Stretch Stretch Stretch Stretch 7 of 13 Tricot single face (LB) 0.00 3.33 1.85 10.00 10.00 2.78 20.00 10.00 3.70 Tricot double face (DB) 0.00 1.52 0.00 6.67 3.03 0.83 6.67 6.06 1.67 Tricot with pillar stitch 0.00 0.95 0.00 0.00 0.95 1.20 1.75 2.86 2.41 Table 2.(TL)Pressure loss after 5, 10 and 15 washing cycles of the fabrics LB, DB, TL and SW mounted at 10%, 20% and 30% stretchTricot on(1 × the 1) rigid(SW) cylinder.3.60 0.00 1.33 7.21 2.70 2.00 10.81 4.05 4.00

Pressure Losing Percentage (%) 3.3. Fabric Elasticity Washes 5× Washes 10× Washes 15× Fabric Name Double knits (DB) are in general more stable and heavier than single knits due to the 10% 20% 30% 10% 20% 30% 10% 20% 30% opposite loop orientation of the double guide bar patterns. Single knit structures are di- Stretch Stretch Stretch Stretch Stretch Stretch Stretch Stretch Stretch mensionally unstable and split easily when damaged. It is also quite evident in the strain Tricot single face (LB)of 0.00 fabrics (Figure 3.33 5) that 1.85 the tricot 10.00 single face 10.00 (LB) has the 2.78 highest strain 20.00 (190 mm) 10.00 at max- 3.70 Tricot double face (DB)imum 0.00 load 1.52of cycle 1, due 0.00 to its looser 6.67 structure 3.03 (Figure 1) 0.83 that makes 6.67 it stretch quite 6.06 easily.1.67 Tricot with pillar stitch (TL)Similarly, 0.00 tricot 0.95 double 0.00 face (DB) 0.00 with the most 0.95 compact 1.20 structure 1.75 exhibited 2.86the lowest 2.41 × Tricot (1 1) (SW)strain 3.60 (116 mm). 0.00 1.33 7.21 2.70 2.00 10.81 4.05 4.00

30 DB TL 25 SW LB 20 Load (N) 15

0 0 25 50 75 100 125 150 175 200 225 Materials 2021, 14, x FOR PEER REVIEW Extension (mm) 8 of 14

Figure 5. 5. Load-extensionLoad-extension curves curves for forthe thefabrics fabrics DB, DB,TL, SW TL, and SW LB. and LB.

3.4.3.4. Theoretical Theoretical Method Method forfor Prediction of of Compression Compression 3.4.1.3.4.1. Interface Interface Pressure Pressure ModelModel

TheThe predictive predictive pressure values values of ofthe thestretc stretchh fabrics fabrics were calculated were calculated by applying by applyingLa- Laplace’splace’s law law [32,33]. [32,33]. In In the the investigation, investigation, the thegeometric geometric form form of the of PVC the PVCcylinder cylinder (Figure (Figure 6) 6 ) was assumed to represent the human body figure (arm). was assumed to represent the human body ﬁgure (arm).

(a) (b)

FigureFigure 6. (a) 6. PVC (a) PVC cylinder cylinder and and fabric fabric interface. interface. (b )(b Forces) Forces present: present: pressure pressure (P)(P) and te tensionnsion (T) (T) for for a acylinder cylinder with with radius radius (r). (r).

Laplace’s law follows from taking the differential circumference, as shown in Figure 7. The derivation is as follows. From the principle of static equilibrium, the pressure applied to a differential area can be given as 𝑃𝑑𝐴 − 2𝐹𝑠𝑖𝑛 =0, with dA being the area on which the pressure works. For small angles, we have 𝑠𝑖𝑛 ≈ , so we can rearrange this as 𝑃𝑑𝐴=𝐹𝑑𝜃.

Figure 7. Differential circumference of fabric on the cylinder.

Materials 2021, 14, x FOR PEER REVIEW 8 of 14

3.4. Theoretical Method for Prediction of Compression 3.4.1. Interface Pressure Model The predictive pressure values of the stretch fabrics were calculated by applying La- place’s law [32,33]. In the investigation, the geometric form of the PVC cylinder (Figure 6) was assumed to represent the human body figure (arm).

(a) (b) Figure 6. (a) PVC cylinder and fabric interface. (b) Forces present: pressure (P) and tension (T) for a cylinder with radius Materials(r).2021 , 14, 4461 8 of 13

Laplace’s law follows from taking the differential circumference, as shown in Figure 7. The derivation is as follows. From the principle of static equilibrium, the pressure ap- pliedLaplace’s to a differential law follows area fromcan be taking given the as differential𝑃𝑑𝐴 − circumference,2𝐹𝑠𝑖𝑛 =0, with dA asshown being the in Figure area on7. The derivation is as follows. From the principle of static equilibrium, the pressure applied which the pressure works. For small angles, we havedθ 𝑠𝑖𝑛 ≈ , so we can rearrange this to a differential area can be given as PdA − 2Fsin 2 = 0, with dA being the area on which as 𝑃𝑑𝐴=𝐹𝑑𝜃. dθ dθ the pressure works. For small angles, we have sin 2 ≈ 2 , so we can rearrange this as PdA = Fdθ.

FigureFigure 7. Differential7. Differential circumference circumference of fabricof fabric on on the the cylinder. cylinder.

Integrating this equation over all angles, constant pressure was assumed:

Z 2π Z 2π w2πrP = PdA = Fdθ = F2π (1) 0 0 This can be rewritten by assuming that the force is the tension force times the width of the sample, as indicated in Equation (2):

F2π Tw2π 2πT T P = = = = (2) w2πr w2πr C r where P is the surface pressure on the PVC cylinder (Pa), F is the internal force of the fabric (N), T the fabric tension applied on the cylindrical surface (N/m), C is the circumference of the fabric (m), w is the fabric width (m), and r is the curvature radius of the cylindrical surface (m). This equation is known as Laplace’s law [11,34]. Stress developed on the fabric due to the surface pressure can be determined as the 2 ratio of applied force to the fabric cross-sectional area, As (m ) (Equation (3)), which can be given as: F Tw T σ = = = (3) As sw s where σ is engineering stress (Pa) of the test fabric, and w and s are fabric width and fabric thickness (in m), respectively. Tension can hence be found as a function of the resisting stress and thickness of the fabric, as T = σs. Substituting it into Equation (2) gives:

σs2π Ees2π P = = (4) C C Here, we assumed that stress and strain obey Hooke’s law (Equation (5)) in the elastic region of the fabric: σ = Ee (5) Materials 2021, 14, 4461 9 of 13

where E is engineering Young’s Modulus (Pa) and e is the engineering strain (m/m). The strain for the cylindrical model can also be determined from the change in circumference, as shown in Equation (6): Cf − Ci ε = (6) Ci

where Cf is the initial test fabric circumferences and the ﬁnal test fabric measured before and after stretching. Prediction model of the fabric stress: The test data obtained from Instron based on EN 14704-1 [17] were related using a third-order polynomial equation (Equation (7)), which provides the best ﬁt:

3 2 σ(ε) = a1ε + a2ε + a3ε + b (7)

where a1, a2, a3 coefﬁcients and b is a constant number. Substituting Equation (6) into Equation (7), it gives:

3 2 Cf − Ci Cf − Ci Cf − Ci σ(ε) = a + a + a + b (8) 1 Ci 2 Ci 3 Ci

The resulting pressure, P (Pa), was calculated using Equations (4) and (8) for stress and compared with the measured pressure:

3 2 ! s2π Cf − Ci Cf − Ci Cf − Ci P = a + a + a + b (9) C 1 C 2 C 3 C Materials 2021, 14, x FOR PEER REVIEW i i i 10 of 14

3.4.2. Inﬂuence of Fabric Mechanical Properties on Interface Pressure The stress–strain3.4.2. Influence curves of were Fabric analyzed, Mechanical and Properties Figure on8 presents Interface Pressure the obtained tension for a speciﬁc strain foundThe throughout stress–strain the curves stress–strain were analyzed, curves. and Figure As presented 8 presents inthe the obtained experimental tension procedure, the knittedfor a specific fabric strain tube found extends throughout when the wornstress–strain (Figure curves.2). TheAs presented applied in tension the exper- at imental procedure, the knitted fabric tube extends when worn (Figure 2). The applied ten- these extension valuession at these was extension obtained values from was the obtained stress–strain from the stress–strain curve. curve.

Figure 8. Stress–strain curvesFigure 8. for Stress–strain the tested curves samples. for the tested samples. 3.5. Measurements and Comparison of the Pressure with Prediction Values In this study, we applied the mathematical model (Equation (9)) and compared it with the experimental pressure (PE). The measured (PE) and predicted values (P) of each fabric with various extension levels (i.e., 1 = 10%, 2 = 20% and 3 = 30%) are shown in Table 3. The pressure results using the developed prediction model derived as (Equation (9)) were statistically analyzed and compared with the experimental pressure measurement results.

Table 3. Measurement of engineering strain, stress, modulus and predicted compression.

Measured Engineering Young’s Experimental Predicted % Error Fabric Engineering Pressure PE (2) Stress 𝜹 Modulus E Pressure PE (3) Pressure P [(|P − PE |) Code Strain 𝜺 (mmHg) (kPa) (kPa) (kPa) (kPa) /PE] × 100 DB 1 3.0 0.111 55.054 495.49 0.40 0.39 2.50 DB 2 6.6 0.249 118.047 473.90 0.88 0.84 4.89 DB 3 12.0 0.429 225.496 526.16 1.60 1.60 0.06 TL 1 5.7 0.111 77.638 698.74 0.76 0.78 2.11 TL 2 10.5 0.249 139.112 558.46 1.40 1.39 0.64

Materials 2021, 14, 4461 10 of 13

3.5. Measurements and Comparison of the Pressure with Prediction Values In this study, we applied the mathematical model (Equation (9)) and compared it with the experimental pressure (PE). The measured (PE) and predicted values (P) of each fabric with various extension levels (i.e., 1 = 10%, 2 = 20% and 3 = 30%) are shown in Table3. The pressure results using the developed prediction model derived as (Equation (9)) were statistically analyzed and compared with the experimental pressure measurement results.

Table 3. Measurement of engineering strain, stress, modulus and predicted compression.

Experimental Measured δ Young’s Predicted % Error (1) Engineering Strain ε Engineering Stress Modulus E (2) [(|P − P |) Fabric Code Pressure PE (kPa) Pressure PE Pressure P E (mmHg) (kPa) (kPa) (kPa) /PE] × 100

DB 1 3.0 0.111 55.054 495.49 0.40 0.39 2.50 DBMaterials2 2021, 14, x FOR6.6 PEER REVIEW 0.249 118.047 473.90 0.88 0.8411 of 14 4.89 DB 3 12.0 0.429 225.496 526.16 1.60 1.60 0.06 TL 1 5.7 0.111 77.638 698.74 0.76 0.78 2.11 TL 2 10.5 0.249 139.112 558.46 1.40 1.39 0.64 TL 3 16.6 0.429 223.967 522.59 2.22 2.24 0.86 SW 1TL 3 16.62.7 0.429 0.111 223.967 35.754522.59 321.79 2.22 0.37 2.24 0.38 0.86 1.89 SW SW2 1 7.42.7 0.111 0.249 35.754 102.405321.79 411.11 0.37 0.99 0.38 1.08 1.89 9.42 SW 3 15.0 0.429 184.584 430.70 2.00 1.95 2.70 SW 2 7.4 0.249 102.405 411.11 0.99 1.08 9.42 LB 1 2.0 0.111 46.010 414.09 0.27 0.24 9.02 LB 2SW 3 15.06.0 0.429 0.249 184.584 161.954430.70 650.16 2.00 0.80 1.95 0.85 2.70 6.74 LB 3 10.8 0.429 263.138 1.420 1.39 1.39 2.29 LB 1 2.0 0.111 46.010 414.09 0.27 0.24 9.02 (1) IndicesLB 2 1/2/3 represent6.0 extension 0.249 levels of 10%/20%/30% 161.954 of the650.16 fabrics DB, TL, SW 0.80 and LB. Values 0.85 correspond 6.74 to PE (mmHg), as (2) indicatedLB 3 in Figure10.84, for 0x washing 0.429 cycles. PE values 263.138 (mmHg) in1.420 the ﬁrst column 1.39 converted to kPa. 1.39 2.29 Indices 1/2/3 represent extension levels of 10%/20%/30% of the fabrics DB, TL, SW and LB. (2) Values correspond to PE (mmHg), as indicated in Figure 4,Linear for 0x washing regression cycles. analysis (3) PE values was (mmHg) used in to the investigate first column howconverted much to kPa. the independent variable (experimental pressure) explained the dependent variable. Figure9 shows the coefﬁcient of determinationLinear regression value analysis (R-squared was used value)to investigate of the how prediction much the model independent (P), explaining vari- 99.5% able (experimental pressure) explained the dependent variable. Figure 9 shows the coef- officient the experimentalof determination results. value (R-squared In addition, value) the of the accuracy prediction of amodel developed (P), explaining prediction model (P)99.5% against of the experimental experimental results. pressure In additi (PE)on, was the assessed. accuracy of The a developed errors between prediction predicted and experimentalmodel (P) against values experimental ranged betweenpressure (PE) 0.06% was andassessed. 9.42%. The The errors absolute between values predicted of their relative errorsand experimental were very values small, ranged except between for SW2 0.06% (9.42%), and 9.42%. LB1 (− The9.02%) absolute and values LB2 (6.74%). of their This can be causedrelative byerrors an extensionwere very small, (%), theexcept radius for SW2 of the (9.42%), area onLB1 which (−9.02%) pressure and LB2 is (6.74%). applied and other mechanicalThis can be caused parameters by an extension of the (%), fabric, the radius such asof the friction area on properties which pressure on theis applied PVC tube [35,36]. and other mechanical parameters of the fabric, such as friction properties on the PVC tube However, according to our analysis, this value is acceptable for the development process of [35,36]. However, according to our analysis, this value is acceptable for the development interfaceprocess of compressioninterface compression pressure. pressure.

2.500 y = 0.9906x + 0.0116 R² = 0.9959 2.000

1.500

Predicted pressure

Predicted pressure P pressure (kPa) Predicted 1.000 (kPa)(P)

0.500

0.000 0.000 0.500 1.000 1.500 2.000 2.500

Experimental pressure PE (kPa)

Figure 9. Comparison of predicted compression pressure (P) with experimental results (PE). Figure 9. Comparison of predicted compression pressure (P) with experimental results (PE). 3.6. The Influence of Extension and Young’s Modulus on Interface Pressure As illustrated in Figure 10, at the same strain value, the interface pressure increases when Young’s modulus increases (Table 3). This variation is related to the mechanical behavior of the sample fabrics. The greater the Young’s modulus (as in the case of knitted fabric TL), the higher the tension needed to obtain the desired strain. Consequently, the interface pressure becomes higher. Furthermore, these results clearly follow Laplace’s law, where pressure applied by a fabric with a greater elastic modulus is higher than that exerted by a fabric with a lower elastic modulus for a similar strain.

Materials 2021, 14, 4461 11 of 13

3.6. The Inﬂuence of Extension and Young’s Modulus on Interface Pressure As illustrated in Figure 10, at the same strain value, the interface pressure increases when Young’s modulus increases (Table3). This variation is related to the mechanical behavior of the sample fabrics. The greater the Young’s modulus (as in the case of knitted fabric TL), the higher the tension needed to obtain the desired strain. Consequently, the Materials 2021, 14interface, x FOR PEER pressure REVIEW becomes higher. Furthermore, these results clearly follow Laplace’s 12 of 14

law, where pressure applied by a fabric with a greater elastic modulus is higher than that exerted by a fabric with a lower elastic modulus for a similar strain.

16 DB 14 TL SW 12 LB 10

Interface pressure (mmHg) pressure Interface 4

10 15 20 25 30 35 40 45

Strain (%) FigureFigure 10. Interface 10. Interface pressure pressure applied applied by the different by the differentsamples at samples determined at determined strain values strain of 10%, values 20% and of 10%,30%. 20% and 30%. With respect to extension, Figure 10 shows that at the same strain value, the interface With respectpressure to extension, values increase Figure 10 when shows Young’s that atmodulus the same increases. strain value,This difference the interface can be related pressure valuesto increase the behavior when of Young’s the knitted modulus samples. increases. This difference can be related to the behavior of the knitted samples. 3.7. Influence of Fabric Thickness on Interface Pressure 3.7. Inﬂuence of FabricThe Thickness mechanical on Interface properties Pressure of textile materials are influenced by fabric thickness, as The mechanicalpresented properties in Table of 1. textile Figure materials10 also shows are inﬂuencedthe effect of by fabric fabric thickness thickness, on the as interface presented in Tablepressure1. Figure applied. 10 also Except shows for fabric the effect SW, it of is fabric clear that thickness the thicker on the the interfacecompression fabric pressure applied.(i.e., Except fabric forTL fabricand DB), SW, the it greater is clear the that interface the thicker pressure. the compression This can be explained fabric by the (i.e., fabric TL andfact DB), that theat the greater same the fabric interface circumference, pressure. if Thisthe other can be parameters explained are by kept the factconstant, the that at the samethickest fabric circumference, knitted fabric generates if the other more parameters internal force are keptto the constant, rigid cylinder. the thickest knitted fabric generates more internal force to the rigid cylinder. 4. Conclusions 4. Conclusions Compression clothing is broadly utilized nowadays in several medical and sports applications. However, there is a lack of studies on the interface pressure applied by the Compression clothing is broadly utilized nowadays in several medical and sports different sized garments at different extension levels. The scope of this investigation was applications. However,to assess thereand predict is a lack the ofinterf studiesace pressure on the interfacegenerated pressureby a knitted applied fabricby on thea rigid tube, different sized garmentsbased on ata pressure different measurement extension levels. device The and scope a revised of this Laplace’s investigation law. In wasorder to to predict assess and predictthe the interface interface pressure pressure generated generated by a by knitted a knitted elastic fabric garment on a rigidon a human tube, based body part, the on a pressure measurementbasic Laplace’s device law is inadequate, and a revised as the Laplace’s tension remains law. In unknown. order to predictTherefore, the it was nec- interface pressureessary generated to present by adjustments a knitted elastic to this garment law to re oncast a humanit in terms body of extension. part, the basicFor predicting the interface pressure, the fabric–cylinder circumference, knitted fabric thickness, applied for extension, and its corresponding strain were included. The theoretical results obtained with this modified Laplace’s law model were presented and compared to that of the experimental pressure value. We found that the prediction of the model was sufficiently

Materials 2021, 14, 4461 12 of 13

Laplace’s law is inadequate, as the tension remains unknown. Therefore, it was necessary to present adjustments to this law to recast it in terms of extension. For predicting the interface pressure, the fabric–cylinder circumference, knitted fabric thickness, applied for extension, and its corresponding strain were included. The theoretical results obtained with this modified Laplace’s law model were presented and compared to that of the experimental pressure value. We found that the prediction of the model was sufficiently good for use in compression garment design. Moreover, to verify the validity of the proposed model, experimental and predicted data were compared and very low error values were found. As limitations, we need to mention that this study included a small number of fabric types and cylinder circumferences. A further analysis should include a higher number of fabrics and more circumference. The interface pressure was measured using a rigid cylinder approach, which is frequently used in the literature, but the human body is not rigid and has anatomical variation. Due to the non-uniformity of body shapes, the pressure exerted by a garment with a given tension is not uniform and is distributed differently over the various areas of the body as well. Nevertheless, the presented results aid in the selection of materials for good-fitting compression garments.

Author Contributions: Conceptualization, Y.T.; Data collection, S.V. and W.E.; Formal analysis, Y.T. and W.E.; Funding acquisition, L.V.L.; Methodology, Y.T. and B.M.; Project administration, L.V.L.; Resources, S.V.; Software, W.E.; Supervision, B.M., T.T. and L.V.L.; Validation, W.E.; Visualization, B.M. and S.V.; Writing—original draft, Y.T.; Writing—review and editing, B.M., T.T., S.V. and L.V.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the CoE project funded through KFW project No. 51235 (PE 479— Higher Education) under grant number MBZ/201166305 in collaboration with the Ethiopian government. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing is not applicable to this article. Acknowledgments: We are thankful to Metekie Limenih for her technical assistance and cooperation during laboratory work. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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