New approach for the prediction of compression pressure using the cut strip method

Abstract

The main objective of the current research is the development of a new mathematical model for the prediction of compression pressure based on the incorporation of some new parameters. These new parameters include deformed width (w_f), true stress ( $σ_{T}$ ), true/logarithm strain ( $ɛ_{T}$ ), true modulus of elasticity ( $E_{T}$ ), along with measurement of engineering stress ( $σ_{E}$ ), engineering strain ( $ɛ_{E}$ ) and engineering modulus of elasticity ( $E_{E}$ ) at ankle position. Various brands of compression socks comprising similar fibrous combinations, as well as knit type, were purchased. Initially they were hand washed, put on the leg for marking, marked in a square, sliced, and cut into rectangular strips. The rectangular cut strips were evaluated for force–elongation characterization at different strain values considering the requisite practical elongation values (circumferential difference between leg and sock at ankle portion). For pressure measurement, a Salzmann MST MK IV pressure measuring device using a standard-sized wooden leg (circumference = 240 mm) was used. For tensile evaluation, a Testometric tensile tester was used. In this research we developed the two mathematical models based on true Young’s modulus and engineering Young’s modulus were compared with Hooke’s law and Laplace’s law. The developed models were also compared with previously existing models statistically.

Keywords

mathematical modeling hand washing compression socks cut strips

Compression socks are a highly acclaimed textile garment for pressure exertion on the lower part of the leg. It is used for the prophylaxis and treatment of venous disorders in the human lower extremities. Venous diseases range from minor asymptomatic incompetence of venous valves to chronic venous ulceration. To reduce and eliminate venous hyper pressure, a technique of compression therapy is recommended.^1,2

Compression therapy is an efficient treatment for chronic venous insufficiency, but its precise mode of action remains unclear. Compression may be achieved by different modalities, such as bandages, compression hosiery and external pneumatic compression pumps.^3–6 The wearing of (graduated) compression stockings is one of the most widely used mechanical compression approaches due to its convenient and economical merits, and sustaining and ambulant therapeutic functions.⁴ One of the main parameters of graduated compression socks is the pressure exertion on the surface of the leg. This is also known as the interface pressure.

Interface pressure (P) is the pressure exerted by the stocking on the surface of the leg and can be divided into static pressure (pressure exerted per unit area of the leg without the presence of additional deformation) and dynamic pressure (pressure exerted per unit area of the leg under the presence of multidimensional deformation). The principles involved in measuring the static and dynamic compression pressure are Laplace’s law and Pascal’s law. These both laws explain how the different modalities of compression therapy provide the deliverance of compression pressure to the limb.^7,8

Here the Laplace’s law describes the relationship between the wall tension and radius of cylinders (e.g. blood vessels) to the pressure difference due to inflation and deflation of two halves of cylindrical vessels.^9–11 The equation can be expressed as

P = \frac{T}{R}

(1)

where P denotes pressure (Pa), T is the tension in the cylinder wall (N/mm) and R is the radius of the cylinder (mm).

This law is now widely used to explain and assess the pressure delivered to a limb of a known radius by a fabric under known tension. Interface pressure (P) on an area of a limb has a positive correlation with fabric tension (T) and a negative correlation with the radius of the limb of curvature (R) at that area. Fabric tensions can be determined by measuring the tensile force (N) using an Instron machine, when elastic fabric is stretched to a specific percentage under certain load and speed.

It is well proven that garment pressure can be calculated by analyzing the tensile behavior of elastic compression garments characterized by its mechanical parameters, including modulus of elasticity which is the ratio of tensile stress to tensile strain. The higher the elasticity modulus, the lower the degree of extensibility of elastic garments. Based on such a behavior of elastic fabric, the targeted pressure between skin and garment can be achieved by calculating the reduction percentage. Reduction percentage is actually the ration of circumferential difference between leg and socks to circumference of leg.

Various studies exist in which tensile properties, necessary for the evaluation of mechanical properties (Young’s modulus of elasticity, Poisson ratio, etc.), of elastic fabrics are analyzed using stress-strain diagrams, force–elongation curves and force–extension diagrams. There are various tensile testing machines which work on the principle of uniaxial directions; these include Instron, Zwick and Testometric tensile testers^12–22 and have been used by different researchers.

Previous research^23–29 on pressure garments is limited to measurements of skin and garment interfacial pressure on the body. Some researchers^30–32 have also investigated the distribution of pressure changes on different body parts.

According to scientific literature review, the most frequent methodologies and techniques used for the prediction of compression pressure include numerical/finite element analysis techniques (characterize the dynamic interaction between compression socks and human leg) and mathematical modeling approaches (characterize the static interaction between compression socks and wooden leg).

Recently, 3D biomechanical mathematical models have been applied as a numerical simulation method to investigate pressure performances exerted by compression interventions based on the finite element analysis.^15,33–42

We found limited studies on prediction of the static distribution of compression pressure on the surface of the wooden leg using mathematical modeling approaches based on the theory of Laplace’s law and its modification to different equations.

Hui et. al.⁴³ developed a model to predict interfacial pressure exerted on a fabric tube with multilayers. All layers exhibited varying tensile properties of the various elastic fabrics. In this study, they formulated a theoretical model to predict interfacial pressure and concluded that if this assumption is still valid for a multilayer fabric tube, the pressure exerted on N layers of the fabric tube is generalized as

P = 2 π ɛ \frac{\sum_{i = 1}^{N} (Ei \times hi)}{C}

(2)

where N is the number of fabric layers sewn together to form the tube,

Ei

is the modulus of elasticity (N/mm²),

hi

is the thickness of the fabric (mm), C is the circumference of the cylindrical fabric (mm) and

ɛ

is the engineering strain.

Ng et. al.⁴⁴ attempted to design a pressure model for the human leg interface pressure by the compression garment. They tried to attribute the factors, especially the contour surface of the human leg; tensile properties of the garment and skin garment interface pressure are given below

Re = \frac{1}{1 + \frac{2 π EI (CF)}{C_{human} P}}

(3)

where EI is the modulus of elasticity (gf/cm²), CF represents the compression factor, R_e is the reduction percentage value (%) and C is the circumference of the cylindrical tube (mm).

Macintyre et al.⁹ modified Laplace’s law to a new equation to predict compression pressure. They established the below equation, equation (4), and concluded that predicted compression pressure exhibits deviations within ±2.1 mmHg. However, Laplace’s law significantly overestimates the pressure exerted on cylinder models and limbs of “small” circumference in some cases.

P = \frac{2 π T}{133.32 C}

(4)

where T is the tension (force per unit width) (N/m), C is the circumference of the leg (mm) and P is the pressure (N/mm²).

Leung et al.⁴⁵ designed a new model for compression pressure based on Laplace’s law. To design their model, they incorporated body circumference, original cross-sectional area, applied strain as a function of circumferential difference, and Young’s modulus values. The objective of their research was to design a pressure prediction model incorporating the above-mentioned factors as given below

P = \frac{2 {π EA}_{o} ɛ}{133.32 ℓ_{o} (1 + ɛ) C}

(5)

where E is the modulus of elasticity (N/mm²), C is the body circumference (mm), A_o is the original cross-sectional area of fabric (mm) and

ɛ

is the applied strain as a function of circumferential difference between socks and leg.

Dubuis et al.⁴⁶ studied a patient-specific finite element model leg under elastic compression and designed a model to evaluate compression pressure. They established a theoretical model that is given below

P = \frac{Stiff ɛ}{R}

(6)

where Stiff is the stiffness of the sock (N/mm), R is the curvature radius of the leg (mm) and ɛ is the strain of the sock in the horizontal plane.

Researchers have incorporated and introduced a variety of mechanical parameters for the prediction of interface static distribution of compression pressure on human legs as well as artificial legs. These parameters include modulus of elasticity or stiffness values, reduction percentage, stretch percentage, radius and circumference of the legs at different positions (ankle to calf), measure of stress and strain values, the tension exerted by fabric around the leg and conversion factors to model the compression pressure.^9,43–46

As per previously developed models, we observed additional mechanical parameters that have not been introduced but can be incorporated for better prediction of compression pressure. These parameters include the concept of true stress, logarithmic strain, true modulus and deformed width.

The main objective of the current research is to measure the effect of multiple machine washes on compression pressure. The main objective of the current research is to modify Laplace’s law to develop a mathematical model for the prediction of compression pressure by incorporating some new parameters, including true stress, true strain, true modulus and deformed width. Finally, we compared newly developed models with recently developed models of Leung’s model (equation (5)) and Dubuis’s model (equation (6)).

Materials and methods

A total of 13 sock samples were purchased from three different countries (Czech Republic, Turkey and Switzerland) exhibiting different compression levels (class I: 2.399–2.799 kPa; class II: 3.06–4.266 kPa; and class III: 4.532–6.132 kPa; where (1 kPa = 7.500 mmHg). While selecting the sock samples it was ensured that the type of knit/structure should be the same (plain/single jersey, 1 × 1 laid-in, 1 × 1 knit miss). All sock samples were selected for a fixed-size standard wooden leg (240 mm circumference at ankle) and exhibiting three compression levels and circumferential range between 14.5 and 21.4 cm at the lower part of the leg (ankle portion). Selected sock samples were composed of only polyamide and polyurethane content.

Total 13 sock samples were evaluated for their built-in physical and technical specifications, as shown in Tables 1 and 2, with great precision and accuracy under standard atmospheric conditions (relative humidity 65 ± 5%, temperature 20 ± 2℃) according to the standard testing protocol provided by RAL-GZ 387/1 (Medical compression hosiery quality assurance) and CEN 15831:2009.

Table 1.

Physical specifications of compression socks

Sr. No.	Brand	Sample code	Circumference at ankle (cm)	Fiber analysis (%) polyurethane/ polyamide	Category
1	Maxis	A1	19.0	30/70	CCL I (2.40–2.80 kPa)
2	Aries	A2	18.6	31/69
3	Variteks	A3	14.4	28/72
4	Variteks	B1	15.6	33/67	CCL II (3.06–4.27 kPa)
5	Variteks	B2	17.8	30/70
6	Variteks	B3	16.4	25/75
7	Sigvaris	C1	16.2	50/50	CCL III (4.53–6.13 kPa)
8	Sigvaris	C2	15.6	45/55
9	Maxis	C3	14.6	38/62
10	Variteks	C4	17.8	28/72
11	Aries	C5	15.6	40/60
12	Maxis	C6	14.6	32/68
13	Aries	C7	16.2	45/55

CCL: compression class level.

Table 2.

Technical specifications of compression socks

Code	Thickness (mm)	Fabric GSM (g/m²)	Course density (per cm)	Wales density (per cm)	Stitch density (stitches/cm²)	Main yarn	Inlaid yarn
Code	Thickness (mm)	Fabric GSM (g/m²)	Course density (per cm)	Wales density (per cm)	Stitch density (stitches/cm²)	Type	Type
A1	0.40	139.44	22.4	19.21	430.43	MF	DCV
A2	0.46	134.00	24.6	16.20	398.52	ACV	DCV
A3	0.54	149.28	18.20	20.00	360.00	MF	DCV
B1	0.90	291.60	22.00	18.00	396.00	ACV	SCV
B2	0.75	298.00	22.60	18.27	412.90	MF + ACV	DCV
B3	0.64	306.08	23.20	22.06	511.79	MF + ACV	DCV
C1	0.69	281.60	20.80	22.41	466.12	MF + ACV	SCV
C2	0.68	265.20	21.80	20.34	443.41	MF + ACV	SCV
C3	0.65	296.00	21.00	23.44	492.24	MF + ACV	DCV
C4	0.86	360.56	19.20	19.00	364.80	MF + ACV	DCV
C5	0.70	298.44	24.00	22.00	528.00	MF + ACV	DCV
C6	0.87	312.80	16.80	24.48	411.26	MF + ACV	DCV
C7	0.72	384.88	22.60	26.00	587.60	MF + ACV	DCV

ACV: air-covered;DCV: double-covered; GSM: grams per square meter; MF: multifilament; SCV: single-covered.

Hand washing

Hand washing and rinsing of samples were carried out using slightly hot water for washing at a temperature of about 35 ± 5℃ as per the specifications given in Table 3.

Table 3.

Hand washing parameters

Parameters	Dipping time	Water temperature	Samples weight	Water quantity
Hand washed	10–15 min	35 ± 5℃	250 g	5 l

All of 13 sock samples were dipped in a bucket for 10–15 min; after that, the socks were dehydrated (using hydroextraction) by placing them flatly between two layers of towels for 24 h under standard atmospheric conditions (relative humidity 65 ± 5%, temperature 20 ± 2℃) for fully drying purposes.

Marking and slicing of loop strips from ankle portion of compression socks

Initially, the dried sock samples were put onto the wooden leg in such a way that the sock samples were not fully stretched in the wale direction (longitudinal direction). When there were no creases on the surface/face of the fabric, a mark of a mean dashed line was made on the sock corresponding to the main grooved line engraved on the face of the wooden leg, as shown in Figure 1. After marking the mean dashed line, all of the sock samples were removed from the leg and allowed to relax for 24 h. After 24 h, a marked square line of 50 × 50 mm was made at the ankle portion on the face of the relaxed fabric, using the mean dashed line as reference. This was done to overcome variation due to repeatedly measuring the compression pressure and from physical handling. The putting on and pulling off of all the hand-washed sock samples was done five times. Keeping the mean of the marked square at the main grooved line around the leg, compression pressure using a Salzmann MST pressure measuring device (MK IV) was measured. It was considered that this method would reduce variability and achieve excellent repeatability of compression pressure values. After marking and pressure measurement, circular strips with a width of almost 50 mm were sliced into loop strips, as shown in Figure 1. The sliced loop strips of all 13 sock samples were put on the leg to measure deformed width (w_f) as shown in Figure 1(d).

Figure 1.

Marking: (a) locating exact grooved line on leg on face of socks; (b) square marking 50 × 50 mm; (c) coinciding mean line and main line over the sensor at ankle on leg surface; (d) deformed width after wearing loop strip; (e) grooved line (ankle portion).

One piece of socks sample from each of 13 pairs of the socks were tested for 20 wearing machine washing levels while second piece of socks sample out 13 pairs were sliced for tensile evaluation and later on pressure prediction measurements.

Pressure measurement of all sock sample was performed on a standard-sized wooden leg provided by Swisslastic, a standard leg producing company, in Switzerland; the leg specifications are given in Table 4. The standard test method for evaluation of compression pressure was developed by German institute of quality assurance and identification (RAL-GZ 387/1) and by CEN 15831.

Table 4.

Standard wooden leg specifications (RAL-GZ 387/1)

Sr. No.	Parameters	Wooden Leg
1	Size	9
2	b level girth circumference	240 mm
3	b1 level girth circumference	300 mm
4	c level girth circumference	375 mm

Measurement of compression pressure

Currently there are two major methods used for determination of compression performance: the direct in vivo method and the indirect in vitro method. In this research we performed the in vitro method for indirect evaluation of compression pressure using Salzmann pressure measuring device available in different models (i.e. the original MST, MST MK II, MST MK III, and MST MK IV models). MST MK IV (Salzmann AG, St Gallen, Switzerland) pressure measuring device was used for this research. It comprises a thin plastic sleeve (4 cm wide, 0.5 mm thick), with four paired electrical contact points connected to an air pump and a pressure transducer. Sensors are located on the medial (inner) side of a wooden leg, placed between the leg and sock as shown in Figure 2. The air pump inflates the envelope until the contacts open (when the inner pressure exerted by the air is just above external pressure due to the compression device). When the contacts open, the transducer reads the pressure at located measuring points and displays it digitally with 1-mmHg resolution. Two lengths of the probe are available. Only the shorter one (34 cm long) with four contact points was used in this study.^47,48 Such evaluation of compression measurement was performed under the standard test method RAL-GZ 387/1.

Figure 2.

MST MK IV pressure measuring device.

Derivation of mathematical model

Theoretical background

The compression pressure is defined by the force which is exerted to an area of 1 m². From Figure 3(a), it is clear that the curvature of the leg plays a deciding role in imparting pressure on the surface of the human leg. This is described by Laplace’s law (equation (1)), stating that the pressure (P) is directly proportional to the tension (T) of compression socks but inversely proportional to the radius (R) of the curvature to which it is applied.

Figure 3.

(a) Front view of leg. (b) Top view of sock position at ankle.

For the prediction of compression pressure exerted by the circular strip, we placed the strips at the ankle portion of the wooden leg as shown in Figure 3(a). To evaluate the intensity of compression pressure on the surface of the wooden leg at the ankle portion, we divided the circular wooden leg into two halves along with the deformed circular cut strip, as shown in Figure 3(b). The width of each half portion of the circular cut strip when stretched and deformed was analyzed and assigned the different notations describing the suppression of cut strip from inner side.

Figure 4 describes the mechanism of force of exertion from the internal side of the circular stretched cut strips per unit area of small arc length (dL = R.dθ) and reversal force of exertion assumed to be interface compression pressure (P). To evaluate the interface pressure (P) we assumed the following limitations of the current model.

Axial force is assumed to be zero.

Friction between sock and leg is neglected.

Circumferential force (F_L) is assumed to be acting around the leg causing the interface pressure (P).

Thickness of circular cut strip after stretching is very small so assumed to be unchanged.

Circumferential extending force (F_L) as causing decrease of width of circular cut strip, as shown in Figure 3(a), is considered as deformed width (w_f).

Figure 4.

Mechanism of suppression of circular cut strip due to wooden leg.

Figure 5.

Measurement of compression pressure.

Laplace’s law is the basic principle that is attributed to characterize the graduated nature of compression hosiery. It describes the tension produced by a pressure gradient acting across the wall of an elastic cylinder. Laplace’s law can be easily derived by considering the case for static equilibrium as the force caused by the internal pressure (P) induced by a medium of width (w) stretched on a cylinder by a force (F_L), as shown in Figure 3(a) and Figure 4. It was found that pressure exerted by a strip on the surface of the human leg has compatibility with Laplace’s law. F_L

circumferential force (N)

diameter of stretched socks/wooden leg (mm)

w_{f}

deformed width of circular strip (mm)

total length of strip (mm)

thickness of compression sock (mm)

radius of wooden leg (mm)

x-coordinate (x-component)/x-axis

y-coordinate (y-component)/y-axis/direction of pressure

segment of total length of circular strip/arc length (mm)

θ

degree angle

pressure exerted per unit area (N/mm²)

Due to static equilibrium condition

\sum F_{y} = \vec{0}

Total sum of forces will become

2 \vec{F} = \int_{0}^{π} P w_{f} Rd θ

$\vec{P} canbereplacedbyP . sin θ$ so the above equation will become

\begin{matrix} 2 F_{L} = {PRw}_{f} \int_{0}^{π} Sin θ . d θ \\ = P . R . w_{f} . - [cos 180^{\circ} - cos 0^{\circ}] \\ 2 F_{L} = {PRw}_{f} . [1 + 1] = 2 {PRw}_{f} \\ F_{L} = {PRw}_{f} \end{matrix}

(7)

Here

F_{L}

is the stretching force (circumferential force) of the cut strip around the leg, P is the pressure (N/m²), R is the radius of wooden leg,

{and w}_{f}

is the deformed width of sock strip around the leg.⁴⁹

Relationship between engineering Young’s modulus of elasticity and compression pressure

Engineering stress

Force and extension data were obtained using theTestometric tensile testing device. The corresponding engineering stresses and strains were calculated using equations (8) and (9).

Stress (σ) is defined as the force per unit area of a material, so engineering stress can be calculated as

σ_{E} = \frac{F_{L}}{A_{0}} = \frac{F_{L}}{twi}

(8)

$F_{L}$ is the stretching force applied to fabric (N), $A_{0}$ is the original cross-sectional area of the fabric (mm²), t is the thickness of fabric (mm), wi is the width of fabric (mm).

Engineering/circumferential/longitudinal strains

In terms of cut strips, strain ( $ɛ_{E}$ ) is defined as extension per unit length, so engineering strain can be defined and calculated using

ɛ_{E} = \frac{Extendedlength}{Originallength} = \frac{Δ ℓ}{ℓ_{o}} = \frac{ℓ - ℓ_{o}}{ℓ_{o}} \frac{ℓ}{ℓ_{o}} - 1

(9)

Equation (9) can be used to measure stretch ratio/draw ratio (λ) which is the reciprocal of elastic coefficients

Draw ratio = \frac{ℓ}{ℓ_{o}} = 1 + ɛ_{E} = λ

(10)

where

ɛ_{E}

is the engineering strain, ℓ is the final length (mm) and

ℓ_{o} is

the original length (mm).

In case of circular loop strip and cylindrical wooden leg, circumferential/longitudinal strains can be calculated using

\begin{matrix} ɛ_{E} = \frac{Extendedcircumferenceofloopstrip}{Originalcircumferenceofloopstrip} \\ = \frac{C_{L} - C_{S}}{C_{S}} = \frac{C_{L}}{C_{S}} - 1 \end{matrix}

(11)

where

ɛ_{E}

is the engineering strain,

C_{L}

is the circumference of the leg (mm) and

C_{S}

is the circumference of the circular sock strip (mm).

Measurement of deformed width

When the sock’s circular cut strips from the ankle portion were put on the wooden leg at the ankle portion as shown in Figure 2(d) at practical elongation, the deformations in the strip widths were measured around the leg using the following equation

\begin{matrix} Deformedwidth (W_{f}) = Finalwidth (mm) \\ - Initial width (mm) \end{matrix}

(12)

Engineering modulus

Young’s modulus, or the modulus of elasticity, is one of the most important measures of the mechanical properties of a material. However, it is difficult to obtain an exact stress-strain diagram on textile fibers even if we use the load-extension diagram as a substitute for the stress-strain diagram. This is because the load-extension diagram does not make a straight line and because the percentage of extension is so high that many difficulties occur in determining Young’s modulus. Generally, Young’s modulus is measured while elongation is kept very small.

Take an ideally elastic material satisfying Hooke’s law and let $σ_{E}$ be the engineering stress and $ɛ_{E}$ be an engineering strain at any point in the straight line region of the stress-strain diagram. Then, Young’s modulus E is defined as the ratio of engineering stress to engineering strain.^9,50–54 So we can write

E_{E} = \frac{Engineeringstress}{Engineeringstrain} = \frac{σ_{E}}{ɛ_{E}} = \frac{F_{L}}{A_{0} ɛ_{E}}

(13)

where

σ

_E is the engineering stress (N/mm²),

ɛ_{E}

is the engineering strain and E_E is the modulus of elasticity (N/mm²).

Comparing equation (12) and equation (7), we can get

\begin{matrix} F_{L} = F_{L} \\ {PRw}_{f} = E_{E} A_{0} ɛ_{E} \\ P_{E} = \frac{E_{E} A_{0} ɛ_{E}}{{Rw}_{f}} \\ P_{E} = \frac{2 {π E}_{E} A_{0} ɛ_{E}}{{Cw}_{f}} \end{matrix}

(14)

where

E_{E}

is the engineering elastic modulus (N/mm²),

ɛ_{E}

is the engineering strain, P_E is the mathematical measurement of compression pressure based on true modulus, C is the circumference (mm) and

w_{f}

is the deformed width (mm). Equation (14) is named as Model 2 with respect to engineering Young’s modulus.

Relationship between true Young’s modulus of elasticity and compression pressure

True stress

The stress is calculated based on the instantaneous area at any instant of load, then it is the true stress.⁵⁵ There could exist a relationship between the true stress and engineering stress once no volume change is assumed in the specimen. Under this assumption

Truestress (σ_{T}) = \frac{F_{L}}{A}

(15)

where A is the actual area of the cross-section corresponding to load

F_{L}

Assuming material volume remains constant

A . ℓ = A_{0} . ℓ_{o}

(16)

On the basis of assumption of equation (16), equation (15) can be written as

σ_{T} = \frac{F_{L} A_{o}}{{AA}_{0}} = \frac{F_{L} A_{0}}{A_{0} A} = \frac{F_{L}}{A_{0}} \frac{ℓ}{ℓ_{o}}

(17)

Using equations (8) and (10) in equation (17) gives

σ_{T} = σ_{E} (1 + ɛ_{E})

(18)

True/logarithm strain

True strain is defined as the instantaneous increase rate in the instantaneous gauge length⁵⁵

\begin{matrix} ɛ_{T} = \int_{ℓ}^{ℓ} \frac{d ℓ}{ℓ} = ln (1 + ɛ_{E}) \\ ɛ_{T} = \ln (1 + ɛ_{E}) \end{matrix}

(19)

True elastic modulus/Young’s logarithm modulus

Using equations (18) and (19)

E_{T} = \frac{Truestress}{Truestrain} = \frac{σ_{T}}{ɛ_{T}} = \frac{σ_{E} (1 + ɛ_{E})}{ln (1 + ɛ_{E})}

(20)

Equations (19) and (8) can be modified to

\begin{matrix} E_{T} \ln (1 + ɛ_{E}) = \frac{F_{L} (1 + ɛ_{E})}{A_{0}} \\ F_{L} = \frac{E_{T} A_{0} ln (1 + ɛ_{E})}{(1 + ɛ_{E})} \end{matrix}

(21)

Equating equation (7) and equation (21), relation will become

\begin{matrix} F_{L} = F_{L} \\ \frac{E_{T} A_{0} ln (1 + ɛ_{E})}{(1 + ɛ_{L})} = {PRw}_{f} \\ P_{T} = \frac{E_{T} A_{0} \ln (1 + ɛ_{E})}{(1 + ɛ_{E}) {Rw}_{f}} \end{matrix}

(22)

Equation (22) can be named as Model 1 with respect to true Young’s modulus.

We know that the circumference of leg (C) is C = $2 π R,$ so the radius can be calculated using $R = \frac{C}{2 π}$ .

Putting the value of radius R in equation (22), we can write

P_{T} = \frac{2 {π E}_{T} A_{0} ln (1 + ɛ_{E})}{(1 + ɛ_{E}) {Cw}_{f}}

(23)

where

E_{T}

is the true Young’s modulus (N/mm²),

A_{0}

is the original cross-sectional area (mm²),

ɛ_{E}

is the engineering strain, R is the radius of leg (mm), P_T is the mathematical pressure measured based on True modulus values, and

w_{f}

is the deformed width (mm).

Calculation of pressure measurement

Experimental measurement of compression

To measure compression pressure induced by all of the 13 sock samples of varying circumferences, we put the sock samples on the wooden leg keeping the MST Salzmann sensors sleeve between the wooden leg and sock samples. We ran the machine and measured the readings of compression pressure values when the data were relatively stable. In this research, we noted five readings of compression pressure taken from the device at the ankle portion and averaged the data recorded as final pressure, see Table 5.

Table 5.

Measurement of compression pressure

Sample code	Pressure (kPa)	Standard deviation
A1	2.24	0.25
A2	2.40	0.40
A3	3.07	0.27
B1	3.65	0.35
B2	3.75	0.31
B3	4.34	0.37
C1	4.71	0.10
C2	4.83	0.30
C3	5.29	0.30
C4	5.33	0.20
C5	5.46	0.15
C6	6.26	0.27
C7	6.46	0.35

Force–extension curve analysis

A Testometric tensile testing machine was used to evaluate force–extension analysis of the compression sock using the cut strip method as shown in Figure 6(a) and (b). Cut strips were clamped between two jaws of Testometric tensile testing device with one of them was static while second moves to stretch the strips up to level of practical elongation for each samples respectively. The specifications were followed as per BS EN 14704-1 standard test method. Test specifications were the following: tensile rate of 100 mm/min; specimen dimension of 146–190 × 40–55 mm (lengthways dimension of all cut strips × widthways dimension of all cut strips); adjusted gauge length of 50 mm. All the cut strips were loaded up to practical elongation (stretch percentage) context to circumference of the wooden leg at the ankle portion for five times and the results were measured on the fifth cycle. After each cycle, each cut strip was relaxed for 15 min. Load-extension data of all the 13 sock samples are shown in Figure 7.

Figure 6.

Cut strips: (a) clamped strip without extension, (b) clamped strip after extension.

Figure 7.

Force–extension curve corresponding to circumferential stretch difference between sock and leg.

As the circumference of the leg is fixed to 240 mm but each of the sock samples exhibits different circumferences at the ankle, so the force of extension at a specific practical elongation (extension level) is different. The load-extension curve was obtained by stretching the sock’s cut strips up to practical elongation. The stretch percentage⁴⁸ can be calculated using the below relation

\begin{matrix} Stretch % = \frac{Circumferenceofleg - Circumferenceofsocks}{Circumferenceofsocks} \\ \times 100 = \frac{C_{L} - C_{S}}{C_{S}} \times 100 \end{matrix}

(24)

Measurements of specifications of all circular and rectangular cut strips as well as wooden leg circumference results are given below in Table 6.

Table 6.

Specifications of cut strips/compression socks/wooden leg

	Force	Extension	Strip width	Thickness	Area	Final length	Strip circumference
Unit	(N)	(mm)	(mm)	(mm)	(mm²)	(mm)	(mm)
Code	F_L	$Δ ℓ$	w_i	t	$A_{o}$	ℓ	$S_{c}$
A1	3.962	13.15	50.0	0.40	20.00	63.15	190
A2	4.720	14.51	46.5	0.46	21.39	64.51	186
A3	3.896	28.95	40.0	0.54	21.44	78.95	144
B1	7.336	24.07	54.0	0.90	48.65	74.07	156
B2	6.266	21.43	50.0	0.75	37.50	71.43	178
B3	7.876	22.29	54.0	0.64	34.56	72.29	164
C1	8.374	24.07	54.0	0.69	37.26	74.07	162
C2	8.336	25.95	48.0	0.68	32.64	75.95	156
C3	9.587	32.19	52.0	0.66	34.32	82.19	146
C4	8.904	18.97	50.0	0.86	43.00	68.97	178
C5	9.672	26.92	51.3	0.68	34.88	76.92	156
C6	11.210	32.19	50.0	0.87	43.50	82.19	146
C7	12.572	24.07	55.0	0.72	39.60	74.07	162

Results and discussion

Measuring results of engineering stress (

σ_{E})

, circumferential/longitudinal/engineering strain (

ɛ_{E}

), engineering modulus

(E_{E})

, theoretical engineering pressure (P_E) and experimental pressure (P_S) calculated using equations (8), (9), (13) and (14) are given in Table 7. Deformed width (

w_{f}

) results, as shown in Figure 2(d), calculated using equation (12) are also given in Table 7.

Table 7.

Measurement of engineering stress, strain, modulus and theoretical compression

Sample code	Engineering strain	Deformed width on leg	Engineering stress	Engineering modulus	Model 2 (EYM)	Experimental pressure
Code	unitless	(mm)	(N/mm²)	(N/mm²)	(kPa)	(kPa)
Code	$ɛ_{E}$	$w_{f}$	$σ_{E}$	$E_{E}$	$P_{E}$	$P_{S}$
A1	0.263	44.3	0.198	0.753	2.34	2.24
A2	0.290	43.0	0.221	0.760	2.87	2.40
A3	0.667	36.0	0.182	0.273	2.83	3.07
B1	0.538	48.0	0.151	0.280	4.00	3.65
B2	0.348	44.0	0.167	0.480	3.73	3.75
B3	0.463	48.0	0.228	0.492	4.29	4.34
C1	0.481	49.0	0.225	0.467	4.47	4.71
C2	0.538	42.0	0.255	0.474	5.19	4.83
C3	0.644	46.5	0.279	0.434	5.39	5.46
C4	0.348	45.0	0.207	0.594	5.18	5.33
C5	0.538	47.8	0.277	0.515	5.29	5.29
C6	0.644	45.0	0.258	0.400	6.52	6.26
C7	0.481	51.5	0.317	0.659	6.39	6.46

EYM: engineering Young’s modulus.

Measured values of true stress (

σ_{T})

, true/logarithmic strain (

ɛ_{T}

), true elastic modulus

(E_{T})

and theoretical pressure with respect to true modulus (P_T) and experimental pressure (P_S) were calculated using equations (18), (19), (20) and (23). Deformed width (

w_{f}

) values were calculated using equation (12). The results for each cut strip sample are shown in Table 8.

Table 8.

Measurement of true stress, strain, modulus and theoretical true pressure

Sample code	Engineering strain	Deformed width on leg	True stress	True strain	True modulus	Model 1 (TYM)	Experimental pressure
Code	unitless	(mm)	(N/mm²)	unitless	(N/mm²)	(kPa)	(kPa)
Code	$ɛ_{E}$	$w_{f}$	$σ_{T}$	$ɛ_{T}$	$E_{T}$	$P_{T}$	$P_{S}$
A1	0.263	44.3	0.250	0.234	1.071	2.340	2.24
A2	0.290	43.0	0.285	0.255	1.117	2.872	2.40
A3	0.667	36.0	0.303	0.511	0.593	2.832	3.07
B1	0.538	48.0	0.232	0.431	0.538	3.999	3.65
B2	0.348	44.0	0.225	0.299	0.754	3.726	3.75
B3	0.463	48.0	0.334	0.381	0.876	4.294	4.34
C1	0.481	49.0	0.333	0.393	0.847	4.472	4.71
C2	0.538	42.0	0.393	0.431	0.912	5.193	4.83
C3	0.644	46.5	0.459	0.497	0.924	5.395	5.46
C4	0.348	45.0	0.279	0.299	0.934	5.178	5.33
C5	0.538	47.8	0.427	0.431	0.990	5.295	5.29
C6	0.644	45.0	0.424	0.497	0.852	6.518	6.26
C7	0.481	51.5	0.470	0.393	1.197	6.388	6.46

TYM: true Young’s modulus.

Statistical analysis of developed pressure prediction models

Statistical analysis of the newly developed pressure prediction models Model 2 (EYM) and Model 1 (TYM), derived as equations (14) and (23) respectively, were compared with the experimental pressure measurements (P_S). In this research we developed two mathematical models named as Model 2 (EYM) measured with respect to engineering Young’s modulus and Model 1 (TYM) with respect to true Young’s modulus. The measured results were statistically analyzed and compared with experimental pressure measuring results (P_S).

The linear regression analysis tool was used to compare Model 2 (EYM) versus experimental pressure (P_S) is shown in Figure 8. Figure 8 shows the coefficient of determination value (R-squared value) of Model 2 (EYM), explaining 97.02% of experimental results.

Figure 8.

Statistical comparison of predicted compression pressure Model 2 (EYM) with experimental results.

Figure 9 shows the effect of theoretical Model 1 (TYM) on experimental compression pressure. The analysis was conducted using the linear regression analysis tool to investigate how much the independent variable (experimental pressure) is explaining the dependent variable, Model 1 (TYM). Figure 9 shows the coefficient of determination value (R-squared value) of Model 1 (TYM), explaining 97.02% of experimental results.

Figure 9.

Statistical comparison of predicted compression pressure Model 1 (TYM) with experimental results.

Comparison of newly developed pressure prediction models with existing models

Table 9 represents the calculated values of compression pressure measured experimentally and compared with the newly developed models, Model 2 (EYM) and Model 1 (TYM), as well as the recently developed or existing models developed by Leung et al. and Dubuis et al. using equations (5) and (6).

Table 9.

Comparison of theoretical and experimental compression pressure values

Sample code	Experimental pressure (kPa)	Model 1 (TYM) (kPa)	Model 2 (EYM) (kPa)	Leung’s model (kPa)	Dubuis’s model (kPa)
A1	2.24	2.34	2.34	1.64	2.07
A2	2.40	2.87	2.87	1.91	2.47
A3	3.07	2.83	2.83	1.22	2.35
B1	3.65	4.00	4.00	2.50	4.29
B2	3.75	3.73	3.73	2.43	2.66
B3	4.34	4.29	4.29	2.82	4.28
C1	4.71	4.47	4.47	2.96	4.38
C2	4.83	5.19	5.19	2.84	4.53
C3	5.46	5.39	5.39	3.05	5.02
C4	5.33	5.18	5.18	3.46	4.28
C5	5.29	5.29	5.29	3.29	5.06
C6	6.26	6.52	6.52	3.57	5.87
C7	6.46	6.39	6.39	4.44	6.58

EYM: engineering Young’s modulus; TYM: true Young’s modulus.

Figure 10 shows the effect of the recently developed model by Leung et al. for the prediction of compression pressure versus experimental compression pressure. The analysis was conducted using the linear regression analysis tool to investigate how much the independent variable (experimental pressure) is explaining the dependent variable, Leung’s model. Figure 10 shows the coefficient of determination value (R-squared value) of Leung’s model, explaining 86.39% of experimental results.

Figure 10.

Leung’s model versus experimental pressure results.

Figure 11 shows the effect of the recently developed model by Dubuis et al. for the prediction of compression pressure versus experimental compression pressure. The analysis was conducted using the linear regression analysis tool to investigate how much the independent variable (experimental pressure) is explaining the dependent variable, Dubuis’s model. Figure 11 shows the coefficient of determination value (R-squared value) of Dubuis’s model, explaining 88.61% of experimental results.

Figure 11.

Dubuis’s model versus experimental pressure results.

Conclusion

In this research

We have modified Laplace’s law by incorporating the new term of deformed width (w_f), so the newly modified Laplace’s law becomes P = 2πF/C w_f, derived using the original Laplace’s law (P = T/R) equation.

We developed two new models, Model 1 (TYM) and Model 2 (EYM), based on the true Young’s modulus and the engineering Young’s modulus. In the development of these models we incorporated some new parameters including true stress ( $σ_{T}$ ), true/logarithm strain ( $ɛ_{T}$ ), true modulus of elasticity ( $E_{T}$ ), deformed width ( $w_{f}$ ), along with measurement of engineering stress ( $σ_{E}$ ), engineering strain ( $ɛ_{E}$ ) and engineering modulus of elasticity ( $E_{E}$ ) at b position (ankle, minimum girth area). Statistical validation of the newly developed models was carried out using linear regression analysis to assess the experimentally measured compression pressure results.

The newly developed models were compared with the previously developed models of Leung et al. and Dubuis et al.; we found that the newly developed models, that is Model 1 (TYM) and Model 2 (EYM), are more precise and accurate than the aforementioned existing models.

Footnotes

Acknowledgements

Author is pleased to mention here the technical support and provision of compression socks by different brands of compression socks manufacturers. These compression socks Manufacturers who supported us are mainly Sigvaris, Maxis, Aries and Variteks.

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: This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic and the European Union’s European Structural and Investment funds in the frames of Operational Program Research, Development and Education—project “Hybrid Materials for Hierarchical Structures” (HyHi, Registration No. CZ.02.1.01/0.0/16_019/0000843).

ORCID iD

Hafiz Faisal Siddique

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