Clothing Pressure Modeling Using the Modified Laplace’s Law

Abstract

Today, various kinds of pressure garments are designed for specific applications in medical and sports fields. Knitted garments are the most used in these applications due to their high extensibility. The objective of the investigation reported in this article was to develop a theoretical relationship based on Laplace’s law, which describes the compression behavior of knitted compression samples in quasi-static deformation from an initially relaxed state to an extended state. Even though several researchers have used Laplace’s law, there is some discord between theoretical and experimental results. So, it is essential to pinpoint the most important parameters that influence the mechanical properties of the compression knitted garment in order to better describe the interface pressure it applies to the human body. Fabric parameters that influenced the interface pressure, such as elasticity modulus, strain, and thickness, were determined and integrated into Laplace’s law.

Keywords

clothing pressure young modulus flexiforce sensor knitted garment strain

Compression therapy has been used for many centuries in the treatment of edema and other venous disorders in the lower limbs (Mikučionienė & Milašiūtė, 2017). The concept of compression therapy stockings is based on a simple mechanical principle: By compressing the limb with graduated compression—strong at the ankle and decreasing as it goes up the leg—the compression stocking helps venous return, decreases venous pressure, prevents venous stasis and deterioration of venous walls, and efficiently relieves aching and heavy legs. In recent years, researchers have brought a new level of interest to the subject of compression clothing performance and compression ability. According to the literature, the degree of pressure applied by compression clothing to the human skin depends on a number of complex factors. There are some factors related to the garment used to generate interface pressure and others factors related to human leg morphology (Barhoumi, Marzougui, & Ben Abdessalem, 2018; Bera, Chattopadhyay, & Gupta, 2016; Isherwood, Robertson, & Rossi, 1975; Maqsood, Nawab, Umar, Umair, & Shaker, 2017).

Although diverse researchers have concentrated on the kind of fabric used in compression hosiery manufacturing (Koncar, 2016; Liu & Zhou, 2019), few of them have demonstrated the performance of woven fabrics in generating garment pressure. Maqsood, Nawab, Umar, Umair, and Shaker (2017) found that bi-stretch woven fabrics had better compression performance after many washes and maintained the durability of their effect after repeated use, while knitted stretchable fabrics lost their compression properties after repeated use. Most scholars in this area demonstrated that knitted garments are the most commonly used fabric in medical and sports hosiery (Mikučionienė & Milašiūtė, 2017; Thomas, 2014). This result is based on the stretch ability and high elasticity of the knitted garment, which makes it more comfortable and efficient (Bera et al., 2016; Loginov, Grishanov, & Harwood, 2002; Liu, Lao, & Wang, 2013; Sarı & Oğlakcıoğlu, 2018). The most outstanding characteristics of elastic compression garments are conserving body shape in addition to their own durability. Furthermore, fabric recovery is as important as fabric extension. In fact, the degree of elasticity in a compression elastic garment determines its application and its end use (Shishoo, 2005).

In order to characterize the mechanical performance of knitted compression fabrics, we have to analyze their extension-tensile behavior. Referring to earlier works (Chen, Liu, Zhang, & Wang, 2013; Liu, Kwok, & Lao, 2010), the extension-tensile property of knitted compression fabrics plays an important role in both their manufacturing and their end use. It has been established that the tensile modulus and fabric elongation of compressive knitted garments influence the pressure distribution throughout the leg. For this reason, elastic knitted garments with various tensile properties were used to evaluate compression ability and quantify the interface pressure. This quantification needs a model that predicts the interface pressure value. The most famous model used in this area is Laplace’s law, presented in Equation 1. This law was used by Cheng et al. (1984) and Macintyre (2007) to predict pressure delivered to a cylinder of known radius by a fabric under known tension:

P = \frac{T}{R},

where P (N/m² or Pa) is the interface pressure, T (N/m) is the garment tension per unit of length, and R (m) is the radius of the body part to which the interface pressure is applied (Pellicer, García-Morales, & Hernández, 2000; Thomas, 2003). Many researchers focused on demonstrating the accuracy of the use of Laplace’s law in predicting interface pressure values (Barhoumi, Ben Abdessalem, & Marzougui, 2018; Gaied, Drapier, & Lun, 2006). They made a comparison between direct measurement using a developed device and indirect measurement using the simplified Laplace’s law, knowing the local radius of the leg’s curvature. They found that there were some differences between the two methods of measurement. This indicates that the parameters integrated in Laplace’s law were insufficient and deficient to predict the interface pressure. In light of this finding, it is critical to develop a model that better predicts the interface pressure applied by the elastic knitted garment.

The aim of our work is to develop a modified Laplace’s law by introducing factors that influence the interface pressure value due to the compression elastic garment in a specific measurement area (point). In this article, we attempt to formulate a theoretical model based on a simplified Laplace’s law for predicting the interface pressure generated by a compressive elastic knitted fabric with various tensile properties. In order to verify the validity of the proposed model, experimental data and theoretical results were compared and error values were analyzed.

Materials and Methods

Theoretical Model

A compressive knitted garment’s radius is smaller than that of the human leg. The garment’s radius extends as it is worn. When the garment extends, it compresses the body by exerting an external interface pressure. In order to be analyzed, this compressive behavior has to be simulated. This phenomenon is presented in Figure 1.

Figure 1.

Interface pressure applied by the compressive garment on a leg model.

Where P is the interface pressure, T is the garment tension per unit of length, R is the leg radius, and e is the garment thickness. Lange and Dean (1967) showed that tension T (N/m) could be expressed in terms of applied load F(N) by the length. In our study, the leg circumference C (m) corresponds to the length. Therefore, we obtain the following equation:

T = \frac{F}{C} .

According to Hui and Ng (2001), the load F (N) is expressed by Equation 3:

F = 2πσ e R,

where R and e are the radius of the cylindrical model and the knitted tube thickness, respectively, expressed in meter (m). σ (N/m²) is the stress applied to the elastic knitted tube when worn. According to Hook’s law, the stress is given by Equation 4:

σ = ∊ E,

where ∊ is the fabric extension and E is the elastic modulus. Substituting Equation 4 onto Equation 3, we obtain:

F = 2 π ∊ E e R .

Then, the new expression of the tension T is given in Equation 6:

T = \frac{2 π ∊ E e R}{C} .

Substituting the expression of the tension T (Equation 6) in the basic formula of Laplace’s law, we derived the new model for interface pressure P, Equation 7:

P = \frac{2 π ∊ E e}{C} .

Cylinder Models

In this investigation, we measure the interface pressure values in six measurement points according to the European standard Medical Compression Hosiery (European Committee for Standardization, 1996). Size S was chosen for the circumference measurement and the knitted samples. In order to simulate the human leg, six wooden cylinder models were used. In fact, the section of a leg is similar to a cylinder with circumference C (Al Khaburi, Dehghani-Sanij, Nelson, & Hutchinson, 2011; Jung et al., 2018; Partsch, Partsch, & Braun, 2006). C_i refers to the circumference code of the cylinder (i represents, respectively, B, B1, C, D, E, and F), as shown in Figure 2.

Figure 2.

Measurement points (European Committee for Standardization, 1996).

Table 1 summarizes the positions of the measurement points and the circumference values in the corresponding points.

Table 1.

Measurement Points and Their Circumference Values.

Length Code	Leg Length (cm)	Circumference Code	Circumference Value (cm)
LB	10	C_B1	27.5
LB₁	19	C_B	29.0
LC	27	C_C	32.5
LD	35	C_D	34.5
LE	41	C_E	36.0
LF	54	C_F	37.5

In order to measure the interface pressure applied by a knitted compression band on the cylinder models, the garment was first laid flat on a table, relaxed, and under no extension; its circumference in a relaxed state, C_k, was measured. Secondly, the elastic compression knit was put over the wooden cylinder model. Its circumference when worn corresponds to C_i. These circumferences measurements were used to calculate the percentage of extension, ∊ in width direction for each sample when worn on the wooden cylinder models using Equation 8:

∊ = \frac{C_{i} - C_{k}}{C_{k}} \times 100.

Experimental Equipment

The FlexiForce® A201 (Mescan, Les Mureaux, France) sensor (Figure 3) was used to measure interface pressure applied by the knitted compressive bands on the wooden models. It is a piezoresistive sensor that is 14 mm wide, 200 mm long, and has a sensing area of 71.1 mm². This FlexiForce sensor acts as a force-sensing resistor in an electrical circuit. When the force is unloaded, the resistance is very high. When applying a force on the sensor’s active area, the resistance decreases. The accuracy of using the FlexiForce sensor in measuring experimental values of the pressure exerted between the human skin and compression garments has been reported in the past (Bachus, DeMarco, Judd, Horwitz, & Brodke, 2006; Barhoumi, Marzougui, & Ben Abdessalem, 2018; Polliack et al., 2000). The advantage of using this sensor is that it has tolerable drift (low frequency change in the sensor with time), repeatability (ability to repeat a measurement in the same condition), linearity (consistency of measurements over the range of measurements), and hysteresis (property of a system whose evolution does not follow the same path). It is so thin that its presence does not affect measured values. Most of force sensors available on the market are not suitable for measuring interface pressure because of their large size and complex equipment, which often result in measurement difficulties. The FlexiForce sensor is considered the best sensor used for this type of application, which needs high sensitivity when compared to piezoelectric, capacitive, and hydrostatic sensors (Buis & Convery, 1979). The calibration of the FlexiForce sensor was performed before use by employing different mass levels: 5, 15, 20, 30, 40, and 50 g. The interface pressure is applied at a surface equal to 71.1 mm². Knowing that 1 g corresponds to 9.8 × 10⁻³ N, the unit used to express interface pressure is the millimeter of mercury (mmHg). Equation 9 is used to convert the mass’s unit (g) into pressure unit (mmHg):

Figure 3.

FlexiForce® Sensor Model A201.

I n t e r f a c e p r e s s u r e (m m H g) = 1.03 \times m a s s (g r a m) .

The FlexiForce sensor was connected to an electric circuit system providing 15 V developed according to the recommended circuit in the sensor datasheet. The circuit output was connected to a multimeter. A linear fitting curve was used to describe the interface pressure (mmHg) in terms of the measured voltage, as shown in Figure 4.

Figure 4.

Relationship between pressure and voltage.

From this curve, the interface pressure value was obtained according to Equation 10:

I n t e r f a c e p r e s s u r e (m m H g) = 7.51 \times V o l t a g e (V) + 0.021 .

Knitted Samples Characteristics

As shown in Figure 5, five samples (G₁, G₂, G₃, G₄, and G₅) of compressive knitted bands with various manufacturing parameters were produced on a Santoni SM8 TOP2 machine.

Figure 5.

The tested elastic bands.

The gauge of this machine was E28 and its diameter was 24 in. The ground yarn used for the construction of our samples was a polyamide 6-6 filament. Polyamide 6-6 is often used for this kind of application because of its moisture resistance, high abrasion resistance, high flexibility and elasticity, and low shrinkage. Elastane yarn was knitted simultaneously with the ground yarn, using a plating technique. The characteristics of tested samples are presented in Table 2.

Table 2.

Characteristics of Tested Samples.

Characteristics	Unit	Sample Code
Characteristics	Unit	G1	G2	G3	G4	G5
Young’s modulus	MPa	0.41	0.63	0.78	0.69	0.73
Polyamide 6-6 count	dTex	77	77	77	77	77
Elastane yarn count	dTex	17	44	44	17	44
Elastane percentage	%	15	8	15	10	10
Breaking strength	N	287.01	221.91	335.66	300.30	268.30
Fabric weight	g/m²	199	232	227	222	244
Fabric thickness	mm	0.80	0.87	0.95	0.74	0.95
Fabric tube circumference	cm	20.60	20.60	20.60	20.60	20.60

During wear, compression knitted garments are extended before reaching their maximum elongation. In order to study properties under these use conditions, the maximum extension was taken as 80% (Baussan, Bueno, Rossi, & Derler, 2010; Chen et al., 2013; Dai, Li, Liu, & Kwok, 2006). As only the course-wise direction is responsible for exerting interface pressure, all tensile tests were carried out in the course direction of the compression knitted bands. Simple tensile tests were carried out according to the conditions of the Standard ISO 13934 (International Organization for Standardization, 2014), which consist of applying a traction effort at a constant speed of 100 mm/min to a rectangular fabric sample (200 × 50 mm²). At least five specimens per sample were tested and the average values were recorded. The induced tension for a particular strain was obtained throughout the load-strain curves presented in Figure 6.

Figure 6.

Tension–strain curves of the tested samples.

When worn, the knitted compression tube extends. The applied tension at this extension value is obtained from the tension–strain curve. The experimental procedure is shown in Figure 7. The sample is carefully worn on the leg in order to cover the active area of the sensor. Then, voltage values are recorded and the interface pressure is calculated using Equation 8.

Figure 7.

Experimental interface pressure measurement.

These measurements were repeated 5 times, and the mean interface pressure values were calculated and discussed.

Results and Discussion

The Effect of Mechanical Parameters on Interface Pressure

Effect of Young’s modulus

As shown in Figure 8, at the same strain value, the interface pressure increases when Young’s modulus increases. This variation is related to the mechanical behavior of the knitted samples.

Figure 8.

Interface pressure exerted by the different samples (G1, G2, G3, G4, and G5) at different strain levels.

The greater the Young’s modulus, the higher the tension needed to obtain the desired strain. Consequently, the interface pressure becomes higher. These results are in agreement with those found by Chen, Liu, Zhang, and Wang (2013), who demonstrated that clothing pressure exerted by fabric with a greater elastic modulus is generally higher than that applied by fabric with a lower elastic modulus for the same elongation. Furthermore, Young’s modulus is sensitive to applied strain. This implies that when the knitted compression bands extend, Young’s modulus will be higher and the knitted band will be more compressive.

Effect of extension

Figure 8 shows that at the same Young’s modulus and thickness (same knitted tube), changes in interface pressure values are dependent on fabric extension. In fact, knitted tube circumference is smaller than leg circumference, and knitted tube extends in order to reach the leg diameter. Consequently, the shape of the leg is variable, calf circumference is greater than that of the ankle, and a distribution of interface pressure is obtained.

Liu et al. (2006) showed the influence of body shape in their research on fabric compression. They concluded that body shape has a significant effect on fabric extension as a result of interface pressure distribution. Moreover, Kawabata, Tanaka, Sakai, and Ishikawa (1988) showed that the variation of fabric extension reflects the change of pressure exerted by fabric.

When the fabric of clothing is tight, some limits are created for the human body. This causes fabric extension, produces fabric tension, and generates interface pressure. Researchers have also found that fabric extension has a significant effect on the applied interface pressure at the same body part circumference (Liu et al., 2006; Wang, Felder, & Cai, 2011; Zhong & Zhang, 2006).

Effect of fabric thickness

Fabric thickness greatly influences the mechanical properties of textile materials (Kar, Fan, & Yu, 2011; Lee & Kim, 2000). Figure 8 presents the effect of this parameter on the interface pressure applied by the compression knitted bands. It is noticeable that the thicker the compression knitted bands, the greater the interface pressure. This result can be explained by the fact that at the same fabric circumference, the thickest compression knitted fabric generates more internal force to the human skin. Indeed, a thicker sample induces a local reduction in the radius curvature of the leg, which affects the local interface pressure value. Al Khaburi, Dehghani-Sanij, Nelson, and Hutchinson (2012) studied the influence of this parameter in pressure applied by compression bandages. They showed that considering bandage thickness while predicting the interface pressure applied by a medical compression bandage involves accurate estimation.

Assessing the Accuracy of a Modified Laplace’s Law Against a Simple Laplace’s Law

Modeling and designing compression garments with Laplace’s law was based on the relationship between interface pressure, fabric tension, and leg radius. In this work, a theoretical model based on this law was developed. Laplace’s law was modified by adding mechanical parameters that have a notable influence on the obtained pressure.

In order to highlight the necessity of this modification, a comparison between the modified and the initial Laplace’s law is needed. The sample G1 was chosen to underline this comparison. The error percentage value was calculated using Equation 11. The obtained results are presented in Table 3.

Error percentage = | \frac{Theoretical pressure value - Experimental pressure value}{Experimental pressure value} | ×100 .

By comparing the theoretical values obtained by Laplace’s law and the theoretical values obtained by a modified and experimental Laplace’s law, we see that the modified Laplace’s law is more accurate. In fact, the errors between theoretical values obtained by the modified and experimental Laplace’s law are in the range of 1.47–3.83%. This is lower than the error between the experimental interface pressure measurement and value obtained by Laplace’s law (between 4.91% and 22.67%). This discrepancy can be explained by the fact that Laplace’s law considers only fabric tension and radius of the area on which interface pressure is applied (Aghajani, Jeddi, & Tehran, 2011; Barhoumi, Marzougui, & Ben Abdessalem, 2018; Gaied et al., 2006). Thus, we can note that the use of tension T and leg radius R remains insufficient. The modified Laplace’s law involves a new relationship between interface pressure and other mechanical parameters in order to be more precise and accurate.

Table 3.

Comparison Between the Interface Pressure Values Predicted by Laplace’s Law and Those Obtained Using the Modified Laplace’s Law.

Experiment Number	Extension (%)	P_ex (mmHg)	P_L (mmHg)	P_m (mmHg)	Err (P_ex − P_L; %)	Err (P_ex − P_m; %)
1	30	15.52	19.30	15.76	16.68	1.55
2	40	20.69	25.38	21.03	22.67	1.64
3	50	25.87	27.14	26.25	4.91	1.47
4	60	31.04	34.53	29.85	11.24	3.83
5	70	36.21	40.53	37.30	11.93	3.01
6	80	41.38	44.40	42.02	7.30	1.55

Note. P_ex = experimental interface pressure; P_L = theoretical interface pressure obtained by Laplace’s law; P_m = theoretical interface pressure obtained by the modified Laplace’s law; Err (P_ex − P_L) = error percentage between experimental interface pressure values and those obtained by Laplace’s law; Err (P_ex − P_m) = error percentage between experimental interface pressure values and those obtained by the modified Laplace’s law.

Validation of a Modified Laplace’s Law

To evaluate the effectiveness of the developed theoretical model, we tested three samples with different combinations. The characteristics of these samples are presented in Table 4.

Table 4.

Characteristics of Tested Samples.

Characteristics	Unit	Sample Code
Characteristics	Unit	F1	F2	F3
Young’s modulus	MPa	0.27	0.39	0.46
Polyamide 6-6 count	dTex	77	77	77
Elastane yarn count	dTex	17	44	17
Elastane percentage	%	8	10	15
Breaking strength	N	205.25	235.17	280.56
Fabric weight	g/m²	169	175	185
Fabric thickness	mm	0.75	0.85	0.92
Fabric tube circumference	cm	20.6	20.6	20.6

Table 5 shows experimental measurements and the corresponding theoretical calculations obtained by Equation 7.

Table 5.

Comparison Between Theoretical and Experimental Interface Pressure.

Sample	Strain (%)	Experimental Interface Pressure (mmHg)		SD	Error Percentage (%)
Sample	Strain (%)	Theoretical Interface Pressure (mmHg)	Average Interface Pressure Value (mmHg)	SD	Error Percentage (%)
F1	30	9.88	9.67	.54	2.18
	40	13.18	13.40	.43	1.68
	50	16.47	16.89	.18	2.49
	60	19.77	20.51	.51	3.63
	70	23.06	23.47	.47	1.77
	80	26.35	26.66	.58	1.18
F2	30	15.25	15.13	.13	0.76
	40	20.34	20.20	.68	0.67
	50	25.42	25.08	.16	1.35
	60	30.51	30.08	.81	1.14
	70	35.59	35.77	.99	0.51
	80	40.68	40.77	.58	0.23
F3	30	21.10	20.38	.78	3.50
	40	28.13	28.05	.46	0.27
	50	35.17	34.67	.36	1.43
	60	42.20	42.46	.43	0.63
	70	49.23	49.13	.37	0.19
	80	56.27	55.62	.69	1.15

It can be seen that the predicted values are close to the average values of the experimental measurements. The standard deviation (SD) is used to quantify the amount of dispersion of experimental interface pressure values (Shiffler & Harsha, 1980). As shown in Table 5, for all the samples, SD values are less than 1. This means that the distribution of the values around the mean interface pressure value is acceptable (Altman, 2005). In addition, the error percentage was calculated using Equation 11. It never exceeds 5%. It is between 1.18% and 3.63% for F1, between 0.23% and 1.35% for F2, and between 0.27% and 3.5% for F3. This error margin is considered acceptable (Taylor, 1997). Thus, there is no discrepancy between theoretical and experimental values as the percentage error value can be evaluated as acceptable (less than 5%). The relationships between measured and calculated values are presented in Figure 9.

Figure 9.

Relationship between experimental and theoretical interface pressure values for samples F1, F2, and F3.

As shown in Figure 9, the correlation coefficient’s R² is .99 for F1, F2, and F3. We found a good correlation between the theoretical and experimental values. Thus, the theoretical calculations closely agree with the experimental pressure measurements. Therefore, the model presented in Equation 7 can be used to calculate the change of the interface pressure due to changes of leg circumference, Young’s modulus, and thickness of the fabric. This could be very useful in selecting the appropriate type of compressive garment according to the needs of patients. For example, by measuring the change in the size of the patient’s leg, clinicians can select appropriate materials that provide the best care for particular patient needs. They can use the developed model to estimate the change of interface pressure that occurs in a compressive garment.

Conclusion

Compression garments are widely used today in several applications in the medical and sport fields. However, there is a lack of information on fabric characteristics and surface pressure. This study was based on the development of a pressure measurement device and the use of a modified Laplace’s law to predict the interface pressure generated by knitted elastic bands. In fact, due to the inadequacy of the basic Laplace’s law to predict the interface pressure directly generated by a knitted elastic garment on a human body, it was necessary to introduce modifications to this law in order to improve its accuracy. The new relationship for predicting interface pressure includes body circumference, thickness of the knitted fabric, applied extension, and its corresponding Young’s modulus. The accuracy of the developed model was shown and compared to that of the basic Laplace’s law. With this new model, it is possible to predict more accurately the choice of fabric parameters according to needed interface pressure value. The modified Laplace’s law could be of great interest to physical therapists and compression garment manufacturers who have to model fabric architecture according to the required interface pressure on the body.

Footnotes

Declaration of Conflicting Interests

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

Funding

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

ORCID iD

H. Barhoumi

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