Precise Measurement of Tension on Curvature Elastic Shells Marzie Aghajani, Mohammad Amani Tehran, PhD, Ali Asghar Asgharian Jeddi, PhD Amir Kabir University, Tehran IRAN Correspondence to: Mohammad Amani Tehran email: amani@aut.ac.ir ABSTRACT Many kinds of pressure garments are currently used for various applications, such as treating of hypertrophic scars, reversing the effect of shock on the body s blood distribution, improving energy saving for athletes and enhancing the aesthetic appearance of the wearer. In order to design the desired pressure garments, predicting the amount of pressure is mandatory. Although many researchers have used the Laplace law as the prediction equation, there has been some discrepancy between predicted and experimental measured pressures. In this study, we focused on one of the parameters which influence this discrepancy. To predict the pressure by this law, the induced tension in the material is measured by a tensile strength device, and then is inserted into the equation while the real tension is induced by extending on curvature shape. We measured the tension induced in the rubber band, which was extended on a cylindrical surface by using a new tensile test assembly. Subsequently, this tension was compared with the tension induced in the flat geometry which is commonly used. The results show that there is a significant difference between the tension in the curvature and flat geometry. Keywords: Pressure Garments, Laplace law, Curvature Tension INTRODUCTION It is known that every elastic material, like elastic shells, when wrapped under tension around a curvature surface, exert a radial pressure on that surface. Pressure induced in elastic cylindrical shells is usually predicted by the Laplace law [1-3]. Air inflated structures, balloons, physiological applications (blood vessel, heart, esophageal varicose) and pressure garments are the examples of pressurized cylindrical shells. Among them, pressure garments have been most widely applied. There are many studies investigating the accuracy of the Laplace law which is used for pressure prediction in pressure garments [1, 4, 5] but only a few of them took the deviation between the experimental and Laplace prediction values into account [1, 5]. In these works, there were no clear descriptions about this phenomenon. The mathematical expression of the Laplace law is presented by Eq. (1). T = R p (1) where T is the tension of shell (N/m), R is the radius of curvature (m) and P is pressure (Pa). Practically in order to predict the interfacial pressure between garment and limb, the tensile force is obtained from linear fabric tensile behavior. The tensile behavior of fabric is usually measured by a tensile strength device. In the tensile strength device, a rectangular sample is extended in flat geometry as shown in Figure 1. In this study, in order to find the origin of the Laplace law deviation, we attended to the method of tension measuring. In a tubular pressure garment, the fabric is extended in cylindrical geometry (Figure 2). Therefore, we decided to investigate the difference between tension induced in flat and cylindrical geometries. It is interesting to investigate whether it is possible to have different tensile behaviors in the case of geometry change of the extending material (curved path or straight). This paper presents a newly designed tensile test method to measure the tension on a curved surface. The ultimate target of this work was to improve the Laplace law to predict the interfacial pressure more precisely. FIGURE 1. A band sample under flat force. Journal of Engineered Fibers and Fabrics 82 http://www.jeffjournal.org
FIGURE 2. Tension induced on the curvature surface. MATERIAL AND METHOD Pressure garments are made from elastic fabrics containing different rubber filaments. Different degrees of pressure are provided by different degrees of elasticity and strength of the elastic material. In this study, in order to eliminate the complexity of the structure of the fabric, two kinds of rubber having different tensile behavior were used. The tensionstrain curves based on ASTM D412 using an Instron tensile device 5566 are presented in Figure 3. We developed a new test method using the same tensile device to simulate rubber extending on a curved surface to study the tensile behavior during this type of deformation (Figure 4). The base was fixed in the lower jaw. The rubber sample with 5 cm width was fastened around a cylinder while the strip ends were fixed by the upper jaw. In this way, the tensioning of the sample during cylindrical wearing was simulated. While moving up the upper jaw, the strip was extended and the load was recorded by the device. A schematic presentation of the provided setup is illustrated in Figure 5. There are two individual strained regions; flat and curvature regions as shown in Figure 5. The measurements were performed on some cylinders with different diameters. FIGURE 4. Photography of suggested setup: The base was fixed in the lower jaw (1); the rubber sample with 5 cm width (2) was fastened around the cylinder (3) while the strip ends were fixed by the upper jaw. FIGURE 5. Schematic view of the provided setup: location of upper jaw before and after displacement is indicated as () and (1) respectively. FIGURE 3. Tensile behavior of tested rubbers. Journal of Engineered Fibers and Fabrics 83 http://www.jeffjournal.org
MEASUREMENT AND CALCULATION RESULTS Geometry of Designed Set Up The geometrical parameters of θ, φ and χ as indicated in Figure 5 are related to each other as follows: R = arcsin( ) R + α + δ φ (2) R χ = (3) tan(φ) π = π - ( - φ) 2 θ (4) Where δ is the movement of the upper jaw and α is the distance of cylinder surface from the upper jaw before movement. Table I demonstrates the values of α for the cylinders. TABLE I. The values of α for the cylinders. Sample Diameter= 9 cm Diameter= 2 cm Rubber1 4.5 cm 8. cm Rubber2 4.5cm 8.5 cm Local Tension Eq. (5) can be utilized to calculate the tension (T) induced by rubber extension, as follows: F T = (N/m) (5) 2.5 cos(φ) where F is the recorded force by Instron device (Figure 5). The width of band rubber is.5 m. Flat Region Strain ε f By using flat tension-strain curve of each rubber, the accordant strain of each tension obtained by Eq. (5) in flat arm was determined. Curvature Region Strain c During the jaw movement, the rubber in both flat and curvature regions is extended; the extended tail of curvature region enters in the flat region. Thus the initial length of flat arm is changed during the extension. The initial length of flat region χ at each extension is obtained via Eq. (6): χ = χ (1+ εf 1) (6) where χ is the extended length of rubber in the flat region, εf is the strain of flat region. The initial length of curvature region c during the extension process is calculated via Eq. (7): = + R χ (7) c θ - where and R θ are the initial length of flat region and curvature region before extension process, respectively. So the strain of curvature region (ε c ) is obtained by Eq. (8) as follows: ε c Rθ - = ( c c ) 1 (8) Where R θ is the extended length of rubber in the curvature region. Analysis of Friction Effect Friction is the force facing to the extension of rubber in curvature region which is necessary to be calculated. An experimental system was developed for measuring the friction coefficient, which allows direct measurement of friction coefficient between rubber and cylinder surface. The friction coefficient of Rubber1 and Rubber2 was obtained.17 and.217 respectively. The relationship between the friction coefficient, tension and angle of contact is famous [6] and expressed as follows: T T1e μβ 2 = (9) Where T 2 and T 1 are tension (T 2 >T 1 ) and β is angle of contact. From the rubber point of flat and curvature regions towards to point S (as shown in Figure 5), the angle of rubber contact (β ) is increased so the friction force facing to the extension rubber will be increased. Thus each point of rubber in curvature region is in various tensions. Tension induced in the curvature region (T c ) is calculated via Eq. (1): Τ c = Τ / e μβ β θ (1) Where T is the tension calculated by Eq. (5). Whereas T c varies by changingβ, each point of rubber in curvature region is under different strain rates. The calculated curvature strain ( εc ) by Eq. (8) is the average of strain in curvature region. Journal of Engineered Fibers and Fabrics 84 http://www.jeffjournal.org
Obviously, there is one point in the curvature region where its strain is equal to this calculated average strain ( εc ). In order to analyze the tensile behavior of rubber in curvature region, we need to know T c at this point (average point). Accordingly, it is necessary to find the location of this point. Also the frictional force at this point is the average of frictional force induced in the whole curvature region. The location of this point can be calculated as follows: (e θ μβ μβ μθ ) = ( dβ dβ) = ( e -1) μθ (11) ave e θ According to Eq. (1) and Eq. (11) the curvature tension in average point (T - c) is obtained as follows: μβ _ Tc = Τ (e ) (12) ave The results of calculations are shown in the Tables II - V. The coefficient of variation of Instron device measurements was 4 %. TABLE II. Results of calculations of Rubber1 for the cylinder with the diameter of 9 cm. δ (mm) θ (rad) T(N/M) εf % χ c (cm ) ε c % T f (N/m) ( e μβ ) ave T - c(n/m) T - c/tf 9 2.4 37 6 8.3 9. 3 25 1.2 39 1.5 3 1.95 155 22 9.1 8.1 9 578 1.19 889 1.6 51 1.89 156 37 9.8 7.5 14 852 1.18 1322 1.5 68 1.86 1886 49 1.1 7.1 18 146 1.18 163 1.5 94 1.82 2262 68 1.6 6.6 24 1297 1.17 1931 1.5 111 1.8 2451 8 1.9 6.4 27 144 1.17 296 1.5 132 1.77 2638 95 11.1 6.1 31 1527 1.17 226 1.5 149 1.76 2766 18 11.3 5.9 34 166 1.17 2373 1.5 17 1.74 296 123 11.5 5.7 37 1672 1.16 2497 1.5 191 1.73 337 139 11.6 5.6 4 1726 1.16 2612 1.5 TABLE III. Results of calculations of Rubber1 for the cylinder with the diameter of 2 cm. δ(mm) θ (rad) T(N/M) εf % χ c (cm ) ε c % T f (N/m) ( e μβ ) ave T - c(n/m) T - c/t f 51 2. 92 17 17.8 18.8 7 459 1.19 756 1.6 89 1.94 1354 3 19.2 17.4 12 746 1.19 1142 1.5 111 1.91 1569 37 19.8 16.7 15 92 1.18 1326 1.5 132 1.89 1759 45 2.4 16.2 17 999 1.18 1491 1.5 17 1.86 245 58 21.3 15.3 22 1218 1.18 1739 1.4 213 1.83 2292 72 22.1 14.5 26 1369 1.17 1954 1.4 251 1.8 2467 85 22.7 13.9 3 1498 1.17 218 1.4 289 1.78 2618 98 23.2 13.4 33 1581 1.17 2241 1.4 332 1.77 2768 112 23.6 12.9 37 1672 1.17 2373 1.4 37 1.75 2894 125 24. 12.6 39 179 1.16 2484 1.5 TABLE IV. Results of calculations of Rubber2 for the cylinder with the diameter of 9 cm. δ(mm) θ (rad) T(N/M) εf % χ c (cm ) ε c % T f (N/m) ( e μβ ) ave T - c(n/m) T - c/t f 68 1.85 949 43 1.6 6.6 26 615 1.23 771 1.3 132 1.77 1598 83 11.9 5.4 49 994 1.22 131 1.3 153 1.76 1858 97 12.1 5.1 56 199 1.22 1527 1.4 17 1.74 285 17 12.4 4.9 61 1174 1.22 1715 1.5 191 1.73 2385 121 12.6 4.6 68 1278 1.21 1965 1.5 Journal of Engineered Fibers and Fabrics 85 http://www.jeffjournal.org
TABLE V. Results of calculations of Rubber2 for the cylinder with the diameter of 2 cm. δ (mm) θ (rad) T(N/M) εf % χ c (cm ) ε c % T f (N/m) ( e μβ ) ave T - c(n/m) T - c/t f 3 2.4 328 9 17.4 19.5 5 23 1.26 26 1.3 51 1.99 48 15 18.5 18.4 9 319 1.25 383 1.2 17 1.85 153 51 22.5 14.5 28 675 1.23 855 1.3 28 1.82 1228 63 23.3 13.6 34 772 1.23 1 1.3 251 1.8 1438 76 24.1 12.9 4 865 1.22 1175 1.4 289 1.78 1646 88 24.7 12.3 45 927 1.22 1348 1.5 331 1.76 19 1 25.3 11.7 51 123 1.22 1559 1.5 DISCUSSION The tensile behavior of flat and curvature regions is shown for two rubbers in Figure 6-7. These figures demonstrate the trend of T - c versus strain for curvature regions in comparison with the experimental values which come from the flat test based on ASTM D412. It can be observed that there is a significant difference in the tensile behavior between flat and curvature regions. This is valid for both cylinder diameters in each rubber. But surprisingly the tensile behavior in the curvature region is similar in both cases. In order to clarify the influence of curvature on tension, we introduce one more tension parameter: T f. By using ε c and tensionstrain curve of rubber obtained based on ASTM D412, the accordant tension of each strain was determined and designated as T f (Table II - V). The difference between T - c and T f was expressed as a ratio of T - c and T f, (Tables II - V). It can be observed that the tension in the curvature region is nearly 1.5 and 1.4 times greater than the one in the flat region for Rubber1 and Rubber2 respectively. This difference could give an explanation for the reported deviation in the Laplace law. The difference between the ratios of T - c to T f in two rubbers may be dependent on the mechanical properties of rubbers. In order to establish the relation between physical properties and deviation of tensile behavior in the curved path, more study should be done. Furthermore, it seems that the curvature size is not a very effective parameter on the quantity of this deviation. However, it is imperative to investigate the effect of curvature size on tensile behavior in a curved path more precisely. FIGURE 6. Comparison of the tensile behavior measured by our designed method for the cylinders with the diameter of 9 and 2 cm and the flat test based on ASTM D412 for Rubber1. FIGURE 7. Comparison of the tensile behavior measured by our designed method for the cylinders with the diameter of 9 and 2 cm and the flat test based on ASTM D412 for Rubber2. Journal of Engineered Fibers and Fabrics 86 http://www.jeffjournal.org
CONCLUSION In this study, a new method was designed to measure tension in a curvature surface. The results demonstrate that tension in the curvature region was more than the flat region at the same strain. As a main conclusion, it should be pointed out that it is possible to have different tensile behaviors if the geometry (straight or curved path) of extending material changes. This is an encouraging result that helps us to find out one of the major sources of the Laplace law deviation in pressure garment applications. Further work has been investigated on establish the relationship between tensile behavior of elastic fabrics on curvature geometry and structural parameters of fabrics and will be published later. REFERENCES [1] Gaied I., Drapier S., Lun B., Experimental assessment and analytical 2D predictions of the stocking pressures induced on a model leg by medical compressive stockings, Journal of Biomechanics, 39, 26, 317-325. [2] Hui, C.L, Ng, S.F., Theoretical analysis of tension and pressure decay of a tubular elastic fabric, Textile Research Journal, 23, 73(3), 268-272. [3] Strazdiene, E., Gutaukas, M., Behavior of stretchable textiles with spatial loading, Textile Research Journal, 23 73(6), 53 534. [4] Macintyre. L., Baird. M, Weedall, P., The study of pressure delivery for hypertrophic scar treatment International, Journal of Clothing Science and Technology, 24 16(1/2), 173-183. [5] Kawabata, H., Tanaka, Y., Sakai, T., Shikawa K., Measurement of garment pressure (part 1)-pressure estimation from local strain of fabric, Sen-I Gakkaishi, 1987, 44(3), 142-148. [6] Meriam, J.L., Statics, 28-29, John Wiley &Sons, Inc. 1971. AUTHORS ADDRESSES Marzie Aghajani Mohammad Amani Tehran, PhD Ali Asghar Asgharian Jeddi, PhD Amir Kabir University Hafez St. Tehran 98 IRAN Journal of Engineered Fibers and Fabrics 87 http://www.jeffjournal.org