Determining effective bending stiffness of fabrics and other materials from tests involving an added weight

Abstract

It is often desired to determine the bending stiffness of a fabric or other material. Some simple tests have been utilized for this purpose. The tests considered here involve a horizontal cantilever, vertical cantilever, hanging ring, heart loop and hanging pear loop. Usually a strip or loop of material is only subjected to its self-weight in these tests. In some cases, an additional weight is applied and causes greater deformation; this problem is analyzed here. In each test, the position of an extreme point is measured under the self-weight and applied load. Polynomial approximations are proposed to obtain the bending stiffness of the material.

Keywords

fabric tests bending stiffness added weight numerical analysis elastica

This paper extends the analysis in Plaut¹ by including an added weight (load) to some simple tests. The objective is to determine the effective bending stiffness of the material being tested, which could be a strip or loop of fabric (e.g. Peirce²), paper (e.g. Hall et al.^3,4), prepreg (e.g. de Bilbao et al.,⁵ Liang et al.,⁶ Alshahrani and Hojjati,^7,8 Boisse et al.⁹), laminate (e.g. Price¹⁰), thin film (e.g. Price¹⁰, Hall et al.^3,4) or other material. Some previous investigations have included such an added load to the cantilever and hanging ring tests.

The added load allows one to examine the response of the material to various amounts of deformation, and to determine how the effective bending stiffness changes with increased deformation. This is useful for predicting whether wrinkling, folds or waves will occur (Dangora¹¹, Liang et al.¹², Boisse et al.⁹). With respect to the cantilever test, Peirce noted that adding a vertical load at the tip “has possible utility in testing materials that are stiff as well as curly.”²

Five tests are considered. The first is the cantilever test, with a horizontal strip subjected to its self-weight and a vertical load (hanging weight) at its tip. In the second, the cantilever is hung vertically with its fixed end at the top, and the tip is deflected horizontally. Thus the self-weight acts vertically in this case. The remaining tests involve a loop of material, with vertical self-weight and a concentrated load applied downward at the bottom of the loop. The loop hangs from a point in the third test. In the fourth and fifth tests, respectively, the loop resembles a heart and a pear.

Different tests are most appropriate for different materials. For testing paper, Hall et al. demonstrated that the horizontal cantilever test is more appropriate than the hanging pear loop test.⁴ Peirce and Bickley stated that the heart loop test is more appropriate for flimsy materials than the horizontal cantilever test.^2,13 Peirce also said that the loop tests are appropriate for “soft, curly material,” and that the hanging ring test is probably the best test for yarns.²

Method

The analysis assumes that the strip or loop is uniform, homogeneous, isotropic, inextensible and linearly elastic, with its bending moment proportional to its curvature at each location. Hence the strip or loop is modeled as an elastica. It has length L, constant weight w per unit length, applied concentrated load F, and constant bending stiffness EI (where, for standard continuous materials, E denotes the modulus of elasticity and I denotes the moment of inertia). The quantity h is the tip deflection for the two cantilever tests, and is a measure of the height for the three loop tests. For the purposes of this investigation, the shape of the strip or loop when it is unstrained is irrelevant.

The total weight W and the nondimensional quantities that will be utilized are

W = wL, f = \frac{F L^{2}}{EI}, \hat{w} = \frac{w L^{3}}{EI}, \hat{h} = \frac{h}{L}

(1)

The quantities L, w and F are known from the test. From the measured value of h, the value of ĥ is known from equation (1). Polynomials will be presented, from which f is computed from ĥ, and then the bending stiffness EI is obtained from the equation for f in equation (1). For each of the polynomials, the applied load must be a certain proportion k of the total weight, i.e.

F = kW, f = k \hat{w}

(2)

This condition is imposed so that simple and accurate polynomials in a single variable, ĥ, could be derived. Each polynomial is valid for a specified range of ĥ.

The “bending length” c can be found from the relationship c = L/[(ŵ^1/3)] (Plaut¹).

The governing equations and boundary conditions can be deduced from papers such as: Bickley,¹³ Lippmann et al.,¹⁴ Watson and Wang,^15,16 Takatera and Shinohara,¹⁷ Bélendez et al.,¹⁸ Santillan et al.,¹⁹ and Kimiaeifar et al.²⁰ Numerical solutions are obtained using a shooting method (e.g. Santillan et al.,¹⁹ Plaut and Virgin,²¹ Virgin et al.²²). Approximating sixth-order polynomials for f in terms of ĥ are derived using numerical results and the subroutine FindFit in Mathematica.

Results and discussion

Cantilever test

The cantilever is depicted in Figure 1. The distributed weight w and applied load F, here and in schematics for the other tests, are drawn on the case with the largest values of F and w. The profiles in Figure 1 correspond to k = 1 (i.e. F = W) in equation (2) and, starting from the horizontal shape, to f = 0, 1, 2 and 3. The nondimensional tip deflections, respectively, are ĥ = 0, 0.3904, 0.5922 and 0.6916.

Figure 1.

Cantilever shapes for f = ŵ = 0, 1, 2 and 3.

The cantilever test without an applied load (i.e. just self-weight) has been used extensively to determine the bending stiffness of a strip of material, starting with Peirce². A cantilever with self-weight w and vertical tip load F was considered in that paper and in papers by Grosberg and Swani²³, Potluri et al.²⁴, Bélendez et al.¹⁸, de Bilbao et al.⁵, Kimiaeifar et al.,²⁰ Liang et al.¹² and Hall et al.,³ among others.

The entire profile of the cantilever was measured in experiments described in Clapp et al.,²⁵ de Bilbao et al.,^26,5 Liang et al.⁶ and Hall et al.³ In the last of these, profiles were obtained for three types of paper and for a thin film, each subjected to several tip loads F. Predictions of the bending stiffness (divided by the strip thickness) determined from these profiles were compared to predictions based simply on the tip deflection.

Results from the present analysis are depicted in Figures 2 and 3. In Figure 2, ŵ is fixed at 0, 2, 4 and 6, and f is plotted as a function of ĥ. As for all curves of f versus ĥ in this paper, the slope increases as f increases. When f = 0 in Figure 1, ĥ = 0, 0.2385, 0.4252 and 0.5539, respectively, for ŵ = 0, 2, 4 and 6. Naturally, for a fixed tip load, the tip deflection increases as the self-weight increases.

Figure 2.

Load versus tip deflection for cantilever test with ŵ = 0, 2, 4 and 6.

Figure 3.

Load versus tip deflection for cantilever test with k = 0.25, 0.5, 0.75 and 1.

In Figure 3, k is fixed in equation (2) at 0.25, 0.5, 0.75 and 1. Since ŵ = f/k, if f is fixed, ŵ and therefore ĥ increase as k decreases. For example, if f = 1, the values of ĥ are 0.5763, 0.4645, 0.4166 and 0.3904 for k = 0.25, 0.5, 0.75 and 1, respectively.

Polynomial approximations have been derived for the curves in Figure 3. For k = 0.25 and 0 < ĥ < 0.81:

\begin{matrix} f = 0.0961 \hat{h} + 18.72 {\hat{h}}^{2} - 109.75 {\hat{h}}^{3} + 296.2 {\hat{h}}^{4} \\ - 367.2 {\hat{h}}^{5} + 175 {\hat{h}}^{6} \end{matrix}

(3)

For k = 0.5 and 0 < ĥ < 0.75:

\begin{matrix} f = 1.335 \hat{h} + 7.089 {\hat{h}}^{2} - 45.4 {\hat{h}}^{3} + 141.1 {\hat{h}}^{4} \\ - 196.6 {\hat{h}}^{5} + 107.9 {\hat{h}}^{6} \end{matrix}

(4)

For k = 0.75 and 0 < ĥ < 0.71:

\begin{matrix} f = 1.776 \hat{h} + 4.425 {\hat{h}}^{2} - 29.28 {\hat{h}}^{3} + 99.8 {\hat{h}}^{4} \\ - 148 {\hat{h}}^{5} + 88.05 {\hat{h}}^{6} \end{matrix}

(5)

For k = 1 and 0 < ĥ < 0.69:

\begin{matrix} f = 2.013 \hat{h} + 3.463 {\hat{h}}^{2} - 23.31 {\hat{h}}^{3} + 84.55 {\hat{h}}^{4} \\ - 130 {\hat{h}}^{5} + 81.19 {\hat{h}}^{6} \end{matrix}

(6)

In Plaut a polynomial approximation for ŵ was derived for the cantilever subjected only to its self-weight.¹ Harrison pointed out that the approximation is not very accurate for very low values of ŵ.²⁷ This is due to inclusion of a constant term in the polynomial. A better approximation for the range 0 < ĥ < 0.8 (with 0 < ŵ < 16.2) is given by

\begin{matrix} \hat{w} = 4.958 \hat{h} + 55.974 {\hat{h}}^{2} - 343.322 {\hat{h}}^{3} + 978.749 {\hat{h}}^{4} \\ - 1264.433 {\hat{h}}^{5} + 631.912 {\hat{h}}^{6} \end{matrix}

(7)

It is noted that Peirce,² Abbott²⁸ and de Bilbao et al.^26,5 tested cantilevers with F = 0 and with different lengths to determine the bending behavior under various amounts of deformation.

Vertical cantilever test

In the study by Soteropoulos et al. the cantilever was hung vertically.²⁹ This eliminates the initial transverse deflection that exists in the horizontal cantilever before the tip load is applied. In experiments, a string was attached to the free end at the bottom. The string was horizontal until it went around a pulley and then hung vertically. A weight was applied at the lower end of the string to cause a horizontal deflection of the tip of the cantilever. The material was a 0^o/90^o biaxial fiberglass non-crimp fabric with polyester stitching. In a finite element analysis, the self-weight of the cantilever was neglected, and an effective bending stiffness was determined.

The same setup was utilized by Dangora et al.^11,30 and Alshahrani and Hojjati.^7,8 Again, experiments were conducted, and finite element analyses were included in the first three of these references. In Dangora et al.¹¹ the tests were conducted on strips of a woven textile and a non-crimp fabric. In Dangora et al.³⁰ a four-level cross-ply thermoplastic laminate was used. Alshahrani and Hojjati tested four stacking sequences of a five-harness satin weave impregnated with an epoxy resin.^7,8

For the present analysis, Figure 4 depicts the vertical cantilever for k = 1 in equation (2), with f = 0, 1.5, 3 and 4.5. The corresponding nondimensional horizontal tip deflections, respectively, are ĥ = 0, 0.3641, 0.5224 and 0.6002. The tip deflection for f = 3 is 24.5% smaller for the vertical cantilever than for the horizontal cantilever (Figure 1, bottom profile).

Figure 4.

Vertical cantilever shapes for f = ŵ = 0, 1.5, 3 and 4.5.

In Figure 5, as in Figure 2, ŵ is fixed at 0, 2, 4 and 6. Here, ĥ = 0 when f = 0, and the curves are higher for higher ŵ. In Figure 6, k is fixed at 0.5, 1, 2 and 4. Unlike Figure 3, the curves become lower as k increases. For the vertical cantilever, the self-weight tends to reduce the tip deflection. If f is fixed and k increases, ŵ decreases and this restraining effect is reduced.

Figure 5.

Load versus tip deflection for vertical cantilever test with ŵ = 0, 2, 4 and 6.

Figure 6.

Load versus tip deflection for vertical cantilever test with k = 0.5, 1, 2 and 4.

Polynomial approximations have been derived for the curves in Figure 6. For k = 0.5 and 0 < ĥ < 0.53:

\begin{matrix} f = 2.3 \hat{h} + 20.8 {\hat{h}}^{2} - 165.2 {\hat{h}}^{3} + 716.3 {\hat{h}}^{4} \\ - 1390.8 {\hat{h}}^{5} + 1086.0 {\hat{h}}^{6} \end{matrix}

(8)

For k = 1 and 0 < ĥ < 0.62:

f = 2.5 \hat{h} + 13.1 {\hat{h}}^{2} - 92.4 {\hat{h}}^{3} + 351.9 {\hat{h}}^{4} - 594.5 {\hat{h}}^{5} + 404.6 {\hat{h}}^{6}

(9)

For k = 2 and 0 < ĥ < 0.67:

\begin{matrix} f = 2.5 \hat{h} + 10.9 {\hat{h}}^{2} - 74.3 {\hat{h}}^{3} + 264.6 {\hat{h}}^{4} \\ - 417.3 {\hat{h}}^{5} + 264.3 {\hat{h}}^{6} \end{matrix}

(10)

For k = 4 and 0 < ĥ < 0.69:

f = 2.5 \hat{h} + 11.0 {\hat{h}}^{2} - 74.1 {\hat{h}}^{3} + 252.4 {\hat{h}}^{4} - 382.3 {\hat{h}}^{5} + 231.2 {\hat{h}}^{6}

(11)

Hanging ring test

In this test, the material is formed into a circular shape and supported at its top point in a vertical plane. The perimeter is L and the height is h. The ring is subjected to its self-weight w and a vertical load F at its bottom point. In Figure 7, where F = W (i.e. k = 1), the four rings, respectively, have f = 0, 100, 200 and 300, with ĥ = 0.3183 ( = 1/π), 0.3785, 0.4064 and 0.4220. The loads are shown for the case f = 300. The hanging rings are symmetric about the vertical line passing through the top support and bottom point.

Figure 7.

Hanging ring shapes for f = ŵ = 0, 100, 200 and 300.

If the self-weight is neglected, the problem consists of a circular ring pulled apart by opposing concentrated loads. This problem has been analyzed by Sonntag,³¹ Frisch-Fay,³² Pan,³³ Owen and Riding,³⁴ Leaf,³⁵ Watson and Wang,¹⁵ Abbott,³⁶ Takatera and Shinohara,¹ Plaut and Virgin,²¹ Kurbak^37,38 and Virgin et al.,²² among others. (Kurbak, at Equation 31, assumes that the increase in curvature at the top of the ring has the same magnitude as the decrease of curvature at the side (where the width is maximum), which allows the curvature to change sign.³⁷ This is not true; the curvature at every point of the ring must have the same sign, as pointed out by Pan.³) The shape of the deformed ring with w = 0 is symmetric about a horizontal line as well as the vertical line passing through the support.

The case of a hanging ring with no bottom load, just self-weight, has been analyzed in Peirce,² Watson and Wang,¹⁶ Takatera and Shinohara¹⁷ and Plaut,¹ among others. Experiments on the hanging ring with no bottom load were described in Peirce² and Plaut and Virgin.²¹ Experiments including a bottom load were discussed in Carlene,³⁹ Abbott,³⁶ Virgin et al.²² and Kurbak,^40,38 among others.

Shapes of rings with opposing concentrated tensile loads (i.e. w = 0) have been depicted in Frisch-Fay³² for f = 0, 22.8, 75.8, 276.3, 664.0 and 1505.7, and in Takatera and Shinohara¹⁷ for f = 0, 32, 128, 288, 512, 800, 1800, 3200 and 5000. Shapes of hanging rings with no bottom load have been shown in Watson and Wang¹⁶ for ŵ = 8, 160, 800 and 4800, in Takatera and Shinohara¹⁷ for ŵ = 0, 64, 216, 512, 1000, 1728, 4096 and 8000, in Plaut and Virgin²¹ for ŵ = 0, 500, 1000 and 1500, and in Plaut¹ for ŵ = 500.

The range of ĥ begins at 0.30 in Figures 8 and 9. In Figure 8, f is plotted versus ĥ for ŵ = 0, 200, 400 and 600. At f = 0, the respective values of ĥ are 0.3183, 0.3627, 0.3876 and 0.4029. In Figure 9, with f = kŵ, the curves are associated with k = 0.25, 0.5, 0.75 and 1 (the same as in Figure 3). All of them start at the circular shape with f = 0 and ĥ = 0.3183.

Figure 8.

Load versus height for hanging ring test with ŵ = 0, 200, 400 and 600.

Figure 9.

Load versus height for hanging ring test with k = 0.25, 0.5, 0.75 and 1.

Polynomial approximations have been derived for the curves in Figure 9. For k = 0.25 and 0.32 < ĥ < 0.44:

\begin{matrix} f = 1.315235 \cdot 10^{6} - 2.1564133 \cdot 10^{7} \hat{h} \\ + 1.4711811 \cdot 10^{8} {\hat{h}}^{2} - 5.3466639 \cdot 10^{8} {\hat{h}}^{3} \\ + 1.0918749 \cdot 10^{9} {\hat{h}}^{4} - 1.18819738 \cdot 10^{9} {\hat{h}}^{5} \\ + 5.38428 \cdot 10^{8} {\hat{h}}^{6} \end{matrix}

(12)

For k = 0.5 and 0.32 < ĥ < 0.43:

\begin{matrix} f = 9.096185 \cdot 10^{5} - 1.5150057 \cdot 10^{7} \hat{h} \\ + 1.0503275 \cdot 10^{8} {\hat{h}}^{2} - 3.8805954 \cdot 10^{8} {\hat{h}}^{3} \\ + 8.0602154 \cdot 10^{8} {\hat{h}}^{4} - 8.925718 \cdot 10^{8} {\hat{h}}^{5} \\ + 4.11838 \cdot 10^{8} {\hat{h}}^{6} \end{matrix}

(13)

For k = 0.75 and 0.32 < ĥ < 0.425:

\begin{matrix} f = 6.37965 \cdot 10^{5} - 1.074108 \cdot 10^{7} \hat{h} \\ + 7.5302 \cdot 10^{7} {\hat{h}}^{2} - 2.8146184 \cdot 10^{8} {\hat{h}}^{3} \\ + 5.917297 \cdot 10^{8} {\hat{h}}^{4} - 6.636123 \cdot 10^{8} {\hat{h}}^{5} \\ + 3.10295 \cdot 10^{8} {\hat{h}}^{6} \end{matrix}

(14)

For k = 1 and 0.32 < ĥ < 0.42:

\begin{matrix} f = 5.67949 \cdot 10^{5} - 9.58951 \cdot 10^{6} \hat{h} + 6.743548 \cdot 10^{7} {\hat{h}}^{2} \\ - 2.529074 \cdot 10^{8} {\hat{h}}^{3} + 5.33668 \cdot 10^{8} {\hat{h}}^{4} \\ - 6.00939 \cdot 10^{8} {\hat{h}}^{5} + 2.8226 \cdot 10^{8} {\hat{h}}^{6} \end{matrix}

(15)

Heart loop test

The heart loop test is considered now. As shown in Figure 10, the ends of the strip are clamped together and upward. The perimeter is L and the vertical distance from the clamp to the bottom of the loop is h. For the shapes in Figure 10, F = W and f = 0, 150, 300 and 450, respectively, with ĥ = 0.1339, 0.1826, 0.2193 and 0.2465. The self-weight w and vertical load F are depicted for the case f = 450.

Figure 10.

Heart loop shapes for f = ŵ = 0, 150, 300 and 450.

For the case F = 0, analyses were presented in Bickley,¹³ Peirce,² Lippmann et al.,¹⁴ Takatera and Shinohara,¹⁷ Zhou and Ghosh⁴¹ and Plaut.¹ Peirce² and Takatera and Shinohara¹⁷ also analyzed the heart loop subjected to the vertical load F, and reported experimental results. Additional experiments were described in Abbott,⁴² Price,¹⁰ Dai et al.⁴³ and Mohamad et al.⁴⁴

For F = 0, Bickley¹³ showed seven shapes, corresponding to different values of tension at the support. Lippmann et al.¹⁴ derived shapes for five of those tension values. Takatera and Shinohara¹⁷ depicted shapes for ŵ = 0, 64, 216, 512, 1000, 1728, 4096 and 8000, and Plaut¹ presented the shape for ŵ = 500. For w = 0, Takatera and Shinohara¹⁷ showed shapes of the heart loop for f = 0, 32, 128, 288, 512, 800, 1800, 3200 and 5000.

In Figure 11, f is plotted versus ĥ for ŵ = 0, 200, 400 and 600 (as in Figure 8). At f = 0, the respective values of ĥ in Figure 11 are 0.1339, 0.2224, 0.2754 and 0.3092. The range of ĥ in the figure begins at 0.13. In Figure 12, with f = kŵ, the curves are associated with k = 1, 1.5, 2 and 3. The curves begin at ĥ = 0.1339, and again the horizontal axis begins at ĥ = 0.13.

Figure 11.

Load versus height for heart loop test with ŵ = 0, 200, 400 and 600.

Figure 12.

Load versus height for heart loop test with k = 1, 1.5, 2 and 3.

Polynomial approximations have been derived for the curves in Figure 12. For k = 1 and 0.134 < ĥ < 0.25:

\begin{matrix} f = - 749.7 + 14, 709 \hat{h} - 143, 691 {\hat{h}}^{2} + 835, 169 {\hat{h}}^{3} \\ - 2.428 \cdot 10^{6} {\hat{h}}^{4} + 3.098 \cdot 10^{6} {\hat{h}}^{5} - 271, 034 {\hat{h}}^{6} \end{matrix}

(16)

For k = 1.5 and 0.134 < ĥ < 0.29:

\begin{matrix} f = 307.43 - 14, 7859 \hat{h} + 211, 097 {\hat{h}}^{2} - 1.4476 \cdot 10^{6} {\hat{h}}^{3} \\ + 5.5915 \cdot 10^{6} {\hat{h}}^{4} - 1.143 \cdot 10^{7} {\hat{h}}^{5} + 9.934 \cdot 10^{6} {\hat{h}}^{6} \end{matrix}

(17)

For k = 2 and 0.134 < ĥ < 0.31:

\begin{matrix} f = - 339 + 5535 \hat{h} - 44, 483 {\hat{h}}^{2} + 222, 932 {\hat{h}}^{3} \\ - 468, 516 {\hat{h}}^{4} + 141, 531 {\hat{h}}^{5} + 782, 508 {\hat{h}}^{6} \end{matrix}

(18)

For k = 3 and 0.134 < ĥ < 0.32:

\begin{matrix} f = 108 - 7, 098 \hat{h} + 105, 496 {\hat{h}}^{2} - 720, 637 {\hat{h}}^{3} \\ + 2.8 \cdot 10^{6} {\hat{h}}^{4} - 5.773 \cdot 10^{6} {\hat{h}}^{5} + 5.112 \cdot 10^{6} {\hat{h}}^{6} \end{matrix}

(19)

Hanging pear loop test

Finally, the hanging pear loop test is considered. The ends of the strip are clamped vertically downward at the top of the loop. The perimeter is L and the height is h. In Figure 13, where F = W, the four loops, respectively, have f = 0, 150, 300 and 450 (as in Figure 10), with ĥ = 0.4243, 0.4504, 0.4608 and 0.4666. The loads are shown for the case f = 450.

Figure 13.

Hanging pear loop shapes for f = ŵ = 0, 150, 300 and 450.

For the case of no vertical load F, Peirce presented an analysis and some experimental results,² Takatera and Shinohara and Plaut conducted analyses,^17,1 and Hall et al. discussed the results of experiments.⁴ Takatera and Shinohara¹⁷ showed shapes for ŵ = 0, 64, 216, 512, 1000, 1728, 4096 and 8000, and Plaut¹ depicted the shape for ŵ = 100. For the case of no self-weight w, shapes for f = 0, 32, 128, 288, 512, 800, 1800, 3200 and 5000 were shown in Takatera and Shinohara.¹⁷

The range for ĥ in Figures 14 and 15 starts at 0.42. In Figure 14, f is plotted versus ĥ for ŵ = 0, 500, 1000 and 1500. At f = 0, the respective values of ĥ are 0.4243, 0.4465, 0.4552 and 0.4600. In Figure 15, with f = kŵ, the curves are associated with k = 0.25, 0.5, 0.75 and 1 (the same as in Figures 3 and 9), and all start at ĥ = 0.4243.

Figure 14.

Load versus height for hanging pear loop test with ŵ = 0, 500, 1000 and 1500.

Figure 15.

Load versus height for hanging pear loop test with k = 0.25, 0.5, 0.75 and 1.

Polynomial approximations have been derived for the curves in Figure 12. For k = 0.25 and 0.4243 < ĥ < 0.472:

\begin{matrix} f = 1.199608497 \cdot 10^{9} - 1.6275262355 \cdot 10^{10} \hat{h} \\ + 9.199949979 \cdot 10^{10} {\hat{h}}^{2} - 2.7734856296 \cdot 10^{11} {\hat{h}}^{3} \\ + 4.703027515 \cdot 10^{11} {\hat{h}}^{4} - 4.253253453 \cdot 10^{11} {\hat{h}}^{5} \\ + 1.602701047 \cdot 10^{11} {\hat{h}}^{6} \end{matrix}

(20)

For k = 0.5 and 0.4243 < ĥ < 0.470:

\begin{matrix} f = 1.02656244 \cdot 10^{9} - 1.39547906 \cdot 10^{10} \hat{h} \\ + 7.904157587 \cdot 10^{10} {\hat{h}}^{2} - 2.387795064 \cdot 10^{11} {\hat{h}}^{3} \\ + 4.057671039 \cdot 10^{11} {\hat{h}}^{4} - 3.677706208 \cdot 10^{11} {\hat{h}}^{5} \\ + 1.388974488 \cdot 10^{11} {\hat{h}}^{6} \end{matrix}

(21)

For k = 0.75 and 0.4243 < ĥ < 0.469:

\begin{matrix} f = 8.60046622 \cdot 10^{8} - 1.172330053 \cdot 10^{10} \hat{h} \\ + 6.65857544 \cdot 10^{10} {\hat{h}}^{2} - 2.017119217 \cdot 10^{11} {\hat{h}}^{3} \\ + 3.4374012138 \cdot 10^{11} {\hat{h}}^{4} - 3.124351956 \cdot 10^{11} {\hat{h}}^{5} \\ + 1.18336322 \cdot 10^{11} {\hat{h}}^{6} \end{matrix}

(22)

For k = 1 and 0.4243 < ĥ < 0.468:

\begin{matrix} f = 4.771939 \cdot 10^{8} - 6.55503855 \cdot 10^{9} \hat{h} \\ + 3.75228117 \cdot 10^{10} {\hat{h}}^{2} - 1.1457021648 \cdot 10^{11} {\hat{h}}^{3} \\ + 1.968045961 \cdot 10^{11} {\hat{h}}^{4} - 1.803308698 \cdot 10^{11} {\hat{h}}^{5} \\ + 6.8861344 \cdot 10^{10} {\hat{h}}^{6} \end{matrix}

(23)

Concluding remarks

This study has analyzed the effect of an added weight (load) on five simple tests to determine the effective bending stiffness of fabrics and other materials. Adding concentrated loads can demonstrate the influence of additional deformation on the effective bending stiffness. The cantilever test, vertical cantilever test, hanging ring test, heart loop test and hanging pear loop test were investigated.

In an experiment, the weight and length of the strip or loop need to be measured. The maximum deflection is then measured for a certain applied load. The effective bending stiffness can be computed from the proposed sixth-order polynomials that give approximations to the analytical results. The main assumption in the analysis is that the material behavior is linearly elastic. Even if the real material is not linearly elastic, the calculated effective bending stiffness can provide useful information regarding possible softening or hardening behavior.

For the range specified, each approximating polynomial gives values of the nondimensional load f that have less than 0.5% error in most cases. Somewhat larger errors sometimes occur for small values of f. With regard to the number of significant digits in the coefficients of each polynomial, the curve-fitting program produced at least 16 digits for each coefficient. The resulting curve was overlaid on the curve from the numerical data, and then the number of digits for each coefficient was reduced to the minimum number (which was as low as two and as high as 11) for which the two curves were indistinguishable. (The measurements of weight, length and other quantities in the physical test do not need to have more than two or three significant digits.)

All the plots and approximating polynomials in this paper are new. The polynomials should be useful in practice to determine the effective bending stiffness of fabrics and other materials.

Footnotes

Acknowledgements

The author is grateful to Benjamin Z. Dymond for preparing the figures, and to the reviewers for their helpful suggestions.

Declaration of conflicting interests

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

Funding

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

References

Plaut

. Formulas to determine fabric bending rigidity from simple tests. Textil Res J 2015; 85: 884–894.

Peirce

. The “handle” of cloth as a measurable quantity. J Textil Inst Trans 1930; 21: T377–T416.

Hall

AKI

O'Connor

McGath

, et al. The bending mechanics of aged paper. J Appl Mech 2018; 85: 071005.

Hall

AKI

Plaut

McGuiggan

. Comparing non-destructive mechanical testing methods for the assessment of brittle papers – the cantilever, hanging pear loop, and clamped fold tests. Restaurator 2019; 40: 15–34.

de Bilbao

Soulat

Hivet

, et al. Experimental study of bending behaviour of reinforcements. Exp Mech 2010; 50: 333–351.

Liang

Hamila

Peillon

, et al. Analysis of thermoplastic prepreg bending stiffness during manufacturing and of its influence on wrinkling simulations. Compos A Appl Sci Manuf 2014; 67: 111–122.

Alshahrani

Hojjati

. Bending behavior of multilayered textile composite prepregs: Experiment and finite element modeling. Mater Design 2017; 124: 211–224.

Alshahrani

Hojjati

. A new test method for the characterization of the bending behavior of textile prepregs. Compos A Appl Sci Manuf 2017; 97: 128–140.

Boisse

Colmars

Hamila

, et al. Bending and wrinkling of composite fiber preforms and prepregs: A review and new developments in the draping simulations. Compos B Eng 2018; 14: 234–239.

10.

Price HL. Techniques for the measurement of the flexural rigidity of thin films and laminates. Washington, DC: National Aeronautics and Space Administration, NASA TN D-3270, 1966.

11.

Dangora

Mitchell

Sherwood

. Predictive model for the detection of out-of-plane defects formed during textile-composite manufacture. Compos A Appl Sci Manuf 2015; 78: 102–112.

12.

Liang

Chaudet

Boisse

. Curvature determination in the bending test of continuous fibre reinforcements. Strain 2017; 53: e12213.

13.

Bickley

. The heavy elastica. Phil Mag 1934; 17: 603–622.

14.

Lippmann

Mahrenholtz

Johnson

. Thin heavy elastic strips at large deflexions. Int J Mech Sci 1961; 2: 294–310.

15.

Watson

Wang

. A homotopy method applied to elastica problems. Int J Solids Struc 1981; 17: 29–37.

16.

Watson

Wang

. Hanging an elastic ring. Int J Mech Sci 1981; 23: 161–167.

17.

Takatera

Shinohara

. A unified analysis to compare conventional methods for estimating bending rigidity of fabrics. J Textil Mach Soc Jpn 1996; 42: 86–92.

18.

Beléndez

Neipp

Beléndez

. Numerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in mechanics of materials. Int J Eng Ed 2003; 19: 885–892.

19.

Santillan

Virgin

Plaut

. Equilibria and vibration of a heavy pinched loop. J Sound Vib 2005; 288: 81–90.

20.

Kimiaeifar

Tolou

Barari

, et al. Large deflection analysis of cantilever beam under end point and distributed loads. J Chinese Inst Engrs 2014; 37: 438–445.

21.

Plaut

Virgin

. Deformation and vibration of upright loops on a foundation and of hanging loops. Int J Solids Struc 2014; 51: 3067–3075.

22.

Virgin

Giliberto

Plaut

. Deformation and vibration of compressed, nested, elastic rings on rigid base. Thin-Wall Struc 2018; 132: 167–175.

23.

Grosberg

Swani

. The mechanical properties of woven fabrics, Part IV: The determination of the bending rigidity and frictional restraint in woven fabrics. Textil Res J 1966; 36: 338–345.

24.

Potluri

Atkinson

Porat

. Large deformation modelling of flexible materials. J Textil Inst 1996; 87: 129–151.

25.

Clapp

Peng

Ghosh

, et al. Indirect measurement of the moment-curvature relationship for fabrics. Textil Res J 1990; 60: 525–533.

26.

de Bilbao

Soulat

Hivet

, et al. Study of bending behaviour of composite reinforcements. Int J Mater Forming 2009; 2(Suppl. 1): 205–208.

27.

Harrison

. Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Compos A Appl Sci Manuf 2016; 81: 145–157.

28.

Abbott

. The measurement of stiffness in textile fabrics, Part II: A study of the Peirce cantilever test for stiffness of textile fabrics. Textil Res J 1951; 21: 441–444.

29.

Soteropoulos D, Fetfatsidis K, Sherwood JA et al. Digital method of analyzing the bending stiffness of non-crimp fabrics. In: ESAFORM 2011: Proceedings of the 14th International Conference on Material Forming (ed G Menary), Melville, NY: American Institute of Physics, 2011, pp.913–917.

30.

Dangora

Mitchell

White

, et al. Characterization of temperature-dependent tensile and flexural rigidities of a cross-ply thermoplastic lamina with implementation into a forming model. Int J Mater Forming 2018; 11: 43–52.

31.

Sonntag

. Die Kreisringfeder: Zur Theorie des geschlossenen Kreisringes mit grosser Formänderung. Ing-Arch 1943; 113: 380–397.

32.

Frisch-Fay R. Flexible bars. Washington, DC: Butterworth, 1962.

33.

Pan

. Non-linear deformation of a flexible ring. Qtly J Mech Appl Math 1962; 15: 401–412.

34.

Owen

Riding

. The weighted-ring stiffness test. J Textil Inst Trans 1964; 55: T414–T417.

35.

Leaf

GAV

. The weighted-ring stiffness test. J Textil Inst Trans 1964; 55: T431–T432.

36.

Abbott

. Yarn-bending and the weighted-ring stiffness test. Textil Res J 1983; 74: 281–286.

37.

Kurbak

. Load-extension properties of glass plain knitted technical fabrics: Part I: Some additions to the extension of the circular ring problem to make it applicable for a knitted loop head curve. Textil Res J 2018; 88: 654–666.

38.

Kurbak

. Load-extension properties of glass plain knitted technical fabrics: Part II: extensions in the course-wise and wale-wise directions. Textil Res J 2018; 88: 667–695.

39.

Carlene

. The relation between fibre and yarn flexural rigidity in continuous filament viscose yarns. J Textil Inst Trans 1950; 41: T159–T172.

40.

Kurbak

. Some works on the bending rigidities of technical yarns (in Turkish). Textil ve Mühendis 2017; 25: 62–65.

41.

Zhou

Ghosh

. On-line measurement of fabric bending behavior, Part II: Effects of fabric nonlinear bending behavior. Textil Res J 1998; 68: 533–542.

42.

Abbott

. The measurement of stiffness in textile fabrics, Part I: A comparison of five methods of laboratory evaluation. Textil Res J 1951; 21: 435–441.

43.

Dai

Zhang

. Simulating anisotropic woven fabric deformation with a new particle model. Textil Res J 2003; 73: 1091–1099.

44.

Mohamad

Cassidy

Brydon

, et al. The measurement of plain weft-knitted fabric stiffness. Meas Sci Technol 2012; 23: 055602.