Analytical modeling on mechanical responses and damage morphology of flexible woven composites under trapezoid tearing

Abstract

This paper describes an analytical model for predicting trapezoid tearing strength of flexible woven composites. The analytical model was established based on the tearing damage behavior of the flexible woven composite sample during experimental tests. It was observed during the test that the tearing force increased gradually with the increase of the tearing delta zone. The analytical model involves the formulation of the strength sustained by the fiber bundles (warp yarns and weft yarns) and coatings, respectively. Based on the analytical model, the tearing deformation at different tearing delta zones has been calculated to derive the tearing strength at different displacements. The model was validated from the good agreement between the calculated results and the experimental results. From the model, it was found that the failure strain and the elastic modulus of the yarn as well as the weaving density of the fabric are the key factors affecting the tearing strength of flexible woven composites. It is expected that the analytical model could be extended to the design of flexible woven composites with a high tearing resistance, such as flexible pipelines and architecture membranes.

Keywords

flexible woven composite analytical model failure criteria tearing delta zone tearing strength

Flexible composites have been widely used in transport pipelines, architecture and civil engineering. The most commonly used flexible composite is coated, woven fabric and it contains at least two components: substrate fabric (base fabric) and a coating. The substrate fabric is used to provide strength and dimensional stability to the coated fabric, whereas the coating is used to protect the substrate fabric against external environment damage. During the service life of a flexible composite, tearing damage or failure is bound to happen and by understanding its tearing failure mechanisms, its properties can be improved during the design stage.

The research in fabric tearing can be traced back to 1915,¹ and a number of applicable methods for fabric tearing testing have been developed. The most commonly used methods include the tongue tear-strength method² used to test the tearing damage of the fabric and the trapezoid tear test used for characterizing the tearing strength of the fabric.

Since 1945, many investigations focusing on the tearing behaviors of woven fabrics have been done. Krook and Fox² pioneered the attempt to explain the tongue tearing behavior. They discussed the fabric tearing behavior and described the del-shaped (▽) opening which could be observed in the damage of the tearing area. The first study on the mechanical model of trapezoid tearing damage was conducted by Hager et al.,³ where an equation related to tearing strength from mechanical parameters in the trapezoid tearing test was derived. Teixeira et al.⁴ investigated the mechanics of woven fabrics undergoing tongue tear in a view to isolate the factors of fabric construction which seem to determine the tear strength. Based on the theory of Hager et al.,³ Turl⁵ found that the trapezoid tear method could be interpreted equally well through a single maximum tearing force for each specimen. By testing cotton and rayon fabrics, Steele and Gruntfest⁶ derived the tearing force when the first yarn ruptures. Taylor⁷ found that the tearing strength can be shown to be dependent mainly on the spacing and the strength of the threads being torn and the force required to make them slip over the crossing threads. Topping⁸ compared four different theories which predicted the burst strength of longitudinally-slit, pressurized fabric cylinders with tests on warp cylinders having various diameters and lengths and containing slits of various lengths and widths.

With a view to understanding the physical mechanisms governing the tearing process of apparel fabrics in service and to enable a meaningful interpretation of the observed tear lengths, Sarma⁹ proposed a theoretical model by formulating the tear length as a waiting-time distribution. Hamkins and Backer¹⁰ emphasized differential yarn extension in the region and the differential yarn slippage across the apex of the tear. Godfrey et al.¹¹ and Godfrey and Rossettos^12–14 did a series of research on coated and uncoated fabric and set up a simple mechanical model of tensile structures. Triki et al.¹⁵ developed a criterion based on the tearing energy for textile structures. Hinkley and Hoogstraten¹⁶ examined the thickness effect for a model polyimide material and detailed the nature of the deformation occurring near the crack. Minami¹⁷ tested four types of coated fabric experimentally for their fracture toughness. Dartman and Shishoo¹⁸ found that the adhesive bonding between a PVC coating and a fabric substrate is affected by the hydrophilic character of the material and the environmental conditions under which the coating process was carried out. Bigaud et al.¹⁹ dealt with the analysis of mechanical behavior of textile-reinforced, soft composites used in tensile structures and focused on the crack propagation strength of polyester fabrics coated with PVC. Ping et al.²⁰ and Wang et al.²¹ reported the tear damage of woven fabric by finite element analysis (FEA) investigation. They compared the trapezoid tearing behaviors of uncoated and coated, woven fabrics in experimental and FEA approaches. However, little investigations have been done on trapezoid tearing damage failure criteria of flexible composites.

The objective of this study is to investigate the tearing damage of flexible woven composites in an analytical way. An analytical model was developed to predict the tearing response of the flexible composites in the evolution of the tearing delta zone. The analytical model results were compared with the experimental results. The effect of weaving density and material elastic modulus on the tearing performance of the flexible composite was also discussed. The change of the weaving angle between the weft and warp yarns during the tearing was also taken into consideration. The analytical method developed in this study could be used for optimizing tearing strength during the design of the flexible, laminated composites.

Materials and testing

Flexible composite

The substrate fabric used was 2/1 twill woven fabric made up of high tenacity polyethylene terephthalate (PET) continuous filament fiber tows. The coating material used was thermoplastic urethane (TPU). The areal density of the TPU was 2950 g/m². The thickness of the flexible woven composite was 2.5 mm. The specifications of the materials are listed in Table 1. The photographs of the flexible composite in surface and cross-section are shown in Figure 1. As an extension of the previous work,²¹ the cross-section of the weft and warp yarns are assumed to be elliptical and the ratio of the major axis to the minor one is set to be 3. For numerical calculation of the length of the major and minor axis, it is reasonable to assume that the PET filaments are compactly packed between the two coating layers. With this assumption, the packing density of bundle of fibers is theoretically assumed to be 90.7%. Most PET filament fiber tows have a density close to 1.38 g/cm³. Then the nominal density of PET yarns is 1.25 g/cm³. Therefore, the length of the minor axis of the ellipse section is given by $b = \sqrt{\frac{Tex}{10^{6} \times ρ_{yarn} \times 10^{3} \times 3 π}} = \sqrt{\frac{Tex}{11775}}$ where b is the length of the minor axis in the ellipse section (mm); Tex, the warp or weft yarn fineness (g/1000 m); $ρ_{yarn}$ , the nominal density of the yarn (g/cm³).

Figure 1.

Photographs of the flexible composite: (a) flexible composite surface; (b) flexible composite cross-section.

Table 1.

Specifications of the flexible composite

Parameters	Weft yarn	Warp yarn	TPU
Weave density (ends/5 cm)	19	28	–
Cross-section (10^-6 m²)	1.218	0.723	–
Modulus (GPa)	6.79	7.32	0.1
Yarn fineness (tex)	700 × 2	454 × 2	–
Fabric thickness (mm)	2.5

Environmental conditions

All tests were done at a relative humidity of 65 ± 4.0% and a temperature of 20 ± 2.0℃ according to the ISO 139-2005 standard.

Trapezoid tearing tests

The trapezoid-shaped specimen of flexible composite with an initial edge slit is shown in Figure 2(a). Figure 2(b) illustrates the geometrical dimension of the tested sample which is based on the ASTM D 5587-96 standard. The pre-existing slit on the top of the specimen is 15 mm in length. The blank area in Figure 2(b) represents the effective dimensions. The trapezoid tearing tests were carried out on a MTS 810.23 tester as shown in Figure 3(a). In Figure 3(b), the inclined clamping lines of the trapezoid specimen were rotated into parallel positions and the clamping areas of the tested samples were inserted into the jaws of MTS machine. Tearing tests were carried out along the weft direction with the speed of the upper jaw set at 100 mm/min.

Figure 2.

Photograph of the specimen and the effective dimensions under the trapezoid tearing tests: (a) photograph of the specimen; (b) geometrical dimension of the specimen.

Figure 3.

Trapezoid tearing tests on the MTS 810.23 materials tester system and the tearing morphology of the specimen at different stages: (a) original stage; (b) middle stage; (c) final stage.

Testing results

The load-displacement curves obtained from the trapezoid tearing experiments are displayed in Figure 4. It can be seen that the fluctuation in the load-displacement curve represents the fracture of yarns during tearing. The increase in the tearing load can be attributed to the evolution and enlargement of the tearing delta zone.

Figure 4.

Load-displacement curves of the flexible composite under the trapezoid tearing tests.

Analytical model

As shown in Figure 3, the weft yarns between the parallel jaws failed mainly due to direct tensile fracture, while the warp yarns gradually failed in a tilt deformation with the movement of the upper jaw. The yarns in the deformation region were divided into two categories. Weft yarns were principal yarns and the warp yarns were secondary yarns. The substrate fabric and the TPU coating layer gradually deformed with the movement of the upper jaw. Since the failure strain of the TPU was much bigger than that of the PET yarns, the maximum strain criterion was defined for the failure of the substrate fabric and the maximum stress criterion was defined for the failure of the TPU coating.

Tearing load sustained by the substrate fabric

The number of yarns in the deformation region are shown in Figure 5, where $W_{i} (i = 1 \dots n)$ represent the weft yarns and $J_{k}$ ( $k = 1 \dots m$ ) represents for the warp yarns. The geometric model of the reinforcement fabric under trapezoid tearing is illustrated in Figure 6.

Figure 5.

Number of the weft and warp yarns in the tearing region.

Figure 6.

Geometric model of the woven fabric under the trapezoid tearing test.

To simplify the analytical model, the hypotheses were given as follows:

Yarns are of uniform strength, extensibility and fineness and the TPU coating is isotropic.

PET filament tows obey Hooke’s Law under trapezoid tearing.

The TPU coating was under its maximum stress when the weft yarns were stretched to the maximum strain.

Friction and slippage between the weft and warp yarns were negligible and the bonding between the substrate fabric and the TPU coating was also assumed to be perfect.

According to the hypotheses mentioned above, the tearing strength of the flexible composite could be calculated as the sum of the breaking force of the weft yarns, the tension force of the warp yarns and the rupture force of the TPU coating layer.

Breaking strength of principal yarns

The deformation evolution of the weft yarns under tearing is shown in Figure 7, where the line thickness indicates the strain value in the yarns. The original state before the tearing tests is shown in Figure 7(a). The weft yarns were gradually fractured as the distance between the two jaws increased and some weft yarns gradually became under tension exhibiting the evolution of the tearing delta zone. Hence, the tearing strength sustained by the weft yarns could be analyzed as a function of the displacement of the upper jaw (x). Here x was defined to be zero when the first weft yarn began to stretch. (Actually, the upper jaw had moved 15 mm before the first weft yarn started to deform).

Figure 7.

Deformation evolution of the weft yarns under tearing: (a) original stage; (b) the first tearing delta zone; (c) number q tearing delta zone; (d) the last tearing delta zone.

Breaking strength in the first tearing delta zone of principal yarns

As displayed in Figure 7, the strain in the first weft yarn was increased from $ɛ_{0}$ (strain equal to zero) to $ɛ_{max}$ (strain equal to maximum), as the upper jaw was moved upward. At the same time, the first delta zone was propagated to the second weft yarn when the distance of the two jaws was equal to its original length.

Based on classic mechanical theory

f = σ \times A

(1)

where

σ = E \times ɛ

This can be expressed as

f = A \times E \times ɛ

(2)

where

ɛ

is the strain,

σ

is the stress,

A

is the area of the fiber bundles and

E

is the Young’s modulus of the fiber bundles.

Based on linear interpolation, when $x = l_{1} ɛ_{w}$ , there are three weft yarns in the first tearing delta zone. The strain of these first three yarns in the first delta zone can be written as

ɛ_{1}^{1} = \frac{x}{l_{1}}, ɛ_{2}^{1} = \frac{\frac{2}{3} x}{l_{1} + d}, ɛ_{3}^{1} = \frac{\frac{1}{3} x}{l_{1} + 2 d}

(3)

where

ɛ_{w}

is the maximum strain of the weft yarn,

l_{1}

is the original length of the first intact weft yarn and

ɛ_{p}^{q}

is the strain of p^th weft yarn in the q^th tearing delta zone.

The tearing strength $f_{1}$ can be calculated from equation (2), as follows

f_{1} = A_{w} E_{w} ɛ_{1}^{1} + A_{w} E_{w} ɛ_{2}^{1} + A_{w} E_{w} ɛ_{3}^{1}

(4)

Substituting equation (3) into equation (4), the tearing strength of the weft yarns in the first delta zone can be expressed as

f_{1} = A_{w} E_{w} [\frac{x}{l_{1}} + \frac{\frac{2}{3} x}{l_{1} + d} + \frac{\frac{1}{3} x}{l_{1} + 2 d}]

(5)

Breaking strength in the q^th tearing delta zone of principal yarns

Similar to the first tearing delta zone, the q ^th tearing delta zone was generated when $l_{q} ɛ_{w} = x < l_{i}$ . At this moment, the distance between the two jaws is smaller than the original length of the i^th yarn. In other words, the number of weft yarns in the q^th tearing delta zone is $i - q$ . At this moment, the q^th weft yarn had already been fractured, with the other $i - q - 1$ weft yarns are in a tension state. The strain of each weft yarn in the q^th delta zone could be expressed based on linear interpolation and the original length of the weft yarns. Accordingly, the tearing force applied on the weft yarns could be obtained as

f_{q} = A_{w} E_{w} {\frac{[x - (q - 1) d]}{l_{1} + (q - 1) d} + \frac{\frac{i - q - 1}{i - q} [x - (q - 1) d]}{l_{1} + qd} + \frac{\frac{i - q - 2}{i - q} [x - (q - 1) d]}{l_{1} + (q + 1) d} + \dots + \frac{\frac{1}{i - q} [x - (q - 1) d]}{l_{1} + (q + i - 2) d}}

(6)

where

d

is the difference in length between adjacent weft yarns.

All principal yarns failure

As analyzed above, if the vertex of the tearing triangle is at the last yarn of the specimen, as shown in Figure 7(d), the weft yarns in this tearing delta zone will assemble together and at this moment the whole specimen is actually in complete failure.

Tearing strength sustained by the secondary yarns

Warp yarns were defined as the secondary yarns herein. As shown in Figure 8(a), some warp yarns are fixed onto the jaws at some points (i.e. P1, P2); while some other warp yarns are bonded together with the TPU coating layer. Part of the warp yarns rotated and were in a tension state due to the fracture of the weft yarns when the upper jaw moved at a constant velocity. In other words, the segments of the warp yarns in the tearing delta zone rotated over an angle and then stretched, while the other part of the warp yarns was perfectly fixed in the TPU coating layer without any deformation. The warp yarns contribute to the tearing strength of the flexible composite in this way.

Figure 8.

Deformation evolution of the warp yarns under tearing: (a) original stage; (b) the first tearing delta zone; (c) number q tearing delta zone; (d) the geometrical model of the deformation region.

When the weft yarn fractured, the intersecting point was damaged; therefore, the warp yarn was loaded relative to the jaws with a tensile strain. After this time, the tearing delta zone moved to the next weft yarn, while the former warp yarn failed. The former warp yarn did not recover within the elastic range but also it did not apply the tensile strain to all of the fabric system.

The geometrical model of the warp yarns in the tearing deformation region was developed to calculate the tearing strength shared by the warp yarns, as shown in Figure 8(d). Based on the observation of the real tearing tests, there were three weft yarns and two warp yarns deformed in the first tearing delta zone. The two warp yarns rotated at an angle $β$ , when $x = l_{1} ɛ_{w}$ and the tearing strength of the warp yarns can be obtained from the following analytical model

Calculation of tearing strength for warp yarn J#1

L_{1} = l_{1} \times (1 + ɛ_{w}), l_{3} = l_{1} + 2 d

(7)

when

l_{3} < L_{1} < l_{4}

, the strain of

l_{4}

is zero

EF = \frac{L_{1} - l_{3}}{2}, DE = \frac{L_{1} - l_{3}}{2} \times \cot θ

(8)

where

θ

is the clamping angle of the specimen as show in Figure 8(d).With the spacing between adjacent weft yarns being

d_{w}

, the following parameters can be obtained

EG = d_{w} - DE = d_{w} - \frac{L_{1} - l_{3}}{2} \times \cot θ

(9)

AB = EG + \frac{x}{2} \times \cot θ

(10)

Then

AB = (d_{w} - \frac{L_{1} - l_{3}}{2} \times \cot θ) + \frac{x}{2} \times \cot θ

(11)

Thus,

β_{1}

could be deduced as

\tan β_{1} = \frac{\frac{x}{2}}{AB} = \frac{\frac{x}{2}}{(d_{w} - \frac{L_{1} - l_{3}}{2} \times \cot θ) + \frac{x}{2} \times \cot θ}

(12)

and

\sin β = \frac{\tan β}{\sqrt{1 + \tan^{2} β}}

The strain of warp yarn J#1 is

ξ_{1}

, as follows

ξ_{1} = \frac{\frac{\frac{x}{2}}{\sin β_{1}} - \frac{\frac{x}{2}}{\tan β_{1}}}{\frac{\frac{x}{2}}{\tan β_{1}}} = \frac{\frac{1}{\sin β_{1}} - \frac{1}{\tan β_{1}}}{\frac{1}{\tan β_{1}}} = \sqrt{1 + \tan^{2} β_{1}} - 1

(13)

where

β_{1}

is the angle between the movement direction of the jaws and the stretching direction of the warp yarns. The tearing strength provided by the warp yarns in the movement direction of the jaws could be written as

t_{1} = 2 A_{j} E_{j} ξ_{1} \sin β_{1}

(14)

Calculation of tearing strength for warp yarn J#2 With the distance between adjacent warp yarns being $d_{j}$ , the following geometric relationship can be calculated

A' C' = \frac{x}{2} - d_{j}

(15)

and

A' B' = (\frac{x}{2} - d_{j}) \times \cot θ + EG = (\frac{x}{2} - d_{j}) \times \cot θ + d_{w} - \frac{L_{1} - l_{3}}{2} \times \cot θ

(16)

The rotation angle of the second warp yarn is

β_{2}

\tan β_{2} = \frac{\frac{x}{2} - d_{j}}{A' B'} = \frac{\frac{x}{2} - d_{j}}{(\frac{x}{2} - d_{j}) \times \cot θ + (d_{w} - \frac{L_{1} - l_{3}}{2} \times \cot θ) + \frac{x}{2} \times \cot θ}

(17)

The strain of the second warp yarn (J#2) can be expressed as

ξ_{2} = \frac{\frac{\frac{x}{2}}{\sin β_{2}} - \frac{\frac{x}{2}}{\tan β_{2}}}{\frac{\frac{x}{2}}{\tan β_{2}}} = \frac{\frac{1}{\sin β_{2}} - \frac{1}{\tan β_{2}}}{\frac{1}{\tan β_{2}}} = \sqrt{1 + \tan^{2} β_{2}} - 1

(18)

Similarly, the tearing strength supported by the second warp yarn (J#2) can be given as

t_{2} = 2 A_{j} E_{j} ξ_{2} \sin β_{2}

(19)

In summary, all of the tearing strength supported by the warp yarns in the first tearing delta zone could be summed up as

T_{1} = t_{1} + t_{2}

(20)

Accordingly, the tearing strength

T_{n}

shared by the warp yarns in the n^th tearing delta zone could be written as

T_{n} = \sum_{m = 1}^{k} 2 \times A_{j} \times E_{j} \times ξ_{m} \times \sin β_{m}

(21)

where

k

is the number of warp yarns that are in the deformation region of the nth tearing delta zone.

Breaking strength of the TPU coatings

The substrate 2/1 twill fabric was sandwiched between two TPU coating layers. The adhesion strength between the coatings and the fabric is critical for the high performance of the flexible composites. Assuming perfect bonding between the coatings and the fabric, the fracture behavior of the TPU layer can be assumed as the same as that of the fabric. However, due to the larger failure strain of the TPU, the maximum stress failure criterion was chosen. And to simplify the model, the TPU coating layer was divided into n strips, with each one being parallel to the weft yarns, as displayed in Figure 7.

Based on the assumption that the mechanical behavior of the TPU coatings is governed by Hooke’s Law, the tearing strength supplied by the n^th TPU strip can be written as

c_{n} = 2 A_{c} E_{c} {\frac{[x - (n - 1) d]}{l_{1} + (n - 1) d} + \frac{\frac{i - n - 1}{i - n} [x - (n - 1) d]}{l_{1} + nd} + \frac{\frac{i - n - 2}{i - n} [x - (n - 1) d]}{l_{1} + (n + 1) d} + \dots + \frac{\frac{1}{i - n} [x - (n - 1) d]}{l_{1} + (n + i - 2) d}}

(22)

As a result, the tearing strength of the whole flexible composite could be summarized as

S = f_{n} + T_{n} + c_{n}

(23)

where

S

is the tearing strength of the flexible composite.

Results and discussion

Load–displacement curves

The load-displacement curves of the coated woven fabric obtained from the trapezoid tearing tests and the analytical model established above were compared in Figure 9. Good agreement can clearly seen between the experimental and theoretical, especially for the average tearing force.

Figure 9.

Comparison of the load-displacement curves between the experimental results and the analytical model.

The relative error for the tearing strength can be expressed as

Relative error = \frac{| experimental - theoretical |}{experimental} \times 100 %

(24)

The equation (24) can be calculated for each point shown in Figure 9. The average relative error is 7.45% for the averaged tear forces.

The fluctuations caused by the fracture of each yarn can also be predicted by the analytical model. It also was found that the maximum tearing strength from the analytical model was slightly higher than that found in the experiment. The primary reason can be attributed to the hypotheses introduced into the analytical model. The yarns were supposed to be continuous and homogeneous in the analytical model, whereas in fact the yarn is composed by thousands of filaments. These filaments break non-simultaneously due to defects or curvatures along their length. Thus, de-bonding between the fabric and the two coating layers can occur easily in the experiment due to different extensibility. Some yarns are pulled out from the coating layers because of the imperfect adhesion of the two components.

Nevertheless, two main differences still existed between the analytical and the experimental results. One was the slow increase of load during the initial stage of the load-displacement curves and the other was the specific oscillation of the load. The reason for the slow increase of the load during the experiment is the indentation of the top coating layer as well as the extension of the crimps in the yarns. In the analytical model, the crimping factor of the yarns was neglected in the calculations. However, the crimps are extended to the straight lines in the experiment. This situation induces the differences seen in the curves in Figure 9. Although specifications for the fabric could be introduced in the analytical model to eliminate the error associated with the crimps, this will not resolve the issue with the analytical method at the yarn level. The analytical method established herein is based at the yarn level. In addition, the tearing behavior of the flexible woven composite is predicted from the behavior of the yarns and resin and the structure of the fabric. The influence of the yarn crimps can only be considered when the analytical model is established at the fabric level, not at the yarn level.

Here, x was defined to be zero when the first weft yarn began to stretch. The upper jaw moved 15 mm when the first weft yarn started to deform. This is because the distance between the jaws before the test was 25 mm, while the length of the first intact yarn was 40 mm. The latter is caused by the asynchronous breakage of the filaments in the yarns when they are subjected to the load. For the second difference, the breakage of the whole yarn was seen to occur simultaneously in the analytical model due to the assumption that all the yarns were assumed to be continuous and homogeneous entities; however, in the specimen, each yarn is assembled by thousands of filaments with defects and curvatures. The non-simultaneous fracture of each filament will cause a small fluctuation of the load-displacement curve. It is very difficult to simulate precisely each fluctuation in the plot.

Tearing damage analysis

Figure 10 shows the evolution of the tearing delta zone of the flexible composite during the trapezoid tearing test. The evolution of the tearing delta zone has a significant influence on the tearing strength. The relationship between the displacement of the upper jaw, the number of the weft and warp yarns in the tearing delta zone, the properties of the yarns and the tearing strength was obtained. As observed from the tearing tests, more and more weft yarns were deformed in the delta zone with the movement of the jaw. In other words, the tearing strength as well as the tearing delta zone was increased during the tearing test. The result implied that the tearing strength would get to the maximum when the vertex of the tearing delta zone reached the last weft yarn of the specimen. Earlier research suggested that the tensile strength of uncoated, woven fabric is greater than that of coated, woven fabric. This is mainly due to the slippage between the yarns being restricted by the coating resin. The damage morphologies of the coated woven fabric have been observed after the tearing failure, as shown in Figure 3(c). Yarns can be well oriented along the direction of tension load on account of the restriction of the coating resin.

Figure 10.

Evolution of the tearing delta zone of the flexible composite during the tearing test.

Energy absorption distribution

Figure 11 shows the energy absorption of each component of the flexible composite. It can be seen that 82.87% of the total energy was absorbed by the principal yarns, 13.53% by the secondary yarns and only with 3.6% by the coating layer. Obviously, the principal yarns play the most important role in the energy absorption ability of the flexible composite. Nevertheless, the energy absorbed by the secondary yarns cannot be ignored, as they contribute to the fluctuation tendency of the load-displacement curve of the flexible composite. In the fabric design, it is suggested to use fibers with high tenacity, modulus and breaking strain to increase the energy absorption by the principal yarns and the deformation of the secondary yarns.

Figure 11.

Energy absorption distribution of each component in the flexible composite.

Although the TPU coatings contribute the least energy absorption of all of the components, it is also an indispensable part for the flexible composite. As a result of the TPU coatings, there were a huge number of consolidation points between the fabric and the coating, especially in the interlacing points of the fabric. The slippage of the yarns during tearing was seriously constraint by these consolidation points. Because of this constraint, the tearing trajectory in the specimen was regular and practically along a straight line, as shown in Figure 3.

Conclusions

The tearing performance of the flexible composite was investigated with the trapezoid tearing test and through an analytical model approach. An analytical model was established by calculating the deformation of the weft and warp yarns and the matrix, respectively. A series of equations were developed to relate the tearing strength of the flexible composite with the extensibility and breaking strength of all of the components of the flexible composite. A good agreement was obtained in the load-displacement curve between the analytical and experimental results, especially in the elastic modulus and the average tearing force. The fluctuation caused by the fracture of each single yarn could also be simulated by the analytical model. It was found that the breaking strain and the elastic modulus of the yarn as well as the weaving density in the fabric are the key factors affecting the tearing force of the flexible composite. Thus, yarns with superior extensibility and higher elastic modulus should be used in the material selection. The tearing performance and deformability of the whole flexible composite should be optimized based on the actual requirement of the end products by adjusting the fabric’s structural parameters (i.e. weaving density, weaving pattern, etc.). The validated analytical model will to some extent assist the designers and manufacturers to obtain the optimum material and specify the construction of the flexible composite.

Nomenclature

$f$

Breaking force

ɛ_{0}

Strain equal to zero

ɛ_{max}

Strain equal to maximum

x

Relative displacement of two jaws

l_{1}

Length of the first intact weft yarn

ɛ_{w}

Breaking strain of weft yarn

ɛ

Strain

σ

Stress

A

Area

E

Young’s modulus of elasticity of the material

l_{n}

Length the number n weft yarn

d

Distance of $l_{n}$ from the shorter one $l_{n - 1}$

ɛ_{p}^{q}

Strain of W p in the number of q tearing delta zone

f_{n}

Breaking force of the number n weft yarn

A_{w}

A_{j}

A_{c}

Cross sectional area of weft yarn, warp yarn and coatings

E_{w}

E_{j}

E_{c}

Elastic modulus of weft yarn, warp yarn and coatings

i

The number of weft yarns staying in the tearing delta zone

β

Weaving angle between weft yarn and warp yarn

L_{1}

Breaking length of the first intact weft yarn

θ

Complement angle of the trapezoid angle

ξ

Strain of warp yarn

m

The number of the warp yarn

k

The number of the warp yarn extension

T_{n}

Warp yarn tensile strength

c_{n}

Tensile force of the $n$ coating bar up to the rupture

S

Tearing strength of the flexible composite

Footnotes

Funding

The authors acknowledge the financial support from the National Science Foundation of China (Grant Numbers 11072058 and 11272087). The financial support from the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD, No. 201056), the Shanghai Rising-Star Program (11QH1400100) and the Fundamental Research Funds for the Central Universities of China are also gratefully acknowledged.

References

Harrison

. The tearing strength of fabics part I: a review of the literature. J Text Inst 1960; 51: T91–T131.

Krook

Fox

. Study of the tongue-tear test. Text Res 1945; 15: 389–396.

Hager

Gagliardi

Walker

. Analysis of tear strength. Text Res 1947; 17: 376–381.

Teixeira

Platt

Hamburger

. Mechanics of elastic performance of textile materials: part XII: relation of certain geometric factors to the tear strength of woven fabrics. Text Res 1955; 25: 838–861.

Turl

. The measurement of tearing strength of textile fabrics. Text Res 1956; 26: 169–176.

Steele

Gruntfest

. An analysis of tearing failure. Text Res 1957; 27: 307–313.

Taylor

. Tensile and tearing strength of cotton cloths. J Text Inst Trans 1959; 50: 161–188.

Topping

. The critical slit length of pressurized coated fabric cylinders. J Ind Text 1973; 3: 96–110.

Sarma

. A study of tear-length distributions in woven apparel fabrics in service. part I: theoretical models. J Text Inst 1975; 66: 375–381.

10.

Hamkins

Backer

. On the mechanisms of tearing in woven fabrics. Text Res 1980; 50: 323–327.

11.

Godfrey

Rossettos

Bosselman

. The onset of tearing at slits in stressed coated plain weave fabrics. J Applied Mech 2004; 71: 879–879.

12.

Godfrey

Rossettos

. Analysis of damage tolerance to tear propagation in stressed structural fabrics. Struct Mat 2000; 6: 149–158.

13.

Godfrey

Rossettos

. A parameter for comparing the damage tolerance of stressed plain weave fabrics. Text Res 1999; 69: 503–511.

14.

Godfrey

Rossettos

. Damage growth in pre-stressed plain weave fabrics. Text Res 1998; 68: 359–370.

15.

Triki

Dolez

Vu-Khanh

. Tear resistance of woven textiles – criterion and mechanisms. Comp Part B: Engr 2011; 42: 1851–1859.

16.

Hinkley

Hoogstraten

. Tearing of thin polyimide films. J Mat Sci 1987; 22: 4422–4425.

17.

Minami

. Strength of coated fabrics with crack. J Ind Text 1978; 7: 269–292.

18.

Dartman

Shishoo

. Studies of adhesion mechanisms between PVC coatings and different textile substrates. J Ind Text 1993; 22: 317–335.

19.

Bigaud

Szostkiewicz

Hamelin

. Tearing analysis for textile-reinforced soft composites under mono-axial and bi-axial tensile stresses. Comp Struct 2003; 62: 129–137.

20.

Wang

Sun

. Finite element modeling of woven fabric tearing damage. Text Res 2011; 81: 1273–1286.

21.

Wang

Sun

. Comparisons of trapezoid tearing behaviors of uncoated and coated woven fabrics from experimental and finite element analysis. Inter J Damage Mech. Epub ahead of print 6 June 2012. DOI: 10.1177/1056789512450524.

Analytical modeling on mechanical responses and damage morphology of flexible woven composites under trapezoid tearing

Abstract

Keywords

Materials and testing

Flexible composite

Environmental conditions

Trapezoid tearing tests

Testing results

Analytical model

Tearing load sustained by the substrate fabric

Breaking strength of principal yarns

Breaking strength in the first tearing delta zone of principal yarns

Breaking strength in the qth tearing delta zone of principal yarns

All principal yarns failure

Tearing strength sustained by the secondary yarns

Breaking strength of the TPU coatings

Results and discussion

Load–displacement curves

Tearing damage analysis

Energy absorption distribution

Conclusions

Nomenclature

Footnotes

Funding

References

Breaking strength in the q^th tearing delta zone of principal yarns