Central crack tearing test and fracture parameter determination of PTFE coated fabric

Construction and Building Materials 208 (2019) 472–481

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Central crack tearing test and fracture parameter determination of PTFE coated fabric Rijin He a, Xiaoying Sun a,b,⇑, Yue Wu a,b a

Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin 150090, China b

h i g h l i g h t s
Central crack tearing tests are conducted to investigate the residual strength of coated fabrics.
Closed form solutions based on classical LEFM are testified to be inapplicable for coated fabrics.
Virtual crack closure technique is introduced to the fracture parameters determination of coated fabrics.
Material non-linearity and orthotropic are considered in the fracture parameters determination.

a r t i c l e

i n f o

Article history: Received 6 December 2018 Received in revised form 5 March 2019 Accepted 5 March 2019

Keywords: Coated fabric Central crack tearing test Tearing property Non-linearity Fracture mechanics Virtual crack closure technique

a b s t r a c t Catastrophic failure caused by crack propagation is the most common failure mode of tensile fabric structure, the tearing resistance of the coated fabric is the key issue of structural design. This study concerns the residual strength and fracture resistance of PTFE coated fabric. Central crack tearing (CCT) tests are conducted, experimental results shown that the existence of initial crack significantly reduces the residual strength. Furthermore, fracture parameters which can be treated as material constants are obtained to estimate the tearing resistance. The available theoretical solutions, based on the classical linear elastic fracture mechanics (LEFM) theory, are adopted to calculate the fracture parameters. Results indicated that these theoretical solutions cannot estimate the fracture parameters properly due to the nonlinear stiffness of the coated fabric. Therefore, the numerical method virtual crack closure technique (VCCT) is introduced to the fracture parameters determination of PTFE coated fabric. Combined with the nonlinear material constitutive model, critical energy release rate GIC obtained from VCCT effectively reducing its dependence on crack length and thus can be treated as material tearing resistance. Finally, the influence of boundary condition and shear modulus are discussed. Results shown that uniform displacement boundary condition should be used in VCCT. When the shear modulus decreases, the GIC increases gradually and the dependence on the crack length is reduced due to the decrease of stress concentration zone. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Woven coated fabrics for tensile fabric structures are widely used for permanent works for various applications such as sport stadiums, transportation and commercial constructions. It can provide excellent architectural expressions and large space. In structural design, large safety factors (usually 5  7 for the fabric strength) are applied to consider material variability, environmental degradation and local damages during construction [1], thus there is a large safety margin for flawless fabric and it is unlikely ⇑ Corresponding author. E-mail address: [email protected] (X. Sun). https://doi.org/10.1016/j.conbuildmat.2019.03.046 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

to be ruptured under tension stress. However, the commonly used PTFE coated glass fiber fabric usually have high tensile strength but low tearing strength, crack propagation can be easily induced even small load is applied, hence catastrophic failure caused by crack propagation is the most common failure mode [2], the tearing resistance of the coated fabric become the key issue of structural design. To ensure the tearing resistance is large enough to prevent the pre-existing small flaws from propagating, large efforts have been devoted to the understanding of the tearing behavior of coated fabrics. In the past decades, two kinds of test methods are proposed to study the tearing behaviors of woven fabrics. One is in-plane tearing test such as trapezoidal tearing method [3–6], central crack

R. He et al. / Construction and Building Materials 208 (2019) 472–481

tearing method [2,7–14] and single edge notch tearing method [15–17]. The other is out-of-plane tearing test such as tongue tearing method [14,18–21]. In the case of coated fabric for structural use, most of the test standards adopt the trapezoidal tearing method (CECS 158:2015 [22], 2015; MSAJ/M-03:2003 [23], 2003; JIS L1096:2010 [24], 2010; ASTM D4851-07 [25], 2015). However, the trapezoidal tearing method were originally intended for the garments industry rather than as appropriate tests for structural fabrics [1]. During trapezoidal tearing test, the external loads were mainly sustained by a few yarns within the tearing delta zone, and the tearing strength would arrive its maximum when the first yarn of the tearing delta zone ruptured [5]. Thus, the tearing strength obtained by the trapezoidal tearing test is far lower than the tensile strength, no conclusion about actual tearing strength can be drawn from this test, this test is only included for the sake of completeness [1]. Since the trapezoidal tearing method is inappropriate for structural fabrics, central crack tearing method is recommended by European Design Guide for Tensile Surface Structures [1] and FAA-P-8110-2, Airship Design Criteria [26] due to its similar tearing characteristics to the actual conditions in terms of the stress distribution and crack opening shape [11]. In addition, available fracture parameter solutions based on fracture mechanics theory can be directly applied to central crack tearing tests. Currently, the evaluation of the woven fabric’s tearing resistance was mostly performed through tearing strength and allowable crack length [2,7,11,15]. The so called tearing strength (or residual strength) is a nominal stress which refers to the residual load bearing capacity of the damaged structure, it can be defined as the peak tearing load divided by the cross section area of the specimen. However, stress in the cracked specimen is not uniformly distributed, the nominal stress can’t reflect the stress concentration intensity near the crack tip. Furthermore, the tearing strength is highly dependent on the specimen geometry, thus it is not a material constant and can’t be used as material tearing resistance. To deal with fracture initiation and failure processes in cracked structures, fracture mechanics has been widely used. According to fracture mechanics, fracture parameters such as stress intensity factor (SIF) KI and strain energy release rates (SERR) GI are needed to be found to identify the loading condition at the crack. Once KI or GI reach the material fracture resistance, i.e. fracture toughness KIC or critical strain energy release rate GIC, crack starts propagation. Moreover, the application of fracture mechanics in residual strength prediction also requires the determination of the fracture parameter KIC or GIC. Analytical solutions to the SIFs for various cracked configurations of isotropic material can be found in handbooks [27]. However, the SIFs for orthotropic materials are not only a function of the crack geometries and boundary conditions, but also related to the material properties [28,29], only few cases of cracked fabric specimen was modeled as orthotropic materials for simplicity. Davidson et al. [30] developed an experimental technique to measure the fracture toughness of fabrics, pneumatic pressure is applied on an expandable fabric cylinder with a crack of known length until the crack propagates. The fracture toughness KIC was derived from linear elastic fracture mechanics (LEFM) solutions for isotropic material. Bigaud et al. [10] adopted the central crack tearing method to measure the fracture toughness of polyester fabrics with PVC coating. The fracture toughness KIC is also derived from solutions for isotropic material. Obviously, the effect of orthotropic and nonlinearity of coated fabric are not estimated in aforementioned investigations. Minami [31] proposed a theoretical formula to calculate the critical SERR of woven coated fabric, it was derived based on Griffith’s energy balance theory and Hedgepeth’s yarn model, the material orthotropic and yarn density are considered. However, the nonlinearity of coated fabric is still ignored.

473

This study focused on the residual strength and fracture toughness of PTFE coated fabric. To this end, central crack tearing tests were conducted, the influence of the crack length on residual strength was investigated. According to the test data, the coated fabric exhibits significant nonlinearity, thus the commonly used theoretical solutions which were derived from LEFM theory may not be applicable. Therefore, the numerical method VCCT is introduced to the fracture parameters determination of PTFE coated fabric. Combined with a nonlinear orthotropic material model, the critical SERR obtained from VCCT effectively reducing the crack length dependences, thus it can be regarded as intrinsic material property which represent the tearing resistance. Finally, the importance of boundary condition and shear modulus on fracture parameter determination are discussed. 2. Experiments In this part, the tensile test and central crack tearing test are conducted. The nonlinear material model is derived from the tensile stress-strain curves and then implanted into finite element analysis, the residual strength and tearing properties are obtained from the central crack tearing test, and these residual strength data is necessary for the fracture parameters determination. 2.1. Material and specimens The material used in these tests is a plain-woven glass fiber fabric with a polytetrafluoroethylene (PTFE) coating (Fig. 1), there are 14 and 11 yarn counts per centimeter in the warp and weft directions, respectively. The thickness is 0.6 mm and the areal density is 1050 g/m2. This material is intended for large-scale tensile fabric structure. Because of the plain-woven structure, it behaves like orthotropic material. The small thickness and the softness of the matrix reduce the bending stiffness, it thus behaves as a membrane. All the units of strength and stress are described as kN/m, which is a common practice in membrane material studies. As shown in Fig. 2(a), the specimen for the uniaxial tension test is tailored to a strip which refers to the standard of MASJ/M-03:2003, the gauge length and the width of the specimen are 200 mm and 50 mm, respectively. The clamped lengths in two sides of the specimens are 40 mm respectively, the clamped area was reinforced with aluminum sheets to prevent failure in the clamped region prior to the crack propagation. These strip specimens were aligned to either the warp or weft direction of the fabric. For each direction, 5 specimens are tested to guarantee the reliability of the results. In order to study tearing properties of the coated fabric with initial crack through uniaxial tensile test, central crack specimens are prepared. The crack was introduced along the yarn orientations using a knife blade, it is guaranteed that equal number of yarns are cut off in the same crack length specimens. Fig. 2(b) illustrates the geometrical dimension of tested sample, which is the same as the tensile test sample. The crack lengths varies from 5 mm to 25 mm with an interval of 5 mm. The layout of the experimental plan is shown in Table 1.

Fig. 1. The PTFE coated fabric.

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Fig. 3. The MTS-810 test device. Fig. 2. Schematic diagram of the (a) tensile specimen and (b) central crack tearing (CCT) specimens. Unit = mm. 2.2. Experiment setup The tests are carried out on a MTS-810 uniaxial hydraulic servo tester (Fig. 3). This tester has equipped with hydraulic clamps, the clamping pressure can be adjusted according to the stiffness of the material. Since the flexibility of the coated fabric, the clamping pressure is set as 4 MPa to prevent the failure and slippage occurring in the clamped region. A digital camera is used to take high resolution images during the loading process. The actuator displacement and load cell data are recorded at a frequency of 40 Hz. The load–displacement response is obtained from the experiments. In order to reduce dynamic effects, the loading rate should be set up at a relatively low speed. Meanwhile, the elongation at break is usually very large for the tearing specimen, thus the loading rate shouldn’t be too low in order to shorten experimental period. Therefore, a medium loading rate 10 mm/min is set so that the dynamic effect can be neglected and test process will not be too long. 2.3. Experimental results 2.3.1. Tensile test results The typical tensile behaviors of PTFE coated fabric subjected to mono-uniaxial loading are shown in Fig. 4. The tensile strength in warp and weft direction are 99.44 kN/m and 94.84 kN/m respectively, the tensile strength of warp direction is only 4.8% higher than that of weft direction, this is due to the similarity of yarn counts in the warp and weft directions, which is 14 and 11 yarn counts per centimeter respectively. The elongation at break in warp and weft direction are 6.16% and 7.73% respectively, the ultimate strain in warp direction is 20.3% lower than that of weft direction, this significant difference could be due to the fact that the woven fabric has a higher level of crimp in the weft direction [32], an applied load in the weft direction must removes the crimp first, and then straightens the weft yarns.

Table 1 Layout of the experiment plan. Test

Specimen dimension

Loading direction

Total number of specimens

Tensile test

2 W = 50 mm; 2L = 200 mm2 W = 50 mm; 2L = 200 mm; 2a = 5 mm, 10 mm, 15 mm, 20 mm, 25 mm

Warp Weft

10

Central crack tearing test

30

Fig. 4. Stress-strain curves in warp and weft direction.

The high non-linearity presented in Fig. 4 can be divided into two parts: material non-linearity and geometric non-linearity. Material non-linearity is evident in the load extension characteristics of both the yarn fibers and the fabric coating. Geometric non-linearity occurs in the finished fabric due to crimp interchange [32]. In order to simplify the stress-strain relation so that can be easily implanted into finite element analysis, both warp and weft stress-strain curves can be approximatively divided into three regions: (I) the low stiffness linear region OA, which corresponds to the cooperation work of coating and woven yarns. (II) The nonlinear region AB, this is due to geometric non-linearity caused by crimp interchange and slack in and between the yarns. (III) The high stiffness linear region BC, the increase in tensile stiffness is due to the completion of crimp interchange and yarn stretching becomes the dominant deformation mechanism. The characteristic points A, B, C can be determined based on the principle that the three regions model fits the nonlinear stress-strain curves best. Young’s modulus of the material can be obtained by calculating the slope of the stress–strain curves in the elastic region. Traditionally, the least square regression coupled with stress-strain curve were applied to determine the modulus. Since the non-linearity of the stress–strain relation, the tensile modulus depends extremely on the strain. Thus, polynomial functions are adopted to fit the stress-strain curves, then the first derivative of the stress-strain curves, i.e. the modulus-strain curves, is obtained to study the non-linearity characteristic of fabric mechanical

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R. He et al. / Construction and Building Materials 208 (2019) 472–481 behavior. The modulus-strain curve contains the same information as the stressstrain curve, but small differences in mechanical behaviors between warp and weft direction stand out more observably in the modulus-strain curve [33,34]. When using the modulus-strain curve, Chen [35] proposed a formula based on the mean value theorem of integrals to calculate modulus. As shown in Eq. (1), this formula can calculate modulus in arbitrary strain range, and it is proven to have equal accuracy with the least square regression method [35].

R e2 E¼

EðeÞde e2  e1

e1

ð1Þ

From Fig. 5, it can be seen that the modulus in warp and weft direction are dramatically different. With the increase of strain, the modulus in warp direction obviously shows two plateau region when the strain ranges from 0  2.5% and 4.0%  6.2%, corresponding to the two linear region OA and BC in Fig. 4, the modulus in these plateau regions are 244 kN/m and 3062 kN/m respectively. Between these two plateau regions (when the strain ranges from 2.5%  4.0%), modulus changes approximately linearly from the lower plateau to the higher plateau, corresponding to the non-linear part AB in Fig. 4. Meanwhile, the modulus-strain curves in weft direction can also be divided into three regions but in different strain range. Modulus in these three regions are shown in Table 2. In order to implement the nonlinear material model into finite element analysis, the user material subroutine (UMAT) in ABAQUS/Standard 6.13 is employed. Based on the plane stress orthotropic material model, the stiffness matrix can be written as:

2 d11 6 ½D ¼ 4 d21 0

2.3.2. Tearing test results The evolution of the tearing force for different initial crack lengths are shown in Fig. 6. The reference curve (black line) corresponds to the flawless specimen. As shown in Fig. 6, nonlinearity is also significant in the P-d relationship. With the increase of initial crack length, the specimen stiffness and the maximum load falls down, this is mainly due to the residual yarns number decrease and the stress concentration coefficient at the crack tips increase. Based on the P-d relationships, two failure modes can be distinguished:

d12 d22 0

0

3

7 0 5 d33

ð2Þ

E t

where d11 ¼ 1tEw tt , d22 ¼ 1t f wf

fw

wf

tfw ,

t Ef t , wf tfw

d12 ¼ 1fwt

t Ew t . wf tfw

d21 ¼ 1wft

mwf and mfw are Poisson

ratios. This study mainly focus on the nonlinearity of tensile modulus, thus fixed values are adopted for the Poisson’s ratio, commonly used values for coated fabric mwf = 0.2 and mfw = 0.4 are adopted [47,48]. d33 = Gwf is the shear stiffness, which is estimated by the ‘rule of thumb’ G = E/20 commonly be used in engineering practice [1], therefore G = 70 kN/m is adopted. The tensile modulus for the material model are divided into three parts: OA, AB and BC presented in Table 2. In order to verify this UMAT code, numerical simulations are performed to simulate the uniaxial tensile tests. The simulated stress-strain curves (dash lines in Fig. 4) are compared with the experimental data (solid lines in Fig. 4), good agreement is observed between FEM results and experimental data.

(1) Progressive failure: For a crack longer than a certain threshold, such as crack length 2a = 10 mm to 25 mm cases, the initial crack propagates alternately on each side of crack tip, the fracture of yarns causes the P-d response exhibits zigzag fluctuation before it reaches peak load. After that, the load decreases gradually until the sample global failure. (2) Brutal failure: For a shorter crack, such as crack length 2a = 5 mm case, catastrophic global failure occurs suddenly without evident crack propagation, the P-d response exhibits little fluctuation and the force decreases rapidly after it reaches peak load, an extreme example is the un-cracked tensile behavior. These two failure mode can be explained from the point of energy. For the brutal failure corresponding to small crack case, specimen can sustain higher load, all the yarns parallel to the loading direction accumulate very high strain energy before it reaches peak load. When the first yarn at the crack tip raptured due to stress concentration, the following yarns can’t sustain extra external loads because it already accumulate very high strain energy, hence the yarns rapture one by one rapidly. For the progressive failure corresponding to large crack case, the strain energy accumulated in the yarns is not very high before it reaches peak load. After the first yarn raptured due to stress concentration, the following yarns still have considerable capacity to sustain extra loads, hence the yarns rapture one by one gradually. For this type of material used in tensioned membrane structures, it is critical to figure out the limiting value of the load when the crack starts propagating, which can be described as the ‘‘crack propagation threshold force’’. For brutal failure, the crack propagation threshold force Pc is almost equal to the maximum force Pu. For progressive failure, Pc is slightly smaller than the maximum force Pu. Since it is very difficult to precisely identify the crack propagation threshold force Pc from the load-displacement curves, and the difference between Pc and the maximum force Pu is relatively small [36], hence the Pc value was provisionally determined as the maximum force Pu [37–39]. In order to eliminate the effect of specimen size and directly using fracture mechanics theory, nominal stress which corresponds to critical tearing load Pu divided by gross cross-section is introduced, we defined the nominal stress as residual strength rres as shown in Eq. (3). The residual strengths of the central crack specimens were displayed in Table 3.

rres ¼

Pu 2W

ð3Þ

When taking the flawless tensile strength as a reference, for central crack specimen with a crack of only 5 mm which accounts for 1/10 of the sample width, the residual strength in the weft direction decreases 30% of the tensile strength, which means that the crack weakening its residual strength remarkably. As a type of composite membrane used for tensile structures, it is meaningful to estimate the crack sensitive property. The sensitivity of a material to crack can be further illustrated by a simple net strength concept. If a material completely insensitive to the stress concentrations at the crack tips, the stress is uniformly distributed in the cross section at the crack, the failure load PN of central crack specimen can be given by:

PN ¼ f t ðW  aÞ ¼

P0 ðW  aÞ W

ð4Þ

where P0 is the un-cracked failure load, i.e. the tensile strength. On the other hand, if the material has more tendency to accommodate stress concentration, the cracked specimen’s failure load PN would fall below the prediction of Eq. (4). To investigate the crack sensitivity of the coated fabric, Fig. 7 shows the nondimensional residual strength PN/P0 versus the crack lengths ratio a/W, the straight-line upper bound represents Eq. (4). Non-dimensional residual strengths obtained from tearing tests fall below the linear relation in both the warp and weft

Fig. 5. Modulus–strain curves.

Table 2 Modulus in warp and weft direction. Region

OA AB BC OC

Warp

Weft

Strain range

Ew∙t (kN/m)

Strain range

Ef ∙t (kN/m)

0  2.5% 2.5%4.0% 4.0%6.2% 0  6.2%

244 4452 + 187840ewarp 3062 1481

0  2.5% 2.5%6.0% 6.0%7.7% 0  7.7%

446 827 + 50932eweft 2229 1197

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Fig. 6. The tearing load-displacement curves of central crack specimens.

Table 3 Residual strength of central crack specimen. Direction

Crack length (mm)

0

5

10

15

20

25

Weft

Residual strength (kN/m) Strength ratio (%)

99.44 100.00

70.78 71.18

61.32 61.67

52.17 52.46

45.49 45.75

37.49 37.70

Warp

Residual strength (kN/m) Strength ratio (%)

94.84 100.00

61.93 65.30

53.13 56.02

48.86 51.52

36.40 38.38

34.30 36.17

implanted in the numerical method virtual crack closure technique (VCCT) to calculate fracture parameters. 3.1. Closed form solutions based on classical LEFM Based on LEFM, many closed form expressions for the SIF solution to a central crack specimen have been proposed. The most commonly used SIF solution was given by Bao et al. [28]:

K I ¼ YðqÞ  Fða=WÞ  r

pffiffiffiffiffiffi pa

ð5Þ

h i 1 þ q1=4 2 3 Y ðqÞ ¼ 1 þ 0:1ðq  1Þ  0:016ðq  1Þ þ 0:002ðq  1Þ  2 ð5aÞ

  a 2  a 4 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa sec F ða=W Þ ¼ 1  0:025 þ 0:060 W W 2W Fig. 7. Cracked strength PN normalized with un-cracked strength P0 against crack length (a/b). direction, illustrating that coated fabric is indeed crack-sensitive. The data points in the weft are lower than those in the warp, showing that the material is more ductile in the warp direction.

3. Methods of fracture parameter determination It is known that the classical strength criteria are inapplicable to crack propagation analysis, thus fracture mechanics has been widely used. Fracture parameters are essential to characterize material’s resistance against crack propagation. However, the commonly used fracture parameter solutions are derived from classical LEFM theory, which may be inapplicable due to the nonlinearity of the coated fabric. Therefore, a nonlinear material model is

q¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Exx  Eyy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  mxy  myx 2Gxy

ð5bÞ

ð5cÞ

where 2a and 2 W are the crack length and the width of the specimen, respectively (shown in Fig. 8). Y(q) is the orthotropic correction factor. F(a/W) is the crack geometry factor. r is the applied remote tensile stress. While for the uniform displacement condition, r = P/(2 wt), and P is the total reaction force on the clamped edge, t is the specimen’s thickness. For central crack specimen, the satisfactory accuracy error is less than 5% when the dimensionless parameter q<4. For the PTFE coated fabric tested in this study, the nominal Young’s modulus in OC region is adopted (Table 2), i.e. Ewt = 1481 kN/m and Eft = 1197 kN/m, Poisson ratios mwf = 0.2 and mfw = 0.4, and the shear stiffness Gwf = 70 kN/m. According to Eq. (5c), q is up to 9.23, which means the accuracy may be not guaranteed.

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R. He et al. / Construction and Building Materials 208 (2019) 472–481 Table 4 Critical energy release rate GIC obtained from analytical solutions (N/mm). Crack length (mm)

Bao’s model

5 10 15 20 25

Minami’s model

Warp

Weft

Warp

Weft

161.91 252.80 292.48 326.00 316.63

138.27 221.22 285.33 235.45 294.81

102.27 153.42 166.38 168.53 143.02

87.51 134.38 162.43 121.77 133.22

end, the numerical model which adopts nonlinear material constitutive are needed to calculate the GIC. 3.2. Numerical method virtual crack closure technique (VCCT)

Fig. 8. Central crack plate subject to uniform tension.

Aiming at dealing with the tearing resistance of coated fabric, Minami [31] proposed a formula to calculate the fracture toughness by using Hedgepeth’s yarn model and the Griffith’s energy balance theory. In the Hedgepeth’s method, the woven coated fabric is discretized into a number of yarns along the loading direction which sustain the tension force, and the perpendicular yarns only provide the shearing stiffness. From the equilibrium of the forces, the displacement and force of each yarn can be solved, and then the SERR can be derived as:

  p 1 GI ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 4a þ r2 nc 8 Ex Gxy

ð6Þ

where, Ex is the elastic modulus in the loading direction. Gxy is the shear modulus. 2a is the crack length. nc is the number of yarns per unit length in the loading direction. In fracture mechanic theory, parameter SIF and SERR are equivalent. For orthotropic material, SIF and SERR can convert to each other by the formula proposed by Sih et al. [40]:

 GI ¼

1þq 2Exx Eyy

1=2

k1=4  K 2I ¼

K 2I E

ð7Þ

Substituting the residual strengths rres given in Table 3 into Eqs. (5) and (6), the fracture toughness KIC and critical energy release rates GIC can be obtained. Since the aforementioned solutions are derived based on classical LEFM theory, constant modulus must be adopted. Therefore, the modulus in OC region (Table 2) are adopted although the coated fabric exhibits significant nonlinearity, thus the GIC obtained from these analytical models are highly dependent on crack length. As shown in Table 4, with the crack length increases from 5 mm to 25 mm, the GIC in warp direction obtained from Bao’s model (Eq. (5)) varies from 161.91 N/mm to 316.63 N/mm, which means that the Bao’s model is especially not suitable for coated fabric. For Minami’s model (Eq. (6)), the GIC in warp direction varies from 102.27 N/mm to 168.53 N/mm which is also highly dependent on crack length. The fracture mechanics properties should be regarded as material constants, it is preferable take the material nonlinearity into account and reduce the dependence of GIC on crack length. To this

There are many ways to calculate the fracture parameters using finite element method, among them the VCCT [41] is the most widely used. VCCT is based on the crack closure integral and can be used with a constant strain finite element and relatively coarse mesh. Based on Irwin’s crack closure integral [42], the energy absorbed in the crack propagation process is equal to the work required to close the crack to its original length if a crack extends by a small amount Da. In terms of the finite element representation, the energy release rate GI for mode I crack opening deformation is one-half the product of the nodal reaction forces at the crack tip and the nodal displacements behind the crack tip. In equation form, the expressions for GI is:

GI ¼ lim

Da!0

1 2BDa

Z 0

Da

ryy Dv dx ffi

F y1 Dv 3;4 2BDa

ð8Þ

where Fy1 is the nodal force at the crack tip node 1 in y-direction, Dv3, 4 is the relative displacement of nodes 3 and 4 in y-direction, which are located at a distance Da behind the crack tip (illustrated in Fig. 9). In the Finite element analysis (FEA), a simplified FEA model of the coated fabric was created in a commercial FEA package ABAQUS/standard 6.13. Due to the symmetry of the specimen, only 1/4 of the test specimen is established as shown in Fig. 10, the global size of the finite element model is 25 mm 100 mm. In a laboratory test, the clamped location in the real specimen is closer to uniform displacement boundary condition, thus ux = 0, uy = d are applied at the clamped edge as shown in Fig. 10, the applied displacement is determined through trial and error until the total reaction force is equal to the maximum force Pu shown in Table 3. Plane stress elements CPS4 are used in the calculation. Fig. 11 shows the FEA mesh used in the calculations, the mesh close to the crack tip is finer, the finest element size is 0.125 mm 0.125 mm and the element size is gradually transferred into larger element to reduce the computational demand. The total crack length 2a was varied from 5 mm to 25 mm with an interval of 5 mm. The nonlinear orthotropic material model proposed in Section 2.3.1 is employed in numerical simulation, stiffness, strains, and stresses can be tracked at the material point of each element. This information is provided by the material model which can be interfaced with the finite element software of ABAQUS through the user defined subroutine UMAT. As a material property, the fracture mechanics parameters, including fracture toughness KIC and critical energy release rate GIC, should be independent from the crack length and specimen configuration. Fig. 12 shows the dependence of the GIC obtained from VCCT on the normalized crack length a = a/W. It can be seen that the GIC approximatively stay constant, except the GIC for

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Fig. 9. Schematic diagram of Finite element representation of VCCT.

Fig. 11. Finite element mesh.

Fig. 10. 1/4 model (shaded in oblique line) and the uniform displacement boundary condition.

a = 0.1 is smaller than others. This means the critical energy release rate can be treated as independent on the crack length when a 0.2 in this study. The smaller value of GIC for a = 0.1 case may result in two reasons: Firstly, the coated fabric come to the plastic regime at low stress level [43], thus the GIC calculated based on the non-linear elastic material model may ignore the plastic part, and thus under-estimate the fracture toughness. In the case of 2a = 5 mm for example, the residual stress is higher than other cases (shown in Table 3), thus the plastic zone near the crack tip is the larger and possibly cannot be neglected. As a result, the critical energy release rate obtained from VCCT which only contains elastic part is relatively smaller as shown in Fig. 12. Secondly, due to the fact that the critical crack propagation load Pc is very difficult to identify and it was usually determined as the

Fig. 12. GIC obtained from VCCT.

maximum load Pu, this brings in inaccuracy when using Pu to calculate the GIC. In the large crack length case which correspond to the progressive failure mode, the maximum load Pu is slightly larger than the crack propagation load Pc which lead to overestimate of the real fracture toughness. In the short crack length case which correspond to the brutal failure mode, the maximum load Pu is almost equal to the crack propagation Pc, the inaccuracy is reduced. Due to these two reasons, the GIC for large a ratio cases are slightly larger than that of small a ratio cases. However, the overestimate of GIC due to the substitution of Pu for Pc is not remarkable, several previous studies [10,11] reported the difference between Pu

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and Pc is really small. Therefore, the GIC obtained from large a values can be considered as material constant. In this study, the critical energy release rate GIC for the coated woven fabric in warp and weft direction are 106.46 N/mm and 129.15 N/mm respectively (as shown in Table 5). When comparing the results obtained by analytical models and numerical model VCCT (Fig. 13), significant difference can be observed. Due to the inherent deficiency of the analytical models which based on LEFM, it is not suitable for nonlinear materials like coated fabric, the GIC obtained from the analytical models are highly dependent on crack length, perhaps special analytical model should be adopted to deal with the nonlinear effects [39,44]. 4. Discussion 4.1. Effect of boundary condition Several previous studies suggested that the boundary conditions have a significant effect on the fracture parameter [45,46]. In order to study the effect of boundary conditions, uniform stress boundary conditions are also simulated and compare with the uniform displacement conditions. The boundary effect on fracture parameter can be represented by SIF correction factor f(a) which follows Eq. (9).

KI f ðaÞ ¼ pffiffiffiffiffiffi r pa

Fig. 13. Comparison of the GIC obtained from different models.

ð9Þ

where KI can be obtained by VCCT and Eq. (7), a is half of the crack length. For uniform stress condition, r refers to the applied stress at the clamped edge. For uniform displacement condition, r = P/(2 wt), where P is the total reaction force at the clamped edge when a certain displacement is applied, t is the thickness. A constant displacement of 1.5 mm was applied to the nodes at the clamped edge of the model for the uniform displacement boundary condition. In the uniform stress boundary conditions, the total reaction force P at the clamped edge should equal to the corresponding uniform displacement case, and then the applied stress was obtained by P/(2 wt). Fig. 14 shows the SIF correction factors f(a) for the displacement and uniform stress boundary conditions obtained by VCCT. For the same crack geometry, the correction factor f(a) obtained from the uniform stress boundary is significantly larger than that from the displacement condition, which indicates that the effect of boundary conditions cannot be ignored. Since the displacement boundary condition is closer to the real experimental clamp condition. Therefore, the displacement boundary condition should be used in VCCT. 4.2. Effect of shear modulus In order to determine the fracture parameters, both residual strength and material properties are needed. There are many standardized method to test the residual strength such as central crack tearing test and single edge crack tearing test, the elastic modulus and tensile strength are usually tested through strip uniaxial tension test. However, very few studies have been done on the shear modulus of architectural coated fabric. The shear behavior of coated fabric not only allows flat fabric patterns develop into complex forms, but also significantly affects

Fig. 14. SIF’s obtained from two different boundary conditions.

the stress concentration level near the crack tip. Take the CCT specimen for example, the tensile stress originally sustained by the severed yarns is transmitted to the yarns near crack tip due to shear stiffness (Fig. 15), which is the fundamental cause of stress concentration near the crack tip region. Due to the fact that the shear modulus of the coated fabrics is much smaller than that of traditional rigid building material such as steel and concrete, the stress concentration zone in cracked coated fabric is much smaller than that of traditional rigid building material. A simple rule currently used by structural engineers to consider the shear modulus of coated woven fabric is equal to 1/20 of the tensile stiffness [1], it is evident that the rough estimation of shear modulus cannot be a satisfying solution when shear stiffness significantly influences the fracture behavior.

Table 5 Critical energy release rate GIC obtained from VCCT. Crack length (mm) GIC (N/mm)

Warp weft

5

10

15

20

25

Average value

88.38 84.14

102.96 118.60

107.26 140.76

110.22 116.98

105.38 140.24

106.46 129.15

480

R. He et al. / Construction and Building Materials 208 (2019) 472–481

Fig. 17. Stress concentration zone varies with shear modulus.

5. Conclusions

Fig. 15. Stress concentration caused by shear modulus.

Fig. 16. Effect of shear modulus on GIC obtained by VCCT.

In order to investigate the effect of shear modulus Gwf on critical SERR GIC, a parameter study of the shear modulus Gwf various from 40 kN/m to 130 kN/m with an interval of 30 kN/m is conducted, results obtained by VCCT are shown in Fig. 16. It can be seen that with the shear modulus increase from 40 kN/m to 130 kN/m, the GIC decreases from 120 N/mm to 87 N/mm. As shown in Fig. 17, larger shear modulus results in larger high stress zone. When the residual strength is given, larger shear modulus means lower tearing resistance. Furthermore, when the shear modulus decreases, the dependence of GIC on the crack length is significantly reduced. This may attribute to the stress concentration zone decreases as the shear modulus decreases, thus the plastic zone within the high stress zone is also smaller. Consequently, the under estimation of GIC in small a/W case due to the neglect of plastic zone is significantly reduced.

In this study, the residual strength and the fracture mechanics properties of the architectural coated fabric are investigated. Central crack tearing tests are conducted to investigate the influence of the crack length on residual strength. Using the tested residual strength, numerical method VCCT is introduced to calculate fracture parameters since the classical fracture mechanic theories are inapplicable for coated fabrics. Finally, the effect of boundary condition and shear modulus are discussed. The following conclusions can be drawn: (1). Tensile tests indicate that the PTFE coated fabric shows significant orthotropic and non-linearity, a three-stage material constitutive model is derived from the experimental stressstrain curves. (2). Central crack tearing test shows the non-linearity of the tearing force-displacement curves is also significant. By analyzing the influence of the crack length on residual strength, coated fabric shows evident crack-sensitive property. The existence of initial crack significantly reduces the residual strength, for an initial crack which accounts for only 1/10 of the sample width, the residual strength decreases 30% of the tensile strength. (3). To estimate the tearing resistance of coated fabric, commonly used classical LEFM theories are adopted to calculate the fracture parameters, results shown the fracture parameters rely heavily on the crack length due to the non-linear stiffness of the coated fabric, thus inapplicable for coated fabrics. Therefore, the nonlinear material constitutive model combined with the VCCT are adopted, crack length dependence of GIC is greatly reduced and can be considered as material constant. (4). The influence of boundary condition and shear modulus on GIC are also investigated. Results shown that uniform displacement boundary condition should be used in VCCT. When the shear modulus decreases, the GIC increases gradually and the dependence on the crack length is reduced due to the decrease of stress concentration zone. Conflict of Interest No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.

R. He et al. / Construction and Building Materials 208 (2019) 472–481

Acknowledgement

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The authors would like to acknowledge the National Natural Science Foundation of China (Grant No. 51678192) for the financial support of the research.

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