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Abstract

This report presents a new critical tearing strength prediction model, the modified stress field model. Firstly, the critical tearing strength of six coated fabrics is obtained using the single-edge notched tests. The results show that the warp and weft strength utilization of different coated fabrics is similar. Based on the test data, the prediction effects, parameter meanings, and basic properties of three models, namely Thiele’s empirical formula, the stress field model, and the modified stress field model, are discussed and compared. Then, the predicted results and properties of the three models in different crack ranges, specifically the small crack range, are compared, and their application scopes are verified. The two commonly used models lack the improved prediction effects in the small crack range that is provided by the modified stress field model. Finally, to reduce the workload while maintaining a certain accuracy, a crack combination of 3 mm + 15 mm is recommended, which is further confirmed using the root-mean-square error index. The prediction results of Thiele’s empirical formula and the modified stress field model under this crack combination are compared. The new model can still maintain good prediction effects even when extrapolating for small cracks using limited test data. Because of its precision and convenience, the proposed model may relatively facilitate the application of single-edge notched or uniaxial central tearing tests.

Keywords

Coated fabric prediction model tearing strength tearing test

Introduction

As new long-span structures, membrane structures are developing rapidly worldwide because of their light and aesthetically appealing appearance. These structures have widespread applications, particularly in public facilities and in open or half-open buildings. However, despite their various advantages, membrane structures occasionally fail. Although membrane structures are flexible, when there is strong wind or heavy snow, existing cracks, holes, and creases in membrane materials may cause stress concentration to arise and lead to tearing, and consequently overall failures.^1,2

Architectural coated fabrics primarily comprise coatings, substrates, and top coatings³ (Figure 1). The coatings mainly include polyvinylchloride (PVC), polytetrafluoroethylene (PTFE), silicone, and polyurethane. Among them, PVC and PTFE coatings are widely used in architectural coated fabrics. PVC coating is often flexible and covered with polyvinylidene fluoride (PVDF) to improve chemical stability, whereas PTFE coating is inert and usually of higher stiffness. The architectural coated fabrics mainly include polyester fiber fabrics with PVC coating and glass fiber fabrics with PTFE coating. Usually, PVC-coated fabrics are ductile, low-cost, and self-cleaning but have low strength, meanwhile, PTFE-coated fabrics have high strength and good durability but poor ductility. In addition, the glass fibers are not resistant to folding. Therefore, PTFE-coated fabrics are more sensitive to creases during construction and transportation, which are major factors leading to stress concentration and tearing.⁴

Figure 1.

Composition of coated fabric.

Thus, regardless of the membrane structure or material, tearing may have serious consequences, and the tearing strength of membrane materials is very important. However, design and calculation are still conducted based on the tensile strength in domestic and international technical standards, and the tearing strength is only proposed as a property parameter.^5–7 Furthermore, different tearing tests have been adopted with different specifications.^8,9 The trapezoidal tearing test has the widest application in specifications for membrane structures because of its simple failure mode and stable results. Many studies have focused on this method. Hager et al.¹⁰ performed experimental studies and listed the factors influencing the trapezoidal tearing strength. Steele et al.,¹¹ Chu et al.,¹² and Wang et al.¹³ conducted theoretical studies and proposed formulas to predict the trapezoidal tearing strength. In recent years, finite element analysis has become a new research method, and Wang et al.¹⁴ adopted it to simulate the trapezoidal tearing process of coated and uncoated fabrics. The trapezoidal tearing test has received considerable attention; however, its tearing strength is determined by the size of the tearing del-zone, which is a triangular zone formed owing to uneven elongation of some longitudinal yarns at the crack tips in specimens.¹⁵ This characteristic indicates that the stress state deviates from those of actual membrane structures, which are mainly subjected to multi-axial stress and have loads carried by whole sections.

Therefore, a tearing test whose stress state is closer to that of membrane structures should be adopted to obtain the tearing strength. Among various tearing tests, the single-edge notched and uniaxial or biaxial central tearing tests can meet this requirement. The biaxial central tearing tests are closer to practical situations, while single-edge notched and uniaxial central tearing tests partly approximate realistic situations, which have loads carried by whole sections but are subjected to uniaxial stress. However, specific equipment is often required for achieving multi-axial stress, whereas uniaxial stress can be attained easily. Therefore, the single-edge notched and uniaxial central tearing tests are also of significance and can be the first step of the study in a practical situation. Recently, numerous studies have been focused on these tests, specifically in the fields of architecture and aviation, to achieve a deeper understanding of their mechanisms. Among these, experimental and theoretical studies have taken up major parts. Maekawa et al.¹⁶ focused on the tearing properties of airship envelope materials based on biaxial central tearing tests and inflatable cylinder tests. They proposed a model in which the stress was exponentially distributed in the crack section. The results showed that the predicted tearing strength of the model agreed well with the test results. Bigaud et al.¹⁷ performed uniaxial and biaxial central tearing tests on PVC-coated fabrics and analyzed the factors affecting the failure modes and tearing strength. They also used the finite element method and fracture mechanics theories to analyze the test results. Liu et al.¹⁸ conducted single-edge notched tests on an airship envelope material and established a single-edge notched tearing strength prediction model, which could acquire results similar to the test results. Zhang et al.¹⁹ studied the uniaxial central tearing properties of a PVC-coated fabric and systematically illustrated the effects of the notch shape, notch size, loading rate, specimen size, and other factors on the tearing behavior. They also established a model for predicting tearing strength based on the stress field model established by Maekawa et al.¹⁶ Sun et al.²⁰ performed uniaxial central tearing tests on PTFE-coated and PVC-coated fabrics. They proposed a new tearing strength prediction model based on the theories of fracture mechanics. Chen et al.^21–23 systematically studied the uniaxial and biaxial central tearing properties of airship envelope materials and considered factors such as the notch length, notch angle, off-axis angle, and stress ratio. They finally proposed formulas to predict the tearing strength with and without the off-axis angle based on the Griffith energy theory and Tsai–Hill criterion, respectively. Wang et al.²⁴ performed uniaxial central tearing tests on a new airship envelope material and proposed a tearing strength prediction model by assuming a stress distribution on yarns. Sun et al.²⁵ investigated the uniaxial central tearing properties of a PTFE-coated fabric, including the tearing process, failure modes, and tearing strength. Godfrey et al.^26,27 proposed a model to predict the uniaxial central tearing strength of uncoated and coated fabrics, and the adopted coatings including silicone, urethane, and PVC.

In recent years, new research perspectives and measuring methods have been adopted in single-edge notched, and uniaxial and biaxial central tearing tests. In contrast to the traditional perspective, which mainly focuses on tearing strength, some researchers have attempted to find inherent parameters using concepts in fracture mechanics. Bao et al.²⁸ conducted uniaxial and biaxial central tearing, single-edge notched, and trapezoidal tearing tests on a PVC-coated fabric. The critical strain energy release rates (G_IC) obtained by theoretical and test methods were introduced and compared. He et al.²⁹ conducted uniaxial central tearing tests on PTFE-coated fabrics. They calculated and analyzed G_IC based on the virtual crack closure technique and noted that G_IC could be adopted as a material constant to measure tearing performance. As a new technology, the digital image correlation (DIC) method has been used gradually for architectural coated fabrics owing to its accuracy and non-contact nature. Bao et al.³⁰ employed this method to study the uniaxial and biaxial central tearing properties of a PVC-coated fabric and obtained the strain distribution. He et al.³¹ conducted detailed DIC parametric studies and revealed the biaxial central tearing properties of a PVC-coated fabric. Chen et al.^32,33 also employed the DIC method to analyze the uniaxial and biaxial central tearing properties of an airship envelope material and conducted theoretical studies on the single-edge notched properties of this material.

From the aforementioned studies, majority are experimental works which focus on the tearing properties of one or two coated fabrics, while theoretical studies are inadequate. Among different studies, Thiele’s empirical formula and the stress field model are still the most widely used, but usually only their prediction effects are examined.^{16,19,20,24,30,32} To date, no detailed analyses reveal their properties, parameter meanings, and application scope. Furthermore, in existing theoretical studies, their proposed models may be established based on different assumptions or theories, but in most cases, all test data are involved in fitting constant acquisition requiring numerous tests. The predicted results are typically compared within the tested crack range, which may be one of the reasons why good prediction effects are always obtained. There are almost no models that can use limited test data to offer good prediction effects; and for the predicted results in different crack ranges, particularly for small cracks in which extrapolation is often needed, the relevant studies are still insufficient. Therefore, intensive research on two commonly used models is required. In addition, a new model that requires fewer test data and has good prediction effects in different crack ranges must be developed. This can have great significance in engineering fields, as it will reduce the amount of workload while maintaining certain prediction accuracy.

In this report, a new critical tearing strength prediction model, which is a modified stress field model, is proposed. To investigate its properties, a comparison among Thiele’s empirical formula, the stress field model, and the new model is conducted. Before theoretical research, single-edge notched tests are performed on six architectural coated fabrics with different types and properties. Based on the test results, the prediction effects, basic properties, and parameter meanings of the three models are first inspected and compared. Then, the prediction effects of the three models in different crack ranges are checked, and the application scopes are assessed. In particular, the predicted results in the small crack range are compared in detail. The modified stress field model has better effects for small cracks, which is what the other two models lack. Finally, the fitting constants of the new model and Thiele’s empirical formula are obtained uniquely based on test data with 3 mm and 15 mm cracks, and this crack combination is confirmed by the root-mean-square error (RMSE). The prediction effects and properties of the two models under this crack combination are discussed and compared. The developed modified stress field model is anticipated to facilitate the adoption of single-edge notched or uniaxial central tearing tests because it reduces the workload and maintains a certain accuracy in different crack ranges.

Tests

Before investigating the prediction effects and properties of different models, basic test data are required. Therefore, in this section, single-edge notched tests are conducted on several coated fabrics. The majority of this study consists of theoretical research; experimental tests are performed to serve the theoretical assumptions. Therefore, the tearing strength is the main parameter displayed and discussed, which can be a basis for the subsequent analysis and discussions on different models.

Materials

This study aims at providing insights into structural engineering; hence, materials widely used in existing membrane structures are firstly considered. Two types of architectural coated fabrics mentioned in Section 1 are both selected, including four PVC-coated fabrics (denoted by P1–P4) and two PTFE-coated fabrics (denoted by G1 and G2), with various properties. The basic properties are listed in Table 1, where CV represents the coefficient of variation. The tensile strength and breaking elongation are the average values of five specimens. The tensile speed is 100 mm/min, width of each tensile specimen is 50 mm, and mearing length is 200 mm.

Table 1.

Fundamental parameters of six coated fabrics.

Type	No.	Thickness (mm)	Areal density (g/m²)	Yarns per centimeter		Tensile strength (N/mm)				Breaking elongation (%)
				Warp	Weft	Warp		Weft		Warp		Weft
				Warp	Weft	Mean	CV (%)	Mean	CV (%)	Mean	CV (%)	Mean	CV (%)
Polyester fiber fabrics with PVC coating	P1	0.81	1,050	5.8	6.5	78.58	1.7	85.04	4.2	19.19	2.0	20.42	4.3
	P2	0.83	1,150	6.3	6.4	105.02	0.9	97.1	1.8	26.51	2.7	33.11	3.6
	P3	0.83	1,000	6.2	6.6	84.15	1.4	78.05	2.2	15.22	1.4	20.41	1.1
	P4	1.1	1,350	6.6	6.5	137.11	1.6	124.9	2.2	20.65	0.4	28.58	0.6
Glass fiber fabrics with PTFE coating	G1	0.84	1,300	10.5	7.5	146.52	0.9	127.58	1.7	5.68	1.2	11.04	1.5
Glass fiber fabrics with PTFE coating	G2	0.82	1,305	9.8	7.4	153.22	1.9	119.88	0.8	5.17	1.3	10.01	1.0

Experimental program

The dimensions of the single-edge notched samples are shown in Figure 2(a). The width and length are 50 mm and 300 mm, respectively. Referring to FAA-P-8110-2-1995,⁹ a measuring length of 76.2 mm is adopted. Crack “a” is at the center of the edge. Crack sizes of 3, 5, 10, 15, and 20 mm are included for all materials. In order to cover more crack sizes and display the tearing properties of various coated fabrics, crack sizes of 8 mm or 25 mm are adopted for different materials. Three samples are prepared for each crack. In this study, critical tearing strength is important. Therefore, it was necessary to ensure the integrity of the first yarn at the crack tip, and intact yarns whose total lengths are closest to the crack sizes are cut. The final adopted crack sizes and numbers of cut yarns are listed in Table 2, where the symbol “×” indicates exclusion. However, for convenience of presentation and comparison, the results are still distinguished by the crack sizes.

Figure 2.

Test specimens (unit: mm) in the (a) single-edge notched and (b) uniaxial central tearing tests.

Table 2.

Adopted crack sizes and numbers of cut yarns in the single-edge notched tests.

Crack size a (mm)	P1		P2		P3		P4		G1		G2
Crack size a (mm)	Warp	Weft	Warp	Weft	Warp	Weft	Warp	Weft	Warp	Weft	Warp	Weft
3	2	2	2	2	2	2	2	2	3	2	3	2
5	4	4	3	4	3	3	4	4	6	4	5	4
8	×	×	×	×	×	×	×	×	9	6	7	6
10	6	7	6	6	6	7	7	7	11	8	10	8
15	9	10	9	9	9	10	10	10	15	11	15	11
20	12	13	12	12	12	13	13	13	21	15	20	15
25	×	×	×	×	15	16	16	16	×	×	×	×

As described in Section 4, the prediction effects of the different models are compared, and additional single-edge notched tests with one yarn and half-yarn cut of material P3 in warp are conducted. For the former, the same dimensions in Figure 2(a) are adopted, and the crack size is a yarn spacing. For the latter, single-edge notched tests with a half-yarn cut are very difficult to perform and uniaxial central tearing tests with a yarn cut are conducted to approximate the results. The sizes of the uniaxial central tearing samples were the same as those of the single-edge notched samples; only the crack location is different, which is at the center, as shown in Figure 2(b). The central crack size “2a” corresponds to a yarn spacing. Based on previous studies,^19,28,32 because the critical tearing stress per unit length is needed rather than the critical tearing load and only one yarn is cut, the residual yarns are plentiful and the approximation is feasible. Careful and delicate operation is required to ensure the integrity of the first yarn at the crack tip and that the cracks are very small. The two additional tests are only used to examine the prediction effects of different models in a small crack range. Three samples are prepared for each crack.

As shown in Figure 3, all tests are conducted using the universal testing machine CMT4204 where the clamping fixture is adopted, and the maximum load is 20 kN so that it can meet the requirements of all tests. Furthermore, the reliable clamping and alignment of samples need a careful operation to ensure the test results. To reduce the dynamic effects, a low constant loading rate of 3 mm/min is implemented, and an initial tensile force of 5 N is set. All tests are conducted at a temperature of 20 ± 2°C and a relative humidity of 65 ± 3%.

Figure 3.

Test setup in the (a) universal testing machine, and (b) general situation.

Experimental results and discussion

To maintain consistency with the unit of all models and eliminate the effect of sample width, a uniform stress away from the crack tip is needed and can be obtained according to stress at the clamped end. The critical tearing stress per unit length rather than the critical load is adopted to denote the tearing strength. The critical tearing strength, σ_c, is defined as the load at the onset of tearing, F_c, divided by the specimen width L, which is 50 mm in this study, as illustrated in equation (1).^19,20,28 The mean values, standard deviations (SD), and coefficient of variation (CV) of σ_c obtained from the single-edge notched tests of all materials in warp and weft are listed in Tables 3–8. The small SD and CV demonstrate good consistency in the tests.

σ_{c} = \frac{F_{c}}{L}

(1)

Table 3.

Critical tearing strength of P1 (N/mm).

Crack size a (mm)	Warp			Weft
Crack size a (mm)	Mean	SD	CV (%)	Mean	SD	CV (%)
3	42.57	0.25	0.6	43.50	0.70	1.6
5	31.67	0.42	1.3	30.33	0.25	0.8
10	24.87	0.68	2.8	21.87	0.26	1.2
15	20.43	0.19	0.9	17.43	0.31	1.8
20	17.07	0.52	3.0	15.73	0.31	2.0

Table 4.

Critical tearing strength of P2 (N/mm).

Crack size a (mm)	Warp			Weft
Crack size a (mm)	Mean	SD	CV (%)	Mean	SD	CV (%)
3	50.25	0.35	0.7	47.90	0.20	0.4
5	35.50	0.90	2.5	32.85	0.38	1.1
10	29.70	0.60	2.0	26.78	0.25	0.9
15	24.85	0.55	2.2	21.93	0.09	0.4
20	21.05	0.65	3.1	18.43	0.11	0.6

Table 5.

Critical tearing strength of P3 (N/mm).

Crack size a (mm)	Warp			Weft
Crack size a (mm)	Mean	SD	CV (%)	Mean	SD	CV (%)
3	43.16	2.20	5.1	41.95	2.52	6.0
5	36.47	0.25	0.7	32.26	2.20	6.8
10	26.98	0.48	1.8	23.85	0.51	2.2
15	22.16	0.17	0.8	18.16	0.49	2.7
20	19.66	0.09	0.5	17.49	0.73	4.2
25	16.34	0.30	5.1	15.55	0.54	6.0

Table 6.

Critical tearing strength of P4 (N/mm).

Crack size a (mm)	Warp			Weft
Crack size a (mm)	Mean	SD	CV (%)	Mean	SD	CV (%)
3	71.87	1.29	1.8	68.39	2.30	3.4
5	50.72	1.47	2.9	50.65	2.40	4.7
10	36.73	0.63	1.7	35.97	1.41	3.9
15	32.11	0.73	2.3	31.97	0.40	1.2
20	28.56	0.74	2.6	28.26	0.20	0.7
25	24.76	1.13	4.6	24.49	0.55	2.2

Table 7.

Critical tearing strength of G1 (N/mm).

Crack size a (mm)	Warp			Weft
Crack size a (mm)	Mean	SD	CV (%)	Mean	SD	CV (%)
3	86.97	1.13	1.3	77.17	1.88	2.4
5	62.27	0.68	1.1	58.37	0.82	1.4
8	50.87	0.78	1.5	47.67	1.31	2.7
10	45.10	0.67	1.5	39.10	1.28	3.3
15	35.27	1.21	3.4	33.00	1.00	3.0
20	28.77	0.38	1.3	25.93	0.49	1.9

Table 8.

Critical tearing strength of G2 (N/mm).

Crack size a (mm)	Warp			Weft
Crack size a (mm)	Mean	SD	CV (%)	Mean	SD	CV (%)
3	84.77	0.93	1.1	78.83	1.35	1.7
5	69.10	1.53	2.2	62.77	0.85	1.4
8	59.90	1.21	2.0	50.17	0.73	1.5
10	50.60	0.59	1.2	43.07	0.62	1.4
15	38.10	0.86	2.3	34.87	0.26	0.8
20	33.43	0.29	0.9	27.60	1.44	5.2

To display the reduction in the tensile strength σ_u, due to the existence of cracks intuitively, diagrams presenting the ratio σ_c/σ_u versus the ratio of the crack size to the sample width L are shown in Figure 4, which also includes the net-section residual strength. It is evident that σ_c is much lower than σ_u and the net-section residual strength. Thus, the existence of cracks not only directly reduces the cross-section but also induces a stress concentration that further reduces the material strength. As the crack sizes increase, σ_c decreases, but the decreasing trend gradually slows. Comparing the σ_c/σ_u values of different coated fabrics in warp and weft, it is shown that the weft ratio is close to that of the warp. This indicates that most materials have similar strength utilization in both directions with different crack sizes. However, in Tables 3–8, the absolute value of σ_c in the warp is larger.

Figure 4.

σ_c/σ_u-a/L diagrams for (a) P1, (b) P2, (c) P3, (d) P4, (e) G1, and (f) G2.

Critical tearing strength prediction models

A modified stress field model is proposed based on the traditional stress field model and by considering the various stress reduction degrees in the crack section when crack sizes are different. This section discusses the comparison of two commonly used models (Thiele’s empirical formula and stress field model) and the proposed model. All three models can be used in architectural coated fabrics, and the plane stress state is typically required. The single-edge notched tests on six coated fabrics can meet these conditions. According to the test results, fitting constants of the three models are obtained. Their prediction effects and basic properties are displayed and analyzed. Furthermore, the rationality of the stress distribution assumption in the modified stress field model is checked.

Thiele’s empirical formula

Thiele’s empirical formula is a classical empirical model that has been widely used. It is based on inflatable cylinder tests and can accurately predict the tearing strength of airship envelope materials.^22,34 The original form is

σ_{c} = \frac{C_{l} C_{s}}{{(2 a)}^{n} (1 + 2 a / r)},

(2)

where σ_c is the critical tearing strength; r is the radius of the cylinder; a is the half-length of the crack; C_s is the uniaxial central tearing stress of samples with length and width of 6 and 4 in, respectively, according to FAA-P-8110-2-1995⁹; and C_l and n are constants.

The single-edge notched test is an in-plane tearing method, where the curvature radius of specimens is positive and infinitely large. Thus, the value of r is close to positive infinity, and the formula can be simplified as

σ_{c} = \frac{C_{k}}{{(2 a)}^{n}},

(3)

where a is the crack size in the single-edge notched tests. Constants C_k and n can be obtained by fitting the test data.

Stress field model

The stress field model assumes that the stress in the crack section is exponentially distributed.¹⁶ At the onset of tearing, the stress is distributed as shown in Figure 5(a) and follows:

σ (x) = σ_{c} + (σ_{u} - σ_{c}) e^{- B x},

(4)

where B is a constant, σ_u is the tensile strength, and σ_c is the critical tearing strength.

Figure 5.

Stress distribution at the onset of tearing in the (a) stress field model and (b) modified stress field model.

According to the stress equilibrium, the relationship between σ_c and the crack size a in the single-edge notched tests can be proposed:

σ_{c} = \frac{σ_{u}}{1 + B \cdot a} .

(5)

Constant B is obtained by fitting the test results.

From the assumption and derivation of the traditional stress field model, equations (4) and (5) and Figure 5(a) demonstrate that B has physical significance, which represents the degree of stress reduction in the crack section. In fact, σ_u and B are the factors determining the tearing strength in this model. σ_u determines the timing of tearing, and B determines how the stress in the crack section decreases. The larger σ_u, the later the failure at the crack tip occurs. The smaller B, the more gradually the stress decreases. Both result in larger σ_c. This model identifies samples with different cracks that share the same degree of stress reduction because of the consistent B value. However, as demonstrated by the results in Section 3.4, this assumption deviates from the actual situation.

Modified stress field model

In contrast to the consistent B value adopted in the traditional stress field model, a modified stress field model is proposed and takes B as a power function of the crack size, with reference to Thiele’s empirical formula. The stress distribution of the crack section at the onset of tearing is shown in Figure 5(b) and follows:

σ (x) = σ_{c} + (σ_{u} - σ_{c}) e^{- k \cdot a^{m} x},

(6)

where a is the crack size in the single-edge notched tests, σ_c is the critical tearing strength, and σ_u is the tensile strength.

The stress equilibrium is also considered. The stress in the crack section should be balanced with remote tensile stress. Because of the stress concentration caused by cracks, the concentrated stress around the crack tip should be balanced with the transferred stress:

σ_{c} \cdot a = \int (σ_{u} - σ_{c}) e^{- k \cdot a^{m} x} \cdot d x .

(7)

By simplification, the relationship between the critical tearing strength and crack size can be obtained as follows:

σ_{c} = \frac{σ_{u}}{1 + k \cdot a^{m + 1}},

(8)

where k and m are constants that can be acquired based on the test data by fitting. These constants can reflect the different degrees of stress reduction when the crack is different:

B = k \cdot a^{m} .

(9)

Results

Based on the results of the single-edge notched tests in Tables 3–8, the values of C_k and n in equation (3), B in equations (5), and k and m in equation (8) obtained by fitting are listed in Table 9. Further, the prediction effects of the three models are displayed in Figures 6–11.

Table 9.

Constants C_k, n, B, k, and m.

Material	Direction	Number	C _k	n	B	k	m
P1	Warp	1	116.933	0.510	0.184	0.312	−0.190
P1	Weft	2	120.422	0.556	0.241	0.387	−0.169
P2	Warp	3	120.977	0.477	0.229	0.498	−0.289
P2	Weft	4	125.236	0.527	0.246	0.439	−0.215
P3	Warp	5	104.889	0.466	0.181	0.420	−0.291
P3	Weft	6	94.273	0.465	0.185	0.430	−0.286
P4	Warp	7	178.716	0.504	0.200	0.412	−0.242
P4	Weft	8	169.305	0.489	0.177	0.361	−0.239
G1	Warp	9	244.667	0.570	0.213	0.264	−0.079
G1	Weft	10	200.537	0.538	0.201	0.272	−0.114
G2	Warp	11	218.328	0.501	0.190	0.331	−0.205
G2	Weft	12	204.568	0.521	0.165	0.199	−0.072

Figure 6.

Prediction effects of P1 in (a) warp and (b) weft.

Figure 7.

Prediction effects of P2 in (a) warp and (b) weft.

Figure 8.

Prediction effects of P3 in (a) warp and (b) weft.

Figure 9.

Prediction effects of P4 in (a) warp and (b) weft.

Figure 10.

Prediction effects of G1 in (a) warp and (b) weft.

Figure 11.

Prediction effects of G2 in (a) warp and (b) weft.

It is evident that Thiele’s empirical formula achieves good prediction effects in the tested crack range, which is similar to the results of previous studies.^{16,19,24,28,32} However, owing to its inversely proportional form, deviations will arise when extrapolating the critical tearing strength for smaller cracks, specifically for cracks close to 0 mm. In most cases, poor predicted results of the traditional stress field model are obtained until the cracks that are larger than 15 mm. However, the prediction effects of the modified stress field model are comparable to those of Thiele’s empirical formula in the tested crack range and better than those of the stress field model. Furthermore, for samples without cracks, σ_c should be equal to σ_u. The modified model can meet this requirement, which is not the case for Thiele’s empirical formula.

Discussion

From the aforementioned results, both Thiele’s empirical formula and the modified stress field model offer a good prediction. However, the fitting constants of the Thiele’s empirical formula lack physical significance.

In contrast, although the stress field model has physical significance, its prediction effects are poor. This can be attributed to the requirement that samples with different crack sizes need to share the same degree of stress reduction due to the consistent B value. However, a special situation is that for samples without cracks (corresponding to uniaxial tensile state), there should be no stress reduction degree. By equation (5), the individual B values for every crack size in all materials are calculated and shown in Figures 12–17. It is evident that B is different when the crack size is different, and generally, B decreases with the increase of crack sizes. Therefore, it is inappropriate to adopt a constant B value, and a decreasing trend of B should be considered when better prediction effects are needed.

Figure 12.

B of P1 in (a) warp and (b) weft.

Figure 13.

B of P2 in (a) warp and (b) weft.

Figure 14.

B of P3 in (a) warp, and (b) weft.

Figure 15.

B of P4 in (a) warp and (b) weft.

Figure 16.

B of G1 in (a) warp and (b) weft.

Figure 17.

B of G2 in (a) warp and (b) weft.

In addition, the modified stress field model can reflect the different degrees of stress reduction for different cracks. According to its assumption, by substituting k and m in Table 9 into equation (9), all the different B values can be calculated, as shown in Figures 12–17. For uniaxial tensile state, equation (9) results in B equal to 0, which can reflect the actual situation. The figures depict that the modified model exhibits a decreasing trend of B with the increase in crack sizes, which is why the modified model offers better prediction effects than the stress field model.

The rationality of the stress distribution assumption in the modified stress field model is explained using the work of Bao.³⁰ By employing the DIC equipment, Bao obtained the stress distribution in the crack section of material P2 in uniaxial central tearing tests with different central cracks, as shown in Figure 18. By substituting k, m, and B of P2 in Table 9 and the different crack sizes into equations (4)–(6), and (8), the predicted stress distribution in the crack section can be obtained using the traditional and modified stress field models, as shown in Figure 18. It is clear that the results of the modified model are similar to the test results, whereas some deviations occur in the stress field model. Therefore, the stress distribution assumption of the modified stress field model is more reasonable than that of the traditional model.

Figure 18.

Stress distribution in the crack section of P2 in (a) warp-3.18, (b) weft-3.9, (c) warp-4.77, (d) weft-5.46, (e) warp-9.54, and (f) weft-10.14.

From the above results, the basic properties of the three models can be assessed: (i) Thiele’s empirical formula offers good prediction results but is an empirical model; (ii) the traditional stress field model has poor prediction because it requires all notched samples to share the same stress reduction degree; (ii) the prediction effects of the modified stress field model are good and similar to those of Thiele’s empirical formula, with the advantage that the fitting constants of the new model can reflect the different stress reduction degrees for various cracks.

Prediction effects of three models in different crack ranges

Background and test results in small crack range

In this study, the minimum crack size is 3 mm. For cracks less than 3 mm, because the integrity of the first yarn at the crack tip must be guaranteed and the cracks are small, the test operation is often difficult. However, for engineering needs, σ_c in different ranges is important. Therefore, a theoretical method to acquire predicted σ_c for small cracks instead of performing testing is worth considering, which leads to the need for model extrapolation and accuracy in the small crack range. Based on the above results, both Thiele’s empirical formula and the modified stress field model have good prediction effects in the tested crack range. Observing the forms of the three models, it is evident that the σ_c values predicted by Thiele’s empirical formula gradually approach infinity with the decrease of crack sizes. Although this situation does not appear in the other models, the prediction effects of the traditional stress field model are not good. Therefore, this section compares the predicted σ_c values of the three models in different ranges, particularly in the small crack range. Accordingly, the properties of the three models can be discussed and inspected in more detail.

Before comparison, test results in a small crack range are required to check the extrapolation effects of the different models. For this purpose, material P3 in the warp is taken as an example. As mentioned in Section 2.2, test results with crack sizes of half-yarn and one yarn are added here, as shown in Table 10.

Table 10.

Critical tearing strength of P3 with half and one yarn cut in warp (N/mm).

Number	Number of cut yarns	Mean value	Standard deviation
1	0.5	66.72	0.96
2	1	54.49	1.66

Comparison of prediction effects in different crack ranges

The comparison is conducted by checking the prediction effects in different ranges in the two cases (listed in Table 11). In Case 1, only the data in Table 5 are employed, which means that extrapolation is needed. In Case 2, the data in Tables 5 and 10 are employed, hence, extrapolation is not necessary. The accuracy in different ranges and the inner mechanisms of the three models in the two cases are assessed to determine their properties. The fitting constants are listed in Table 12, and the prediction effects are presented in Figure 19.

Table 11.

Data employed in different cases.

Case number	Employed data
1	Only data in Table 5
2	Data in Tables 5 and 12

Table 12.

C_k, n, B, k, and m in both cases.

Case number	C _k	n	B	k	m
1	104.889	0.466	0.181	0.420	−0.291
2	87.497	0.410	0.181	0.353	−0.221

Figure 19.

Prediction effects of three models for P3 in warp in (a) Case 1, (b) local enlarged view of Case 1, (c) Case 2, and (d) local enlarged view of Case 2.

For cracks smaller than 5 mm, Figure 19(b) and (d) indicate that the deviations of the stress field model are large. However, as B values in Table 12 are approximately the same, the two σ_c-a curves appear similar. In this crack range, the prediction effects of Thiele’s empirical formula are not accurate enough. In Case 1, the test data in Table 10 are not involved in the fitting. Because of the inversely proportional form of Thiele’s empirical formula, the deviations are large for tests with only one yarn and half-yarn cut when extrapolating. However, for the remaining cracks, the deviations are sufficiently small. In Case 2, the test data in Table 10 are included. By utilizing these data, the predicted σ_c values are adjusted for tests with the minimum crack and one or two subsequent cracks in Thiele’s empirical formula, as shown in Figure 19(d). Therefore, the predicted σ_c values deviate slightly. However, as shown in Figure 19(b) and (d), regardless of the test data in Table 10 are involved in fitting or not, the deviation caused by extrapolation or adjustment in Thiele’s empirical formula does not appear in the modified stress field model. The prediction effects of the modified model are good, except for a slight deviation for the minimum crack in Case 1.

For cracks larger than 5 mm but smaller than 15 mm, in both cases (as shown in Figure 19(a) and (c)) the prediction effects of Thiele’s empirical formula and the modified stress field model closely approach and fit well the test data. However, in this crack range, the deviations of the stress field model remain large.

For cracks larger than 15 mm, the prediction effects of the three models are similar, and the results are all close to the test data (see Figure 19(a) and (c)), which is why the stress field model showed good prediction effects when it was first proposed.¹⁶ In that study, the minimum crack size was 10 mm (corresponding to 5 mm in the single-edge notched tests), and the other adopted cracks were larger. Therefore, the deviations of Thiele’s empirical formula and the stress field model were both small in that work.

Discussion

According to the comparison results, the properties of the three models in different crack ranges are clear. The prediction effects of Thiele’s empirical formula in the small crack range may not be accurate because of extrapolation or adjustment, but as the crack size increases, this formula becomes more accurate, particularly in the tested crack range. The prediction effects of the stress field model are not good, but in the large crack range, the deviations become smaller. The modified stress field model can offer good prediction effects in different crack ranges, the inaccuracy caused by extrapolation or adjustment in Thiele’s empirical formula does not appear, and the predicted results for small cracks can remain undistorted even when extrapolating. These properties of the modified model, specifically in the small crack range, are what the two commonly used models lack and can be regarded as supplements and improvements to them.

Prediction effects based on only two cracks

Currently, the trapezoidal tearing test is widely adopted in many specifications. Usually, the crack size of its samples is fixed, and the number of tests is between three and five. This can offer a stable tearing strength. However, the tearing strength is determined by the size of the tearing del-zone and the stress state deviates from that of the membrane structure. Therefore, the trapezoidal tearing strength is simply adopted as a material parameter in the present study.

The failure modes and stress states of the single-edge notched and uniaxial central tearing tests are close to those of the membrane structures. However, they are not widely adopted in different specifications, partly because the number of tests is often large. To obtain the comprehensive single-edge notched or uniaxial central tearing properties, the critical tearing strengths at different crack sizes are required, which is the σ_c-a curve. The currently adopted operation involves conducting tests with several cracks and obtaining a series of test data, which are some points on the σ_c-a curve. Then, using different models, a continuous σ_c-a curve can be obtained. Based on the curve, the design and calculations can be conducted, such as the determination of the allowable crack size.^19,24,30,32

Therefore, if a model can employ fewer test data to obtain an effective σ_c-a curve, it will reduce the amount of test work and facilitate the use of single-edge notched or uniaxial central tearing tests. Both the modified stress field model and Thiele’s empirical formula have two fitting constants and their σ_c-a curves can, in theory, be obtained based on only two test data. Therefore, in this section, we examine the prediction effects of these two models based on only two crack sizes in order to acquire the most suitable crack combination. The properties of the two models are, as well, further discussed. Although the stress field model has only one fitting constant, its σ_c-a curve deviates significantly. Therefore, no further consideration is given to it.

Acquisition of k and m based on only two cracks

In terms of fitting, if the selected crack sizes are close, larger deviations will appear. A more reasonable combination should include a small crack and a large one. All 21 crack combinations are listed in Table 13, where “description” refers to the employed test data corresponding to both crack sizes. The prediction effects of all combinations are compared with the assistance of the RMSE, which indicates the general deviation for all tested cracks:

R M S E = \sqrt{\frac{\sum_{i} {(σ_{c_t e s t} - σ_{c_p r e d i c t e d})}^{2}}{p}},

(10)

where for a material in a particular direction, p is the total number of cracks adopted in the single-edge notched tests, σ_{c_test} is the critical tearing strength obtained from the tests, and σ_{c_predicted} is the predicted critical tearing strength. For one combination, based on the two test data, two equations could be established according to equations (8), and k and m could then be calculated. By substituting the other crack sizes into equation (8), the remaining σ_{c_predicted} values could be obtained and the RMSE could be calculated accordingly.

Table 13.

Crack combinations.

Combination	1	2	3	4	5	6	7
Description	3 + 5	3 + 8	3 + 10	3 + 15	3 + 20	3 + 25	5 + 8
Combination	8	9	10	11	12	13	14
Description	5 + 10	5 + 15	5 + 20	5 + 25	8 + 10	8 + 15	8 + 20
Combination	15	16	17	18	19	20	21
Description	8 + 25	10 + 15	10 + 20	10 + 25	15 + 20	15 + 25	20 + 25

Thus, the RMSEs of all crack combinations for the six coated fabrics in the two directions are calculated and shown in Figure 20(a). According to the figure, the maximum RMSE is less than 10. A large crack and a small one can induce reasonable fitting results, whereas adjacent cracks, specifically adjacent large cracks, exhibit certain deviations. Small RMSEs with sufficient data occur for Combination 4, which includes crack sizes of 3 mm and 15 mm. In this combination, all RMSE values are less than 2. Although Combinations 6 and 14 also have small RMSEs, they lack sufficient data and thus are not investigated.

Figure 20.

Obtained RMSE distributions using the (a) modified stress field model and (b) Thiele’s empirical formula.

The results based on Combination 4 are displayed in more detail, and the suffix “Com4” is used to denote this combination. The k and m values obtained using Combination 4 are listed in Table 14. The corresponding σ_c-a curves are shown as solid lines in Figures 21–26. The σ_c-a curves obtained from all test data in Tables 3–8 using the modified stress field model are also presented as dotted lines for comparison, with the suffix “all data.” It is obvious that adopting Combination 4 can lead to σ_c-a curves that are close to the test data and similar to those obtained using all data, while greatly reducing the number of tests.

Table 14.

C_k, n, k, and m values only based on Combination 4.

Material	Direction	Number	C_k_Com4	n_Com4	k_Com4	m_Com4
P1	Warp	1	109.394	0.488	0.311	−0.194
P1	Weft	2	122.202	0.568	0.359	−0.129
P2	Warp	3	119.477	0.468	0.473	−0.279
P2	Weft	4	124.022	0.520	0.413	−0.199
P3	Warp	5	98.598	0.443	0.410	−0.282
P3	Weft	6	107.088	0.520	0.341	−0.165
P4	Warp	7	178.438	0.500	0.371	−0.204
P4	Weft	8	162.578	0.473	0.339	−0.218
G1	Warp	9	230.810	0.561	0.253	−0.051
G1	Weft	10	177.478	0.498	0.280	−0.133
G2	Warp	11	208.026	0.496	0.324	−0.181
G2	Weft	12	177.081	0.479	0.211	−0.094

Figure 21.

Prediction effects based on Combination 4 of P1 in (a) warp and (b) weft.

Figure 22.

Prediction effects based on Combination 4 of P2 in (a) warp and (b) weft.

Figure 23.

Prediction effects based on Combination 4 of P3 in (a) warp and (b) weft.

Figure 24.

Prediction effects based on Combination 4 of P4 in (a) warp and (b) weft.

Figure 25.

Prediction effects based on Combination 4 of G1 in (a) warp and (b) weft.

Figure 26.

Prediction effects based on Combination 4 of G2 in (a) warp and (b) weft.

Acquisition of C_k and n based on only two cracks

The acquisition of the σ_c-a curve using Thiele’s empirical formula based on only two test data is similar to the process described above. The RMSEs of all combinations in Table 13 for the six coated fabrics based on Thiele’s empirical formula are calculated and shown in Figure 20(b). It is also evident that a reasonable crack combination should include a small crack and a large one. Combination 3 (3 mm + 10 mm) and Combination 4 (3 mm + 15 mm) have small RMSEs with sufficient data. In the two combinations, the maximum RMSE is slightly greater than 3. For the modified model, the maximum RMSE for Combination 4 is slightly less than 2. Therefore, both models offer good prediction effects based on Combination 4. This finding also indicates that the recommended crack combination is similar, although the model adopted is different.

The fitting constants C_k and n acquired based on Combination 4, are listed in Table 14. Then, the σ_c-a curves are obtained and shown as dashed lines in Figures 21–26. For most PVC-coated fabrics, the prediction effects of the modified stress field model and Thiele’s empirical formula are similar. However, in the remaining cases, particularly for the PTFE-coated fabrics, the modified model provides better prediction effects. The deviations of Thiele’s empirical formula are also caused by adjustments when the test data corresponding to smaller cracks are involved in the fitting. As shown in Figures 25 and 26, the deviations are mainly located in the subsequent two or three tests whose cracks are adjacent to 3 mm, and as the crack size increases, the deviations gradually decrease. This adjustment method is the same as discussed in Section 4.2 (Case 2 in Table 11). However, because only two test data are adopted to obtain the fitting constants, the weight of the test data with a 3 mm crack is increased, the predicted σ_c being adjusted is greater, and the deviations increase accordingly.

Discussion

The discussion on RMSE indicates that a crack combination of 3 mm + 15 mm is preferable. This combination can induce reasonable predicted results in the modified stress field model, and the results are close to those obtained by fitting involving all data, which means that the number of tests can be greatly reduced. However, under this condition, Thiele’s empirical formula may deviate in certain cases because of its adjustment.

In addition, the extrapolated prediction effects of the two models for small cracks based on the crack combination of 3 mm + 15 mm are compared. The comparison is also conducted according to the data in Tables 5 and 10, and the results are presented in Figure 27. The large deviations of Thiele’s empirical formula are evident in Figure 27(b). However, the modified model still achieves good prediction effects, even when extrapolation is needed. Therefore, even if only two test data are employed for small cracks, the prediction effects of the modified model remain undistorted.

Figure 27.

Prediction effects of two models based on Combination 4 for P3 in warp in (a) overall σ_c-a curve, and (b) local enlarged view.

Conclusion

In this study, a modified stress field is proposed by combining the characteristics of two commonly used models: Thiele’s empirical formula and the stress field model. Based on the single-edge notched test results of six coated fabrics, the prediction effects and properties of the three models are discussed and compared. The conclusions are as follows:

(1). Single-edge notched tests of four PVC-coated and two PTFE-coated fabrics are conducted. The results show that the existence of cracks significantly reduces the tearing strength. Specifically, the critical tearing strength σ_c is much lower than the uniaxial tensile strength σ_u and net-section residual strength, and σ_c in the weft is less than that in the warp. However, the strength utilization in both directions is similar.

(2). Using the test results, the prediction effects and basic properties of two commonly used models are obtained. Thiele’s empirical formula offers better prediction but lacks physical significance. Fitting constant B of the stress field model represents the degree of stress reduction in the crack section. This model requires samples with different cracks to share the same stress reduction degree, which does not accurately represent a realistic situation. Thus, its predicted results are often poor.

(3). A modified stress field model is proposed by assuming that B is a power function of the crack size, which refers to Thiele’s empirical formula. In the tested crack range, its good prediction effects are similar to those of Thiele’s empirical formula. Furthermore, the new model can display the decreasing stress reduction degree as crack sizes increase.

(4). The predicted results and properties of the three models in different crack ranges are further discussed and compared, particularly in the small crack range. Because of the form of Thiele’s empirical formula, its prediction effects in the small crack range may deviate because of extrapolation or adjustment; outside this range, its predicted results are good. The prediction effects of the stress field model are not good until the crack size is large. The modified stress field model can provide accurate predictions in different crack ranges even when extrapolating for small cracks, an aspect that lacks in the other two models.

(5). To reduce the number of tests and facilitate the application of single-edge notched or uniaxial central tearing tests, a crack combination of 3 mm + 15 mm is recommended, which is further confirmed using the RMSE index. A comparison between Thiele’s empirical formula and the new model is conducted using this combination. Although only two test data are involved, the modified model can still achieve good prediction, which is close to that obtained using all the test data involved in fitting. In addition, when extrapolation is needed for smaller cracks, its results remain undistorted. Under such condition, Thiele’s empirical formula deviates in certain cases because of its adjustment and inaccurate extrapolation.

Abbreviations

DIC

digital image correlation

PTFE

polytetrafluoroethylene

PVC

polyvinylchloride

RMSE

root-mean-square error

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China through research project No. 51778458.

ORCID iD

Xubo Zhang

References

Kim

. On the membrane collapse of Jeju World Cup Stadium by typhoon. Proc IASS-APCS 2003; 2003: 60–61.

Umezawa

Agari

. Report of snow damage of Kumagaya Dome. Proc IASS Ann Symp 2016; 2016: 1–6.

Zhang

. Advances in mechanical properties of coated fabrics in civil engineering. J Ind Text 2016; 48: 255–271.

Zhang

Zhou

, et al. Mechanical properties of PTFE coated fabrics. J Reinf Plast Comp 2010; 29: 3624–3630.

CECS158 . Technical specification for design of membrane structure. China Association for Engineering Construction Standardization; 2015.

ASCE/SEI 55-10 . Tensile membrane structure. American Society of Civil Engineers; 2010.

MSAJ/M-03 . Test methods for membrane materials (coated fabrics)-qualities and performances. Membrane Structures Association of Japan; 2003.

DG/TJ08-2019-2007 . Technical specification for inspection of membrane structure. Shanghai Construction and Transportation Commission; 2007.

FAA-P-8110-2-1995 . Airship design criteria. Federal Aviation Administration of USA; 1995.

10.

Hager

Gagliardi

Walker

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

11.

Steele

Gruntfest

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

12.

Chu

Chen

. Tearing failure mechanism of woven fabrics and comparison between tear test methods. Text Res J 1992; 13: 196–200.

13.

Wang

Sun

. Analytical modeling on mechanical responses and damage morphology of flexible woven composites under trapezoid tearing. Text Res J 2013; 83: 1297–1309.

14.

Wang

Sun

. Comparisons of trapezoid tearing behaviors of uncoated and coated woven fabrics from experimental and finite element analysis. Int J Damage Mech 2013; 22: 464–489.

15.

Krook

Fox

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

16.

Maekawa

Shibasaki

Kurose

, et al. Tear propagation of a high-performance airship envelope material. J Aircr 2008; 45: 1546–1553.

17.

Bigaud

Szostkiewicz

Hamelin

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

18.

Liu

Xiao

. Tear strength characteristics of laminated envelope composites based on single edge notched film experiment. Eng Fract Mech 2014; 127: 21–30.

19.

Zhang

Zhou

, et al. Central tearing behaviors of PVC coated fabrics with initial notch. Compos Struct 2019; 208: 618–633.

20.

Sun

. A novel tearing residual strength model for architectural coated fabrics with central crack. Constr Build Mater 2020; 263: 120133.

21.

Chen

Zhao

, et al. Mechanical responses and damage morphology of laminated fabrics with a central slit under uniaxial tension: a comparison between analytical and experimental results. Constr Build Mater 2015; 101: 488–502.

22.

Chen

Zhou

, et al. Central tearing characteristics of laminated fabrics: effect of slit parameter, off-axis angle, and loading speed. J Reinf Plast Comp 2017; 36: 921–941.

23.

Chen

Zhou

, et al. Fracture failure analysis and bias tearing strength criterion for a laminated fabric. J Ind Text 2017; 47: 1–32.

24.

Wang

Chen

Liu

, et al. Investigation of the tearing properties of a new airship envelope fabric based on experimental and theoretical methods. J Ind Text 2019; 48: 1327–1347.

25.

Sun

, et al. Uniaxial tearing properties and the tearing residual strength models of PTFE coated fabric. Structures 2021; 33: 1354–1364.

26.

Godfrey

Rossettos

. The onset of tear propagation at slits in stressed uncoated plain weave fabrics. J Appl Mech 1999; 66: 926–933.

27.

Godfrey

Rossettos

Bosselman

. A model for the onset of tearing at slits in stressed coated woven fabrics. Proc ASME-IMECE 2004; 2004: 1–10.

28.

Bao

Zhang

. Study on tearing tests and the determination of fracture toughness of PVC-coated fabric. J Ind Text 2021; 51: 977–1006.

29.

Sun

. Central crack tearing test and fracture parameter determination of PTFE coated fabric. Constr Build Mater 2019; 208: 472–481.

30.

Bao

Zhang

. Tearing analysis of PVC coated fabric under uniaxial and biaxial central tearing tests. J Ind Text 2020; 51: 900–925.

31.

Sun

, et al. Biaxial tearing properties of woven coated fabrics using digital image correlation. Compos Struct 2021; 272: 114206.

32.

Chen

Ding

, et al. Investigation of tear strength of an airship envelope fabric by theoretical method and uniaxial tear test. J Eng Fiber Fabr 2019; 14: 1–11.

33.

Chen

. Tear strength of a laminated fabric for stratospheric airship under uniaxial and biaxial tests. J Eng Fiber Fabr 2021; 16: 155892502110018.

34.

Thiele

. Approximation of envelope critical envelope critical slit length. AIAA Light-Than-Air Syst Technol Conf 1995; 1995: 15–18.

Modified stress field model for critical tearing strength of architectural coated fabrics

Abstract

Keywords

Introduction

Tests

Materials

Experimental program

Experimental results and discussion

Critical tearing strength prediction models

Thiele’s empirical formula

Stress field model

Modified stress field model

Results

Discussion

Prediction effects of three models in different crack ranges

Background and test results in small crack range

Comparison of prediction effects in different crack ranges

Discussion

Prediction effects based on only two cracks

Acquisition of k and m based on only two cracks

Acquisition of C k and n based on only two cracks

Discussion

Conclusion

Abbreviations

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Acquisition of C_k and n based on only two cracks