Experimental Investigations into Failures and Nonlinear Behaviors of Structural Membranes with Open Cuttings

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Failure modes of structural membrane with open cuttings have been identified. The new insights can be applicable to structural health monitoring of the thin-walled shell structures.The nonlinear relationship between cutting angle and the force strongly relates to the ultimate limit state of the structural membranes.A new formula has been proposed to help engineering design and evaluate the ultimate capacity of the structural membranes with and without open cuttings.

Abstract

Reportedly, structural failures in membrane structures have occurred frequently, mostly originating from localized damage caused by intense loads on the membrane surface. It is thus necessary to investigate the nonlinear behaviors and load-carrying capacity of membranes with local damage. This study has conducted uniaxial tensile tests for membranes with a variety of original defects by using a specialized experimental setup and photogrammetry technique. The nonlinear relationship between the mechanical properties and the deforming angle of membranes, which portrays the principal axis, tensor, tensile stress, and position of the original defects, is investigated. The entire process of membrane failure has been recorded, and the strain and stress during each test specimen are compared. The new results indicate that the membranes exhibit predominantly elastic deformation before failure but surprisingly impart brittle fracture upon failure. Finally, a novel approach for estimating the load-bearing capacity of initially damaged membranes was proposed through the analysis of the load-bearing capacity of the damaged membranes under various conditions, positions, angles, and other influential factors.

1. Introduction

Membrane materials have found wide-ranging applications in various engineering fields such as civil engineering, aerospace, biotechnology, and medical devices due to their light weight, flexibility, and ability to separate different mediums [1,2,3]. The typical tensile membrane structure is shown in Figure 1, which is the famous architecture at Tokyo university [4]. However, in practical applications, membranes are often subjected to various types of damage, including the formation of slits or cuts, which can significantly affect their mechanical behavior and load-carrying capacity [5,6,7].

Understanding the strength characteristics of a tensile membrane with initial damage is of great significance to ensure its reliability and durability in civil engineering. To understand the research status of scholars in this field and provide reference for follow-up experiments, the recent research is analyzed. Recent studies have employed experimental tests, numerical simulations, and analytical models to explore various aspects of membrane strength. Laura et al. [8] investigated the tearing behavior of fabric by high-speed imaging analysis. Their studies showed that the high-speed imaging technique is a suitable way to observe the performance of the test of fabric. D. Bigaud et al. [9] have conducted an analysis for textile-reinforced soft composites for a tensile structure by bi-axial tensile tests. They observed and analyzed the failure mechanisms of a ripped sample. Han Bao, Minger Wu et al. [10,11] have studied the tearing test and the determination of fracture toughness of PVC-coated fabric. They used the tearing test and found that the uniaxial central tearing test could be replaced by the corresponding single-edge notched tearing test to minimize the usage of test materials and they gave the appreciate value of the fracture toughness. Cheng Jianwen [12,13,14,15] ran a lot of tests. The mechanical principle and properties of fiber were found on the basis of the experiments. And Rijin He [16,17,18] and Zhang Yingying [19,20] studied the same topic; they used the theoretical method to deduce a formula to calculate the mechanical properties of the membrane.

Recent research on the strength of membranes with slits in civil engineering structures has contributed to advancing our understanding of their behaviour and performance. Experimental investigations, numerical simulations, and analytical models have provided valuable insights into the effects of the slit configuration, loading case, and durability considerations. Further research is warranted to optimize the design guidelines and develop innovative solutions for enhancing the structural integrity and longevity of slit membrane structures.

This research aims to investigate the strength characteristics of membranes with slit defects, focusing on elucidating how the presence of slits influences their mechanical behavior under different loading cases. Through the uniaxial tear test of the damaged membrane structure, a three-dimensional optical scanner was used to photograph and scan, and the three-dimensional deformation map of the membrane was obtained when the membrane was stretched and finally damaged, so as to better show the deformation of the membrane. Through the summary of the test data, the calculation formula of the ultimate stress of the slit of the membrane is obtained.

2. The Setting of the Experiment

This study’s experimental design follows the standards for uniaxial tensile testing. The selected dimensions of the membrane materials are as follows: 50 mm × 400 mm with 0.8 mm thickness.

The data for the membrane materials are presented in

Table 1. The parameters of membrane materials.

Modulus	Yarns	Thickness	Tensile Strength	Tear Strength	Adhesion Strength	Surface Treatment
420–435 MPa	1500 D	0.8 mm	3500 N/5 cm	1300/1200 N	120 N/5 cm	PVDF and acrylic

. The specific parameters for the initial defects of the membrane materials and the regions of clamping at both ends are illustrated in Figure 2. Both clamping regions at the ends are 50 mm long. Due to welding process errors, there are slight differences in the lengths of the membrane materials. Therefore, the calibration points pasted on the membrane surface serve as reference positions, as shown in Figure 3.

In this experiment, a homemade testing apparatus was used for staged loading, with each stage set at 200 N. The schematic diagram of the apparatus is shown in Figure 4. Q7 three-dimensional scanners and S-type force gauges are utilized to collect force and displacement data. The specific parameters are listed in

Table 2. Instrument parameters.

Item	Information
Optical scanner	Scanning range (mm): 60 × 45–600 × 450 Data precision: ±0.01 mm, data acquisition speed: <0.1 s
S-type load cell DYLY-103	Sampling rate: 10–80 points/s, range: 100 kg Measurement precision: 0.05%

. Subsequently, stress–strain curves are generated from the collected images and tensile force data, and various analyses are performed on the membrane specimens.

To simulate the damaged effects of the membrane materials, a batch of membrane materials with original defects were prepared for this experiment. Additionally, the influence of the cutting defects at different angles and positions on the mechanical properties of the membrane materials was considered. The specific data are presented in

Table 3. Pore names and parameters.

Case	Position	Angle	Shape	Size
1	Middle	0°	Cut	30 mm
2	Middle	30°	Cut	30 mm
3	Middle	45°	Cut	30 mm
4	Middle	75°	Cut	30 mm
5	Middle	90°	Cut	30 mm
6	Edge	0°	Cut	30 mm
7	Edge	30°	Cut	30 mm
8	Edge	45°	Cut	30 mm
9	Edge	75°	Cut	30 mm
10	Edge	90°	Cut	30 mm
11	No cuts

. Case 11 serves as the control group, where no treatment is applied, and the membrane materials undergo direct tensile testing.

Because the mechanical properties of the membrane are different under different loading rates, the loading rate of 100 mm/min is adopted according to the tensile strength test standard [21,22] and the application situation of scholars from various countries [23,24]. The final failure of the test is defined as the failure of the membrane material when it is completely broken or when the test fails to load its critical tensile load.

3. Test Results

3.1. Method of Test

The experimental results indicate that the membrane materials exhibit brittle fracture upon failure, with the fracture surfaces being smooth and perpendicular to the direction of tension, as shown in Figure 5. By selecting calibration points on the same edge and measuring the distance between the centers of the two points as the distance between the upper and lower edges, taking Case 3 as an example, when the load is 0 N, one central position calibration point is not identified. The length changes in the upper and lower edges of Case 3 are illustrated in Figure 6, with the specific data presented in

Table 4. The length of each stage at the upper and lower edges of Case 3 (middle case).

Stage (N)	Upper Edge Length (mm)	Down Edge Length (mm)
0	81.2889	72.7552
200	83.4735	73.2701
400	84.8250	73.4476
600	85.9289	73.6244
800	87.2999	74.2634
1000	88.8738	74.9993
1200	90.2581	76.1812
1400	91.1664	77.0246

. Subsequently, the specimen underwent sudden brittle fracture during subsequent stages.

The various stages of tension collected by the S-type force gauge are organized. By dividing each stage of tension by the cross-sectional area of the membrane material (the width of the membrane material is taken as the effective length after removing the cut seam), the stress of each stage of the membrane material under different cases can be obtained. Considering the differences in distance between calibration points and the difficulty in achieving an ideal centering state during clamping of the specimen, or the significant differences in length changes between the two sides due to the position of pre-fabricated voids not being centered, the average value of the lengths on both sides is taken as the length of the membrane material for each stage to calculate the displacement of each stage. Using the displacement divided by the initial length as the result of each stage’s strain, stress–strain curves are then plotted. Taking Case 3 as an example, at this time, the width of the membrane material is approximately 28.79 mm. The specific calculation data are shown in

Table 5. The displacement (mm) and stress (MPa) vs. strain (%) data for Case 3.

Stage (N)	Stress	Mean Length	Displacement	Strain
0	6.94	77.02	0	0.00
200	6.95	78.37	1.35	1.75
400	13.9	79.13	2.11	2.75
600	20.84	79.78	2.75	3.58
800	27.79	80.78	3.76	4.88
1000	34.73	81.94	4.91	6.38
1200	41.68	83.15	6.13	7.96
1400	48.63	84.10	7.07	9.18

.

3.2. The Results of the Middle Cases

The data of all the middle void specimens are summarized and presented in Figure 7. Among them, only the specimen of Case 1 did not experience failure, while the remaining specimens all suffered from failure. Case 5 exhibited the smallest ultimate stress, while Case 2 exhibited the largest ultimate stress. Overall, the ultimate stress appears to be inversely proportional to the angle. Except for Case 5, which exhibited a period of minimal deformation before failure, all the specimen curves were linear, with very small differences in the slope. Cases 2–4 maintained consistency before failure, and the difference in ultimate stress at failure was not significant. At the point of failure, the strain ranged from 0.07 to 0.09 for all specimens.

The specific data of Figure 7 are shown in

Table 6. Stress (MPa) and strain (%) data for various cases (middle case).

Case	Stress	Strain	Case	Stress	Strain
1	0.0	0.00	2	0.0	0
	17.0	2.39		5.7	0.71
	34.1	5.40		11.4	1.27
	51.2	8.81		17.1	2.27
	60.2	11.04		22.9	3.66
3	0.0	0.00		28.6	5.19
	7.1	1.75		34.3	6.45
	14.3	2.75		40.0	7.77
	21.4	3.58		51.4	8.98
	28.6	4.88	4	0.0	0.00
	35.7	6.38		9.5	0.65
	42.9	7.96		19.0	3.45
	50.0	9.18		28.6	4.43
5	0.0	0.00		38.1	6.58
	10.0	2.21		47.6	8.97
	20.0	3.87	/
	30.0	7.20
	40.0	9.28

. The stress–strain curves are approximately linear, allowing us to analyze the slope of the stress–strain curve to determine the elastic modulus of the membrane material tested in the experiment, which is approximately 417.3 MPa. This value is close to the elastic modulus of the membrane material (425.0 MPa) obtained from previous data, which were obtained from the company. Combining the above data, it can be concluded that when the edge of the membrane material is damaged, it has a reliable load-bearing capacity within a strain range of 0.0–0.1; when the middle of the membrane material is damaged, it has a reliable load-bearing capacity within a strain range of 0.00–0.07. The data indicate that as the angle of the void increases, the load-bearing capacity of the membrane material significantly decreases, and the ultimate stress slightly decreases. This may be because the increasing angle of the void gradually reduces the effective width of the membrane material, thereby weakening its load-bearing capacity. Additionally, the reduction in the effective width may lead to the damage of the fiber fabric, thereby reducing the overall stiffness and tear resistance of the membrane material.

3.3. Test Results of Edge Cases

The data of all edge void specimens are summarized and presented in Figure 8 and

Table 7. Stress (MPa) and strain (%) data for various Cases 6–10 (edge case).

Case	Stress	Strain	Case	Stress	Strain
7	0.0	0.00	6	0.0	0.00
	5.7	0.37		10.0	1.63
	11.4	1.23		20.0	4.12
	17.1	2.99		30.0	7.56
	22.9	4.41		40.0	10.10
	28.6	5.97		50.0	12.46
	34.3	8.01		60.0	15.61
	40.0	10.99	8	0.0	0.00
	45.7	12.40		7.1	−0.24
	51.4	14.16		14.3	0.97
	57.1	15.15		21.4	3.23
	/	/		28.6	6.35
				35.7	9.80
				42.9	13.64
9	0.0	0.00	10	0.0	0.00
	4.5	3.60		2.5	3.02
	9.5	7.14		6.5	6.93
	19.0	10.18		10.5	8.75
	28.6	13.95		20.5	12.02
	33.7	15.10		32.4	15.52

. Among them, only Case 6 did not experience failure, while the remaining specimens all suffered from failure. Among these five specimens, Cases 9–10 exhibited the smallest ultimate stress, and their stress–strain curves maintained a high degree of consistency. Case 6 was least affected by the angle, and Cases 6, 7, and 8 were essentially identical at the starting position. Case 7 exhibited the highest ultimate stress. Overall, the ultimate stress decreases with the increasing angle. For all specimens, the strain exceeded 0.1 at the point of failure.

4. Analysis of Strength

The ultimate strength of each case is shown in

Table 8. Ultimate strength of each case.

Case	1	2	3	4	5	6	7	8	9	10	11
Angle	0	30	45	75	90	0	30	45	75	90	0
Strength	89	57	49	44	41	91	58	42	34	32	98

. By plotting the curve of the angle along the cut seam and the tension force against the ultimate stress, as shown in Figure 9, we can study the relationship between the ultimate stress and the angle of the cut seam. As the angle increases, the ultimate stress gradually decreases. This relationship is nonlinear, and when the angle between the cut seam and the tension force is 90°, the load-bearing capacity is the smallest, which can be considered the most unfavorable case during failure.

It can be inferred that as the angle of the cut seam increases, the load-bearing capacity of the membrane material decreases due to the reduction in the effective width of the membrane material. Additionally, the stiffness and ultimate stress of the membrane material decrease because the fabric in the membrane material’s coating is damaged. We can analyze the relationship between these factors by organizing the data, as shown in Table 5. Here, we mainly compare the tearing strength of Case 11 with Cases 1–5 and Cases 6–10. When the cut seam angle is 0°, the ultimate stress of the membrane material slightly decreases, but the decrease is not significant. When the cut seam angle is 30°, the effective length of the cut seam is 15 mm, accounting for 30% of the membrane material width. Compared with the result of 1, the ultimate stress of Cases 2 and 6 decreases by 42% and 41%, respectively. When the slice angle is 45°, the effective length of the cut is approximately 21.2 mm, accounting for 42% of the membrane width. At this point, the ultimate stress of Cases 3 and 8 decreases by 50% and 57%, respectively. When the cut angle is 90°, the effective length of the cut seam is about 30 mm, accounting for 60% of the membrane material width. At this point, the ultimate stress of Cases 5 and 10 is the smallest, and it decreases by 58% and 67%, respectively.

Therefore, when the cut seam appears, the ultimate stress of the membrane material immediately decreases. This is because the cut seam damages the fibers in the membrane, reducing the overall stiffness and tear resistance of the membrane material. The rate of decrease in the load-bearing capacity largely depends on the effective length of the cut seam (i.e., the projection of the cut seam perpendicular to the direction of tension). This is because an increase in the effective length damages more fibers, reducing the ultimate stress of the membrane material while also decreasing the width at the most unfavorable location.

Consequently, after the fiber failure at the slit point, the entire membrane material subsequently fails. Combined with the example in Figure 10, it can be inferred that at lower stress levels, the mechanical performance of the membrane material can be considered as that of a homogeneous material, as depicted in Figure 11a; as the stress increases, the cracks in the membrane material extend to the yarns, as shown in Figure 11b; after the baseline ruptures, the cracks rapidly expand, leading to overall membrane failure, as illustrated in Figure 11c,d. It should be noted that the pink and green areas represent areas of yarn stretching deformation. The pink area deforms more than the green area.

5. Discussion

The present study assumes that the defects are approximately elliptical during tensile deformation. According to Charles E. Inglis’s elastic solution in 1913, the stress at the location of a defect crack arises from two components: uniform stress in the absence of defects and enhanced stress beneath the defect, calculated as Equation (1).

$${\sigma }_c=\sigma (1+{\displaystyle \frac{2a}{b}})=\sigma +\sigma {\displaystyle \frac{2a}{b}}$$

(1)

where ${\sigma }_c$ is the strength of the membrane and $\sigma$ is the external stress. The lengths of 2a and 2b represent the major and minor axes of the ellipse, respectively. At the tip of the defect, the stress can be approximated as the superposition of the uniform tensile stress and enhanced stress at the defect location. Thus, the stress can be expressed as shown in Equation (2).

$${\sigma }_c=\sigma +\sigma \sin (\theta )\cdot {\displaystyle \frac{2a}{b}}$$

(2)

During the stretching process, the length of the defect changes very little, while the width varies and cannot be quantified. Thus, the formula for calculating the bearing capacity can be approximated, as shown in Equation (3), where $\kappa$ is the parameter of the tip stress, which should be determined by a test.

$$\sigma \le {\displaystyle \frac{{\sigma }_s}{1+\sin (\theta )\cdot \frac{2a}{b}}}={\displaystyle \frac{{\sigma }_s}{1+\kappa \sin (\theta )}}$$

(3)

The curve of the ultimate stress versus angle is depicted in Figure 12. It can be observed that as the angle increases, the ultimate stress gradually decreases. The angle plays the main role in the strength of the membrane with slits. Moreover, when the angle between the defect and the tension force is 90 degrees, the bearing capacity is minimal, representing the most unfavorable condition during failure. By conducting fitting analysis on the data represented in Figure 11, it can be observed that the data points align very well with the fitted curve.

6. Conclusions

This paper focuses on the mechanical properties of damaged membrane materials. Through staged uniaxial tensile tests using a self-made loading experimental device and a three-dimensional scanner, the influence of different parameters such as the angle, position, and shape on the mechanical properties of the membrane material was obtained. Through a comparison of the experimental results and analysis, the main conclusions are as follows:

• Defective membrane materials exhibit brittle failure. This is because the stress on the membrane material is mainly borne by the fibers. In ideal cases, the coating and base fabric work together, exhibiting certain plastic properties. When there are defects, stress is concentrated at the defect location, and the base fabric fibers continuously fracture while the coating does not act.
• After the membrane material is damaged, it still has load-bearing capacity within a certain strain range. This range depends on the position of the cut seam and is not significantly related to the angle of the cut seam.
• The angle between the cut seam and the direction of force has a significant impact on the ultimate stress of the membrane material, and the ultimate stress is also affected by the position factor. It can be approximately calculated using Equation (3).