A high efficient constitutive model of coated plain-weave fabrics under uniaxial tensile loads in arbitrary directions

Abstract

The complex mechanical properties of coated fabrics make it difficult to obtain accurate constants for design and analysis. A high efficient constitutive model of coated plain-weave fabrics is proposed to realize the prediction of mechanical behaviours in arbitrary directions. The coupling mechanism between macro and meso deformation of coated fabrics is analyzed firstly, and the constitutive relation of the model is established based on mesoscopic parameters with clear physical interpretation. Then, the material property values of specimens are obtained by using a multi-objective optimization algorithm according to the experimental data. Unlike the previous constitutive relations which need to use finite element methods to get numerical solutions, the constitutive relation established in this paper can be mapped to a quartic equation with analytical solution, and the calculation efficiency can be greatly improved. The deviations of model predictions with experimental results, e. g., 0.052 (tension), 0.009 (width), 0.014 (length, warp), 0.025 (length, weft), 0.027 (angle, warp), 0.030 (angle, weft), show that the model predictions are in good agreement with the experimental data.

Keywords

Coated fabrics constitutive model large deformation uniaxial tensile anisotropic

Introduction

The inflatable structure made of coated fabrics has the characteristics of light weight, small, strong bearing capacity, fast deployment, etc. It has been widely used in emergency safety assurance, such as inflatable aircraft evacuation slides,¹ marine evacuation system (MES),^2,3 etc. Coated fabrics are fiber reinforced composite materials with quite complex mechanical behavior, such as strong geometric nonlinearity, large deformation, stress relaxation, creep characteristics, etc. The components of large inflatable structure may be as long as several meters, or even more than 10 m, and the change range of length before and after inflation can reach more than 0.5 m. The mechanical behaviours of coated fabrics used in the manufacture of inflatable products determine the geometric scale changes of the components under different internal gas pressures. The anisotropy of coated fabrics can be used to synchronize the deformation of inflatable components with different scales. For instance, the stiffness of coated fabrics is selected by cutting direction, so that two inflatable columns with same diameter/stiffness ratio but different diameters can be deformed synchronously during inflation.

The physical structure of coated fabrics determines their mechanical behavior.⁴ According to different modeling methods, the constitutive models of coated fabrics can be divided into two categories: continuum models and meso-structural models.^5,6

Continuum models are based on three assumptions: homogeneous, anisotropic and continuum. The stress-strain curves or response surfaces are obtained through a large amount of experiments, then mathematical expressions are established by using increments, polynomials,^7,8 etc., and finally the methods such as multi-linear and response surfaces are used to establish the constitutive model to describe the macroscopic characteristics of materials, such as geometric nonlinearity,^9–11 large deformation,¹² stress relaxation,¹³ creep characteristics,¹⁴ viscoelastic deformation,¹⁵ plastic deformation,¹⁶ etc. The advantage of continuum models is that they can be easily implemented in the commercial software, e. g., Dinh et al.¹⁶ proposed a new constitutive relation to express the nonlinearity based on the elasto-plastic assumptions, which is implemented in ABAQUS as a user material subroutine. However, continuum models can only represent the macro mechanical properties, but cannot explain the micro mechanical behavior. The accuracy of the model is based on the test curve, and a large amount of experiments are required to use this kind of model. Moreover, the continuum model regards the material as a homogeneous continuum composed of anisotropic materials, and there are always several parameters without clear physical interpretation in its mathematical expression, so it is difficult to use a unified expression to describe the complex deformation relationship of the material. Although the calculation amount of continuum models is lower than that of meso-structural models, they still need to be simplified before engineering application.

Meso-structural models predict macro mechanical behavior based on fabric structure and deformation of coated fabrics. The internal mechanical mechanism is explained at meso level. The physical interpretation of model parameters is clear, and the description of the constitutive relation is more accurate. Researchers have established constitutive relations based on the microstructures of yarns, coatings and their interactions^17–19 through the combination of mechanical concepts and mathematical analysis, and have proposed Tensioned Fabric Structures,²⁰ hyperelastic constitutive model,²¹ micro-mesa unit cell model,²² yarn model,²³ digital element methods^24–26 and multi-layer solid elements method,¹⁹ etc. Among the studies, Gade et al.¹⁷ established a meso-scale model with regard to the coating stiffness, by performing an inverse process of parameter identification using Levenberg-Marquardt algorithm, and the model is implemented as a material law into a finite element program. The meso-structural models describe the constitutive relation based on the microstructure of the real coated fabric, so the prediction of its macro mechanical behavior is more accurate. Although the prediction accuracy is high, the current meso-structural models of coated fabrics are already very complicated, and the complex mathematical expressions lead to a huge workload. With further development, the meso-structural models will become more complicated, which will seriously affect their popularization and use.

At present, the research on the mechanical properties of coated fabrics lags behind conventional materials, resulting in many uncertainties in the analysis.⁵ Therefore, it is necessary to establish a simple, efficient, accurate and reliable constitutive model of coated fabrics. During the development of the large inflatable structures, our research group has conducted a large number of experiments and analyses on the mechanical properties of coated fabrics. In this paper, a new model for predicting the mechanical properties of coated plain-weave fabrics in arbitrary directions is proposed, which has the following advantages: (1) The model is established based on parameters with clear physical interpretation, and the constitutive relation can be mapped to a quartic equation with analytical solution. (2) The modeling method is intuitive and easy to understand, and the model can predict the mechanical properties of coated fabrics at both macro and meso levels. (3) The key material properties of coated fabrics can be obtained by using only 3 groups of tensile test data from weft, warp, and other arbitrary directions.

It is a simple and accurate constitutive model to express the complex response under various loadings, which is suitable for the needs of engineering design and analysis, and can solve the problem of design and material selection for large inflatable structures.

Mechanism analysis of mechanical model of coated fabrics

Coated plain-weave fabrics are composite materials that cover both sides of an underlying woven structure with flexible impermeable coatings. The underlying woven structure is made of high-strength warp yarns (hereinafter single warp yarn is abbreviated as A-FB) and weft yarns (hereinafter single weft yarn is abbreviated as E-FB). The coating, as the physical protective layer for the underlying woven structure, has the characteristics of liquid sealing, gas sealing, wear resistance, corrosion resistance,⁴ etc. The underlying woven structure and the coatings jointly determine the mechanical properties of coated fabrics. Coated fabrics have excellent tensile strength and shear strength. The thickness of coated fabrics used for manufacturing inflatable structures is usually within a millimeter. Each single yarn (hereinafter abbreviated as FB) is composed of multiple fibers with small diameter, and it has weak compression resistance and bending resistance. As shown in Figure 1, A-S is the section of A-FB, which can display the micro morphology of A-FB, and, E-S is the section of E-FB, which can display the micro morphology of E-FB.

Figure 1.

Macro and micro morphologies of coated fabrics.

The coated fabrics are a kind of large deformation, nonlinear and anisotropic material. Its ultimate tensile strain can reach more than 0.5. Its nonlinearity is represented by the significant nonlinearity of the mechanical behaviours at macro level, and at micro level, the tension distribution in E-FB, A-FB and the coating changes dynamically due to the change of weaving angles. Its anisotropy shows that there is obvious variation in the mechanical behaviours in different tensile directions at macro level, and the variation law of weaving angles in different tensile directions is also different at micro level. Generally, A-FB and E-FB are made of the same materials, but the mechanical behaviours curves in the warp and weft directions are different due to the difference in the quantity of fibers within A-FB and E-FB, density (the amount of A-FB and E-FB per unit width) and the weaving undulations.

As shown in Figure 2, if a coated fabric specimen is tensed in the Y direction, it will become longer in the Y direction and shorter in the X direction perpendicular to Y. Since the coating on its surface has certain compressive resistance in the X direction, even if the deformation reaches the limit and fracture occurs, the width of the specimen in the X direction will not tend to zero.

Figure 2.

Tensile deformation of a coated fabric specimen.

It can be seen from Figure 2 that there are three cross-sections which can establish constitutive relations, namely, X-S, A-S and E-S. The force is transferred in tension shear coupling at A-S and E-S sections. Shear is the dominative deformation mode for woven fabrics in forming. In order to avoid shear analysis and simplify the constitutive relation, X-S section is selected to establish the constitutive relation, and the shear deformation²⁷ can be equivalent to compressive deformation. Since the tension stiffness of the coating is much smaller than that of FB, it can be added onto FB. In the balance state, the tension is transferred through A-FB and E-FB on the X-S section, and the components of A-FB and E-FB in the X direction are balanced with rebound force of the coating. The deformation of the specimen in the X direction consists of two parts: (1) the deformation driven by the Poisson’s effect of the coating; (2) the deformation driven by the tension component of FB in the X direction. During the deformation of the specimen, the weaving angles change dynamically, leading to the dynamic change of the number of A-FB and E-FB crossing X-S section, as shown in Figure 2. The proposed model in this paper contains four key indexes of material parameter: tension stiffness of A-FB, tension stiffness of E-FB, Poisson ratio and compressive stiffness of the coating, and the physical interpretation of each parameter is clear.

Membrane materials are always in biaxial stress state, but it is difficult to achieve perfect biaxial stress failure in the laboratory.⁵ In design specifications, a reduction factor is always used to reflect the relationship between biaxial tensile strength and uniaxial tensile strength. Uniaxial tension methods are widely used and recommended in design specifications. In this paper, the constitutive relation of the model will be established based on uniaxial tensile methods. Generally, the design of civil engineering focuses on the prediction of the damage of coated fabrics caused by the ultimate tension.²⁸ However, the gas-sealing of coated fabrics also needs to be considered in the design of inflatable structures. Tiny gaps will appear in the coating under the condition of large deformation, resulting in gas leakage. Therefore, the model established in this paper does not consider the ultimate damage, and its predetermined working range is 0–2/3 ultimate tension.

Establishment of the constitutive relation of coated fabrics

Definition of geometric and physical parameters of the model

θ_T, θ_A, θ_E, A, C_A, C_E, B_A, B_E, L are geometric parameters related to the initial state of a specimen, all of which are in the measurement area. θ_T, I_A, I_E are geometric properties of coated fabrics, and their values can be directly measured by microscope. D_Y, A, L are geometric parameters of a specimen, and their values are determined by the contour design of the specimen. After the design of the specimen, θ_A, θ_E, C_A, C_E, B_A, B_E can be obtained by geometric measurement of the specimen or trigonometric function calculation with D_Y, A, L as input. Under the action of tension in the D_Y direction, the geometry of the specimen will change and the above parameters will change to θ_t, θ_a, θ_e, a, c_a, c_e, b_a, b_e, l. The geometric description of these parameters is shown in Figure 3.

Figure 3.

Definition of geometric parameters of coated fabrics and the comparison between before and after deformation.

It can be seen from Figure 3 that the widths of E-FB and A-FB in the X direction are always consistent during uniaxial tension of coated fabric specimen, which is the basis to solve the geometric deformation of E-FB and A-FB. k_a, k_e, k_pc, μ_p are inherent material properties of coated fabrics, and their values can be obtained by performing an inverse process of parameter identification using a multi-objective optimization algorithm.²⁹ The others are process parameters.

Establishment of constitutive relation

The constitutive relation of coated fabrics is established based on the following 3 assumptions:

(1) The specimen has no constraint in the X direction and can deform freely, ignoring the influence of fixture.

(2) Deformation and tension are in the same plane, each E-FB or A-FB in the measurement area of a specimen has exactly the same geometric and physical state, and E-FB or A-FB always keeps a straight line during deformation.

(3) During the test, the specimen is kept in a non-destructive state without irreversible material damage such as fiber pull out.

The above assumptions are analyzed as follows: (1) Dumbbell shaped specimens should be prepared according to GB/T 528-2009 standard, and a transition zone shall be formed between the two ends of the measurement area to eliminate the constraint of the X-direction deformation. (2) Since the coated fabrics are very thin, usually within 1 mm, which is far less than the length and width of the specimen, it belongs to a plane stress problem. Ignoring the occasional defects in the processing of coated fabrics, it is considered that all the E-FBs or A-FBs in the specimen are completely the same and in a straight line state. (3) The maximum tension of coated fabrics used for inflatable structures shall be lower than the value that creates sealing damage. The sealing damage occurs before the structural damage. This model is applicable to the stage with good gas tightness, ignoring the irreversible material damage.

Under the above assumptions, the accidental factors in the manufacturing and testing process that impact the model accuracy can be ignored.

When θ_A≠0 and θ_A≠θ_T, under the action of F_Y, the width of the specimen will change from A to a, and then ε_x=ε_μ+ε_s, in which ε_μ represents the strain due to the influence of Poisson’s ratio and ε_s represents the strain due to coating compression by tension component of FB in the X direction.

Taking E-FB passing through X-S section as an example for deformation analysis, formula (1) can be obtained.

ε_{x} = ε_{μ} + ε_{s} = μ_{p} ε_{e y} + f_{e x} / k_{p c} = a / A - 1

(1)

Since $\sin θ_{e} = a / c_{e}$ , $ε_{e y} = b_{e} / B_{E} - 1$ , formula (1) can be rewritten as formula (2).

μ_{p} (b_{e} / B_{E} - 1) - (c_{e} / C_{E} - 1) (a / c_{e}) (k_{e} / k_{p c}) = a / A - 1

(2)

Let $M_{e} = μ_{p} / K_{e p} + a / C_{E} - 1 / K_{e p} + a / (A K_{e p})$ , $N_{e} = - μ_{p} / (B_{E} K_{e p})$ , formula (2) can be simplified as formula (3).

c_{e} = a / (N_{e} b_{e} + M_{e})

(3)

As shown in Figure 3, a, b_e, c_e form a right triangle and satisfy Pythagorean theorem (formula (4)).

a^{2} + b_{e}^{2} = c_{e}^{2}

(4)

Substitute formula (3) into formula (4) to eliminate c_e, then formula (5) can be obtained.

N_{e}^{2} b_{e}^{4} + 2 M_{e} N_{e} b_{e}^{3} + (M_{e}^{2} + a^{2} N_{e}^{2}) b_{e}^{2} + 2 a^{2} M_{e} N_{e} b_{e} + {a^{2} M_{e}}^{2} - a^{2} = 0

(5)

Formula (5) is a quartic equation about b_e, in which 5 coefficients are known. b_e can be solved directly by “Ferrari” or “adjoint matrix of polynomial”. Then, c_e can be solved by substituting b_e into formula (3). b_a and c_a can be solved similarly. When θ_A≠0 and θ_A≠θ_T, due to the influence of coating stabilization and friction between FBs, the tension is transferred in the coated fabrics in a tension-shear coupling manner (Figure 2, sections A-S, E-S). ε_y is determined by b_e and b_a, and its mathematical expression is as formula (6).

ε_{y} = (b_{e} + b_{a}) / (B_{E} + B_{A}) - 1 = l / L - 1

(6)

Formula (6) corresponds to the mechanical behaviours when θ_A≠0 and θ_A≠θ_T. There is no E-FB or A-FB through both ends of the specimen, and the tension needs to be transmitted in alternating between E-FB and A-FB. Therefore, the deformation is composed of the deformations of E-FB and A-FB.

As shown in Figure 2, under the action of F_Y, the specimen may lengthen in the Y direction and shorten in the X direction, resulting in the changes of the number of E-FBs (n_e) and the number of A-FBs (n_a) passing through X-S section. It can be seen from Figure 4 that θ_a and θ_e determine n_e and n_a. Taking the calculation of n_e as an example, $N_{E} = A / (I_{E} \sin θ_{A})$ can be calculated from θ_A and A, and N_E remains unchanged during the deformation of the specimen. Since the specimen lengthens in the Y direction, A→a, when n_e can be obtained as formula (7).

n_{e} = N_{E} c_{e}^{'} / c_{e}

(7)

c_a^′, c_e^′, a form a triangle, and formula (8) can be obtained according to the sine theorem.

c_{e}^{'} / \cos θ_{a} = a / \sin (θ_{e} + θ_{a})

(8)

Figure 4.

Schematic diagram of solving the number of warp and weft yarns passing through arbitrary X-S section.

Divide both ends of formula (8) by c_e, and substitute a/c_e=sinθ_e, then get formula (9).

c_{e}^{'} / c_{e} = \cos θ_{a} \sin θ_{e} / \sin (θ_{e} + θ_{a})

(9)

Substitute formula (9) into formula (7), and calculate n_e by formula (10).

n_{e} = A c o s θ_{a} \sin θ_{e} / (I_{E} \sin θ_{A} \sin (θ_{e} + θ_{a}))

(10)

Similarly, calculate n_a by formula (11).

n_{a} = A c o s θ_{e} \sin θ_{a} / (I_{A} \sin θ_{E} \sin (θ_{e} + θ_{a}))

(11)

Then the tension in the Y direction (f_y) can be calculated by formula (12).

f_{y} = k_{a} ε_{a} n_{a} \cos θ_{a} + k_{e} ε_{e} n_{e} \cos θ_{e}

(12)

All the variables (ε_a, n_a, θ_a, ε_e, n_e, θ_e, ε_y) are functions of a. For each given value of a, the corresponding ε_x, ε_y, f_y can be calculated. So far, the constitutive relation of coated fabrics in arbitrary directions (D_Y) has been established.

The following is about the identification of the property values of the specimen material (coated fabrics): k_a, k_e, k_pc, μ_p.

Identification of property values of coated fabrics

According to formulas (1)–(6), μ_p, K_ap, K_ep, ε_x→ε_y, ε_a, ε_e, θ_a, θ_e, which is called geometric transformation (G-T) of coated fabrics. According to formula (12), ε_a, ε_e, θ_a, θ_e, k_a, k_e→f_y, which is called mechanical transformation (F-T) of coated fabrics. Since (G-T) → (F-T), the values of material properties μ_p, K_ap, K_ep need to be determined firstly.

The coating is relative thin, and ε_x is mainly from ε_s, ε_μ≪ε_s, leading to a relative small μ_p. Let k_a=k_A−FB+k_pca and k_e=k_E−FB+k_pce, in which k_A−FB and k_E−FB represent the tension stiffnesses of A-FB and E-FB, respectively, k_pca and k_pce represent the tension stiffnesses of the coatings on A-FB and E-FB, respectively. In order to simplify the model, k_a and k_e are used to describe the tensile properties of A-FB and E-FB in the subsequent analysis. $K_{a p} (k_{a} / k_{p c})$ and $K_{e p} (k_{e} / k_{p c})$ are not independent. When θ_A = 0, $k_{a} \leftrightarrow F_{Y} / ε_{a}$ , and when θ_E = 0, $k_{e} \leftrightarrow F_{Y} / ε_{e}$ . The tensile testing machine can be used to test the two specimens with θ_A = 0 and θ_E = 0 , and then determine the value range of k_a/k_e, i.e., $R_{k_{a} / k_{e}}$ , according to the two F-T curves. Since $k_{a} / k_{e} = K_{a p} / K_{e p} \to R_{K_{a p} / K_{e p}} = R_{k_{a} / k_{b}}$ , the solution interval can be narrowed down.

When ε_a or ε_e reaches a certain threshold, FB shows yield characteristics. Define the yield adjustment function of E-FB as $s (ε_{e}) = α_{e} e^{γ} / {(1 + e^{γ})}^{2}$ , where $γ = (| ε_{e} - ε_{s e} | + ε_{e} - ε_{s e}) / 2$ , α_e is yield correction factor of E-FB, and ε_se is yield strain calibration point of E-FB. Similarly, define the yield adjustment function of A-FB as $s (ε_{a})$ , and α_a, ε_sa are yield correction factor and yield strain calibration point of A-FB, respectively. Thus, this constitutive model can describe the yielding effect of coated fabrics under large deformation.

The conventional constitutive relation is usually solved by numerical method, while the constitutive relation established in this paper can directly obtain the analytic solution and has high calculation efficiency, making it possible to find the material property values of the specimen in a wide range and with high precision. Therefore, the acquisition of key material property values of the specimen can be transformed into the solution of the classical multi-objective optimization problem.^29–31 For the seven dimensions of f_y, θ_a, θ_e, a, c_a, c_e, ε_y, the objective is to minimize their deviations between the experimental data and the calculated results from this relation to find the optimal k_a, k_e, k_pc, μ_p and α_a, ε_sa, α_e, ε_se. The acquisition algorithm for key material property values consists of the following three stages.

(1) According to the G-T relation, constrained by the geometric deviations of all the 6 dimensions (θ_a, θ_e, c_a, c_e, ε_x, ε_y) less than the threshold β_G, narrow intervals of μ_p, k_e/k_pc, ck_a/k_p.

(2) Based on (1), according to the F-T relation, constrained by the deviation of f_y less than the threshold β_F, obtain several groups of candidate vectors (μ_p, k_pc, k_a, k_e, α_a, ε_sa, α_e, ε_se).

(3) Set a certain range of floating interval for μ_p, k_pc, k_a, k_e, α_a, ε_sa, α_e, ε_se, target minimum weighted deviation of seven dimensions (θ_a, θ_e, c_a, c_e, ε_x, ε_y, f_y) to obtain the optimal combination of μ_p, k_pc, k_a, α_a, ε_sa, k_e, α_e, ε_se.

In this paper, the MCTA algorithm proposed in the literature²⁹ is used to calculate the key material property values of coated fabrics.

Experiment and verification

Experimental specimens and conditions

In order to verify the correctness and effectiveness of the constitutive model of coated fabrics under uniaxial tensile loads in arbitrary directions, 7 specimens are prepared according to GB/T 528-2009 standard. The coated fabrics (TPU840-060, thickness: ca. 0.60 mm) have plain-weave nylon 66 textile (thickeness: ca. 0.30 mm) as the substrate and polyurethane coating (thickness: ca. 0.15 mm) on both sides by thermoplastic method to realize gas sealing and substrate protection. The membrane material is uniform for all the 7 specimens only with different cutting directions. Taking the E-FB direction as the reference datum, the cutting angles of the specimens are 0 (weft direction), 15, 29, 44, 59, 73 and 88 (warp direction), marked as TS00, TS15, TS29, TS44, TS59, TS73, TS88, respectively. The effective measurement area is 30 mm × 230 mm (Figure 5(a)), tested on STM3000 tensile testing machine, as displayed in Figure 5(b). In order to simulate the tensile deformation during inflation, the tensile speed of 90 mm/min is selected. The coated fabrics, as polymer composite material, have obvious stress relaxation, creep and other complex characteristics, and the change rate gradually decreases with time. In order to predict the elongation of the inflatable component in the stable state, it is measured after the tension keeps constant for 10 min, and the complex effects such as stress relaxation and creep which tend to be basically stable are added onto the stiffness of FB. Considering the low inflation and deflation frequency of MES, the experiment adopts single loading, and the time interval between two tests of the same specimen is more than 120 min, so as to eliminate the residual effects of the previous tests as much as possible. Under the working condition, the tension of membrane material of MES is between 200 and 300 N/30 mm, and the gas tightness of the membrane may be reduced significantly when it exceeds 500 N/30 mm. The test range is set as 25–600 N to cover the underload and overload working range of MES. Each specimen was tested for 14 times under different tension conditions in the test range, and 7 physical and geometric quantities (f_y, θ_a, θ_e, a, c_a, c_e, ε_y) were recorded in each test.

Figure 5.

(a) Specimens with effective measurement area of 30 mm × 230 mm and (b) uniaxial tensile test.

In theory, the key material properties of a specimen can be obtained by using 3 groups of test data from TS00, TS88 and any other specimen. In practice, 4 groups of test data (TS00, TS29, TS73, TS88) are input into the model to improve the stability of prediction. The obtained key material parameters are μ_p=0.05, k_pc=15.02 (N), k_a=74.3 (N), k_e=115.6 (N), α_a=1, ε_sa=0.21, α_e=9, ε_se=0.12. Since ε_x↔f_y is adopted to describe the constitutive relation and ε_y→f_y cannot be obtained directly, PID (Proportional Integral Derivative) method is first used to get ε_x corresponding to ε_y after only a few iterations, and then ε_y→f_y is obtained. The subsequent comparative analysis of model output data and experimental data is based on the same ε_y.

Relative deviation between model output data (o_j,kⁱ) and experimental data (t_j,kⁱ) is $d_{j, k}^{i} = t_{j, k}^{i} / o_{j, k}^{i} - 1$ , where i∈{TS00, TS15, ⋯, TS88} is the index of specimens, $j \in {f_{y}, c_{e}, θ_{e}, c_{a}, θ_{a}, a}$ is the index of physical and geometric parameter values obtained by experiments, and $k \in {F_{y}} = {25,50, \dots, 600}$ is the index of tensions. Mean deviation of parameter (j) of a specimen (i) is $d_{j}^{i} = A v e r a g e (| d_{j, k}^{i} |)$ , and the total deviation of a specimen (i) is $d^{i} = A v e r a g e (d_{j}^{i})$ . Mean deviation of each parameter (j) of the model is $d_{j} = A v e r a g e (d_{j}^{i})$ , and the total deviation of the model is $d = A v e r a g e (d^{i}) = A v e r a g e (d_{j})$ .

Experimental data analysis

Macro analysis

If the black box is used to shield the mesoscopic structure of the coated fabrics, and the specimen is regarded as a homogeneous anisotropic body, the macroscopic mechanical characteristics can be manifested as deformation-tension and deformation-width relationships. The comparison between the prediction curves output by the model and the experimental data is shown in Figures 6(a) and (b).

Figure 6.

The comparison between the model curves and the experimental data for (a) ε_y↔f_y; (b) ε_y↔a; (c) ε_y↔c; (d) ε_y↔θ.

It can be seen fromFigures 6(a) and (b) that the change trend of model curves (ε_y↔f_y, ε_y↔a) is consistent with that of the experimental results, and the model data is in good agreement with the experimental data. The deviation analysis results ( $d_{f_{y}} = 0.052$ , d_a=0.009) indicate that the model can accurately predict the macro mechanical properties of the coated fabrics.

As shown in Figure 6(a), the value of $\max (ε_{y})$ of the 7 specimens is 0.53, which indicates that the coated fabrics are large deformation materials. When f_y value is equal, the stiffness characteristics (f_y/ε_y) of specimens have a great difference, presenting a saddle shape, i.e. high at both ends (TS00, TS88) and low in the middle (TS44), indicating that the coated fabrics has significant anisotropy. The difference of stiffness between TS44 and TS88 can reach more than 3 times. The ε_y↔f_y curve of specimens (TS15-TS73) shows obvious nonlinearity. The model curves of TS88 and TS73 show obvious yield effect when f_y is large (500 N), and the curve change matches well with the experimental data, indicating that the model has the ability to describe this yield effect. In Figure 6(b), a decreases with the increase of ε_y, but ε_x/ε_y is not a constant, which demonstrates that the Poisson’s ratio of the coated fabrics also has anisotropic characteristics at macro level.

Meso analysis

If the black box shielding is removed, the mechanical behaviours of the coated fabrics can be presented as the change of c and θ at meso level. The comparisons between the model curves and the experimental data for ε_y↔c, ε_y↔θ are shown in Figures 6(c) and (d), respectively.

Since both ends of the FBs of TS00 and TS88 are clamped by the fixture, the morphological change of FBs is consistent with ε_y, which belongs to constraint deformation at both ends. The deformation state of the 2 specimens is simple (only length changes), so the detailed analysis is not required. For the other 5 specimens, only a small amount of FBs (located at the end of the measurement area) is restrained by the fixture at one end and the other end is free, and most of FBs with two free ends have complex mesoscopic morphological changes (length changes and deflection occurs). The change law of the morphology of FBs can be revealed by our constitutive model. It is observed from Figures 6(c) and (d) that the change trend of model curves (ε_y↔c, ε_y↔θ) is highly consistent with that of the experimental results. The deviation analysis results ( $d_{c_{a}} = 0.014$ , $d_{θ_{a}} = 0.027$ , $d_{c_{e}} = 0.025$ , $d_{θ_{e}} = 0.030$ ) indicate that the model can accurately predict the change of mesoscopic geometry of the coated fabrics.

Additionally, as shown in Figures 6(c) and (d), in order to adapt to the increase of ε_y, for TS15 and TS29, the morphological change of E-FBs is due to mainly c_e supplemented by θ_e, and the change of A-FBs is due to mainly θ_a supplemented by c_a, which is opposite for TS73 and TS59, while the influence of θ and c is balanced for TS44. This indicates that D_Y has a great impact on the mesoscopic geometry of the coated fabrics. The above phenomena can be explained geometrically as follows: ∆θ→ε_x^θ, ε_y^θ, ∆c→ε_x^c, ε_y^c, ε_x=ε_x^θ+ε_x^c, ε_y=ε_y^θ+ε_y^c, ε_x^θ=∆θcosθ, ε_y^θ=−∆θsinθ. When θ>π/4→ε_y^θ>ε_x^θ, for any ∆a, there is ∆θ, let $ε_{x} / ε_{y} = - μ_{p}$ and ∆c=0, ε_x=∆a/a=ε_x^θ, ε_y=ε_y^θ, only Poisson effect can produce ∆a. As shown in Figure 6(c), c_a of TS15 and c_e of TS73 keep almost unchanged in the tension process. When θ<π/4, Poisson effect cannot adapt to ∆a alone, and additional compressive deformation ε_s (from formula (1)) is required, which needs FB to be elongated, i.e. ∆c>0. Since ε_x^c induces pressure in the X direction and ε_y^c induces tension in the Y direction, as Figure 6(c) illustrates, c_e of TS15 and c_a of TS73 change dramatically in the tension process.

Analysis of model deviation and solution efficiency

Except ε_y, the deviations of the other 6 parameters (f_y, θ_a, θ_e, a, c_a, c_e) are shown as Table 1.

Table 1.

Deviation analysis of the constitutive model.

F _Y	θ _E	f _y	c _a	θ _a	c _e	θ _e	a	θ _E	f _y	c _a	θ _a	c _e	θ _e	a	θ _E	f _y
25	15	0.052	0.033	−0.012	0.019	−0.009	0.008	29	−0.027	0.006	−0.010	0.022	−0.026	−0.005	00	0.003
50		−0.108	0.016	−0.015	0.005	−0.012	0.004		0.022	0.003	−0.010	0.027	−0.045	0.002		−0.331
75		0.021	0.016	0.000	0.030	0.006	0.002		−0.003	0.000	−0.011	0.031	−0.045	−0.008		−0.140
100		0.021	0.015	−0.001	0.036	−0.007	−0.003		−0.004	0.011	−0.010	0.036	−0.051	−0.003		−0.109
150		−0.007	0.013	0.008	0.037	0.006	0.006		−0.021	0.005	−0.020	0.045	−0.025	0.004		−0.140
200		0.070	−0.004	0.009	0.030	−0.001	0.005		0.077	0.001	−0.003	0.039	−0.031	−0.010		0.003
250		0.041	−0.006	0.011	0.024	0.004	0.011		0.043	−0.005	0.011	0.018	−0.041	−0.012		−0.045
300		0.041	0.008	−0.015	0.027	−0.042	0.013		0.050	0.005	−0.014	0.023	−0.018	−0.003		−0.037
350		0.034	0.006	−0.009	0.015	−0.003	−0.006		0.044	0.028	−0.006	0.020	−0.021	0.006		−0.032
400		0.072	0.005	−0.006	0.024	−0.023	−0.012		0.051	0.023	−0.007	0.018	−0.032	−0.012		0.003
450		0.055	0.003	−0.043	0.018	−0.032	0.007		0.050	0.033	−0.010	0.015	−0.018	−0.009		0.003
500		0.059	0.017	−0.058	0.023	−0.050	0.002		0.064	0.028	−0.020	0.014	−0.042	−0.011		0.029
550		0.022	0.014	−0.068	0.034	−0.059	0.003		0.037	0.037	−0.036	0.034	−0.023	−0.006		−0.019
600		0.023	0.029	−0.088	0.024	−0.082	−0.003		0.010	0.044	−0.041	0.039	−0.039	−0.001		−0.037
d _j ⁱ		0.045	0.013	0.025	0.025	0.024	0.006		0.036	0.016	0.015	0.027	0.033	0.007		0.066
25	44	−0.207	0.009	−0.030	0.018	−0.048	−0.017	59	−0.034	−0.013	−0.051	−0.042	−0.021	−0.025	88	0.222
50		−0.097	0.004	−0.033	0.031	−0.054	−0.005		0.011	−0.019	−0.049	−0.033	−0.040	−0.016		0.222
75		−0.096	−0.002	−0.012	0.021	−0.047	−0.010		−0.020	−0.008	−0.012	−0.039	−0.033	−0.006		0.222
100		−0.096	0.005	−0.046	0.034	−0.060	−0.018		−0.026	−0.006	−0.047	−0.031	−0.032	−0.017		−0.022
150		−0.039	0.020	−0.039	0.019	−0.070	−0.014		0.016	−0.008	−0.068	−0.027	−0.044	−0.015		0.048
200		−0.044	−0.002	−0.004	0.024	−0.019	−0.008		0.066	−0.009	−0.044	−0.035	−0.016	−0.001		0.087
250		0.074	0.006	−0.022	0.038	−0.040	−0.010		0.045	−0.012	−0.032	−0.045	−0.028	−0.022		0.019
300		0.033	0.019	−0.025	0.031	−0.027	−0.010		0.035	−0.007	−0.026	−0.028	−0.013	−0.008		−0.022
350		0.052	0.024	−0.017	0.019	−0.019	−0.026		0.038	0.000	−0.050	−0.024	−0.019	−0.020		0.007
400		0.080	0.029	−0.033	0.028	−0.038	−0.024		0.054	0.007	−0.061	−0.032	−0.022	−0.016		0.029
450		0.076	0.033	−0.010	0.035	−0.013	0.006		0.060	0.014	−0.025	−0.027	−0.026	−0.013		0.000
500		0.066	0.024	−0.023	0.031	−0.028	−0.013		0.037	0.018	−0.056	−0.037	−0.024	−0.004		−0.022
550		0.052	0.038	−0.020	0.036	−0.004	−0.008		0.014	0.022	−0.038	−0.023	−0.009	0.004		−0.032
600		0.036	0.052	−0.039	0.040	−0.048	−0.030		0.012	0.034	−0.056	−0.007	−0.002	0.006		−0.030
d _j ⁱ		0.075	0.019	0.025	0.029	0.037	0.014		0.033	0.013	0.044	0.031	0.024	0.012		0.070
25	73	0.029	−0.009	−0.003	−0.014	−0.035	−0.016
50		0.089	−0.004	−0.010	−0.016	−0.023	−0.002
75		0.076	0.000	−0.015	−0.018	−0.020	−0.004
100		−0.022	0.002	0.015	−0.005	−0.025	0.000
150		0.029	0.011	−0.006	−0.009	−0.009	0.008
200		0.077	0.011	−0.003	−0.013	−0.001	−0.005
250		0.050	−0.002	−0.025	−0.017	0.068	0.005
300		0.016	0.003	−0.015	−0.022	0.064	−0.004
350		0.041	0.013	−0.020	−0.011	−0.021	−0.003
400		0.055	0.007	−0.026	−0.015	−0.031	−0.003
450		0.033	0.013	−0.025	−0.020	−0.014	0.001
500		0.011	0.013	−0.064	−0.025	−0.042	0.005
550		−0.002	0.018	−0.067	−0.001	−0.051	0.009
600		0.013	0.020	−0.080	−0.005	−0.047	−0.015
d _j ⁱ		0.039	0.009	0.027	0.014	0.032	0.006

The total deviation of the model (d =0.026) can be obtained from Table 1, indicating that the model data match well with the experimental results on the whole. The deviation of each specimen is d^TS00 = 0.066, d^TS15 = 0.023, d^TS29 = 0.022, d^TS44 =0.033, d^TS59 = 0.026, d^TS73 = 0.021, d^TS88 = 0.070. Since the data of TS29 and TS73 are input as the model parameter, their matching degrees are slightly better than those of TS15, TS44 and TS59. The small deformation range of d^TS00 and d^TS88 due to no change of their weaving angles, results in their higher sensitivity of mechanical behaviours than other specimens, therefore, a greater relative deviation will be produced under the same measurement noise.

It can be seen from the column chart in Table 1 that the maximum value of d_j is $d_{f_{y}}$ = 0.052. The reason is that the weaving undulations of A-FB and E-FB are different in plain weaving mode (Figure 1). The parameter to describe this undulation is not introduced into the model for the sake of simplification, resulting in a large tension deviation in the undulation release stage, e.g. d^TS00=0.18 within the tension (F_Y) range of 50–150 N. Similarly, the complex effects such as stress relaxation, creep and plasticity are added onto the stiffness of FB to simplify the model, leading to a large relative deviation in the small deformation stage, e.g. d^TS88=0.22 within the tension (F_Y) range of 25–75 N. However, it is worthwhile to ignore or blur the above factors and exchange the loss of relatively small deviation for model simplification. In addition, the random defects in the manufacturing process of the coated fabrics, the restriction of the end clamping on the free deformation of the specimen, and the noise and error in the measurement process all have a certain impact on the model matching degree with the experimental results.

The deviation analysis shows that the model proposed in this paper can accurately describe the mechanical behaviours of coated fabrics, and the prediction accuracy can meet the needs of large inflatable structure in the engineering application. The model can not only describe the macroscopic mechanical properties of the coated fabrics, but also explain the geometric and mechanical mechanism at meso level, which can provide inspiration for material design. Moreover, the model has high calculation efficiency, and the single calculation time for ε_x↔f_y is 2.4×10⁻⁵ seconds. The calculation efficiency of ε_y↔f_y depends on PID adjustment accuracy of ε_x↔ε_y. When the accuracy is less than 1×10⁻⁴, the single calculation time for ε_y↔f_y is ca. 5×10⁻⁴ seconds. The high calculation efficiency of the model will facilitate large-scale engineering analysis.

Conclusion

A novel constitutive model of coated plain-weave fabrics under uniaxial tensile loads is established in this paper. This model can not only describe the macroscopic mechanical properties of the coated fabrics, but also the mesoscopic changes of FBs and weaving angles. Only four key material parameters with clear physical interpretation are needed for this model, and the constitutive relation can be mapped to a quartic equation with analytic solution, which has extremely high efficiency. The experimental data show that the model can well predict the mechanical behaviours and internal geometric changes of the coated fabrics. If the characteristics of fibers (FB composition) are considered, the model will have the ability of microscopic description. If viscoelasticity, relaxation, creep and other factors of FB are considered, the model can more accurately describe the complex mechanical properties of coated fabrics. The geometric and physical interpretation of the constitutive relation of this model is intuitive and clear, which is easier for engineers and technicians to understand and apply than the complex mathematical description methods such as tensor and partial differential equation. Due to the extremely fast solution speed of the constitutive relation, it can support large-scale engineering structure performance analysis and optimization, and can solve the deformation simulation problem of large inflatable structures, which has good engineering application potential.

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 Key R&D Program of China (grant number 2022YFC3006004); the Ministry of Industry and Information Technology High tech Ship Project (grant number MC-201902-C01); and the National Natural Science Foundation of China (grant number 51305052).

ORCID iD

Changle Sun

Appendix

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