2.1 Elastic parameters based on least-squares method

In the MSAJ standard, the mechanical performance parameters are obtained based on the least-squares method. As the thickness of plain-weave materials is minimal, it is generally considered that their mechanical properties are orthotropic elastic and thus that the corresponding stress–strain relationship can be approximated using an orthotropic plane model:

(1) \begin{equation}\left\{ {\begin{array}{*{20}{c}}{{\varepsilon _x}}\\[3pt]{{\varepsilon _y}}\end{array}} \right\} = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{{E_x}}}}&{ - \frac{{{\nu _{xy}}}}{{{E_y}}}}\\[3pt]{ - \frac{{{\nu _{{\rm{yx}}}}}}{{{E_x}}}}&{\frac{1}{{{E_y}}}}\end{array}} \right]\left\{ {\begin{array}{*{20}{c}}{{\sigma _x}}\\[3pt]{{\sigma _y}}\end{array}} \right\}\end{equation}

where $\varepsilon $ is the strain, $\sigma $ is the stress, E is the elastic modulus, $\upsilon $ is the Poisson’s ratio, and the subscripts x and y denote the warp and weft directions, respectively. The above formula can be simplified to

(2) \begin{equation}{E_{{\rm{11}}}} = \frac{1}{{{E_x}}},\ {E_{{\rm{22}}}} = \frac{1}{{{E_y}}},\,{E_{{\rm{12}}}} = - \frac{{{\upsilon _{xy}}}}{{{E_y}}},\ {E_{{\rm{21}}}} = - \frac{{{\upsilon _{yx}}}}{{{E_x}}}\end{equation}

For orthotropic elastic materials, the Poisson’s ratio and elastic modulus are constrained by the Maxwell relationship:

(3) \begin{equation}\frac{{{\upsilon _{xy}}}}{{{E_y}}} = \frac{{{\upsilon _{yx}}}}{{{E_x}}}\end{equation}

From Equations (2) and (3), it can be determined that ${E_{{\rm{21}}}} = {E_{{\rm{12}}}}$ . Under different stress ratio loads, the sum of the squares of the strain difference can be obtained as

(4) \begin{equation}{\rm{S = }}\sum {\left[ {{{({E_{{\rm{11}}}}{\sigma _{xi}} + {E_{12}}{\sigma _{yi}} - {\varepsilon _{xi}})}^2}} \right]} + \sum {\left[ {{{({E_{{\rm{12}}}}{\sigma _{xi}} + {E_{22}}{\sigma _{yi}} - {\varepsilon _{yi}})}^2}} \right]} \end{equation}

where i represents the experimental data obtained at the i-th stress ratio. Using the principle of the minimum sum of squares of strain residuals, the derivative of the unknown variables is equal to 0:

(5) \begin{equation}\frac{{\partial {\rm{S}}}}{{\partial {E_{{\rm{11}}}}}} = \frac{{\partial {\rm{S}}}}{{\partial {E_{{\rm{12}}}}}} = \frac{{\partial {\rm{S}}}}{{\partial {E_{22}}}} = 0\end{equation}

Combining Equations (4) and (5) yields

(6) \begin{equation}\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\sum {\sigma _{xi}^2} }&{\sum {{\sigma _{xi}}{\sigma _{yi}}} }&0\\[3pt]{\sum {{\sigma _{xi}}{\sigma _{yi}}} }&{\sum {\!\left( {\sigma _{xi}^2 + \sigma _{yi}^2} \right)} }&{\sum {{\sigma _{xi}}{\sigma _{yi}}} }\\[4pt]0&{\sum {{\sigma _{xi}}{\sigma _{yi}}} }&{\sum {\sigma _{yi}^2} }\end{array}} \right]\left\{ {\begin{array}{*{20}{c}}{{E_{11}}}\\[3pt]{{E_{12}}}\\[3pt]{{E_{22}}}\end{array}} \right\} = \left\{ {\begin{array}{c}{\sum {{\sigma _{xi}}{\varepsilon _{xi}}} }\\[3pt]{\sum {({\sigma _{yi}}{\varepsilon _{xi}} + {\sigma _{xi}}{\varepsilon _{yi}}} )}\\[3pt]{\sum {{\sigma _{yi}}{\varepsilon _{yi}}} }\end{array}} \right\}\end{equation}

From Equations (2) and (6), the Poisson’s ratio ( ${\upsilon _{yx}},{\upsilon _{xy}}$ ) and elastic modulus ( ${E_x},{E_y}$ ) can be obtained.

According to this theoretical background, the method for calculating the elastic modulus and Poisson’s ratio can now be established in principle, by applying the following fitting methods. There are some limitations when using the least-squares method in response surface modelling, as the algorithm cannot deal with some nonlinear effects. When the relationship between the independent and dependent variables of the prototype to be modelled is not simple linear or quasi-linear, the prediction performance will be significantly reduced.

2.2 RBF model in Isight

The RBF model in Isight follows the method of Hardy^{(Reference Hardy26)} described by Kansa^{(Reference Kansa27)}. Let ${x_1},...,{x_N} \in \Omega \subset {\Re ^N}$ be a given set of nodes. Let ${g_i}(x) = g(\| {x - {x_j}} \|) \in \Re $ , $j = 1,2,...,N$ be a set of any radial basis functions. Here, $\| {x - {x_j}} \|$ is the Euclidian distance given by ${\left( {x - {x_j}} \right)^T}\left( {x - {x_j}} \right)$ . We are given the interpolation values ${y_1},...,{y_N} \subset \Re $ at data locations ${x_1},...,{x_N} \in \Omega \subset {\Re ^N}$ .

By solving the system of N + 1 linear equations with N + 1 unknown expansion coefficients ${\alpha _j}$ :

(7) \begin{equation}\sum\limits_{j = 1}^N {{\alpha _j}} {g_j}({x_i}) + {\alpha _{N + 1}} = {y_i},\ \ i = 1,...,N\end{equation}

(8) \begin{equation}\ \sum\limits_{j = 1}^N {{\alpha _j}} = 0\end{equation}

the RBF interpolant is obtained as

(9) \begin{equation}F(x) = \sum\limits_{j = 1}^N {{\alpha _j}} {g_j}(x) + {\alpha _{N + 1}}\end{equation}

Introducing the notation: $p = \left[ {\begin{array}{*{20}{c}}1\\ \vdots \\1\end{array}} \right] \in {\Re ^N}$ , $G = \left( {\begin{array}{*{20}{c}}{{g_1}({x_1})}& \cdots &{{g_N}({x_1})}\\ \vdots & \ddots & \vdots \\{{g_N}({x_1})}& \cdots &{{g_N}({x_N})}\end{array}} \right) \in {\Re ^{N \times N}}$ , $H = \left[ {\begin{array}{*{20}{c}}G&p\\{{p^T}}&0\end{array}} \right] \in {\Re ^{(N + 1) \times (N + 1)}}$ , $\alpha = {({\alpha _1},...,{\alpha _{N + 1}})^T}$ and $y = {({y_1},...,{y_N},0)^T} \in {\Re ^{N \times N}}$ . The system of Equations (7) and (8) can be rewritten in matrix form as

(10) \begin{equation}H\alpha = y\end{equation}

The interpolation expansion coefficients are then given by

(11) \begin{equation}\alpha = {H^{ - 1}}y\end{equation}

One can then easily find the derivatives of the interpolant at any node ${x_i}$ , $i = 1,2,...,N$ .

Isight uses a function with variable power splines and a radial basis that can be tuned to approximate a large number of other functions:

(12) \begin{equation}g(x) = {\| {x - {x_j}} \|^c}\end{equation}

where c is a shape function variable between 0.2 and 3.

3.1 Experimental materials and preparation

The UN-5100 used in this study is a type of plain-weave material previously applied in stratospheric airships. The properties of the warp and weft yarns are the same, and it exhibits low air permeability, strong environmental corrosion resistance, high strength and low density. The fibre used in UN-5100 is Vectran, with polyethene terephthalate (PET) for the thermal bonding and helium barrier layers. The material properties of UN-5100 are presented in Table 1.

Table 1

Parameters of UN-5100 material

According to the manufacturer, the uniaxial tensile strength of UN-5100 is 568/540cm/N (at room temperature) in the warp and weft direction, respectively. Its micromorphology and material properties are shown in Fig. 1.

Figure 1.

The micromorphology of UN-5100 material.

Biaxial testing was carried out using a self-developed biaxial tensile testing machine (SJTU-I), shown in Fig. 2. The strain measurement range of the test machine is 0–20%, the standard tensile rate of the fixture is 2–4mm/min, and the real-time control is 1–5μs. On the power device, a precise servohydraulic cylinder is used, with precise flow control being realised using a proportional valve and overflow valve. Moreover, using force sensor closed-loop feedback and a proportional–integral–differential (PID) controller, a load with any proportion in two directions can be obtained. The biaxial specimen parameters according to standard ISO1421-1998 are shown in Fig. 3. PE rods were fixed on the extension arms to apply longitudinal and latitudinal stress to the sample. The stress through the specimen was kept as moderate as possible under the action of the slits. Digital image correlation (DIC) technology is used instead of extensometers to obtain higher-precision strains. DIC is a non-contact full-field measurement method, which offers the advantages of a wide strain range, high precision, simple equipment, low requirement regarding the test environment and smooth operation.

Figure 2.

SJTU-I biaxial tensile testing machine.

Figure 3.

Biaxial specimen (units: millimetres)

3.2 Test procedure

The experiment was carried out at humidity of 60 $ \pm $ 4.0% and temperature of 23 $ \pm $ 2.0°C. To remove residual strains and avoid high initial levels of creep, pre-stress of 200N was applied in the warp and weft direction, respectively, as shown in Fig. 4. This method also offers the advantage of preventing the specimen from being slack in its initial state. The application of this initial strain means that the tested stress–strain curves will not pass through the origin, which is compensated for in the data analysis.

Figure 4.

The initial state of the specimen.

The stress loading was carried out according to the following test protocol, as shown in Fig. 5.

In envelope materials under typical stress states, the ratio between the warp and weft stress ranges from 1:2 to 2:1, but may reach 4:1 in some areas. Therefore, seven loading ratios of 1:1, 1:2, 2:1, 1:3, 3:1, 1:4 and 4:1 were selected. In Fig. 5, ‘Pre 1:1’ indicates the pre-loading with a stress ratio of 1:1. In contrast to the pre-stress, the purpose of this pre-loading is to avoid the influence of residual deformation before and after loading at different stress ratios.

Figure 5.

The test protocol.

According to the results of uniaxial testing, 20–25% of the uniaxial strength is generally accepted as the maximum value in biaxial tensile loading tests. According to the code for the structural design of airships (FAA-P-8110-2⁽²⁸⁾), the safety factor for strength is ≥4. Considering the difference between the warp and weft strength values, the lowest value of 1/4 is broadly employed. Table 2 presents the ratio of the maximum test loading value to the ultimate strength in the warp and weft directions.

Table 2

Loading history of biaxial tensile test