Nonlinear Out-of-Plane Mechanical Behaviours of Rectangular Woven Composite Flexible Skin Under Uniform Normal Pressure

Abstract

This paper seeks to investigate nonlinear out-of-plane mechanical behaviours of woven composite flexible skin (WCFS) through experimental and theoretical methods. Firstly, quasi-static experiments are carried out on rectangular WCFSs with different aspect ratios under uniform normal pressure to measure out-of-plane deformation and failure loads. Afterwards, a new three-parameter geometric model is presented to describe 3D geometry of pressurized WCFS, and thus, a nonlinear mechanical model is deduced for depicting the relationship between pressure and out-of-plane displacement. The application of the aforementioned models for experimental results shows that the new models have adequately and logically depicted deformation geometry and nonlinear mechanical characteristics.

Appendices

Appendix A: Feature Point Coordinates of Planes $\pi_{1m}$ and $\pi_{2n}$

From Fig. 4b, the X-, Y-, and Z-coordinates of points 1 and 2 are, respectively,

$$ x_{1m} = \frac{m - 1}{{n_{1} - 1}}a $$

(A1)

$$ y_{1m} = 0 $$

(A2)

$$ z_{1m} = 0 $$

(A3)

and

$$ x_{2m} = \frac{m - 1}{{n_{1} - 1}}a $$

(A4)

$$ y_{2m} = b $$

(A5)

$$ z_{2m} = 0 $$

(A6)

With the aid of Eq. (9), the X-, Y-, and Z-coordinates of point 3, respectively, satisfy

$$ \int_{0}^{{x_{3m} }} {\sqrt {1 + \left[ {\delta \left( {\ln p} \right)\left( {1 - q^{b} } \right)p^{x} } \right]^{2} } } {\text{d}} x = \left( {m - 1} \right)\Delta l_{2} $$

(A7)

$$ y_{3m} = b $$

(A8)

$$ z_{3m} = \delta \left( {1 - p^{{x_{3m} }} } \right)\left( {1 - q^{b} } \right) $$

(A9)

Likewise, from Fig. 4c, the X-, Y-, and Z-coordinates of points 4 and 5 are, respectively,

$$ x_{4n} = 0 $$

(A10)

$$ y_{4n} = \frac{n - 1}{{n_{2} - 1}}b $$

(A11)

$$ z_{4n} = 0 $$

(A12)

and

$$ x_{5n} = a $$

(A13)

$$ y_{5n} = \frac{n - 1}{{n_{2} - 1}}b $$

(A14)

$$ z_{5n} = 0 $$

(A15)

From Eq. (8), one has

$$ x_{6n} = a $$

(A16)

$$ \int_{0}^{{y_{6n} }} {\sqrt {1 + \left[ {\delta \left( {\ln q} \right)\left( {1 - p^{a} } \right)q^{y} } \right]^{2} } } {\text{d}} y = \left( {n - 1} \right)\Delta l_{1} $$

(A17)

$$ z_{6n} = \delta \left( {1 - p^{a} } \right)\left( {1 - q^{{y_{6n} }} } \right) $$

(A18)

Appendix B: Coordinates of Intersection Points

On the basis of Eqs. (16) and (17), one has

$$ \delta \left( {1 - q^{y} } \right)\left[ {1 - p^{{\frac{{k_{2n} }}{{k_{1m} }}\left( {y - \frac{n - 1}{{n_{2} - 1}}b} \right) + \frac{m - 1}{{n_{1} - 1}}a}} } \right] = k_{2n} \left( {y - \frac{n - 1}{{n_{2} - 1}}b} \right) $$

(B1)

Letting

$$ F\left( y \right) = \delta \left( {1 - q^{y} } \right)\left[ {1 - p^{{\frac{{k_{2n} }}{{k_{1m} }}\left( {y - \frac{n - 1}{{n_{2} - 1}}b} \right) + \frac{m - 1}{{n_{1} - 1}}a}} } \right] - k_{2n} \left( {y - \frac{n - 1}{{n_{2} - 1}}b} \right) $$

(B2)

Equation (B1) turns into

$$ F\left( y \right) = 0 $$

(B3)

It is clear that coordinate $y_{m,n}$ of the intersection point could be solved by combining Eqs. (B2) and (B3).

Subsequently, coordinates $z_{m,n}$ and $x_{m,n}$ of the intersection point are deduced as

$$ z_{m,n} = k_{2n} \left( {y_{m,n} - \frac{n - 1}{{n_{2} - 1}}b} \right) $$

(B4)

$$ x_{m,n} = \frac{{z_{m,n} }}{{k_{1m} }} + \frac{m - 1}{{n_{1} - 1}}a $$

(B5)

Appendix C: Slopes of Threads

In light of Eqs. (25) and (26), the thread slopes at boundary line OB are obtained as

$$ \left( {\frac{\partial z}{{\partial x}}} \right)_{1m} = k_{1m} \, \left( {x = \frac{m - 1}{{n_{1} - 1}}a,y = 0,z = 0} \right) $$

(C1)

$$ \left( {\frac{\partial z}{{\partial y}}} \right)_{1m} = - \delta \left( {\ln q} \right)\left[ {1 - p^{{\frac{m - 1}{{n_{1} - 1}}a}} } \right] \, \left( {x = \frac{m - 1}{{n_{1} - 1}}a,y = 0,z = 0} \right) $$

(C2)

Likewise, the thread slopes at boundary line OC can be solved according to Eqs. (28) and (29), namely

$$ \left( {\frac{\partial z}{{\partial x}}} \right)_{2n} = - \delta \left( {\ln p} \right)\left[ {1 - q^{{\frac{n - 1}{{n_{2} - 1}}b}} } \right] \, \left( {x = 0,y = \frac{n - 1}{{n_{2} - 1}}b,z = 0} \right) $$

(C3)

$$ \left( {\frac{\partial z}{{\partial y}}} \right)_{2n} = k_{2n} \, \left( {x = 0,y = \frac{n - 1}{{n_{2} - 1}}b,z = 0} \right) $$

(C4)