Computation of the effective nonlinear mechanical response of lattice materials considering geometrical nonlinearities

Abstract

The asymptotic homogenization technique is presently developed in the framework of geometrical nonlinearities to derive the large strains effective elastic response of network materials viewed as repetitive beam networks. This works extends the small strains homogenization method developed with special emphasis on textile structures in Goda et al. (J Mech Phys Solids 61(12):2537–2565, 2013). A systematic methodology is established, allowing the prediction of the overall mechanical properties of these structures in the nonlinear regime, reflecting the influence of the geometrical and mechanical micro-parameters of the network structure on the overall response of the chosen equivalent continuum. Internal scale effects of the initially discrete structure are captured by the consideration of a micropolar effective continuum model. Applications to the large strain response of 3D hexagonal lattices and dry textiles exemplify the powerfulness of the proposed method. The effective mechanical responses obtained for different loadings are validated by FE simulations performed over a representative unit cell.

Appendices

Appendix 1: Small strains homogenization: expressions of forces, moments and virtual translation and rotation velocities

The first order normal and transverse forces and the second order moment about $\text {x}^{\prime }$, $\text {y}^{\prime }$, and $\text {z}^{\prime }$ at the beam extremities can be successively expressed versus the kinematical nodal variables as

$$\begin{aligned} F_x^{\varepsilon {b}}= & {} \frac{E_s^b A^{\varepsilon {b}}}{l^{\varepsilon {b}}}\left( {\mathbf{e}_x \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) } \right) \end{aligned}$$

(29)

$$\begin{aligned} F_y^{\varepsilon {b}}= & {} \frac{12E_s^b I_z^{\varepsilon {b}} }{\left( {l^{\varepsilon {b}}} \right) ^{3}}\left( \mathbf{e}_y \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) \right. \nonumber \\&-\frac{l^{\varepsilon {b}}}{2}\Bigg ( \mathbf{e}_z \cdot \Bigg ( {{\varvec{\upvarphi }} }_0^{O_{R(b)} } +{{\varvec{\upvarphi }} }_0^{E_{R(b)} } +\varepsilon \Bigg ( {{\varvec{\upvarphi }} }_1^{O_{R(b)} } +{{\varvec{\upvarphi }} }_1^{E_{R\left( b \right) } }\nonumber \\&\left. \left. \left. +\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib} \Bigg ) \right) \right) \right) \end{aligned}$$

(30)

$$\begin{aligned} F_z^{\varepsilon {b}}= & {} \frac{12E_s^b I_y^{\varepsilon {b}} }{\left( {l^{\varepsilon {b}}} \right) ^{3}}\Bigg ( \mathbf{e}_z \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) \nonumber \\&+\frac{l^{\varepsilon {b}}}{2}\Bigg ( \mathbf{e}_y \cdot \Bigg ( {{\varvec{\upvarphi }} }_0^{O_{R(b)} } +{{\varvec{\upvarphi }} }_0^{E_{R(b)} } +\varepsilon \Bigg ( {{\varvec{\upvarphi }} }_1^{O_{R(b)} } +{{\varvec{\upvarphi }} }_1^{E_{R\left( b \right) } }\nonumber \\&\left. \left. \left. \left. +\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib} \right) \right) \right) \right) \end{aligned}$$

(31)

$$\begin{aligned} M_x^{O(b)\varepsilon }= & {} \frac{G_s^b J^{\varepsilon {b}}}{l^{\varepsilon {b}}}\varepsilon \left( {\mathbf{e}_x \cdot \left( {{{\varvec{\upvarphi }} }_1^{O_{R(b)} } -\left( {\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib}+{{\varvec{\upvarphi }} }_1^{E_{R(b)} } } \right) } \right) } \right) \nonumber \\ M_x^{E(b)\varepsilon }= & {} \frac{G_s^b J^{\varepsilon {b}}}{l^{\varepsilon {b}}}\varepsilon \left( {\mathbf{e}_x \cdot \left( {{{\varvec{\upvarphi }} }_1^{E_{R(b)} } -{{\varvec{\upvarphi }} }_1^{O_{R(b)} } +\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib}} \right) } \right) \nonumber \\ \end{aligned}$$

(32)

$$\begin{aligned} M_y^{O(b)\varepsilon }= & {} \frac{6E_s^b I_y^{\varepsilon {b}} }{\left( {l^{\varepsilon {b}}} \right) ^{2}}\left( {\mathbf{e}_z \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) } \right) \nonumber \\&\,+\frac{E_s^b I_y^{\varepsilon {b}} }{l^{\varepsilon {b}}}\Bigg ( \mathbf{e}_y \cdot \Bigg ( 4{{\varvec{\upvarphi }} }_0^{O_{R(b)} } +2{{\varvec{\upvarphi }} }_0^{E_{R(b)} } +\varepsilon \Bigg ( 4{{\varvec{\upvarphi }} }_1^{O_{R(b)} }\nonumber \\&\left. \left. \left. \,+2{{\varvec{\upvarphi }} }_1^{E_{R\left( b \right) }} +2\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib} \right) \right) \right) \nonumber \\ M_y^{E(b)\varepsilon }= & {} \frac{6E_s^b I_y^{\varepsilon {b}} }{\left( {l^{\varepsilon {b}}} \right) ^{2}}\left( {\mathbf{e}_z \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) } \right) \nonumber \\&\,+\frac{E_s^b I_y^{\varepsilon {b}} }{l^{\varepsilon {b}}}\Bigg ( \mathbf{e}_y \cdot \Bigg ( 2{{\varvec{\upvarphi }} }_0^{O_{R(b)} } +4{{\varvec{\upvarphi }} }_0^{E_{R(b)} } +\varepsilon \Bigg ( 2{{\varvec{\upvarphi }} }_1^{O_{R(b)} }\nonumber \\&\left. \left. \left. +\,4{{\varvec{\upvarphi }} }_1^{E_{R\left( b \right) } } +4\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib} \right) \right) \right) \end{aligned}$$

(33)

$$\begin{aligned} M_z^{O(b)\varepsilon }= & {} \frac{6E_s^b I_z^{\varepsilon {b}} }{\left( {l^{\varepsilon {b}}} \right) ^{2}}\left( {-\mathbf{e}_y \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) } \right) \nonumber \\&+\frac{E_s^b I_z^{\varepsilon {b}} }{l^{\varepsilon {b}}}\Bigg ( \mathbf{e}_z \cdot \Bigg ( 4{{\varvec{\upvarphi }} }_0^{O_{R(b)} } +2{{\varvec{\upvarphi }} }_0^{E_{R(b)} } +\varepsilon \Bigg ( 4{{\varvec{\upvarphi }} }_1^{O_{R(b)} }\nonumber \\&\left. \left. \left. +2{{\varvec{\upvarphi }} }_1^{E_{R\left( b \right) } } +2\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib} \right) \right) \right) \nonumber \\ M_z^{E(b)\varepsilon }= & {} \frac{6E_s^b I_z^{\varepsilon {b}} }{\left( {l^{\varepsilon {b}}} \right) ^{2}}\left( {-\mathbf{e}_y \cdot \left( {\varepsilon \Delta \mathbf{U}_{1}^\mathrm{b} +\varepsilon ^{{2}}\Delta \mathbf{U}_{2}^\mathrm{b} } \right) } \right) \nonumber \\&+\frac{E_s^b I_z^{\varepsilon {b}} }{l^{\varepsilon {b}}}\Bigg ( \mathbf{e}_z \cdot \Bigg ( 2{{\varvec{\upvarphi }} }_0^{O_{R(b)} } +4{{\varvec{\upvarphi }} }_0^{E_{R(b)} } +\varepsilon \nonumber \\&\times \left. \left. \left( 2{{\varvec{\upvarphi }} }_1^{O_{R(b)} } +4{{\varvec{\upvarphi }}}_1^{E_{R\left( b \right) } }+4\frac{\partial {{\varvec{\upvarphi }} }_0 }{\partial \beta ^{i}}\delta ^\textit{ib} \right) \right) \right) \nonumber \\ \end{aligned}$$

(34)

where $E_s^b$ and $G_s^b$ the tensile and shear modulus of the bulk material.

The asymptotic development of the virtual velocity and rotation rate are next expressed. For any virtual velocity field $\mathbf{v}^{\upvarepsilon }(\upbeta )$, a Taylor series expansion leads to

$$\begin{aligned} \mathbf{v}^{\upvarepsilon }\left( {\hbox {O}\left( \hbox {b} \right) } \right) -\mathbf{v}^{\upvarepsilon }\left( {\hbox {E}\left( \hbox {b} \right) } \right) \approx \upvarepsilon \frac{\partial \mathbf{v}\left( {\upbeta ^{\upvarepsilon }} \right) }{\partial \upbeta ^\mathrm{i}}{{\updelta }^{\mathrm{ib}}} \end{aligned}$$

(35)

The rotation rate field is similarly expanded taking into account the central node of the beam, so that a change of curvature of any beam can be captured:

$$\begin{aligned} \mathbf{w}^{\mathrm{O}(\mathrm{b})\upvarepsilon }\left( \upbeta \right)= & {} \mathbf{w}\left( \upbeta \right) ;\quad \mathbf{w}^{\mathrm{E}(\mathrm{b})\upvarepsilon }\left( {\upbeta +\upvarepsilon \delta ^\mathrm{i}} \right) =\mathbf{w}\left( \upbeta \right) \nonumber \\&+\,\upvarepsilon \frac{\partial \mathbf{w}\left( \upbeta \right) }{\partial \upbeta ^\mathrm{i}}{\updelta }^\mathrm{i} \end{aligned}$$

(36)

Appendix 2: Computation of the tangent stiffness matrix for the DH scheme

The variations of the beam orientation and length are obtained after straightforward computations as follows:

$$\begin{aligned} \updelta \mathbf{e}_\mathrm{x}^\mathrm{b}= & {} \mathbf{C}\cdot \mathbf{P}\cdot \updelta \mathbf{B}^{\mathrm{b}}/\mathrm{l}^{\mathrm{b}}, \updelta \mathbf{e}_\mathrm{y}^\mathrm{b} ={{\varvec{\Omega }} }_\mathrm{z} \cdot \updelta \mathbf{e}_\mathrm{x}^\mathrm{b}, \updelta \mathbf{e}_\mathrm{z}^\mathrm{b} ={{\varvec{\Omega }} }_\mathrm{y} \cdot \updelta \mathbf{e}_\mathrm{x}^\mathrm{b}\nonumber \\ \updelta \hbox {l}^{\mathrm{b}}= & {} \mathbf{B}^{\mathrm{b}}\cdot \left[ {\mathbf{I}+\mathbf{C}\cdot \mathbf{P}} \right] \cdot \updelta \mathbf{B}^{\mathrm{b}}/\mathrm{l}^{\mathrm{b}} \end{aligned}$$

(37)

In (37), we have introduced the projection operators $\mathbf{P}$ and $\mathbf{C}$ expressing as

$$\begin{aligned} \mathbf{P}=\left( {\mathbf{I}-\mathbf{e}_\mathrm{x}^\mathrm{b} \otimes \mathbf{e}_\mathrm{x}^\mathrm{b} } \right) ,\quad \mathbf{C}=\left( {\mathbf{I}-\frac{1}{2}{} \mathbf{e}_\mathrm{x}^\mathrm{b} \otimes \mathbf{e}_\mathrm{x}^\mathrm{b} } \right) \end{aligned}$$

(38)

In the present large strains regime, since the beam length is changing, one has to expand it versus the asymptotic parameter $\upvarepsilon$ as for all other kinematic variables (these expansions are not repeated in this subsection),

$$\begin{aligned} \hbox {l}^\mathrm{b}=\hbox {l}_0^\mathrm{b} +\upvarepsilon \hbox {l}_1^\mathrm{b} +{\upvarepsilon }^{2}\hbox {l}_2^\mathrm{b} +\cdots +{\upvarepsilon }^\mathrm{p}\hbox {l}_\mathrm{p}^\mathrm{b} \end{aligned}$$

(39)

The induced perturbations of the forces and moments are then obtained as

Table 4 Plain weave and twill fabric configuration parameter

Table 5 Elastic properties of weft and warp yarns

Table 6 Mechanical properties of weft, warp and contact beams

(40)

We introduced in (40) the two orthogonal transformations ${{\varvec{\Omega }} }_\mathrm{y},{{\varvec{\Omega }} }_\mathrm{z}$ elaborated as

$$\begin{aligned} \begin{array}{lll} {{\varvec{\Omega }} }_\mathrm{y} \left( {\mathbf{e}_\mathrm{x}^\mathrm{b},\mathbf{e}_\mathrm{z}^\mathrm{b},\mathrm{y}} \right) =\left[ {{\begin{array}{lll} {\cos \left( {\frac{\uppi }{2}} \right) }&{} 0&{} {\sin \left( {\frac{\uppi }{2}} \right) } \\ 0&{} 1&{} 0 \\ {-\sin \left( {\frac{\uppi }{2}} \right) }&{} 0&{} {\cos \left( {\frac{\uppi }{2}} \right) } \\ \end{array} }} \right] \\ {{\varvec{\Omega }} }_\mathrm{z} \left( {\mathbf{e}_\mathrm{x}^\mathrm{b},\mathbf{e}_\mathrm{y}^\mathrm{b},\mathrm{z}} \right) =\left[ {{\begin{array}{lll} {\cos \left( {\frac{\uppi }{2}} \right) }&{} {-\sin \left( {\frac{\uppi }{2}} \right) }&{} 0 \\ {\sin \left( {\frac{\uppi }{2}} \right) }&{} {\cos \left( {\frac{\uppi }{2}} \right) }&{} 0 \\ 0&{} 0&{} 1 \\ \end{array} }} \right] \\ \end{array} \end{aligned}$$

(41)

The linear stiffness, the initial displacement stiffness and initial stress stiffness express successively as

$$\begin{aligned} \mathbf{K}^{\mathrm{b}}_{\mathrm{So}}= & {} \frac{\hbox {E}_\mathrm{S}^\mathrm{b} \mathrm{A}^{\mathrm{b}}}{\mathrm{l}^{\mathrm{b}}}\left( {\mathbf{e}_\mathrm{x}^\mathrm{b} \otimes \mathbf{e}_\mathrm{x}^\mathrm{b} } \right) +\left( {\frac{12\hbox {E}_\mathrm{S}^\mathrm{b} \mathrm{I}_\mathrm{y}^\mathrm{b} }{\left( {\mathrm{l}^{\mathrm{b}}} \right) ^{3}}} \right) \left( {\mathbf{e}_\mathrm{y}^\mathrm{b} \otimes \mathbf{e}_\mathrm{y}^\mathrm{b} } \right) \nonumber \\&+\left( {\frac{12\hbox {E}_\mathrm{S}^\mathrm{b} \mathrm{I}_\mathrm{z}^\mathrm{b} }{\left( {\hbox {l}^{\mathrm{b}}} \right) ^{3}}} \right) \left( {\mathbf{e}_\mathrm{z}^\mathrm{b} \otimes \mathbf{e}_\mathrm{z}^\mathrm{b} } \right) \end{aligned}$$

(42)

$$\begin{aligned} \mathbf{K}^{\mathrm{b}}_\mathrm{u}= & {} \frac{\hbox {E}_\mathrm{S}^\mathrm{b} \hbox {A}^{\mathrm{b}}}{\mathrm{L}}\left[ \left( {\frac{1}{\hbox {l}^{\mathrm{b}}}} \right) \left( {\mathbf{e}_\mathrm{x}^\mathrm{b} \otimes \left( {\mathbf{B}^{\mathrm{b}}-\mathbf{B}_\mathrm{o}^\mathrm{b} } \right) } \right) \cdot \left( {\mathbf{C.P}} \right) \right. \nonumber \\&\left. -\frac{\left( {\mathbf{B}^{\mathrm{b}}-\mathbf{B}_\mathrm{o}^\mathrm{b} } \right) \cdot \mathbf{e}_\mathrm{x}^\mathrm{b} }{\left( {\hbox {l}^{\mathrm{b}}} \right) ^{2}}\left( {\left( {\mathbf{e}_\mathrm{x}^\mathrm{b} \otimes \mathbf{B}^{\mathrm{b}}} \right) \cdot \left[ {\mathbf{\mathrm{I}+C.P}} \right] } \right) \right] \nonumber \\&+\left( {\frac{12\hbox {E}_\mathrm{S}^\mathrm{b} \mathrm{I}_\mathrm{y}^\mathrm{b} }{\left( {\hbox {l}^{\mathrm{b}}} \right) ^{3}}} \right) \left[ \left( \left( {\frac{1}{\hbox {l}^{\mathrm{b}}}} \right) \left( {\mathbf{e}_\mathrm{y}^\mathrm{b} \otimes \left( {\mathbf{B}^{\mathrm{b}}-\mathbf{B}_\mathrm{o}^\mathrm{b} } \right) } \right) \cdot \left( {{{\varvec{\Omega }} }_z \cdot \mathbf{C.P}} \right) \right. \right. \nonumber \\&\left. \left. -\frac{\left( {\mathbf{B}-\mathbf{B}_\mathrm{o}^\mathrm{b} } \right) \cdot \mathbf{e}_\mathrm{y}^\mathrm{b} }{\left( {\hbox {l}^{\mathrm{b}}} \right) ^{2}}\left( {\left( {\mathbf{e}_\mathrm{y}^\mathrm{b} \otimes \mathbf{B}^\mathrm{b}} \right) \cdot \left[ {\mathrm{I}+\mathbf{C.P}} \right] } \right) \right) \right] \nonumber \\&+\left( {\frac{12\hbox {E}_\mathrm{S}^\mathrm{b} \mathrm{I}_\mathrm{z}^\mathrm{b} }{\left( {\hbox {l}^\mathrm{b}} \right) ^{3}}} \right) \left[ {\left( {\begin{array}{l} \left( {\frac{1}{\hbox {l}^{\mathrm{b}}}} \right) \left( {\mathbf{e}_\mathrm{z}^\mathrm{b} \otimes \left( {\mathbf{B}^\mathrm{b}-\mathbf{B}_\mathrm{o}^\mathrm{b} } \right) } \right) \cdot \left( {{{\varvec{\Omega }} }_\mathrm{y} \cdot \mathbf{C.P}} \right) - \\ \frac{\left( {\mathbf{B}-\mathbf{B}_\mathrm{o}^\mathrm{b} } \right) \cdot \mathbf{e}_\mathrm{z}^\mathrm{b} }{\left( {\hbox {l}^\mathrm{b}} \right) ^{2}}\left( {\left( {\mathbf{e}_\mathrm{z}^\mathrm{b} \otimes \mathbf{B}^\mathrm{b}} \right) \cdot \left[ {\mathrm{I}+\mathbf{C.P}} \right] } \right) \\ \end{array}} \right) } \right] \nonumber \\ \end{aligned}$$

(43)

$$\begin{aligned} \mathbf{K}^{\mathrm{b}}_\upsigma= & {} \left[ {\left( {\frac{\hbox {F}_\mathrm{x}^\mathrm{b} }{\mathrm{l}^{\mathrm{b}}}} \right) \mathbf{C.P}} \right] +\left[ {\left( {\frac{2\hbox {F}_\mathrm{y}^\mathrm{b} }{\left( {\hbox {l}^\mathrm{b}} \right) ^{2}}} \right) \left( {\left( {\mathbf{e}_\mathrm{y}^\mathrm{b} \otimes \mathbf{B}^\mathrm{b}} \right) \cdot \left[ {\mathbf{\mathrm{I}}+\mathbf{C.P}} \right] } \right) } \right] \nonumber \\&+\left[ {\left( {\frac{2\hbox {F}_\mathrm{z}^\mathrm{b} }{\left( {\hbox {l}^\mathrm{b}} \right) ^{2}}} \right) \left( {\left( {\mathbf{e}_\mathrm{z}^\mathrm{b} \otimes \mathbf{B}^\mathrm{b}} \right) \cdot \left[ {\mathbf{\mathrm{I}}+\mathbf{C.P}} \right] } \right) } \right] \nonumber \\&+\left[ {\left( {\frac{\hbox {F}_\mathrm{y}^\mathrm{b} }{\hbox {l}^\mathrm{b}}} \right) \left( {{{\varvec{\Omega }} }_\mathrm{z} \cdot \mathbf{C.P}} \right) } \right] +\left[ {\left( {\frac{\hbox {F}_\mathrm{z}^\mathrm{b} }{\hbox {l}^\mathrm{b}}} \right) \left( {{{\varvec{\Omega }} }_\mathrm{y} \cdot \mathbf{C.P}} \right) } \right] \end{aligned}$$

(44)

The tangent couple stress stiffness matrix therein express as

$$\begin{aligned} \mathbf{K}^{\mathrm{b}}_{\mathrm{mo}}= & {} \left( {\frac{-6\hbox {E}_\mathrm{s}^\mathrm{b} \hbox {I}_\mathrm{y}^\mathrm{b} }{\left( {\hbox {l}^\mathrm{b}} \right) ^{2}}} \right) \left( {\mathbf{e}_\mathrm{y}^\mathrm{b} \otimes \mathbf{e}_\mathrm{z}^\mathrm{b} } \right) +\left( {\frac{-6\hbox {E}_\mathrm{s}^\mathrm{b} \hbox {I}_\mathrm{z}^\mathrm{b} }{\left( {\hbox {l}^\mathrm{b}} \right) ^{2}}} \right) \left( {\mathbf{e}_\mathrm{z}^\mathrm{b} \otimes \mathbf{e}_\mathrm{y}^\mathrm{b} } \right) \nonumber \\&+\left( {\frac{\hbox {E}_\mathrm{s}^\mathrm{b} \mathrm{J}^\mathrm{b} }{\mathrm{l}^\mathrm{b}}} \right) \left( {\mathbf{e}_\mathrm{x}^\mathrm{b} \otimes \mathbf{e}_\mathrm{x}^\mathrm{b} } \right) +\left( {\frac{\hbox {E}_\mathrm{s}^\mathrm{b} \mathrm{I}_\mathrm{y}^\mathrm{b} }{\mathrm{l}^{\mathrm{b}}}} \right) \left( {\mathbf{e}_\mathrm{y}^\mathrm{b} \otimes \mathbf{e}_\mathrm{y}^\mathrm{b} } \right) \nonumber \\&+\left( {\frac{\hbox {E}_\mathrm{s}^\mathrm{b} \mathrm{I}_\mathrm{z}^\mathrm{b} }{\mathrm{l}^\mathrm{b}}} \right) \left( {\mathbf{e}_\mathrm{z}^\mathrm{b} \otimes \mathbf{e}_\mathrm{z}^\mathrm{b} } \right) \end{aligned}$$

(45)

$$\begin{aligned} \mathbf{K}^\mathrm{b}_\mathrm{m}= & {} \left[ {\left( {\frac{\hbox {M}_\mathrm{x}^{\mathrm{E}\left( \mathrm{b} \right) } }{\hbox {l}^\mathrm{b}}} \right) \mathbf{C}\cdot \mathbf{P}+\left( {\frac{\hbox {M}_\mathrm{x}^{\mathrm{O}\left( \mathrm{b} \right) } }{\hbox {l}^\mathrm{b}}} \right) \mathbf{C}\cdot \mathbf{P}} \right] \nonumber \\&+\left[ {\left( {\frac{\hbox {M}_\mathrm{y}^{\mathrm{E}\left( \mathrm{b} \right) } }{\hbox {l}^\mathrm{b}}} \right) {{\varvec{\Omega }} }_\mathrm{z} \cdot \mathbf{C}\cdot \mathbf{P}+\left( {\frac{\hbox {M}_\mathrm{y}^{\mathrm{O}\left( \mathrm{b} \right) } }{\hbox {l}^{\mathrm{b}}}} \right) {{\varvec{\Omega }} }_\mathrm{z} \cdot \mathbf{C}\cdot \mathbf{P}} \right] \nonumber \\&+\left[ {\left( {\frac{\hbox {M}_\mathrm{z}^{\mathrm{E}\left( \mathrm{b} \right) } }{\hbox {l}^\mathrm{b}}} \right) {{\varvec{\Omega }} }_\mathrm{y} \cdot \mathbf{C}\cdot \mathbf{P}+\left( {\frac{\hbox {M}_\mathrm{z}^{\mathrm{O}\left( \mathrm{b} \right) } }{\hbox {l}^\mathrm{b}}} \right) {{\varvec{\Omega }} }_\mathrm{y} \cdot \mathbf{C}\cdot \mathbf{P}} \right] \nonumber \\ \end{aligned}$$

(46)

Appendix 3: Geometrical and mechanical parameters for the considered textile performs

The geometrical parameters for plain weave and twill are given in Table 3.

Mechanical properties of weft and warp made of PET are given in Table 4; we intentionally choose very different moduli to represent an unbalanced fabric, leading to an expected anisotropic behavior.

The mechanical properties of the yarns are the same for both unit cells. The tensile, flexural, and torsion rigidities of the beam segments are given in Table 5.

Furthermore, the geometric and material parameters for the contact beam are

$$\begin{aligned} L_{c_{1,2} }= & {} Lf_1 \textit{Sin}\theta _f,Lp_1 \textit{Sin}\theta _p, r_c =\frac{r_f +r_p }{2}, G_{sc}\\= & {} \frac{G_{sf} +G_{pf} }{2},\; \hbox {and}\;E_{sc} =\frac{E_{sf} +E_{pf} }{2} \end{aligned}$$

where $L_{c1,2}$, $r_{c}$, $G_{sc}$, and $E_{sc}$, represent the lengths, radius, shear and Young’s modulus of the contact beams respectively (beams connecting the warp and weft yarns at their crossing points). As an assumption, we take the contact beam with radius and mechanical modulus as average values from the weft and warp corresponding values.