## Appendix B  
Here, we give the detailed derivations of constitutive tensors in Equations (20)-(25). The fourth-order tensor \(\mathbb{C}^{4}\) (\(C_{ijkl}^{4}\)) can be obtained as follows using the recurrence relations in Appendix A. \[ \begin{array}{ll}C_{ijkl}^{4}&=l^{2}N_{p}\int_{0}^{\pi\pi}\int_{0}^{2\pi}K_{ ik}^{a}n_{i}n_{i}\xi\sin\gamma\mathrm{d}\gamma\mathrm{d}\varphi\\ &=l^{2}N_{p}\left(\delta_{pq}+P_{pq}\right)\frac{1}{4\pi}\iint\left[(K_{n}^{a}- K_{s}^{a})n_{i}n_{k}+K_{s}^{a}\delta_{ik}\right]n_{i}n_{j}n_{i}n_{p}n_{q}\sin \gamma\mathrm{d}\gamma d\varphi\\ &=l^{2}N_{p}\left(\delta_{pq}+F_{pq}\right)\left[\frac{1}{105}(K_{n}^{a}-K_{s} ^{a})l_{ijklpq}^{6}+\frac{1}{15}K_{s}^{a}\delta_{ik}l_{jlpq}^{4}\right]\\ &=l^{2}N_{p}\left(\delta_{pq}+F_{pq}\right)\left[c_{1}\delta_{ik}l_{jlpq}^{4} +c_{2}\left(\delta_{ij}l_{klpq}^{4}+\delta_{il}l_{jlpq}^{4}+\delta_{ip}l_{jklq} ^{4}+\delta_{iq}l_{jklp}^{4}\right)\right]\\ &=l^{2}N_{p}\left[5c_{1}\delta_{ik}\delta_{jl}+c_{2}\left(5\delta_{ij}\delta_ {il}+5\delta_{il}\delta_{jk}+2l_{ijkl}^{4}\right)+2c_{1}\delta_{ik}F_{jl}+c_{2 }\left(2\delta_{ij}F_{kl}+2\delta_{il}F_{jk}+2F_{il}l_{ijkl}^{4}\right)\right] \end{array}  \tag{16}\]  
It is noting that the [PERSON] summation convention is not used for \(F_{ii}\) in Equation (16) nor in what follows. Similarly, the sixth-order tensor \(\mathbb{H}^{6}\) (\(H_{ijklmn}^{6}\)) is derived as follows:\[H^{6}_{ijklmn} =\frac{t^{4}N_{\rm s}}{4}\iint K^{6}_{il}n_{j}n_{k}m_{n}n_{r}\zeta \sin\gamma\mathrm{d}\gamma d\varphi \tag{100}\] \[=\frac{t^{4}N_{\rm s}}{4}\left(\beta_{pq}+F_{pq}\right)\frac{1}{4 \pi}\iint\left[(K^{6}_{\rm n}-K^{6}_{\rm s})n_{i}n_{l}+K^{6}_{\rm s}\delta_{il} \right]n_{j}n_{k}m_{n}n_{n}n_{p}n_{q}\sin\gamma\mathrm{d}\gamma d\varphi\] \[=\frac{t^{4}N_{\rm s}}{4}\left(\delta_{pq}+F_{pq}\right)\left[ \frac{1}{945}(K^{6}_{\rm n}-K^{6}_{\rm s})I^{8}_{ijklmnp}+\frac{1}{105}K^{6}_{ \rm s}\delta_{il}I^{6}_{jjkmnpq}\right]\] \[=\frac{t^{4}N_{\rm s}}{4}\left(\delta_{pq}+F_{pq}\right)\left[h_{ 1}\delta_{1}I^{6}_{jkmnpq}+h_{2}\left(\delta_{ij}I^{6}_{jkmnp}+\delta_{ik}I^{6 }_{jkmnp}+\delta_{il}I^{6}_{jkmnp}+\delta_{il}I^{6}_{jkmnp}+\delta_{il}I^{6}_{ jklmnp}+\delta_{il}I^{6}_{jklmnp}\right)\right]\] \[=\frac{t^{4}N_{\rm s}}{4}\left[ \begin{array}{c}7h_{1}\delta_{il} I^{4}_{kmnn}+h_{2}\left(\delta_{ij}I^{4}_{kmnn}+\delta_{ik}I^{4}_{jmnn}+ 7\delta_{il}I^{4}_{jkm}+7\delta_{il}I^{4}_{jklm}+2I^{6}_{jklmn}\right)\\ +2h_{1}\delta_{il}Y^{4}_{jkmm}+2h_{2}\left(\delta_{ij}Y^{4}_{klmn}+\delta_{ik}Y^ {4}_{jlmm}+\delta_{il}Y^{4}_{jklm}+\delta_{il}Y^{4}_{jklm}+F_{li}I^{6}_{jklmm} \right)\end{array} \right]\]  
where the fourth-order tensor \(Y^{4}_{jkmm}\) is the function of fabric and the Kronecker symbol, given by  
\[Y^{4}_{jkmm}=F_{jk}\delta_{mn}+F_{jm}\delta_{kl}+F_{jn}\delta_{mk}+F_{km} \delta_{jn}+F_{kn}\delta_{jm}+F_{mn}\delta_{kj} \tag{101}\]  
Other constitutive tensors can be obtained in the same way. Due to limited space, only their final expressions are given here. \[E^{4}_{ijkl}=l^{2}N_{p}\Big{[}5e_{1}\delta_{ik}\delta_{jl}+e_{2} \Big{(}5\delta_{ij}\delta_{kl}+5\delta_{il}\delta_{jk}+2I^{4}_{ijkl}\Big{)}+2 e_{1}\delta_{ik}F_{jl}+e_{2}\Big{(}2\delta_{ij}F_{kl}+2\delta_{il}F_{jk}+2F_{ii}I^{4}_{ ijkl}\Big{)}\Big{]}, \tag{102}\] \[I^{4}_{ijkl}=l^{2}N_{p}\Big{[}5b_{1}\delta_{ik}\delta_{jl}+b_{2} \Big{(}5\delta_{ij}\delta_{kl}+5\delta_{il}\delta_{jk}+2I^{4}_{ijkl}\Big{)}+2b _{1}\delta_{ik}F_{jl}+b_{2}\Big{(}2\delta_{ij}F_{kl}+2\delta_{il}F_{jk}+2F_{ii} I^{4}_{ijkl}\Big{)}\Big{]},\] (103) \[M^{4}_{ijkl}=l^{2}N_{p}\Big{[}5p_{1}\delta_{ik}\delta_{jl}+p_{2} \Big{(}5\delta_{ij}\delta_{kl}+5\delta_{il}\delta_{jk}+2I^{4}_{ijkl}\Big{)}+2p _{1}\delta_{ik}F_{jl}+p_{2}\Big{(}2\delta_{ij}F_{kl}+2\delta_{il}F_{jk}+2F_{ii} I^{4}_{ijkl}\Big{)}\Big{]},\] (104) \[N^{6}_{ijklmn}=\frac{l^{4}N_{p}}{4}\left[