Table A1.

Algorithm for the computation of stress and strain energy. Here n is the iteration number, and N is the number of fiber families. Initially, n = 0 and ${}^0{I}_4^i{=}^0{\tilde{\beta }}_i{=}^0{I}_{1,ini}^i{=}^0{\tilde{\beta }}_i^{ini}{=}^0{\gamma }_i=0$.

(I) Do for i = 1..N

(i) Compute $I_4^i$, I1 using eqs. 9–11

(ii) Check initial damage state

if λ < λD,ini, then go to (vi)

else

(a) Read history ${}^n{I}_4^i$, ${}^n{\tilde{\beta }}_i$, ${}^n{I}_4^{i,ini}$, ${}^n{\tilde{\beta }}_i^{ini}$

(b) Compute ${\tilde{\beta }}_i{=}^n{\tilde{\beta }}_i+\langle I_4^i{-}^n{I}_4^i\rangle$

(c) if first time in damage protocol, then

set $I_4^{i,ini}=I_4^i$, ${\tilde{\beta }}_i^{ini}={\tilde{\beta }}_i$

else set from history $I_4^{i,ini}{=}^n{I}_4^{i,ini}$, ${\tilde{\beta }}_i^{ini}{=}^n{\tilde{\beta }}_i^{ini}$

(iii) Read history nγt and compute maximal damage saturation value

(a) Trial criterion ${\varphi }_i^{trial}=\langle I_4^i-I_4^{i,ini}\rangle {-}^n{\gamma }_i$

(b) Check algorithmic saturation criterion

if ${\varphi }_i^{trial}>0$, then set ${\gamma }_i=\langle I_4^i-I_4^{i,ini}\rangle$

else set from history γi = nγi

(c) Compute damage saturation value

$D_{s,i}=D_{\infty }\left[1-\text{exp}\left(\frac{\text{ln}\left(1-r_{\infty }\right)}{{\gamma }_{\infty }}{\gamma }_i\right)\right]$

(iv) Compute internal variable ${\beta }_i=\langle {\tilde{\beta }}_i-{\tilde{\beta }}_i^{ini}\rangle$

(v) Compute damage function $D_i=D_{s,i}\left[1-\text{exp}\left(\frac{\text{ln}\left(1-r_s\right)}{{\beta }_s}{\beta }_i\right)\right]$

(vi) Compute transversely isotropic stress tensor

(a) Compute effective stress tensor $S_i^{eff}=2{\partial }_C{I}_4^i$

if λ > λD,ini, then $S_i^{ti}=m^{\prime }\left(1-D_i\right)S_i^{eff}$

else $S_i^{ti}=m^{\prime }{S}_i^{eff}$ where $m^{\prime }=\frac{c_1^i}{2}\left[\left(1-D_i\right)I_4^i-1\right]\cdot e^{C_2^i{\langle \left(1-D_i\right)I_4^i-1\rangle }^2}$

(b) Compute the anisotropic Cauchy stress tensor

$t_i^{ti}=F\cdot S_i^{ti}\cdot F^T$

(vii) Compute the anisotropic strain energy $W_i^{ti}=\frac{C_1^i}{N{C}_2^i}\left\{e^{\left(C_2^i{\langle \left(1-D_i\right)I_4^i-1\rangle }^2\right)}-1\right\}$

(II) Calculate isotropic Cauchy stress tensor $t^{iso}=2F\cdot \frac{\partial W_{gr}}{\partial C}\cdot F^T$

(III) Compute isotropic strain energy $W_{gr}=\frac{C_{gr}}{2}\left(I_1-3\right)$

(IV) Compute total strain energy $W=W_{gr}+{\sum }_{i=1}^N{W}_i^{ti}$

(V) Compute total Cauchy stress $t=t^{iso}+{\sum }_{i=1}^N{t}_i^{ti}$