## 2 Constitutive Model
### Invariant-Based Formula  
From the continuum theory of fiber-reinforced composites [27], the strain energy function could be expressed as an invariant deformation tensor and a fiber orientation. For example, consider a composite material composed of an isotropic hyper-elastic matrix reinforced with fibers. In the reference configuration,the orientation of a fiber is characterized by a set of unit vectors. Assuming that the strain energy depends on the fiber orientation, as well as the deformation, it can be expressed as follows:  
\[U=U(\mathbf{C},\ \mathbf{A}_{\alpha});\alpha=1,\ldots,N \tag{7}\]  
The strain energy of the material should not change when both the matrix and fibers in the reference configuration are rigid-body rotations. Subsequently, the strain energy can be expressed as an isotropic function of a non-reducible set of scalar invariants formed based on tensor \(\mathbf{C}\) and vectors \(\mathbf{A}_{\alpha}\):  
\[U=U\Big{(}\overline{I}_{1},\ \overline{I}_{2},\ J,\ \overline{I}_{4(\alpha \beta)},\ \overline{I}_{5(\alpha\beta)},\ \zeta_{\alpha\beta}\Big{)};\ \alpha=1,\ \ldots,\ N;\ \beta=1,\ \ldots,\ \alpha \tag{8}\]  
where \(\overline{I}_{1}\) and \(\overline{I}_{2}\) are the first and second deviatoric strain invariants; \(\overline{I}_{4(\alpha\beta)}\) and \(\overline{I}_{5(\alpha\beta)}\) are the pseudo-invariants of \(\overline{\mathbf{C}}\), \(\mathbf{A}_{\alpha}\) and \(\mathbf{A}_{\beta}\); and \(J\) is the third strain invariant, as follows:  
\[\overline{I}_{4(\alpha\beta)}=\mathbf{A}_{\alpha}\overline{\mathbf{C}} \mathbf{A}_{\beta},\ \text{and}\ \overline{I}_{5(\alpha\beta)}=\mathbf{A}_{\alpha} \overline{\mathbf{C}}^{2}\mathbf{A}_{\beta};\ \alpha=1,\ \ldots,\ N;\ \beta=1,\ \ldots,\ \alpha \tag{9}\]  
On the other hand, \(\zeta_{\alpha\beta}\) is a geometric constant, such as the angular cosine between the fiber orientations in the reference configuration, namely,  
\[\zeta_{\alpha\beta}=\mathbf{A}_{\alpha}\mathbf{A}_{\beta};\ \alpha=1,\ \ldots,\ N;\ \beta=1,\ \ldots,\ \alpha \tag{10}\]  
In the invariant-based formulation, the fiber orientations need not be orthogonal in the initial configuration [22; 23; 24]. From Equation (8), the variation in \(U\) is expressed as follows:  
\[\delta U = \frac{\partial U}{\partial\overline{I}_{1}}\delta\overline{I}_{1} +\frac{\partial U}{\partial\overline{I}_{2}}\delta\overline{I}_{2}+\frac{ \partial U}{\partial\overline{J}}\delta J+\sum\limits_{\alpha=1}^{N}\sum \limits_{\beta=1}^{\alpha}\Bigg{(}\frac{\partial U}{\partial\overline{I}_{4( \alpha\beta)}}\delta\overline{I}_{4(\alpha\beta)}+\frac{\partial U}{\partial \overline{I}_{5(\alpha\beta)}}\delta\overline{I}_{5(\alpha\beta)}\Bigg{)} \tag{11}\]  
From the principle of virtual work, Equation (4), the stress components for a compressible material can be expressed as follows:  
\[\begin{split}\mathbf{S}=\frac{2}{J}\mathrm{DEV}&\Bigg{[} \bigg{(}\frac{\partial U}{\partial\overline{I}_{1}}+\overline{I}_{1}\frac{ \partial U}{\partial\overline{I}_{2}}\bigg{)}\overline{\mathbf{B}}-\frac{ \partial U}{\partial\overline{I}_{2}}\overline{\mathbf{B}}\overline{\mathbf{ B}}\bigg{]}+\sum\limits_{\alpha=1}^{N}\sum\limits_{\beta=1}^{\alpha}\frac{ \partial U}{\partial\overline{I}_{4(\alpha\beta)}}\mathrm{DEV}& \Big{(}\overline{\mathbf{a}}_{\alpha}\overline{\mathbf{a}}_{\beta}+\overline {\mathbf{a}}_{\beta}\overline{\mathbf{a}}_{\alpha}\Big{)}\\ &+\sum\limits_{\alpha=1}^{N}\sum\limits_{\beta=1}^{\alpha}\frac{ \partial U}{\partial\overline{I}_{5(\alpha\beta)}}\mathrm{DEV}& \Big{(}\overline{\mathbf{a}}_{\alpha}\overline{\mathbf{a}}_{\beta}^{ \prime}+\overline{\mathbf{a}}_{\alpha}^{\prime}\overline{\mathbf{a}}_{\beta}+ \overline{\mathbf{a}}_{\beta}\overline{\mathbf{a}}_{\alpha}^{\prime}+ \overline{\mathbf{a}}_{\beta}^{\prime}\overline{\mathbf{a}}_{\alpha}\Big{)} \end{split} \tag{12}\]  
\[p=-\frac{\partial U}{\partial J} \tag{13}\]  
where \(\overline{\mathbf{a}}_{\alpha}\overline{\mathbf{E}}\mathbf{A}_{\alpha}\) and \(\overline{\mathbf{a}}_{\alpha}^{\prime}\overline{\mathbf{B}}\mathbf{a}_{\alpha}\).