# Geometrical and Mechanical Modeling of Polymeric Multi-Ply Yarns
## 2 Materials and Methods
### Anisotropic Viscoelastic-Viscoplastic Model  
The transversal isotropic elasto-viscoplastic model developed in [13] is here assumed. Isotropic hardening is also included in the present work. Figure 5 shows schematically the assumption of strain additivity between elastic, viscoelastic and viscoplastic contributions. The governing equations of the model read  
\[\mathbf{\sigma}=\mathbf{C}:(\mathbf{\epsilon}-\mathbf{\epsilon}^{vc}-\mathbf{\epsilon}^{vp}) \tag{8}\] \[\chi=h\,\gamma\] (9) \[\mathbf{X}_{\ell}=H_{\ell}\,\mathbf{\alpha}_{\ell}\] (10) \[\mathbf{X}_{n\ell}=H_{n\ell}\,\mathbf{\alpha}_{n\ell}\] (11) \[\dot{\mathbf{\epsilon}}^{vv}=\mathbf{\Xi}^{-1}:(\mathbf{\sigma}-\bar{\mathbf{C}} :\mathbf{\epsilon}^{vc})\] (12) \[\dot{\mathbf{\epsilon}}^{vp}=\frac{\langle\varphi\rangle}{\eta}\, \text{sign}((\mathbf{\sigma}-\mathbf{X}_{\ell}-\mathbf{X}_{n\ell}):(\mathbf{a}\otimes\mathbf{a}))( \mathbf{a}\otimes\mathbf{a})\] (13) \[\dot{\gamma}=\frac{\langle\varphi\rangle}{\eta}\] (14) \[\dot{\mathbf{\alpha}}_{\ell}=\dot{\mathbf{\epsilon}}^{vp}\] (15) \[\dot{\mathbf{\alpha}}_{n\ell}=\dot{\mathbf{\epsilon}}^{vp}-\frac{\langle \varphi\rangle}{\eta}\,\frac{B}{H_{n\ell}}\mathbf{X}_{n\ell}\] (16) \[\varphi=|(\mathbf{\sigma}-\mathbf{X}_{\ell}-\mathbf{X}_{n\ell}):(\mathbf{a} \otimes\mathbf{a})|-\sigma_{y}-\chi \tag{17}\]  
Equations (8)-(11) express the linear state laws relating the static variables (stress \(\mathbf{\sigma}\), static internal variables of isotropic hardening \(\chi\), and of linear and non-linear kinematic hardening \(\mathbf{X}_{\ell}\) and \(\mathbf{X}_{n\ell}\)) to the conjugate kinematic ones (elastic strain \(\mathbf{\epsilon}^{e}=\mathbf{\epsilon}-\mathbf{\epsilon}^{vc}-\mathbf{\epsilon}^{vp}\), kinematic internal variables of isotropic hardening \(\gamma\), and of linear and non-linear kinematic hardening \(\mathbf{\alpha}_{\ell}\) and \(\mathbf{\alpha}_{n\ell}\)). In Equation (8), \(\mathbf{C}\) is the elastic stiffness of the transversely isotropic material:  
\[\chi=h\,\gamma\]0 \[\chi=h\,\gamma\]1  
where \(\mathbf{a}\) is the vector defining the fiber direction, \(\mathbf{I}\) is the second-order unit tensor, \(\mathbf{I}\) is the fourth-order unit tensor and \(\mathbf{A}\) is a fourth-order tensor whose components are given by the following:  
\[\chi=h\,\gamma\]2  
The stiffness matrix depends on five material parameters: \(\mu_{L}\), \(\mu_{P}\) are the shear moduli, respectively, along the fiber direction \(\mathbf{a}\) (index \(L\)) and in the plane of isotropy (orthogonal to \(\mathbf{a}\), index \(P\)), and \(\lambda,\alpha\) and \(\beta\) are three material constants that can be derived from the more usual engineering elastic constants \(E_{L}\), \(E_{P}\), \(\
u_{P}\) and \(\
u_{PL}\), as shown in [13]. The material parameter \(h\) governs the isotropic hardening, assumed linear for simplicity, while \(H_{\ell}\) and \(H_{n\ell}\) define the linear and non-linear kinematic hardening. \[\chi=h\,\gamma\]3 Scheme of the assumed constitutive model: elastic, viscoelastic (Voigt type) and viscoplastic (Bingham-type with non-linear hardening) elements connected in series. Equation (12) expresses the evolution law of the viscoelastic strain \(\varepsilon^{\text{re}}\) having assumed linear viscoelasticity in the Voigt form, with \(\bar{\mathcal{C}}\) having the same form as \(\mathcal{C}\), but with five different material parameters. Even though in principle one can identify all five parameters independently, to reduce the number of parameters, we here assume proportionality between the two elasticity tensors  
\[\chi=h\,\gamma\]4  
Similarly, \(\Xi\) is a fourth-order tensor, depending on five retardation times (cp. [13]). Here, a single retardation time is considered for simplicity, hence assuming  
\[\chi=h\,\gamma\]5  
Equations (13)-(16) express the evolution laws for the viscoplastic variables. All of them evolve together and are proportional to the positive part \(\langle\varphi\rangle\) of the yield function \(\varphi;\eta\) is a positive viscosity parameter.