A micromechanical analysis of inter-fiber failure in long reinforced composites based on the phase field approach of fracture combined with the cohesive zone model

Abstract

The nature of failure in long fiber-reinforced composites is strongly affected by damage at the micro-scale. The presence of different phases at different length scales leads to a significant complexity in the failure progression. At the micro-scale, the complexity is due to the presence of different points of initiation for damage and the presence of cracks propagating both in the matrix and along the fiber–matrix interfaces. This scenario gives also the opportunity to improve the material design by modifying the properties of the different constituents in order to inhibit or delay some failure mechanisms. In view of this complexity, the development of predictive numerical tools with high capabilities in terms of reliability becomes of notable importance. In order to address this aspect, the combination of the phase-field approach for fracture and the cohesive zone model is herein exploited to demonstrate its capability and accuracy for the study of failure initiation at the micro-scale. Single-fiber problems subjected to transverse loading are considered as benchmark for the prediction of the sequence of stages of failure initiation, the size effect of the fiber radius on the apparent strength, the effect of a secondary tensile transverse load and the effect of a secondary neighbouring fiber. Numerical predictions are found to be in very good agreement with experimental trends and finite fracture mechanics predictions available in the literature.

Appendices

Appendix A: Numerical implementation of the phase field approach of fracture in the bulk

In the sequel, the main aspects of the numerical implementation of the PF approach of fracture for triggering failure events within the bulk are described within the FEM framework.

The displacements and the phase field variable are interpolated using an isoparametric approach using standard Lagrangian shape functions $N^{I}({\varvec{\xi }})$ defined in the parametric domain ${\varvec{\xi }} = \{\xi ^{1}, \xi ^{2} \}$ and arranged in the vector ${\mathbf {N}}$. Therefore, the interpolation of these fields, their variation and their linearization adopt the following scheme:

$$\begin{aligned} {\mathbf {x}}\cong & {} \sum _{I=1}^{n} N^{I} \widetilde{{\mathbf {x}}}_{I} = {\mathbf {N}} \widetilde{{\mathbf {x}}}; \quad {\mathbf {u}} \cong \sum _{I=1}^{n} N^{I} {\mathbf {d}}_{I} = {\mathbf {N}} {\mathbf {d}}; \nonumber \\ \delta {\mathbf {u}}\cong & {} \sum _{I=1}^{n} N^{I} \delta {\mathbf {d}}_{I} = {\mathbf {N}} \delta {\mathbf {d}}; \nonumber \\ \varDelta {\mathbf {u}}\cong & {} \sum _{I=1}^{n} N^{I} \varDelta {\mathbf {d}}_{I} = {\mathbf {N}} \delta {\mathbf {d}}, \end{aligned}$$

(A.17)

$$\begin{aligned} {\mathfrak {d}}\cong & {} \sum _{I=1}^{n} N^{I} {\mathfrak {\overline{d}}}_{I} = {\mathbf {N}} {\mathfrak {\overline{d}}}; \quad \delta {\mathfrak {d}} \cong \sum _{I=1}^{n} N^{I} \delta {\mathfrak {\overline{d}}}_{I} = {\mathbf {N}} \delta {\mathfrak {\overline{d}}}; \nonumber \\ \varDelta {\mathfrak {d}}\cong & {} \sum _{I=1}^{n} N^{I} \varDelta {\mathfrak {\overline{d}}}_{I} = {\mathbf {N}} \varDelta {\mathfrak {\overline{d}}}. \end{aligned}$$

(A.18)

In Eqs. (A.17), (A.18), ${\mathbf {d}}_{I}$ and ${\mathfrak {\overline{d}}}_{I}$ are the nodal displacement vector and phase field values, respectively, which are arranged at the element level in the vectors ${\mathbf {d}}$ and ${\mathfrak {\overline{d}}}$

$$\begin{aligned} {\varvec{\varepsilon }} \cong {\mathbf {B}}_{{\mathbf {d}}} {\mathbf {d}}; \quad \delta {\varvec{\varepsilon }} \cong {\mathbf {B}}_{{\mathbf {d}}} \delta {\mathbf {d}}; \quad \varDelta {\varvec{\varepsilon }} \cong {\mathbf {B}}_{{\mathbf {d}}} \varDelta {\mathbf {d}} \end{aligned}$$

(A.19)

Moreover, the spatial gradient of the phase field variable can be approximated as follows:

$$\begin{aligned} \nabla _{{\mathbf {x}}} {\mathfrak {d}} \cong {\mathbf {B}}_{{\mathfrak {d}}} {\mathfrak {\overline{d}}}; \quad \nabla _{{\mathbf {x}}} (\delta {\mathfrak {d}}) \cong {\mathbf {B}}_{{\mathfrak {d}}} \delta {\mathfrak {\overline{d}}}; \quad \nabla _{{\mathbf {x}}} (\varDelta {\mathfrak {d}}) \cong {\mathbf {B}}_{{\mathfrak {d}}} \varDelta {\mathfrak {\overline{d}}},\nonumber \\ \end{aligned}$$

(A.20)

being ${\mathbf {B}}_{{\mathfrak {d}}}$ a phase field compatibility-like operator.

The discrete form of Eq. (6) at the element level, being denoted by the superscript el, renders

$$\begin{aligned}&\delta {\tilde{\varPi }}_{b}^{el} ({\mathbf {d}}, \delta {\mathbf {d}}, {\mathfrak {\overline{d}}} , \delta {\mathfrak {\overline{d}}}) \nonumber \\&\quad = \delta {\mathbf {d}}^{\text {T}} \left\{ \int _{{\mathcal {B}}^{el}} \left[ \left( \left( 1 - {\mathfrak {d}} \right) ^{2} + {\mathcal {K}} \right) {\mathbf {B}}_{{\mathbf {d}}}^{\text {T}} {\varvec{\sigma }} \right. \right. \nonumber \\&\qquad \left. \left. + \, {\mathbf {B}}_{{\mathbf {d}}}^{\text {T}} {\varvec{\sigma }}_{-} \right] \,\mathrm {d}\varOmega - \int _{ \partial {\mathcal {B}}^{el}} {\mathbf {N}}^{\text {T}} \overline{{\mathbf {t}}} \,\mathrm {d} \partial \varOmega - \int _{{\mathcal {B}}^{el}} {\mathbf {N}}^{\text {T}} {\mathbf {f}}_{v} \,\mathrm {d}\varOmega \right\} \nonumber \\&\qquad +\delta {\mathfrak {\overline{d}}}^{\text {T}} \left\{ \int _{ {\mathcal {B}}^{el}} - 2(1- {\mathfrak {d}}) {\mathbf {N}}^{\text {T}} \psi ^{e}_{+} ({\varvec{\varepsilon }}) \,\mathrm {d}\varOmega \right. \nonumber \\&\qquad \left. + \int _{ {\mathcal {B}}^{el}} {\mathcal {G}}_{c}^{b} l\left( {\mathbf {B}}_{{\mathfrak {d}}}^{\text {T}} \nabla _{{\mathbf {x}}}{\mathfrak {d}} + \frac{1}{l^{2}} {\mathbf {N}}^{\text {T}} {\mathfrak {d}} \right) \,\mathrm {d}\varOmega \right\} \nonumber \\&\quad = \delta {\mathbf {d}}^{\text {T}} {\mathbf {f}}_{{\mathbf {d}}}^{b} + \delta {\mathfrak {\overline{d}}}^{\text {T}} {\mathbf {f}}_{{\mathfrak {d}}}^{b} \end{aligned}$$

(A.21)

where

$$\begin{aligned} {\mathbf {f}}_{{\mathbf {d}},\text {int}}^{b}= & {} \int _{{\mathcal {B}}^{el}} \left[ \left( \left( 1 - {\mathfrak {d}} \right) ^{2} + {\mathcal {K}} \right) {\mathbf {B}}_{{\mathbf {d}}}^{\text {T}} {\varvec{\sigma }}_{+} + {\mathbf {B}}_{{\mathbf {d}}}^{\text {T}} {\varvec{\sigma }}_{-}\right] \,\mathrm {d}\varOmega ,\nonumber \\ \end{aligned}$$

(A.22)

$$\begin{aligned} {\mathbf {f}}_{{\mathbf {d}},\text {ext}}^{b}= & {} \int _{ \partial {\mathcal {B}}^{el}} {\mathbf {N}}^{\text {T}} \overline{{\mathbf {t}}} \,\mathrm {d} \partial \varOmega + \int _{{\mathcal {B}}} {\mathbf {N}}^{\text {T}} {\mathbf {f}}_{v} \,\mathrm {d}\varOmega , \end{aligned}$$

(A.23)

$$\begin{aligned} {\mathbf {f}}_{{\mathfrak {d}}}^{b}= & {} \int _{ {\mathcal {B}}^{el}} - 2(1- {\mathfrak {d}}) {\mathbf {N}}^{\text {T}} \psi ^{e}_{+} ({\varvec{\varepsilon }}) \,\mathrm {d}\varOmega \nonumber \\&+ \int _{ {\mathcal {B}}^{el}} {\mathcal {G}}_{c}^{b} l\left[ {\mathbf {B}}_{{\mathfrak {d}}}^{\text {T}} \nabla _{{\mathbf {x}}} + {\mathfrak {d}} \frac{1}{l^{2}} {\mathbf {N}}^{\text {T}} {\mathfrak {d}} \right] \,\mathrm {d}\varOmega . \end{aligned}$$

(A.24)

The vectors ${\mathbf {f}}_{{\mathbf {d}},\text {int}}^{b}$ and ${\mathbf {f}}_{{\mathbf {d}},\text {ext}}^{b}$ correspond to the internal and residual vectors associated with the displacement field, whilst ${\mathbf {f}}_{{\mathfrak {d}}}^{b}$ stands for the residual vector of the phase field.

The linearization of the previous residual vector yields to the following system (Nguyen et al. 2016b, a; Ambati et al. 2015):

$$\begin{aligned} \begin{bmatrix} {\mathbf {K}}_{{\mathbf {d}}{\mathbf {d}}}^{b}&{\mathbf {0}} \\ {\mathbf {0}}&{\mathbf {K}}_{ {\mathfrak {d}} {\mathfrak {d}}}^{b} \end{bmatrix} \begin{bmatrix} \varDelta {\mathbf {d}} \\ \varDelta {\mathfrak {d}} \end{bmatrix} =\begin{bmatrix} {\mathbf {f}}_{{\mathbf {d}},\text {ext}}^{b} \\ 0 \end{bmatrix} - \begin{bmatrix} {\mathbf {f}}_{{\mathbf {d}},\text {int}}^{b} \\ {\mathbf {f}}_{ {\mathfrak {d}}}^{b} \end{bmatrix}. \end{aligned}$$

(A.25)

The system given in Eq. (A.25) is solved in Jacobi-type solution scheme in line with (Miehe et al. 2010a), though the fully coupled system can be computed via the incorporation of a viscous term in order to guarantee the convexity of the corresponding formulation. It is worth noting that the modification of the formulation in order to incorporate this viscous term is not addressed in the current paper since a staggered solution scheme has been adopted. Moreover, the previous formulation is equipped with a history variable into a modified version of the residual vector of the phase field in line with (Miehe et al. 2010b; Reinoso et al. 2017c):

$$\begin{aligned} {\mathbf {f}}_{{\mathfrak {d}}}^{b}= & {} \int _{ {\mathcal {B}}^{el}} - 2(1- {\mathfrak {d}}) {\mathbf {N}}^{\text {T}} \left[ \psi ^{e}_{+} ({\varvec{\varepsilon }})\right. \nonumber \\&\left. - \frac{\eta }{n \varDelta t} \left\langle {\dot{\overline{\mathfrak {d}}}} \right\rangle _{-}^{n}\right] \,\mathrm {d}\varOmega \nonumber \\&+ \int _{ {\mathcal {B}}^{el}} {\mathcal {G}}_{c}^{b} l\left[ {\mathbf {B}}_{{\mathfrak {d}}}^{\text {T}} \nabla _{{\mathbf {x}}} + {\mathfrak {d}} \frac{1}{l^{2}} {\mathbf {N}}^{\text {T}} {\mathfrak {d}} \right] \,\mathrm {d}\varOmega . \end{aligned}$$

(A.26)

where $\eta$ is a penalty parameter which is set $n=2$ (Msekh et al. 2015), which was previously examined by the authors in Reinoso et al. (2017c, 2017a) with respect to experimental data leading to satisfactory predictions, see (Linse et al. 2017) for a very comprehensive and detailed assessment regarding the convergence of PF methods; $\left\langle {\dot{\overline{\mathfrak {d}}}}\right\rangle _{-}^{n}= \left\langle {\overline{\mathfrak {d}}}^{t+1} - {\overline{\mathfrak {d}}}^{t}\right\rangle _{-}^{n}$ and the operator $\left\langle \right\rangle _{-}$ for a variable a renders:

$$\begin{aligned} \left\langle a\right\rangle _{-} = \left\{ \begin{aligned} - a\text {, }&\text { } a < 0, \\ 0\text {, }&\text { } a \ge 0. \end{aligned} \right. \end{aligned}$$

(A.27)

Finally the particular form of the tangent matrices ${\mathbf {K}}_{{\mathbf {d}}{\mathbf {d}}}^{b}$ and ${\mathbf {K}}_{ {\mathfrak {d}}{\mathfrak {d}}}^{b}$ are not included in the present manuscript for conciseness reasons.

Appendix B: Variational formulation and numerical implementation of the interface element compatible with the PF approach of fracture

This Appendix presents the variational formulation regarding the interface contribution to the total functional of the system given in Eq. (10). Complying with a standard Galerkin procedure, the weak form of the interface contribution reads

$$\begin{aligned}&\delta \varPi _{\varGamma _{i}} ({\mathbf {u}}, \delta {\mathbf {u}}, {\mathfrak {d}}, \delta {\mathfrak {d}}) \nonumber \\&\quad = \int _{\varGamma _{i}} \left( \dfrac{\partial {\mathcal {G}}^{i}({\mathbf {u}},{\mathfrak {d}})}{\partial {\mathbf {u}}}\delta {\mathbf {u}}+\dfrac{\partial {\mathcal {G}}^{i}({\mathbf {u}},{\mathfrak {d}})}{\partial {\mathfrak {d}}}\delta {\mathfrak {d}} \right) \,\mathrm {d}\varGamma ,\nonumber \\ \end{aligned}$$

(B.28)

In Eq. (B.28), $\delta {\mathbf {u}}$ denotes the admissible kinematic field vector function (${\mathfrak {V}}^{u} = \big \{\delta {\mathbf {u}}\, | \, {\mathbf {u}} = \overline{{\mathbf {u}}} \text{ on } \partial {\mathcal {B}}_{u} , {\mathbf {u}} \in {\mathcal {H}}^{1} \big \}$), whereas $\delta {\mathfrak {d}}$ identifies the approximation functions of the phase field variable (${\mathfrak {V}}^{{\mathfrak {d}}} = \big \{\delta {\mathfrak {d}} \, | \, \delta {\mathfrak {d}} = 0 \text{ on } \varGamma _{b} , {\mathfrak {d}} \in {\mathcal {H}}^{0} \big \}$).

The discretization of the previous variational form can be carried out within the context of the FEM through the use of standard isoparametric elements. Without loss of generality, we adopt a first-order interpolation scheme for the kinematic and the phase field variables. Thus, ${\mathbf {d}}$ represents the nodal-based displacement field and $\bar{{\mathfrak {d}}}$ denotes the nodal-based phase field, both vectors being defined at the element level for the corresponding numerical implementation.

Accordingly, the discrete form of Eq. (B.28) at the element level $\varGamma _{i}^{el}$ ($\varGamma _i \sim \bigcup \varGamma _{i}^{el}$) adopts the form:

$$\begin{aligned}&\delta {\tilde{\varPi }}^{el}_{\varGamma _i} ({\mathbf {d}}, \delta {\mathbf {d}}, \bar{{\mathfrak {d}}}, \delta \bar{{\mathfrak {d}}})\nonumber \\&\quad =\int _{\varGamma _i^{el}} \left( \dfrac{\partial {\mathcal {G}}^{i} ({\mathbf {d}},\bar{{\mathfrak {d}}})}{\partial {\mathbf {d}}}\delta {\mathbf {d}}+\dfrac{\partial {\mathcal {G}}^{i} ({\mathbf {d}},\bar{{\mathfrak {d}}})}{\partial \bar{{\mathfrak {d}}}}\delta \bar{{\mathfrak {d}}} \right) \,\mathrm {d}\varGamma ,\nonumber \\ \end{aligned}$$

(B.29)

where ${\mathcal {G}}^{i}={\mathcal {G}}_{I}^{i}+{\mathcal {G}}_{II}^{i}$:

The displacement vector between the crack flanks along the interface is characterized by the gap vector, ${\mathbf {g}}$. The discrete form of the gap vector for each point of $\varGamma _i^{el}$ can be computed as the difference between the displacements of opposite points along the interface, which can be obtained from the nodal displacements ${\mathbf {d}}$ multiplied by the average matrix ${\mathbf {L}}$ and the interpolation matrix ${\mathbf {N}}$:

$$\begin{aligned} {\mathbf {g}}={\mathbf {N}}{\mathbf {L}}{\mathbf {d}}= \hat{{\mathbf {B}}}_{{\mathbf {d}}} {\mathbf {d}}, \end{aligned}$$

(B.30)

identifying $\hat{{\mathbf {B}}}_{{\mathbf {d}}}={\mathbf {N}}{\mathbf {L}}$ the interface compatibility operator.

The constitutive law at the interface requires the consideration of a local setting, which is defined by the normal and tangential unit vectors (Paggi and Wriggers 2012). Correspondingly, the gap vector in the global setting (Eq. (B.30)) is multiplied by rotation matrix ${\mathbf {R}}$ in order to obtain the gap vector in the local setting ${\mathbf {g}}_{\text {loc}}$:

$$\begin{aligned} {\mathbf {g}}_{\text {loc}} \cong {\mathbf {R}}{\mathbf {g}}={\mathbf {R}}\hat{{\mathbf {B}}}_{{\mathbf {b}}} {\mathbf {d}}. \end{aligned}$$

(B.31)

In a similar manner, the discrete average phase field variable ${\mathfrak {d}}$ at the interface ($\varGamma ^{el}_i$) can be computed at the element level as follows:

$$\begin{aligned} {\mathfrak {d}} \cong {\mathbf {N}}_{{\mathfrak {d}}} {\mathbf {M}}_{{\mathfrak {d}}} \bar{{\mathfrak {d}}} ={\hat{{\mathbf {B}}}}_{{\mathfrak {d}}} \bar{{\mathfrak {d}}}, \end{aligned}$$

(B.32)

where ${\mathbf {M}}_{{\mathfrak {d}}}$ is an matrix for determining the average value of the phase field variable between the interface flanks, and ${\hat{{\mathbf {B}}}}_{{\mathfrak {d}}}={\mathbf {N}}_{{\mathfrak {d}}} {\mathbf {M}}_{{\mathfrak {d}}}$ identifies the compatibility operator for the phase field. The particular form of such matrices are derived in Reinoso and Paggi (2014) and Paggi and Reinoso (2017), being omitted here for the sake of brevity.

Then, the discrete variational form renders

$$\begin{aligned} \begin{aligned}&\delta {\tilde{\varPi }}_{\varGamma _i}^{el} ({\mathbf {d}}, \delta {\mathbf {d}}, \bar{{\mathfrak {d}}}, \delta \bar{{\mathfrak {d}}}) \\&\quad =\delta {\mathbf {d}}^{\mathrm {T}}\int _{\varGamma _i^{el}} \left( \dfrac{\partial {\mathcal {G}}^{i} ({\mathbf {d}}, \bar{{\mathfrak {d}}})}{\partial {\mathbf {d}}}\right) ^{\mathrm {T}}\,\mathrm {d}\varGamma \\&\qquad + \delta \bar{{\mathfrak {d}}}^{\mathrm {T}}\int _{\varGamma _i^{el}} \left( \dfrac{\partial {\mathcal {G}}^{i} ({\mathbf {d}}, \bar{{\mathfrak {d}}})}{\partial \bar{{\mathfrak {d}}}}\right) ^{\mathrm {T}}\,\mathrm {d}\varGamma \\&\quad =\delta {\mathbf {d}}^{\mathrm {T}}\int _{\varGamma _i^{el}} \hat{{\mathbf {B}}}_{{\mathbf {d}}}^{\mathrm {T}}{\mathbf {R}}^{\mathrm {T}}\left( \dfrac{\partial {\mathcal {G}}^{i} ({\mathbf {d}}, \bar{{\mathfrak {d}}}) }{\partial {\mathbf {g}}_{\text {loc}}}\right) ^{\mathrm {T}}\,\mathrm {d}\varGamma \\&\qquad + \delta \bar{{\mathfrak {d}}}^{\mathrm {T}}\int _{\varGamma _i^{el}} {\hat{{\mathbf {B}}}}_{{\mathfrak {d}}}^{\mathrm {T}}\left( \dfrac{\partial {\mathcal {G}}^{i}({\mathbf {d}}, \bar{{\mathfrak {d}}})}{\partial \bar{{\mathfrak {d}}}}\right) ^{\mathrm {T}}\,\mathrm {d}\varGamma \end{aligned} \end{aligned}$$

(B.33)

leading to the residual vectors

$$\begin{aligned} {\mathbf {f}}_{{\mathbf {d}}}^{i}&=\int _{\varGamma _i^{el}}\hat{{\mathbf {B}}}_{{\mathbf {d}}}^{\mathrm {T}}{\mathbf {R}}^{\mathrm {T}}\left( \dfrac{\partial {\mathcal {G}}^{i} ({\mathbf {d}}, \bar{{\mathfrak {d}}}) }{\partial {\mathbf {g}}_{\text {loc}}}\right) ^{\mathrm {T}}\,\mathrm {d}\varGamma , \end{aligned}$$

(B.34a)

$$\begin{aligned} {\mathbf {f}}_{{\mathfrak {d}}}^{i}&=\int _{\varGamma _i^{el}}{\hat{{\mathbf {B}}}}_{{\mathfrak {d}}}^{\mathrm {T}}\left( \dfrac{\partial {\mathcal {G}}^{i}({\mathbf {d}}, \bar{{\mathfrak {d}}})}{\partial {\mathfrak {d}}}\right) ^{\mathrm {T}}\,\mathrm {d}\varGamma . \end{aligned}$$

(B.34b)

Finally, the consistent linearization of the previous residual vectors allows the computation of the element tangents of the proposed interface formulation:

$$\begin{aligned} {\mathbf {K}}_{{\mathbf {dd}}}^{i}&=\dfrac{\partial {\mathbf {f}}_{{\mathbf {d}}}}{\partial {\mathbf {d}}}=\int _{\varGamma _i^{el}}\hat{{\mathbf {B}}}_{{\mathbf {d}}}^{\mathrm {T}}{\mathbf {R}}^{\mathrm {T}}{\mathbb {C}}_{{\mathbf {dd}}}^{i}{\mathbf {R}}\hat{{\mathbf {B}}}_{{\mathbf {d}}}\,\mathrm {d}\varGamma , \end{aligned}$$

(B.35a)

$$\begin{aligned} {\mathbf {K}}_{{\mathbf {d}}{\mathfrak {d}}}^{i}&=\dfrac{\partial {\mathbf {f}}_{{\mathbf {d}}}}{\partial {\mathfrak {d}}}=\int _{\varGamma _i^{el}}\hat{{\mathbf {B}}}_{{\mathbf {d}}} ^{\mathrm {T}}{\mathbf {R}}^{\mathrm {T}}{\mathbb {C}}_{{\mathbf {d}}{\mathfrak {d}}}^{i}{\hat{{\mathbf {B}}}}_{{\mathfrak {d}}}\,\mathrm {d}\varGamma , \end{aligned}$$

(B.35b)

$$\begin{aligned} {\mathbf {K}}_{{\mathfrak {d}}{\mathbf {d}}}^{i}&=\dfrac{\partial {\mathbf {f}}_{\mathfrak {d}}}{\partial {\mathbf {d}}}=\int _{\varGamma _i^{el}}{\hat{{\mathbf {B}}}}_{{\mathfrak {d}}}^{\mathrm {T}}{\mathbb {C}}_{{\mathfrak {d}}d}^{i}{\mathbf {R}}\hat{{\mathbf {B}}}_{{\mathbf {d}}}\,\mathrm {d}\varGamma , \end{aligned}$$

(B.35c)

$$\begin{aligned} {\mathbf {K}}_{{\mathfrak {d}}{\mathfrak {d}}}^{i}&=\dfrac{\partial {\mathbf {f}}_{\mathfrak {d}}}{\partial {\mathfrak {d}}}=\int _{\varGamma _i^{el}}{\hat{{\mathbf {B}}}}_{{\mathfrak {d}}} ^{\mathrm {T}}{\mathbb {C}}_{{\mathfrak {d}}{\mathfrak {d}}}^{i}{\hat{{\mathbf {B}}}}_{{\mathfrak {d}}}\,\mathrm {d}\varGamma , \end{aligned}$$

(B.35d)

where the interface operators are given by

$$\begin{aligned} {\mathbb {C}}_{dd}^{i}&=\left[ \begin{array}{cc} {\hat{\alpha }} k_{n} &{} 0 \\ 0 &{} {\hat{\beta }} k_{t} \\ \end{array} \right] , \end{aligned}$$

(B.36a)

$$\begin{aligned} {\mathbb {C}}_{{\mathbf {d}}{\mathfrak {d}}}^{i}&=\left[ g_n k_{n}\dfrac{\partial {\hat{\alpha }}}{\partial {\mathfrak {d}}},g_t k_{t}\dfrac{\partial {\hat{\beta }}}{\partial {\mathfrak {d}}}\right] , \end{aligned}$$

(B.36b)

$$\begin{aligned} {\mathbb {C}}_{{\mathfrak {d}}{\mathbf {d}}}^{i}&=\left[ \begin{array}{c} g_n k_{n}\dfrac{\partial {\hat{\alpha }}}{\partial {\mathfrak {d}}} \\ g_t k_{t}\dfrac{\partial {\hat{\beta }}}{\partial {\mathfrak {d}}}\\ \end{array} \right] , \end{aligned}$$

(B.36c)

$$\begin{aligned} {\mathbb {C}}_{{\mathfrak {d}}{\mathfrak {d}}}^{i}&=\dfrac{1}{2}g_{n}^2 k_{n}\dfrac{\partial ^2{\hat{\alpha }}}{\partial {\mathfrak {d}}^2}+\dfrac{1}{2}g_{t}^2k_{t}\dfrac{\partial ^2 {\hat{\beta }}}{\partial {\mathfrak {d}}^2}. \end{aligned}$$

(B.36d)

In the previous expressions ${\hat{\alpha }}$ y ${\hat{\beta }}$ are particularized as follows:

$$\begin{aligned} {\hat{\alpha }}&= \dfrac{g_{nc,0}^2}{\left[ (1-{\mathfrak {d}})g_{nc,0}+ {\mathfrak {d}} g_{nc,1}\right] ^2}, \end{aligned}$$

(B.37a)

$$\begin{aligned} {\hat{\beta }}&= \dfrac{g_{tc,0}^2}{\left[ (1-{\mathfrak {d}})g_{tc,0}+ {\mathfrak {d}} g_{tc,1}\right] ^2}. \end{aligned}$$

(B.37b)

In line with Eq. (A.25), the final derivation endows a coupled system of equations.