The stored thermoelastic energy density of the composite material is defined by the functional
$$\begin{aligned} \Psi= & {} \zeta \Psi _{\mathrm {mat}}^{\mathrm {e},\theta }(\tilde{\bar{{\varvec{F}}}}^{\mathrm {e}}, \tilde{J}^{\mathrm {e}},\theta ) +\frac{1-\zeta }{2}\Psi _{\mathrm {fib}}^{\mathrm {e},\theta }(\tilde{\lambda }_{\mathrm {L}}, \tilde{\lambda }_{\mathrm {M}},\tilde{\phi },\tilde{{{\varvec{\kappa }}}}_{\mathrm {L}}, \tilde{\varvec{\kappa }}_{\mathrm {M}},\theta )\nonumber \\= & {} \zeta \left( \Psi _{\mathrm {mat}}^{\mathrm {e}}(\tilde{\bar{{{\varvec{F}}}}}^{\mathrm {e}}, \tilde{J}^{\mathrm {e}},\theta )+\Psi _{\mathrm {mat}}^{\theta }(\theta )\right) \nonumber \\&+\,\frac{1-\zeta }{2}\left( \Psi _{\mathrm {fib}}^{\mathrm {e}}(\tilde{\lambda }_{\mathrm {L}}, \tilde{\lambda }_{\mathrm {M}},\tilde{\phi },\tilde{{{\varvec{\kappa }}}}_{\mathrm {L}}, \tilde{{{\varvec{\kappa }}}}_{\mathrm {M}},\theta )+\Psi _{\mathrm {fib}}^{\theta }(\theta )\right) , \end{aligned}$$
(26)
where
\(\zeta \in [0,1]\) is the volume fraction of the matrix material. The elastic contribution to the stored energy function of the matrix material is decomposed into volumetric and deviatoric parts
$$\begin{aligned} \Psi _{\mathrm {mat}}^{\mathrm {e}}= & {} \Psi _{\mathrm {mat}}^{\mathrm {e,iso}}\left( \tilde{\bar{{\varvec{F}}}}^{\mathrm {e}}(\lambda _1^{\mathrm {e}}, \lambda _2^{\mathrm {e}},\lambda _3^{\mathrm {e}},{\mathfrak {s}}),\theta \right) \nonumber \\&+\Psi _{\mathrm {mat}}^{\mathrm {e,vol}}\left( \tilde{J}^{\mathrm {e}} (\lambda _1^{\mathrm {e}},\lambda _2^{\mathrm {e}},\lambda _3^{\mathrm {e}},{\mathfrak {s}}),\theta \right) . \end{aligned}$$
(27)
As a representative non-linear constitutive law, a modified Ogden material model with the associated strain energy density function
$$\begin{aligned} \Psi _{\mathrm {mat}}^{\mathrm {e,iso}}=\sum \limits _a\sum \limits _b\frac{\mu _b}{\alpha _b} \left( (\tilde{\bar{\lambda }}_a^{\mathrm {e}})^{\alpha _b}-1\right) \end{aligned}$$
(28)
and
$$\begin{aligned} \Psi _{\mathrm {mat}}^{\mathrm {e,vol}}= & {} \frac{\kappa }{\beta ^2} \left( \beta \ln (\tilde{J}^{\mathrm {e}})+(\tilde{J}^{\mathrm {e}})^{-\beta }-1\right) \nonumber \\&-3\frac{\epsilon \kappa }{\gamma }(\theta -\theta _0)\left( (\tilde{J}^{\mathrm {e}})^{\gamma }-1\right) \end{aligned}$$
(29)
is used for the numerical examples. The parameters
\(\mu _b\) and
\(\alpha _b\) with
\(b=\{1,\hdots ,N\}\) are related to the shear modulus and the parameters
\(\kappa \) and
\(\beta \) are related to the bulk modulus. Moreover,
\(\theta _0\) is a reference temperature and the parameters
\(\epsilon \) and
\(\gamma \) are related to the thermal expansion coefficient. Assuming that the fiber portion in both directions is identical, the corresponding elastic contribution of the fiber material is defined by
$$\begin{aligned} \Psi _{\mathrm {fib}}^{\mathrm {e}}= & {} \frac{1}{2}a\left( (\tilde{\lambda }_{\mathrm {L}}-1)^2+(\tilde{\lambda }_{\mathrm {M}}-1)^2\right) +b\,\tilde{\phi }^2\nonumber \\&+\,\frac{1}{2}\left( \tilde{\varvec{\kappa }}_{\mathrm {L}}\cdot \varvec{c}\,\tilde{\varvec{\kappa }}_{\mathrm {L}}+\tilde{\varvec{\kappa }}_{\mathrm {M}}\cdot \varvec{c}\, \tilde{\varvec{\kappa }}_{\mathrm {M}}\right) \nonumber \\&+a\upsilon (\theta -\theta _0)\left( (\tilde{\lambda }_{\mathrm {L}}-1)+(\tilde{\lambda }_{\mathrm {M}}-1)\right) \end{aligned}$$
(30)
where
a and
b are stiffness parameters related to stretch and shear of the fiber material and
\(\upsilon \) denotes the thermal expansion coefficient. Moreover, the stiffness tensor related to fiber curvature is given as
$$\begin{aligned} \varvec{c}=c_{\#}(\tilde{\varvec{l}}\otimes \tilde{\varvec{l}}+\tilde{\varvec{m}}\otimes \tilde{\varvec{m}})+c_{\perp }\tilde{\varvec{n}}\otimes \tilde{\varvec{n}}\quad \text {with}\quad \tilde{\varvec{n}}=\tilde{\varvec{l}}\times \tilde{\varvec{m}}\nonumber \\ \end{aligned}$$
(31)
taking into account a geometric dependency via the stiffness parameters
\(c_{\#}\) and
\(c_{\perp }\), which can be interpreted as the in-plane and out-of-plane bending stiffness, see Section
5.1.1, [
7,
62], and [
20] for details. Next, the purely thermal contributions to the stored energy of the matrix and the fiber material are defined by
$$\begin{aligned} \Psi _{\mathrm {mat}}^{\theta }=c_{\mathrm {mat}}\left( \theta -\theta _0-\theta \ln \left( \frac{\theta }{\theta _0}\right) \right) \end{aligned}$$
(32)
and
$$\begin{aligned} \Psi _{\mathrm {fib}}^{\theta }=2c_{\mathrm {fib}}\left( \theta -\theta _0-\theta \ln \left( \frac{\theta }{\theta _0}\right) \right) , \end{aligned}$$
(33)
respectively. Therein,
\(c_{\mathrm {mat}}\) and
\(c_{\mathrm {fib}}\) are constant parameters representing specific heat capacities of the respective material.
The evolution of the stored thermoelastic energy is given by
$$\begin{aligned} \frac{\,\mathrm {d}}{\,\mathrm {d}t}\Psi= & {} \zeta \left( \sum \limits _a\frac{\partial \Psi _{{\mathrm {mat}}}^{\mathrm {e}}}{\partial \lambda _a^{\mathrm {e}}}\dot{\lambda }_a^{\mathrm {e}} +\frac{\partial \Psi _{{\mathrm {mat}}}^{{\mathrm {e}}}}{\partial {\mathfrak {s}}}\dot{{\mathfrak {s}}} +\frac{\partial (\Psi _{{\mathrm {mat}}}^{{\mathrm {e}}}+\Psi _{{\mathrm {mat}}}^{\theta })}{\partial \theta } \dot{\theta }\right) \nonumber \\&+\,\frac{1-\zeta }{2}\left( \frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \varvec{F}}\dot{{{\varvec{F}}}}+\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \nabla {{\varvec{F}}}}\nabla \dot{{\varvec{F}}} +\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial {\mathfrak {s}}_{\mathrm {L}}} \dot{{\mathfrak {s}}}_{{\mathrm {L}}}\right. \nonumber \\&\left. +\,\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial {\mathfrak {s}}_{{\mathrm {M}}}}\dot{{\mathfrak {s}}}_{{\mathrm {M}}} +\frac{\partial (\Psi _{{\mathrm {fib}}}^{{\mathrm {e}}}+\Psi _{{\mathrm {fib}}}^{\theta })}{\partial \theta }\dot{\theta }\right) . \end{aligned}$$
(34)
Regarding the partial derivatives therein, we introduce first relations related to the Kirchhoff stress
\(\varvec{\tau }=\varvec{\tau }_{\mathrm {mat}}+\varvec{\tau }_{\mathrm {fib}}\) as
$$\begin{aligned} \varvec{\tau }_{\mathrm {mat}}= & {} \varvec{\tau }^{\mathrm {dev}}_{\mathrm {mat}}+\varvec{\tau }^{\mathrm {vol}}_{\mathrm {mat}}\nonumber \\= & {} \sum \limits _a\left( \tau _{\mathrm {mat},a}^{\mathrm {dev}}+\tau _{\mathrm {mat},a}^{\mathrm {vol}}\right) \varvec{n}_{a}\otimes \varvec{n}_{a}\nonumber \\= & {} \zeta \sum \limits _a\lambda _a^{\mathrm {e}}\left( \frac{\partial \Psi _{\mathrm {mat}}^{\mathrm {e,iso}}}{\partial \lambda _a^{\mathrm {e}}}+\frac{\partial \Psi _{\mathrm {mat}}^{\mathrm {e,vol}}}{\partial \lambda _a^{\mathrm {e}}}\right) \varvec{n}_{a}\otimes \varvec{n}_{a} \end{aligned}$$
(35)
and
$$\begin{aligned} \varvec{\tau }_{\mathrm {fib}}= & {} \frac{1-\zeta }{2}\left( \frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\lambda }_{\mathrm {L}}}\frac{\partial \tilde{\lambda }_{\mathrm {L}}}{\partial \varvec{F}} +\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\lambda }_{\mathrm {M}}}\frac{\partial \tilde{\lambda }_{\mathrm {M}}}{\partial \varvec{F}} +\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\phi }}\frac{\partial \tilde{\phi }}{\partial \varvec{F}} \right. \nonumber \\&\left. +\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\varvec{\kappa }}_{\mathrm {L}}}\frac{\partial \tilde{\varvec{\kappa }}_{\mathrm {L}}}{\partial \varvec{F}}+\,\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\varvec{\kappa }}_{\mathrm {M}}}\frac{\partial \tilde{\varvec{\kappa }}_{\mathrm {M}}}{\partial \varvec{F}}\right) \varvec{F}^{\mathrm {T}}, \end{aligned}$$
(36)
the higher-order stress of the fiber material as
$$\begin{aligned} \varvec{\mathfrak {P}}_{\mathrm {fib}}=\frac{1-\zeta }{2}\left( \frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\varvec{\kappa }}_{\mathrm {L}}}\frac{\partial \tilde{\varvec{\kappa }}_{\mathrm {L}}}{\partial \nabla \varvec{F}} +\frac{\partial \Psi _{\mathrm {fib}}^{\mathrm {e}}}{\partial \tilde{\varvec{\kappa }}_{\mathrm {M}}}\frac{\partial \tilde{\varvec{\kappa }}_{\mathrm {M}}}{\partial \nabla \varvec{F}}\right) , \end{aligned}$$
(37)
the driving force of the respective crack phase-field as
$$\begin{aligned}&\mathcal {H}=-\zeta \frac{\partial \Psi ^{\mathrm {e}}_{\mathrm {mat}}}{\partial {\mathfrak {s}}},\quad \mathcal {H}_{\mathrm {L}}=-\frac{1-\zeta }{2}\frac{\partial \Psi ^{\mathrm {e}}_{\mathrm {fib}}}{\partial {\mathfrak {s}}_{\mathrm {L}}},\nonumber \\&\quad \mathcal {H}_{\mathrm {M}}=-\frac{1-\zeta }{2}\frac{\partial \Psi ^{\mathrm {e}}_{\mathrm {fib}}}{\partial {\mathfrak {s}}_{\mathrm {M}}} \end{aligned}$$
(38)
and the specific entropy as
$$\begin{aligned} \eta= & {} \eta _{\mathrm {mat}}+\eta _{\mathrm {fib}}\nonumber \\= & {} -\zeta \frac{\partial (\Psi _{\mathrm {mat}}^{\mathrm {e}}+\Psi _{\mathrm {mat}}^{\theta })}{\partial \theta }-\frac{1-\zeta }{2}\frac{\partial (\Psi _{\mathrm {fib}}^{\mathrm {e}}+\Psi _{\mathrm {fib}}^{\theta })}{\partial \theta }. \end{aligned}$$
(39)
Moreover, we introduce a dissipation function
$$\begin{aligned} \mathcal {D}_{\mathrm {int}}=\nu _{\mathrm {p_{mat}}}\varvec{\tau }_{\mathrm {mat}}:\varvec{d}^{\mathrm {p}}+\nu _{\mathrm {f_{mat}}}\mathcal {H}\dot{{\mathfrak {s}}}+\nu _{\mathrm {f_{fib}}}(\mathcal {H}_{\mathrm {L}}\dot{{\mathfrak {s}}}_{\mathrm {L}}+\mathcal {H}_{\mathrm {M}}\dot{{\mathfrak {s}}}_{\mathrm {M}}),\nonumber \\ \end{aligned}$$
(40)
to account for a transfer of dissipated energy due to plastification and fracture into the thermal field, where
\(\varvec{d}^{\mathrm {p}}\) denotes the Eulerian plastic rate of deformation tensor. The above relations are derived in a thermodynamically consistent manner by assuming that the dissipated energy is completely transfered into the thermal field, i.e. by setting
\(\nu _{\mathrm {p_{mat}}}=\nu _{\mathrm {f}_{\mathrm {mat}}}=\nu _{\mathrm {f}_{\mathrm {fib}}}=1\). Note, however, that the plastic dissipation factor
\(\nu _{\mathrm {p}_{\mathrm {mat}}}\) is typically chosen in the range of 85–95% in the context of thermoplasticity, see e.g. [
45,
67,
82]. In addition, based on experimental observations it is reasonable to set fracture dissipation factors to
\(\nu _{\mathrm {f}_{\mathrm {mat}}}<1\) and
\(\nu _{\mathrm {f}_{\mathrm {fib}}}<1\), see the discussion related to an energy transfer into the thermal field in [
26,
63] and the references therein.
To model the plastic and fracture mechanical response, we introduce an auxiliary functional as
$$\begin{aligned} \widehat{\Psi }= & {} \zeta \left( \widehat{\Psi }_{\mathrm {mat}}^{\mathrm {p}}(\alpha ,\nabla \alpha ,\theta ) +\widehat{\Psi }_{\mathrm {mat}}^{\mathrm {f}}({\mathfrak {s}},\nabla {\mathfrak {s}},\alpha )\right) \nonumber \\&+\frac{1-\zeta }{2}\widehat{\Psi }_{\mathrm {fib}}^{\mathrm {f}}({\mathfrak {s}}_{\mathrm {L}},\nabla {\mathfrak {s}}_{\mathrm {L}},{\mathfrak {s}}_{\mathrm {M}},\nabla {\mathfrak {s}}_{\mathrm {M}}). \end{aligned}$$
(41)
The plastic contribution
\(\widehat{\Psi }_{\mathrm {mat}}^{\mathrm {p}}\) describes the response of isotropic strain-gradient hardening related to the matrix material. To be specific, we focus on the equivalent plastic strain
\(\alpha \) and its gradient
\(\nabla \alpha \) with the particular form
$$\begin{aligned} \widehat{\Psi }_{\mathrm {mat}}^{\mathrm {p}}(\alpha ,\nabla \alpha ,\theta )=\int \limits _{0}^{\alpha }y(\bar{\alpha },\theta )\,\mathrm {d}\bar{\alpha }+y_0(\theta )\frac{l^2_{\mathrm {p}}}{2}\Vert \nabla \alpha \Vert ^2. \end{aligned}$$
(42)
Here,
\(l_{\mathrm {p}}\) is a plastic length scale related to a strain-gradient hardening effect and accounts for size effects to overcome the nonphysical mesh sensitivity of the localized plastic deformation in softening materials, as outlined in [
1]. Moreover,
\(y(\alpha ,\theta )\) is an isotropic local hardening function taken from Cayzac and Saï [
16] and Selles [
66] and adapted to thermoplasticity following Simo and Miehe [
67], Reis et al. [
59] and Da Costa Mattos et al. [
18]. In particular, we use the saturation-type function
$$\begin{aligned} y(\alpha ,\theta )= & {} y_{0}(\theta ) + y_{1}(\theta )\mathrm {exp}[\omega _{\mathrm {p1}}\alpha ]\nonumber \\&+ y_{2}(\theta )(1-\mathrm {exp}[-\omega _{\mathrm {p2}}\alpha ]) , \end{aligned}$$
(43)
with the three temperature-dependent material parameters
\(y_0>0\),
\(y_1\ge 0\) and
\(y_{2}\ge 0\) defined as
$$\begin{aligned} y_{0}(\theta )= & {} y_{0}(\theta _{\mathrm {ref}})(1-\omega _{\mathrm {t0}}(\theta -\theta _{\mathrm {ref}})),\nonumber \\ y_{1}(\theta )= & {} y_{1}(\theta _{\mathrm {ref}})(1-\omega _{\mathrm {t1}}(\theta -\theta _{\mathrm {ref}})),\nonumber \\ y_{2}(\theta )= & {} y_{2}(\theta _{\mathrm {ref}})(1-\omega _{\mathrm {t2}}(\theta -\theta _{\mathrm {ref}})). \end{aligned}$$
(44)
Note that this formulation is typically applied for polyamide which is often used as matrix material of composite structures. Therein, the initial yield stress
\(y_0+y_1\) determines the threshold of the effective elastic response,
\(y_{2}(\theta )(1-\mathrm {exp}[-\omega _{\mathrm {p2}}\alpha ])\) describes an initial hardening stage and
\(y_{1}(\theta )\mathrm {exp}[\omega _{\mathrm {p1}}\alpha ]\) allows for the simulation of large stretches of fibrils which leads to an abrupt increase of stress. This phenomenon is often called rheo-hardening. Moreover,
\(\omega _{\mathrm {p1}}\) and
\(\omega _{\mathrm {p2}}\) are saturation parameters and
\(\omega _{\mathrm {t}0}\),
\(\omega _{\mathrm {t}1}\) and
\(\omega _{\mathrm {t}2}\) are thermal hardening/softening parameters. Note that since we are only interested in the mean mechanical effects of the semi-crystalline matrix material, we consider a unified elastoplastic model with averaged material parameters taken from the multimechanism model in [
16,
66]. Next, we formulate fracture contributions for the matrix as well as the fiber material. Therefore, we approximate a sharp crack surface
\(\Gamma _{\bullet }\) by a regularized functional
1$$\begin{aligned} \widehat{\Gamma }_{\bullet }({\mathfrak {s}}_{\bullet },\nabla {\mathfrak {s}}_{\bullet })= & {} \int \limits _{\mathcal {B}_0}\widehat{\gamma }_{\bullet }({\mathfrak {s}}_{\bullet },\nabla {\mathfrak {s}}_{\bullet })\,\mathrm {d}V\quad \text {with}\nonumber \\ \widehat{\gamma }_{\bullet }({\mathfrak {s}}_{\bullet },\nabla {\mathfrak {s}}_{\bullet })= & {} \frac{1}{2l_{\mathrm {f}_{\bullet }}}({\mathfrak {s}}_{\bullet }^2+l_{\mathrm {f}_{\bullet }}^2\Vert \nabla {\mathfrak {s}}_{\bullet }\Vert ^2), \end{aligned}$$
(45)
based on a specific crack regularization profile
\(\widehat{\gamma }_{\bullet }\) defined per unit volume of the reference configuration and the fracture length scale
\(l_{\mathrm {f}_{\bullet }}\) which controls the regularization. Concerning ductile fracture of the matrix material, we require that
\(l_{\mathrm {p}}\ge l_{\mathrm {f}}\) such that the regularized crack zone lies inside of the plastic zone. Using the regularization given in (
45), the approximated fracture energy of the composite material reads
$$\begin{aligned} W^{\mathrm {f}}\approx & {} \int \limits _{\mathcal {B}_0}\zeta g_{\mathrm {c}}(\alpha )\widehat{\gamma }({\mathfrak {s}},\nabla {\mathfrak {s}})+\frac{1-\zeta }{2}\left( g_{\mathrm {cL}}\widehat{\gamma }_{\mathrm {L}}({\mathfrak {s}}_{\mathrm {L}},\nabla {\mathfrak {s}}_{\mathrm {L}})\right. \nonumber \\&\left. +\,g_{\mathrm {cM}} \widehat{\gamma }_{\mathrm {M}}({\mathfrak {s}}_{\mathrm {M}},\nabla {\mathfrak {s}}_{\mathrm {M}})\right) \,\mathrm {d}V. \end{aligned}$$
(46)
Here,
\(g_{\mathrm {c}_{\bullet }}\) denotes the Griffith-type critical energy density required to create fracture within the respective material. For the matrix material, the critical energy density is decomposed additively into elastic and plastic contributions as
$$\begin{aligned} g_{\mathrm {c}}(\alpha )=g_{\mathrm {c,p}}+g_{\mathrm {c,e}}\exp (-\omega _{\mathrm {f}}\alpha ), \end{aligned}$$
(47)
where
\(\omega _{\mathrm {f}}\) is a modeling parameter. Summarized, the phase-field fracture contributions are given in terms of crack density functions as
$$\begin{aligned} \widehat{\Psi }^{\mathrm {f}}_{\mathrm {mat}}= & {} g_{\mathrm {c}}(\alpha )\widehat{\gamma }({\mathfrak {s}},\nabla {\mathfrak {s}})\nonumber \\= & {} \frac{g_{\mathrm {c}}(\alpha )}{2l_{\mathrm {f}}}({\mathfrak {s}}^2+l^2_{\mathrm {f}}\Vert \nabla {\mathfrak {s}}\Vert ^2) \end{aligned}$$
(48)
and
$$\begin{aligned} \widehat{\Psi }^{\mathrm {f}}_{\mathrm {fib}}= & {} g_{\mathrm {cL}}\widehat{\gamma }_{\mathrm {L}}({\mathfrak {s}}_{\mathrm {L}},\nabla {\mathfrak {s}}_{\mathrm {L}})+g_{\mathrm {c_M}}\widehat{\gamma }_{\mathrm {M}}({\mathfrak {s}}_{\mathrm {M}},\nabla {\mathfrak {s}}_{\mathrm {M}})\nonumber \\= & {} \frac{g_{\mathrm {cL}}}{2l_{\mathrm {fL}}}({\mathfrak {s}}_{\mathrm {L}}^2+l^2_{\mathrm {fL}}\Vert \nabla {\mathfrak {s}}_{\mathrm {L}}\Vert ^2) +\frac{g_{\mathrm {cM}}}{2l_{\mathrm {fM}}}({\mathfrak {s}}_{\mathrm {M}}^2+l^2_{\mathrm {fM}}\Vert \nabla {\mathfrak {s}}_{\mathrm {M}}\Vert ^2).\nonumber \\ \end{aligned}$$
(49)
Eventually, we obtain dissipative resistance forces of the plastic field and the respective crack phase-field via the variational derivatives of
\(\widehat{\Psi }\) with respect to
\(\alpha \) and
\({\mathfrak {s}}_{\bullet }\) as
$$\begin{aligned} r^{\mathrm {p}}=\zeta \delta _{\alpha }\widehat{\Psi }^{\mathrm {p}}_{\mathrm {mat}}=\zeta (\partial _{\alpha }\widehat{\Psi }^{\mathrm {p}}_{\mathrm {mat}}-\nabla \cdot (\partial _{\nabla \alpha }\widehat{\Psi }^{\mathrm {p}}_{\mathrm {mat}})) \end{aligned}$$
(50)
and
$$\begin{aligned} r^{\mathrm {f}}_{\bullet }=\delta _{{\mathfrak {s}}_{\bullet }}\widehat{\Psi } = \partial _{{\mathfrak {s}}_{\bullet }}\widehat{\Psi }-\nabla \cdot (\partial _{\nabla {\mathfrak {s}}_{\bullet }}\widehat{\Psi }). \end{aligned}$$
(51)