Appendix 1: Basic relations
Consider a displacement field \(\mathbf {u}_\mathrm{m}\) which is perturbed producing \(\mathbf {u}_{m,\tau }=\mathbf {u}_\mathrm{m}+\tau \delta \mathbf {u}_\mathrm{m}\). With this perturbation we have the deformation gradient, originally given by \(\mathbf {F}_\mathrm{m}=\mathbf {I}+\nabla _\mathrm{m}\mathbf {u}_\mathrm{m}\), results in \(\mathbf {F}_{m,\tau }=\mathbf {I}\,+\,\nabla _\mathrm{m}\mathbf {u}_{m,\tau }=\mathbf {I}\,+\,\nabla _\mathrm{m}\left( \mathbf {u}_\mathrm{m}+\tau \delta \mathbf {u}_\mathrm{m}\right) \). Let us compute the expressions of the derivatives of several quantities involving \(\mathbf {F}_{m,\tau }\), with respect to \(\tau \). This will be employed in the linearization procedures whenever the material configuration is known. Then, we have
$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}\tau }\mathbf {F}_{m,\tau }\right| _{\tau =0}= & {} \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}=\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) _\mathrm{m}\mathbf {F}_\mathrm{m}, \end{aligned}$$
(45)
$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}\tau }\mathbf {F}_{m,\tau }^{-1}\right| _{\tau =0}= & {} -\mathbf {F}_\mathrm{m}^{-1}\left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \mathbf {F}_\mathrm{m}^{-1}\nonumber \\= & {} -\mathbf {F}_\mathrm{m}^{-1}\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) _\mathrm{m}, \end{aligned}$$
(46)
$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}\tau }\det \mathbf {F}_{m,\tau }\right| _{\tau =0}= & {} \det \mathbf {F}_\mathrm{m}(\mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m})\nonumber \\= & {} \det \mathbf {F}_\mathrm{m}(\hbox { div }_\mathrm{s}\delta \mathbf {u}_\mathrm{s})_\mathrm{m}, \end{aligned}$$
(47)
$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}\tau }\mathbf {E}_{m,\tau }\right| _{\tau =0}= & {} (\mathbf {F}_\mathrm{m}^{T}(\nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}))^{S}\nonumber \\= & {} \mathbf {F}_\mathrm{m}^{T}\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) _\mathrm{m}^{S}\mathbf {F}_\mathrm{m}. \end{aligned}$$
(48)
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}\tau }\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| \bigg |_{\tau =0}=\frac{\mathrm{d}}{\mathrm{d}\tau }\sqrt{\mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\cdot \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}}\bigg |_{\tau =0}\nonumber \\&\quad =\frac{1}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\left( -\mathbf {F}_\mathrm{m}^{-T}\left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) ^{T}\mathbf {F}_\mathrm{m}^{-T}\right) \mathbf {n}_{0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\nonumber \\&\quad =-\left[ \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right] _\mathrm{m}\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| \end{aligned}$$
(49)
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}\tau }\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| ^{-1}\bigg |_{\tau =0}\nonumber \\&\quad =-\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| ^{-3}\left( -\mathbf {F}_\mathrm{m}^{-T}\left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) ^{T}\mathbf {F}_\mathrm{m}^{-T}\right) \mathbf {n}_{0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\nonumber \\&\quad =\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| ^{-1}\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) _\mathrm{m}^{T}\left( \mathbf {n}_\mathrm{s}\right) _\mathrm{m}\cdot \left( \mathbf {n}_\mathrm{s}\right) _\mathrm{m}\end{aligned}$$
(50)
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}\tau }\frac{\mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\bigg |_{\tau =0}=-(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})_\mathrm{m}^{T}\frac{\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\nonumber \\&\qquad +\frac{\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| ^{2}}\frac{(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})_\mathrm{m}\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\nonumber \\&\quad = -\left[ \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\right] _\mathrm{m}+\left[ \left[ \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right] \mathbf {n}_\mathrm{s}\right] _\mathrm{m}\nonumber \\&\quad = +\left\{ \left[ -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}+\left( \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right) \mathbf {I}\right] \mathbf {n}_\mathrm{s}\right\} _\mathrm{m} \end{aligned}$$
(51)
Now, consider the displacement field \(\mathbf {u}_\mathrm{s}\) which is perturbed producing \(\mathbf {u}_{s,\tau }=\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s}\). Then \(\mathbf {F}_\mathrm{s}^{-1}=\mathbf {I}-\nabla _\mathrm{s}\mathbf {u}_\mathrm{s}\) results in \(\mathbf {F}_{s,\tau }^{-1}=\mathbf {I}-\nabla _\mathrm{s}\mathbf {u}_{s,\tau }=\mathbf {I}-\nabla _\mathrm{s}\left( \mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s}\right) \). The derivatives of several quantities involving \(\mathbf {F}_{s,\tau }\) with respect to \(\tau \) are
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\mathbf {F}_{s,\tau }^{-1}\right| _{\tau =0}=-\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}, \end{aligned}$$
(52)
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\mathbf {F}_{s,\tau }\right| _{\tau =0}=\mathbf {F}_\mathrm{s}\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \mathbf {F}_\mathrm{s},\end{aligned}$$
(53)
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\det \mathbf {F}_{s,\tau }\right| _{\tau =0}=\det \mathbf {F}_\mathrm{s}\left( \mathbf {F}_\mathrm{s}^{T}\cdot \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ,\end{aligned}$$
(54)
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\mathbf {E}_{s,\tau }\right| _{\tau =0}=\mathbf {F}_\mathrm{s}^{T}(\mathbf {F}_\mathrm{s}(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}))^{S}\mathbf {F}_\mathrm{s}. \end{aligned}$$
(55)
Appendix 2: Linearization procedures
1.1 Solid preload problem
For the preload problem is presented the linearization of the variational expressions (7)–(8). We recall that for this case \(\varOmega _\mathrm{s}\) is fixed and for each Newton–Raphson iteration a new material configuration \(\varOmega _\mathrm{m}^{k}\) is obtained, with points \(\mathbf {x}_\mathrm{m}^{k}=\mathbf {x}_\mathrm{s}-\mathbf {u}_\mathrm{s}^{k}\). In compact form the Problem 1 reads: find \(\left( \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}\right) \in {\mathcal {V}}_\mathrm{s}\times {\mathcal {L}}_\mathrm{s}\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} \langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s},\lambda _\mathrm{s}),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}=0 &{} \quad \forall \hat{\mathbf {u}}_\mathrm{s}\in {\mathcal {V}}_\mathrm{s}\\ \langle {\mathcal {J}}_\mathrm{s}(\mathbf {u}_\mathrm{s}),\hat{\lambda }_s\rangle _{\varOmega _\mathrm{s}}=0 &{} \quad \forall \hat{\lambda }_s\in {\mathcal {L}}_\mathrm{s} \end{array}\right. } \end{aligned}$$
(56)
The Newton–Raphson linearization applied to the above expression at the point \(\left( \mathbf {u}_\mathrm{s}^{k},\lambda _\mathrm{s}^{k}\right) \in {\mathcal {U}}_\mathrm{s}\times {\mathcal {L}}_\mathrm{s}\) and the increment/perturbation \(\left( \delta \mathbf {u}_\mathrm{s},\delta \lambda _\mathrm{s}\right) \in {\mathcal {V}}_\mathrm{s}\times {\mathcal {L}}_\mathrm{s}\) gives:
$$\begin{aligned}&\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s}^k,\lambda _\mathrm{s}^k),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\nonumber \\&\quad +\,\frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s}^k+\tau \delta \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}^k+\tau \delta \lambda _\mathrm{s}),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\bigg |_{\tau =0}=0\quad \forall \hat{\mathbf {u}}_\mathrm{s}\in {\mathcal {V}}_\mathrm{s} \end{aligned}$$
(57)
$$\begin{aligned}&\langle {\mathcal {J}}_\mathrm{s}(\mathbf {u}_\mathrm{s}^k),\hat{\lambda }_s\rangle _{\varOmega _\mathrm{s}}\nonumber \\&\quad +\,\frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {J}}_\mathrm{s}(\mathbf {u}_\mathrm{s}^k+\tau \delta \mathbf {u}_\mathrm{s}),\hat{\lambda }_s\rangle _{\varOmega _\mathrm{s}}\bigg |_{\tau =0}=0\quad \forall \hat{\lambda }_s\in {\mathcal {L}}_\mathrm{s} \end{aligned}$$
(58)
As shown in Appendix 1, to denote the presence of the perturbation \((\tau \delta \mathbf {u}_\mathrm{s})\) into the quantities that depend on \(\mathbf {u}_\mathrm{s}\), we introduce the additional index \(\tau \), i.e., \(\mathbf {F}_\mathrm{s}^{-1}=\mathbf {I}-\nabla _\mathrm{s}\mathbf {u}_\mathrm{s}\) results in \(\mathbf {F}_{s,\tau }^{-1}=\mathbf {I}-\nabla _\mathrm{s}\mathbf {u}_{s,\tau }=\mathbf {I}-\nabla _\mathrm{s}\left( \mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s}\right) \). For the sake of clarity, we omit index k on the fields \(\mathbf {u}_s\) and \(\lambda _s\) (and also on quantities that are updated at each iteration such as \(\mathbf {F}_s\)) on the further developments. Analyzing the expanded expression for the perturbed residual in the material configuration, contribution of the second term of Eq. (57), takes the form
$$\begin{aligned}&\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}+\tau \delta \lambda _\mathrm{s}),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{s}}\left[ -\left( \lambda _\mathrm{s}+\tau \delta \lambda _\mathrm{s}\right) \text {div}\hat{\mathbf {u}}_\mathrm{s}+\varvec{\sigma }_{s,\tau }\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s})\right] \, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}-{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}t_\mathrm{s}^{W,n}\mathbf {n}_\mathrm{s}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -\int \limits _{\partial \varOmega _\mathrm{s}^{N}}\mathbf {t}_\mathrm{s}^{N}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{N}\quad \forall \hat{\mathbf {u}}_\mathrm{s}\in {\mathcal {V}}_\mathrm{s}, \end{aligned}$$
(59)
where the perturbated expression for the Cauchy stress tensor \(\varvec{\sigma }_{s,\tau }\) reads
$$\begin{aligned} \varvec{\sigma }_{s,\tau }=\left( \frac{1}{\det \mathbf {F}_{s,\tau }}\mathbf {F}_{s,\tau }\mathbf {S}_{s,\tau }\mathbf {F}_{s,\tau }^{T}\right) . \end{aligned}$$
(60)
Analogously, for the second term of (58) , the perturbated residuals is written as
$$\begin{aligned} \langle {\mathcal {J}}_\mathrm{s}(\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s}),\hat{\lambda }_s\rangle _{\varOmega _\mathrm{s}} ={\displaystyle \int \limits _{\varOmega _\mathrm{s}}}[1-\det \mathbf {F}_{s,\tau }^{-1}]\hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s}=0\quad \forall \hat{\lambda }_s\in {\mathcal {L}}_\mathrm{s}.\nonumber \\ \end{aligned}$$
(61)
Using Eqs. (52)–(55) from “Appendix 1”, we obtain the derivatives of the perturbed equations
$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}+\tau \delta \lambda _s),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\bigg |_{\tau =0}\nonumber \\&\quad = -\int \limits _{\varOmega _\mathrm{s}}\left( \mathbf {F}_\mathrm{s}^{T}\cdot \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \varvec{\sigma }_\mathrm{s}\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{s}}2(\mathbf {F}_\mathrm{s}(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})\varvec{\sigma }_\mathrm{s})\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{s}}\mathbf {D}_\mathrm{s}(\mathbf {F}_\mathrm{s}\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})^{S}\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad -\int \limits _{\varOmega _\mathrm{s}}\delta \lambda _\mathrm{s}\text {div}\hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\varOmega _\mathrm{s}, \end{aligned}$$
(62)
and
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {J}}_\mathrm{s}(\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s}),\hat{\lambda }_s\rangle _{\varOmega _\mathrm{s}}\bigg |_{\tau =0}=-\int \limits _{\varOmega _\mathrm{s}}\left( \mathbf {F}_\mathrm{s}^{T}\cdot \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s} \nonumber \\ \end{aligned}$$
(63)
where S denotes the symmetric part, and \(\mathbf {D}_\mathrm{s}(\mathbf {F}_\mathrm{s}\nabla _{{s}}\delta \mathbf {u}_\mathrm{s})^{S}\) stands for
$$\begin{aligned} \mathbf {D}_\mathrm{s}(\mathbf {F}_\mathrm{s}\nabla _{{s}}\delta \mathbf {u}_\mathrm{s})^{S} =\frac{1}{\det \mathbf {F}_\mathrm{s}}\mathbf {F}_\mathrm{s}\left[ \left( \frac{\partial \mathbf {S}_\mathrm{m}}{\partial \mathbf {E}_\mathrm{m}}\right) _\mathrm{s}\mathbf {F}_\mathrm{s}^{T}(\mathbf {F}_\mathrm{s}\nabla _{{s}}\delta \mathbf {u}_\mathrm{s})^{\mathcal {S}}\mathbf {F}_\mathrm{s}\right] \mathbf {F}_\mathrm{s}^{T}.\nonumber \\ \end{aligned}$$
(64)
Then, \(\mathbf {D}_\mathrm{s}\) is a fourth-order tensor, whose components in a Cartesian system are given by
$$\begin{aligned} \left[ \mathbf {D}_\mathrm{s}\right] _{ijkl}= \frac{1}{\text {det }\mathbf {F}_\mathrm{s}}\left[ \mathbf {F}_\mathrm{s}\right] _{ia}\left[ \mathbf {F}_\mathrm{s}\right] _{jb}\left[ \mathbf {F}_\mathrm{s}\right] _{kc}\left[ \mathbf {F}_\mathrm{s}\right] _{ld}\left[ \left( \frac{\partial \mathbf {S}_\mathrm{m}}{\partial \mathbf {E}_\mathrm{m}}\right) _\mathrm{s}\right] _{abcd}. \nonumber \\ \end{aligned}$$
(65)
Collecting Eqs. (62) and (63) we can now formulate the linearized problem: given \(\left( \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}\right) \) (displacement and pressure fields on the previous iteration—omitted k index) find \(\left( \delta \mathbf {u}_\mathrm{s},\delta \lambda _\mathrm{s}\right) \in {\mathcal {V}}_\mathrm{s}\times {\mathcal {L}}_\mathrm{s}\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} d_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})+e_\mathrm{s}(\delta \lambda _\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})=g_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s}) &{} \forall \hat{\mathbf {u}}_\mathrm{s}\in {\mathcal {V}}_\mathrm{s}\\ f_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\lambda }_s)=h_\mathrm{s}\left( \hat{\lambda }_s\right) &{} \forall \hat{\lambda }_s\in {\mathcal {L}}_\mathrm{s} \end{array}\right. } \end{aligned}$$
(66)
with
$$\begin{aligned} d_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})= & {} -\int \limits _{\varOmega _\mathrm{s}}\left( \mathbf {F}_\mathrm{s}^{T}\cdot \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \varvec{\sigma }_\mathrm{s}\cdot \varvec{\varepsilon }_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&+\int \limits _{\varOmega _\mathrm{s}}2(\mathbf {F}_\mathrm{s}(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})\varvec{\sigma }_\mathrm{s})\cdot \varvec{\varepsilon }_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&+\int \limits _{\varOmega _\mathrm{s}}\mathbf {D}_\mathrm{s}(\mathbf {F}_\mathrm{s}\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})^{S}\cdot \varvec{\varepsilon }_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s},\end{aligned}$$
(67)
$$\begin{aligned} e_\mathrm{s}(\delta \lambda _\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})= & {} -\int \limits _{\varOmega _\mathrm{s}}\delta \lambda _\mathrm{s}\hbox { div }\hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\varOmega _\mathrm{s}, \end{aligned}$$
(68)
$$\begin{aligned} g_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s})= & {} -\int \limits _{\varOmega _\mathrm{s}}\left[ -\lambda _\mathrm{s}\hbox { div }\hat{\mathbf {u}}_\mathrm{s}+\varvec{\sigma }_\mathrm{s}\cdot \varvec{\varepsilon }_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s})\right] \, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&+{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}+{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}t_\mathrm{s}^{W,n}\mathbf {n}_\mathrm{s}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&+\int \limits _{\partial \varOmega _\mathrm{s}^{N}}\mathbf {t}_\mathrm{s}^{N}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{N} \end{aligned}$$
(69)
$$\begin{aligned} f_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\lambda }_s)= & {} -\int \limits _{\varOmega _\mathrm{s}}\left( \mathbf {F}_\mathrm{s}^{T}\cdot \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s} \end{aligned}$$
(70)
$$\begin{aligned} h_\mathrm{s}\left( \hat{\lambda }_s\right) =-{\displaystyle \int \limits _{\varOmega _\mathrm{s}}}[1-\det \mathbf {F}_\mathrm{s}^{-1}]\hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s} \end{aligned}$$
(71)
1.2 Solid direct problem
In this case the linearization of the variational expressions (10)–(11) is carried out taking into account that the material configuration \(\varOmega _\mathrm{m}\) is fixed. In abstract form the above expressions reads: find \((\mathbf {u}_\mathrm{m},\lambda _\mathrm{m})\in {\mathcal {U}}_\mathrm{m}\times {\mathcal {L}}_\mathrm{m}\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} \langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m},\lambda _\mathrm{m}),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}=0 &{} \quad \forall \hat{\mathbf {u}}_\mathrm{m}\in {\mathcal {V}}_\mathrm{m}\\ \langle {\mathcal {J}}_\mathrm{m}(\mathbf {u}_\mathrm{m}),\hat{\lambda }_m\rangle _{\varOmega _\mathrm{m}}=0 &{} \quad \forall \hat{\lambda }_m\in {\mathcal {L}}_\mathrm{m} \end{array}\right. } \end{aligned}$$
(72)
The Newton–Raphson linearization applied to the above expression at the point \((\mathbf {u}_\mathrm{m}^{k},\lambda _\mathrm{m}^{k})\in {\mathcal {U}}_\mathrm{m}\times {\mathcal {L}}_\mathrm{m}\) and the increment/perturbation \((\delta \mathbf {u}_\mathrm{m},\delta \lambda _\mathrm{m})\)
\(\in {\mathcal {V}}_\mathrm{m}\times {\mathcal {L}}_\mathrm{m}\) gives:
$$\begin{aligned}&\langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m}^{k},\lambda _\mathrm{m}^{k}),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\nonumber \\&\qquad +\,\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m}^{k}+\tau \delta \mathbf {u}_\mathrm{m},\lambda _\mathrm{m}^{k}),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}\nonumber \\&\qquad +\,\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m}^{k},\lambda _\mathrm{m}^{k}+\tau \delta \lambda ),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}\nonumber \\&\qquad =0\quad \forall \hat{\mathbf {u}}_\mathrm{m}\in {\mathcal {V}}_\mathrm{m} \end{aligned}$$
(73)
$$\begin{aligned}&\langle {\mathcal {J}}_\mathrm{m}(\mathbf {u}_\mathrm{m}^{k}),\hat{\lambda }_m\rangle _{\varOmega _\mathrm{m}}\nonumber \\&\quad +\,\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \langle {\mathcal {J}}_\mathrm{m}(\mathbf {u}_\mathrm{m}^{k}+\tau \delta \mathbf {u}_\mathrm{m}),\hat{\lambda }_m\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}=0\quad \forall \hat{\lambda }_m\in {\mathcal {L}}_\mathrm{m} \nonumber \\ \end{aligned}$$
(74)
As shown in “Appendix 1”, to denote the presence of the perturbation \(\left( \tau \delta \mathbf {u}_\mathrm{m}\right) \) into the quantities that depend on \(\mathbf {u}\), we introduce the additional index \(\tau \), i.e., \(\mathbf {F}_{m,\tau }=\mathbf {I}+\nabla _\mathrm{m}\left( \mathbf {u}_\mathrm{m}+\tau \delta \mathbf {u}_\mathrm{m}\right) =\mathbf {F}_\mathrm{m}+\tau \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\). As before, we omit index k indicating quantities evaluated at the previous iteration. Analyzing the expanded expression for the perturbed residual in the material configuration, contribution of the second term of Eq. (73), takes the form
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m}+\tau \delta \mathbf {u}_\mathrm{m},\lambda _\mathrm{m}),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}\nonumber \\&\quad = -\int \limits _{\varOmega _\mathrm{m}}\frac{\mathrm{d}}{\mathrm{d}\tau }\left[ \lambda _\mathrm{m}\left( \mathbf {F}_{m,\tau }^{-T}\cdot \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_{m,\tau }\right] \bigg |_{\tau =0}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{m}}\frac{\mathrm{d}}{\mathrm{d}\tau }[\mathbf {S}_\mathrm{m}(\mathbf {E}_{m,\tau })\cdot \dot{\mathbf {E}}_{\tau }(\hat{\mathbf {u}}_\mathrm{m})]\bigg |_{\tau =0}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \left( t_\mathrm{m}^{W,n}\mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_{m,\tau }\right| _{\tau =0}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {P}_{m,\tau }\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_{m,\tau }\right| _{\tau =0}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{N}}}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \left( \mathbf {t}_\mathrm{m}^{N}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_{m,\tau }\right| _{\tau =0}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{N},\nonumber \\ \end{aligned}$$
(75)
where \(\dot{\mathbf {E}}_{\tau }(\hat{\mathbf {u}}_\mathrm{m})=\frac{1}{2}[\mathbf {F}_{m,\tau }^{T}(\nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m})+(\nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m})^{T}\mathbf {F}_{m,\tau }]\). Using Eqs. (45)–(51) detailed in “Appendix 1” to perform the derivation with respect to \(\tau \) (and evaluate at \(\tau =0\)) the following expression is obtained
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m}+\tau \delta \mathbf {u}_\mathrm{m},\lambda _\mathrm{m}),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{m}}\lambda _\mathrm{m}\left[ \left( \mathbf {F}_\mathrm{m}^{-T}(\nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m})^{T}\mathbf {F}_\mathrm{m}^{-T}\right) \cdot (\nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m})\right] \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad -\int \limits _{\varOmega _\mathrm{m}}\lambda _\mathrm{m}\left[ \left( \mathbf {F}_\mathrm{m}^{-T}\cdot (\nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m})\right) \left( \mathbf {F}_\mathrm{m}^{-T}\cdot (\nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m})\right) \right] \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{m}}\left( \frac{\partial \mathbf {S}_\mathrm{m}}{\partial \mathbf {E}_\mathrm{m}}((\nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m})^{T}\mathbf {F}_\mathrm{m})^{S}\right) \cdot \dot{\mathbf {E}}\left( \hat{\mathbf {u}}_\mathrm{m}\right) \, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{m}}\mathbf {S}_\mathrm{m}(\mathbf {E}_\mathrm{m})\cdot ((\nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m})^{T}(\nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}))^{S}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left( t_\mathrm{m}^{W,n}\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_\mathrm{m}\left( \mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left[ t_\mathrm{m}^{W,n}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \left( \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right) \right| _{\tau =0}\cdot \hat{\mathbf {u}}_\mathrm{m}\right] \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {P}_{m,\tau }\right| _{\tau =0}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{m}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\left( \mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{m}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| _{\tau =0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{N}}}\left( \mathbf {t}_\mathrm{m}^{N}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\left( \mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \, \mathrm{d}\partial \varOmega _\mathrm{m}^{N}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{N}}}\left( \mathbf {t}_\mathrm{m}^{N}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \frac{\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \left( \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right) \right| _{\tau =0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| }\det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{N}\nonumber \\ \end{aligned}$$
(76)
where \(\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| _{\tau =0}=\left( -\mathbf {F}_\mathrm{m}^{-T}\left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) ^{T}\mathbf {F}_\mathrm{m}^{-T}\right) \mathbf {n}_{0}\) and
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {P}_{m,\tau }\right| _{\tau =0}=\left[ \frac{\mathrm{d}}{\mathrm{d}\tau }\frac{\mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\bigg |_{\tau =0}\otimes \frac{\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\right] ^{S}, \end{aligned}$$
with \(\frac{\mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\bigg |_{\tau =0}\) given by expression (51) provided in “Appendix 1”. Similarly, for the contribution of the third term of (73), a perturbation \((\tau \delta \lambda _\mathrm{m})\) is introduced, and the expression is derived. The calculation of this block is straightforward since \(\langle {\mathcal {R}}_\mathrm{m},\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\) is linear in \(\lambda _\mathrm{m}\), then
$$\begin{aligned}&{\displaystyle \left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{m}(\mathbf {u}_\mathrm{m},\lambda _\mathrm{m}+\tau \delta \lambda _\mathrm{m}),\hat{\mathbf {u}}_\mathrm{m}\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}}\nonumber \\&\quad = -{\displaystyle \int \limits _{\varOmega _\mathrm{m}}}\delta \lambda _\mathrm{m}\left( \mathbf {F}_\mathrm{m}^{-T}\cdot \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m}. \end{aligned}$$
(77)
Finally, the second term of (74) evaluated at \(\tau =0\) yields
$$\begin{aligned}&{\displaystyle \left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {J}}_\mathrm{m}(\mathbf {u}_\mathrm{m}+\tau \delta \mathbf {u}_\mathrm{m}),\hat{\lambda }_m\rangle _{\varOmega _\mathrm{m}}\right| _{\tau =0}}\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{m}}\det \mathbf {F}_\mathrm{m}(\mathbf {F}_\mathrm{m}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m})\hat{\lambda }_m\, \mathrm{d}\varOmega _\mathrm{m}. \end{aligned}$$
(78)
With these blocks we are able to write the linear problem: given \((\mathbf {u}_\mathrm{m},\lambda _\mathrm{m})\) (displacement and pressure fields at previous iteration—omitted k index), find \(\left( \delta \mathbf {u}_\mathrm{m},\delta \lambda _\mathrm{m}\right) \in {\mathcal {V}}_\mathrm{m}\times {\mathcal {L}}_\mathrm{m}\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} a_m\left( \delta \mathbf {u}_\mathrm{m},\hat{\mathbf {u}}_\mathrm{m}\right) + b_m\left( \delta \lambda _\mathrm{m},\hat{\mathbf {u}}_\mathrm{m}\right) =l_m\left( \hat{\mathbf {u}}_\mathrm{m}\right) &{} \forall \hat{\mathbf {u}}_\mathrm{m}\in {\mathcal {V}}_m\\ c_m\left( \delta \mathbf {u}_\mathrm{m},\hat{\lambda }_m\right) =m_m\left( \hat{\lambda }_m\right) &{} \forall \hat{\lambda }_m\in {\mathcal {L}}_m \end{array}\right. } \end{aligned}$$
(79)
where the linear and bilinear forms are obtained through
$$\begin{aligned}&l_m\left( \hat{\mathbf {u}}_\mathrm{m}\right) ={\displaystyle \int \limits _{\varOmega _\mathrm{m}}}\lambda _\mathrm{m}\left( \mathbf {F}_\mathrm{m}^{-T}\cdot \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\quad -{\displaystyle \int \limits _{\varOmega _\mathrm{m}}}\left( \varvec{\Sigma }_\mathrm{m}^{R}+\mathbf {S}_\mathrm{m}\left( \mathbf {E}_\mathrm{m}\right) \right) \cdot \dot{\mathbf {E}}\left( \hat{\mathbf {u}}_\mathrm{m}\right) \, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\quad +{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left( t_\mathrm{m}^{W,n}\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\quad +{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{m}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\frac{\det \mathbf {F}_\mathrm{m}}{\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| }\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\quad +{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{N}}}\left( \mathbf {t}_\mathrm{m}^{N}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{N}, \end{aligned}$$
(80)
$$\begin{aligned}&a_m\left( \left( \delta \mathbf {u}_\mathrm{m},\delta \lambda _\mathrm{m}\right) ,\hat{\mathbf {u}}_\mathrm{m}\right) \nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{m}}\lambda _\mathrm{m}\left[ \left( \mathbf {F}_\mathrm{m}^{-T}\left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) ^{T}\mathbf {F}_\mathrm{m}^{-T}\right) \cdot \left( \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \right] \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad -\int \limits _{\varOmega _\mathrm{m}}\lambda _\mathrm{m}\left[ \left( \mathbf {F}_\mathrm{m}^{-T}\cdot \left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \right) \left( \mathbf {F}_\mathrm{m}^{-T}\cdot \left( \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \right) \right] \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{m}}\left( \frac{\partial \mathbf {S}_\mathrm{m}}{\partial \mathbf {E}_\mathrm{m}}\left( \left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) ^{T}\mathbf {F}_\mathrm{m}\right) ^{S}\right) \cdot \dot{\mathbf {E}}\left( \hat{\mathbf {u}}_\mathrm{m}\right) \, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{m}}\mathbf {S}_\mathrm{m}\left( \mathbf {E}_\mathrm{m}\right) \cdot \left( \left( \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) ^{T}\left( \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \right) ^{S}\, \mathrm{d}\varOmega _\mathrm{m}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left( t_\mathrm{m}^{W,n}\mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_\mathrm{m}\left( \mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left[ t_\mathrm{m}^{W,n}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \left( \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right) \right| _{\tau =0}\cdot \hat{\mathbf {u}}_\mathrm{m}\right] \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {P}_{m,\tau }\right| _{\tau =0}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{m}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\left( \mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{m}\mathbf {t}_\mathrm{m}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{m}\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| _{\tau =0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right| }\det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{N}}}\left( \mathbf {t}_\mathrm{m}^{N}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| \det \mathbf {F}_\mathrm{m}\left( \mathbf {F}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \, \mathrm{d}\partial \varOmega _\mathrm{m}^{N}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{N}}}\left( \mathbf {t}_\mathrm{m}^{N}\cdot \hat{\mathbf {u}}_\mathrm{m}\right) \frac{\frac{\mathrm{d}}{\mathrm{d}\tau }\left. \left( \mathbf {F}_{m,\tau }^{-T}\mathbf {n}_{0}\right) \right| _{\tau =0}\cdot \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}}{\left| \mathbf {F}_\mathrm{m}^{-T}\mathbf {n}_{0}\right| }\det \mathbf {F}_\mathrm{m}\, \mathrm{d}\partial \varOmega _\mathrm{m}^{N},\end{aligned}$$
(81)
$$\begin{aligned}&b_m\left( \delta \lambda _\mathrm{m},\hat{\mathbf {u}}_\mathrm{m}\right) = -{\displaystyle \int \limits _{\varOmega _\mathrm{m}}}\delta \lambda _\mathrm{m}\left( \mathbf {F}_\mathrm{m}^{-T}\cdot \nabla _\mathrm{m}\hat{\mathbf {u}}_\mathrm{m}\right) \det \mathbf {F}_\mathrm{m}\, \mathrm{d}\varOmega _\mathrm{m},\end{aligned}$$
(82)
$$\begin{aligned}&c_m\left( \delta \mathbf {u}_\mathrm{m},\hat{\lambda }_m\right) =\int \limits _{\varOmega _\mathrm{m}}\det \mathbf {F}_\mathrm{m}\left( \mathbf {F}_\mathrm{m}^{-T}\cdot \nabla _\mathrm{m}\delta \mathbf {u}_\mathrm{m}\right) \hat{\lambda }_m\, \mathrm{d}\varOmega _\mathrm{m}, \end{aligned}$$
(83)
and
$$\begin{aligned} m_m\left( \hat{\lambda }_m\right) =-\int \limits _{\varOmega _\mathrm{m}}\left( \det \mathbf {F}_\mathrm{m}-1\right) \hat{\lambda }_m\, \mathrm{d}\varOmega _\mathrm{m} \end{aligned}$$
(84)
The above linear problem is convenient to be rewritten (evaluated) in terms of the variables now defined in an updated configuration \(\varOmega _\mathrm{s}^{k}\) (written as \(\varOmega _\mathrm{s}\) for sake of simplicity), with points \(x_\mathrm{s}^{k}=x_\mathrm{m}+u_\mathrm{m}^{k}\). To do that as a first step we seek for the spatial expression of the tangent components. With the expressions provided in “Appendix 1” in mind, we can write the spatial version of (76) as
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s}^{k}+\tau \delta \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\right| _{\tau =0}\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{s}}\lambda _\mathrm{s}((\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})^{T}\cdot (\nabla _\mathrm{s}\hat{\mathbf {u}}_\mathrm{s})-(\text {div}\delta \mathbf {u}_\mathrm{s})(\text {div}\hat{\mathbf {u}}_\mathrm{s}))\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{s}}(\mathbf {D}_\mathrm{s}\varvec{\varepsilon }(\delta \mathbf {u}_\mathrm{s})\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s})+(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})\varvec{\sigma }_\mathrm{s}\cdot (\nabla _\mathrm{s}\hat{\mathbf {u}}_\mathrm{s}))\, \mathrm{d}\varOmega _\mathrm{s},\nonumber \\&\qquad {\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left[ t_\mathrm{s}^{W,n}\left( \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}-\mathbf {I}\right) \mathbf {n}_\mathrm{s}\cdot \hat{\mathbf {u}}_\mathrm{s}\right] \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\left( \mathbf {H}\left( \delta \mathbf {u}_\mathrm{s}\right) \mathbf {n}_\mathrm{s}\otimes \mathbf {n}_\mathrm{s}\right) ^{S}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\left( \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\left( -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right) \, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{N}}}\left( \mathbf {t}_\mathrm{s}^{N}\cdot \hat{\mathbf {u}}_\mathrm{s}\right) \left[ \left( \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right] \, \mathrm{d}\partial \varOmega _\mathrm{s}^{N}\nonumber \\ \end{aligned}$$
(85)
where \(\mathbf {H}\left( \delta \mathbf {u}_\mathrm{s}\right) \) stands for
$$\begin{aligned} \mathbf {H}\left( \delta \mathbf {u}_\mathrm{s}\right) = \left[ -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}+\left( \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right) \mathbf {I}\right] , \end{aligned}$$
(86)
and recalling (65), \(\mathbf {D}_\mathrm{s}\varvec{\varepsilon }_\mathrm{s}(\delta \mathbf {u}_\mathrm{s})\) is given by
$$\begin{aligned} \mathbf {D}_\mathrm{s}\varvec{\varepsilon }_\mathrm{s}(\delta \mathbf {u}_\mathrm{s})= \frac{1}{\det \mathbf {F}_\mathrm{s}}\mathbf {F}_\mathrm{s}\left[ \left( \frac{\partial \mathbf {S}_\mathrm{m}}{\partial \mathbf {E}_\mathrm{m}}\right) _\mathrm{s}\mathbf {F}_\mathrm{s}^{T}\varvec{\varepsilon }(\delta \mathbf {u}_\mathrm{s})\mathbf {F}_\mathrm{s}\right] \mathbf {F}_\mathrm{s}^{T}. \end{aligned}$$
(87)
Now we add and subtract the term \(\lambda _\mathrm{s}(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})\cdot (\nabla _\mathrm{s}\hat{\mathbf {u}}_\mathrm{s})\) into Eq. (85) and consider the operation with second order tensors
$$\begin{aligned} \lambda _\mathrm{s}(\hbox { div }\delta \mathbf {u}_\mathrm{s})(\hbox { div }\hat{\mathbf {u}}_\mathrm{s})=\lambda _\mathrm{s}(\mathbf {I}\otimes \mathbf {I})\varvec{\varepsilon }(\delta \mathbf {u}_\mathrm{s})\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s}), \end{aligned}$$
(88)
leading to
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\right| _{\tau =0}\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{s}}(\mathbf {D}_\mathrm{s}+\lambda _\mathrm{s}(2{\mathbb {I}}-(\mathbf {I}\otimes \mathbf {I})))\varvec{\varepsilon }(\delta \mathbf {u}_\mathrm{s})\cdot \varvec{\varepsilon }(\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\quad +\int \limits _{\varOmega _\mathrm{s}}(\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})\varvec{\sigma }_\mathrm{s}\cdot (\nabla _\mathrm{s}\hat{\mathbf {u}}_\mathrm{s})\, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\quad +{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\left[ t_\mathrm{s}^{W,n}\left( \left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}-\mathbf {I}\right) \mathbf {n}_\mathrm{s}\cdot \hat{\mathbf {u}}_\mathrm{s}\right] \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\quad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\left( \mathbf {H}\left( \delta \mathbf {u}_\mathrm{s}\right) \mathbf {n}_\mathrm{s}\otimes \mathbf {n}_\mathrm{s}\right) ^{S}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\quad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\left( \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\quad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\left( -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right) \, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\quad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{N}}}\left( \mathbf {t}_\mathrm{s}^{N}\cdot \hat{\mathbf {u}}_\mathrm{s}\right) \left[ \left( \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right] \, \mathrm{d}\partial \varOmega _\mathrm{s}^{N}\nonumber \\ \end{aligned}$$
(89)
where \({\mathbb {I}}\) and \(\mathbf {I}\) are the fourth and second order identity tensors, respectively. Analogously, the spatial expressions (77) and (78) result in
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {R}}_\mathrm{s}(\mathbf {u}_\mathrm{s},\lambda _\mathrm{s}+\tau \delta \lambda _\mathrm{s}),\hat{\mathbf {u}}_\mathrm{s}\rangle _{\varOmega _\mathrm{s}}\right| _{\tau =0}\nonumber \\&\quad = -\int \limits _{\varOmega _\mathrm{s}}\delta \lambda _\mathrm{s}\hbox { div }\hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\varOmega _\mathrm{s}, \end{aligned}$$
(90)
and
$$\begin{aligned}&\left. \frac{\mathrm{d}}{\mathrm{d}\tau }\langle {\mathcal {J}}_\mathrm{s}(\mathbf {u}_\mathrm{s}+\tau \delta \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}),\hat{\lambda }_s\rangle _{\varOmega _\mathrm{s}}\right| _{\tau =0}\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{s}}\hbox { div }\delta \mathbf {u}_\mathrm{s}\hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s}. \end{aligned}$$
(91)
Hence, from Eqs. (85), (90) and (91), the spatial form of the linearized problem for incompressible materials is formulated as follows: given \(\left( \mathbf {u}_\mathrm{s},\lambda _\mathrm{s}\right) \) (displacement and pressure fields at previous Newton–Raphson iteration-omitted k index-) find \(\left( \delta \mathbf {u}_\mathrm{s},\delta \lambda _\mathrm{s}\right) \in {\mathcal {V}}_\mathrm{s}\times {\mathcal {L}}_\mathrm{s}\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} a_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})+b_\mathrm{s}(\delta \lambda _\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})=l_\mathrm{s}(\hat{\mathbf {u}}_\mathrm{s}) &{} \forall \hat{\mathbf {u}}_\mathrm{s}\in {\mathcal {V}}_\mathrm{s}\\ c_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\lambda }_s)=m_\mathrm{s}(\hat{\lambda }_s) &{} \forall \hat{\lambda }_s\in {\mathcal {L}}_\mathrm{s} \end{array}\right. } \end{aligned}$$
(92)
where the bilinear and linear forms are given by
$$\begin{aligned}&a_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})\nonumber \\&\quad =\int \limits _{\varOmega _\mathrm{s}}\left[ \mathbf {D}_\mathrm{s}+\lambda _\mathrm{s}\left( 2{\mathbb {I}}-\left( \mathbf {I}\otimes \mathbf {I}\right) \right) \right] \varvec{\varepsilon }\left( \delta \mathbf {u}_\mathrm{s}\right) \cdot \varvec{\varepsilon }\left( \hat{\mathbf {u}}_\mathrm{s}\right) \, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad +\int \limits _{\varOmega _\mathrm{s}}\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \varvec{\sigma }_\mathrm{s}\cdot \left( \nabla _\mathrm{s}\hat{\mathbf {u}}_\mathrm{s}\right) \, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\qquad +{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\left[ t_\mathrm{s}^{W,n}((\nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s})^{T}-\mathbf {I})\mathbf {n}_\mathrm{s}^{W}\cdot \hat{\mathbf {u}}_\mathrm{s}\right] {\hbox { div }_\mathrm{s}\delta \mathbf {u}_\mathrm{s}}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\left( \mathbf {H}\left( \delta \mathbf {u}_\mathrm{s}\right) \mathbf {n}_\mathrm{s}\otimes \mathbf {n}_\mathrm{s}\right) ^{S}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\left( \text {div}_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) \, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{m}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W,t}\cdot \hat{\mathbf {u}}_\mathrm{s}\left( -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right) \, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\qquad -{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{N}}}\left( \mathbf {t}_\mathrm{s}^{N}\cdot \hat{\mathbf {u}}_\mathrm{s}\right) \left[ \left( \hbox { div }_\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) -\left( \nabla _\mathrm{s}\delta \mathbf {u}_\mathrm{s}\right) ^{T}\mathbf {n}_\mathrm{s}\cdot \mathbf {n}_\mathrm{s}\right] \, \mathrm{d}\partial \varOmega _\mathrm{s}^{N},\end{aligned}$$
(93)
$$\begin{aligned}&b_\mathrm{s}(\delta \lambda _\mathrm{s},\hat{\mathbf {u}}_\mathrm{s})=-\int \limits _{\varOmega _\mathrm{s}}\delta \lambda _\mathrm{s}\hbox { div }\hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\varOmega _\mathrm{s},\end{aligned}$$
(94)
$$\begin{aligned}&l_\mathrm{s}\left( \hat{\mathbf {u}}_\mathrm{s}\right) =-{\displaystyle \int \limits _{\varOmega _\mathrm{s}}}\left[ -\lambda _\mathrm{s}\hbox { div }\hat{\mathbf {u}}_\mathrm{s}+\varvec{\sigma }_\mathrm{s}\cdot \varvec{\varepsilon }_\mathrm{s}\left( \hat{\mathbf {u}}_\mathrm{s}\right) \right] \, \mathrm{d}\varOmega _\mathrm{s}\nonumber \\&\quad +{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}\mathbf {P}_\mathrm{s}\mathbf {t}_\mathrm{s}^{W}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}+{\displaystyle \int \limits _{\partial \varOmega _\mathrm{s}^{W}}}t_\mathrm{s}^{W,n}\mathbf {n}_\mathrm{s}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{W}\nonumber \\&\quad +\int \limits _{\partial \varOmega _\mathrm{s}^{N}}\mathbf {t}_\mathrm{s}^{N}\cdot \hat{\mathbf {u}}_\mathrm{s}\, \mathrm{d}\partial \varOmega _\mathrm{s}^{N},\end{aligned}$$
(95)
$$\begin{aligned}&c_\mathrm{s}(\delta \mathbf {u}_\mathrm{s},\hat{\lambda }_s)={\displaystyle \int \limits _{\varOmega _\mathrm{s}}}\hbox { div }\delta \hat{\mathbf {u}}_\mathrm{s}\,\hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s}, \end{aligned}$$
(96)
and
$$\begin{aligned} m_\mathrm{s}\left( \hat{\lambda }_s\right) =-{\displaystyle \int \limits _{\varOmega _\mathrm{s}}}\left( 1-\det \mathbf {F}_\mathrm{s}^{-1}\right) \hat{\lambda }_s\, \mathrm{d}\varOmega _\mathrm{s}. \end{aligned}$$
(97)
1.3 Fluid problem
In order to solve numerically the variational problem set by Eqs. (30)–(32), a fixed-point method is used where the velocity of the reference domain at each iteration is calculated as \(\mathbf {w}^{k}=\frac{\mathbf {d}^{k}-\mathbf {d}^{n}}{\Delta t}\) (other choices are perfectly viable), with superscripts k and n referring to the solution at the previous fixed-point iteration and at the previous time step, respectively. The proposed fixed-point method reads: until convergence is achieved, for every iteration \(k=0,1,\ldots \) find \(\left( \mathbf {v}^{k+1},p^{k+1},\mathbf {d}^{k+1}\right) \in {\mathcal {V}}^{k}_t\times {\mathcal {P}}^{k}_t\times {\mathcal {D}}^{k}_t\) such that
$$\begin{aligned}&\int \limits _{\varUpsilon _t^k} \rho \frac{\partial \mathbf {v}^{k+1}}{\partial t}\cdot \hat{\mathbf {v}}\, \mathrm{d}\varUpsilon _t^k +\int \limits _{\varUpsilon _t^k}\rho \nabla \mathbf {v}^{k+1}\left( \mathbf {v}^k-\mathbf {w}^k\right) \cdot \hat{\mathbf {v}}\, \mathrm{d}\varUpsilon _t^k\nonumber \\&\qquad -\int \limits _{\varUpsilon _t^k}p^{k+1}\hbox { div }\hat{\mathbf {v}}\, \mathrm{d}\varUpsilon _t^k +\int \limits _{\varUpsilon _t^k}\varvec{\bar{\sigma }}\left( \mathbf {v}^{k+1}\right) \cdot \varvec{\varepsilon }\left( \hat{\mathbf {v}}\right) \, \mathrm{d}\varUpsilon _t^k\nonumber \\&\quad =\sum _{i=1}^C \int \limits _{\partial \varUpsilon _t^{k,A,i}}t_t^i\mathbf {n}^i\cdot \hat{\mathbf {v}}\, \mathrm{d}\partial \varUpsilon _t^{ k,A,i}\quad \forall \hat{\mathbf {v}}\in {\mathcal {V}}^{k,*}_t, \end{aligned}$$
(98)
$$\begin{aligned}&\int \limits _{\varUpsilon _t^k}\hat{p}\hbox { div }\mathbf {v}^{ k+1}\, \mathrm{d}\varUpsilon _t^k=0\quad \forall \hat{p}\in {\mathcal {P}}_t^k, \end{aligned}$$
(99)
$$\begin{aligned}&\int \limits _{\varUpsilon _t^k}\nabla \mathbf {d}^{k+1}\cdot \nabla \hat{\mathbf {d}}\, \mathrm{d}\varUpsilon _t^{ k}=0\quad \forall \hat{\mathbf {d}}\in {\mathcal {D}}^{k,*}_t, \end{aligned}$$
(100)
where \({\mathcal {V}}^{k,*}_t, {\mathcal {P}}_t^k\) and \({\mathcal {D}}^{k,*}_t\) are the counterparts of \({\mathcal {V}}^{*}_t, {\mathcal {P}}_t\) and \({\mathcal {D}}^{*}_t\) but with functions defined over \(\varUpsilon _t^k\). Considering a Crank–Nicolson scheme for the time integration, the time derivative is approximated as follows
$$\begin{aligned} \frac{\partial \mathbf {v}^{k+1}}{\partial t} = \frac{\mathbf {v}^{k+1} - \mathbf {v}^n}{\Delta t}, \end{aligned}$$
(101)
and all other terms in (98), except for the coupling with the pressure field, are evaluated at \(t_{n+\frac{1}{2}}=\frac{1}{2}(t_{n+1}+t_n)\).
