If inside the surface term of Eq. (
47), one makes explicit the dependence of the projectors on the normal vector components, the transport of the higher order contributions can be investigated without the recourse to the extended divergence theorem. By comparing the detailed expressions of the Eulerian and the Lagrangian terms in Eq. (
19), one has
$$\begin{aligned}&\int _{\Sigma }\left\{ -\frac{\partial }{\partial x^{c}} \left( T_{2\,i}^{ck}\,\right) \,n_{k}- T_{2\,i}^{ck} \frac{\partial \,n_{k}}{\partial x^{c}} +\right. \nonumber \\&\quad \left. +\frac{\partial }{\partial x^{r}} \left( T_{2\,i}^{jk}\right) \,n_{k}n^{r}n_{j} +T_{2\,i}^{jk}\,n_{k}n_{j}\frac{\partial n^{r}}{\partial x^{r}} \right\} \, \delta \chi ^{i}d\Sigma =\nonumber \\&=\int _{\Sigma _{\star }} \left\{ -\frac{\partial P_{2\,i}^{AB}}{\partial X^{B}} N_{A}-P_{2\,i}^{AB} \frac{\partial N_{A}}{\partial X^{B}}+\right. \nonumber \\&\quad \left. +\frac{\partial P_{2\,i}^{AB}}{\partial X^{C}}\,N_{B}N^{C}N_{A} +P_{2\,i}^{AB} N_{B}N_{A}\frac{\partial N^{D}}{\partial X^{D}} \, \right\} \delta \chi ^{i\,}d\Sigma _{\star }\,+\left( \lhd \right) \end{aligned}$$
(50)
where symbol
\(\left( \lhd \right) \) indicates the residual term previously discussed, in the form provided by Eq. (
41). Now, for the sake of simplicity, let us group the four addends at lhs of Eq. (
50) into two derivatives, and let us convert the Eulerian hyperstresses in their arguments through the formulae Eq. (
29), as follows
$$\begin{aligned}&\int _{\Sigma }\left\{ -\frac{\partial }{\partial x^{c}} \left( T_{2\,i}^{ck}n_{k}\,\right) +\frac{\partial }{\partial x^{r}} \left( T_{2\,i}^{jk}\,n_{k}n^{r}n_{j}\right) \right\} \, \delta \chi ^{i}d\Sigma =\nonumber \\ =\quad&\int _{\Sigma }\Bigg \{\underbrace{-\frac{\partial }{\partial x^{c}} \left( J^{-1}\,P_{2\,i}^{AB}\,F_{A}^{c}F_{B}^{k}\,n_{k}\right) }_{=\left( \bowtie \right) }\,+\underbrace{\frac{\partial }{\partial x^{r}} \left( J^{-1}\,P_{2\,i}^{AB}\,F_{A}^{j}F_{B}^{k}\,n_{k}n^{r}n_{j}\right) }_{=\left( \curlyvee \right) } \, \Bigg \}\, \delta \chi ^{i}\,d\Sigma \end{aligned}$$
(51)
After the change of variables, by applying Piola’s transformation and utilizing the transport formula for the covariant normal vector components, one finds for the former addend
$$\begin{aligned} \left( \bowtie \right)&=\int _{\Sigma _{\star }}-J^{-1}\underbrace{F_{A}^{c}\frac{\partial }{\partial x^{c}} }_{=\partial (\cdot )\,/\partial X^{A}} \left( \,P_{2\,i}^{AB}\frac{N_{B}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \delta \chi ^{i}\Vert J{\mathbf {F}}^{-T}{\mathbf {N}}\Vert \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\delta \chi ^{i}\,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \,\left\{ -\frac{\partial \, P_{2\,i}^{AB}}{\partial X^{A}} \frac{N_{B}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }-P_{2\,i}^{AB}\frac{\partial \,N_{B}}{\partial X^{A}} \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }+\right. \nonumber \\&\quad \left. -P_{2\,i}^{AB}N_{B}\frac{\partial }{\partial X^{A}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \right\} \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\delta \chi ^{i}\,\left\{ -\frac{\partial \, P_{2\,i}^{AB}}{\partial X^{A}}N_{B} -P_{2\,i}^{AB}\frac{\partial \,N_{B}}{\partial X^{A}}\right\} \,d\Sigma _{\star }+\nonumber \\&\quad \underbrace{-\int _{\Sigma _{\star }}\delta \chi ^{i}\,P_{2\,i}^{AB}N_{B}\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \frac{\partial }{\partial X^{A}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \,d\Sigma _{\star }}_{=\mathrm {res}_{1}} \end{aligned}$$
(52)
In the above equation, we have successfully retrieved the first two Lagrangian addends at rhs of Eq. (
50), plus a residual addend
\(\mathrm {res}_{1}\) which will be considered later. As for the second addend of Eq. (
51), utilizing the novel transport formula for the contravariant normal vector
\(n^{r}\) proposed in [Part I, Eq. (66)], one finds
$$\begin{aligned} \left( \curlyvee \right)&=\int _{\Sigma _{\star }}\,\delta \chi ^{i}\,\Vert J{\mathbf {F}}^{-T}{\mathbf {N}}\Vert \, \left\{ \frac{\partial }{\partial x^{r}} \left( J^{-1}\,P_{2\,i}^{AB}\,\frac{N_{A}N_{B}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\,\frac{g^{\star \,E}_{\,\,S}F^{r}_{E}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \, \right\} \, \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\,\delta \chi ^{i}\,\Vert J{\mathbf {F}}^{-T}{\mathbf {N}}\Vert \, \Bigg \{ J^{-1}\underbrace{F^{r}_{E}\frac{\partial }{\partial x^{r}}}_{=\partial (\cdot )\,/\partial X^{E}} \left( \,P_{2\,i}^{AB}\,\frac{N_{A}N_{B}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\,\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \, \Bigg \}\, \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\,\delta \chi ^{i}\,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \, \left\{ \frac{\partial }{\partial X^{E}} \left( \,P_{2\,i}^{AB}\,N_{A}N_{B}\,\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\right) \, \right\} \, \,d\Sigma _{\star }= \end{aligned}$$
(53)
Differentiating the product within parentheses by the Leibniz rule and simplifying whenever possible the surface element norm, one finds
$$\begin{aligned}&=\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, \left\{ \frac{\partial \, P_{2\,i}^{AB}}{\partial X^{E}} \, \,\,N_{A}N_{B}\,\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\,\, \right\} \, \,d\Sigma _{\star }+\nonumber \\&\quad +\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, \left\{ P_{2\,i}^{AB}\,\frac{\partial }{\partial X^{E}} \left( \,N_{A}N_{B}\right) \,\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\, \right\} \, \,d\Sigma _{\star }+\nonumber \\&\quad +\int _{\Sigma _{\star }}\,\delta \chi ^{i}\,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \, \left\{ P_{2\,i}^{AB}\,N_{A}N_{B}\,\frac{\partial }{\partial X^{E}} \left( \,\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\right) \, \right\} \, \,d\Sigma _{\star } \end{aligned}$$
(54)
As already discussed in [Part I, Eqs. (67)–(68)],the Lagrangian vector
\(g^{\star \,E}_{\,\,S}N^{S}\) possesses non-vanishing components in both the tangent and the normal space, namely
$$\begin{aligned}&\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}= N^{E}+\left( \frac{n^{t}\left( {\mathbf {F}}^{-1}\right) ^{E}_{t}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }-N^{E}\right) ; \end{aligned}$$
Hence, since the above pullback metric tensor appears once in each addend, also Equation (
54) can be decomposed in the form
\(\left( \curlyvee \right) =\left( \curlyvee _{\perp }\right) +\left( \curlyvee _{\parallel }\right) \). Firstly, let us develop the orthogonal contribution
$$\begin{aligned} \left( \curlyvee _{\perp }\right)&=\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, \left\{ \frac{\partial \, P_{2\,i}^{AB}}{\partial X^{E}} \,N_{A}N_{B}\,N^{E}\, \right\} \, \,d\Sigma _{\star }+\nonumber \\&\quad +\underbrace{\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, \left\{ P_{2\,i}^{AB}\,\frac{\partial }{\partial X^{E}} \left( \,N_{A}N_{B}\right) \,N^{E}\, \right\} \, \,d\Sigma _{\star }}_{=0}+\nonumber \\&\quad +\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, P_{2\,i}^{AB}\,N_{A}N_{B}\,\frac{\partial \, N^{E}}{\partial X^{E}} \,d\Sigma _{\star }+\nonumber \\&\quad +\underbrace{ \int _{\Sigma _{\star }}\,\delta \chi ^{i}\,P_{2\,i}^{AB}\,N_{A}N_{B}N^{E}\frac{\partial }{\partial X^{E}} \left( \,\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \,d\Sigma _{\star }}_{\mathrm {res}_{2}}; \end{aligned}$$
(55)
thus retrieving two Lagrangian addends, plus one additional term
\(\mathrm {res}_{2}\) to be discussed later. For the parallel contribution
\(\left( \curlyvee _{\parallel }\right) \) instead, one has:
$$\begin{aligned} \left( \curlyvee _{\parallel }\right)&=\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, \left\{ \frac{\partial \, P_{2\,i}^{AB}}{\partial X^{E}} \,N_{A}N_{B}\,\left( \frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}-N^{E}\right) \, \right\} \, \,d\Sigma _{\star }+\nonumber \\&\quad +\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, \left\{ P_{2\,i}^{AB}\,\frac{\partial }{\partial X^{E}} \left( \,N_{A}N_{B}\right) \,\left( \frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}-N^{E}\right) \, \right\} \, \,d\Sigma _{\star }+\nonumber \\&\quad +\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, P_{2\,i}^{AB}\,N_{A}N_{B}\,\frac{\partial }{\partial X^{E}} \left( \frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}-N^{E}\right) \,d\Sigma _{\star }+\nonumber \\&\quad +\underbrace{\int _{\Sigma _{\star }}\,\delta \chi ^{i}\, P_{2\,i}^{AB}\,N_{A}N_{B} \,\left( \frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}-N^{E}\right) \, \frac{\partial }{\partial X^{E}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert d\Sigma _{\star }}_{=0} \end{aligned}$$
(56)
The Lagrangian surface term still unbalanced, which was indicated at rhs of Eq. (
50), can be now recalled in the suggestive form of Eq. (
41), finding that
$$\begin{aligned} \left( \lhd \right)&=- \int _{\Sigma _{\star }} \frac{\partial }{\partial X^{E}}\left( P_{2\,i}^{AB}\,N_{A}N_{B}\right) \left\{ \frac{g^{\star \,ES}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}} N_{S}-N^{E}\right\} \delta \chi ^{i}\,d\Sigma _{\star }+\nonumber \\&\quad -\int _{\Sigma _{\star }} P_{2\,i}^{AB}\,N_{A}N_{B} \frac{\partial }{\partial X^{L}} \left( \frac{g^{\star \,LS}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}} N_{S}-N^{L}\right) \delta \chi ^{i}\,d\Sigma _{\star }+\nonumber \\&\quad +\underbrace{\int _{\Sigma _{\star }} P_{2\,i}^{AB}\,N_{A}N_{B}N_{L}N^{E}\frac{\partial }{\partial X^{E}}\left( \frac{g^{\star \,LS}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}} N_{S}-N^{L}\right) \, \delta \chi ^{i}\,d\Sigma _{\star }}_{=0} \end{aligned}$$
(57)
One can observe that the first two addends of
\(\left( \lhd \right) \) in Eq. (
57), after differentiation of the vector
\(P_{2\,i}^{AB}\,N_{A}N_{B}\) by the product rule, cancel out with the first three terms in
\(\left( \curlyvee _{\parallel }\right) \) Eq. (
56). To proceed further, we are going to prove that the last addend in
\(\left( \curlyvee _{\parallel }\right) \) and the third addend in
\(\left( \lhd \right) \) vanish, as annotated
\((=0)\) below the relevant expressions (next points i-ii), and then we will discuss the two residual terms res
\(_{1}\) and res
\(_{2}\) (at point iii).
(i) About the last addend in Eq. (
56), recalling formulae for the derivative of the surface element norm [Part I, Eqs. (32)–(35)], we notice that
$$\begin{aligned}&-N^{E} \frac{\partial }{\partial X^{E}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) +\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}} \frac{\partial }{\partial X^{E}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) =\nonumber \\&\quad =-N^{E}\left( -\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\,g^{\star \,RM}N_{R}\,N_{M|E}\right) +\nonumber \\&\qquad +\frac{g^{\star \,E}_{\,\,S}N^{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\left( -\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\,g^{\star \,RM}N_{R}\,N_{M|E}\right) =\nonumber \\&\quad =+\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\,g^{\star \,RM}N_{R}\,N_{M|E}N^{E} -\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{5}}\,g^{\star \,ES}N_{S}\,g^{\star \,RM}N_{R}\,N_{M|E}=\nonumber \\&\quad =-\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\,g^{\star \,RM}N_{R}\,\Gamma _{ME}^{S}N_{S}N^{E} +\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}\,g^{\star \,RM}N_{R}\,\Gamma _{ME}^{S}N_{S}N^{E}=0; \end{aligned}$$
(58)
In the last row of Eq. (
58), to simplify and scale the second addend, including the covariant derivative
\(N_{M|E}\) contracted with two pullback metric tensors, we utilized the remarkable relationship [Part I, Eq. (43)]. Such relationship among Lagrangian variables was proven starting from the condition of vanishing normal derivative for the normal vector in the Eulerian configuration, see [Part I, Eqs. (41)–(44)]. Moreover, in the last passage we have indicated explicitly the Christoffel symbols
\(\Gamma _{ME}^{S}\) (see [Part I, Eqs. (30)–(31)] to emphasize the fact that, after multiplying by
\(N^{E}\) the covariant derivative
\(N_{M|E}\), the addend including the (ordinary) derivative of the normal along the normal direction vanishes, see also Eq. (
20).
(ii) As for the vanishing addend of Eq. (
57), a few preliminary steps are herein outlined for the reader’s convenience. Firstly, let us consider the derivative of
\(g^{\star \,LS}N_{S}\) alone. Since the metrics
\(g^{tj}\) in the Eulerian configuration equals herein the unit tensor, one finds
$$\begin{aligned}&\frac{\partial }{\partial X^{E}}\left( g^{\star \,LS}N_{S}\right) =\frac{\partial }{\partial X^{E}}\left( g^{tj}\left( {\mathbf {F}}^{-1}\right) ^{L}_{t}\left( {\mathbf {F}}^{-1}\right) ^{S}_{j}N_{S}\right) =\nonumber \\&=g^{tj}\frac{\partial \left( {\mathbf {F}}^{-1}\right) ^{L}_{t}}{\partial X^{E}}\left( {\mathbf {F}}^{-1}\right) ^{S}_{j}N_{S}+ g^{tj}\left( {\mathbf {F}}^{-1}\right) ^{L}_{t}\frac{\partial \left( {\mathbf {F}}^{-1}\right) ^{S}_{j}}{\partial X^{E}}N_{S}+\nonumber \\&\quad +g^{tj}\left( {\mathbf {F}}^{-1}\right) ^{L}_{t}\left( {\mathbf {F}}^{-1}\right) ^{S}_{j}\frac{\partial \, N_{S}}{\partial X^{E}}= \end{aligned}$$
(59)
and, taking into account the formula (see, e.g. [
36])
$$\begin{aligned}&\frac{\partial \, }{\partial X^{E}}\left( {\mathbf {F}}^{-1}\right) ^{A}_{a}=\frac{\partial \, }{\partial F^{i}_{M}}\left( {\mathbf {F}}^{-1}\right) ^{A}_{a}\,\frac{\partial \, F^{i}_{M}}{\partial X^{E}}=-\left( {\mathbf {F}}^{-1}\right) ^{A}_{i}\left( {\mathbf {F}}^{-1}\right) ^{M}_{a}F^{i}_{ME}; \end{aligned}$$
utilizing the definition of covariant derivative with the Christoffel symbols [Part I, Eqs. (
30)–(
31)], from Eq. (
59) one has
$$\begin{aligned}&=-g^{tj}\left( {\mathbf {F}}^{-1}\right) ^{L}_{i}\left( {\mathbf {F}}^{-1}\right) ^{M}_{t}F^{i}_{ME}\,\left( {\mathbf {F}}^{-1}\right) ^{S}_{j}N_{S}+\nonumber \\&\quad -g^{tj}\left( {\mathbf {F}}^{-1}\right) ^{L}_{t}\left( {\mathbf {F}}^{-1}\right) ^{S}_{i}\left( {\mathbf {F}}^{-1}\right) ^{M}_{j}F^{i}_{ME}\,N_{S}+ g^{tj}\left( {\mathbf {F}}^{-1}\right) ^{L}_{t}\left( {\mathbf {F}}^{-1}\right) ^{S}_{j}\frac{\partial \, N_{S}}{\partial X^{E}}=\nonumber \\&=-g^{\star \,MS}\Gamma ^{L}_{ME} N_{S}-g^{\star \,LM}\Gamma ^{S}_{ME}N_{S}+g^{\star \,LS}\frac{\partial \, N_{S}}{\partial X^{E}}=\nonumber \\&=-g^{\star \,MS}\Gamma ^{L}_{ME} N_{S}+g^{\star \,LM}\,N_{M|E}; \end{aligned}$$
(60)
By adding the surface element norm squared to the denominator in Eq. (
59), the above derivative becomes
$$\begin{aligned}&\frac{\partial }{\partial X^{E}}\left( \frac{g^{\star \,LS}N_{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\right) =\left( g^{\star \,LM}\,N_{M|E}-g^{\star \,MS}\Gamma ^{L}_{ME} N_{S}\right) \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}+\nonumber \\&\quad +g^{\star \,LS}N_{S}\left( -\frac{2}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{4}}g^{\star \,RM}\,N_{R}\,N_{M|E}\right) ; \end{aligned}$$
(61)
where the norm derivative was computed by the formula [Part I, Eq. (
34)]. Moreover, multiplying Eq. (
61) by
\(N_{L}N^{E}=[M_{\perp }]_{L}^{E}\) (i.e. by the orthogonal projector), one finds
$$\begin{aligned}&N_{L}N^{E}\frac{\partial }{\partial X^{E}}\left( \frac{g^{\star \,LS}N_{S}}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\right) =\nonumber \\&=N_{L}N^{E}\left( g^{\star \,LM}\,N_{M|E}-g^{\star \,MS}\Gamma ^{L}_{ME} N_{S}\right) \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}+\nonumber \\&\quad +\underbrace{\left( N_{L}g^{\star \,LS}N_{S}\right) }_{=\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\, N^{E}\left( -\frac{2}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{4}}g^{\star \,RM}\,N_{R}\,N_{M|E}\right) =\nonumber \\&=\left( g^{\star \,LM}N_{L}\,N_{M|E}N^{E}-g^{\star \,MS}N_{S}\,\Gamma ^{L}_{ME}N_{L}N^{E}+\right. \nonumber \\&\quad \left. -2\,g^{\star \,RM}\,N_{R}\,N_{M|E}N^{E}\right) \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}= \end{aligned}$$
(62)
As already noticed, the product between
\(N^{E}\) and the above covariant derivative
\(N_{M|E}\) cancels in the latter the addend with the ordinary directional derivative of the normal, see Eq. (
20). Therefore, one can indicate explicitly the Christoffel symbols, finding that:
$$\begin{aligned}&=\left( -g^{\star \,LM}N_{L}\,\Gamma ^{S}_{ME}N_{S}N^{E}-g^{\star \,MS}N_{S}\,\Gamma ^{L}_{ME}N_{L}N^{E}+\right. \nonumber \\&\quad \left. +2\,g^{\star \,RM}N_{R}\,\Gamma ^{S}_{ME}N_{S}N^{E}\right) \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}=0; \end{aligned}$$
(63)
In the same equation (
57), the derivative of
\(N^{L}\) does not affect the final result, being
\(N_{L}N^{E}\frac{\partial N^{L}}{\partial X^{E}}=0\).
(iii) To complete the transport of the boundary face equations from the Eulerian to the Lagrangian configuration, two terms remain still unbalanced (from Eqs.
56 and
57), and their sum must vanish. By simple manipulations, recalling the covariant derivative of the norm [Part I, Eq. (
28)], one has
$$\begin{aligned} \mathrm {res}_{1}+\mathrm {res}_{2}&=-\int _{\Sigma _{\star }}\delta \chi ^{i}\,P_{2\,i}^{AB}N_{B}\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \frac{\partial }{\partial X^{A}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \,d\Sigma _{\star }+\nonumber \\&\quad +\int _{\Sigma _{\star }}\,\delta \chi ^{i}\,P_{2\,i}^{AB}\,N_{A}N_{B}N^{E}\frac{\partial }{\partial X^{E}} \left( \,\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\delta \chi ^{i}\,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \,P_{2\,i}^{AB}N_{B}\left\{ -\frac{\partial }{\partial X^{A}}\left( \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) +\right. \nonumber \\&\quad \left. +N_{A}N^{E}\frac{\partial }{\partial X^{E}} \left( \,\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert }\right) \right\} \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\delta \chi ^{i}\,\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert \,P_{2\,i}^{AB}N_{B}\left\{ +\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}g^{\star \,RM}N_{R}\,N_{M|A}+\right. \nonumber \\&\quad \left. +N_{A}\left( -\frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{3}}g^{\star \,RM}N_{R}\,N_{M|E}N^{E}\right) \right\} \,d\Sigma _{\star }=\nonumber \\&=\int _{\Sigma _{\star }}\delta \chi ^{i}\, \frac{1}{\Vert {\mathbf {F}}^{-T}{\mathbf {N}}\Vert ^{2}}\,P_{2\,i}^{AB}N_{B} \left\{ +g^{\star \,RM}N_{R}\,N_{M|A}+\right. \nonumber \\&\quad \left. +N_{A}\left( g^{\star \,RM}N_{R}\,\Gamma ^{S}_{ME}N_{S}N^{E}\right) \right\} \,d\Sigma _{\star } \end{aligned}$$
(64)
Condition for the above surface integral to vanish (
\(\forall \, \delta \chi ^{i}\),
\(F^{i}_{A}\),
\(P_{2\,i}^{AB}\)) is that the two addends in curly brackets satisfy the following equality
$$\begin{aligned}&+g^{\star \,RM}N_{R}\,N_{M|A}=-N_{A}\left( g^{\star \,RM}N_{R}\,\Gamma ^{S}_{ME}N_{S}N^{E}\right) ; \end{aligned}$$
(65)
By multiplying both sides of the above equation by the contravariant component
\(N^{A}\), one finds
$$\begin{aligned}&N^A \,g^{\star \,RM}N_{R}\,N_{M|A}=-g^{\star \,RM}N_{R}\,\Gamma ^{S}_{MA}N_{S}N^{A}=\nonumber \\&=-\underbrace{N^{A}N_{A}}_{=1}\left( g^{\star \,RM}N_{R}\,\Gamma ^{S}_{ME}N_{S}N^{E}\right) ; \end{aligned}$$
(66)
The change of sign at lhs was due to the fact that, for the covariant derivative of the covariant vectors, Christoffel symbols are subtracted. Hence, the residual addends in Eq. (
64) cancel out. The transformation from the Eulerian to the Lagrangian form of all the contributions to the inner virtual work was finally completed. The transformation of the external work terms will be investigated elsewhere.