Nonlinear Elasticity in a Deforming Ambient Space

Abstract

In this paper, we formulate a nonlinear elasticity theory in which the ambient space is evolving. For a continuum moving in an evolving ambient space, we model time dependency of the metric by a time-dependent embedding of the ambient space in a larger manifold with a fixed background metric. We derive both the tangential and the normal governing equations. We then reduce the standard energy balance written in the larger ambient space to that in the evolving ambient space. We consider quasi-static deformations of the ambient space and show that a quasi-static deformation of the ambient space results in stresses, in general. We linearize the nonlinear theory about a reference motion and show that variation of the spatial metric corresponds to an effective field of body forces.

Appendices

Appendix 1: Geometry of Riemannian Submanifolds

In the following, we tersely review a few elements of the geometry of embedded submanifolds. Here we mainly follow do Carmo (1992), Capovilla and Guven (1995), Spivak (1999), and Kuchař (1976). Let us consider a Riemannian manifold $\mathcal {S}$ embedded in another Riemannian manifold $\mathcal {Q}$ and assume that $\dim \mathcal {S}<\dim \mathcal {Q}\,$. We consider a time-dependent embedding $\psi _t:\mathcal {S}\rightarrow \mathcal {Q}\,$. The metric $\varvec{h}$ on $\mathcal {Q}$ induces a metric ${\varvec{g}_t}=\psi _t^*\varvec{h}$ on $\mathcal {S}$ (the first fundamental form). At any given point p of $\mathcal {S}\,$, the tangent space $T_{p}\mathcal {S}_t$ has an orthogonal complement $\left( T_{p}\mathcal {S}_t\right) ^{\perp }\subset T_{}\mathcal {Q}$ such that

$$\begin{aligned} T_{p}\mathcal {Q} = T_{p}\mathcal {S}_t \oplus \left( T_{p}\mathcal {S}_t\right) ^{\perp }. \end{aligned}$$

(4.1)

Note that such a decomposition is smooth in the sense that any smooth vector field $\varvec{u}$ on $\mathcal {S}_t$ can be smoothly decomposed into a vector field $\varvec{u}_\parallel$ tangent to $\mathcal {S}_t$ and a vector field $\varvec{u}_\perp$ normal to $\mathcal {S}_t\,$, so that $p\rightarrow (\varvec{u}_\parallel )_p= (\varvec{u}_p)_\parallel$ and $p\rightarrow (\varvec{u}_\perp )_p = (\varvec{u}_p)_\perp$ are smooth. We write $\varvec{u} = \varvec{u}_\parallel + \varvec{u}_\perp \,$. The orientation of $\varvec{\eta }^t_i$, for $i\in \{1,\ldots ,k\}\,$, is chosen such that the orientations of $\mathcal {S}_t$ and $\mathcal {Q}$ are consistent in the sense that the orientation induced from $\mathcal {S}_t$ along with the ordered sequence $\{\varvec{\eta }^t_i\}_{i\in \{1,\ldots ,k\}}\,$ is equivalent to the orientation of $\mathcal {Q}\,$. Let $\dim \mathcal {S} = n$ and $\dim \mathcal {Q} = n + k = m\,$. Following the smoothness of the decomposition (4.1), one can take a set of smooth vector fields $\{\varvec{\eta }^t_i\}_{i=1,\ldots ,k}$ normal to $\mathcal {S}_t$ such that they form an orthonormal basis for $\mathfrak {X}^\perp (\mathcal {S}_t)\,$, the set of vector fields normal to $\mathcal {S}_t\,$. Let $\{\chi ^\alpha \}_{\alpha =1,\ldots ,n+k}$ be a local coordinate chart for $\mathcal {Q}$ such that at any point of $\mathcal {S}_t, \{\chi ^1,\ldots ,\chi ^{n}\}$ is a local coordinate chart for $\mathcal {S}_t\,$, and such that the ith unit normal vector field $\varvec{\eta }^t_i$ for $i\in \{1,\ldots ,k\}$ is tangent to the coordinate curve $\chi ^{n+i}\,$. Hence, every vector field $\varvec{u}$ on $\mathcal {Q}$ along $\mathcal {S}_t$ can be written as $\varvec{u} = \varvec{u}_\parallel + \sum _{i=1}^k u_\perp ^i \varvec{\eta }^t_i\,$.¹⁹ Note that, for $i,j\in \{1,\ldots ,k\}\,$, one has $\left\langle \!\left\langle \varvec{\eta }^t_i,\varvec{\eta }^t_j \right\rangle \!\right\rangle _{\varvec{h}} = \delta _{ij}$ and $\left\langle \!\left\langle \varvec{\eta }^t_i,\varvec{u}_\parallel \right\rangle \!\right\rangle _{\varvec{h}} = 0\,$, where the Kronecker delta symbol $\delta _{ij}$ is defined as: $\delta _{ij}=1$ if $i=j$ and $\delta _{ij}=0$ if $i\ne j$. Note that at any point of $\mathcal {S}_t\,$, one has $h_{\alpha (n+i)} = \delta _{\alpha (n+i)}\,$, for $i\in \{1,\ldots ,k\}$ and $\alpha \in \{1,\ldots ,n+k\}\,$. We denote the connection coefficients for the Levi–Civita connections $\nabla ^{\varvec{h}}$ and $\nabla ^{{\varvec{g}_t}}$ corresponding to the metrics $\varvec{h}$ and $\varvec{g}_t$ by $\tilde{\gamma }^{\alpha }_{\beta \gamma }$ and $\gamma ^a_{bc}\,$, respectively. We denote by $D^{\varvec{h}}_t$ and $D^{\varvec{g}_t}_t$ the covariant derivatives along $\tilde{\varphi }_X$ and $\varphi _X\,$, respectively. For a vector field $\varvec{u}$ on $\mathcal {Q}$ along $\mathcal {S}_t\,$, we write $D^{\varvec{h}}_t \varvec{u} = \frac{\partial u^\alpha }{\partial t}\tilde{\partial }^t_\alpha + \frac{\partial u_\perp ^i}{\partial t}\varvec{\eta }^t_i+ \nabla ^{\varvec{h}}_{\varvec{\Upsilon }}\varvec{u}\,$ and for a vector field $\varvec{w}$ on $\mathcal {S}, D^{\varvec{g}_t}_t \varvec{w} = \frac{\partial w^a}{\partial t}\partial ^t_a + \nabla ^{\varvec{g}_t}_{\varvec{V}}\varvec{w}\,$, where $\{\tilde{\partial }^t_\alpha \}_{\alpha =1,\ldots ,n}$ and $\{\partial ^t_a\}_{a=1,\ldots ,n}$ denote local coordinate bases for $\mathcal {S}_t$ and $\mathcal {S}\,$, respectively.

Note that for vector fields $\varvec{X}$ and $\varvec{Y}$ defined on $\mathcal {S}_t$ and $\mathcal {Q}\,$, respectively, such that $\varvec{Y}$ is everywhere tangent to $\mathcal {S}_t, \nabla ^{{\varvec{g}_t}}_{\psi _t^*\varvec{X}}\psi _t^*\varvec{Y} =\psi _t^* (\nabla ^{\varvec{h}}_{\varvec{X}}\varvec{Y})_{\parallel }$.²⁰ As a corollary, given a curve c in $\mathcal {S}_t$ and $\varvec{X}$ a vector field along c tangent to $\mathcal {S}_t$ everywhere, $D^{{\varvec{g}_t}}_c\psi _t^*\varvec{X}=(D^{\varvec{h}}_c\varvec{X})_{\parallel }\,$. For $i\in \{1,\ldots ,k\}\,$, the ith second fundamental form of $\mathcal {S}_t$ along $\varvec{\eta }^t_i$ is a $\left( {\begin{array}{c}0\\ 2\end{array}}\right)$-tensor $\varvec{\kappa }_i^t$ on $\mathcal {S}_t$ defined as (do Carmo 1992; Capovilla and Guven 1995)

$$\begin{aligned} \varvec{\kappa }_i^t(\varvec{u},\varvec{w}) =\left\langle \!\left\langle \nabla ^{\varvec{h}}_{\varvec{u}}\varvec{\eta }^t_i,\varvec{w} \right\rangle \!\right\rangle _{\varvec{h}},\quad \forall ~\varvec{u},\varvec{w}\in T_{\chi }\mathcal {S}_t. \end{aligned}$$

(4.2)

It is known that $\varvec{\kappa }_i^t$ is a symmetric tensor and can equivalently be written as

$$\begin{aligned} \varvec{\kappa }_i^t=(\nabla ^{\varvec{h}}\varvec{\eta }^t_i)^{\flat }_\parallel ,\quad i=1,\ldots ,k. \end{aligned}$$

On $\mathcal {S}\,$, we define, for $i\in \{1,\ldots ,k\}\,$, the ith second fundamental form as $\varvec{k}_i^t=\psi _t^*\varvec{\kappa }_i^t\,$. For vector fields $\varvec{u},\varvec{w}\in T_{\varvec{x}}\mathcal {S}$, one can write $\nabla _{\psi _*\varvec{u}}^{\varvec{h}}\psi _*\varvec{w} = \psi _*\nabla _{\varvec{u}}^{{\varvec{g}_t}}\varvec{w} + \sum _{i=1}^{k}h^i(\varvec{u},\varvec{w})\varvec{\eta }^t_i\,$, where $h^i(.,.)$ is a bilinear form. Therefore

$$\begin{aligned} h^i(\varvec{u},\varvec{w})=\left\langle \!\left\langle \nabla _{\psi _*\varvec{u}}^{\varvec{h}}\psi _*\varvec{w}, \varvec{\eta }^t_i\right\rangle \!\right\rangle _{\varvec{h}},\quad i=1,\ldots ,k. \end{aligned}$$

Knowing that $\left\langle \!\left\langle \psi _*\varvec{w},\varvec{\eta }^t_i\right\rangle \!\right\rangle _{\varvec{h}}=0$, one concludes that

$$\begin{aligned} \left\langle \!\left\langle \nabla _{\psi _*\varvec{u}}^{\varvec{h}}\psi _*\varvec{w}, \varvec{\eta }^t_i\right\rangle \!\right\rangle _{\varvec{h}} =-\left\langle \!\left\langle \nabla _{\psi _*\varvec{u}}^{\varvec{h}}\varvec{\eta }^t_i, \psi _*\varvec{w}, \right\rangle \!\right\rangle _{\varvec{h}},\quad i=1,\ldots ,k. \end{aligned}$$

Hence

$$\begin{aligned} h^i(\varvec{u},\varvec{w})= & {} -\left\langle \!\left\langle \nabla _{\psi _*\varvec{u}}^{\varvec{h}}\varvec{\eta }^t_i, \psi _*\varvec{w}, \right\rangle \!\right\rangle _{\varvec{h}}=-(\nabla ^{\varvec{h}}\varvec{\eta }^t_i)^{\flat }(\psi _*\varvec{u},\psi _*\varvec{w}) =-\varvec{k}_i^t(\varvec{u},\varvec{w}),\\ i= & {} 1,\ldots ,k. \end{aligned}$$

Therefore, we obtain Gauss’s equation

$$\begin{aligned} \nabla _{\psi _*\varvec{u}}^{\varvec{h}}\psi _*\varvec{w}=\psi _*\nabla _{\varvec{u}}^{{\varvec{g}_t}}\varvec{w} -\sum _{i=1}^{k}\varvec{k}_i^t(\varvec{u},\varvec{w})\varvec{\eta }^t_i. \end{aligned}$$

On the other hand, for $i,j\in \{1,\ldots ,k\}\,$, the projection of $\nabla ^{\varvec{h}}\varvec{\eta }^t_i$ along $\varvec{\eta }^t_j$ defines $\varvec{\omega }_{ij}^t\,$, the normal fundamental 1-form of $\mathcal {S}_t$ relative to the unit normals $\varvec{\eta }^t_i$ and $\varvec{\eta }^t_j\,$. For any vector $\varvec{w}$ tangent to $\mathcal {S}_t\,$, the 1-form $\varvec{\omega }_{ij}^t$ is defined by (Capovilla and Guven 1995)

$$\begin{aligned} \varvec{\omega }_{ij}^t \cdot \varvec{w}= \left\langle \!\left\langle \nabla ^{\varvec{h}}_{\varvec{w}} \varvec{\eta }^t_i, \varvec{\eta }^t_j \right\rangle \!\right\rangle _{\varvec{h}}. \end{aligned}$$

Note that, for $i,j\in \{1,\ldots ,k\}$, the normal fundamental 1-form $\varvec{\omega }_{ij}^t$ is such that $\varvec{\omega }_{ij}^t = -\varvec{\omega }_{ji}^t$. On $\mathcal {S}\,$, one defines the normal fundamental 1-forms, for $i,j\in \{1,\ldots ,k\}\,$, as $\varvec{o}_{ij}^t=\psi _t^*\varvec{\omega }_{ij}^t\,$. Note that, for a tangent vector field $\varvec{w}$ on $\mathcal {S}_t$, one can write the following²¹

$$\begin{aligned} \nabla ^{\varvec{h}}_{\varvec{w}}\varvec{\eta }^t_i = \varvec{h}^\sharp \cdot \varvec{\kappa }_i^t \cdot \varvec{w} + \sum _{j=1}^k \left( \varvec{\omega }_{ij}^t \cdot \varvec{w}\right) \varvec{\eta }^t_j. \end{aligned}$$

(4.3)

One needs to be careful in calculating time derivatives in $\left( \mathcal {S},\varvec{g}_t\right) \,$, since the induced metric ${\varvec{g}_t}$ itself depends on time. In particular, when calculating the derivative of the inner product $\left\langle \!\left\langle \varvec{u},\varvec{w}\right\rangle \!\right\rangle _{{\varvec{g}_t}}$ of two vector fields $\varvec{u}$ and $\varvec{w}$ along a time-parametrized curve $c\,$, the usual formula

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t} \left\langle \!\left\langle \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}} = \left\langle \!\left\langle D_{t}^{{\varvec{g}_t}} \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}} + \left\langle \!\left\langle \varvec{u},D_{t}^{{\varvec{g}_t}}\varvec{w}\right\rangle \!\right\rangle _{{\varvec{g}_t}}, \end{aligned}$$

(4.4)

is no longer valid when the metric ${\varvec{g}_t}$ is t dependent. One instead has²²

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {dt}}} \left\langle \!\left\langle \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}}=\left\langle \!\left\langle D_{t}^{{\varvec{g}_t}} \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}} + \left\langle \!\left\langle \varvec{u},D_{t}^{{\varvec{g}_t}}\varvec{w}\right\rangle \!\right\rangle _{{\varvec{g}_t}} + \left\langle \!\left\langle \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{\frac{\partial \varvec{g}_t}{\partial t}}, \end{aligned}$$

(4.5)

where

$$\begin{aligned} \left\langle \!\left\langle \varvec{u},\varvec{w}\right\rangle \!\right\rangle _{\frac{\partial \varvec{g}_t}{\partial t}} = u^av^b \frac{\partial {g_t}_{ab}}{\partial t}. \end{aligned}$$

This can be written in terms of the inner product with respect to ${\varvec{g}_t}$ as

$$\begin{aligned} \left\langle \!\left\langle \varvec{u},\varvec{w}\right\rangle \!\right\rangle _{\frac{\partial \varvec{g}_t}{\partial t}} = \left\langle \!\left\langle \varvec{u}, {\varvec{g}_t^\sharp }\cdot \frac{\partial {\varvec{g}}}{\partial t}\cdot \varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}}, \end{aligned}$$

where ${\varvec{g}_t^\sharp }$ denotes the “inverse metric,” with components $g_t^{ab}\,$. Therefore²³

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t} \left\langle \!\left\langle \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}}=\left\langle \!\left\langle D_{t}^{{\varvec{g}_t}} \varvec{u},\varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}} + \left\langle \!\left\langle \varvec{u},D_{t}^{{\varvec{g}_t}}\varvec{w}\right\rangle \!\right\rangle _{{\varvec{g}_t}} + \left\langle \!\left\langle \varvec{u},{\varvec{g}_t^\sharp }\cdot \frac{\partial \varvec{g}_t}{\partial t}\cdot \varvec{w} \right\rangle \!\right\rangle _{{\varvec{g}_t}}. \end{aligned}$$

(4.7)

Using the Levi–Civita connection for the metric ${\varvec{g}_t}$ to calculate covariant derivatives, the symmetry lemma of classical Riemann geometry (Lee 1997; Nishikawa 2002) still holds.²⁴

Lemma 3.1

For a Riemannian manifold with a time-dependent metric ${\varvec{g}_t}$,

$$\begin{aligned} D_{\epsilon }^{{\varvec{g}_t}} \frac{\partial c(t,\epsilon )}{\partial t} = D_{t}^{{\varvec{g}_t}} \frac{\partial c(t,\epsilon )}{\partial \epsilon }. \end{aligned}$$

The velocity of the time-dependent embedding $\psi _t$ is defined as

$$\begin{aligned} \varvec{\zeta }=\frac{\partial \psi (t,\varvec{x})}{\partial t} = \varvec{\zeta }_{\parallel } + \sum _{i=1}^{k}\zeta _\perp ^i\varvec{\eta }^t_i, \end{aligned}$$

where $\varvec{\zeta }_{\parallel }$ is the tangential velocity of the embedding. We also define $\varvec{Z} := \psi _{t}^*\varvec{\zeta }_\parallel \circ \varphi _t\,$.

Lemma 3.2

For an arbitrary embedding $\psi _t\,$, the following relation holds

$$\begin{aligned} \frac{\partial \varvec{g}_t}{\partial t} = \mathfrak {L}_{\varvec{Z}}{\varvec{g}_t} + 2\sum _{i=1}^{k}\zeta _\perp ^i\varvec{k}_i^t, \end{aligned}$$

(4.8)

where $\mathfrak {L}$ denotes the autonomous Lie derivative.²⁵ For a transversal embedding, i.e., when $\varvec{Z}=\varvec{0}\,$, (4.8) reduces to

$$\begin{aligned} \frac{\partial \varvec{g}_t}{\partial t} = 2\sum _{i=1}^{k}\zeta _\perp ^i\varvec{k}_i^t. \end{aligned}$$

(4.9)

Proof

First, we note that

$$\begin{aligned} \varvec{L}_{\varvec{\zeta }}\varvec{h}= & {} \left[ \frac{{\hbox {d}}}{{\hbox {d}}t}\left( \psi _{t} \circ \psi _{s}^{-1}\right) ^* \varvec{h} \right] _{s=t} = \left[ \frac{{\hbox {d}}}{{\hbox {dt}}} \psi _{s*}\psi _t^* \varvec{h} \right] _{s=t} = \left[ \frac{{\hbox {d}}}{{\hbox {dt}}} \psi _{s*}\varvec{g}_t \right] _{s=t}\nonumber \\= & {} \psi _{t*} \left[ D_t^{\varvec{g}_t} \varvec{g}_t \right] _{s=t} = \psi _{t*}\frac{\partial \varvec{g}_t}{\partial t}. \end{aligned}$$

(4.10)

On the other hand, we also have

$$\begin{aligned} \varvec{L}_{\varvec{\zeta }}\varvec{h} = \mathfrak {L}_{\varvec{\zeta }}\varvec{h} = \mathfrak {L}_{\varvec{\zeta }_\parallel }\varvec{h} + \sum _{i=1}^{k}\zeta _\perp ^i \mathfrak {L}_{\varvec{\eta }^t_i}\varvec{h}. \end{aligned}$$

However, for $i\in \{1,\ldots ,k\}$ one has

$$\begin{aligned} \left( \mathfrak {L}_{\varvec{\eta }^t_i}\varvec{h} \right) _{\alpha \beta } = (\varvec{\eta }^t_i)_{\alpha |\beta } + (\varvec{\eta }^t_i)_{\beta |\alpha } = 2\kappa _{(i)\alpha \beta }. \end{aligned}$$

(4.11)

We observe that $\mathfrak {L}_{\varvec{\zeta }_\parallel }\varvec{h} = \mathfrak {L}_{\psi _{t*}\varvec{Z}}\psi _{t*}\varvec{g}_t\,$, and following (Marsden and Hughes 1983, p. 98), we have $\mathfrak {L}_{\psi _{t*}\varvec{Z}}\psi _{t*}\varvec{g}_t = \psi _{t*}\mathfrak {L}_{\varvec{Z}}\varvec{g}_t\,$. Thus

$$\begin{aligned} \varvec{L}_{\varvec{\zeta }}\varvec{h} = \psi _{t*}\left( \mathfrak {L}_{\varvec{Z}}\varvec{g}_t + 2\sum _{i=1}^{k}\zeta _\perp ^i\varvec{k}_i^t\right) . \end{aligned}$$

(4.12)

Finally, it follows from (4.10) and (4.12) that

$$\begin{aligned} \frac{\partial \varvec{g}_t}{\partial t} = \mathfrak {L}_{\varvec{Z}}{\varvec{g}_t} + 2\sum _{i=1}^{k}\zeta _\perp ^i\varvec{k}_i^t. \end{aligned}$$

$\square$

Appendix 2: An Alternative Derivation of the Tangent Balance of Linear Momentum

In this section, we provide an alternate proof for the tangential balance of linear momentum in the particular case of a transversal evolution of the ambient space. This derivation is a generalized version, for arbitrary co-dimension $k = \dim \mathcal {Q} - \dim \mathcal {S}_t$, of a theorem appearing in Marsden and Hughes (1983), p. 129. The generalized version can be stated as follows (see § 2.1 and Fig. 2 to recall the notation):

Theorem 3.1

Assume that given scalar functions a and b, and a vector field $\varvec{c}$ satisfy the following master balance law for any open set $\mathcal {U}$ with $C^1$ piecewise boundary:

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {dt}}}\int _{\varphi _t(\mathcal {U})}a{\hbox {d}}v = \int _{\varphi _t(\mathcal {U})}b{\hbox {d}}v + \int _{\partial \varphi _t(\mathcal {U})}\left\langle \!\left\langle \varvec{c} , \varvec{\mathsf n} \right\rangle \!\right\rangle _{\varvec{g}_t} da, \end{aligned}$$

(5.1)

where $\varvec{\mathsf n}$ is the unit normal vector to $\partial \varphi _t\left( \mathcal {U}\right)$ in $\mathcal {S}\,$. Localization of (5.1) gives one

$$\begin{aligned} \frac{{\hbox {d}}a}{{\hbox {d}}t} + a\, \mathrm{div}_{\varvec{g}_t}\varvec{v} + a \sum _{i=1}^{k}\zeta _\perp ^i \mathrm{tr}\left( \varvec{k}_i^t\right) = b + \mathrm{div}_{\varvec{g}_t} \varvec{c}, \end{aligned}$$

(5.2)

where we recall that $\varvec{v}$ is the velocity field of $\varphi _t\, and \varvec{\zeta } = \sum _{i=1}^{k}\zeta _\perp ^i\varvec{\eta }^t_i$ is the velocity field of $\psi _t\,$ (we have $\varvec{\zeta }_\parallel =\varvec{0}$ since we are assuming transversal evolution).²⁶

Proof

Note that

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\int _{\varphi _t(\mathcal {U})}a{\hbox {d}}v = \frac{{\hbox {d}}}{{\hbox {d}}t}\int _{\mathcal {U}}aJ{\hbox {d}}V = \int _{\mathcal {U}}\frac{{\hbox {d}}}{{\hbox {d}}t}(aJ){\hbox {d}}V = \int _{\mathcal {U}}\left( \frac{{\hbox {d}}a}{{\hbox {dt}}}J+a\frac{{\hbox {d}}J}{{\hbox {d}}t}\right) {\hbox {d}}V. \end{aligned}$$

However

$$\begin{aligned} \frac{{\hbox {d}}J}{{\hbox {d}}t} =\left( {\text {div}}_{{\varvec{g}_t}}\varvec{v}\right) J + \frac{1}{2}J{\text {tr}}\left( \frac{\partial \varvec{g}_t}{\partial t}\right) , \end{aligned}$$

and following (4.8), one has

$$\begin{aligned} \frac{\partial \varvec{g}_t}{\partial t} = 2\sum _{i=1}^{k}\zeta _\perp ^i\varvec{k}_i^t. \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\int _{\varphi _t(\mathcal {U})}a{\hbox {d}}v&= \int _{\mathcal {U}}\left( \frac{{\hbox {da}}}{{\hbox {d}}t}+a\, \mathrm{div}_{\varvec{g}_t}\varvec{v} + a \sum _{i=1}^{k}\zeta _\perp ^i \mathrm{tr}\left( \varvec{k}_i^t\right) \right) J{\hbox {d}}V \\ {}&= \int _{\varphi _t(\mathcal {U})}\left( \frac{{\hbox {da}}}{{\hbox {d}}t}+a\, \mathrm{div}_{\varvec{g}_t}\varvec{v} + a \sum _{i=1}^{k}\zeta _\perp ^i \mathrm{tr}\left( \varvec{k}_i^t\right) \right) {\hbox {d}}v. \end{aligned} \end{aligned}$$

(5.3)

On the other hand, by using Stokes’ theorem, one can write

$$\begin{aligned} \int _{\partial \varphi _t(\mathcal {U})}\left\langle \!\left\langle \varvec{c} , \varvec{\mathsf n} \right\rangle \!\right\rangle _{\varvec{g}_t} da = \int _{\varphi _t(\mathcal {U})}\mathrm{div}_{\varvec{g}_t} \varvec{c}\, {\hbox {d}}v. \end{aligned}$$

(5.4)

Therefore, by using (5.3) and (5.4), (5.1) transforms to

$$\begin{aligned} \int _{\varphi _t(\mathcal {U})}\left( \frac{{\hbox {d}}a}{{\hbox {d}}t}+a\, \mathrm{div}_{\varvec{g}_t}\varvec{v} + a \sum _{i=1}^{k}\zeta _\perp ^i \mathrm{tr}\left( \varvec{k}_i^t\right) \right) {\hbox {d}}v = \int _{\varphi _t(\mathcal {U})}\left( b + \mathrm{div}_{\varvec{g}_t} \varvec{c}\right) {\hbox {d}}v. \end{aligned}$$

Thus, by arbitrariness of the subset $\mathcal {U}\,$, one finds that

$$\begin{aligned} \frac{{\hbox {d}}a}{{\hbox {d}}t} + a\, \mathrm{div}_{\varvec{g}_t}\varvec{v} + a \sum _{i=1}^{k}\zeta _\perp ^i \mathrm{tr}\left( \varvec{k}_i^t\right) = b + \mathrm{div}_{\varvec{g}_t} \varvec{c}. \end{aligned}$$

(5.5)

$\square$

First, the localized conservation of mass is derived. In integral form, conservation of mass reads

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\int _{\varphi _t(\mathcal {U})} \rho {\hbox {d}}v = 0. \end{aligned}$$

Hence, using the above theorem ($a=\rho , b=0\,$, and $c=0$), the conservation of mass in localized form reads²⁷

$$\begin{aligned} \frac{{\hbox {d}}\rho }{{\hbox {d}}t} + \rho \, \mathrm{div}_{\varvec{g}_t}\varvec{v} + \rho \sum _{i=1}^{k} \zeta _\perp ^i \mathrm{tr}\left( \varvec{k}_i^t\right) = 0. \end{aligned}$$

(5.6)

Next, we look at the balance of linear momentum, which in integral form reads

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\int _{\varphi _t(\mathcal {U})}\rho \varvec{v} {\hbox {d}}v = \int _{\varphi _t(\mathcal {U})} \rho \varvec{B} {\hbox {d}}v + \int _{\partial \varphi _t(\mathcal {U})} \varvec{\sigma } \cdot \varvec{\mathsf n}{\hbox {d}}a, \end{aligned}$$

(5.7)

where $\varvec{B}$ is the body force per unit mass, $\varvec{\sigma }$ is the Cauchy stress tensor, and $\varvec{\mathsf n}$ is the unit normal vector to $\partial \varphi _t\left( \mathcal {U}\right) \,$.

Remark 3.3

Note that (5.7) makes sense only when the ambient space $\mathcal {S}_t$ is endowed with a linear structure. In a general manifold, integrating a vector field does not make sense. Therefore, the proof given in this appendix is only valid for linear ambient spaces. However, the resulting localized tangential balance of linear momentum (5.8) still holds in the case of a general manifold as shown in § 2.1 using a Lagrangian field theory, see Eq. (2.17).

In order to use the above theorem, we contract the balance of linear momentum (5.7) with an arbitrary time-independent covariantly constant vector field $\varvec{u}$ tangent to $\varphi _t\left( \mathcal {U}\right) \,$, i.e.

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\int _{\varphi _t(\mathcal {U})} \left\langle \!\left\langle \rho \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} {\hbox {d}}v = \int _{\varphi _t(\mathcal {U})} \left\langle \!\left\langle \rho \varvec{B}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} {\hbox {d}}v + \int _{\partial \varphi _t(\mathcal {U})} \left\langle \!\left\langle \varvec{\sigma } \cdot \varvec{\mathsf n}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}}{\hbox {d}}a. \end{aligned}$$

We can then use the above theorem for $a = \left\langle \!\left\langle \rho \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}}, b = \left\langle \!\left\langle \rho \varvec{B}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}}\,$, and $\varvec{c} = \varvec{\sigma } \cdot \varvec{u}\,$. Hence, it follows that

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\left\langle \!\left\langle \rho \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} + \left\langle \!\left\langle \rho \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} \mathrm{div}_{\varvec{g}_t}\varvec{v} + \left\langle \!\left\langle \rho \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} v_n \mathrm{tr}\varvec{k} = \left\langle \!\left\langle \rho \varvec{B}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} + \mathrm{div}_{\varvec{g}_t}\left( \varvec{\sigma } \cdot \varvec{u}\right) . \end{aligned}$$

Note that

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}\left\langle \!\left\langle \rho \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} = \frac{{\hbox {d}}\rho }{{\hbox {d}}t}\left\langle \!\left\langle \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} +\rho \left\langle \!\left\langle D_t^{\varvec{h}}\varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}}, \end{aligned}$$

where $D_t^{\varvec{h}}$ denotes the covariant time derivative with respect to the metric ${\varvec{h}}\,$. Therefore, it follows that

$$\begin{aligned} \left( \frac{{\hbox {d}}\rho }{{\hbox {d}}t} + \rho \mathrm{div}_{\varvec{g}_t}\varvec{v} + \rho v_n \mathrm{tr}\varvec{k} \right) \left\langle \!\left\langle \varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} +\rho \left\langle \!\left\langle D_t^{\varvec{h}}\varvec{v}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} = \left\langle \!\left\langle \rho \varvec{B}, \varvec{u}\right\rangle \!\right\rangle _{\varvec{h}} + \mathrm{div}_{\varvec{g}_t}\left( \varvec{\sigma } \cdot \varvec{u}\right) . \end{aligned}$$

The first term vanishes following the conservation of mass (5.6). Thus, by arbitrariness of $\varvec{u}$ one concludes that

$$\begin{aligned} \rho \left( D_t^{\varvec{h}}\varvec{v}\right) _\parallel = \rho \varvec{B} + \mathrm{div}_{\varvec{g}_t}\varvec{\sigma }. \end{aligned}$$

(5.8)

Note that $\left( D_t^{\varvec{h}}\varvec{v}\right) _\parallel$ is different from $D_t^{\varvec{g}}\varvec{v}\,$. In fact, we can write following Proposition 2.1 that

$$\begin{aligned} \begin{aligned} \left( D_t^{\varvec{h}}\varvec{v}\right) _\parallel&= D^{\varvec{g}_t}_t \varvec{v} + 2\sum _{i=1}^k \zeta _\perp ^i \varvec{g}_t^\sharp \cdot \varvec{k}^t_i \cdot \varvec{v} - \sum _{i=1}^k \zeta _\perp ^i \left( {\hbox {d}}\zeta _\perp ^i\right) ^\sharp - \sum _{i,j=1}^k \zeta _\perp ^i\zeta _\perp ^j \varvec{o}^{t\sharp }_{ij} \\ {}&= D^{\varvec{g}_t}_t \varvec{v} + \varvec{g}_t^\sharp \cdot \frac{\partial \varvec{g}_t}{\partial t} \cdot \varvec{v} - \sum _{i=1}^k \zeta _\perp ^i \left( {\hbox {d}}\zeta _\perp ^i\right) ^\sharp - \sum _{i,j=1}^k \zeta _\perp ^i\zeta _\perp ^j \varvec{o}^{t\sharp }_{ij}. \end{aligned} \end{aligned}$$

(5.9)