Nonlinear mechanics of accretion

Abstract

We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory, the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material is added to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.

Appendices

Appendices

We tersely review some elements of Riemannian geometry, geometry of surfaces and of foliations, and the geometric theory of nonlinear elasticity and anelasticity. For more detailed discussions, see Marsden and Hughes (1983), Camacho and Neto (2013), Yavari (2010), Yavari and Goriely (2012a).

Riemannian Geometry

For a smooth n-manifold $\mathcal {M}$, the tangent space to $\mathcal {M}$ at a point $p \in \mathcal {M}$ is denoted $T_p\mathcal {M}$. A smooth vector field $\mathbf {W}$ on $\mathcal {M}$ assigns a vector $\mathbf {W}_p$ to every $p\in \mathcal {M}$ and $p\mapsto \mathbf {W}_p\in T_p\mathcal {M}$ varies smoothly. A 1-form at $p \in \mathcal {M}$ is a linear map $\varvec{\lambda }: T_{p}\mathcal {M} \rightarrow \mathbb {R}$. In a local coordinate chart $\{X^A\}$ for $\mathcal {M}$, $\langle \varvec{\lambda },\mathbf {U} \rangle = \lambda _K U^K$, where $\mathbf {U} \in T_{p} \mathcal {M}$. The vector space of 1-form at $p \in \mathcal {M}$ is denoted with $T^*_{p}\mathcal {M}$. A smooth differential 1-form assigns a 1-form $\varvec{\lambda }_p$ to every $p\in \mathcal {M}$ and $\varvec{\lambda }_p\in T^*_p\mathcal {M}$ varies smoothly. A type $r\atopwithdelims ()s$-tensor at $p \in \mathcal {M}$ is a multilinear map

$$\begin{aligned} T: \overbrace{T^*_{p}\mathcal {M} \times \cdots \times T^*_{p}\mathcal {M}}^{r~\text {times}} \times \overbrace{T_{p}\mathcal {M} \times \cdots \times T_{p}\mathcal {M}}^{s~\text {times}} \rightarrow \mathbb {R} . \end{aligned}$$

(A.1)

In a local chart one has

$$\begin{aligned} \mathbf {T}(\varvec{\lambda }^1,\ldots ,\varvec{\lambda }^r,\mathbf {U}_1,\ldots ,\mathbf {U}_s)= T^{I_1 \ldots I_r}{}_{J_1 \ldots J_s}\lambda ^1_{I_1}\ldots \lambda ^r_{I_r} U^{J_1}_1 \ldots U^{J_s}_s . \end{aligned}$$

(A.2)

An $r\atopwithdelims ()s$-tensor field is a smooth map $p \mapsto \mathbf {T}_p$. A Riemannian manifold is a pair $(\mathcal {M},\mathbf {G})$ where $\mathbf {G}$, the metric tensor field, is a field of positive-definite $0\atopwithdelims ()2$-tensors. If $\mathbf {U}$ and $\mathbf {W}$ are vector fields on $\mathcal {M}$, then $p \mapsto \mathbf {G}_p(\mathbf {U}_p,\mathbf {W}_p) =: \langle \!\langle \mathbf {U}_p,\mathbf {W}_p \rangle \!\rangle _{\mathbf {G}_p}$ is a smooth function. A metric tensor can be used to raise and lower indices. Given a 1-form $\varvec{\lambda }$, we indicate with $\varvec{\lambda }^{\sharp }$ the vector with components $G^{IJ}\lambda _J$ (the $G^{IJ}$’s being the components of the inverse metric, i.e., $G_{IJ}G^{JK}=\delta ^K_I$), while given a vector ${\mathbf {U}}$, we indicate with ${\mathbf {U}}^{\flat }$ the 1-form with components $G_{IJ} U^J$.

Suppose $\mathcal {N}$ is another n-manifold and $\psi :\mathcal {M}\rightarrow \mathcal {N}$ is a smooth and invertible map. If $\mathbf {W}$ is a vector field on $\mathcal {M}$, then $\psi _*\mathbf {W} = T\psi \cdot \mathbf {W} \circ \psi ^{-1}$ is a vector field on $\psi (\mathcal {M})$ called the push-forward of $\mathbf {W}$ by $\psi$. Similarly, if $\mathbf {w}$ is a vector field on $\psi (\mathcal {M}) \subset \mathcal {N}$, then $\psi ^*\mathbf {w}=T(\psi ^{-1}) \cdot \mathbf {w} \circ \psi$ is a vector field on $\mathcal {M}$ that is called the pull-back of $\mathbf {w}$ by $\psi$. Let us denote $\mathbf {F}=T\psi$. In the local charts $\{X^I\}$ and $\{x^i\}$ for $\mathcal {M}$ and $\mathcal {N}$, respectively, $\mathbf {F}$ (a two-point tensor) has the following representation (when $\psi$ is a deformation mapping, $\mathbf {F}$ is the so-called deformation gradient of nonlinear elasticity)

$$\begin{aligned} \mathbf {F}=F^i{}_I\frac{\partial }{\partial x^i}\otimes \mathrm{d}X^I,\quad F^i{}_I=\frac{\partial \psi ^i}{\partial X^I}, \end{aligned}$$

(A.3)

where $\left\{ \frac{\partial }{\partial X^I}\right\}$ is a basis for $T_p\mathcal {M}$ and $\{\mathrm{d}X^I\}$ is a basis for $T^*_{\psi (p)}\psi (\mathcal {M})$, the co-tangent space, i.e., the dual space of $T_{\psi (p)}\psi (\mathcal {M})$, or the space of co-vectors (1-forms). The push-forward and pull-back of vectors have the following coordinate representations $(\psi _*\mathbf {W})^a=F^a{}_A W^A$, and $(\psi ^*\mathbf {w})^A=(F^{-1})^A{}_a w^a$. The push-forward and pull-back of 1-forms are defined as $(\psi _*\varvec{\Lambda })_a=(F^{-1})^A{}_a \Lambda _A$, and $(\psi ^*\varvec{\lambda })_A=F^a{}_A \lambda _a$. We sometimes indicate them with $({\mathbf{F}}^{-1})^{{\star }} {\varvec{\Lambda }}$ and ${\mathbf {F}}^{\star }\varvec{\lambda }$, where ${\mathbf F}^{\star }$ is the dual of $\mathbf {F}$ and is defined as

$$\begin{aligned} {\mathbf {F}}^{\star }=F^i{}_I \mathrm{d}X^I \otimes \frac{\partial }{\partial x^i}. \end{aligned}$$

(A.4)

The push-forward and pull-back of an $r\atopwithdelims ()s$-tensor field are given by

$$\begin{aligned} \begin{aligned} (\psi _*\mathbf {T})^{i_1 \ldots i_r}{}_{j_1 \ldots j_s}&= F^{i_1}{}_{I_1} \ldots F^{i_r}{}_{I_r} ~T^{I_1 \ldots I_r}{}_{J_1 \ldots J_s}~ (F^{-1})^{J_1}{}_{j_1} \ldots (F^{-1})^{J_s}{}_{j_s} , \\ (\psi ^*\mathbf {t})^{I_1 \ldots I_r}{}_{J_1 \ldots J_s}&= (F^{-1})^{I_1}{}_{i_1} \ldots (F^{-1})^{I_r}{}_{i_r} ~t^{i_1 \ldots i_r}{}_{j_1 \ldots j_s}~ F^{j_1}{}_{J_1} \ldots F^{j_s}{}_{J_s} . \end{aligned} \end{aligned}$$

(A.5)

Suppose $(\mathcal {M},\mathbf {G})$ and $(\mathcal {N},\mathbf {g})$ are Riemannian manifolds and $\psi :\mathcal {M}\rightarrow \mathcal {N}$ is a diffeomorphism (smooth map with smooth inverse). Push-forward of the metric $\mathbf {G}$ is a metric on $\psi (\mathcal {M})\subset \mathcal {N}$, which is denoted by $\psi _*\mathbf {G}$ and is defined as

$$\begin{aligned} (\psi _*\mathbf {G})_{\psi (p)}\left( \mathbf {u}_{\psi (p)},\mathbf {w}_{\psi (p)}\right) :=\mathbf {G}_p\left( (\psi ^*\mathbf {u})_p,(\psi ^*\mathbf {w})_p\right) . \end{aligned}$$

(A.6)

In components, $(\psi _*\mathbf {G})_{ij} = (F^{-1})^I{}_i (F^{-1})^J{}_j G_{IJ}$. Similarly, pull-back of the metric $\mathbf {g}$ is a metric in $\psi ^{-1}(\mathcal {N})\subset \mathcal {M}$, which is denoted by $\psi ^*\mathbf {g}$ and is defined as

$$\begin{aligned} \left( \psi ^*\mathbf {g}\right) _p\left( \mathbf {U}_p,\mathbf {W}_p\right) :=\mathbf {g}_{\psi (p)}\left( \left( \psi _*\mathbf {U}\right) _{\psi (p)},\left( \psi _*\mathbf {W}\right) _{\psi (p)}\right) . \end{aligned}$$

(A.7)

In components

$$\begin{aligned} (\psi ^*\mathbf {g})_{AB} = F^a{}_A F^b{}_B g_{ab}. \end{aligned}$$

(A.8)

If $\mathbf {g}=\psi _*\mathbf {G}$, or equivalently, $\mathbf {G}=\psi ^*\mathbf {g}$, $\psi$ is called an isometry and the Riemannian manifolds $(\mathcal {M},\mathbf {G})$ and $(\mathcal {N},\mathbf {g})$ are isometric. Note that an isometry preserves distances.

A linear (affine) connection on a manifold $\mathcal {B}$ is an operation $\nabla :\mathcal {X}(\mathcal {B})\times \mathcal {X}(\mathcal {B})\rightarrow \mathcal {X}(\mathcal {B})$, where $\mathcal {X}(\mathcal {B})$ is the set of vector fields on $\mathcal {B}$, such that $\forall ~\mathbf {X},\mathbf {Y},\mathbf {X}_1,\mathbf {X}_2,\mathbf {Y}_1,\mathbf {Y}_2\in \mathcal {X}(\mathcal {B}),\forall ~f,f_1,f_2\in C^{\infty }(\mathcal {B}),\forall ~a_1,a_2\in \mathbb {R}$:

$$\begin{aligned}&i) \quad \nabla _{f_1\mathbf {X}_1+f_2\mathbf {X}_2}\mathbf {Y}=f_1\nabla _{\mathbf {X}_1}\mathbf {Y} +f_2\nabla _{\mathbf {X}_2}\mathbf {Y}, \end{aligned}$$

(A.9)

$$\begin{aligned}&ii) \quad \nabla _{\mathbf {X}}(a_1\mathbf {Y}_1+a_2\mathbf {Y}_2)=a_1\nabla _{\mathbf {X}}(\mathbf {Y}_1)+a_2\nabla _{\mathbf {X}}(\mathbf {Y}_2), \end{aligned}$$

(A.10)

$$\begin{aligned}&iii) \quad \nabla _{\mathbf {X}}(f\mathbf {Y})=f\nabla _{\mathbf {X}}\mathbf {Y}+(\mathbf {X}f)\mathbf {Y}. \end{aligned}$$

(A.11)

$\nabla _{\mathbf {X}}\mathbf {Y}$ is called the covariant derivative of $\mathbf {Y}$ along $\mathbf {X}$. In a local chart $\{X^I\}$, $\nabla _{\partial _I}\partial _J=\Gamma ^K{}_{IJ}\partial _K$, where $\Gamma ^K{}_{IJ}$ are Christoffel symbols of the connection and $\partial _K=\frac{\partial }{\partial x^K}$ are natural bases for the tangent space corresponding to a coordinate chart $\{x^A\}$. In components the covariant derivative of a tensor field reads

$$\begin{aligned} ( \nabla _{\mathbf {U}}\mathbf {W} )^I = \frac{\partial W^I}{\partial X^H}U^H+\Gamma ^I{}_{HK}W^KU^H , \end{aligned}$$

(A.12)

while for a field of 1-forms one has

$$\begin{aligned} ( \nabla _{\mathbf {U}}\varvec{\lambda } )_I = \frac{\partial \lambda _I}{\partial X^H}U^H-\Gamma ^K{}_{HI}\lambda _K U^H . \end{aligned}$$

(A.13)

The previous relations can be generalized to tensors as

$$\begin{aligned} ( \nabla _{\mathbf {U}}\mathbf {T} )^I= & {} \frac{\partial T^{I_1\ldots I_r}{}_{J_1\ldots J_s}}{\partial X^H}U^H + \Gamma ^{I_1}{}_{HK} T^{K\ldots I_r}{}_{J_1\ldots J_s} U^H + \cdots \nonumber \\&\quad - \Gamma ^K{}_{H J_1} T^{I_1\ldots I_r}{}_{K\ldots J_s} U^H . \end{aligned}$$

(A.14)

A linear connection is said to be compatible with a metric $\mathbf {G}$ of the manifold if

$$\begin{aligned} \nabla _{\mathbf {X}}\left\langle \!\left\langle \mathbf {Y},\mathbf {Z}\right\rangle \!\right\rangle _{\mathbf {G}} =\left\langle \!\left\langle \nabla _{\mathbf {X}}\mathbf {Y},\mathbf {Z}\right\rangle \!\right\rangle _{\mathbf {G}} +\left\langle \!\left\langle \mathbf {Y},\nabla _{\mathbf {X}}\mathbf {Z}\right\rangle \!\right\rangle _{\mathbf {G}}, \end{aligned}$$

(A.15)

where $\left\langle \!\left\langle .,. \right\rangle \!\right\rangle _{\mathbf {G}}$ is the inner product induced by the metric $\mathbf {G}$. It can be easily shown that $\nabla$ is compatible with $\mathbf {G}$ if and only if $\nabla \mathbf {G}=\mathbf {0}$. A linear connection is said to be symmetric if $\nabla _{\mathbf {U}}\mathbf {W}-\nabla _{\mathbf {W}}\mathbf {U}-[\mathbf {U},\mathbf {W}]=\mathbf {0}$, with $[\mathbf {U},\mathbf {W}]$ indicating the Lie brackets of $\mathbf {U}$ and $\mathbf {W}$, i.e., $([\mathbf {U},\mathbf {W}])^I = W^I,_J U^J - U^I,_J W^J$. The symmetric connection that is compatible with the metric is called the Levi–Civita connection and its Christoffel coefficients read

$$\begin{aligned} \Gamma ^I{}_{JK} = G^{IH} \left( \frac{\partial G_{HJ}}{\partial X^K}+\frac{\partial G_{HK}}{\partial X^J}-\frac{\partial G_{JK}}{\partial X^H} \right) . \end{aligned}$$

(A.16)

The Riemann curvature $\varvec{\mathcal {R}}$ associated with $(\mathcal {M},\nabla )$ is a $1\atopwithdelims ()3$-tensor defined as

$$\begin{aligned} \varvec{\mathcal {R}}(\mathbf {X},\mathbf {Y})\mathbf {Z} = \nabla _{\mathbf {X}} \nabla _{\mathbf {Y}} \mathbf {Z} - \nabla _{\mathbf {Y}} \nabla _{\mathbf {X}} \mathbf {Z} - \nabla _{[\mathbf {X},\mathbf {Y}]} \mathbf {Z} \quad \forall \mathbf {X} , \mathbf {Y} , \mathbf {Z}\in T\mathcal {M} , \end{aligned}$$

(A.17)

where $\varvec{\mathcal {R}}(\mathbf {X},\mathbf {Y})\mathbf {Z}$ stands for the contraction $R^I{}_{JLM} X^LY^MZ^J$. In a local coordinate chart $\{X^A\}$, the components of the Riemman curvature tensor read

$$\begin{aligned} \mathcal {R}^I{}_{JLM}=\frac{\partial \Gamma ^I{}_{MJ}}{\partial X^L} -\frac{\partial \Gamma ^I{}_{LJ}}{\partial X^M} +\Gamma ^I{}_{LH}\Gamma ^H{}_{MJ}-\Gamma ^I{}_{MH}\Gamma ^H{}_{LJ} . \end{aligned}$$

(A.18)

The Ricci curvature $\mathbf {R}$ is defined as $R_{IJ}=\mathcal {R}^H{}_{IHJ}$ and is a symmetric $0\atopwithdelims ()2$-tensor. The scalar curvature R is defined as $R=R_{HK}G^{HK}$. In dimension three only six components of $\varvec{\mathcal {R}}$ are independent and the Ricci curvature fully determines the Riemannian curvature. In dimension two $\varvec{\mathcal {R}}$ contains only one independent component so it is fully determined by the scalar curvature. A metric whose Levi–Civita connection has zero curvature is said to be flat and is locally isometric to $\mathbb {R}^n$ (see the different versions of the Test Case Theorem in Spivak 1999).

Geometry of Surfaces

For our purposes, a surface is a two-dimensional submanifold $\Omega$ of a three-dimensional manifold $\mathcal {M}$ (for details see do Carmo 1992). The ambient manifold $\mathcal {M}$ is endowed with a metric $\mathbf {G}$ that, in general, is not flat. The surface $\Omega$ inherits a metric $\tilde{\mathbf {G}}$ — the first fundamental form — from $\mathbf {G}$, such that

$$\begin{aligned} \tilde{\mathbf {G}} ( \mathbf {Y} , \mathbf {Z} ) = \mathbf {G} ( \mathbf {Y} , \mathbf {Z} ), \quad \forall \mathbf {Y} , \mathbf {Z}\in T\Omega . \end{aligned}$$

(B.1)

On an open set one can define the unit vector $\mathbf {N}$ normal to $\Omega$, i.e., a vector field such that i) $\mathbf {G}(\mathbf {Y},\mathbf {N})=0$ for each tangent vector $\mathbf {Y}$, ii) $\mathbf {G}(\mathbf {N},\mathbf {N})=1$. It is globally defined only when $\Omega$ is orientable. We also define the associated 1-form $\mathbf {N}^{\flat }$ such that $\langle \mathbf {N}^{\flat } ,\mathbf {X} \rangle = 0$ for any $\mathbf {X}\in T\Omega$, and $\langle \mathbf {N}^{\flat } , \mathbf {N} \rangle = 1$, which can be obtained by lowering the index of $\mathbf {N}$, i.e., $N_I = G_{IJ} N^J$.

The two Levi–Civita connections $\nabla$ on $(\mathcal {M},\mathbf {G})$, and $\tilde{\nabla }$ on $(\Omega ,\tilde{\mathbf {G}})$ satisfy the Gauss formula

$$\begin{aligned} {\tilde{\nabla }}_{\mathbf {Y}} \mathbf {Z} = \nabla _{\mathbf {Y}} \mathbf {Z} - \mathbf {G}(\nabla _{\mathbf {Y}}\mathbf {Z},\mathbf {N})\mathbf {N} ,\quad \forall \mathbf {Y}, \mathbf {Z} \in T\Omega . \end{aligned}$$

(B.2)

This means that ${\tilde{\nabla }}$ is the tangent projection of $\nabla$. The second fundamental form $\mathbf {K}$ is a $0\atopwithdelims ()2$-tensor defined as

$$\begin{aligned} \mathbf {K} (\mathbf {Y},\mathbf {Z}) = \mathbf {G}(\nabla _{\mathbf {Y}}\mathbf {Z},\mathbf {N}) , \quad \forall \mathbf {Y}, \mathbf {Z}\in T\Omega . \end{aligned}$$

(B.3)

Then, Eq. (B.2) can be rewritten as

$$\begin{aligned} \nabla _{\mathbf {Y}} \mathbf {Z} - {\tilde{\nabla }}_{\mathbf {Y}} \mathbf {Z} = \mathbf {K}(\mathbf {Y},\mathbf {Z})\mathbf {N} . \end{aligned}$$

(B.4)

It can be easily shown that $\mathbf {K}$ is well defined and symmetric. Moreover, since $\nabla _{\mathbf {Y}}\left( \mathbf {G}(\mathbf {Z},\mathbf {N}) \right) =\nabla _{\mathbf {Y}}\mathbf {0}=\mathbf {0}$ for any pair $\mathbf {Y}, \mathbf {Z}\in T\Omega$, one has

$$\begin{aligned} \mathbf {K} (\mathbf {Y},\mathbf {Z}) = \mathbf {G}(\nabla _{\mathbf {Y}}\mathbf {Z},\mathbf {N}) = - \mathbf {G}(\mathbf {Z},\nabla _{\mathbf {Y}}\mathbf {N}) = - \langle \nabla _{\mathbf {Y}}\mathbf {N}^{\flat } , \mathbf {Z} \rangle , \end{aligned}$$

(B.5)

which is known as the Weingarten formula, and in short it reads $\mathbf {K} = -\nabla {\mathbf {N}}^{\flat }$.

An adapted chart $\Xi$ on $\mathcal {U}\subset \mathcal {M}$ has the property $\Xi (\mathcal {U}\cap \Omega )= V \times \{0\}$ with $V\subset \mathbb {R}^2$. In other words, the surface is locally given by $\Xi ^3=0$ and $(\Xi ^1,\Xi ^2)$ are coordinate functions on $\Omega \cap \mathcal {U}$. We use letters from A to G to denote indices that span $\{1,2\}$ (layer indices), and letters from H to Z for indices that span $\{1,2,3\}$ (3D indices). The components of the second fundamental form can now be obtained using the Weingarten formula:

$$\begin{aligned} K_{AB} = -\frac{\partial N_A}{\partial X^B} + \Gamma ^I{}_{AB} N_I = N_I \Gamma ^I{}_{AB} = \frac{1}{N^3} \Gamma ^3{}_{AB} , \end{aligned}$$

(B.6)

where all quantities are referred to an adapted chart.

Fig. 20

Foliation charts. Left: sketch of two overlapping local charts in 2D. Right: sketch of a foliation chart in a 3-manifold

The Riemann curvature tensor $\varvec{\mathcal {R}}$ on $(\mathcal {M},\mathbf {G})$ was introduced in Appendix A. One can define in the same way the curvature tensor $\tilde{\varvec{\mathcal {R}}}$ for $(\Omega ,\tilde{\mathbf {G}})$. In components one has

$$\begin{aligned} {\tilde{R}}^A{}_{BCD}&= \frac{\partial {\tilde{\Gamma }}^A{}_{DB}}{\partial X^C} - \frac{\partial {\tilde{\Gamma }}^A{}_{CB}}{\partial X^D} +{\tilde{\Gamma }}^A{}_{CE} {\tilde{\Gamma }}^E{}_{DB} - {\tilde{\Gamma }}^A{}_{DE} {\tilde{\Gamma }}^E{}_{CB} . \end{aligned}$$

(B.7)

The curvature tensor has 16 components but, as mentioned earlier, only one is independent. Hence, it can be expressed by the scalar curvature R, or by the Gaussian curvature $g=\det (\tilde{G}^{AC}K_{CB})$. It can be shown that $R=2g$ and $g\det \tilde{\mathbf {G}}=\frac{1}{2}R\det \tilde{\mathbf {G}}=R_{1212}$. The Guass equation relates the tangent part of $\varvec{\mathcal {R}}$ to $\tilde{\varvec{\mathcal {R}}}$ and $\mathbf {K}$. In components, it reads

$$\begin{aligned} R_{ABCD} = \tilde{R}_{ABCD} + K_{AD} K_{BC} - K_{AC} K_{BD} . \end{aligned}$$

(B.8)

When the ambient space $(\mathcal {M},\mathbf {G})$ is Euclidean the Gauss equation provides an expression for the curvature of the surface, viz. $\tilde{R}_{ABCD} = K_{AC} K_{BD} - K_{AD} K_{BC}$. The Codazzi–Mainardi equations relate $\varvec{\mathcal {R}}$ to $\tilde{\nabla } \mathbf {K}$. In components, they are given by

$$\begin{aligned} N^H \mathcal {R}_{HBCD} = N_H \mathcal {R}^H{}_{BCD} = \tilde{\nabla }_D K_{BC} - \tilde{\nabla }_C K_{BD} . \end{aligned}$$

(B.9)

When the ambient space $(\mathcal {M},\mathbf {G})$ is Euclidean the Codazzi–Mainardi equations enforce the symmetry of the tensor $\tilde{\nabla }_A K_{BC}$ with respect to all permutations.

Geometry of Foliations

Let $\mathcal {M}$ be an n-dimensional manifold. An $(n-1)$-foliation (or foliation of codimension 1) is an atlas of charts $(\mathcal {U}_a,\Xi _a)$ (a in some index set $\mathcal {I}$) such that $\Xi _a(\mathcal {U}_a)= V_a \times I_a$ with $V_a \subset \mathbb {R}^{n-1}$ and $I_a \subset \mathbb {R}$ open sets (see Fig. 20). For further details see Camacho and Neto (2013). The coordinate charts $\Xi _a$ are called the foliation charts. When this property holds, one obtains a partition of $\mathcal {M}$ as a collection $\{\Omega _t \}_{t\in \mathbb {R}}$ of embedded submanifolds of dimension two — the leaves of the foliation. In particular, a Riemannian material manifold $(\mathcal {M},\mathbf {G})$ is partitioned into $(n-1)$-dimensional Riemannian submanifolds $(\Omega _{\tau },\tilde{\mathbf {G}}_{\tau })$ with $\tau \in \mathbb {R}$. The metric $\tilde{\mathbf {G}}_{\tau }$ on the layer $\Omega _{\tau }$ is inherited from $\mathbf {G}$, i.e., it is defined as

$$\begin{aligned} \tilde{\mathbf {G}}_{\tau } ( \mathbf {Y}_{\tau } , \mathbf {Z}_{\tau } ) = \mathbf {G} ( \mathbf {Y}_{\tau } , \mathbf {Z}_{\tau } ), \quad \forall \mathbf {Y}_{\tau } , \mathbf {Z}_{\tau }\in T\Omega _{\tau } , \end{aligned}$$

(C.1)

and constitutes the first fundamental form of the layer.