Nonlinear and Linear Elastodynamic Transformation Cloaking

Abstract

In this paper we formulate the problems of nonlinear and linear elastodynamic transformation cloaking in a geometric framework. In particular, it is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak using a cloaking transformation. We show that nonlinear elastodynamic transformation cloaking is not possible while nonlinear elastostatic transformation cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. In the case of classical linear elastodynamics, in agreement with the previous observations in the literature, we show that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. We prove that elastodynamic transformation cloaking is not possible regardless of the shape of the hole and the cloak. It is shown that the small-on-large theory, i.e., linearized elasticity with respect to a pre-stressed configuration, does not allow for transformation cloaking either. However, elastodynamic cloaking of a cylindrical hole is possible for in-plane deformations while it is not possible for anti-plane deformations. We next show that for a cavity of any shape elastodynamic transformation cloaking cannot be achieved for linear gradient elastic solids; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in dimension two for any shape of the hole and the cloak. In particular, in dimension two transformation cloaking cannot be achieved in linear Cosserat elasticity. We show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat elasticity. We conjecture that this result is true for a cavity of any shape. It should be emphasized that in this paper we do not consider the so-called metamaterials [70, 72].

Appendix A Riemannian Geometry

To make the paper self-contained, in this appendix some basic concepts of Riemannian geometry are tersely reviewed. It should be emphasized that only those geometric concepts that have been used in the paper are discussed here.

For a smooth n-dimensional manifold $\mathcal {B}$, the tangent space of $\mathcal {B}$ at a point $X \in \mathcal {B}$ is denoted by $T_X\mathcal {B}$. Assume that $\mathcal {S}$ is another n-dimensional manifold and $\varphi :\mathcal {B}\rightarrow \mathcal {S}$ is a diffeomorphism (smooth and invertible map with a smooth inverse) between the two manifolds. A smooth vector field $\mathbf {W}$ on $\mathcal {B}$ assigns a vector $\mathbf {W}_X\in T_X\mathcal {B}$ for every $X\in \mathcal {B}$ such that the mapping $X\mapsto \mathbf {W}_X$ is smooth. If $\mathbf {W}$ is a vector field on $\mathcal {B}$, then the push-forward of $\mathbf {W}$ by $\varphi$ is a vector field on $\varphi (\mathcal {B})$ defined as $\varphi _*\mathbf {W} = T\varphi \cdot \mathbf {W} \circ \varphi ^{-1}$. Similarly, if $\mathbf {w}$ is a vector field on $\varphi (\mathcal {B}) \subset \mathcal {S}$, the pull-back of $\mathbf {w}$ by $\varphi$ is a vector field on $\mathcal {B}$ defined as $\varphi ^*\mathbf {w}=T(\varphi ^{-1}) \cdot \mathbf {w} \circ \varphi$. Let us denote the tangent map of $\varphi$ by $\mathbf {F}$, i.e., $\mathbf {F}=T\varphi$. Let $\{X^A\}$ and $\{x^a\}$ be the local charts for $\mathcal {B}$ and $\mathcal {S}$, respectively. More specifically, a local chart for $\mathcal {B}$ at $X\in \mathcal {B}$ is a homeomorphism from an open subset $\mathcal {U}\subset \mathcal {B}$ ($X\in \mathcal {U}$) to an open subset $\mathcal {V}\subset {{\mathbb {R}}}^n$. $\{X^A\}$ are components of this map. The derivative map $\mathbf {F}$ is a two-point tensor with the following representation in the local charts: $\mathbf {F}=F^a{}_A\frac{\partial }{\partial X^A}\otimes dx^a$, $F^a{}_A=\frac{\partial \varphi ^a}{\partial X^A}$, where $\{\frac{\partial }{\partial X^A}\}$ and $\{dx^a\}$ are bases for $T_X\mathcal {B}$ and $T^*_{\varphi (X)}\varphi (\mathcal {B})$, respectively. Recall that $T^*_{\varphi (X)}\varphi (\mathcal {B})$ denotes the cotangent space (or the dual space) of $T_{\varphi (X)}\varphi (\mathcal {B})$. The push-forward and pull-back of vectors have the following coordinate representations: $(\varphi _*\mathbf {W})^a=F^a{}_A W^A$, and $(\varphi ^*\mathbf {w})^A=(F^{-1})^A{}_a w^a$.

A type $({}^{0}_{2})$-tensor at $X \in \mathcal {B}$ is a bilinear map $\mathbf {T}: T_{X}\mathcal {B}\times T_{X}\mathcal {B} \rightarrow {\mathbb {R}}$, where in a local coordinate chart $\{X^A\}$ for $\mathcal {B}$ it reads $\mathbf {T}(\mathbf {U},\mathbf {V})= T_{AB}U^{A}V^{B}, ~\forall \, \mathbf {U},\mathbf {V} \in T_{X} \mathcal {B}$. A Riemannian manifold $(\mathcal {B},\mathbf {G})$ is a smooth manifold $\mathcal {B}$ endowed with an inner product $\mathbf {G}_X$ (a symmetric $({}^{0}_{2})$-tensor field) on the tangent space $T_X\mathcal {B}$ that smoothly varies in the sense that if $\mathbf {U}$ and $\mathbf {V}$ are smooth vector fields on $\mathcal {B}$, then $X \mapsto \mathbf {G}_X(\mathbf {U}_X,\mathbf {V}_X) =: \langle \!\langle \mathbf {U}_X,\mathbf {V}_X \rangle \!\rangle _{\mathbf {G}_X}$, is a smooth function. Let $(\mathcal {B},\mathbf {G})$ and $(\mathcal {S},\mathbf {g})$ be Riemannian manifolds and let $\varphi :\mathcal {B}\rightarrow \mathcal {S}$ be a diffeomorphism (smooth map with smooth inverse). The push-forward of the metric $\mathbf {G}$ is a metric on $\varphi (\mathcal {B})\subset \mathcal {S}$, which is denoted by $\varphi _*\mathbf {G}$ defined as

$$\begin{aligned} (\varphi _*\mathbf {G})_{\varphi (X)}\left( \mathbf {u}_{\varphi (X)},\mathbf {v}_{\varphi (X)}\right) :=\mathbf {G}_X\left( (\varphi ^*\mathbf {u})_X,(\varphi ^*\mathbf {v})_X\right) . \end{aligned}$$

(A.1)

In components, $(\varphi _*\mathbf {G})_{ab} = (F^{-1})^A{}_a (F^{-1})^B{}_b G_{AB}$. Similarly, the pull-back of the metric $\mathbf {g}$ is a metric in $\mathcal {B}$, which is denoted by $\varphi ^*\mathbf {g}$ defined as

$$\begin{aligned} (\varphi ^*\mathbf {g})_X(\mathbf {U}_X,\mathbf {V}_X) :=\mathbf {g}_{\varphi (X)}((\varphi _*\mathbf {U})_{\varphi (X)},(\varphi _*\mathbf {V})_{\varphi (X)}) . \end{aligned}$$

(A.2)

In components, $(\varphi ^*\mathbf {g})_{AB} = F^a{}_A F^b{}_B g_{ab}$. The diffeomorphism $\varphi$ is an isometry between two Riemannian manifolds $(\mathcal {B},\mathbf {G})$ and $(\mathcal {S},\mathbf {g})$ if $\mathbf {g}=\varphi _*\mathbf {G}$, or equivalently, $\mathbf {G}=\varphi ^*\mathbf {g}$. An isometry, by definition, preserves distances.

Affine connections, and their torsion and curvature tensors.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 $\forall ~\mathbf {X},\mathbf {Y},\mathbf {X}_1,\mathbf {X}_2,\mathbf {Y}_1,\mathbf {Y}_2\in \mathcal {X}(\mathcal {B}),\forall ~f,f_1,f_2\in C^{\infty }(\mathcal {B}),\forall ~a_1,a_2\in {\mathbb {R}}$: i) $\nabla _{f_1\mathbf {X}_1+f_2\mathbf {X}_2}\mathbf {Y}=f_1\nabla _{\mathbf {X}_1}\mathbf {Y} +f_2\nabla _{\mathbf {X}_2}\mathbf {Y}$, ii) $\nabla _{\mathbf {X}}(a_1\mathbf {Y}_1+a_2\mathbf {Y}_2)=a_1\nabla _{\mathbf {X}}(\mathbf {Y}_1)+a_2\nabla _{\mathbf {X}}(\mathbf {Y}_2)$, and iii) $\nabla _{\mathbf {X}}(f\mathbf {Y})=f\nabla _{\mathbf {X}}\mathbf {Y}+(\mathbf {X}f)\mathbf {Y}$. $\nabla _{\mathbf {X}}\mathbf {Y}$ is called the covariant derivative of $\mathbf {Y}$ along $\mathbf {X}$. In a local coordinate chart $\{X^A\}$, $\nabla _{\partial _A}\partial _B=\varGamma ^C{}_{AB}\partial _C$, where $\varGamma ^C{}_{AB}$ are Christoffel symbols of the connection, and $\partial _A=\frac{\partial }{\partial x^A}$ are the natural bases for the tangent space corresponding to a coordinate chart $\{x^A\}$. A linear connection is said to be compatible with a metric $\mathbf {G}$ on the manifold if

(A.3)

where $\left\langle \!\left\langle .,. \right\rangle \!\right\rangle _{\mathbf {G}}$ is the inner product induced by the metric $\mathbf {G}$. It can be shown that $\nabla$ is compatible with $\mathbf {G}$ if and only if $\nabla \mathbf {G}=\mathbf {0}$, or in components

$$\begin{aligned} G_{AB|C}=\frac{\partial G_{AB}}{\partial X^C}-\varGamma ^S{}_{CA}G_{SB}-\varGamma ^S{}_{CB}G_{AS}=0. \end{aligned}$$

(A.4)

Suppose $\mathbf {V},\mathbf {W}\in \mathcal {X}(\mathcal {B})$ are vector fields and $\alpha :I\rightarrow \mathcal {B}$ is a smooth curve. The restriction of the vector fields to $\alpha$, i.e., $\mathbf {V}\circ \alpha$ and $\mathbf {W}\circ \alpha$ are called vector fields along the curve $\alpha$. The set of all vector fields along $\alpha$ is denoted by $\mathcal {X}(\alpha )$. Covariant derivative along the curve $\alpha$ is a map $D_t:\mathcal {X}(\alpha )\rightarrow \mathcal {X}(\alpha )$ with the following properties: $D_t(\mathbf {V}+\mathbf {W})=D_t\mathbf {V}+D_t\mathbf {W}$, and $D_t(f\mathbf {W})=\frac{df}{dt}\mathbf {W}+fD_t\mathbf {W}$. If $\mathbf {W}\in \mathcal {X}(\alpha )$ is the restriction of $\widetilde{\mathbf {W}}\in \mathcal {X}(\mathcal {B})$ to $\alpha$, then, $D_t\mathbf {W}=\nabla _{\alpha '(t)}\widetilde{\mathbf {W}}$. If the connection $\nabla$ is $\mathbf {G}$-compatible, then

(A.5)

The covariant derivative of a two-point tensor $\mathbf {T}$ is given by

$$\begin{aligned}&T^{AB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{g\cdots q|K} =\frac{\partial }{\partial X^k}T^{AB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{g\cdots q}\nonumber \\&\qquad +T^{RB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{g\cdots q}\varGamma ^A{}_{RK} +\mathrm {(all\,\,upper\,\, referential\,\, indices)}\nonumber \\&\qquad -T^{AB\cdots F}{}_{R\cdots Q}{}^{ab\cdots f}{}_{g\cdots q}\varGamma ^R{}_{GK} -\mathrm {(all\,\,lower\,\, referential\,\, indices)}\nonumber \\&\qquad +T^{RB\cdots F}{}_{G\cdots Q}{}^{lb\cdots f}{}_{g\cdots q}\gamma ^a{}_{lr}F^r{}_K +\mathrm {(all\,\,upper\,\, spatial\,\, indices)}\nonumber \\&\qquad -T^{AB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{l\cdots q}\gamma ^l{}_{gr}F^r{}_K -\mathrm {(all\,\,lower\,\, spatial\,\, indices)}\,. \end{aligned}$$

(A.6)

The torsion of a connection is defined as $\varvec{T}(\mathbf {X},\mathbf {Y})=\nabla _{\mathbf {X}}\mathbf {Y}-\nabla _{\mathbf {Y}}\mathbf {X}-[\mathbf {X},\mathbf {Y}]$, where $[\mathbf {X},\mathbf {Y}](F)=\mathbf {X}(\mathbf {Y}(F))-\mathbf {Y}(\mathbf {X}(F)),~\forall ~F\in C^{\infty }(\mathcal {S})$, is the commutator of $\mathbf X$ and $\mathbf Y$. In components, in a local chart $\{X^A\}$, $T^A{}_{BC}=\varGamma ^A{}_{BC}-\varGamma ^A{}_{CB}$, and $[\mathbf {X},\mathbf {Y}]^a=\frac{\partial Y^a}{\partial x^b}X^b-\frac{\partial X^a}{\partial x^b}Y^b$. $\nabla$ is symmetric if it is torsion-free, i.e., $\nabla _{\mathbf {X}}\mathbf {Y}-\nabla _{\mathbf {Y}}\mathbf {X}=[\mathbf {X},\mathbf {Y}]$. On any Riemannian manifold $(\mathcal {B},\mathbf {G})$ there is a unique linear connection $\nabla ^{\mathbf {G}}$ that is compatible with $\mathbf {G}$ and is torsion-free. This is the Levi-Civita connection. If the Levi-Civita connection $\nabla ^{\mathbf {G}}$ is used, the covariant time derivative is denoted by $D^{\mathbf {G}}_t$. In a manifold with a connection the curvature is a map $\varvec{\mathcal {R}}:\mathcal {X}(\mathcal {B})\times \mathcal {X}(\mathcal {B})\times \mathcal {X}(\mathcal {B})\rightarrow \mathcal {X}(\mathcal {B})$ defined by $\varvec{\mathcal {R}}(\mathbf {X},\mathbf {Y},\mathbf {Z})=\nabla _{\mathbf {X}}\nabla _{\mathbf {Y}}\mathbf {Z}-\nabla _{\mathbf {Y}}\nabla _{\mathbf {X}}\mathbf {Z}-\nabla _{[\mathbf {X},\mathbf {Y}]}\mathbf {Z}$, or in components $\mathcal {R}^A{}_{BCD}=\frac{\partial \varGamma ^A{}_{CD}}{\partial X^B}-\frac{\partial \varGamma ^A{}_{BD}}{\partial X^C}+\varGamma ^A{}_{BM}\varGamma ^M{}_{CD}-\varGamma ^A{}_{CM}\varGamma ^M{}_{BD}$. The Riemannian curvature is the curvature tensor of the Levi-Civita connection $\nabla ^{\mathbf {G}}$ and is denoted by $\varvec{\mathcal {R}}_{\mathbf {G}}$. The Ricci identity for a vector field $\mathbf {U}$ with components $W^A$ reads $U^A{}_{|BC}-U^A{}_{|CB}=\mathcal {R}^A{}_{BCD}U^D$. Ricci identity for a 1-form $\varvec{\alpha }$ with components $\alpha _A$ reads $\alpha _A{}_{|BC}-\alpha _A{}_{|CB}=\mathcal {R}^D{}_{BCA}\alpha _D$. The Ricci curvature $\varvec{\mathsf {Ric}}$ is defined as $\mathsf {Ric}_{CD}=\mathcal {R}^A{}_{ACD}$, and is a symmetric tensor. The Ricci curvature of the Levi-Civita connection $\nabla ^{\mathbf {G}}$ is denoted by $\varvec{\mathsf {Ric}}_{\mathbf {G}}$.

Vector bundles.Suppose $\mathcal {E}$ and $\mathcal {B}$ are sets and consider a map $\pi :\mathcal {E}\rightarrow \mathcal {B}$. The fiber over $X\in \mathcal {B}$ is the set $\mathcal {E}_X:=\pi ^{-1}(X)\subset \mathcal {E}$. For an onto map $\pi$ fibers are non-empty and $\mathcal {E}=\sqcup _{X\in \mathcal {B}}\mathcal {E}_X$, where $\sqcup$ denoted disjoint union of sets. Now suppose $\mathcal {E}$ and $\mathcal {B}$ are manifolds and assume that for any $X\in \mathcal {B}$, there exists a neighborhood $\mathcal {U}\subset \mathcal {B}$ of X, a manifold $\mathcal {F}$, and a diffeomorphism $\psi :\pi ^{-1}(\mathcal {U})\rightarrow \mathcal {U}\times \mathcal {F}$ such that $\pi ={\text {pr}}_1\circ \psi$, where ${\text {pr}}_1:\mathcal {U}\times \mathcal {F}\rightarrow \mathcal {U}$ is projection onto the first factor. $(\mathcal {E},\pi ,\mathcal {B})$ is called a fiber bundle and $\mathcal {E}$, $\pi$, and $\mathcal {B}$ are called the total space, the projection, and the base space, respectively. If for any $X\in \mathcal {B}$, $\pi ^{-1}(X)$ is a vector space, $(\mathcal {E},\pi ,\mathcal {B})$ is called a vector bundle. The set of sections of this bundle $\varGamma (\mathcal {E})$ is the set of all smooth maps $\sigma :\mathcal {B}\rightarrow \mathcal {E}$ such that $\sigma (X)\in \mathcal {E}_X,~\forall ~X\in \mathcal {B}$. An important example of a vector bundle is the tangent bundle of a manifold for which $\mathcal {E}=T\mathcal {B}$.

Induced bundle and connection.Consider a map between Riemannian manifolds $\varphi :\mathcal {B}\rightarrow \mathcal {S}$. The tangent bundles of $\mathcal {B}$ and $\mathcal {S}$ are denoted by $T\mathcal {B}=\sqcup _{X\in \mathcal {B}}T_X\mathcal {B}$ and $T\mathcal {S}=\sqcup _{x\in \mathcal {S}}T_x\mathcal {S}$, respectively. We define an induced vector bundle $\varphi ^{-1}T\mathcal {S}$, which is a vector bundle over $\mathcal {B}$ whose fiber over $X\in \mathcal {B}$ is $T_{\varphi (X)}\mathcal {S}$ [76]. The connection $\nabla ^{\mathbf {g}}$ induces a unique connection $\nabla ^{\varphi }$ on $\varphi ^{-1}T\mathcal {S}$ defined as

$$\begin{aligned} \nabla ^{\varphi }_{\mathbf {W}}\mathbf {w}\circ \varphi =\nabla ^{\mathbf {g}}_{\varphi _*\mathbf {W}}\mathbf {w} ,~~~\mathbf {W}\in T_X\mathcal {B},~\mathbf {w}\in \varGamma (T\mathcal {S})\,. \end{aligned}$$

(A.7)

$\nabla ^{\varphi }$ is called the induced connection. It can be shown that its connection coefficients with respect to the coordinate charts $\{X^A\}$ and $\{x^a\}$ of $\mathcal {B}$ and $\mathcal {S}$, respectively, are $\frac{\partial \varphi ^b}{\partial X^A}\gamma ^a{}_{bc}$. In particular, the variation field $\delta \varphi$ defined in §3 is a section of $\varGamma (\varphi ^{-1}T\mathcal {S})$, i.e., $\delta \varphi$ defines a vector field in $\mathcal {S}$ along the map $\varphi$. For a two-point tensor, e.g., deformation gradient, covariant derivative involves both $\nabla ^{\mathbf {g}}$ and $\nabla ^{\mathbf {G}}$: $F^a{}_{A|B}=\frac{\partial F^a{}_A}{\partial X^B}+(F^b{}_B\gamma ^a{}_{bc})F^c{}_A-\varGamma ^C{}_{AB}F^a{}_C=\frac{\partial F^a{}_A}{\partial X^B}+\gamma ^a{}_{bc}F^b{}_BF^c{}_A-\varGamma ^C{}_{AB}F^a{}_C$. We denote the covariant derivative of the deformation gradient by $\nabla \mathbf {F}=F^a{}_{A|B}dX^B\otimes dX^A\otimes \frac{\partial }{\partial x^a}$. It is straightforward to show that [76]

$$\begin{aligned} \nabla ^{\varphi }\mathbf {F}(\mathbf {X},\mathbf {Y})=\nabla ^{\varphi }_{\mathbf {X}}\varphi _*\mathbf {Y} -\varphi _*\nabla ^{\mathbf {G}}_{\mathbf {X}}\mathbf {Y},~~~ \nabla ^{\varphi }_{\mathbf {X}}\varphi _*\mathbf {Y}-\nabla ^{\varphi }_{\mathbf {Y}}\varphi _*\mathbf {X} =\varphi _*[\mathbf {X},\mathbf {Y}]. \end{aligned}$$

(A.8)

The metrics $\mathbf {G}$ and $\mathbf {g}$ induce an inner product $\langle , \rangle _X$ in $T_{\varphi (X)}\mathcal {S}\otimes T^*_X\mathcal {B}$. This is defined first for the basis $\left\{ \frac{\partial }{\partial x^a}\otimes dX^A, ~ 1\le a\le n,~1\le A\le n \right\}$ as $\langle \frac{\partial }{\partial x^a}\otimes dX^A,\frac{\partial }{\partial x^b}\otimes dX^B \rangle _X=g_{ab}G^{AB}$, and then one extends it linearly to arbitrary elements in $T_{\varphi (X)}\mathcal {S}\otimes T^*_X\mathcal {B}$. $\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B}$ is the vector bundle whose fiber at $X\in \mathcal {B}$ is $T_{\varphi (X)}\mathcal {S}\otimes T^*_X\mathcal {B}$. The two-point tensor $\mathbf {F}=T\varphi :\mathcal {B}\rightarrow \varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B}$, is a section of this bundle, i.e., $\mathbf {F}\in \varGamma (\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B})$. One can define a fiber metric on $\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B}$ using the inner product $\langle , \rangle _X$ in $T_{\varphi (X)}\mathcal {S}\otimes T_X^*\mathcal {B}$ as follows: for $\sigma , \tau \in \varGamma (\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B})$, define . One can define a connection $\nabla$ in $\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B}$ using the Levi-Civita connections $\nabla ^{\mathbf {G}}$ and $\nabla ^{\mathbf {g}}$: consider a section $\mathbf {W}\otimes \varvec{\alpha }\in \varGamma (\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B})$ and let $\nabla (\mathbf {W}\otimes \varvec{\alpha })=\nabla ^{\varphi }\mathbf {W}\otimes \varvec{\alpha }+\mathbf {W}\otimes \nabla ^{\mathbf {G}}\varvec{\alpha }$. This connection is compatible with the fiber metric in $\varphi ^{-1}T\mathcal {S}\otimes T^*\mathcal {B}$.

The Piola transform.The Piola transform of a vector ${\mathbf {w}}\in T_{\varphi (X)}\mathcal {S}$ is a vector $\mathbf {W}\in T_X\mathcal {B}$ given by $\mathbf {W}=J\varphi ^*\mathbf {w}=J\mathbf {F}^{-1}\mathbf {w}$. In coordinates, $W^A=J(F^{-1})^A{}_bw^b$, where $J=\sqrt{\frac{\det \mathbf {g}}{\det \mathbf {G}}}\det \mathbf {F}$ is the Jacobian of $\varphi$ with $\mathbf {G}$ and $\mathbf {g}$ the Riemannian metrics of $\mathcal {B}$ and $\mathcal {S}$, respectively. It can be shown that ${\text {Div}}\mathbf {W}=J({\text {div}}\mathbf {w})\circ \varphi$. In coordinates, $W^A{}_{|A}=Jw^a{}_{|a}$; this is also known as the Piola identity. Another way of writing the Piola identity is in terms of the unit normal vectors of a surface in $\mathcal {B}$ and its corresponding surface in $\mathcal {S}$ and the area elements. It is written as $\hat{\mathbf {n}}da=J\mathbf {F}^{-\star }\hat{\mathbf {N}}dA$, or in components, $n_ada=J(F^{-1})^A{}_aN_AdA$. In the literature of continuum mechanics, this is called Nanson’s formula.

Lie derivative.Let $\mathbf {w}:\mathcal {U}\rightarrow T\mathcal {S}$ be a vector field, where $\mathcal {U} \subset \mathcal {S}$ is open. A curve $\alpha :I \rightarrow \mathcal {S}$, where I is an open interval, is an integral curve of $\mathbf {w}$ if $\frac{d \alpha (t)}{dt}=\mathbf {w}(\alpha (t)),~\forall ~t \in I$. For a time-dependent vector field $\mathbf {w}:\mathcal {S}\times I\rightarrow T\mathcal {S}$, where I is some open interval, the collection of maps $\psi _{\tau ,t}$ is the flow of $\mathbf {w}$ if for each t and x, $\tau \mapsto \psi _{\tau ,t}(x)$ is an integral curve of $\mathbf {w}_t$, i.e., $\frac{d}{d\tau }\psi _{\tau ,t}(x)=\mathbf {w}(\psi _{\tau ,t}(x),\tau )$, and $\psi _{t,t}(x)=x$. Let $\mathbf {t}$ be a time-dependent tensor field on $\mathcal {S}$, i.e., $\mathbf {t}_t(x)=\mathbf {t}(x,t)$ is a tensor. The Lie derivative of $\mathbf {t}$ with respect to $\mathbf {w}$ is defined as $\mathbf {L}_{\mathbf {w}}\mathbf {t}=\frac{d}{d \tau }\psi _{\tau ,t}^* \mathbf {t}_{\tau } \big |_{\tau =t}$. Note that $\psi _{\tau ,t}$ maps $\mathbf {t}_t$ to $\mathbf {t}_{\tau }$. Hence, to calculate the Lie derivative one drags $\mathbf {t}$ along the flow of $\mathbf {w}$ from $\tau$ to t and then differentiates the Lie dragged tensor with respect to $\tau$. The autonomous Lie derivative of $\mathbf {t}$ with respect to $\mathbf {w}$ is defined as $\mathfrak {L}_{\mathbf {w}}\mathbf {t}=\frac{d}{d \tau }\psi _{\tau ,t}^* \mathbf {t}_{t} \big |_{\tau =t}$. Thus, $\mathbf {L}_{\mathbf {w}}\mathbf {t}=\partial \mathbf {t}/\partial t+\mathfrak {L}_{\mathbf {w}}\mathbf {t}$. For a scalar f, $\mathbf {L}_{\mathbf {w}}f=\partial f/\partial t+\mathbf {w}[f]$. In a coordinate chart $\{x^a\}$ this reads, $\mathbf {L}_{\mathbf {w}}f=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x^a}w^a$. For a vector $\mathbf {u}$, one can show that $\mathbf {L}_{\mathbf {w}}\mathbf {u}=\frac{\partial \mathbf {w}}{\partial t}+[\mathbf {w},\mathbf {u}]$. If $\nabla$ is a torsion-free connection, then $[\mathbf {w},\mathbf {u}]=\nabla _{\mathbf {w}}\mathbf {u}-\nabla _{\mathbf {u}}\mathbf {w}$. Thus, $\mathbf {L}_{\mathbf {w}}\mathbf {u}=\frac{\partial \mathbf {w}}{\partial t}+\nabla _{\mathbf {w}}\mathbf {u}-\nabla _{\mathbf {u}}\mathbf {w}$.

When linearizing nonlinear elasticity one starts with a one-parameter family of motions $\varphi _{t,\epsilon }:\mathcal {B}\rightarrow \mathcal {S}$. By definition of the variation field $\mathbf {U}_t=\delta \varphi _t$, $\varphi _{t,\epsilon }$ is the flow of the variation field. Given a tensor field $\mathbf {t}$ in $\mathcal {S}$, $\bar{\mathbf {T}}_{\epsilon }=\varphi _{t,\epsilon }^*\mathbf {t}\circ \varphi _{t,\epsilon }$ is a vector field on $\mathcal {B}$. Its linearization is defined as

$$\begin{aligned} \begin{aligned} \delta \bar{\mathbf {T}}&=\frac{d}{d\epsilon }\bar{\mathbf {T}}_{\epsilon }\Big |_{\epsilon =0} =\left( \frac{d}{d\epsilon }\varphi _{t,\epsilon }^*\mathbf {t}\circ \varphi _{t,\epsilon }\right) \Big |_{\epsilon =0} \\&=\left( \varphi _{t,\epsilon }^*\mathbf {L}_{\mathbf {U}_{t,\epsilon }}\mathbf {t}\circ \varphi _{t,\epsilon }\right) \Big |_{\epsilon =0} =\mathring{\varphi }_t^*\left( \mathbf {L}_{\mathbf {U}_{t}}\mathbf {t}\circ \mathring{\varphi }_t\right) . \end{aligned} \end{aligned}$$

(A.9)

Thus, $\delta \mathbf {t}=\mathbf {L}_{\mathbf {u}_t}\mathbf {t}$, where $\mathbf {u}_t=\mathbf {U}_t\circ \mathring{\varphi }_t^{-1}$.