数理解析研究所講究録
第 1692 巻 2010 年 101-111

101

A Continuum Dynamics on Vector Bundle
理化学研究所 山岡英孝 (Hidetaka Yamaoka)1
Computational Cell Biomechanics Team,
VCAD System Research Program,

RIKEN
Abstract
The Cosserat theory for continua with microstructure can be geometrically interpreted as a continuum dynamics on vector bundle. In this talk, we begin with geometrical settings for the continua with microstructure and construct its dynamics
as continuum dynamics on vector bundle. As an example, we deal with a Cosseratrod, which is considered as a one-dimensional continuum with microstructure. In
addition, we suggest that we can describe lower microstructure of the elastic rods, byextending the base manifold of the vector bundle to a three-dimensional continuum.
Such rod model well represents biomolecules involving various interacting factors, sothat the model can be applied to analysis of deformation behavior of biomolecules.
We expect that our geometrical method would contribute to development of themolecular biomechanics.

1

Introduction

We attempt to construct geometrical foundations and dynamical frameworks of a directed
medium based on the fiber bundle theory. The directed medium is a continuum withmicrostructures that is described by a deformable vector, called a director. Studies on the
directed medium were actively pursued in the $1960s$ , for example, by Ericksen [1], Toupin[2], and Eringen and \S uhubi [3] and in recent years, have been investigated from many
point of views such as elast-plasticity [4, 5], advanced materials [6], and biomechanics[7, 8]. In contrast, since about the $1960s$ , the elastic theory has been reconstructed using
differential geometry, for example, by Green and Rivilin [9], Noll [10], and Wang [11].
A modern text by Marsden and Hughes [12] helps us to consider geometrical settings ofelasticity (see also an early textbook [13]). In this study, we develop the dynamics of thedirected medium based on the fiber bundle theory in differential geometry.
In geometric continuum mechanics, an elastic body is viewed as a differentiable manifold, while a directed medium is viewed as a vector bundle whose fiber denotes a collectionof the deformable directors. Hence, the mechanical behaviors of the directed medium
should be described as the continuum dynamics on a tangent bundle of a vector bundle.
Thus, we begin with a geometrical setting of the continuum dynamics on a vector bundle,
and derive a weak form and equations of motion for the directed medium. For futureapplications, we use elasticity notations to provide a framework of continuum dynamics
on the vector bundle and present some figures for better understanding. Moreover, welhyamaoka@riken.jp

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apply our resultant equations to a Cosserat rod, as an example, and find that the derived
equations of motion coincide with the balance laws of large deformable rods. It is simple
to prove such a coincidence if the equations of motion are restricted to the special Cosserat
rod with undeformed cross-section.
We can use our description to examine such macro-micro interactive mechanisms, if
we have to consider only the geometrical structures of objects using the mechanisms,
i. e., the corresponding base manifolds and fiber spaces. Such geometrical considerations
help us to improve our understanding of the complicated mechanical behaviors of various
structures associated with the macro-micro interactive mechanisms.

2

Geometry and kinematics

In the geometric continuum mechanics, an elastic body is viewed as an m-dimen-sional
Riemannian manifold , and deforms in an ambient space $S$ , an n-dimen-sional Riemannian manifold $(m\leq n)[12]$ . In contrast, when we consider a deformation of a continuumwith microstructure, we must replace these manifolds with principal fiber bundles, denoting by
and 2, respectively, over the manifolds
and $S$ . The microstructure isoften expressed by a r-dimensional vector $(r\leq n)$ , called director. In this case, the spaceconsisting of the directors, 7, is exactly the fiber of the bundle, and thencan be considered as the real vector bundle.
A configuration of the continuum with microstructure is given by a smooth embedding:, then the configuration space is a space of all embeddings,$\mathcal{B}$$\mathcal{P}$$\mathcal{B}$$\mathcal{P}arrow \mathcal{B}$$\mathcal{P}$$g\simeq \mathcal{B}\cross 7$$\Phi$$\mathcal{P}arrow 2$$=$ {$\Phi$:$\mathcal{P}arrow 2$, smooth embedding}.

Indeed, for an arbitrary point
the embedding through embedding of the base manifold
onto the fiber :;$p=(b, v)\in \mathcal{P}\simeq \mathcal{B}\cross\gamma/(b\in \mathcal{B}$$\Phi$$\varphi$$\phi$(2.1)

and $v\in\gamma)$ , we can define:
and projection$\mathcal{B}arrow S$$\mathcal{B}\cross 7arrow \mathbb{R}^{r}$$\Phi(p)=(\phi(b), \varphi(b, v))$

.

(2.2)

It is easy to verify that this map is the embedding. We note here that the ambient bundle
J2 is the vector bundle $9arrow S$ with fiber , i. e.,, as shown in Fig. 1.
$9arrow S\cross \mathbb{R}^{r}$$\mathbb{R}^{r}$

For sake of simplicity, we take the ambient space as the n-dimensional Euclidean
space;, and we embed the fiber space
into the same Euclidean space.
Then we denote position vectors of points in the reference bodyand current body
by $X=\phi_{0}(b_{0})(b_{0}\in \mathcal{B}_{0})$ and $x=\phi(b)(b\in \mathcal{B})$ , respectively. Also, we denotereference and current directors associated with their points by $\Xi=\varphi_{0}(b_{0}, v_{0})(v_{0}\in 7_{0}’)$and $\xi=\varphi(b, v)(v\in Y)$ , as shown Fig. 1. Accordingly, three deformation gradient tensors
can be defined as$S\simeq \mathbb{R}^{n}$$\mathbb{R}^{r}$$\mathbb{R}^{n}$$\mathcal{B}_{0}$$\mathcal{B}$$F= \frac{\partial x}{\partial X}$,$\mathfrak{F}=\frac{\partial\xi}{\partial X}$,$\mathcal{F}=\frac{\partial\xi}{\partial_{-}^{-}-}$,

called the macro, mixture, and micro deformation gradient, respectively.

(2.3)

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Figure 1: Illustration of the continuum with microstructure,, and its local coordinates in 2.
the ambient bundle,$\mathcal{P}\simeq \mathcal{B}\cross 7$, embedded into$\ovalbox{\tt\small REJECT}\simeq S\cross \mathbb{R}^{r}$

Since the trivialization of bundle 2 is expressed as$9\simeq S\cross \mathbb{R}^{r}$

(2.4)

,

the flat connection is defined on the bundle 2. Accordingly, the tangent bundle T.2 is;decomposed into the tangent bundles $TS$ and$T\mathbb{R}^{r}$$T2\simeq TS\oplus T\mathbb{R}^{r}$

.

(2.5)

Also, the cotangent bundle $T^{*}9$ is decomposed by the flat connection, so that one-form$dq\in T_{q}^{*}2$ is put in the form
(2.6)

$dq=dx+d\xi$ ,and expressed, in terms of the reference coordinates $q_{0}=(X, \Xi)$ , as$dq=(F+\mathfrak{F})dX+\mathcal{F}d\Xi$

From Eq. (2.7), the quadratic form$dq^{2}$.

(2.7)

is calculated as$dq^{2}=dX^{T}(F+\mathfrak{F})^{T}(F+\mathfrak{F})dX$$+dX^{T}(F+\mathfrak{F})^{T}\mathcal{F}d\Xi+d\Xi^{T}\mathcal{F}^{T}(F+\mathfrak{F})dX$$+d\Xi^{T}\mathcal{F}^{T}\mathcal{F}d\Xi$

where

$T$

(2.8)

,

denotes the transposition of tensors. Here, we set$C=(F+\mathfrak{F})^{T}(F+\mathfrak{F})$$C=\mathcal{F}^{T}\mathcal{F}$(2.9a)(2.9b),$C=\mathcal{F}^{T}(F+\mathfrak{F})$,,(2.9c)

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called macro deformation, micro deformation, and (macro-micro) mixture deformation,
respectively. The reference deformations put in the form
$C_{0}=(I+\mathfrak{F}_{0})^{T}(I+\mathfrak{F}_{0})$(2.10a)(2.10b),$C_{0}=I$ ,$C_{0}=I^{T}(I+\mathfrak{F}_{0})$(2.10c),where is n-th order identity tensor, and where
the difference of the current and reference quadratic forms,$I$$\mathfrak{F}_{0}=\partial\Xi/\partial X$, as well. Thus, we have$dq^{2}-dq_{0}^{2}=dX^{T}(C-C_{0})dX$$+dX^{T}(C-C_{0})^{T}d\Xi+d\Xi^{T}(C-C_{0})dX$(2.11)$+d\Xi^{T}(C-C_{0})d\Xi$ ,so that we define the strains as$E= \frac{1}{2}(C-C_{0})$$\mathcal{E}=\frac{1}{2}(C-C_{0})$(2.12a),,$\not\subset=\frac{1}{2}(C-C_{0})$(2.12b).

(2.12c)Then we call them macro strain, micro strain, and (macro-micro) mixture strain, respectively. Here, we comment on the terminologies used by Eringen’s textbook for the
microcontinuum [14]. In the textbook, our mixture deformation
is decomposed into
, called the Cosserats deformation tensor, when the director is deformed
andrigidly, “micropolar continua” according to the textbook, and the wryness tensor, respectively. Additionally, the micro deformation tensor is defined in the same manner, while
the macro deformation tensor is linearized to $F^{T}F$ .
$C$$\mathcal{F}^{T}F$3$\mathcal{F}^{T}\mathfrak{F}$Dynamics

Now, we consider a Lagrangian, whereis the kinetic energy, defined througha metric on , andis a potential function on . Then the dynamics of the continuumwith microstructure is described on the tangent bundle $T$ of the configuration space ,that is, the Lagrangian is defined as a function of the tangent bundle $T$ to ,$\mathcal{L}=\mathcal{T}-\mathcal{W}$$\mathcal{T}$$\mathcal{W}$$\mathcal{L}$$\mathbb{R}$$\mathcal{L}(\Phi,\dot{\Phi})=\mathcal{T}(\Phi,\dot{\Phi})-\mathcal{W}(\Phi)$.

Usually, the strain energy
is expressed as a functional offunction of the deformation gradient tensors, and :$\mathcal{W}(\Phi)$$\psi$, which is difined as a$\mathcal{F}$$F_{:}\mathfrak{F}$.

$\mathcal{W}(\Phi)=\int_{p}\psi(F, \mathfrak{F}, \mathcal{F})dV$

Here, $dV$ is the volume form of the rnaterial bundle(3.1)$\mathcal{P}$

.

(3.2)

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and ,9, respectively,
We also denote the Lagrangian and kinetic enegy densities by, and, for simplicity, we consider those densities as
andi. e.,. In this case, theon the tangent bundle
functions of the local coordinatesis expressed as follows:
Hamilton’s principle for any time interval$\mathcal{L}$$\mathcal{L}=\int \mathcal{L}dV$$\mathcal{T}=\int\ovalbox{\tt\small REJECT} dV$$(x, \xi,\dot{x},\dot{\xi})$$T\ovalbox{\tt\small REJECT}$$[t_{0}, t_{1}]$$\int_{t_{0}}^{t_{1}}\int_{\Phi(\mathcal{P})}(\frac{\partial \mathcal{L}}{\partial x}$.

$\delta x+\frac{\partial \mathcal{L}}{\partial\dot{x}}$$+ \frac{\partial \mathcal{L}}{\partial F}$:.

$\delta\dot{x}+\frac{\partial \mathcal{L}}{\partial\xi}$$\delta F+\frac{\partial \mathcal{L}}{\partial \mathfrak{F}}$:.

$\delta\xi+\frac{\partial \mathcal{L}}{\partial\dot{\xi}}$$\delta \mathfrak{F}+\frac{\partial \mathcal{L}}{\partial \mathcal{F}}$.

$\delta\dot{\xi}$: $\delta \mathcal{F})\Phi(dV)dt=0$ ,(3.3)

where we have denoted, by. and :, the inner product of vectors and double contractionof tensors, respectively, or equivalently the simple-dot and double-dot products in the
dyadics. Then, by performing partial integration and using the divergence theorem, we
obtain the weak form for the continuum with microstructure,$\int_{t_{0}}^{t_{1}}\int_{\Phi(9)}[(\frac{\partial}{\partial t}(\frac{\partial ff}{\partial\dot{x}})-\frac{\partial F}{\partial x}-\frac{\partial}{\partial X}\cdot(\frac{\partial\Psi}{\partial F}))\cdot\delta x$$+( \frac{\partial}{\partial t}(\frac{\partial F}{\partial\dot{\xi}})-\frac{\partial\ovalbox{\tt\small REJECT}}{\partial\xi}-\frac{\partial}{\partial X}\cdot(\frac{\partial\psi}{\partial \mathfrak{F}})-\frac{\partial}{\partial_{-}^{-}-}\cdot(\frac{\partial\psi}{\partial \mathcal{F}}))\cdot\delta\xi]\Phi(dV)dt$,$+ \int_{t_{0}}^{t_{1}}\int_{\Phi(\partial 9)}[N\cdot\frac{\partial\Psi}{\partial F}\cdot\delta x+N\cdot(\frac{\partial\psi}{\partial \mathfrak{F}}+\frac{\partial\psi}{\partial \mathcal{F}})\cdot\delta\xi]\Phi(dA)dt=0$(3.4), and $N$ and $dA$ are the unit
denotes the boundary of the material bundle
where, respectively.
and the area form of
normal to
has a compact support
Finally, we assume that the Lagrangian density
and that the variations are fixed at the end points, $\delta x=\delta\xi=0(t=t_{0}, t_{1})$ . Then we
have equations of motion for the continuum with microstructure$\mathcal{P}$$\partial \mathcal{P}$$\partial \mathcal{P}$$\partial \mathcal{P}$$\mathcal{L}=\ovalbox{\tt\small REJECT}-\psi$$\frac{\partial}{\partial t}(\frac{\partial F}{\partial\dot{x}})=\frac{\partial ff}{\partial x}+\frac{\partial}{\partial X}\cdot(\frac{\partial\Psi}{\partial F})$

(3.5a),$\frac{\partial}{\partial t}(\frac{\partial ff}{\partial\dot{\xi}})=\frac{\partial ff}{\partial\xi}+\frac{\partial}{\partial X}\cdot(\frac{\partial\Psi}{\partial \mathfrak{F}})+\frac{\partial}{\partial_{-}^{-}-}\cdot(\frac{\partial\Psi}{\partial \mathcal{F}})$.

(3.5b)

By introducing the generalized Piola-Kirchhoff stress tensors,$P= \frac{\partial\psi}{\partial F}$,$\mathfrak{P}=\frac{\partial\Psi}{\partial \mathfrak{F}}$,$\mathcal{P}=\frac{\partial\psi}{\partial \mathcal{F}}$

(3.6)

,

we obtain the equations of motion, in terms of the generalized stress tensors,$\frac{\partial}{\partial t}(\frac{\partial g}{\partial\dot{x}})=\frac{\partial ff}{\partial x}+P\cdot(\frac{\partial}{\partial X})$

(3.7a),$\frac{\partial}{\partial t,}(\frac{\partial ff}{\partial\dot{\xi}})=\frac{\partial F}{\partial\xi}+\mathfrak{P}\cdot(\frac{\partial}{\partial X})+\mathcal{P}\cdot(\frac{\partial}{\partial_{-}^{--}})$.

Thus, the balance laws for the continuum with microstructure are reconstituted.

(3.7b)

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4

An example: Cosserat rods

As an example, we consider a Cossrat rod laid in the three-dimensional Euclidean space. In this case, the bundle
is the Cosserat rod, a one-dimensional continuum
expresses the center axis of rod, and the director attached to each point ofis thethree-dimensional vector to describe points in the cross-section at each point on the axis$(m=1, r=3)$ . Then, it is enough that the vector bundle $e2$ is taken as.
We parameterize a position vector $r(s, t)$ of an arbitrary point onby the arclength parameter and the time parameter . We define a right-handed orthonormal
basis, $\{d_{1}(s, t), d_{2}(s, t), d_{3}(s, t)\}$ , along
at with, and introduce thecurvature vector
in the current body through$\mathbb{R}^{3}(n=3)$$\mathcal{P}$$\mathcal{B}$$\mathcal{B}$$\mathbb{R}^{3}\cross \mathbb{R}^{3}$$\phi(\mathcal{B})$

$s$

$t$

$\phi(\mathcal{B})$$d_{1}=\partial r/\partial s$$s$$\kappa(s, t)$$\frac{\partial d_{k}}{\partial s}=\kappa\cross d_{k}$

.

(4.1)

The componentof gives the torsion ofin the current configuration;
the two components,$\alpha=2,3$ , are components of the current
curvature ofand are related to the geometric curvature of the current axial curve through theformula. Then the Cosserat rod is provided as$\kappa^{1}=\langle d_{1},$$\kappa\}$$\kappa^{\alpha}=\langle d_{\alpha},$$\kappa$$\phi(\mathcal{B})$$\kappa\rangle,$$\phi(\mathcal{B})$$\tilde{\kappa}$$(\tilde{\kappa})^{2}=(\kappa^{2})^{2}+(\kappa^{3})^{2}$$x(s, t)=r(s, t)$ ,$\xi(s, t)=\xi^{k}d_{k}(s, t)$

.

(4.2)

It is illustrated as Fig. 2. We use curvilinear coordinates with respect to. Throughout
this article, the summation convention is used for repeated indices, with Latin indicestaking the values {1, 2, 3} and Greek indices taking the values {2, 3}.
$\{d_{k}\}$Figure 2: Illustration of the reference and current configurations of the Cosserat rod.

Additionally, we denote quantities for reference body by the capital letters of thecorresponding ones for current body, that is, $R(S, 0)$ is the position vector of an arbitrary point onby the arc-length parameterassociated with, and$\phi_{0}(\mathcal{B}_{0})$$S$$\phi_{0}(\mathcal{B}_{0})$107
at suchis a right-handed orthonormal basis along
is similarly defined by. Then, the reference curvature vector$\{D_{1}(S,$ $0),$ $D_{2}(S, 0),$ $D_{3}(S, 0)\}$that$D_{1}=\partial R/\partial S$$\phi_{0}(\mathcal{B}_{0})$$S$$\mathcal{K}_{0}(S, 0)$$\frac{\partial D_{k}}{\partial S}=\mathcal{K}_{0}\cross D_{k}$

(4.3)

.

in the reference configof
gives the torsion of
The component$\alpha=2,3$
components,of the referenceareuration; the two components,of the reference axial
and are related to the geometric curvature
curvature of. Thus, the reference configuration iscurve through the formulaprovide as$\mathcal{K}_{0}^{1}=\langle D_{1},$$\phi_{0}(\mathcal{B}_{0})$$\mathcal{K}_{0}$$\mathcal{K}_{0}\}$$\mathcal{K}_{0}^{\alpha}=\langle D_{\alpha},$$\mathcal{K}_{0}\rangle,$$\tilde{\mathcal{K}}_{0}$$\phi_{0}(\mathcal{B}_{0})$$(\tilde{\mathcal{K}}_{0})^{2}=(\mathcal{K}_{0}^{2})^{2}+(\mathcal{K}_{0}^{3})^{2}$$X(S, 0)=R(S, 0)$ ,$\Xi(S, 0)=\Xi^{\alpha}D_{\alpha}(S, 0)$

(4.4)

.

It is shown as Fig. 2. Here, we consider that the reference configuration is unstressedstate, and then the cross-sections of the reference filament is as sumed to be normal to its
axial curve. If the cross-sections of the current filament remain normal to the current axial
curve, we may constrain $\xi^{1}=0$ . When we suppose the special Cosserat rod, in which it
is assumed that the cross-sections of the current filament remain plane, undeformed, and.
normal to the current axial curve, we have to append the constraints $\xi^{1}=0$ andFurther, the extensionof the axial curve can be defined through$\xi^{\alpha}=\Xi^{\alpha}$$\epsilon(s, t)$$\frac{\partial s}{\partial S}=(1+\epsilon)$

(4.5)

.

Using the above the deference relation, we obtain the deformation gradients$F=(1+\epsilon)d_{1}\otimes D^{1}$(4.6a),$\mathfrak{F}=(1+\epsilon)(\frac{\partial\xi^{k}}{\partial s}+\xi^{k}R(\kappa))d_{k}\otimes D^{1}$$\mathcal{F}=\frac{\partial\xi^{k}}{\partial_{-}^{-\alpha}-}d_{k}\otimes D^{\alpha}$(4.6b),(4.6c),

where $R(a)$ is the skew symmetric tensor associated with a polar vector
current and reference deformations are calculated as, respectively,$C=(1+ \epsilon)^{2}\Vert\frac{\partial}{\partial s}(r+\xi)\Vert^{2}D^{1}\otimes D^{1}$$(1+ \epsilon)\frac{\partial}{\partial s}(r+\xi)\rangle D^{\alpha}\otimes D^{1}$$C= \langle\frac{\partial\xi}{\partial_{-}^{-\alpha}-},$$\frac{\partial\xi}{\partial_{-}^{-\beta}-}\}D^{\alpha}\otimes D^{\beta}$. Then the(4.7a),$\not\subset=\langle\frac{\partial\xi}{\partial_{-}^{-\alpha}-},$$a$(4.7b),(4.7c),

and$C_{0}= \Vert\frac{\partial}{\partial S}(R+\Xi)\Vert^{2}D^{1}\otimes D^{1}$Co$=\langle D_{\alpha},$$\frac{\partial}{\partial S}(R+\Xi)\}D^{\alpha}\otimes D^{1}=\epsilon_{1\alpha\beta}\mathcal{K}_{0}^{1}\Xi^{\alpha}D^{\beta}\otimes D^{1}$$C_{0}=\delta_{\alpha\beta}D^{\alpha}\otimes D^{\beta}$,(4.8a),,(4.8b)(4.8c)

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denotes the standard inner product on theis the Edinton’s epsilon, andwhere
Euclidean spaces. Thus, we obtain the strains as follows:$\Vert\cdot\Vert$$\epsilon_{klm}$$E= \frac{1}{2}[(1+\epsilon)^{2}\Vert d_{1}+\xi^{\alpha}R(\kappa)d_{\alpha}\Vert^{2}-\Vert D_{1}+\Xi^{\alpha}R(\mathcal{K}_{0})D_{\alpha}\Vert^{2}]D^{1}\otimes D^{1}$$\not\subset=\frac{1}{2}[\langle\frac{\partial\xi}{\partial_{-}^{-\alpha}-},$$(1+ \epsilon)\frac{\partial}{\partial s}(r+\xi)\rangle-\langle D_{\alpha},$ $\frac{\partial}{\partial S}(R+\Xi)\rangle]D^{\beta}\otimes D^{1}$$\mathcal{E}=\frac{1}{2}[\langle\frac{\partial\xi}{\partial_{-}^{-\alpha}-},$$\frac{\partial\xi}{\partial_{-}^{-\beta}-}\}-\delta_{\alpha\beta}]D^{\alpha}\otimes D^{\beta}$(4.9a),(4.9b),.

(4.9c)

Because of using the moving frame, we must rewrite the variational formulation. To, associated with the orthonomal basis,
this end, we begin with defining the variation
through$\delta k$$\delta d_{k}=\delta k\cross d_{k}$so that the variation of the director$\xi$(4.10),

is expressed as$\delta\xi=\delta k\cross\xi$(4.11),and the variations of the deformation gradients $F,$$\mathfrak{F}$, and$\delta F=\frac{\partial}{\partial R}(\delta r)+R(\delta k)F$$\mathcal{F}$become(4.12a),,(4.12b).

(4.12c)$\delta \mathfrak{F}=\frac{\partial}{\partial R}(\delta\xi)+R(\delta k)\mathfrak{F}$$\delta \mathcal{F}=\frac{\partial}{\partial_{-}^{-}-}(\delta\xi)+R(\delta k)\mathcal{F}$Then the weak form is rewretten as$\int_{t_{0}}^{t_{1}}\int_{\Phi(9)}[(\frac{\partial}{\partial t}(\frac{\partial F}{\partial\dot{r}})-\frac{\partial ff}{\partial r}-\frac{\partial}{\partial R}\cdot(\frac{\partial\psi}{\partial F}))\cdot\delta r$$-( \frac{\partial\psi}{\partial F}$:$R( \delta k)F+\frac{\partial\Psi}{\partial \mathfrak{F}}$:$R( \delta k)\mathfrak{F}+\frac{\partial\Psi}{\partial \mathcal{F}}$:$R(\delta k)\mathcal{F})$$+( \xi\cross(\frac{\partial}{\partial t}(\frac{\partial ff}{\partial\dot{\xi}})-\frac{\partial ff}{\partial\xi}-\frac{\partial}{\partial R}\cdot(\frac{\partial\Psi}{\partial \mathfrak{F}})-\frac{\partial}{\partial_{-}^{-}-}\cdot(\frac{\partial\Psi}{\partial \mathcal{F}})))\cdot\delta k]\Phi(dV)dt$.

$+ \int_{t_{0}}^{t_{1}}\int_{\Phi(\partial p)}[N\cdot\frac{\partial\psi}{\partial F}\cdot\delta r+\xi\cross(N\cdot(\frac{\partial\psi}{\partial \mathfrak{F}}+\frac{\partial\ovalbox{\tt\small REJECT}’}{\partial \mathcal{F}}))\cdot\delta k]\Phi(dA)dt=0$(4.13)

Hence, under the fixed end points conditons, $\delta r=\delta k=0$ , the equations of motion for
the Cosserat rod is derived as$\frac{\partial}{\partial t}(\frac{\partial F}{\partial\dot{r}})=\frac{\partial\ovalbox{\tt\small REJECT}}{\partial r}+\frac{\partial}{\partial R}\cdot(\frac{\partial\psi}{\partial F})$(4.14a),$\xi\cross\frac{\partial}{\partial t}(\frac{\partial F}{\partial\dot{\xi}})=\xi\cross(\frac{\partial\ovalbox{\tt\small REJECT}}{\partial\xi}+\frac{\partial}{\partial R}\cdot(\frac{\partial\Psi}{\partial \mathfrak{F}}I+\frac{\partial}{\partial_{-}^{-}-}\cdot(\frac{\partial\psi}{\partial \mathcal{F}}))$$+F^{x} \frac{\partial\Psi}{\partial F}+\mathfrak{F}^{\cross}\frac{\partial\Psi}{\partial \mathfrak{F}}+\mathcal{F}^{x}\frac{\partial\psi}{\partial \mathcal{F}}$,(4.14b)

109
where

$\cross$

denotes the cross-dot product in the dyadics, and it is defined as$(a_{1}\otimes a_{2})^{\cross}(a_{3}\otimes a_{4})=(a_{1}\cross a_{3})(a_{2}\cdot a_{4})$(4.15),

for any vectors.
In terms of the local coordinates, the kinetic enegy density is expressed as$a_{i}\in \mathbb{R}^{3}$,$eZ(r, \xi,\dot{r},\dot{\xi})=\frac{1}{2}\rho(r)(\Vert\dot{r}\Vert^{2}+\Vert\dot{\xi}\Vert^{2})$(4.16)

whereis a mass density of the body at in the current configuration. Thus,we obtain the equations of motion for the special Cosserat rods expressed in terms of thegeneralized stress tensors,$\rho(r)$$r$$\frac{\partial}{\partial t}(\rho\dot{r})=\frac{1}{2}\frac{\partial\rho}{\partial r}\Vert\dot{r}\Vert^{2}+\frac{\partial}{\partial S}(P\cdot D^{1})$$\phi(\mathcal{B})$(4.17a),$\xi\cross\frac{\partial}{\partial t}(\rho\dot{\xi})=\xi\cross(\frac{\partial}{\partial S}(\mathfrak{P}\cdot D^{1}+\mathcal{P}\cdot\Xi^{\alpha}R(\mathcal{K}_{0})^{T}D^{\alpha})+\frac{\partial}{\partial_{-}^{-\alpha}-}(\mathcal{P}\cdot D^{\alpha}))$$+ \frac{\partial r}{\partial S}\cross(P\cdot D^{1})+\frac{\partial\xi}{\partial S}\cross(\mathfrak{P}\cdot D^{1}+\mathcal{P}\cdot D^{\alpha})+\frac{\partial\xi}{\partial_{-}^{-\alpha}-}\cross(\mathcal{P}\cdot D^{\alpha})$.

(4.17b)

By these expressions, it is well to reconstruct the balance laws for the Cosserat rod.
At the last in this section, we reduce the above equations to those for the special
Cosserat rod, that is, we impose the constraints $\xi^{1}=0$ and. In this case, themicro deformation gradient becomes$\xi^{\alpha}=\Xi^{\alpha}$$\mathcal{F}=\delta_{\beta}^{\alpha}d_{\alpha}\otimes D^{\beta}$(4.18),

so that the generalized micro stress tensor vanishes;, because of the assumption
about the undeformation of the cross-sections. Indeed, the micro strain vanishes, i. e.,. Here, we note that the linearlized macro and mixture strains become$\mathcal{P}=0$$\mathcal{E}=0$$E_{1inear}=\epsilon D^{1}\otimes D^{1}$(4.19a),$C_{linear}=\frac{1}{2}\epsilon_{1\alpha\beta}(\kappa^{1}-\mathcal{K}_{0}^{1})\Xi^{\alpha}D^{\beta}\otimes D^{1}$.

(4.19b)Then we obtain the well-known equations of motion for the special Cosserat rods expressedin terms of the generalized stress tensors,$\frac{\partial}{\partial t}(\rho\dot{r})=\frac{1}{2}\frac{\partial\rho}{\partial r}\Vert\dot{r}\Vert^{2}+\frac{\partial}{\partial S}(P\cdot D^{1})$,$\xi\cross\frac{\partial}{\partial t}(\rho\dot{\xi})=\frac{\partial}{\partial S}(\xi\cross(\mathfrak{P}\cdot D^{1}))+\frac{\partial r}{\partial S}\cross(P\cdot D^{1})$We comment thatandthe center axis, respectively.
$P\cdot D^{1}$$\xi\cross(\mathfrak{P}\cdot D^{1})$(4.20a).

(4.20b)is exactly the stress and couple-stress along

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5

Summary

In this study, we developed formulations for continuum dynamics on a tangent bundle ofa vector bundle that accurately describes the mechanical behavior of a directed medium.
Indeed, the dynamics of the one-dimensional continuum with a director are well expressedas one of the Cosserat rod, in which the cross-sectional structure is considered as the
microstructure of the rod. For future developments, it is important to examine geometricalstructures of various continua with microstructures. Especially, in the case where we
consider a classification of microcontinua, it is necessary to investigate group actions
on the bodies and microstructures. For example, the group structures correspond to
Eringen’s classification, i. e., micromorphic, microstretch, and micropolar continua [14].
Moreover, we can extend the Cosserat rod to a model describing smaller microstructure of the elastic rod. Then the expressions for the smaller microstructure to analyze
deformation behavior of filaments including biopolymers. When a biopolymer expresses acertain function within a living organism, its conformation is an important factor that determes the function. Therefore, we believe to obtain a new knowledge of the interactions
between the dynamical situations and the biological circumstances of biopolymers, whichhave been investigated recently by considering the deformation behavior of biopolymers
together with their microstructures.

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