Poincaré’s Equations for Cosserat Media: Application to Shells

Abstract

In 1901, Henri Poincaré discovered a new set of equations for mechanics. These equations are a generalization of Lagrange’s equations for a system whose configuration space is a Lie group which is not necessarily commutative. Since then, this result has been extensively refined through the Lagrangian reduction theory. In the present contribution, we apply an extended version of these equations to continuous Cosserat media, i.e. media in which the usual point particles are replaced by small rigid bodies, called microstructures. In particular, we will see how the shell balance equations used in nonlinear structural dynamics can be easily deduced from this extension of the Poincaré’s result. In future, these results will be used as foundations for the study of squid locomotion, which is an emerging topic relevant to soft robotics.

Appendices

Appendix 1: Proof of (44) Through Direct Application of the Hamilton Principle

As evoked in Remark 5.2, Eqs. (43) and (44) can be derived directly by applying the Hamilton principle to an action defined in terms of the different definitions of the reduced Lagrangian of Sect. 2.6. In this “Appendix”, we apply this calculus to the deformed configuration. This can be achieved using Stokes theorem applied to differential forms or alternatively using the divergence theorem and vector analysis. We will follow the latter approach and will denote, according to the context of Sect. 2.4, $\textit{dX}^{1}{} \textit{dX}^{2}\ldots \textit{dX}^{p}=d{\mathcal {D}}$ and $|\overline{H}|^{1/2}dY^{1}dY^{2}$ $\ldots dY^{p-1}=d\partial {\mathcal {D}}$ the material volumes of ${\mathcal {D}}$ and $\partial {\mathcal {D}}$, which are assumed to be two manifolds consistently oriented according to the outward unit normal convention. We start from (19) and (20) in which we replace $\mathfrak {L}$ by $\sqrt{|h|}\mathfrak {L}_{t}$ and $(F_\mathrm{ext},\overline{F}_\mathrm{ext})$ by $(F_\mathrm{ext}|h|^{1/2},\overline{F}_\mathrm{ext}|\overline{h}|^{1/2})$. In this new formulation of Hamilton principle, $\delta$ being achieved at fixed time and material parameters according to Sect. 2.6, this enables us to shift it under the integral. Then, since $\delta (\mathfrak {L}_{t}\sqrt{|h|})=\delta \mathfrak {L}_{t}\sqrt{|h|}+\mathfrak {L}_{t}\delta \sqrt{|h|}$, where $\sqrt{|h|}$ is configuration dependent through the invariant fields $\xi _{\alpha }$ (see Sect. 2.6), we have:

$$\begin{aligned} \int _{t_{1}}^{t_{2}}\int _{{\mathcal {D}}}\left( \left\langle \frac{\partial \mathfrak {L}_{t}}{\partial \eta },\delta \eta \right\rangle +\left\langle \left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t},\delta \xi _{\alpha }\right\rangle \right) \sqrt{|h|}d{\mathcal {D}} dt=-\int _{t_{1}}^{t_{2}}\delta W_\mathrm{ext}dt, \end{aligned}$$

(102)

where we used the notation $[\partial \mathfrak {L}/\partial \xi _{\alpha }]_{t}=\partial \mathfrak {L}_{t}/\partial \xi _{\alpha }$ $+|h|^{-\frac{1}{2}}(\partial |h|^{\frac{1}{2}}/\partial \xi _{\alpha })$ $\mathfrak {L}_{t}$, as it is introduced by (42). Then invoking (23) and applying a by-part time integration with fixed extreme times condition allows the left-hand side of (102) to be rewritten as:

$$\begin{aligned}&\int _{t_{1}}^{t_{2}}\int _{{\mathcal {D}}}\left\langle \frac{1}{\sqrt{|h|}}ad_{\eta }^{*}\left( \sqrt{|h|}\frac{\partial \mathfrak {L}_{t}}{\partial \eta }\right) -\frac{1}{\sqrt{|h|}}\frac{\partial }{\partial t}\left( \sqrt{|h|}\frac{\partial \mathfrak {L}_{t}}{\partial \eta }\right) ,\delta \zeta \right\rangle \sqrt{|h|}d{\mathcal {D}}dt\nonumber \\&\quad +\int _{t_{1}}^{t_{2}}\int _{{\mathcal {D}}}\left\langle \left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t},\frac{\partial \delta \zeta }{\partial X^{\alpha }}+ad_{\xi _{\alpha }}(\delta \zeta ) \right\rangle \sqrt{|h|}d{\mathcal {D}} dt. \end{aligned}$$

(103)

Now let us remark that:

$$\begin{aligned}&\int _{{\mathcal {D}}}\left\langle \left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t},\frac{\partial \delta \zeta }{\partial X^{\alpha }}\right\rangle \sqrt{|h|}d{\mathcal {D}}\nonumber \\&\quad = \int _{{\mathcal {D}}}\frac{\partial }{\partial X^{\alpha }}\left( \sqrt{|h|}\left\langle \left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t},\delta \zeta \right\rangle \right) d{\mathcal {D}}-\int _{{\mathcal {D}}}\left\langle \frac{\partial }{\partial X^{\alpha }}\left( \sqrt{|h|}\left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t}\right) ,\delta \zeta \right\rangle d{\mathcal {D}},\nonumber \\ \end{aligned}$$

(104)

whose first right-hand side term is merely the divergence of a vector field of contravariant components $v^{\alpha }=<[\partial \mathfrak {L}/\partial \xi _{\alpha }]_{t},\delta \zeta>$ in the convected basis $\{h_{\alpha }\}_{\alpha =1,\ldots p}$. Applying the divergence theorem to this term gives:

$$\begin{aligned} \int _{{\mathcal {D}}}\frac{\partial }{\partial X^{\alpha }}\left( \sqrt{|h|}\left\langle \left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t},\delta \zeta \right\rangle \right) d{\mathcal {D}}=\int _{\partial {\mathcal {D}}}\left\langle \left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t},\delta \zeta \right\rangle \nu _{t,\alpha } \mid \overline{h}\mid ^{1/2}d\partial {\mathcal {D}},\nonumber \\ \end{aligned}$$

(105)

where we introduce the metric volume element $|\overline{h}|^{1/2}d\partial {\mathcal {D}}$ on $(\varPhi _{t}\circ e)(\partial {\mathcal {D}})$, and $\nu _{t,\alpha } h^{\alpha }$ is the unit outward normal to the tangent planes of $(\varPhi _{t}\circ e)(\partial {\mathcal {D}})$ which, from (12), is related to the outward unit normal $\nu _{\alpha }E^{\alpha }$ by $\nu _{t,\alpha } |\overline{h}|^{1/2}d\partial {\mathcal {D}}=\nu _{\alpha } \sqrt{|h|}d\partial {\mathcal {D}}$. Then, inserting (105) into (104) and the result into (103) whose the last term is dualized, gives, with (20), a balance of two integral components, one over ${\mathcal {D}}$ with metric volume $\sqrt{|h}|d{\mathcal {D}}$ and the second over $\partial {\mathcal {D}}$, whose metric volume is $|\overline{h}|^{1/2}d\partial {\mathcal {D}}$. This balance being satisfied for any variation $\delta \zeta \in \mathfrak {g}$, it gives the set of Eq. (44), where due to (21), $|\overline{h}|^{1/2}=1$ in the case of beams.

Appendix 2: Proof of (62)

The general expression (106) is stated in Simo et al. (1988) in the case of geometrically exact beams and plates, in this “Appendix” we prove it for shells. Since the medium ${\mathcal {B}}$ is classical (not micropolar), we have:

$$\begin{aligned} \delta W_\mathrm{int}=\int _{{\mathcal {B}}}P:\delta F \sqrt{|g_{o}|} d{\mathcal {B}}=\frac{1}{2}\int _{{\mathcal {B}}}\sigma ^{ij}\text { }\delta g_{ij} \sqrt{|g|} d{\mathcal {B}}, \end{aligned}$$

(106)

where $g_{ij}(g^{i}\otimes g^{j})$ is the fundamental metric tensor in the convected basis of $\varPhi _{t}({\mathcal {B}})$, with determinant |g|, and $\sigma ^{ij}(g_{i}\otimes g_{j})$ is the Cauchy stress tensor in the same basis. Then introducing the Cosserat shell kinematics (64) into $g_{ij}=g_{i}.g_{j}$ with $g_{i}=\partial \varPhi _{t}/\partial X^{i}$ gives:

$$\begin{aligned} g_{\alpha \beta }\simeq \varGamma _{\alpha } \cdot \varGamma _{\beta }+E_{3} \cdot (\varGamma _{\alpha }\times K_{\beta }+\varGamma _{\beta }\times K_{\alpha }) X^{3} , g_{\alpha 3}= \varGamma _{\alpha } \cdot E_{3}, g_{33}=1, \end{aligned}$$

(107)

where, consistently with the first-order Cosserat kinematics (8), the $X^{3}$-quadratic terms are neglected in the expression of $g_{\alpha \beta }$. In these conditions, we recognize in (107) the time-dependent components of the strain measures (86–88). Now, since expressions (107) depend on the medium configuration through the left-invariant fields $\varGamma _{\alpha }$ and $K_{\alpha }$ of (65) only, one can state:

$$\begin{aligned} \delta g_{ij}=\left( \frac{\partial g_{ij}}{\partial \varGamma _{\alpha }}\right) \delta \varGamma _{\alpha }+\left( \frac{\partial g_{ij}}{\partial K_{\alpha }}\right) \delta K_{\alpha }, \end{aligned}$$

(108)

in which the partial derivatives depend on the mid-surface coordinates through the left-invariant fields, and on $X^{3}$, in a linear way. Then, introducing (108) into (106), and integrating the result along the $X^{3}$-thickness dimension, gives a general expression of the virtual work of internal forces similar to (62), along with the explicit expressions of the resultant of stress and couple stress $N^{\alpha }_{t}$ and $M^{\alpha }_{t}$ in terms of the three-dimensional Cauchy stress tensor:

$$\begin{aligned} M_{t}^{\alpha }=\frac{1}{\sqrt{|h|}}\int _{{\mathcal {M}}}\left( \sigma ^{ij}\frac{\partial g_{ij}}{\partial K_{\alpha }}\sqrt{|g|}\right) \textit{dX}^{3} , \quad N_{t}^{\alpha }=\frac{1}{\sqrt{|h|}}\int _{{\mathcal {M}}}\left( \sigma ^{ij}\frac{\partial g_{ij}}{\partial \varGamma _{\alpha }}\sqrt{|g|}\right) \textit{dX}^{3},\nonumber \\ \end{aligned}$$

(109)

where $i<j$. $\square$