Journal of Nonlinear Science

, Volume 27, Issue 1, pp 1–44 | Cite as

Poincaré’s Equations for Cosserat Media: Application to Shells

Article

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.

Keywords

Cosserat media Euler-Poincaré reduction Geometrically exact shells 

List of symbols

t

Time

\({\mathcal {E}}\)

Three-dimensional geometric space of classical mechanics

\({\mathcal {B}}\)

Three-dimensional material space of a classical continuous medium

\({\mathcal {D}}\)

Material (p-dimensional, \(p<3\)) reference subspace

\({\mathcal {M}}\)

Rigid microstructure

\({\mathcal {B}}={\mathcal {D}}\times {\mathcal {M}}\)

Material space of a Cosserat medium

\((O,E_{1},E_{2},E_{3})\)

Material frame attached to \({\mathcal {B}}\)

\((o,e_{1},e_{2},e_{3})\)

Spatial frame attached to \({\mathcal {E}}\)

\(x=x^{i}e_{i}\)

Points of geometric space

\(X=X^{i}E_{i}\)

Material points of \({\mathcal {B}}\)

\(\overline{X}=X^{\alpha }E_{\alpha }\)

Material points of \({\mathcal {D}}\)

\(\varPhi _{t}\)

Transformation at time t from material to geometric space

\(\varPhi _{t}({\mathcal {B}})\)

Deformed configuration of \({\mathcal {B}}\)

\(\varPhi _{o}({\mathcal {B}})\)

Reference configuration of \({\mathcal {B}}\)

\((\varPhi _{t}\circ e)({\mathcal {D}})\)

Deformed configuration of \({\mathcal {D}}\)

\((\varPhi _{o}\circ e)({\mathcal {D}})\)

Reference configuration of \({\mathcal {D}}\)

\(r(\overline{X})\)

Position of \((\varPhi _{t}\circ e)(\overline{X})\)

\(R(\overline{X})\in SO(3)\)

Rotation tensor mapping \((E_{1},E_{2},E_{3})\) onto \((t_{1},t_{2},t_{3})(\overline{X})\)

\((g_{1},g_{2},g_{3})(X)\)

Convected basis on \(\varPhi _{t}({\mathcal {B}})\) at \(\varPhi _{t}(X)\)

\((h_{1}, \ldots h_{p})(\overline{X})\)

Convected basis on \((\varPhi _{t}\circ e)({\mathcal {D}})\) at \(r(\overline{X})\)

\((t_{1},t_{2},t_{3})(\overline{X})\)

Orthonormal spatial basis attached to the X-microstructure

\((g_{ij}g^{i}\otimes g^{j})(X)\)

Euclidean metric tensor in the convected basis of \(\varPhi _{t}({\mathcal {B}})\)

\((h_{\alpha \beta }h^{\alpha }\otimes h^{\beta })(\overline{X})\)

Euclidean metric induced on \((\varPhi _{t}\circ e)({\mathcal {D}})\) in its convected basis

\(\nu \), \(\nu _{o}\), \(\nu _{t}\)

Oriented unit normal vector to the material, reference and deformed surface element of \({\mathcal {D}}\)

\(\textit{dS}\), \(\textit{dS}_{o}\), \(\textit{dS}_{t}\)

Area of the material, reference and deformed surface element of \({\mathcal {D}}\)

\({\mathcal {C}}\)

Configuration space of a Cosserat medium \({\mathcal {D}}\times {\mathcal {M}}\)

G and \(\texttt {g}\)

Group of transformation and transformation of microstructure

\(\mathfrak {g}\), \(\mathfrak {g}^{*}\)

Lie algebra of G and its dual

Ad and \(Ad^{*}\)

Adjoint and co-adjoint action map of G on \(\mathfrak {g}\) and \(\mathfrak {g}^{*}\)

ad and \(ad^{*}\)

Adjoint and co-adjoint action map of \(\mathfrak {g}\) on \(\mathfrak {g}\) and \(\mathfrak {g}^{*}\)

\(\eta \) and \(\xi _{\alpha }\)

Left-invariant fields along time and space variables

\(\mathfrak {L}\), \(\mathfrak {L}_{o}\) and \(\mathfrak {L}_{t}\)

Density of left-reduced Lagrangian of a Cosserat medium per unit of its material, reference and deformed volume

\(\frac{\partial \mathfrak {L}}{\partial \eta }\),\(\frac{\partial \mathfrak {L}_{o}}{\partial \eta }\) and \(\frac{\partial \mathfrak {L}_{t}}{\partial \eta }\)

Densities of material t-conjugate (kinetic) momentum, per unit of material, reference and deformed volume

\(\frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\),\(\frac{\partial \mathfrak {L}_{o}}{\partial \xi _{\alpha }}\) and \(\left[ \frac{\partial \mathfrak {L}}{\partial \xi _{\alpha }}\right] _{t}\)

Densities of material \(X^{\alpha }\)-conjugate (stress) momentum, per unit of material, reference and deformed volume

\(F_\mathrm{ext}\), \(F_{\mathrm{ext},o}\) and \(F_{\mathrm{ext},t}\)

Densities of material external forces per unit of material, reference and deformed volume

\(\overline{F}_\mathrm{ext}\), \(\overline{F}_{\mathrm{ext},o}\) and \(\overline{F}_{\mathrm{ext},t}\)

Densities of external forces per unit of material, reference and deformed boundary volume

\({\mathcal {D}}\times {\mathbb {R}}^{+}\)

Space–time of a p-dimensional Cosserat medium

\(X^{0}\frac{\partial }{\partial t}+X^{\alpha }\frac{\partial }{\partial X^{\alpha }}\)

Point in space–time with \(t=X^{0}\)

\(\varUpsilon \)

Space–time 1-form field with value in \(\mathfrak {g}\)

\(\varLambda \), \(\varLambda _{o}\) and \(\varLambda _{t}\)

Density of a space–time vector field with value in \(\mathfrak {g}^{*}\), per unit of material, reference and deformed volume

\(\langle .,. \rangle \) and (., .)

Duality product in \(\mathfrak {g}\) and space–time

\(Ad^{*}_{\texttt {g}^{-1}}\left( \sqrt{|h|}\left( \frac{\partial \mathfrak {L}_{t}}{\partial \eta }\right) \right) \)

Densities of spatial (in the fixed frame) kinetic wrench, per unit of deformed volume

\(Ad^{*}_{\texttt {g}^{-1}}\left( \sqrt{|h|}\left( \frac{\partial \mathfrak {L}_{t}}{\partial \xi _{\alpha }}\right) \right) \)

Densities of spatial (in the fixed frame) stress wrench, per unit of deformed volume

\(\textit{SE}(3)\)

Special Euclidean group in \({\mathbb {R}}^{3}\) with Lie algebra \(\textit{se}(3)\)

(Rr)

Transformation of \(\textit{SE}(3)\)

\((\varOmega ^{T},V^{T})^{T}\in se(3)\)

Material time rate of transformation (velocity) of the microstructure frames

\((\omega ^{T},v^{T})^{T}\in se(3)\)

Spatial time rate of transformation (velocity) of the microstructure frames

\((\varSigma _{t}^{T},P_{t}^{T})^{T} \in se(3)^{*}\)

Density of material kinetic wrench per unit of deformed volume

\((\sigma _{t}^{T},p_{t}^{T})^{T}\in se(3)^{*}\)

Density of spatial (in the microstructure frame) kinetic wrench per unit of deformed volume

\((K_{\alpha }^{T},\varGamma _{\alpha }^{T})^{T}\in \textit{se}(3)\)

Material \(X^{\alpha }\)-rate of transformation of the microstructure frames

\((k_{\alpha }^{T},\gamma _{\alpha }^{T})^{T}\in \textit{se}(3)\)

Spatial \(X^{\alpha }\)-rate of transformation of the microstructure frames

\((M_{\alpha ,t}^{T},N_{\alpha ,t}^{T})^{T} \in \textit{se}(3)^{*}\)

Density of material stress wrench per unit of deformed volume

\((m_{\alpha ,t}^{T},n_{\alpha ,t}^{T})^{T}\in \textit{se}(3)^{*}\)

Density of spatial stress wrench per unit of deformed volume

\((\overline{\rho },\overline{\rho }_{o},\overline{\rho }_{t})\) and \((\overline{J},\overline{J}_{o},\overline{J}_{t})\)

Densities of mass and of material angular inertia tensor per unit of material, reference and deformed volume

\((\overline{I},\overline{I}_{o},\overline{I}_{t})\)

Densities of spatial inertia tensor per unit of material, reference and deformed volume

\(\epsilon _{\alpha \beta },\rho _{\alpha \beta },\tau _{\alpha }\)

Effective strain measures (stretching, bending, transverse shearing) of a classical shell

\({\mathcal {N}}_{t}^{\alpha \beta },{\mathcal {M}}_{t}^{\alpha \beta },{\mathcal {Q}}_{t}^{\alpha }\)

Densities of effective stress of a classical shell per unit of deformed volume

Mathematics Subject Classification

74A60 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.EMNIRCCyNNantes Cedex 3France
  2. 2.Khalifa University, KURIAbu DhabiUAE

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