Journal of Nonlinear Science

, Volume 27, Issue 1, pp 1–44

# Poincaré’s Equations for Cosserat Media: Application to Shells

• Frederic Boyer
• Federico Renda
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

74A60

## References

1. Antman, S.S.: Nonlinear problems of elasticity. In: Mathematical Sciences, vol. 107. Springer, New York (2005)Google Scholar
2. Arnold, V.I.: Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluides parfaits. Ann. Inst. J. Fourier 16(1), 319–361 (1966)
3. Arnold, V.I.: Mathematical Methods in Classical Mechanics, 2nd edn. Springer, New-York (1988)Google Scholar
4. Boyer, F., Primault, D.: The Poincaré–Chetayev equations and flexible multibody systems. J. Appl. Math. Mech. 69(6), 925–942 (2005). http://hal.archives-ouvertes.fr/hal--00672477
5. Boyer, F., Porez, M., Khalil, W.: Macro-continuous computed torque algorithm for a three-dimensional eel-like robot. IEEE Trans. Robot. 22(4), 763–775 (2006)
6. Boyer, F., Porez, M., Leroyer, A., Visonneau, M.: Fast dynamics of an eel-like robot-comparisons with Navier–Stokes simulations. IEEE Trans. Robot. 24(6), 1274–1288 (2008)
7. Boyer, F., Porez, M., Leroyer, A.: Poincaré–Cosserat equations for the Lighthill three-dimensional large amplitude elongated body theory: Application to robotics. J. Nonlinear Sci. 20, 47–79 (2010)
8. Boyer, F., Ali, S., Porez, M.: Macro-continuous dynamics for hyper-redundant robots: application to kinematic locomotion bio-inspired by elongated body animals. IEEE Trans. Robot. 28(2), 303–317 (2012)
9. Castrillón López, M., Ratiu, T.S., Shkoller, S.: Reduction in principal fiber bundles: covariant Euler–Poincaré equations. Proc. Am. Math. Soc. 128(7), 2155–2164 (2000)
10. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann, Paris (1909)
11. Demoures, F., Gay-Balmaz, F., Kobilarov, M., Ratiu, T.S.: Multisymplectic lie group variational integrator for a geometrically exact beam in r3. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3492–3512 (2014)
12. Ebin, D.G., Marsden, J.E.: Groups of diffeomorphism and the motion of an incompressible fluid. Ann. Math 92, 102–163 (1970)
13. Ellis, D.C.P., Gay-Balmaz, F., Holm, D.D., Putkaradze, V., Ratiu, T.S.: Symmetry reduced dynamics of charged molecular strands. Arch. Ration. Mech. Anal. 197(3), 811–902 (2010)
14. Eringen, A.C.: Microcontinuum Field Theories I: Foundations and Solids. Springer, New York (1998)
15. Fox, D.D., Simo, J.C.: A drill rotation formulation for geometrically exact shells. Comput. Methods Appl. Mech. Eng. 98, 329–343 (1992)
16. Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: Variational principles for spin systems and the Kirchhoff rod. J. Geom. Mech. 1(4), 417–444 (2009)
17. Green, A.E., Naghdi, P.M.: Non-isothermal theory of rods, plates and shells. Int. J. Solids Struct. 6, 209–244 (1970)
18. Green, A.E., Naghdi, P.M.: On the derivation of shell theories by direct approach. J. Appl. Mech. 41(1), 173–176 (1974)Google Scholar
19. Green, A.E., Zerna, W.: Theoretical Elasticity. Clarendon Press, Oxford (1960). end ed. edition
20. Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)
21. Holm, D.D., Putkaradze, V.: Nonlocal orientation-dependent dynamics of charged strands and ribbons. C. R. Acad. Sci. Paris Ser. I 347, 1093–1098 (2009)
22. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)
23. Lichnerowicz, A.: Elements de Calcul Tensoriel. Jacques Gabay, Paris (1987)
24. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, New-Jersey (1969)
25. Marle, C.-M.: On Henri Poincaré’s note: “sur une forme nouvelle des equations de la mécanique”. J. Geom. Symmetry Phys. 29, 1–38 (2013)
26. Marsden, J.E., Montgomery, R., Ratiu, T.S.: Reduction, symmetry, and phases in mechanics. In: Memoirs of the American Mathematical Society, vol. 88 (436). American Mathematical Society (1990)Google Scholar
27. Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity, 1st edn. Dover, Mineola (1994)
28. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, New York (1999)
29. Milne-Thomson, L.M.: Theoretical Hydrodynamics. Macmillan, London (1938)
30. Poincaré, H.: Sur une forme nouvelle des équations de la mécanique. Compte Rendu de l’Académie des Sciences de Paris 132, 369–371 (1901)
31. Pommaret, J.F.: Partial Differential Equations and Group Theory, 1st edn. Springer, Netherlands (1994)
32. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, 69–76 (1945)
33. Renda, F., Giorelli, M., Calisti, M., Cianchetti, M., Laschi, C.: Dynamic model of a multibending soft robot arm driven by cables. IEEE Trans. Robot. 30(5), 1109–1122 (2014)
34. Simmonds, J.G., Danielson, D.A.: Nonlinear shell theory with finite rotation and stress-function vectors. J. Appl. Mech. 39, 1085–1090 (1972)
35. Simo, J.C., Fox, D.D.: On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Comput. Methods Appl. Mech. Eng. 72(3), 267–304 (1989)Google Scholar
36. Simo, J.C., Marsden, J.E., Krishnaprasad, P.S.: The hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates. Arch. Ration. Mech. Anal. 104, 125–183 (1988)
37. Simo, J.C., Rifai, M.S., Fox, D.D.: On a stress resultant geometrically exact shell model. part vi: conserving algorithms for non-linear dynamics. Int. J. Numer. Methods Eng. 34, 117–164 (1992)
38. Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66(2), 125–161 (1988)
39. Spencer, D.C.: Overdetermined systems of partial differential equations. Bull. Am. Math. Soc. 75, 1–114 (1965)
40. Thomas, J.R., Hughes, Brezzi, F.: On drilling degrees of freedom. Comput. Methods Appl. Mech. Eng. 72, 105–121 (1989)
41. Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Rational Mech. Anal. 17, 85–112 (1964)
42. Verl, A., Albu-Schaeffer, A., Brock, O. (eds). Soft Robotics: Transferring Theory to Application. Springer, New York (2015)Google Scholar
43. Vu-Quoc, L.: On the algebra of two point tensors and their applications. Z. Angew. Math. Mech.: ZAMM 76(9), 540–541 (1996)
44. Weymouth, G.D., Triantafyllou, M.S.: Ultra-fast escape of a deformable jet-propelled body. J. Fluid Mech. 721, 367–385 (2013)