Poincaré's equations for Cosserat shells - application to locomotion of cephalopods

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14356

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In 1901 Henri Poincaré proposed a new set of equations for mechanics. These equations are a generalization of Lagrange equations to a system whose configuration space is a Lie group, which is not necessarily commutative. Since then, this result has been extensively refined by the Lagrangian reduction theory. In this article, we show the relations between these equations and continuous Cosserat media, i.e. media for which the conventional model of point particle is replaced by a rigid body of small volume named microstructure. In particular, we will see that the usual shell balance equations of nonlinear structural dynamics can be easily derived from the Poincaré’s result. This framework is illustrated through the simulation of a simplified model of cephalopod swimming.

Poincaré's equations for Cosserat shells - application to locomotion of cephalopods

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application/pdf Poincaré's equations for Cosserat shells - application to locomotion of cephalopods Frederic Boyer, Federico Renda

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In 1901 Henri Poincaré proposed a new set of equations for mechanics. These equations are a generalization of Lagrange equations to a system whose configuration space is a Lie group, which is not necessarily commutative. Since then, this result has been extensively refined by the Lagrangian reduction theory. In this article, we show the relations between these equations and continuous Cosserat media, i.e. media for which the conventional model of point particle is replaced by a rigid body of small volume named microstructure. In particular, we will see that the usual shell balance equations of nonlinear structural dynamics can be easily derived from the Poincaré’s result. This framework is illustrated through the simulation of a simplified model of cephalopod swimming.

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Poincaré equations for Cosserat shells : application to cephalopod locomotion Frédéric Boyer*, Federico Renda° Biorobotics group of Ecole des Mines de Nantes/IRCCyN (*), Khalifa University (o) 1 Geometric Science of Information GSI’15 Ecole Polytechnique – Paris/Saclay 28th-30th October 2015 Motivations of these works: soft bio-inspired robotics Design a new generation of soft robots inspired from soft animals as cephalopods Cephalopods exploit the elastodynamic interactions of a soft body with the fluid • We need new models and algorithms suited to the locomotion of soft animals • Able to encompass both finite (net) transformations and finite deformations Due to the complexity (geometric nonlinearities) we propose to exploit the symmetries of these systems by extending the Poincaré approach… 3 Poincaré equations of a classical mechanical system We consider a system free of external forces with configuration space With a Lagrangian: If is left invariant: Poincaré applies the Hamilton’s principle to with a variation imposed at any fixed time: From variational calculus: “Poincaré” [Poincaré 1901] or “Euler-Poincaré” equations of a mechanical system [Marsden 90, Marle 2013]. Idea: use a “field theory inspired” approach [Pommaret 94, Castrillón López 00] to extend this picture to Cosserat media… 4 What is a Cosserat medium? This is a set of small rigid bodies continuously stacked along some of the material dimensions of a medium. The configuration space: The Lagrangian of a Cosserat medium takes the general form: Density related to the metric volume of reference configuration Subspace of the material space [0,1] (beam), [0,1]² (shell)… Material space = x Reference configuration Deformed configuration Microstructure o 1  ot 1E pE [Cosserat, 1909] 5 Reduction of the Lagrangian density The Lagrangian density is left invariant by , since: (Left) reduced Lagrangian: With the left invariant fields of : : defines the velocity field of the medium : used to define its strain fields does not depend on the choice of the inertial observer is invariant by rigid overall transformations (frame indifference) 6 Reduction of the Lagrangian density Applying the extended Hamilton’s principle to the (left reduced) action: For any variations applied while the time and the material labels are maintained fixed, i.e., s.t.: Gives (from Stokes theorem) a set of partial differential equations standing for the dynamic balance of the Cosserat medium. , 7 Poincaré-Cosserat equations Fields equations: Boundary conditions: Metric on and normal to the boundaries of the reference configuration pulled back in material space External forces Remarks: Removing the - dependence gives the classical Poincaré o.d.e.s They are equations of densities on .. They require reconstruction eq.: Applying them to = [0,1] they are the p.d.e.s of Reissner beams [Boyer 05]. Taking a Lagrangian density / material or def. volume gives 2 other formulations 8 Application to Cosserat shells • Kinematics of Reissner shells [Reissner 45]: • Configuration space of a Cosserat shell: • From which we deduce the left invariant fields: • The first is the velocity field which appears in the density of kinetic energy: • The second and third entirely parameterize the fields of “effective strains”: GSI 2015 9 Considering a hyper-elastic material, i.e., a , Poinc.-Coss. eq. will make the densities of conjugate momentums or “Cosserat stress” appear: Which represents the wrench of stress exerted across the shell cross section constant per unit of length, and: Application to Cosserat shells Which represents the wrench of stress exerted across the shell cross section constant per unit of length. M D 1E 2 dX 21 dXNho 21 dXMho On the material space 10 Finally developing the Poincaré-Cosserat eq. gives: Equilibrium eq. of the shell in the mater (densities of wrenches / ref. volume) Which are the geometrically exact p.d.e.’s of Reissner shells [Simo, 89] Application to Cosserat shells Once pushed forward in space, and related to the def. config. we find: 11 These equations describe the dynamics of a micro-polar shell, i.e. a shell with spin and couple stress along their microstructures. To remove this artifact, we introduce the fields of effective stress defined as the dual of the effective strains: Using duality, we get the relation between the Cosserat and effective stress as: Then removing the inertia momentum and couple stress around the microstructures changes the last (scalar) Poincaré-Cosserat equation into the constitutive constraint: Which is equivalent to the symmetry of the effective stress tensor, a condition naturally satisfied by its definition (*) and the symmetry of the effective strains. (*) Application to Cosserat shells Finally the Poincaré-Cosserat equations of a micropolar shell with the constitutive law (*) hold for classical shells. However, the angular velocity along the microstructures is dynamically undeterminate. To determine it, we use the polar decomposition of the transformation gradient restricted to the reference (mid) surface and find the kinematic relation: That is introduced in the reconstruction equation to close the model of classical shells Application to Cosserat shells Deformed configurationMaterial space Microstructure 2 D R 1E 2E 3E  X The squid is modeled by an axi-symmetric Reissner shell, whose each microstructure is parameterized by a transformation of : Application to the squid jet-propelling Applying the above construction, gives the p.d.e.’s: 2 D R o XE E 3 3E e M 3E • With the following left invariant fields: Application to the squid jet-propelling • The def. of the effective strains and the reduced Hooke’s constitutive laws: • And a model of the external forces deduced from a crude balance of impulse [Renda 15]: Another result with Cosserat beams Thank you for your attention… 16