Thesis defense : Perla AZZI

Geometry of isotropy strata applied to the mechanics of materials

This study revolves around linear representations of compact Lie groups and the study of the isotropy classes of such representations. The initial motivation of this work lays down in solving some problems in mechanics of materials where representations of compact groups, such as subgroups of O(3), are involved.
Isotropy classes of group representations are known to be semialgebraic sets characterized by polynomial invariants inequalities. In this thesis, we prove that, in fact, for any representation of compact Lie group, closed isotropy classes are algebraic sets. This result is motivated by several examples in mechanics. For instance, isotropy classes of SO(3)-representation on the space of second and fourth order harmonic tensors in mechanics are given by polynomial equations, which make them algebraic sets. Furthermore, we prove a theorem in invariant theory, about the rationality of invariants for compact Lie group representations. More specifically, for any representation of a compact Lie group 𝐺 on a real vector space 𝑉 and for any isotropy group 𝐻, any polynomial invariant of the representation of the normalizer 𝑁(𝐻) on the fixed point set 𝑉 𝐻 is indeed the restriction on 𝑉 𝐻 of a rational invariant of the initial representation of 𝐺 on 𝑉 . The latter assertion was already known under some restrictive conditions, such as in the case of algebraically closed fields and generic isotropies. In this work, the results are proven over R and with no further conditions.
In a second time, an effective method for determining the isotropy classes for reducible representations is proposed. This method is based on a binary operation between conjugacy classes of closed subgroups of a compact group, which has been introduced to obtain the isotropy classes of a direct sum of representations, if the isotropy classes are known for the irreducible factors. In the specific case of the three dimensional orthogonal group O(3), clips between some types of subgroups of O(3) have already been calculated. However, until now, clips between type II and type III subgroups were missing. Those are encountered in 3D coupled constitutive laws. In this thesis, we complete the clips tables by computing the missing clips. As an application, we obtain 25 isotropy classes for the standard O(3)-representation on the full 3D Piezoelectric law, which involves the three elasticity, piezoelectricity and permittivity constitutive tensors.
Another problem that has been encountered many times in mechanics is solved in this work using algebraic methods: given an experimental material with no specific isotropy, it is interesting to determine the closest isotropy of a contitutive tensor for this material. To solve such a problem, we used methods of polynomial optimization taking advantage of the characterization of the isotropy classes by polynomial covariants. Two polynomial optimization methods are presented to solve this question and applied to some constitutive laws in mechanics such as elasticity and piezoelectricity laws: one makes use of some effective algorithm based on real algebraic geometry and theory of moments and the other one is an explicit calculation of critical points by an algebraic method called Euler-Lagrange method.

 

Composition of the jury

• Irina Kogan, North Carolina State University -- rapporteur
• Nicolas Dutertre, University of Angers -- rapporteur
• Nicolas Auffray, Sorbonne University -- examiner
• Jacky Cresson, University of Pau -- examiner
• Julien Grivaux, Sorbonne University -- director
• Rodrigue Desmorat, University of Paris-Saclay -- co-director
• Boris Kolev, University of Paris-Saclay -- supervisor

Mots-clés

Représentation de groupe compact, classes d’isotropie, loi de comportement, optimisation poly- nomiale, mécanique des milieux continus.