Eugène Cosserat & Elasticity Theory birth in Toulouse
A reserved, kindly man and a diligent worker, Eugène Cosserat was one of the moving forces in the University of Toulouse for thirty five years. He studied the deformation of surfaces which led him to a “theory of elasticity”. The Cosserat brothers, following a suggestion by Duhem (1893), developed a theory for continuous oriented bodies that consist not just of particles (or material points), but also of directions associated with each particle. Eugène Cosserat died in his home at the Observatory in Toulouse.
See Cosserat biography by J J O'Connor and E F Robertson
http://wwwgroups.dcs.stand.ac.uk/history/Biographies/Cosserat_Francois.html
At the age of 17 he took the competitive entrance examinations for the two major Paris Institutions, the École Polytechnique and the École Normale Supérieure, and was offered a place at both. Unlike his two brothers who both studied at the École Polytechnique, he chose to study at the École Normale Supérieure which he entered in 1883. During three years of study at the École Normale, Cosserat attended lectures by leading mathematicians including Paul Appell, Gaston Darboux, Gabriel Koenigs and Émile Picard. Among his fellow students were several who would make major contributions to mathematics, including Jacques Hadamard and Paul Painlevé. Cosserat graduated in 1886 and spent a short time teaching at the Lycée in Rennes before he was appointed as an assistant astronomer at the Observatory in Toulouse towards the end of 1886.
Even before the award of his doctorate in 1889, Cosserat had begun teaching mathematics courses at the Faculty of Science at Toulouse. In 1896 he became professor of differential and integral calculus there, replacing Thomas Stieltjes who had died on 31 December 1894, and, from that time on, he divided his work between the Faculty of Science and the Observatory. In 1908 Cosserat was appointed to the chair of astronomy at Toulouse, becoming director of the Observatory there for the rest of his life. In this latter role he replaced Édouard Benjamin Baillaud who had left Toulouse to become director of the Paris Observatory. The role of director of the Observatory was a demanding one, and Cosserat became almost totally occupied with administrative tasks from the time of his appointment and so was forced to essentially give up mathematical research from this time on.
Although he was not living in Paris, Cosserat was elected to the Académie des Sciences as a corresponding member on 19 June 1911 and a full member on 31 March 1919. Four years later, he was elected to the Bureau de Longitude. Because he was in Toulouse rather than Paris, he was made a nonresident member of both these organisations. In 1889 he was awarded the Poncelet Prize by the Académie des Sciences.
In mathematics, we have already noted his early work on geometry. In his later work, Cosserat studied the deformation of surfaces which led him to a theory of elasticity. This work was carried out in collaboration with his brother, François Cosserat, who was an engineer. He began his collaboration with his brother in 1896 with the publication Théorie de l'élasticité. This first work studied broad questions relating to the foundations of mechanics but later their work turned towards the physical theory. By the early 1900s, Cosserat had stopped working on the type of geometrical problems that had interested him at the start of his career and all his research efforts were directed towards working on mechanics with his brother. Their most important joint publications are: Note sur la cinématique d'un milieu continu (1897); Note sur la dynamique du point et du corps invariable (1906); Note sur la théorie de l'action euclidienne (1909); and the book Théorie des corps déformables (1909). The first of these was published as an addition to Gabriel Koenigs Leçons de Cinématique professées à la Sorbonne: cinématique théorique. A review of this work by E O Lovett in the Bulletin of the American Mathematical Society in 1900 singles out the Cosserats' contribution:
The introduction of this note is peculiarly fortunate for it is high time that kinematics should comprehend the study of deformation and of deformable spaces. The authors have included in their extract certain generalities on curvilinear coordinates, the deformation of a continuous medium in general, infinitely small deformation, use of the mobile trieder, and the case where the nondeformed medium is referred to any curvilinear coordinates.
This innovative work on mechanics (21 joint publications on this topic are listed in [2]) ended with the François Cosserat's death in 1914, after which time his brother Eugène Cosserat published nothing further on the topic. Jacques Levy describes the two Cosserats' contributions to this area [1]:
The most practical results concerning elasticity were the introduction of the systematic use of the movable trihedral and the proposal and resolution, before Fredholm's studies, of the functional equations of the sphere and ellipsoid. Cosserat's theoretical research, designed to include everything in theoretical physics that is directly subject to the laws of mechanics, was founded on the notion of Euclidean action [least action] combined with Lagrange's ideas on the principle of extremality and Lie's ideas on invariance in regard to displacement groups. The bearing of this original and coherent conception was diminished in importance because at the time it was proposed, fundamental ideas were already being called into question by both the theory of relativity and progress in physical theory.
The authors of [5] write:
The Cosserat brothers, following a suggestion by Duhem (1893), developed a theory for continuous oriented bodies that consist not just of particles (or material points), but also of directions associated with each particle. Thus, in addition to the field of position vectors of a continuum in a given configuration, one also admits vector fields ... which may be chosen so as to represent pertinent features of materials. ... The Cosserats themselves recognised the value of oriented twodimensional continua (i.e., curves and surfaces endowed with additional structure in the form of directors) for representing the deformations of rods and shells respectively. ... [However their] ideas on the subject [were] ignored for half a century.
Another aspect of Eugène Cosserat's work which we should mention is his contributions to the Annales de la faculté des sciences de Toulouse. This journal began publication in 1887 and, two years later, Cosserat joined the editorial board. The two other mathematicians who served on this board at this time were Henri Andoyer and Thomas Jan Stieltjes. In 1896 Cosserat became secretary to the editorial board of the Annals and he continued to hold this role until 1930. In fact, he continued to undertake editorial work up to the time of his death, sending Henri Poincaré a letter on an editorial matter just a few days before his death.
Eugène Cosserat died in his home at the Observatory in Toulouse [4]:
The funeral took place on 2 June, on a morning with gentle sun; a long procession descended from the Observatory along the slopes which, although close to the city, still retained some greenery. It seemed that Nature had staged a scene both bright and calm ... calm as he was in his way.

J R Levy, Biography in Dictionary of Scientific Biography (New York 19701990).
http://www.encyclopedia.com/doc/1G22830900999.html
Books:

J F Pommaret, Lie pseudogroups and mechanics (Taylor & Francis, 1988).
Articles:

M Brocato and K Chatzis, Les Frères Cosserat. Brève Introduction à Leur Vie et à Leurs Travaux en Mécanique.

A Buhl, Eugène Cosserat. Annales de la faculté des sciences de Toulouse 23 (1931), vviii.

J Casey and M J Crochet, Paul M Naghdi (19241994) in J Casey and M J Crochet (eds.), Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids: a collection of papers in honor of Paul M Naghdi (Birkhäuser, 1995), S1S32.

P Caubet, E Cosserat: set vues générales sur I'astronomie de position, Journal des observateurs, 14 (1931), 139143

L Montangerand, Eloge de E Cosserat lu à la séance du 30 juin 1932 de l'Académie des Sciences, Inscriptions et BellesLettres de Toulouse, Ann. de l'Observatoire de Toulouse 10 (1933), xxxxx.

Cosserat, E.; Cosserat, F. (1909). Théorie des Corps deformables. Paris: A, Hermann et Fils