Quelques problèmes de stabilité et de bifurcation des solides visqueux

Farid Abed-Meraim 1
1 MeNu
LPMM - Laboratoire de physique et mécanique des matériaux, LEM3 - Laboratoire d'Etude des Microstructures et de Mécanique des Matériaux
Abstract : This PhD thesis is composed essentially of two main parts, of unequal sizes, which are summarized as follows: Part I: It is concerned with stability and bifurcation issues relating to strain-rate-independent solids and structures (i.e., elastic or elasto-plastic). A thorough and comprehensive review of the various investigations in this field allowed us to propose an original and compact presentation of the theory of stability and bifurcation. An illustration of this theory is then shown through an industrial collaboration with SNECMA. More specifically, numerical simulations of blade forming (for new aircraft engines) are carried out along with the analysis of the associated critical buckling states. These investigations reveal clearly that the material rate sensitivity needs to be taken into account, which motivates the second part of this work. Part II: This part contains the main contributions of the thesis. It deals with stability issues of strain-rate-dependent solids and structures (e.g., visco-elastic or elasto-viscoplastic). The objective of this study is twofold: First, theoretical modeling of the stability of viscous solids is proposed. In this approach, the stability issue of the quasi-static evolution of the structure under consideration is addressed. Indeed, the absence of equilibrium states for such viscous solids naturally suggests considering the stability of their quasi-static evolutions. The strain-rate dependency of these quasi-static evolutions is another source of difficulty in such an analysis. This new approach of stability, which considers the non-stationary response, generalizes the more classical concept of equilibrium stability. On the other hand, stability criteria for these quasi-static evolutions are investigated in close relation with the mathematical results related to non-autonomous differential equations. Note that the non-autonomous nature of the governing equations represents here the main difficulty. Sufficient conditions of stability are then derived using two main methods. In the first approach, the linearization technique is used or a Lyapunov functional, appropriate to the physics of the problem, is constructed. The second approach is more elaborate, but it can be applied to both elasto-plastic and elasto-viscoplastic solids and structures. Finally, to illustrate this methodology and to demonstrate the range of applicability of the resulting criteria, the proposed approach is applied to structural problems (beams, columns ...), corresponding to various elasto-plastic, elasto-viscoplastic or visco-elastic behavior models.
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Farid Abed-Meraim. Quelques problèmes de stabilité et de bifurcation des solides visqueux. Mécanique des solides [physics.class-ph]. Ecole Polytechnique X, 1999. Français. <tel-00655820>



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