<p> This thesis establishes the general Principle of Complementary Potential Energy for the finite deformations of an elastic continuum, in which the Lagrange stress tensor is employed as the stress variable. It is demonstrated that constitutive relations, formulated in terms of the Lagrange stress tensor and the deformation gradient, will admit inversion. Consequently, the present theorem and the theorem proposed by LEVINSON are established as valid principles. The complementary strain energy density of the present theorem, however, is shown to be Independent of rigid displacements, in contrast to that of the LEVINSON formulation. The general Principle is reduced to the form appropriate to finite elastic systems, and it is established that the present theorem reduces to, and therefore contains as a special case, the LIBOVE Theorem.</p> / Thesis / Doctor of Philosophy (PhD)

In this paper we continue the considerations of [5] (CSC/09-05). A numerical study for the parameter identification problem with linear elastic material and large deformations is presented. We discuss the numerical implementation in MATLAB and illustrate some results for a 2D test problem.

In this paper we discuss the identification of parameter functions in material models for elastic large deformations. A model of the the forward problem is given, where the displacement of a deformed material is found as the solution of a n onlinear PDE. Here, the crucial point is the definition of the 2nd Piola-Kirchhoff stress tensor by using several material laws including a number of material parameters. In the main part of the paper we consider the identification of such parameters from measured displacements, where the inverse problem is given as an optimal control problem. We introduce a solution of the identification problem with Lagrange and SQP methods. The presented algorithm is applied to linear elastic material with large deformations.

In this paper we continue the considerations of [5] (CSC/09-05). A numerical study for the parameter identification problem with linear elastic material and large deformations is presented. We discuss the numerical implementation in MATLAB and illustrate some results for a 2D test problem.

Les mécanismes hyperstatiques sont souvent utilisés dans l’industrie pour garantir une bonne tenue mécanique du système et une bonne robustesse aux écarts de fabrication des surfaces. Même si ces assemblages sont très courants, les méthodologies d’analyse de tolérances de ces mécanismes sont difficiles à mettre en oeuvre.En fonction de ses écarts de fabrication, un assemblage hyperstatique peut soit présenter des interférences de montage, soit être assemblé avec jeu. Dans ces travaux de thèse, nous avons appliqué la méthode des polytopes afin de détecter les interférences de montage. Pour un assemblage donné, le polytope résultant du mécanisme est calculé. Si ce polytope est non vide, l’assemblage ne présente pas d’interférence. Si ce polytope est vide, l’assemblage présente des interférences de montage. En fonction du résultat obtenu, deux méthodes d’analyse distinctes sont proposées.Si l’assemblage est réalisable sans interférence le polytope résultant du mécanisme permet de conclure sur sa conformité au regard de l’exigence fonctionnelle. Si l’assemblage présente des interférences de montage, une analyse prenant en compte la raideur des pièces est réalisée. Cette approche est basée sur une réduction de modèle avec des super-éléments. Elle permet de déterminer rapidement l’état d’équilibre du système après assemblage. Un effort de montage est ensuite estimé à partir de ces résultats pour conclure sur la faisabilité de l’assemblage. Si l’assemblage est déclaré réalisable, la propagation des déformations dans les pièces est caractérisée pour vérifier la conformité du système au regard de l’exigence fonctionnelle.La rapidité de mise en oeuvre de ces calculs nous permet de réaliser des analyses de tolérances statistiques par tirage de Monte Carlo pour estimer les probabilités de montage et de respect d’une Condition Fonctionnelle. / Over-constrained mechanisms are often used in industries to ensure a good mechanical strength and a good robustness to manufacturing deviations of parts. The tolerance analysis of such assemblies is difficult to implement.Indeed, depending on the geometrical deviations of parts, over-constrained mechanisms can have assembly interferences. In this work, we used the polytope method to check whether the assembly has interferences or not. For each assembly, the resulting polytope of the mechanism is computed. If it is non empty, the assembly can be performed without interference. If not, there is interferences in the assembly. According to the result, two different methods can be implemented.For an assembly without interference, the resulting polytope enables to check directly its compliance. For an assembly with interferences, a study taking into account the stiffness of the parts is undertaken. This approach uses a model reduction with super elements. It enables to compute quickly the assembly with deformation. Then, an assembly load is computed to conclude on its feasibility. Finally, the spreading of deformation through the parts is calculated to check the compliance of the mechanism.The short computational time enables to perform stochastic tolerance analyses in order to provide the rates of compliant assemblies.

In this paper we discuss the identification of parameter functions in material models for elastic large deformations. A model of the the forward problem is given, where the displacement of a deformed material is found as the solution of a n onlinear PDE. Here, the crucial point is the definition of the 2nd Piola-Kirchhoff stress tensor by using several material laws including a number of material parameters. In the main part of the paper we consider the identification of such parameters from measured displacements, where the inverse problem is given as an optimal control problem. We introduce a solution of the identification problem with Lagrange and SQP methods. The presented algorithm is applied to linear elastic material with large deformations.