Analysis of objectives:
Towards a conceptual unification of exising approaches to structural analysis,
including aspects of application in engineering and mathematical theory aspects
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Geometric non linearities
The problem of elasticity, where non linearities derive from geometric effects of the combination of deformations and stresses, is studied through a tensorial tridimensional approach.
The effects of deformations are represented through Cristoffel symbols on the deformed media with respect to the elastic media, following a Lagrangian approach to structural behaviour, in which movements due to external loads are assumed as an unknown variable. This can be achieved by representing the fundamental tensor of the deformed media by means of the fundamental tensor of the initial meadia and of the displacement vector, expressed in the deformed space.
The non linearities of equilibrium equations are fully accounted for in a system of equations which can be simplified as the need arises, until they are reduced to common linearised equations. The interaction between deformations and stresses is particularly important in those structures where thin and wide components are present.
Let us consider, without using the approximations of infinitesimal theory, the deformations of a tridimensional elastic medium in a Euclidean space, from the initial "non deformed" position, with reference to a curvilinear coordinate, to the final "deformed" position, described through a tensor field with reference to an own curvilinear coordinate. For the present purposes, it is useful to select as a reference systm in the deformed medium the transform of the rederence systm of the non deformed medium, so that the transformed point has the same value of each coordinate of the initial point, thus realizing a Lagrangian approach to the problem.
In such a way, it is possible to adopt as unknown variables of the problem the displacements due to external loads. This can be done by representing the fundamental tensor of the deformed medium by the the fundamental tensor of the initial medium and of the displacement vector in the deformed space.