Schneider, P. ; Kienzler, R. (2019)
A-priori error estimation for energy based dimension-reduction techniques.
5th International Conference on Mechanics of Composites. Lissabon (1. - 4. Juli 2019)
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Kurzbeschreibung (Abstract)

The consistent approximation approach is a dimension-reduction technique for the derivation of lower-dimensional analytical theories for thin structural members from the three-dimensional theory of elasticity. It was successfully applied to a variety of problems by numerous authors. The approach is based on a structured truncation of the elastic energy. On the one hand, it has been shown that leading-order approximations for isotropic material deliver the known classical theories, like the KircThe consistent approximation approach is a dimension-reduction technique for the derivation of lower-dimensional analytical theories for thin structural members from the three-dimensional theory of elasticity. It was successfully applied to a variety of problems by numerous authors. The approach is based on a structured truncation of the elastic energy. On the one hand, it has been shown that leading-order approximations for isotropic material deliver the known classical theories, like the Kirchhoff plate theory, or the Euler-Bernoulli beam theory, without invoking a-priori assumptions. On the other hand, the approach also allows for the derivation of extended refined theories and/or theories for anisotropic materials in a systematic way. This was already demonstrated by the authors for the case of a Reissner-type plate theory for monoclinic material. In the talk, we show how the approach can be extended towards a simultaneous truncation of the dual energy, which allows for a structured derivation of compatible boundary conditions. Furthermore, using a duality principle, it is possible to provide an a-priory error estimate stating that the error of solutions of the arising hierarchy of approximating theories to the exact three-dimensional solution will decrease very fast with the order of approximation, if the structural member is sufficiently thin. Thus, the approximating character of the approach is proven. The procedure is demonstrated at the example of a one-dimensional, isotropic structural member with rectangular cross-section. hhoff plate theory, or the Euler-Bernoulli beam theory, without invoking a-priori assumptions. On the other hand, the approach also allows for the derivation of extended refined theories and/or theories for anisotropic materials in a systematic way. This was already demonstrated by the authors for the case of a Reissner-type plate theory for monoclinic material. In the talk, we show how the approach can be extended towards a simultaneous truncation of the dual energy, which allows for a structured derivation of compatible boundary conditions. Furthermore, using a duality principle, it is possible to provide an a-priory error estimate stating that the error of solutions of the arising hierarchy of approximating theories to the exact three-dimensional solution will decrease very fast with the order of approximation, if the structural member is sufficiently thin. Thus, the approximating character of the approach is proven. The procedure is demonstrated at the example of a one-dimensional, isotropic structural member with rectangular cross-section.

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