Schneider, P. ; Kienzler, R. (2020)
A priori estimation of the systematic error of consistently derived theories for thin structures.
In: International Journal of Solids and Structures, 190
doi: 10.1016/j.ijsolstr.2019.10.010
Artikel, Bibliographie
Kurzbeschreibung (Abstract)
The consistent approximation approach is a dimension reduction technique for the derivation of hierarchies of lower-dimensional analytical theories for thin structural members from the three-dimensional theory of elasticity. It is based on the uniform truncation of the potential energy after a certain power of a geometric scaling factor that describes the relative thinness of the structural member. The truncation power defines the approximation order N of the resulting theory. In the contribution, we deal with the exemplary case of a slender, beam-like continuum with rectangular cross-section. We introduce an extension of the consistent approximation approach towards a simultaneous, uniform truncation of the complementary energy. We show that compatible displacement boundary conditions can be derived from the Euler-Lagrange equations of the complementary energy, whereas, field equations and stress boundary conditions follow from the Euler-Lagrange equations of the potential energy. Thus fully defined boundary value problems are obtained for all orders of approximation N. We provide an a priori-error estimate for the systematic error of the Nth-order theory. It states that the norm difference of the solution of the so derived boundary value problem of order N to the exact solution of the three-dimensional theory of elasticity declines like the geometric scaling factor to the power of N+1. Thus the approximation character of the approach is proved. Furthermore, we outline why the approach is to prefer over the - in engineering mostly used - approach of a fixed displacement field ansatz with unknown coefficients. We show that the later approach leads in general to theories of higher complexity without increasing the approximation accuracy compared to a consistent approximation.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2020 |
| Autor(en): | Schneider, P. ; Kienzler, R. |
| Art des Eintrags: | Bibliographie |
| Titel: | A priori estimation of the systematic error of consistently derived theories for thin structures |
| Sprache: | Englisch |
| Publikationsjahr: | 1 Mai 2020 |
| Verlag: | Elsevier |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | International Journal of Solids and Structures |
| Jahrgang/Volume einer Zeitschrift: | 190 |
| DOI: | 10.1016/j.ijsolstr.2019.10.010 |
| URL / URN: | http://www.sciencedirect.com/science/article/pii/S0020768319... |
| Kurzbeschreibung (Abstract): | The consistent approximation approach is a dimension reduction technique for the derivation of hierarchies of lower-dimensional analytical theories for thin structural members from the three-dimensional theory of elasticity. It is based on the uniform truncation of the potential energy after a certain power of a geometric scaling factor that describes the relative thinness of the structural member. The truncation power defines the approximation order N of the resulting theory. In the contribution, we deal with the exemplary case of a slender, beam-like continuum with rectangular cross-section. We introduce an extension of the consistent approximation approach towards a simultaneous, uniform truncation of the complementary energy. We show that compatible displacement boundary conditions can be derived from the Euler-Lagrange equations of the complementary energy, whereas, field equations and stress boundary conditions follow from the Euler-Lagrange equations of the potential energy. Thus fully defined boundary value problems are obtained for all orders of approximation N. We provide an a priori-error estimate for the systematic error of the Nth-order theory. It states that the norm difference of the solution of the so derived boundary value problem of order N to the exact solution of the three-dimensional theory of elasticity declines like the geometric scaling factor to the power of N+1. Thus the approximation character of the approach is proved. Furthermore, we outline why the approach is to prefer over the - in engineering mostly used - approach of a fixed displacement field ansatz with unknown coefficients. We show that the later approach leads in general to theories of higher complexity without increasing the approximation accuracy compared to a consistent approximation. |
| Freie Schlagworte: | Linear elasticity, Anisotropic material, Thin-walled structures, Energy methods, Dimension reduction, Consistent approximation, A priori error estimation, Higher-order theories, Refined theories |
| Fachbereich(e)/-gebiet(e): | 16 Fachbereich Maschinenbau 16 Fachbereich Maschinenbau > Fachgebiet für Konstruktiven Leichtbau und Bauweisen-KLuB (2023 umbenannt in Leichtbau und Strukturmechanik (LSM)) |
| Hinterlegungsdatum: | 21 Nov 2019 12:26 |
| Letzte Änderung: | 19 Feb 2021 10:01 |
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