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**Schneider, P. and Kienzler, R.** (2019):

*A priori estimation of the systematic error of consistently derived theories for thin structures.*

In: International Journal of Solids and Structures, Elsevier, ISSN 0020-7683,

DOI: 10.1016/j.ijsolstr.2019.10.010,

[Online-Edition: http://www.sciencedirect.com/science/article/pii/S0020768319...],

[Article]

## 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.

Item Type: | Article |
---|---|

Erschienen: | 2019 |

Creators: | Schneider, P. and Kienzler, R. |

Title: | A priori estimation of the systematic error of consistently derived theories for thin structures |

Language: | English |

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. |

Journal or Publication Title: | International Journal of Solids and Structures |

Publisher: | Elsevier |

Uncontrolled Keywords: | Linear elasticity, Anisotropic material, Thin-walledÂ structures, Energy methods, Dimension reduction, Consistent approximation, A priori error estimation, Higher-order theories, RefinedÂ theories |

Divisions: | 16 Department of Mechanical Engineering 16 Department of Mechanical Engineering > Institute for Lightweight Construction and Design (KluB) |

Date Deposited: | 21 Nov 2019 12:26 |

DOI: | 10.1016/j.ijsolstr.2019.10.010 |

Official URL: | http://www.sciencedirect.com/science/article/pii/S0020768319... |

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