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

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-walled structures, Energy methods, Dimension reduction, Consistent approximation, A priori error estimation, Higher-order theories, Refined 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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