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Publications in Scientific Journals:

H. Irschik, R. Heuer:
"Analogies for simply supported nonlocal Kirchhoff plates of polygonal planform";
Acta Mechanica, 229 (2018), 867 - 879.



English abstract:
In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin,
isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation
form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform,
we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we
extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the
local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson
boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context,
it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the
nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal
correction of the deflection turns out to be governed by a membrane-type boundary value problem again.
From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be
derived alternatively from both the local deflection and the local moment sum, which represents a substantial
simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed
form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where
the singular behavior of Greenīs function of the nonlocal Kirchhoff theory can be clarified. Then we discuss
the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best
knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The
present paper has been strongly influenced by a former contribution on analogies between simply supported
polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored
together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of
this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the
framework of various other fields, we finally show in the Appendix that the deflections of simply supported
nonlocal Kirchhoff plates can be considered as deflections of a local "background" Kirchhoff plate under the
action of additional fictitious eigenstrain loadings, such as thermal moments.

Keywords:
Nonlocal Kirchhoff plates, Simply supported polygonal plates , Membrane-type analogies, Plate-type analogies, Nonlocal triangular plates

Created from the Publication Database of the Vienna University of Technology.