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Geometric-material analogy for multiscale modelling of twisted plates

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journal contribution
posted on 2018-03-21, 16:39 authored by A. Kordolemis, Paul M. Weaver
It is well known that the macroscopic behaviour of many engineering materials is strongly affected by the role of underlying microstructure. Currently though, mathematical expressions linking behaviour of large scale structures to the geometry of their microscopic structure are largely lacking. In this respect, establishing quantitative links across different material lengthscales may offer new pathways for engineering design. In the present work an analogy between cross sectional geometrical properties, representing macrostructure, and a material length parameter, representing microstructure, is presented. The analogy is established through the study of a thin plate subject to axial loading undergoing finite displacements from two alternative perspectives. First, we consider a thin elastic plate with a pretwist about the loading axis where a warping term is introduced accounting for the out-of-plane deformation of the cross section. The coupled governing differential equations and the corresponding coupled boundary conditions are explicitly derived employing a classical structural mechanics approach utilising an energy variational statement. Secondly, an axially loaded thin flat plate (i.e. with no pretwist) is studied with strain gradient elasticity theory incorporating only one material length parameter representing the microstructure, in addition to the two classical Lamé stiffness constants. The ensuing analogy emerges by comparison of the governing equations of the two formulations which shows a mathematical expression can be identified, which incorporates both geometric and material length variables, that formalises the link between microscale and macroscale. This mathematical expression, which constitutes the kernel of the proposed multiscale approach, admits a twofold interpretation depending on the assumed independent variable. On the one hand, the proposed multiscale modelling approach suggests that a plate with complex global geometry can be substituted by a structurally - equivalent, flat plate with constitutive relations given by a non - local, strain gradient theory. On the other hand, the material length parameter can be interpreted on a physical basis because for the first time it has been identified as a known function of geometrical features of the structure through simple algebraic relationships for various cross sectional profiles.


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International Journal of Solids and Structures;110-111, pp. 24-35





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Royal Society, SFI


This is the author’s version of a work that was accepted for publication in International Journal of Solids and Structures. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in International Journal of Solids and Structures, 2017, 110-111, pp. 24-35,



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