TY - JOUR

T1 - Disarrangements and instabilities in augmented one-dimensional hyperelasticity

AU - Palumbo, Stefania

AU - Deseri, Luca

AU - Owen, David R.

AU - Fraldi, Massimiliano

N1 - Funding Information:
Data accessibility. This article has no additional data. However, further data can be obtained by replacing other parameters in the proposed model. Authors’ contributions. All the authors contributed equally to the paper. In particular, S.P. developed the model, performed the analysis and wrote the paper; M.F. conceived the idea, advised the model development, wrote and reviewed the paper, with L.D. and S.P. translating the results in terms of structured deformations; L.D. and D.R.O. provided the theory of structured deformations, reviewed and edited the final draft. Competing interests. We have no competing interests. Funding. The work was supported by a grant from the Italian Ministry of Education, Universities and Research (MIUR) ARS01-01384-PROSCAN, and by a grant from the University of Napoli ‘Federico II’, E62F17000200001-NAPARIS, and partially supported by grant ERC-2013-ADG-340561-INSTABILITIES. Acknowledgements. M.F. gratefully acknowledges the support of the grants from MIUR (PROSCAN) and the University of Napoli ‘Federico II’ (NAPARIS). L.D. gratefully acknowledges the partial support of grant ERC-2013-ADG-340561-INSTABILITIES.
Publisher Copyright:
© 2018 The Author(s) Published by the Royal Society. All rights reserved.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - In the present work, the overall nonlinear elastic behaviour of a one-dimensional multi-modular structure incorporating possible imperfections at the discrete (microscale) level is derived with respect to both tensile and compressive applied loads. The model is built up through the repetition of n units, each one comprising two rigid rods having equal lengths, linked by means of pointwise constraints capable of elastically limiting motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analysed by varying the number n of the elemental moduli, as well as in the limit case of an infinite number of infinitesimal constituents, in light of the theory of (first-order) structured deformations (SDs), which interprets the deformation of any continuum body as the projection, at the macroscopic scale, of geometrical changes occurring at the level of its sub-macroscopic elements. In this way, a wide family of nonlinear elastic behaviours is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler’s elastica under compressive loads resulting as special cases in the so-called continuum limit—say when n ? 8. Finally, by plotting the results in terms of the first Piola–Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SD-based one-dimensional models can be used to generalize some standard hyperelastic behaviours by additionally taking into account instability phenomena and concealed defects.

AB - In the present work, the overall nonlinear elastic behaviour of a one-dimensional multi-modular structure incorporating possible imperfections at the discrete (microscale) level is derived with respect to both tensile and compressive applied loads. The model is built up through the repetition of n units, each one comprising two rigid rods having equal lengths, linked by means of pointwise constraints capable of elastically limiting motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analysed by varying the number n of the elemental moduli, as well as in the limit case of an infinite number of infinitesimal constituents, in light of the theory of (first-order) structured deformations (SDs), which interprets the deformation of any continuum body as the projection, at the macroscopic scale, of geometrical changes occurring at the level of its sub-macroscopic elements. In this way, a wide family of nonlinear elastic behaviours is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler’s elastica under compressive loads resulting as special cases in the so-called continuum limit—say when n ? 8. Finally, by plotting the results in terms of the first Piola–Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SD-based one-dimensional models can be used to generalize some standard hyperelastic behaviours by additionally taking into account instability phenomena and concealed defects.

KW - Compressive buckling

KW - Generalized hyperelasticity

KW - One-dimensional models

KW - Structured deformations

KW - Tensile

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U2 - 10.1098/rspa.2018.0312

DO - 10.1098/rspa.2018.0312

M3 - Article

AN - SCOPUS:85055580586

VL - 474

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2218

ER -