Nineteenth century molecular models with a glance at modern discrete—continuum theories

Patrizia Trovalusci,

Corresponding Author

Patrizia Trovalusci

[email protected]

Department of Structural and Geotechnical Engineering, Sapienza, University of Rome

phone +00 39 3204271930, fax +00 39 06 3229441Search for more papers by this author

Patrizia Trovalusci,

Corresponding Author

Patrizia Trovalusci

[email protected]

Department of Structural and Geotechnical Engineering, Sapienza, University of Rome

phone +00 39 3204271930, fax +00 39 06 3229441Search for more papers by this author

First published: 21 October 2015

https://doi.org/10.1002/pamm.201510344

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Abstract

The origins of multiscale modelling, lying in the classical molecular models of Nineteenth century originally formulated to provide explanations per causas of elasticity, are investigated with the aim of drawing suggestions for the modelling of new complex materials. Conceptual guidelines for deriving scale dependent continuum formulations, here continuum models with additional degrees of freedom (multifield continua), are defined starting from non-classical corpuscular descriptions and the historical and epistemological settings of the current modelling proposals are highlighted. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

References

[1] I. Newton. Opticks or a treatise of the reflections, refractions, inflections and colours of ligth, Quest XXXI, 2nd ed. (1717), pp. 242–264.
Google Scholar
[2] C.-L.-M.-H. Navier. Mémoire sur le lois de l'équilibre et du mouvement des corps solides élastiques (1821). In Mémoires de l'Academie des Sciences de l'Institut de France, volume 7 of II (1827) pp. 375–393.
Google Scholar
[3] A.-L. Cauchy. Sur l'équilibre et le mouvement d'un système de points matériels sollicités par des forces d'attraction ou de répulsion mutuelle. Exercices de Mathématiques, 3, (1822, 1827), pp. 188–213 . Œuvres 2 (8)(1828), pp. 227–252.
Google Scholar
[4] S.-D. Poisson. Mémoire sur l'équilibre et le mouvement des corps élastiques. In Mémoires de l'Académie des Sciences de l'Institut de France, volume 8, Lu à l'Académie 1828, (1829), pp. 357–405.
Google Scholar
[5] C.-L.-M.-H. Navier. Résumé des Leçons données a l'École des Ponts et Chaussées sur l'Application de la Méchanique [...]. Troisiéme édition avec des notes et des appendices, par M. Barré de Saint–Venant. Dunod, Paris, 1864.
Google Scholar
[6] I. Thodhunder and K. Pearson. A History of the Theory of Elasticity and of the Strength of Materials: from Galilei to the Present Time. Cambridge University Press, London, 1886–1889.
Google Scholar
[7] W. Voigt. Theoretische Studien über die Elasticitätsverhältnisse der Kristalle. In Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, XXXIV, (1887).
Google Scholar
[8] H. Poincaré. Leçons sur la théorie de l'élasticité. Georges Carré, Paris (1892).
Google Scholar
[9] W. Voigt. Lehrbuch der Kristallphysik. B. G. Teubner, Leipzig, (1910).
Google Scholar
[10] G. Capriz. Continua with Microstructure. Springer-Verlag, Berlin, 1989.
Google Scholar
[11] A. C. Eringen, Microcontinuum Field Theories, Springer–Verlag, New York, (1999).
Google Scholar
[12] P. Trovalusci, D. Capecchi, and G. Ruta. Genesis of the multiscale approach for materials with microstructure. Arch. Appl. Mech., 79, 981–997 (2009).
Google Scholar
[13] D. Capecchi, G. Ruta, and P. Trovalusci, From classical to Voigt's molecular models in elasticity, Arch. Hist. Exact Sci.. 64, 525–559 (2010).
Google Scholar
[14] D. Capecchi, G. Ruta, and P. Trovalusci. Voigt and Poincaré's mechanistic–energetic approaches to linear elasticity and suggestions for multiscale modelling, Arch. Appl. Mech., 81, 1573–1584 (2011).
Google Scholar
[15] P. Trovalusci, Molecular approaches for multifield continua: origins and current developments, in T. Sadowsky and P. Trovalusci (eds.), Multiscale Modelling of Complex Materials: phenomenological, theoretical and computational aspects, CISM Courses and Lectures, 556, Springer, Wien,(2014), pp. 211–278.
Google Scholar
[16] P. Trovalusci, Discrete to scale−dependent sontinua for complex materials. A generalised Voigt approach using the virtual power equivalence, in P. Trovalusci (ed.), Materials with Internal Structure. Multiscale and Multifield Modelling and Simulation, SpringerBriefs Series, Springer, Berlin. (In press).
Google Scholar