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Size-Dependent Continuum Mechanics Approaches pp 417–452Cite as

Gradient Extension of Classical Material Models: From Nuclear & Condensed Matter Scales to Earth & Cosmological Scales

Part of the Springer Tracts in Mechanical Engineering book series (STME)

Abstract

The various mathematical models developed in the past to interpret the behavior of natural and manmade materials were based on observations and experiments made at that time. Classical laws (such as Newton’s for gravity, Hooke’s for elasticity, Navier-Stokes for fluidity, Fick’s/Fourier’s for diffusion/heat transfer, Coulomb’s for electricity, as well as Maxwell’s for electromagnetism and Einstein’s for relativity) formed the basis for shaping our current technology and civilization. The discovery of new phenomena with the aid of recently developed experimental probes have led to various modifications of these laws across disciplines and scales: from subatomic and elementary particle physics to cosmology and from atomistic and nano/micro to macro/giga scales. The emergence of nanotechnology and the further advancement of space technology are ultimately connected with the design of novel tools for observation and measurements, as well as with the development of new methods and approaches for quantification and understanding. This chapter first reviews the author’s previously developed weakly nonlocal or gradient models for elasticity, diffusion and plasticity within a unifying internal length gradient (ILG) framework. It then proposes a similar extension for fluids and Maxwell’s equations of electromagnetism. Finally, it ventures a gradient modification of Newton’s law of gravity and examines its implications to some problems of elementary particle physics, also relevant to cosmology. Along similar lines, it suggests an analogous extension of London’s quantum mechanical potential to include both an “attractive” and a “repulsive” branch. It concludes with some comments on a fractional generalization of the ILG framework.

Dedicated to the unforgettable memory of my mentor James Serrin and my mentee Hussein Zbib. And to the inspiring work of my classmate Constantinos Vayenas and my daughter Katerina.

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Fig. 1

Notes

  1. 1.

    An elaboration of this analogy is given in a forthcoming article by H. Hatzikirou and E.C. Aifantis: On the similarities between the W-A model for dislocations and the GoG model for cancer cells (in preparation).

  2. 2.

    Podolsky [B. Podolsky, A generalized electrodynamics Part I” Non-quantum. Phys. Rev. 62, 68–71 (1942); B. Podolsky, P. Schwed, Review of a generalized electrodynamics. Rev. Mod. Phys. 20, 40–50(1948)] has derived a generalization of Maxwell’s equations through a variational principle, leading to the appearance of \(\,{{\nabla }^{2}}\mathbf {B}\) in addition to \(\,{{\nabla }^{2}}\mathbf {E}\). This is also possible through the aforementioned analogy by replacing \(\mathbf {u}\) with \(\mathbf {u}-{{\ell }^{2}}{{\nabla }^{2}}\mathbf {u}\).

  3. 3.

    In fact, the question of exploring the consequences of such generalization to gravitation emerged during initial discussions with my daughter K.E. Aifantis during my visit in February 2019 to the University of to Florida at Gainesville and follow-up discussions with my former classmate C. Vayenas of the Academy of Athens during his visit in June 2019 to Thessaloniki. The initial numerical calculations reported herein started with the help of KEA’s students in Gainesville and completed with the assistance of my Ph.D. student K. Parisis in Thessaloniki. The same holds for the results on gradient interatomic potentials listed in Sect. 8.

  4. 4.

    For completeness, however, we may refer to the paper by Reid (R.V. Reid, Local phenomenological nucleon–nucleon potentials, Annal. Phys. 50, 411–448 (1968)), where the following expression, among others, is proposed \({{V}_{C}}=h\left[ {{e}^{-x/3}}-13.8\,{{e}^{-3x}}+138\,{{e}^{-6x}} \right] /x\), with \(h=10.463\) MeV and \(x=\mu r,\,\mu =mc/\hbar \approx 0.7\,\mathrm{f}{{\mathrm{m}}^{-1}}\).

  5. 5.

    The fact that solutions of Eq. (46) can be obtained in terms of solutions of Eq. (47) was first observed in [78] for the non-fractional GradEla and was extended later for more general fractional/fractal elastic materials in [114]

  6. 6.

    In fact, two preprints (by C. G. Vayenas, D. Tsousis, D. Grigoriou, K. Parisis and E. C. Aifantis) on Kaons and Deuteron are available and scheduled for arXiv and journal publication. Two more articels on gradient London’s potential (by K. Parisis, F. Shuang, B. Wang, P. Hu, A. Glannakoudakis and A. Konstantinidis: J. Appl. Math. Phys.) and its fractional extension (by K. Parisis and E. C. Aifantis: TMS Proc. 2021) are forthcoming

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Acknowledgements

The work was greatly benefited from the RISE/FRAMED project no. 734485 (https://cordis.europa.eu/project/rcn/207050/factsheet/en) for which Aristotle University of Thessaloniki (AUTh) acts as coordinator. In this connection, thanks are extended to all beneficiary nodes and international partners of FRAMED. The gradient fluids section is a topic of the RISE/ATM2BT project no. 824022 (https://cordis.europa.eu/project/rcn/219192/factsheet/en) for which AUTh is a beneficiary. This section was included in anticipation of follow-up joint work between AUTh, Akita Univeristy/Japan and Aston University/UK (which acts as project’s ATM2BT coordinator). The remaining of the sections were benefited from discussions with my former classmate C. Vayenas, my daughter K.E. Aifantis and my Ph.D. student K. Parisis, as well as by the continuous support of my former Ph.D. students A. Konstantinidis (successor of my Lab in Thessaloniki) and I. Tsagrakis (current collaborator in Crete). Finally, the author is also gratefully acknowledging the support of Deutsche Forschungsgemeinschaft/DFG grant No. 377472739/GRK 2423/1-2019 at Friedrich-Alexander University (FAU) where he has recently been appointed as a Mercator Fellow.

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Authors and Affiliations

  1. School of Engineering, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    Elias C. Aifantis

  2. Michigan Technological University, Houghton, MI, 49931, USA

    Elias C. Aifantis

  3. Mercator Fellow Friedrich-Alexander University, Erlangen-Nüremberg, 90762, Fürth, Germany

    Elias C. Aifantis

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  1. School of Mechanical Engineering, Shiraz University, Shiraz, Iran

    Dr. Esmaeal Ghavanloo

  2. School of Mechanical Engineering, Shiraz University, Shiraz, Iran

    Prof. S. Ahmad Fazelzadeh

  3. Department of Structures for Engineering and Architecture, University of Naples Federico II, Naples, Italy

    Prof. Francesco Marotti de Sciarra

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Aifantis, E.C. (2021). Gradient Extension of Classical Material Models: From Nuclear & Condensed Matter Scales to Earth & Cosmological Scales. In: Ghavanloo, E., Fazelzadeh, S.A., Marotti de Sciarra, F. (eds) Size-Dependent Continuum Mechanics Approaches. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-63050-8_15

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