Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Higher Gradient Theories and Their Foundations

  • Living reference work entry
  • First Online:
Encyclopedia of Continuum Mechanics

Synonyms

Generalized continua; Higher gradient continua; Second gradient continua

Definitions

Elastic continua whose general deformation energy density depends on second and possibly higher gradients of the displacement.

Introduction

Continuum mechanics always supplies approximate models for physical systems, in which a more fundamental (possibly discrete or inhomogeneous) microstructure may be somehow neglected. Indeed, Cauchy (or Cauchy-Navier) continuum theory describes efficiently, at a macroscopic level, the behavior of a mechanical system only when the inhomogeneities which the model does not take into account have a characteristic length scale much smaller than the macroscale where phenomena are observed.

Therefore, it is now widely accepted that in some circumstances, it is necessary to add to the placement field some extra kinematical fields, to take into account, at a macroscopic level, some aspects of the mechanical behavior of materials having complex microscopic...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Alibert J-J, Seppecher P, dell’Isola F (2003) Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids 8(1):51–73

    Google Scholar 

  • Auffray N, dell’Isola F, Eremeyev VA, Madeo A, Rosi G (2014) Analytical continuum mechanics la HamiltonPiola least action principle for second gradient continua and capillary fluids. Math Mech Solids. https://doi.org/1081286513497616

  • Banfi C, Marzocchi A, Musesti A (2006) On the principle of virtual powers in continuum mechanics. Ricerche di Matematica 55:299–310

    Google Scholar 

  • Bleustein JL (1967) A note on the boundary conditions of Toupin’s strain-gradient theory. Int J Solids Struct 3(6):1053–1057

    Google Scholar 

  • Casal P (1961) La capillarité interne, Cahier du groupe Français de rhéologie, CNRS VI, 3, pp 31–37

    Google Scholar 

  • Casal P (1972) La théorie du second gradient et la capillarité. C R Acad Sci Paris, t. 274, Série A 1571–1574

    Google Scholar 

  • Cauchy AL (1827) De la pression ou tension dans un corps solide. Ex de Math 2:4256. Available in Gallica.bnf.fr

    Google Scholar 

  • Chesnais C, Boutin C, Hans S (2012) Effects of the local resonance on the wave propagation in periodic frame structures: generalized Newtonian mechanics. J Acoust Soc Am 132(4), Pt. 2

    Google Scholar 

  • Chesnais C, Boutin C, Hans C (2013) Wave propagation and non-local effects in periodic frame materials: generalized continuum mechanics. Math Mech Solids 20(8):929–958

    Google Scholar 

  • Collin F, Chambon R, Charlier R (2006) A finite element method for poro mechanical modelling of geotechnical problems using local second gradient models. Int J Numer Methods Eng 65:1749–1772

    Google Scholar 

  • Cosserat E, Cosserat F (1908) Note sur la théorie de l’action euclidienne. Gauthier-Villars, Paris

    Google Scholar 

  • Cosserat E, Cosserat F (1909) Sur la Théorie des Corps Déformables. Herman, Paris

    Google Scholar 

  • Coussy O (2004) Poromechanics. Wiley, Chichester

    Google Scholar 

  • Degiovanni M, Marzocchi A, Musesti A (2006) Edge-force densities and second-order powers. Annali di Matematica 185:81–103

    Google Scholar 

  • dell’Isola F, Seppecher P (1995) The relationship between edge contact forces, double force and interstitial working allowed by the principle of virtual power. Comptes Rendus de l’Academie de Sciences – Serie IIb: Mecanique, Physique, Chimie, Astronomie, 321:303–308

    Google Scholar 

  • dell’Isola F, Seppecher P (1997) Edge contact forces and quasi-balanced power. Meccanica 32:33–52

    Google Scholar 

  • dell’Isola F, Seppecher P (2011) Hypertractions and hyperstresses convey the same mechanical information. Continuum Mech Thermodyn (2010) 22:163–176. By Prof. Podio Guidugli and Prof. Vianello and some related papers on higher gradient theories. Continuum Mech Thermodyn 23(5):473–478

    Google Scholar 

  • dell’Isola F, Sciarra G, Vidoli S (2009) Generalized Hooke’s law for isotropic second gradient materials. Proc R Soc Lond Ser A Math Phys Eng Sci 465(2107):2177–2196

    Google Scholar 

  • dell’Isola F, Seppecher P, Madeo A (2012a) How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach á la D’Alembert. Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP) 63(6):1119–1141

    Google Scholar 

  • dell’Isola F, Madeo A, Placidi L (2012b) Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. ZAMM J Appl Math Mech/Zeitschrift fur Angewandte Mathematik und Mechanik 92(1):52–71

    Google Scholar 

  • dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R, Forest S (2014) The complete works of Gabrio Piola: volume I. Advanced structured materials, vol 38. Springer International, Cham

    Google Scholar 

  • dell’Isola F, Seppecher P, Della Corte A (2015a) The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. In: Proceedings of the Royal Society A, vol 471, No 2183. The Royal Society, p 20150415

    Google Scholar 

  • dell’Isola F, Andreaus U, Placidi L (2015b) At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Mech Math Solids 20(8):887–928

    Google Scholar 

  • dell’Isola F, Steigmann D, Della Corte A (2016a) Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl Mech Rev 67(6):060804–060804-21

    Google Scholar 

  • dell’Isola F, Madeo A, Seppecher P (2016b) Cauchy tetrahedron argument applied to higher contact interactions. Arch Ration Mech Anal 219(3):1305–1341

    Google Scholar 

  • Del Piero G (2014) Non-classical continua, pseudobalance, and the law of action and reaction. Math Mech Complex Syst 2(1):7110

    Google Scholar 

  • Dunn JE (1986) Interstitial working and a non classical continuum thermodynamics. In: Serrin J (ed) New perspectives in thermodynamics. Springer, Berlin, pp 187–222

    Google Scholar 

  • Dunn JE, Serrin J (1985) On the thermomechanics of interstitial working. Arch Ration Mech Anal 88(2):95–133

    Google Scholar 

  • Forest S (1998) Mechanics of generalized continua: construction by homogenization. J Phys IV Fr 8:Pr4 39–Pr4 48

    Google Scholar 

  • Forest S (2005) Generalized continua. In: Buschow KHJ, Cahn RW, Flemings MC, Ilschner B, Kramer EJ, Mahajan S (eds) Encyclopedia of materials: science and technology updates. Elsevier, Oxford, pp 1–7

    Google Scholar 

  • Forest S (2006) Milieux continus généralisés et matériaux hétéérogènes. Les Presses de l’Ecole des Mines de Paris. isbn:2-911762-67-3

    Google Scholar 

  • Forest S, Amestoy M, Cantournet S, Damamme G, Kruch S (2015) Mécanique des Milieux Continus, ECOLE DES MINES DE PARIS Année 2005–2006

    Google Scholar 

  • Fosdick R (2011) On the principle of virtual power for arbitrary parts of a body. Continuum Mech Thermodyn 23:483–489

    Article  MathSciNet  MATH  Google Scholar 

  • Fried E, Gurtin ME (2006) Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small length scales. Arch Ration Mech Anal 182:513–554

    Article  MathSciNet  MATH  Google Scholar 

  • Germain P (1972) Sur l’application de la méthode des puissances virtuelles en mécanique des milieux continus. C R Acad Sci Paris Série A-B 274:A1051–A1055

    MATH  Google Scholar 

  • Germain P (1973a) La méthode des puissances virtuelles en mécanique des milieux continus. Première partie. Théorie du second gradient. J Mécanique 12:235–274

    MATH  Google Scholar 

  • Germain P (1973b) Cours de Mècanique des Milieux Continus, tome I. Masson, Paris

    MATH  Google Scholar 

  • Germain P (1973c) The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J Appl Math 25:556–575

    Article  MATH  Google Scholar 

  • Green AE, Rivlin RS (1964a) Multipolar continuum mechanics. Arch Ration Mech Anal 17:113–147

    Article  MathSciNet  MATH  Google Scholar 

  • Green AE, Rivlin RS (1964b) Simple force and stress multipoles. Arch Ration Mech Anal 16:325–353

    MathSciNet  MATH  Google Scholar 

  • Green AE, Rivlin RS (1964c) On Cauchy’s equations of motion. Z Angew Math Phys (ZAMP) 15:290–292

    Article  MathSciNet  MATH  Google Scholar 

  • Green AE, Rivlin RS (1965) Multipolar continuum mechanics: functional theory I. Proc R Soc Ser A 284:303–324

    Article  MathSciNet  Google Scholar 

  • Marzocchi A, Musesti A (2003) Balanced virtual powers in continuum mechanics. Meccanica 38:369–389

    Article  MathSciNet  MATH  Google Scholar 

  • Mindlin RD (1962) Influence of couple-stresses on stress concentrations main features of cosserat theory are reviewed by lecturer and some recent solutions of the equations, for cases of stress concentration around small holes in elastic solids, are described. Exp Mech 3(1):1–7

    Article  MathSciNet  Google Scholar 

  • Mindlin RD (1963) Complex representation of displacements and stresses in plane strain with couple-stresses. Appl Theory Funct Contin Mech (Proc Int Symp, Tbilisi), vol I (Mech Solids, Russian), pp 256–259 Izdat. Nauka”, Moscow

    Google Scholar 

  • Mindlin RD (1965a) Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct 1(4):417–438

    Article  Google Scholar 

  • Mindlin RD (1965b) Stress functions for a Cosserat continuum. Int J Solids Struct 1(3):265–271

    Article  Google Scholar 

  • Mindlin RD (1965c) On the equations of elastic materials with micro-structure. Int J Solids Struct 1(1):73–78

    Article  Google Scholar 

  • Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124

    Article  MATH  Google Scholar 

  • Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448

    Article  MathSciNet  MATH  Google Scholar 

  • Noll W (1959) The foundations of classical mechanics in the light of recent advances in continuum mechanics. In: Proceeding of the Berkeley symposium on the axiomatic method, Amsterdam, pp 226–281

    MATH  Google Scholar 

  • Noll W, Virga EG (1990) On edge interactions and surface tension. Arch Ration Mech Anal 111(1):1–31

    Article  MathSciNet  MATH  Google Scholar 

  • Pideri C, Seppecher P (1997) A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin Mech Thermodyn 9(5):241–257

    Article  MathSciNet  MATH  Google Scholar 

  • Piola G (1825) Sull’applicazione de’ principj della meccanica analitica del Lagrange ai principali problemi. Memoria di Gabrio Piola presentata al concorso del premio e coronata dall’I.R. Istituto di Scienze, ecc. nella solennita del giorno 4 ottobre 1824, Milano, Imp. Regia stamperia

    Google Scholar 

  • Piola G (1846) Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, Mod- ena, Tipi del R.D. Camera

    Google Scholar 

  • Polizzotto C (2007) Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions. Eur J Mech A Solids 26(2):189–211

    Article  MathSciNet  MATH  Google Scholar 

  • Salençon J (2005) Mécanique des milieux continus: concepts généraux (vol 1). Editions Ecole Polytechnique

    MATH  Google Scholar 

  • Schwartz L (1973) Théorie des Distributions. Hermann, Paris

    Google Scholar 

  • Sciarra G, dell’Isola F, Coussy O (2007) Second gradient poromechanics. Int J Solids Struct 44(20):6607–6629

    Google Scholar 

  • Seppecher P, Alibert JJ, dell’Isola F (2011) Linear elastic trusses leading to continua with exotic mechanical. J Phys Conf Ser 319(1):012018

    Google Scholar 

  • Steigmann DJ, Pipkin AC (1991) Equilibrium of elastic nets. Philos Trans R Soc Lond A: Math Phys Eng Sci 335(1639):419–454

    Article  Google Scholar 

  • Toupin RA (1962) Elastic materials with couple-stresses. Arch Rat Mech Anal 11:385–414

    Article  MathSciNet  MATH  Google Scholar 

  • Toupin RA (1964) Theories of elasticity with couple-stress. Arch Ration Mech Anal 17:85–112

    Article  MathSciNet  MATH  Google Scholar 

  • Triantafyllidis N, Aifantis EC (1986) A gradient approach to localization of deformation. I. Hyperelastic materials. J Elast 16(3):225–237

    MathSciNet  MATH  Google Scholar 

  • Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Fliigge’s Encyclopedia of Physics, vol III/3. Springer, Berlin, pp 1–662

    Google Scholar 

  • Truesdell C, Toupin RA (1960) The classical field theories. Handbuch der Physik III/1. Springer, Berlin

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DISG, University of Rome La Sapienza, Rome, Italy

    Francesco dell’Isola

  2. International Research Center M&MoCS, L’Aquila, Italy

    Francesco dell’Isola, Pierre Seppecher & Alessandro Della Corte

  3. IMATH, Université de Toulon, La Garde, France

    Pierre Seppecher

  4. DIMA, University of Rome La Sapienza, Rome, Italy

    Alessandro Della Corte

Authors
  1. Francesco dell’Isola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Seppecher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alessandro Della Corte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco dell’Isola .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. Centre des Matéériaux UMR 7633, Mines ParisTech CNRS, 10 rue Henri Desbruères, BP 87, 91003, Evry Cedex, France

    Samuel Forest

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

dell’Isola, F., Seppecher, P., Corte, A.D. (2018). Higher Gradient Theories and Their Foundations. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_151-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53605-6_151-1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation