Skip to main content
SpringerLink
Log in

Generalized Contact Actions

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

Synonyms

Boundary conditions for generalized continua; Double forces; Hyperstress; Nth order stresses

Definitions

Physical quantities of tensorial nature arising from terms dual in energy of the Nth gradient of the displacement.

Introduction

When gradients of the displacement higher than the first enter into the deformation energy density of a body (see “Strain Gradient Theories”), a special treatment has to be devoted in order to get the correct form for the boundary conditions, and a variational approach is the most suitable in this connection (see among the others Piola 2014; Germain 1973a,b; Mindlin and Eshel 1968; Mindlin 1965; Washizu 1975; Polizzotto 2001; dell’Isola et al. 2015a, 2016; dell’Isola and Placidi 2012). Indeed, the deformation energy density tells us the ways in which a body can store energy as well as the kinds of external actions that are sustainable by that body. Even if many authors, especially in review report and during discussions after seminars, following...

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 JJ, Della Corte A (2015) Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für angewandte Mathematik und Physik 66(5):2855–2870

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Andreaus U, dell’Isola F, Giorgio I, Placidi L, Lekszycki T, Rizzi NL (2016) Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity. Int J Eng Sci 108:34–50

    Article  MathSciNet  Google Scholar 

  • Boutin C, Soubestre J, Dietz MS, Taylor C (2013) Experimental evidence of the high-gradient behaviour of fiber reinforced materials. Eur J Mech-A/Solids 42:280–298

    Article  Google Scholar 

  • Bridgman P (1927) The logic of modern physics. Macmillan, New York

    MATH  Google Scholar 

  • Chang CS, Misra A (1990) Packing structure and mechanical properties of granulates. J Eng Mech 116(5):1077–1093

    Article  Google Scholar 

  • dell’Isola F, Placidi L (2012) Variational principles are a powerful tool also for formulating field theories. Springer, Vienna, pp 1–15. https://doi.org/10.1007/978-3-7091-0983-0_1

    MATH  Google Scholar 

  • dell’Isola F, Seppecher P, Madeo A (2012) How contact interactions may depend on the shape of Cauchy cuts in nth gradient continua: approach “á la D’Alembert”. Zeitschrift für angewandte Mathematik und Physik 63(6):1119–1141

    Article  MathSciNet  Google Scholar 

  • dell’Isola F, Seppecher P, Corte AD (2015a) The postulations á la d’alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc R Soc Lond A Math Phys Eng Sci 471(2183). https://doi.org/10.1098/rspa.2015.0415

    Article  Google Scholar 

  • dell’Isola F, Steigmann D, Della Corte A (2015b) Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl Mech Rev 67(6):060,804

    Article  Google Scholar 

  • dell’Isola F, Madeo A, Seppecher P (2016) Cauchy tetrahedron argument applied to higher contact interactions. Arch Ration Mech Anal 219(3):1305–1341. https://doi.org/10.1007/s00205-015-0922-6

    Article  MathSciNet  Google Scholar 

  • dell’Isola F, Cuomo M, Greco L, Della Corte A (2017) Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J Eng Math 103(1):127–157

    Article  MathSciNet  Google Scholar 

  • Eringen A (1968) Mechanics of micromorphic continua. In: Mechanics of generalized continua. Springer, New York, pp 18–35

    Chapter  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(3):513–554. https://doi.org/10.1007/s00205-006-0015-7

    Article  MathSciNet  Google Scholar 

  • Germain P (1973a) La méthode des puissances virtuelles en mécanique des milieux continus. J Mécanique 12:236–274

    MATH  Google Scholar 

  • Germain P (1973b) The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J Appl Math 25(3):556–575

    Article  Google Scholar 

  • Giorgio I, Della Corte A, dell’Isola F (2017) Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn 88(1):21–31

    Article  Google Scholar 

  • Gurtin ME (1965) Thermodynamics and the possibility of spatial interaction in elastic materials. Arch Ration Mech Anal 19(5):339–352

    Article  MathSciNet  Google Scholar 

  • Milton GW (1992) Composite materials with poisson’s ratios close to 1. J Mech Phys Solids 40(5):1105–1137

    Article  MathSciNet  Google Scholar 

  • Milton GW (2002) The theory of composites. In: Milton GW (ed) The theory of composites. Cambridge University Press, Cambridge, p 748. ISBN:0521781256

    Chapter  Google Scholar 

  • Milton GW, Kohn RV (1988) Variational bounds on the effective moduli of anisotropic composites. J Mech Phys Solids 36(6):597–629

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Misra A, Poorsolhjouy P (2016) Granular micromechanics based micromorphic model predicts frequency band gaps. Contin Mech Thermodyn 28(1–2):215

    Article  MathSciNet  Google Scholar 

  • Misra A, Poorsolhjouy P (2017) Grain-and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model. Mech Res Commun 81:1–6

    Article  Google Scholar 

  • Piola G (2014) The complete works of Gabrio Piola vol I: commented English translation (vol 38). (Translated and edited by dell’Isola F, Maier G, Perego U, Andreaus U, Esposito R and Forest S). Springer, Cham

    Google Scholar 

  • Polizzotto C (2001) Nonlocal elasticity and related variational principles. Int J Solids Struct 38(42):7359–7380

    Article  MathSciNet  Google Scholar 

  • Rahali Y, Giorgio I, Ganghoffer J, dell’Isola F (2015) Homogenization à la piola produces second gradient continuum models for linear pantographic lattices. Int J Eng Sci 97:148–172

    Article  MathSciNet  Google Scholar 

  • Rajagopal K (2011a) Conspectus of concepts of elasticity. Math Mech Solids 16(5):536–562. https://doi.org/10.1177/1081286510387856

    Article  MathSciNet  Google Scholar 

  • Rajagopal K (2011b) Non-linear elastic bodies exhibiting limiting small strain. Math Mech Solids 16(1):122–139

    Article  MathSciNet  Google Scholar 

  • Reiher JC, Giorgio I, Bertram A (2016) Finite-element analysis of polyhedra under point and line forces in second-strain gradient elasticity. J Eng Mech 143(2):04016,112

    Article  Google Scholar 

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

    Google Scholar 

  • Spagnuolo M, Barcz K, Pfaff A, dellIsola F, Franciosi P (2017) Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech Res Commun 83:47–52

    Article  Google Scholar 

  • Spivak M (1975) A comprehensive introduction to differential geometry, vol 1–2. Publish or Perish, Berkeley

    MATH  Google Scholar 

  • Turco E, dell’Isola F, Cazzani A, Rizzi NL (2016) Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4):1–28

    Google Scholar 

  • Washizu K (1975) Variational methods in elasticity and plasticity. Pergamon Press, Oxford

    MATH  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, University of L’Aquila, L’Aquila, Italy

    Francesco dell’Isola, Alessandro Della Corte & Antonio Battista

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

    Alessandro Della Corte

  4. DISIM, University of L’Aquila, L’Aquila, Italy

    Antonio Battista

Authors
  1. Francesco dell’Isola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessandro Della Corte
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Battista
    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. DISG, University of Rome La Sapienza, Rome, Italy

    Francesco dell’Isola

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

    Francesco dell’Isola

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., Corte, A.D., Battista, A. (2018). Generalized Contact Actions. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_229-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53605-6_229-1

  • 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