A Note on Reduced Strain Gradient Elasticity

  • Victor A. EremeyevEmail author
  • Francesco dell’Isola
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


We discuss the particular class of strain-gradient elastic material models which we called the reduced or degenerated strain-gradient elasticity. For this class the strain energy density depends on functions which have different differential properties in different spatial directions. As an example of such media we consider the continual models of pantographic beam lattices and smectic and columnar liquid crystals.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ablowitz MA, Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society lecture note series, vol 149. Cambridge University Press, CambridgeGoogle Scholar
  2. Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaGoogle Scholar
  3. Aifantis EC (1992) On the role of gradients in the localization of deformation and fracture. Int J Engng Sci 30(10):1279–1299Google Scholar
  4. Aifantis EC (2003) Update on a class of gradient theories. Mech Materials 35(3):259–280Google Scholar
  5. Aifantis EC (2014) Gradient material mechanics: perspectives and prospects. Acta Mech 225(4-5):999–1012Google Scholar
  6. Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48(13):1962–1990Google Scholar
  7. Askes H, Gitman I (2017) Reducible and irreducible forms of stabilised gradient elasticity in dynamics. Math Mech Complex Systems 5(1):1–17Google Scholar
  8. Bertram A (2016) Compendium on Gradient Materials . OvGU, MagdeburgGoogle Scholar
  9. Bertram A, Glüge R (2016) Gradient materials with internal constraints. Math Mech Complex Systems 4(1):1–15Google Scholar
  10. Boutin C, dell’Isola F, Giorgio I, Placidi L (2017) Linear pantographic sheets: Asymptotic micromacro models identification. Math Mech Complex Systems 5(2):127–162Google Scholar
  11. Chandrasekhar S (1977) Liquid Crystals. Cambridge University Press, Cambridge, UKGoogle Scholar
  12. Chatzigeorgiou G, Meraghni F, Javili A (2017) Generalized interfacial energy and size effects in composites. J Mech Phys Solids 106:257–282Google Scholar
  13. Cordero NM, Forest S, Busso EP (2016) Second strain gradient elasticity of nano-objects. J Mech Phys Solids 97:92–124Google Scholar
  14. d’Agostino MV, Giorgio I, Greco L, Madeo A, Boisse P (2015) Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements. Int J Solids Struct 59:1–17Google Scholar
  15. dell’Isola F, Steigmann D (2015) A two-dimensional gradient-elasticity theory for woven fabrics. J Elast 118(1):113–125Google Scholar
  16. dell’Isola F, Giorgio I, Pawlikowski M, Rizzi N (2016a) Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium. Proc Roy Soc London A 472(2185):20150,790Google Scholar
  17. dell’Isola F, Steigmann D, della Corte A (2016b) Synthesis of fibrous complex structures: Designing microstructure to deliver targeted macroscale response. Appl Mech Rev 67(6):060,804–060,804–21Google Scholar
  18. dell’Isola F, Della Corte A, Giorgio I (2017) Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math Mech Solids 22(4):852–872Google Scholar
  19. Eastham JF, Peterson JS (2004) The finite element method in anisotropic Sobolev spaces. Computers & Mathematics with Applications 47(10):1775–1786Google Scholar
  20. Engelbrecht J, Berezovski A (2015) Reflections on mathematical models of deformation waves in elastic microstructured solids. Math Mech Complex Systems 3(1):43–82Google Scholar
  21. Eremeyev VA, Pietraszkiewicz W (2006) Local symmetry group in the general theory of elastic shells. J Elast 85(2):125–152Google Scholar
  22. Eremeyev VA, Pietraszkiewicz W (2012) Material symmetry group of the non-linear polar-elastic continuum. Int J Solids Struct 49(14):1993–2005Google Scholar
  23. Eremeyev VA, Pietraszkiewicz W (2016) Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math Mech Solids 21(2):210–221Google Scholar
  24. Eremeyev VA, dell’Isola F, Boutin C, Steigmann D (2017) Linear pantographic sheets: existence and uniqueness of weak solutions. J Elast
  25. Forest S, Cordero N, Busso EP (2011) First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput Materials Sci 50(4):1299–1304Google Scholar
  26. de Gennes G P, Prost J (1993) The Physics of Liquid Crystals, 2nd edn. Clarendon Press, OxfordGoogle Scholar
  27. Giorgio I, Rizzi N, Turco E (2017) Continuum modelling of pantographic sheets for outof- plane bifurcation and vibrational analysis. Proc Roy Soc A 473(2207):21 pages
  28. Grimmett G (2016) Correlation inequalities for the Potts model. Math Mech Complex Systems 4(3):327–334Google Scholar
  29. Harrison P (2016) Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Composites A 81:145–157Google Scholar
  30. Healey TJ, Krömer S (2009) Injective weak solutions in second-gradient nonlinear elasticity. ESAIM: Control, Optimisation and Calculus of Variations 15(4):863–871Google Scholar
  31. Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersing media. Sov Phys Doklady 15(6):539–541Google Scholar
  32. Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor Analysis with Applications in Mechanics. World Scientific, New JerseyGoogle Scholar
  33. Mareno A, Healey TJ (2006) Global continuation in second-gradient nonlinear elasticity. SIAM J Math Analysis 38(1):103–115Google Scholar
  34. de Masi A, Merola I, Presutti E, Vignaud Y (2008) Potts models in the continuum. uniqueness and exponential decay in the restricted ensembles. J Stat Phys 133(2):281–345Google Scholar
  35. de Masi A, Merola I, Presutti E, Vignaud Y (2009) Coexistence of ordered and disordered phases in Potts models in the continuum. J Stat Phys 134(2):243–306Google Scholar
  36. Maugin GA (1999) Nonlinear Waves in Elastic Crystals. Oxford University Press, OxfordGoogle Scholar
  37. Maugin GA (2010) Generalized continuum mechanics: what do we mean by that? In: Maugin GA, Metrikine AV (eds) Mechanics of Generalized Continua. One Hundred Years after the Cosserats, Springer, pp 3–13Google Scholar
  38. Maugin GA (2011) A historical perspective of generalized continuum mechanics. In: Altenbach H, Erofeev VI, Maugin GA (eds) Mechanics of Generalized Continua. From the Micromechanical Basics to Engineering Applications, Springer, Berlin, pp 3–19Google Scholar
  39. Maugin GA (2013) Generalized Continuum Mechanics: Various Paths, Springer, Dordrecht, pp 223–241Google Scholar
  40. Maugin GA (2016) Continuum Mechanics Through Ages. From the Renaissance to the Twentieth Century. Springer, ChamGoogle Scholar
  41. Maugin GA (2017) Non-Classical Continuum Mechanics: A Dictionary. Springer, SingaporeGoogle Scholar
  42. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Analysis 16(1):51–78Google Scholar
  43. Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124Google Scholar
  44. Misra A, Chang CS (1993) Effective elastic moduli of heterogeneous granular solids. Int J Solids Struct 30:2547–2566Google Scholar
  45. Oswald P, Pieranski P (2006) Smectic and Columnar Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments. The Liquid Crystals Book Series (eds GW Gray, JW Goodby, and A Fukuda), Taylor & Francis, Boca RatonGoogle Scholar
  46. Placidi L, Barchiesi E, Turco E, Rizzi NL (2016) A review on 2D models for the description of pantographic fabrics. ZAMP 67(5):121Google Scholar
  47. Placidi L, Andreaus U, Giorgio I (2017) Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J Engng Math 103(1):1–21Google Scholar
  48. Pouget J (2005) Non-linear lattice models: complex dynamics, pattern formation and aspects of chaos. Phil Magazine 85(33–35):4067–4094Google Scholar
  49. Rahali Y, Giorgio I, Ganghoffer JF, dell’Isola F (2015) Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int J Engng Sci 97:148–172Google Scholar
  50. Simmonds JG (1994) A Brief on Tensor Analysis, 2nd edn. Springer, New YourkGoogle Scholar
  51. Soubestre J, Boutin C (2012) Non-local dynamic behavior of linear fiber reinforced materials. Mech Materials 55:16–32Google Scholar
  52. Timoshenko SP, Woinowsky-Krieger S (1985) Theory of Plates and Shells. McGraw Hill, New YorkGoogle Scholar
  53. Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Analysis 11(1):385–414Google Scholar
  54. Wood HG, Morton JB (1980) Onsager’s pancake approximation for the fluid dynamics of a gas centrifuge. J Fluid Mech 101(1):1–31Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Gdańsk University of TechnologyGdańskPoland
  2. 2.International Research Center on Mathematics and Mechanics of Complex System (M&MOCS)Università di Roma “La Sapienza”, Universitá degli Studi dell’AquilaL’AquilaItaly

Personalised recommendations