Continuum Mechanics and Thermodynamics

, Volume 30, Issue 4, pp 689–708 | Cite as

Visualising elastic anisotropy: theoretical background and computational implementation

  • J. NordmannEmail author
  • M. Aßmus
  • H. Altenbach
Review Article


In this article, we present the technical realisation for visualisations of characteristic parameters of the fourth-order elasticity tensor, which is classified by three-dimensional symmetry groups. Hereby, expressions for spatial representations of Young’s modulus and bulk modulus as well as plane representations of shear modulus and Poisson’s ratio are derived and transferred into a comprehensible form to computer algebra systems. Additionally, we present approaches for spatial representations of both latter parameters. These three- and two-dimensional representations are implemented into the software MATrix LABoratory. Exemplary representations of characteristic materials complete the present treatise.


Linear elasticity Anisotropy Visualisation MATLAB 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Weng, Y., Dong, H., Dong, H., Gan, Y. (eds.): Advanced Steels. Springer, Berlin (2011). Google Scholar
  2. 2.
    Altenbach, H., Altenbach, J., Kissing, W.: Mechanics of Composite Structural Elements. Springer, Berlin (2004). CrossRefGoogle Scholar
  3. 3.
    Langston, L.: Each blade a single crystal. Am. Sci. 103(1), 30 (2015). CrossRefGoogle Scholar
  4. 4.
    Altenbach, H.: Kontinuumsmechanik: Einführung in die materialunabhängigen und materialabhängigen Gleichungen, 3rd edn. Springer, Berlin (2015). CrossRefzbMATHGoogle Scholar
  5. 5.
    Lai, W., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics, 4th edn. Butterworth-Heinemann, Oxford (2009)zbMATHGoogle Scholar
  6. 6.
    Skrzypek, J.J., Ganczarski, A.W. (eds.): Mechanics of Anisotropic Materials. Springer, Berlin (2015). zbMATHGoogle Scholar
  7. 7.
    Bertram, A.: Elasticity and Plasticity of Large Deformations: An Introduction, 3rd edn. Springer, Berlin (2012). CrossRefzbMATHGoogle Scholar
  8. 8.
    Nye, J.F.: Physical Properties of Crystals. Oxford University Press, Oxford (1985)zbMATHGoogle Scholar
  9. 9.
    Voigt, W.: Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Vieweg+Teubner Verlag, Berlin (1928). zbMATHGoogle Scholar
  10. 10.
    Fedorov, F.I.: Theory of Elastic Waves in Crystals. Plenum Press, New York (1968). CrossRefGoogle Scholar
  11. 11.
    Cowin, S.C.: Properties of the anisotropic elasticity tensor. Quart. J. Mech. Appl. Math. 42(2), 249–266 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coleman, B.D., Noll, W.: Material symmetry and thermostatic inequalities in finite elastic deformations. Arch. Ration. Mech. Anal. 15(2), 87–111 (1964). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ting, T.C.T.: Generalized Cowin–Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight. Int. J. Solids Struct. 40(25), 7129–7142 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Xiao, H., Bruhns, O.T., Meyers, A.: Existence and uniqueness of the integrable-exactly hypoelastic equation \(\overset{\circ }{\varvec {\tau }}^{\star }=\lambda ({{\rm tr}}\varvec {D})\varvec {I} + 2\mu \varvec {D}\) and its significance to finite inelasticity. Acta Mech. 138(1), 31–50 (1999). CrossRefGoogle Scholar
  15. 15.
    Weyl, H.: The Classical Groups: Their Invariants and Representations, 2nd edn. Princeton University Press, Princeton (1997). 15. print and 1. paperback printzbMATHGoogle Scholar
  16. 16.
    Ting, T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, Oxford (1996)zbMATHGoogle Scholar
  17. 17.
    Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005). CrossRefGoogle Scholar
  18. 18.
    Böhlke, T., Brüggemann, C.: Graphical representation of the generalized Hooke‘s law, Technische Mechanik 21(2):145–158 (2001). Accessed 7 Dec 2017
  19. 19.
    Rychlewski, J.: Unconventional approach to linear elasticity. Arch. Mech. 47(2), 149–171 (1995)MathSciNetzbMATHGoogle Scholar
  20. 20.
    He, Q.-C., Curnier, A.: A more fundamental approach to damaged elastic stress–strain relations. Int. J. Solids Struct. 32(10), 1433–1457 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Reuss, A.: Berechnung der Fliegrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. J. Appl. Math. Mech. 9(1), 49–58 (1929). zbMATHGoogle Scholar
  22. 22.
    Hill, R.: The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. Sect. A 65(5), 349 (1952). ADSCrossRefGoogle Scholar
  23. 23.
    Mehrabadi, M.M., Cowin, S.C.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math. 43(1), 15–41 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mandel, J.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math. 1, 3–30 (1962)Google Scholar
  25. 25.
    Thomson, W.: Mathematical theory of elasticity: elasticity. Encycl. Br. 7, 819–825 (1878)Google Scholar
  26. 26.
    Sutcliffe, S.: Spectral decomposition of the elasticity tensor. J. Appl. Mech. 59(4), 762–773 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Golub, G.H., Loan, C.F.V.: Matrix Computations (Johns Hopkins Studies in Mathematical Sciences), 3rd edn. Johns Hopkins University Press, Baltimore (1996)Google Scholar
  28. 28.
    Marmier, A., Lethbridge, Z.A., Walton, R.I., Smith, C.W., Parker, S.C., Evans, K.E.: ElAM: a computer program for the analysis and representation of anisotropic elastic properties. Comput. Phys. Commun. 181(12), 2102–2115 (2010). ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Gaillac, R., Pullumbi, P., Coudert, F.-X.: ELATE: an open-source online application for analysis and visualization of elastic tensors. J. Phys. Condens. Matter 28(27), 275201 (2016). CrossRefGoogle Scholar
  30. 30.
    dell’Isola, F., Sciarra, G., Vidoli, S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2107), 2177–2196 (2009). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Auffray, N., Quang, H.L., He, Q.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61(5), 1202–1223 (2013). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Aßmus, M., Nordmann, J., Naumenko, K., Altenbach, H.: A homogeneous substitute material for the core layer of photovoltaic composite structures. Compos. Part B Eng. 112, 353–372 (2017). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Engineering Mechanics, Institute of Mechanics, Faculty of Mechanical EngineeringOtto von Guericke University MagdeburgMagdeburgGermany

Personalised recommendations