Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Frameworks for Material Modeling

  • Rainer GlügeEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_263-1



Frameworks for material modeling set the limits within which different material phenomena like elasticity, plasticity, and viscosity are modeled.

Classification of Material Models

After establishing a universal framework for material modeling based on general principles, a classification of materials with respect to the material phenomena that occur is reasonable.

Any textbook on material modeling somehow classifies material models, however the works of Noll (1972) and Krawietz (1986) provide the most systematic approaches. Krawietz (1986) regards phenomenological material modeling by defining process classes and output functionals. The output functional is the material model, which assigns a dependent variable (stresses and forces) to the independent, process-controlled variable (strains and displacements).

We have already established the use of a finite-dimensional material state variable Z (see section “ Principles...
This is a preview of subscription content, log in to check access.


  1. Abeyaratne R, Knowles J (2006) Evolution of phase transitions – a continuum theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  2. Baker M, Ericksen J (1954) Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids. J Wash Acad Sci 44:33–35MathSciNetGoogle Scholar
  3. Ball J (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 63:337–403MathSciNetCrossRefGoogle Scholar
  4. Bertram A (2012) Elasticity and plasticity of large deformations – an introduction, 3rd edn. Springer, BerlinCrossRefGoogle Scholar
  5. Bertram A, Böhlke T, Šilhavý M (2007) On the rank 1 convexity of stored energy functions of physically linear stress-strain relations. J Elast 86:235–243MathSciNetCrossRefGoogle Scholar
  6. Bona A, Bucataru I, Slawinski M (2004) Material symmetry of elasticity tensors. Q J Mech Appl Math 57:583–598CrossRefGoogle Scholar
  7. Ciarlet P (1988) Mathematical elasticity volume 1: three-dimensional elasticity. Studies in mathematics and its applications, vol 20. Elsevier Science Publishers B.V, AmsterdamGoogle Scholar
  8. Dunne F, Petrinic N (2009) Introduction to computational plasticity. Oxford University Press, OxfordzbMATHGoogle Scholar
  9. Forte S, Vianello M (1996) Symmetry classes for elasticity tensors. J Elast 43(2):81–108MathSciNetCrossRefGoogle Scholar
  10. Hosford WF (1972) A generalized isotropic yield criterion. J Appl Mech 39:607–609CrossRefGoogle Scholar
  11. Krawietz A (1986) Materialtheorie. Springer, BerlinCrossRefGoogle Scholar
  12. Landau L, Lifshitz E, Kosevich A, Pitaevskii L (1986) Theory of elasticity. Course of theoretical physics, vol 7. Butterworth-Heinemann, OxfordGoogle Scholar
  13. Morrey C (1952) Quasi-convexity and the lower semicontinuity of multiple integrals. Pac J Math 2(1):25–53MathSciNetCrossRefGoogle Scholar
  14. Noll W (1972) A new mathematical theory of simple materials. Arch Ration Mech Anal 48:1–50MathSciNetCrossRefGoogle Scholar
  15. Reiner M (1945) A mathematical theory of dilatancy. Am J Math 67(3):350–362MathSciNetCrossRefGoogle Scholar
  16. Roters F, Eisenlohr P, Hantcherli L, Tjahjanto D, Bieler T, Raabe D (2010) Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications. Acta Mater 58(4):1152–1211CrossRefGoogle Scholar
  17. Saccomandi G, Ogden R (2004) Mechanics and thermomechanics of rubberlike solids. CISM international centre for mechanical sciences. Springer, ViennaCrossRefGoogle Scholar
  18. Schröder J (2010) Anisotropic polyconvex energies. In: Schröder J, Neff P (eds) Poly-, quasi- and rank-one convexity in applied mechanics. CISM international centre for mechanical sciences, vol 516. Springer, Vienna, pp 53–105CrossRefGoogle Scholar
  19. Simo J, Hughes T (1998) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  20. Taber L (2004) Nonlinear theory of elasticity: applications in biomechanics. World Scientific, SingaporeCrossRefGoogle Scholar
  21. Ting T (1996) Anisotropic elasticity: theory and applications. Oxford University Press, New YorkzbMATHGoogle Scholar
  22. Truesdell CA, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Encyclopedia of physics, vol III/3. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universität MagdeburgMagdeburgGermany

Section editors and affiliations

  • Rainer Glüge
    • 1
  1. 1.Universität MagdeburgMagdeburgGermany