Plasticity of Pressure-Sensitive Materials pp 1-47

Part of the Engineering Materials book series (ENG.MAT.) | Cite as

Basic Equations of Continuum Mechanics



The modeling of the behavior of pressure-sensitive materials is embedded in the general continuum mechanics. The basic equations of continuum mechanics can be split into the material-independent and the material-dependent equations. The starting point is the introduction of the kinematics based on pure mathematical considerations. In addition, the velocities and the accelerations of the relevant kinematical variables are presented. The next section is devoted to the introduction of the action on the continuum and the inner reaction. Starting with such properties like forces and stresses finally the static equilibrium is stated. The last part of the material-independent equations is the introduction of the balances. Limiting our discussions by thermo-mechanical actions only, the balance of mass, momentum, moment of momentum, energy and entropy are deduced. The specific properties and features of the pressure-sensitive materials are presented in the next sections. Within this chapter the general ideas of material modeling (deductive approach) are given. Finally, some examples of special constitutive equations for incompressible and compressible materials are presented. These examples are mostly related to rubber-like materials.


  1. 1.
    Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  2. 2.
    Altenbach, H. (ed.): Holzmann Meyer Schumpich Technische Mechanik Festigkeitslehre, 10th edn. Springer Vieweg, Stuttgart (2012)Google Scholar
  3. 3.
    Gross, D., Hauger, W., Schröder, J., Wall, W.A.: Technische Mechanik, vol. 2. Elastostatik, 9th edn. Springer, Berlin (2008)Google Scholar
  4. 4.
    Eremeyev, V., Lebedev, L., Altenbach, H.: Foundations of Micropolar Mechanics. Springer-Briefs in Applied Sciences and Technologies. Springer, Heidelberg (2013)Google Scholar
  5. 5.
    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group of the non-linear polar-elastic continuum. J. Solids Struct. 49(14), 1993–2005 (2012)CrossRefGoogle Scholar
  6. 6.
    Eringen, A.C.: Microcontinuum Field Theory, Foundations and Solids, vol. 2. Springer, New York (1999a)Google Scholar
  7. 7.
    Eringen, A.C.: Microcontinuum Field Theory, Fluent Media, vol. 2. Springer, New York (1999b)Google Scholar
  8. 8.
    Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46(3–4), 774–787 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Müller I (1973) Thermodynamik: die Grundlagen der Materialtheorie. Bertelsmann UniversitätsverlagGoogle Scholar
  10. 10.
    Palmov, V.A.: Vibrations of Elasto-plastic Bodies. Foundations of Engineering Mechanics. Springer, Berlin (1998)CrossRefGoogle Scholar
  11. 11.
    Salençon, J.: Handbbok of Continuum Mechanics. Springer, Berlin (2001)Google Scholar
  12. 12.
    Willner, K.: Kontinuums—und Kontaktmechanik: Synthetische und analytische Darstellung. Springer, Berlin (2003)CrossRefGoogle Scholar
  13. 13.
    Altenbach, H., Eremeyev, V. (eds.): Generalized Continua— from the Theory to Engineering Applications, CISM International Centre for Mechanical Sciences, vol. 541. Springer, Wien (2013)Google Scholar
  14. 14.
    Altenbach, H., Naumenko, K., Zhilin, P.: A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Continuum Mech Thermodyn 15, 539–570 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Altenbach, H., Maugin, G.A., Erofeev, V. (eds.): Mechanics of Generalized Continua, Advanced Structured Materials, vol. 7. Springer, Heidelberg (2011)Google Scholar
  16. 16.
    Altenbach, H., Forest, S., Krivtsov, A. (eds.): Generalized Continua as Models for Materials with Multi-scale Effects or Under Multi-field Actions, Advanced Structured Materials, vol. 22. Springer, Heidelberg (2013)Google Scholar
  17. 17.
    Maugin, G.A.: Continuum Mechanics Through the Twentieth Century. Springer, Heidelberg (2013)CrossRefMATHGoogle Scholar
  18. 18.
    Maugin, G.A., Metrikine, A. (eds.): Mechanics of Generalized Continua - One Hundred Years After the Cosserats, Advances in Mechanics and Mathematics 21. Springer, Berlin (2010)Google Scholar
  19. 19.
    Rubin, M.B.: Cosserat Theories: Shells Rods and Points. Kluwer, Dordrecht (2000)CrossRefMATHGoogle Scholar
  20. 20.
    Krawietz, A.: Materialtheorie. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  21. 21.
    Noll, W.: The Foundations of Mechanics and Thermodynamics. Springer, Berlin (1974)CrossRefMATHGoogle Scholar
  22. 22.
    Bertram, A.: What is the general constitutive equation? In: Beiträge Festschrift zum 65 . Geburtstag von Rudolf Trostel, TU Berlin. Berlin, pp. 28–37 (1994)Google Scholar
  23. 23.
    Giesekus, H.: Phänomenologische Rheologie : eine Einführung. Springer, Berlin (1994)CrossRefMATHGoogle Scholar
  24. 24.
    Reiner, M.: Rheologie. Fachbuchverlag, Leipzig (1968)Google Scholar
  25. 25.
    Truesdell, C.: A First Course in Rational Continuum Mechanics, Pure and Applied Mathematics, vol. 1. Academic Press, New York (1977)Google Scholar
  26. 26.
    Bertram, A., Forest, S.: The thermodynamics of gradient elastoplasticity. Continuum Mech. Thermodyn. 25, 20 (2013)Google Scholar
  27. 27.
    dell’Isola F, Sciarra G, Vidoli S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R Soc. A 495, 2177–2196 (2009)Google Scholar
  28. 28.
    Forest, S., Trinh, D.K.: Generalized continua and non-homogeneous boundary conditions in homogenisation methods. ZAMM 91(2), 90–109 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Podio-Guidugli, P., Vianello, M.: Hypertractions and hyperstresses convey the same mechanical information. Continuum Mech. Thermodyn. 22, 163–176 (2010)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Backhaus, G.: Zum evolutionsgesetz der kinematischen verfestigung in objektiver darstellung. ZAMM 72(9), 397–406 (1992)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lemaitre, J., Chaboche, J.L.: Mechanics of Solid Materials. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  32. 32.
    Lurie, A.I.: Nonliner Theory of Elasticity. North-Holland, Amsterdam (1990)Google Scholar
  33. 33.
    Ogden, R.: Non-Linear Elastic Deformations. Ellis Horwood, Chichester (1984)Google Scholar
  34. 34.
    Treloar, L.: The Physics of Rubber Elasticity, 3rd edn. Oxford University Press, Oxford (1975)Google Scholar
  35. 35.
    Mooney, M.: A theory of large elastic deformations. J. Appl. Phys. 11, 582–592 (1940)CrossRefMATHGoogle Scholar
  36. 36.
    Rivlin, R.S., Saunders, D.W.: Large elastic deformations of isotropic materials. Phil. Trans. Roy. Soc. A 243, 251–288 (1951)CrossRefMATHGoogle Scholar
  37. 37.
    Biderman V.L.: On the calculation of rubber-like materials (in Russ.). In: Strengths Analysis (in Russ.), 3. Mashgiz, pp. 40–87 (1958)Google Scholar
  38. 38.
    Hart-Smith, L.J., Crisp, J.D.C.: Large elastic deformations in thin rubber membranes. Int. J. Eng. Sci. 5(1), 1–24 (1967)CrossRefMATHGoogle Scholar
  39. 39.
    Alexander, H.: A constitutive relation for rubber-like materials. Int. J. Eng. Sci. 6(9), 549–563 (1968)CrossRefGoogle Scholar
  40. 40.
    Hutchinson, W.D., Becker, G.W., Landel, R.F. Determination of the stored energy function of rubber-like materials. In: Bull 4th Meeting Interagency Chemical Rocket Propulsion Group—Working Group Mechanical Behavior, vol 1. No. 94U, pp 141–152. CPIAGoogle Scholar
  41. 41.
    Ogden, R.W.: Large deformation isotropic elasticity I: on the correlation of theory and experiment for incompressible rubberlike solids. Proc. Roy. Soc. London A 326, 565–584 (1972)CrossRefMATHGoogle Scholar
  42. 42.
    Chernykh, K.F., Shubina, I.M.: The laws of elasticity for isotropic compressible materials (in Russ.). In: Mekhanika Elastomerov. Krasnodar, vol I, 54–65 (1977)Google Scholar
  43. 43.
    FCK (1986) Nonlinear Theory of Elasticity in Engineering Calculations (in Russ.). Mashinostroenie, MoscowGoogle Scholar
  44. 44.
    Bartenev G.M, Khazanovich T.N: On a constitutive law for hyperelastic strans of cross-linked polymer (in russ.). Polym. Sci. 2(1), 20–28 (1960)Google Scholar
  45. 45.
    Fu Y.B, Ogden R.W (eds.): Nonlinear elasticity. Theory and applications. No. 283 in Lecture Note Series. Cambridge University Press, Cambridge (2001)Google Scholar
  46. 46.
    Arruda, E., Boyce, M.: A three-dimensional constitutive model the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41(2), 389–412 (1993)CrossRefGoogle Scholar
  47. 47.
    Blatz, P.J., Ko, W.L.: Application of finite elasticity theory to the deformation of rubbery materials. Tran. Soc. Rheol. 6, 223–251 (1962)CrossRefGoogle Scholar
  48. 48.
    Signorini, A.: Solidi incompribili. Ann. Mat. Pura Appl. 39, 147–201 (1955)Google Scholar
  49. 49.
    Murnaghan, F.D.: Finite Deformation of an Elastic Solid. Wiley, New York (1951)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für MaschinenbauOtto-von-Guericke-Universtät MagdeburgMagdeburgGermany
  2. 2.South Scientific Center of RASci and South Federal UniversityRostov on DonRussia

Personalised recommendations