Abstract
Material theory in a strict sense is not older than just half a century and from the beginning characterized by fundamental disputes and changes of paradigms. A number of different suggestions have been made and - often - forgotten soon after their publication. In this article the main existing theories shall be briefly described and critically discussed. Some important demands for an adequate theory are raised and applied to these suggestions.
Originally, two totally different lines of approaches existed, namely that of history functionals and that of inner variables or state theories. Since both approaches suffer from fundamental deficiencies, neither of them really achieved global acceptance. And for a long time it remained unclear how the two approaches could be mutually related.
This was changed in 1972 by Noll’s New Theory of Simple Materials [19], in which a third approach was suggested that made it possible to compare the two preceding formats. The gain in generality was, however, accompanied by a loss of simplicity. Consequently, this theory has been used only by very few groups within the scientific community.
Since then the majority of the papers in the field of material modeling are pragmatic and do not claim for general validity. However, there remains a clear need for a general theory, as we will finally demonstrate within the context of thermodynamics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bertram, A.: Material Systems a framework for the description of material behavior. Arch. Rat. Mech. Anal. 80(2), 99–133 (1982)
Bertram, A.: Axiomatische Einführung in die Kontinuumsmechanik. BI Wissenschaftsverlag. Mannheim, Wien, Zürich (1989)
Bridgman, P.W.: The Thermodynamics of plastic deformation and generalized entropy. Rev. Modern Physics 22(1), 56–63 (1950)
Casey, J., Naghdi, P.M.: A prescription for the identification of finite plastic strain. Int. J. Engng. Sci. 30(10), 1257–1278 (1992)
Coleman, B.D., Gurtin, M.E.: Thermodynamics with internal state variables. J. Chemical Physics 47, 597 (1967)
Coleman, B.D.: Thermodynamics of materials with memory. Archive Rat. Mech. Anal. 17(1), 1–46 (1964)
Coleman, B.D., Owen, D.: On the Thermodynamics of materials with memory. Archive Rat. Mech. Anal. 36(4), 245–269 (1970)
Eckart, C.: The thermodynamics of irreversible processes. II. Fluid Mixtures. Physical Rev. 58, 269–275 (1940)
Eckart, C.: The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity. Physical Rev. 73(4), 373–382 (1948)
Fabrizio, M., Lazzari, B.: On the notion of state for a material system. Meccanica 14(4), 175–180 (1981)
Frischmuth, K., Kosinski, W., Perzyna, P.: Remarks on mathematical theory of materials. Arch. Mech. 38(1-2), 59–69 (1986)
Green, A.E., Naghdi, P.M.: A general theory of an elastic-plastic continuum. Arch. Rational Mech. Anal. 18(4), 251–281 (1965)
Green, A.E., Rivlin, R.S.: The mechanics of non-linear materials with memory. Archive Rat. Mech. Anal. 1(1), 1–21 (1957)
Haupt, P.: Viskoelastizität und Plastizität. Thermomechanisch konsistente Materialgleichungen. Springer, Berlin (1977)
Horstemeyer, M.F., Bammann, D.J.: Historical review of internal state variable theory of inelasticity. I. J. Plasticity 26, 1310–1334 (2010)
Lee, E.H., Liu, D.T.: Finite-strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38(1), 19–27 (1967)
Mandel, J.: Plasticité classique et viscoplasticité. CISM course No. 97. Springer, Wien (1971)
Noll, W.: A mathematical theory of the mechanical behavior of continuous media. Archive Rat. Mech. Anal. 2, 197–226 (1958)
Noll, W.: A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48, 1–50 (1972)
Maugin, G.A., Muschik, W.: Thermodynamics with Internal Variables, part I and II. J. Non-Equilib. Thermodyn. 19, 217–289 (1994)
Muschik, W., Papenfuss, C., Ehrentraut, H.: A sketch of continuum thermodynamics. J. Non-Newtonian Fluid Mech. 96, 255–290 (2001)
Onat, E.T.: The notion of state and its implications in thermodynamics of inelastic solids. In: Parkus, Sedov (eds.)IUTAM 1966. Irreversible Aspects of Continuum Mechanics, pp. 292–313. Springer, Wien (1968)
Onat, E.T.: Representation of inelastic mechanical behavior by means of state variables. In: Boley (ed.) IUTAM 1968, pp. 213–225. Springer, Berlin (1970)
Perzyna, P., Kosiński, W.: A mathematical theory of materials. Bull. Acad. Polon. Sci., Ser. Sci. Techn. 21(12), 647–654 (1973)
Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997)
Truesdell, C.A., Noll, W.: The non-linear field theories of mechanics. The Encyclopedia of Physics III(3) (1965)
Valanis, K.C.: A theory of viscoplasticity without a yield surface, part I and II. Archive of Mechanics 23, 517–533 (1971)
Wang, C.C.: Stress relaxation and the principle of fading memory. Arch. Rat. Mech. Anal. 18, 117–126 (1965)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bertram, A. (2014). On the History of Material Theory – A Critical Review. In: Stein, E. (eds) The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering. Lecture Notes in Applied Mathematics and Mechanics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39905-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-39905-3_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39904-6
Online ISBN: 978-3-642-39905-3
eBook Packages: EngineeringEngineering (R0)