Abstract
In solid mechanics representing scalar-valued tensor functions or second-order tensor-valued tensor functions is of major concern. For instance, the plastic potential is scalar-valued, whereas constitutive equations are tensor-valued.
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.
References
GRACE, J.H. and YOUNG, A., The Algebra of Invariants, Cambridge Univ. Press. London and New York 1903.
GUREVICH, G.B., Foundations of the Theory of Algebraic Invariants, P. Noordhoff, Groningen 1964.
RIVLIN, R.S., An Introduction to Non-linear Continuum Mechanics, in: Non-linear Continuum Theories in Mechanics and Physics and their applications (Ed. R.S. Rivlin ), Edizioni Cremonese, Rome 1970.
SCHUR, I., Vorlesungen über Invariantentheorie, bearbeitet und herausgegeben von H. Grunsky, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 143, Springer-Verlag, Berlin/Heidelberg/New York 1968.
WEITZENBÖCK, R., Invariantentheorie, P. Noordhoff, Groningen 1923.
WEYL, H., The Classical Groups, Their Invariants and Representation, Princeton Univ. Press, Princeton and New Jersey 1946.
SPENCER, A.J.M., Theory of Invariants, in: Continuum Physics (ed. A.C. Eringen ), Academic Press, New York 1971, 239–353.
TRUESDELL, C. and NOLL, W., The Non-Linear Field Theories of Mechanics, in: Handbuch der Physik (Ed. S. Flügge), Vol. III/3, Springer-Verlag, Berlin/Heidelberg/New York 1965.
BETTEN; J., Ein Beitrag zur Invariantentheorie in der Plastomechanik anisotroper Stoffe, Z. Angew. Math. Mech. (ZAMM), 56 (1976), 557–559
BETTEN, J., Theory of Invariants in Creep Mechanics of Anisotropic Solids (Ed. J.P. Boehler ), Martinus Nijhoff Publishers, The Hague/ Boston/London 1982, 65–80.
BETTEN, J., Creep Theory of Anisotropic Solids, Journal of Rheology, 25 (1981), 565–581.
SPENCER, A.J.M. and RIVLIN, R.S., The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua, Arch. Rational Mech. Anal. 2 (1958/59), 309–336.
SMITH, G.F., On the Yield Condition for Anisotropie Materials, Quart. Appl. Math. 20 (1962), 241–247.
PIPKIN, A.C. and RIVLIN, R.S., The Formulation of Constitutive Equations in Continuum Physics, Arch. Rational Mech. Anal. 4 (1959), 129–144.
PIPKIN, A.C. and WINEMAN, A.S., Material Symmetry Restrictions on Non-Polynomial Constitutive Equations, Arch. Rational Mech. Anal. 12 (1963), 420–426.
BETTEN, J., Elementare Tensorrechnung für Ingenieure, Vieweg-Verlag, Braunschweig 1977.
WINEMAN, A.S. and PIPKIN, A.C., Material Symmetry Restrictions on Constitutive Equations, Arch. Rational Mech. Anal. 17 (1964), 1842 14
SMITH, G.F., SMITH, M.M. and RIVLIN, R.S., Integrity Bases for a Symmetric Tensor and a Vector–The Crystal Classes, Arch. Rational Mech. Anal. 12 (1963), 93–133.
BETTEN, J., Integrity Basis for a Second-Order and a Fourth-Order Tensor, Internat. J. Math. & Math. Sci. 5 (1982), 87–96.
RIVLIN, R.S. and SMITH, G.F., Orthogonal Integrity Basis for N Symmetric Matrices, in: Contributions to Mechanics (Ed. D. Abir ), Pergamon Press, Oxford/…/Braunschweig 1969, 121–141.
BETTEN, J., Zur Anstellung von Stoffgleichungen in der Kriechmechanik anisotroper Körper, Rheol. Acta 20 (1981), 527–535.
BETTEN, J., Elastizitäts-und Plastizitätslehre, Vieweg-Verlag, Braunschweig/Wiesbaden 1985.
BETTEN, Berkeley 1985, to be published in the proceedings: Mathematical Modelling in Science and Technology (eds. X.J.R. Avula, G. Leitmann, C.D. Mote, Jr. and E.Y. Rodin), Pergamon Press, New York 1986.
BETTEN, J., Representation of Constitutive Equations in Creep Mechanics of Isotropic and Anisotropic Materials, in: Creep in Structures (Ed. A.R.S. Ponter and D.R. Hayhurst ), Springer-Verlag, Berlin/Heidelberg/New York 1981, 179–201.
BETTEN, J., Pressure-dependent Yield Behaviour of Isotropic and An-isotropic Materials, IUTAM-Symposium on Deformation and Failure of Granular Materials, Delft 1982, published in the proceedings (eds. P.A. Vermeer and H.J. Luger ), A.A. Balkema, Rotterdam 1982, 81–89.
BETTEN, J., Formulation of Failure Criteria for Anisotropic Materials under Multi-Axial States of Stress, Colloque International du CNRS n° 351 on Failure Criteria of Structured Media, Grenoble 1983, to be published in the proceedings (ed. J.P. Boehler).
BETTEN, J., The Eigenvalue Problem of a Fourth-Order Tensor, in preparation.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Wien
About this chapter
Cite this chapter
Betten, J. (1987). Invariants of Fourth-Order Tensors. In: Boehler, J.P. (eds) Applications of Tensor Functions in Solid Mechanics. International Centre for Mechanical Sciences, vol 292. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2810-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2810-7_11
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-81975-3
Online ISBN: 978-3-7091-2810-7
eBook Packages: Springer Book Archive