Summary
This paper is concerned with the calculation of the elementary work in a continuum. Whence follows a method of associating a definite stress tensor with a given strain tensor. This association is intimately connected with the choice of the definition of the rate of change of the strain tensor. It turns out that very few of the well-known stress and strain tensors are associated. However, additional associations are possible when the material is isotropic in the initial state.
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Literaturverzeichnis
T. C. Doyle, J. L. Ericksen,Nonlinear Elasticity, Advances in Applied Mechanics, Vol. IV (Academic Press, New York 1956).
J. F. Besseling,A Thermodynamic Approach to Rheology, Laboratory for Applied Mechanics, Mechanical Engineering Department, Technological University Delft, Lab. Report No. 335 (1966).
A. Cauchy,CEuvres Complètes, IIe série, tome VII, 82–93 (Gauthier-Villars et Fils 1889).
M. Navier,Mémoire sur les lois de l'équilibre et du mouvement des corps solides élastiques, Mém. Acad. Sci. Inst. Fr.7, 375–393 (1824).
S. D. Poisson,Mémoire sur l'équilibre des corps cristallisés, Mém. Acad. Sci. Inst. Fr.18, 1–152 (1839).
W. Thomson, P. G. Tait,Treatise on Natural Philosophy, C. U. P. (1879).
A. Cauchy,Sur l'équilibre et le mouvement d'un système de points matériels sollicités par des forces d'athaction ou de répulsion mutuelle, CEuvres Complètes, IIe série, tome VIII, 227–252 (Gauthier-Villars et Fils 1890).
G. Green,On the Propagation of Light in Cristallized Media, Trans. Camb. Phil. Soc. (1839). Enthalten in Mathematical Papers, 293–311 (Macmillan and Co. 1872).
E. Almansi,Sulle deformazioni finite dei solidi elastici isotropi, Note I, Atti Accad. naz. Lincei. Rc., Serie quinta,201, 705–714 (1911).
Z. Karni, M. Reiner,Measures of Deformation in the Strained and in the Unstrained State, Bull. Res. Coun. Israel8c, 89 (1960).
L. Brillouin,Les tenseurs en mécanique et en élasticité. (Masson et Cie., Paris 1946).
W. Prager,Einführung in die Kontinuumsmechanik (Birkhäuser Verlag, Basel 1960).
M. A. Biot,Mechanics of Incremental Deformations (John Wiley and Sons, Inc., New York 1965).
A. Signorini,Sulla mecanica dei sistemi continui, Atti Accad. naz. Lincei, serie 6,12, 312–316, 411–416 (1930).
G. Piola,Nuova analisi per tutte le questioni della meccanica molecolare, Memorie Mat. Fis. Soc. ital. Sci. Modena,21, 155–321 (1836).
C. Truesdell,The Non-linear Field Theories of Mechanics, Handbuch der Physik, Band III/3 (Springer-Verlag 1965).
C. Truesdell,The Elements of Continuum Mechanics (Springer-Verlag, 1966).
J. L. Ericksen andR. S. Rivlin,Large Elastic Deformations of Homogeneous Anisotropic Materials. J. rat. Mech. Analysis,3, 281–301 (1954).
J. L. Ericksen andR. S. Rivlin,Stress-deformation Relations for Isotropic Materials, J. rat. Mech. Analysis4 (1955).
J. G. Oldroyd,On the Formulation of the Rheological Equations of State, Proc. R. Soc. [A],200, 523–541 (1950).
Z.-H. Guo,Time Derivatives of Tensor Fields in Non-linear Continuum Mechanics, Archwm. Mech. Stosow.15, no. 1 (1963).
W. Prager,An Elementary Discussion of Definitions of Stress Rate, Tech. Report No. 53, Division of Applied Mathematics, Brown University (1961).
J. Finger,Das Potential der inneren Kräfte..., Sber Akad. Wiss. Wien (2a),103, 163–200 (1894).
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Macvean, D.B. Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren. Journal of Applied Mathematics and Physics (ZAMP) 19, 157–185 (1968). https://doi.org/10.1007/BF01601465
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DOI: https://doi.org/10.1007/BF01601465