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Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates. (English) Zbl 0984.74004
The authors consider the long-standing problem of the choice of objective time rate of symmetric Eulerian tensors in continuum mechanics. The most well-known proposals are those made by Zaremba and Jaumann, Oldroyd, Naghdi and Wainright – and the general notion of Lie derivative. Here, accounting for the fact that the objectivity is not the only requirement, but identification with the deformation rate may be a useful constraint, the authors show the need to use corotational rates. It is finally proved that only Hencky strain can have an objective corotational rate. The spin involved then is the logarithmic spin as defined in a former paper [H. Xiao, O. T. Bruhns and A. Meyers, Acta Mech. 124, No. 1-4, 89-105 (1997; Zbl 0909.73006)].
Reviewer: G.A.Maugin (Paris)

MSC:
74A05 Kinematics of deformation
74A20 Theory of constitutive functions in solid mechanics
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[1] Bertram, A.: Axiomatische Einführung in die Kontinuumsmechanik. Mannheim: Wissenschaftsverlag 1989. · Zbl 0706.73001
[2] Bowen, R. M., Wang, C.-C.: Introduction to vectors and tensors, vol. 1+2. New York: Plenum Press 1976. · Zbl 0329.53008
[3] Dafalias, Y. F.: A missing link in the macroscopic constitutive formulation of large plastic deformations. In: Plasticity today (Sawczuk, A., Bianchi, G., eds.), pp. 135-151, Udine, Italy, 27-30 June, 1983. Proc. International Centre for Mechanical Sciences London: Elsevier 1985.
[4] Eringen, A. C.: Nonlinear theory of continuous media. New York: McGraw-Hill 1962.
[5] Guo, Z.-H.: Time derivatives of tensor fields in nonlinear continuum mechanics. Arch. Mech. Stosow.15, 131-163 (1963). · Zbl 0116.15604
[6] Hencky, H.: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys.9, 214-247 (1928). · JFM 54.0851.03
[7] Hill, R.: On constitutive inequalities for simple materials?I. J. Mech. Phys. Solids16, 229-242 (1968). · Zbl 0162.28702
[8] Jaumann, G.: Geschlossenes System physikalischer und chemischer Differentialgesetze. Akad. Wiss. Wien Sitzber.IIa, 385-530 (1911). · JFM 42.0854.05
[9] Lee, E. H.: Finite deformation effects in plasticity analysis. In: Plasticity today (Sawczuk, A., Bianchi, G., eds.), pp. 61-74, Udine, Italy, 27-30 June, 1983. Proc. International Centre for Mechanical Sciences London: Elsevier 1985.
[10] Lehmann, Th.: Zur Beschreibung zeitabhängiger Vorgänge in der klassischen Kontinuumsmechanik. ZAMM42, T108-T110 (1962). · Zbl 0112.38905
[11] Lehmann, Th.: Anisotrope plastische Formänderungen. Romanian J. Techn. Sci. Appl. Mech.17, 1077-1086 (1972).
[12] Marsden, J. E., Hughes, T. R. J.: Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall 1983. · Zbl 0545.73031
[13] Naghdi, P. M., Wainwright, W. L.: On the time derivatives of tensors in mechanics of continua. Quart. Appl. Math.19, 95-109 (1961). · Zbl 0104.18503
[14] Naghdi, P. M.: Recent developments in finite deformation plasticity. In: Plasticity today (Sawczuk, A., Bianchi, G., eds.), pp. 75-83, Udine, Italy, 27-30 June, 1983. Proc. International Centre for Mechanical Sciences London: Elsevier 1985.
[15] Ogden, R. W.: Non-linear elastic deformations. Mineola, New York: Dover 1997. · Zbl 0938.74014
[16] Oldroyd, J. G.: Finite strains in an anisotropic elastic continuum. Proc. R. Soc. London (A)202, 345-358 (1950). · Zbl 0041.52905
[17] Prager, W.: An elementary discussion of definitions of stress rate. Q. Appl. Math.18, 403-407 (1960). · Zbl 0097.39601
[18] Schieße, P.: Einige Bemerkungen zur Kettenregel bei Zeitableitungen isotroper Tensorfunktionen. ZAMM78, 419-425 (1998).
[19] Stumpf, H., Hoppe, U.: The application of tensor algebra on manifolds to nonlinear continuum mechanics-Invited survey article. ZAMM77, 327-339 (1997). · Zbl 0900.73148
[20] Tsakmakis, C.: Über inkrementelle Materialgleichungen zur Beschreibung großer inelastischer Deformationen. PhD thesis. VDI-Fortschrittsberichte Reihe 18, Nr. 36 Düsseldorf: VDI-Verlag 1987.
[21] Wegener, K.: Zur Berechnung großer plastischer Deformationen mit einem Stoffgesetz vom Überspannungstyp. PhD thesis. Techn. Univ. Braunschweig 1991.
[22] Xiao, H.: Unified explicit basis-free expressions for time rate and conjugate stress of an arbitrary Hill’s strain. Int. J. Solids Struct.32, 3327-3340 (1995). · Zbl 0866.73008
[23] Xiao, H., Bruhns, O., Meyers, A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech.124, 89-105 (1997). · Zbl 0909.73006
[24] Zaremba, S.: Sur une forme perfectionée de la théorie de la relaxation. Bull. Int. Acad. Sci. Cracovie124 594-614 (1903).
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