On objective surface rates
Authors:
H. Stumpf and J. Badur
Journal:
Quart. Appl. Math. 51 (1993), 161-181
MSC:
Primary 73B99; Secondary 53A05, 73K15
DOI:
https://doi.org/10.1090/qam/1205944
MathSciNet review:
MR1205944
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Abstract: In this paper we derive objective, in the sense of surface, rates of tensors and we give a correct formulation of a two-dimensional continuum. Furthermore we present objective rates of tensors on the boundary line enclosing the surface under consideration. It can be considered as a first step to derive objective rates for generalized continua as Cosserat continua and Kirchhoff-Love type nonlinear shell theories.
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F. Bampi and A. Morro, Objectivity and objective time derivatives in continuum physics, Found. Phys. 10, 905–920 (1980)
R. M. Bowen and C.-C. Wang, On displacement derivatives, Quart. Appl. Math. 29, 29–39 (1971)
E. Cosserat and F. Cosserat, Sur la théorie de l’élasticité, Ann. Toulouse 10, 1–116 (1896)
B. A. Cotter and R. S. Rivlin, Tensors associated with time-dependent stress, Quart. Appl. Math. 13, 177–182 (1955)
D. Durban and M. Baruch, Natural stress rate, Quart. Appl. Math. 35, 55–61 (1977)
C. Eckart, The thermodynamics of irreversible process. IV: The theory of elasticity and anelasticity, Phys. Rev. 73, 373–382 (1948)
A. E. Green and B. C. McInnis, Generalized hypo-elasticity, Proc. Roy. Soc. Edinburgh Sect. A 67, 220–230 (1967)
Guo Zhong-Heng, Time derivatives of tensor fields in non-linear continuum mechanics, Arch. Mech. (Arch. Mech. Stos.) 15, 131–163 (1963)
M. E. Gurtin, Multiphase thermomechanics with interfacial structure. 1. Heat conducting and the capillary balance law, Arch. Rational. Mech. Anal. 104, 195–221 (1988)
M. E. Gurtin and A. I. Murdoch, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57, 291–323 (1974)
D. Y. Hsieh, Lagrangian formulation of bubble dynamics, Quart. Appl. Math. 33, 115–130 (1975)
G. Jaumann, Geschlossenes System physikalischer und chemischer Differentialgesetze, Sitzungsber. Berlin. Akad. Wiss. Wien (IIa) 120, 385–530 (1911)
G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. Reine Angew. Math. 56, 285–313 (1859)
W. Kosinski, Field singularities and wave analysis in continuum mechanics, Ellis Horwood Ltd, Chichester, 1986
Th. Lehmann, Formänderungen eines klassischen Kontinuums in vierdimensionaler Darstellung, Proc. 19 Internat. Congress Appl. Mech., H. Görtler, ed., Springer-Verlag, Berlin, 1964, pp. 376–382
D. B. Macveau, Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren, Z. Angew. Math. Phys. 19, 157–185 (1968)
E. F. Masur, On the definition of stress rate, Quart. Appl. Math. 19, 160–163 (1961)
F. D. Murnaghan, Finite deformations of an elastic solid, Amer. J. Math. 59, 235–260 (1937)
P. M. Naghdi and W. L. Wainwright, On the time derivative of tensors in mechanics of continua, Quart. Appl. Math. 19, 95–109 (1961)
L. Natanson, On the laws of viscosity, Philos. Mag. 2, 342–356 (1901)
W. Noll, On the continuity of solid and fluid states, J. Rational Mech. Anal. 4, 3–81 (1955)
J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London Ser. A 200, 523–541 (1950)
Iu. Z. Povstenko and Ia. S. Podstigach, Time differentiation of tensors defined on a surface moving through a three-dimensional Euclidean space, Prikl. Mat. Mekh. 47, 1038–1044 (1983)
W. Prager, An elementary discussion of definitions of stress rate, Quart. Appl. Math. 18, 403–407 (1961)
L. I. Sedov, Concepts of different rates of change of tensors, Prikl. Mat. Mekh. 14, 393–398 (1960)
H. Stumpf, General concept of the analysis of thin elastic shells, Z. Angew. Math. Mech. 66, 337–350 (1986)
H. Stumpf and K. Ch. Le, Variational principles of nonlinear fracture mechanics, Acta Mech. 83, 25–37 (1990)
H. Stumpf, On the balance equations of a surface via descent from 3-D continuum (to appear)
---, On the shakedown analysis in finite elastoplasticity, Internat. J. Plasticity (to appear)
T. Y. Thomas, Kinematically preferred coordinate systems, Proc Nat. Acad. Sci. U. S. A. 41, 762–770 (1955)
C. Truesdell, The mechanical foundation of elasticity and fluid dynamics, J. Rational Mech. Anal. 1, 125–300 (1952); errata 2, 593–616 (1953)
---, The simplest rate theory of pure elasticity, Comm. Pure Appl. Math. 8, 123–132 (1955)
C. Truesdell and R. A. Tupin, The classical field theories, S. Flügge, ed., Handbuch der Physik, Vol. III/I, Springer-Verlag, Berlin, 1960, pp. 226–793
L. M. Truskinovskii, Equilibrium phase interfaces, Sov. Phys. Dokl. 27, 551–553 (1982)
C. E. Weatherburn, Differential geometry of three dimensions, Cambridge Univ. Press, Cambridge, vol. 1 (1927), vol. 2 (1928)
S. Zaremba, Sur une généralisation de la théorie classique de la viscosité, Bull. Internat. l’Acad. Sci. Lett. Cracovie (CI. Sci. Math. Natur.), A 380–403 (1903)
A. Zmitrowicz, Mathematical description of anisotropic friction, Internat. J. Solids Structures 25, 837–862 (1989)
H. Zorski, Rigid motion of relativistic surfaces, Continuum Mechanics and Related Problems of Analysis, N. I. Muskhelishvili annv., Nauka, Moscow, 1972, pp. 203–207
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© Copyright 1993
American Mathematical Society
