Abstract
The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated from a general point of view. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tensor-valued function defining a general class of strain measures. By means of this formula and the chain rule as well as Sylvester's formula for eigenprojections, explicit basis-free expressions for the material time rate of an arbitrary Lagrangean strain measure can be derived in terms of the right Cauchy–Green tensor and the material time rate of any chosen Lagrangean strain measure, e.g. Hencky's logarithmic strain measure. These results provide a new derivation of Carlson–Hoger's general gradient formula for an arbitrary generalized strain measure and supply a new, rigorous proof for Carlson–Hoger's conjecture concerning the n-dimensional case. Next, by virtue of the aforementioned gradient formula, a general fact for objective corotational rates and their defining spin tensors is disclosed: Let Ω = ϒ ( B, D, W) be any spin tensor that is continuous with respect to B, where B, D and B are the left Cauchy–Green tensor, the stretching tensor and the vorticity tensor. Then the corotational rate of an Eulerian strain measure defined by Ω is objective iff Ω = W + υ ( B, D), where Υ is isotropic. By means of this fact and certain necessary or reasonable requirements, it is further found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis- free expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbitrary Eulerian strain measure are established in terms of the left Cauchy–Green tensor B and the stretching tensor B as well as the introduced antisymmetric function. By choosing several particular forms of the latter, all commonly-known spin tensors and corresponding corotational rates are shown to be incorporated into these general formulas in a natural way. In particular, with the aid of these general formulae it is proved that an objective corotational rate of the Eulerian logarithmic strain measure ln V is identical with the stretching tensor D and moreover that in all possible strain tensor measures only ln V enjoys this property.
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References
R. Hill, On constitutive inequalitites for simple materials-I, J. Mech. Phys. Solids 16(1968) 229-242.
R. Hill, Constitutive inequalities for isotropic elastic solids under finite strain, Proc. R. Soc. London A326(1970) 131-147.
R. Hill, Aspects of invariance in solid mechanics, Advances in Applied Mechanics 18(1978) 1-75.
C. Truesdell and R. Toupin, The Classical Field Theories. In: S. Flügge (ed.), Handbuch der PhysikVol. III/1, pp. 226-858, Berlin, Göttingen, Heidelberg: Springer-Verlag (1960).
C.C. Wang and C. Truesdell, Introduction to Rational Elasticity, Leyden: Noordhoff (1973).
R.W. Ogden, Non-Linear Elastic Deformations, Chichester: Ellis Horwood (1984).
M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media.Berlin, Heidelberg, New York: Springer-Verlag (1997).
M.E. Gurtin, An Introduction to Continuum Mechanics.New York: Academic Press (1981).
J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity.Englewood Cliffs, New Jersey: Prentice-Hall (1983).
T.C. Doyle and J.L. Ericksen, Nonlinear elasticity, Advances in Applied Mechanics 4(1956) 53-115.
B.R. Seth, Generalized strain measures with applications to physical problems. In: M. Reiner and D. Abir (eds), Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics.Oxford: Pergamon Press (1964) pp. 162-172.
H. Hencky, Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen, Z. Techn. Phys. 9(1928) 214-247.
S. Zaremba, Sur une forme perfectionée de la théorie de la relaxation, Bull. Intl. Acad. Sci. Cracovie, pp. 594-614 (1903).
G. Jaumann, Geshlossenes System physikalischer und chemischer Differenzialgesetze, Sitzber. Acad. Wiss. Wien (IIa) 120(1911) 385-530.
A.E. Green and P.M. Naghdi, A general theory of an elastic-plastic continuum, Arch. Rat.Mech. Anal. 18(1965) 251-281.
J.K. Dienes, On the analysis of rotation and stress rate in deforming bodies, Acta Mechanica 32(1979) 217-232.
J.K. Dienes, A discussion of material rotation and stress rate, Acta Mechanica 65(1987) 1-11.
D.R. Metzger and R.N. Dubey, Corotational rates in constitutive modelling of elastic-plastic deformation, Int. J. Plasticity 4(1987) 341-368.
M. Scheidler, The tensor equation AX+ XA= Φ( A ; H ), with applications to kinematics of continua, J. Elasticity 36(1994) 117-153.
M.E. Gurtin and K. Spear, On the relationship between the logarithmic strain rate and the stretching tensor, Int. J. Solids Structures 19(1983) 437-444.
R.N. Dubey, Choice of tensor-rates-a methodology, SM Archives 12(1987) 233-244.
J.G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London A200(1950) 523-541.
B.A. Cotter and R.S. Rivlin, Tensors associated with time-dependent stress, Quart. Appl. Math. 13(1955) 177-182.
C. Truesdell, Hypo-elasticity, J. Rat. Mech. Anal. 4(1955) 83-133.
W. Noll, On the continuity of the solid and fluid state, J. Rat. Mech. Anal. 4(1955) 3-81.
T.Y. Thomas, On the structure of stress-strain relation, Proc. Nat. Acad. Sci. U.S.A. 41(1955) 716-720.
W. Prager, An elementary discussion of definitions of stress rate, Quart. Appl. Math. 18(1960) 403-407.
Y.F. Dafalias, Corotational rates for kinematical hardening at large plastic deformations, ASME J. Appl. Mech. 50(1983) 561-565.
J. Stickforth and K. Wegener, A note on Dienes' and Aifantis' co-rotational derivatives, Acta Mechanica 74(1988) 227-234.
L. Szabó and M. Balla, Comparison of some stress rates, Int. J. Solids Structures 25(1989) 279-297.
W. Yang, L. Cheng and K.C. Hwang, Objective corotational rates and shear oscillations, Int. J. Plasticity 8(1992) 643-656.
S. Nemat-Nasser, On finite deformation elasto-plasticity, Int. J. Solids Structures 18(1982) 857-872.
Z.H. Guo and R.N. Dubey, Basic aspects of Hill's method in solid mechanics, SM Archives 9(1984) 353-380.
M. Scheidler, Time rates of generalized strain tensors, Part I: Component formulas, Mech. Mater. 11(1991) 199-210.
M. Scheidler, Time rates of generalized strain tensors, Part II: Approximate basis-free formulas. Mech. Mater. 11(1991) 211-219.
Z.H. Guo, Rates of stretch tensors, J. Elasticity 14(1984) 263-267.
A. Hoger and D.E. Carlson, On the derivative of the square root of a tensor and Guo's rate theorem, J. Elasticity 14(1984) 329-336.
M.M. Mehrabadi and S. Nemat-Nasser, Some basic kinematical relations for finite deformations of continua, Mech. Mater. 6(1987) 127-138.
Z.H. Guo, Th. Lehmann and H.Y. Liang, Further remarks on rates of stretch tensors, Trans. CSME 15(1991) 161-172.
Th. Lehmann, Z.H. Guo and H.Y. Liang, The conjugacy between Cauchy stress and logarithm of the left stretch tensor, Eur. J. Mech., A/Solids 10(1991) 395-404.
Th. Lehmann and H. Y. Liang, The stress conjugate to logarithmic strain ln V, Zeits. Angew. Math. Mech. 73(1993) 357-363.
L. Wheeler, On the derivatives of the stretch and rotation with respect to the deformation gradient, J. Elasticity 24(1990) 129-133.
Y.C. Chen and L. Wheeler, Derivatives of the stretch and rotation tensors, J. Elasticity 32(1993) 175-185.
W.D. Reinhardt and R.N. Dubey, Eulerian strain-rate as a rate of logarithmic strain, Mech. Res. Commun. 22(1995) 165-170.
W.D. Reinhardt and R. N. Dubey, Coordinate-independent representation of spins in continuum mechanics, J. Elasticity 42(1996) 133-144.
J. E. Fitzgerald, A tensorial Hencky measure of strain and strain rate for finite deformation, J. Appl. Phy. 51(1980) 5111-5115.
A. Hoger, The material time derivative of logarithmic strain tensor, Int. J. Solids Struct. 22(1986) 1019-1032.
D.E. Carlson and A. Hoger, The derivative of a tensor-valued function of a tensor, Q. Appl. Math. 44(1986) 409-423.
M. Scheidler, Time rates of generalized strain tensors with applications to elasticity. In: S.C. Chou (ed.), Proc. 12th Army Symp. on Solid Mechanics (1992) pp. 59-71.
W.B. Wang and Z.P. Duan, On the invariant representation of spin tensors with applications, Int. J. Solids Struct. 27(1991) 329-341.
C.S. Man and Z.H. Guo, A basis-free formula for time rate of Hill's strain tensors, Int. J. Solids Struct. 30(1993) 2819-2842.
H. Xiao, Unified explicit basis-free expressions for time rate and conjugate stress of an arbitrary Hill's strain, Int. J. Solids Struct. 32(1995) 3327-3340.
E.H. MacMillan, On the spin of tensors, J. Elasticity 27(1992) 69-84.
Z.H. Guo, Th. Lehmann, H.Y. Liang and C.S. Man, Twirl tensors and the tensor equation AX- XA= C, J. Elasticity 27(1992) 227-242.
J.M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51(1984) 699-728.
C.C. Wang, A new representation theorem for isotropic functions, Part II. Arch. Rat. Mech. Anal. 36(1970) 198-223.
A.J.M. Spencer, Theory of Invariants. In: A.C. Eringen (ed.), Continuum Physics, Vol. I. New York: Academic Press (1971).
H. Xiao, O.T. Bruhns and A. Meyers, Logarithmic strain, logarithmic spin and logarithmic rate, Acta Mechanica 124(1997) 89-105.
H. Xiao, O.T. Bruhns and A. Meyers, Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures, Arch. Mech. (1998) (accepted).
H. Xiao, O.T. Bruhns and A. Meyers, Hypo-elasticity model based upon the logarithmic stress rate, J. Elasticity 47(1997) 51-68.
C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics. In: S. Flügge (ed.), Handbuch der Physik, Vol. III/3, Berlin, New York etc.: Springer-Verlag (1965).
F. Rellich, Perturbation Theory of Eigenvalue Problems.New York: Gordon and Breach (1969).
T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Berlin, New York etc.: Springer-Verlag (1982).
G. Birkhoff and G.-C. Rota, Ordinary Differential Equations.the fourth edition. New York etc.: John Wiley & Sons (1989).
Jr. W.F. Donoghue, Monotone Matrix Functions and Analytic Continuation. Berlin: Springer-Verlag (1974).
K. Sawyers, Comments on the paper determination of the stretch and rotation in the polar decomposition of the deformation gradient, Quart. Appl. Math. 44(1986) 309-311.
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Xiao, H., Timme Bruhns, O. & Meyers, A.T.M. Strain Rates and Material Spins. Journal of Elasticity 52, 1–41 (1998). https://doi.org/10.1023/A:1007570827614
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DOI: https://doi.org/10.1023/A:1007570827614