A unified formulation is developed for deformation-related spins, and for objective rates based on them. The approach generalizes the underlying concepts, and allows new rates to be constructed. Mathematical and thermodynamical restrictions on these are shown. As a result, it can be demonstrated that the Eulerian strain rate is an objective rate of logarithmic strain, based on a spin easily derivable from the general form. Interrelations between other known spins and objective rates emerge very clearly. Consequences of the proposed formalism are explored in hypoelastic and in rigid-plastic constitutive relations, the latter involving purely isotropic and purely kinematic hardening. The application of the resulting models to the simple shear deformation is shown.

Issue Section:
Research Papers
Topics:
Constitutive equations, Deformation, Hardening, Kinematics, Modeling, Rotation, Shear deformation, Solids
1.
Bammann
D. J.
, and
Aifantis
E. C. A.
,
1987
, “
A Model for Finite-Deformation Plasticity
,”
Acta Mechanica
, Vol.
69
, pp.
97
–
117
.
2.
Dafalias
Y. F.
,
1983
, “
Corotational Rates for Kinematic Hardening at Large Plastic Deformation
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
50
, pp.
561
–
565
.
3.
Dienes
J. K.
,
1979
, “
On the Analysis of Rotation and Stress Rate in Deforming Bodies
,”
Acta Mechanica
, Vol.
32
, pp.
217
–
232
.
4.
Dubey
R. N.
,
1987
, “
Choice of Tensor Rates: A Methodology
,”
Solid Mechanics Archives
, Vol.
24
, No.
4
, pp.
233
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244
.
5.
Dubey, R. N., and Bedi, S., 1989, “Some Implications of the MDB Flow Rule in Plasticity,” Advances in Plasticity, Proceedings of PLASTICITY ’89, A. S. Khan and N. Tokuda, eds., Pergamon Press, New York, pp. 291–295.
6.
Dubey
R. N.
, and
Bedi
S.
,
1991
, “
On Flow Rules in Plastic Deformation
,”
Archives of Mechanics
, Vol.
43
, No.
1
, pp.
129
–
137
.
7.
Green
A. E.
, and
McInnis
B. C.
,
1967
, “
Generalized Hypo-Elasticity
,”
Proceedings of the Royal Society Edinburgh
, Vol.
A67
, pp.
220
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230
.
8.
Halleux, J. P., and Donea, J., 1986, “A Discussion of Cauchy Stress Formulations for Large Strain Analysis,” Finite Element Method for Nonlinear Problems, Proc. Europe-US Symp. Trondheim, Norway., P. G. Bergan, K. J. Bathe, and W. Wunderlich, eds., Springer, Berlin.
9.
Hill, R., 1978, “Basic Aspects of Invariance in Solid Mechanics,” Advances in Applied Mechanics, Vol. 18, C. S. Yih, ed., Academic Press, New York, pp. 1–75.
10.
Lee
E. H.
,
Mallet
R. L.
, and
Wertheimer
T. B.
,
1983
, “
Stress Analysis for Anisotropic Hardening in Finite Deformation Plasticity
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
50
, pp.
554
–
560
.
11.
Mehrabadi
M. M.
, and
Nemat-Nasser
S.
,
1987
, “
Some Basic Kinematical Relations for Finite Deformations of Continua
,”
Mechanics of Materials
, Vol.
6
, pp.
127
–
138
.
12.
Metzger
D. R.
, and
Dubey
R. N.
,
1987
, “
Corotational Rates in Constitutive Modelling of Elastic-Plastic Deformation
,”
International Journal of Plasticity
, Vol.
4
, pp.
341
–
368
.
13.
Nagtegaal, J. C., and De Jong, J. E., 1982, “Some Aspects of Nonisotropic Work Hardening in Finite-Strain Plasticity,” Proceedings of the Workshop on Plasticity of Metals at Finite Strain: Theory, Experiment and Computation, E. H. Lee, and R. L. Mallet, eds., R. P. I. Troy, pp. 65–102.
14.
Ogden, R. W., 1984, Non-linear Elastic Deformations, (Ellis Horwood Series in Mathematics and its Applications), Halsted Press, New York.
15.
Truesdell, C., and Noll, W., 1965, The Non-linear Field Theories in Mechanics, Handbuch der Physik, Bd. III/3, S. Flu¨gge, ed., Springer-Verlag, Berlin.
16.
Zbib, H. M., and Aifantis, E. C., 1987, “The Concept of Relative Spin and its Implications to Large Deformation Theories,” Mechanics of Microstructures, MM Report No. 13, Michigan Technological University, Houghton, MI.
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