A class of motions with constant stretch history
Author:
R. R. Huilgol
Journal:
Quart. Appl. Math. 29 (1971), 1-15
DOI:
https://doi.org/10.1090/qam/99767
MathSciNet review:
QAM99767
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Abstract: The purpose of this paper is to examine the kinematics and dynamics of a class of motions with constant stretch history. A kinematical result is announced to indicate the velocity field such a motion may have and two examples, viz. helical-torsional flow and the helical flow combined with the axial motion of fanned planes, are discussed in detail. The helical-torsional flow is found to be experimentally realizable, albeit approximately, and it is shown how an apparatus may be built to measure the material functions occurring in such flows. Two nonlinear differential equations are derived to determine the velocity profile when the motion under study is treated as a nearly viscometric flow. In addition, restrictions on the proper numbers of the first Rivlin-Ericksen tensor are arrived at so that the motion with constant stretch history is completely determined by the first two or first three Rivlin-Ericksen tensors. This permits a reduction in the number of terms occurring in the full expansion of the constitutive equation.
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R. R. Huilgol, On the construction of motions with constant stretch history II: Motions superposable on simple extension and various simplified constitutive equations for constant stretch histories, M.R.C. Technical Report No. 975, Univ. of Wisconsin, Madison, Wis., 1969; parts of this report appear in Trans. Soc. Rheol. 14, 425–437 (1970)
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A. C. Pipkin, Controllable viscometric flows, Quart. Appl. Math. 26, 87–100 (1968)
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R. B. Bird and E. K. Harris, Jr., Analysis of steady state shearing and stress relaxation in the Maxwell orthogonal rheometer, A. I. Ch. E. J. 14, 758–761 (1968)
R. R. Huilgol, On the properties of the motion with constant stretch history occuring in the Maxwell rheometer, Trans. Soc. Rheology (to appear)
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W. Noll, Motions with constant stretch history, Arch. Rational Mech. Anal. 11, 97–105 (1962)
B. D. Coleman, Kinematical concepts with applications in the mechanics and thermodynamics of incompressible viscoelastic fluids, Arch. Rational Mech. Anal. 9, 273–300 (1962)
C.-C. Wang, A representation theorem for the constitutive equation of a simple material in motions with constant stretch history, Arch. Rational Mech. Anal. 20, 329–340 (1965)
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965, pp. 1–602
W.-L. Yin and A. C. Pipkin, Kinematics of viscometric flow, Div. Appl. Math., Brown Univ. Technical Report No. 1, 1969
A. C. Pipkin and D. R. Owen, Nearly viscometric flows, Phys. Fluids 10, 836–843 (1967)
R. R. Huilgol, On the construction of motions with constant stretch history II: Motions superposable on simple extension and various simplified constitutive equations for constant stretch histories, M.R.C. Technical Report No. 975, Univ. of Wisconsin, Madison, Wis., 1969; parts of this report appear in Trans. Soc. Rheol. 14, 425–437 (1970)
W. Noll, A mathematical theory of the mechanical behavior of continuous media, Arch. Rational Mech. Anal. 2, 198–226 (1958)
B. D. Coleman and W. Noll, Steady extension of incompressible simple fluids, Phys. Fluids 5, 840–843 (1962)
G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, New York, (1934), 2nd ed., 1952
W. Noll, On the continuity of the solid and fluid states, J. Rational Mech. Anal. 4, 3–81 (1955)
A. C. Pipkin, Controllable viscometric flows, Quart. Appl. Math. 26, 87–100 (1968)
R. R. Huilgol, On the construction of motions with constant stretch history. I: Superposable viscometric flows, M.R.C. Technical Report 954, Univ. of Wisconsin, Madison, Wis., 1968
J. G. Oldroyd, Some steady flows of the general elastico-viscous liquid, Proc. Roy. Soc. London, Ser. A 283, 115–133 (1965)
B. D. Coleman and W. Noll, Helical flow of general fluids, J. Appl. Phys. 30, 1508–1512 (1959)
B. Bernstein, E. A. Kearsley and L. J. Zapas, A study of stress relaxation with finite strain, Trans. Soc. Rheology 7, 391–410 (1963)
R. S. Rivlin, Further remarks on the stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4, 681–702 (1955)
B. Maxwell and R. P. Chartoff, Studies of a polymer melt in an orthogonal rheometer, Trans. Soc. Rheology 9, 41–52 (1965)
R. B. Bird and E. K. Harris, Jr., Analysis of steady state shearing and stress relaxation in the Maxwell orthogonal rheometer, A. I. Ch. E. J. 14, 758–761 (1968)
R. R. Huilgol, On the properties of the motion with constant stretch history occuring in the Maxwell rheometer, Trans. Soc. Rheology (to appear)
A. J. M. Spencer and R. S. Rivlin, The theory of matrix polynomials and its application to the mechanics of isotropic continua, Arch. Rational Mech. Anal. 2, 309–336 (1958/59)
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Article copyright:
© Copyright 1971
American Mathematical Society