On the time dependence of viscoelastic variational solutions
Author:
R. A. Schapery
Journal:
Quart. Appl. Math. 22 (1964), 207-215
DOI:
https://doi.org/10.1090/qam/99955
MathSciNet review:
QAM99955
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Abstract: Thermodynamic operational-variational principles are employed in a study of the transient response of linear viscoelastic media with an arbitrary degree of anisotropy. Assuming displacements in the form of a series of products of space-dependent functions and time-dependent generalized coordinates, the (approximate) response is calculated by minimizing a functional which is analogous to the potential energy of an elastic body. Similarly, a principle analogous to the principle of minimum complementary energy of elasticity is used to deduce transient behavior of (approximate) stresses. The displacements are not required to satisfy equilibrium or stress boundary conditions, nor are stresses calculated from the complementary principle required to satisfy compatibility or displacement boundary conditions. It is found that when applied loads and displacements are step-functions of time, the transient component of stresses and displacements is given in most cases by a series of exponentials with negative, real arguments.
R. A. Schapery, Irreversible thermodynamics and variational principles with applications to viscoelasticity., Aero. Research Labs., Wright-Patterson Air Force Base, ARL 62–418 (1962)
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- M. E. Gurtin, Variational principles in the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 13 (1963), 179–191. MR 214321, DOI https://doi.org/10.1007/BF01262691
- Ruel V. Churchill, Operational mathematics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958. 2nd ed. MR 0108696
- I. S. Sokolnikoff, Mathematical theory of elasticity, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. 2d ed. MR 0075755
- R. A. Schapery, Approximate methods of transform inversion for viscoelastic stress analysis, Proc. 4th U.S. Nat. Congr. Appl. Mech. (Univ. California, Berkeley, Calif., 1962) Amer. Soc. Mech. Engrs., New York, 1962, pp. 1075–1085. MR 0153175
R. A. Schapery, Irreversible thermodynamics and variational principles with applications to viscoelasticity., Aero. Research Labs., Wright-Patterson Air Force Base, ARL 62–418 (1962)
M. A. Biot, Linear thermodynamics and the mechanics of solids, Proc. 3rd U. S. Natl. Cong. Appl. Mech., ASME, New York, 1–18 (1958)
M. A. Biot, Variational principles in irreversible thermodynamics with application to viscoelasticity Phys. Rev. 97, 1463–1469 (1955)
M. A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27, 240–253 (1956)
M. E. Gurtin, Variational principles in the linear theory of viscoelasticity, Arch. Ratl. Mech. Anal. 13, 179–191 (1963)
R. V. Churchill, Operational mathematics, McGraw-Hill Book Company, Inc., New York, 1958
I. S. Sokolnikoff, Mathematical theory of elasticity, McGraw-Hill Book Company, Inc., New York, 1956
R. A. Schapery, Approximate methods of transform inversion for viscoelastic stress analysis, Proc. 4th U. S. Natl. Congr. Appl. Mech., ASME, New York, 1075–1085 (1962)
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Article copyright:
© Copyright 1964
American Mathematical Society