Variational principles in the linear theory of viscoelastic media with microstructure
Author:
Kenneth A. Kline
Journal:
Quart. Appl. Math. 28 (1970), 69-80
MSC:
Primary 73.49
DOI:
https://doi.org/10.1090/qam/264888
MathSciNet review:
264888
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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
Hu Hai-Chang, On some variational principles in the theory of elasticity and the theory of plasticity, Sci. Sinica 4, 33–54 (1955)
K. Washizu, On the variational principles of elasticity and plasticity, ASRL TR 25-18, Mass. Inst. Tech., Cambridge, Mass., 1955
- R. D. Mindlin, Micro-structure in linear elasticity, Arch. Rational Mech. Anal. 16 (1964), 51–78. MR 160356, DOI https://doi.org/10.1007/BF00248490
- R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal. 17 (1964), 85–112. MR 169425, DOI https://doi.org/10.1007/BF00253050
E. S. Suhubi and A. C. Eringen, Nonlinear theory of microelastic solids. II, Internat. J. Engrg. Sci. 2, 389–404 (1964)
N. Fox, A continuum theory of dislocations for polar elastic materials, Quart. J. Mech. Appl. Math. 19, 343 (1966)
A. E. Green, P. M. Naghdi and R. S. Rivlin, Directors and multipolar displacements in continuum mechanics, Internat. J. Engrg. Sci. 2, 611–620 (1965)
- M. E. Gurtin and Eli Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11 (1962), 291–356. MR 147047, DOI https://doi.org/10.1007/BF00253942
C. N. DeSilva and K. A. Kline, Nonlinear constitutive equations for directed viscoelastic materials with memory, Z. Angew. Math. Phys. 19, 128–139 (1968)
K. A. Kline, On thermodynamics of directed continuous media, Z. Angew. Math. Phys. 19, 793–802 (1968)
- J. L. Ericksen and C. Truesdell, Exact theory of stress and strain in rods and shells, Arch. Rational Mech. Anal. 1 (1958), 295–323. MR 99135, DOI https://doi.org/10.1007/BF00298012
- W. T. Koiter, Couple-stresses in the theory of elasticity. I, II, Nederl. Akad. Wetensch. Proc. Ser. B 67 (1964), 17-29; ibid. 67 (1964), 30–44. MR 0163469
- P. M. Naghdi, On a variational theorem in elasticity and its application to shell theory, Trans. ASME Ser. E. J. Appl. Mech. 31 (1964), 647–653. MR 173390
W. Nowacki, Couple-stresses in the theory of thermoelasticity. II, Bull. Acad. Polon. Sci. Sér. Sci. Tech. 14, 203–212 (1966)
E. Reissner, On the foundations of the theory of elastic shells, Proc. 11th Internat. Congr. of Appl. Mech. (Munich, 1964), Springer-Verlag, Berlin, 1966
W. Günther, Zur Statik und Kinematik des Cosseratschen Kontinuums, Abh. Braunschweig. Wiss. Ges. 10, 195–213 (1958)
W. Nowacki, Couple-stresses in the theory of thermoelasticity. III, Bull. Acad. Polon. Sci. Sér. Sci. Tech. 14, 505–513 (1966)
E. Reissner and F. Y. M. Wan, A note on Günther’s analysis of couple stress, Proc. IUTAM Sympos. (1967), Mechanics of generalized continua, Springer-Verlag, Berlin, 1968
- Eric Reissner, On variational principles in elasticity, Calculus of variations and its applications. Proceedings of Symposia in Applied Mathematics, Vol. VIII, McGraw-Hill Book Co., Inc., New orl-Toronto-London, for the American Mathematical Society, Providence, R.I., 1958, pp. 1–6. MR 0096419
- J. L. Ericksen, Anisotropic fluids, Arch. Rational Mech. Anal. 4 (1960), 231–237 (1960). MR 110394, DOI https://doi.org/10.1007/BF00281389
K. A. Kline, S. J. Allen and C. N. DeSilva, A continuum approach to blood flow, Biorheology 5, 111–118 (1968)
O. D. Kellogg, Foundations of potential theory, Dover, New York, 1953
S. J. Allen, C. N. DeSilva and K. A. Kline, Theory of simple deformable directed fluids, Phys. Fluids 10, 2551–2555 (1967)
- Albert E. Green and Ronald S. Rivlin, On Cauchy’s equations of motion, Z. Angew. Math. Phys. 15 (1964), 290–292 (English, with German summary). MR 176650, DOI https://doi.org/10.1007/BF01607019
M. E. Gurtin, Variational principles in the linear theory of viscoelasticity, Arch. Rat. Mech. Anal. 13, 179–191 (1963)
Hu Hai-Chang, On some variational principles in the theory of elasticity and the theory of plasticity, Sci. Sinica 4, 33–54 (1955)
K. Washizu, On the variational principles of elasticity and plasticity, ASRL TR 25-18, Mass. Inst. Tech., Cambridge, Mass., 1955
R. D. Mindlin, Microstructure in linear elasticity, Arch. Rat. Mech. Anal. 16, 51–78 (1964)
R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rat. Mech. Anal. 17, 85–112 (1964)
E. S. Suhubi and A. C. Eringen, Nonlinear theory of microelastic solids. II, Internat. J. Engrg. Sci. 2, 389–404 (1964)
N. Fox, A continuum theory of dislocations for polar elastic materials, Quart. J. Mech. Appl. Math. 19, 343 (1966)
A. E. Green, P. M. Naghdi and R. S. Rivlin, Directors and multipolar displacements in continuum mechanics, Internat. J. Engrg. Sci. 2, 611–620 (1965)
M. E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rat. Mech. Anal. 11, 291–356 (1962)
C. N. DeSilva and K. A. Kline, Nonlinear constitutive equations for directed viscoelastic materials with memory, Z. Angew. Math. Phys. 19, 128–139 (1968)
K. A. Kline, On thermodynamics of directed continuous media, Z. Angew. Math. Phys. 19, 793–802 (1968)
J. L. Ericksen and C. Truesdell, Exact theory of stress and strain in rods and shells, Arch. Rat. Mech. Anal. 1, 295–323 (1958)
W. T. Koiter, Couple-stresses in the theory of elasticity. II, Nederl. Akad. Wetensch. Proc. Ser. B 67, 30–44 (1964)
P. M. Naghdi, On a variational theorem in elasticity and its application to shell theory, Trans. ASME Ser. F, J. Appl. Mech. 31, 647–653 (1964)
W. Nowacki, Couple-stresses in the theory of thermoelasticity. II, Bull. Acad. Polon. Sci. Sér. Sci. Tech. 14, 203–212 (1966)
E. Reissner, On the foundations of the theory of elastic shells, Proc. 11th Internat. Congr. of Appl. Mech. (Munich, 1964), Springer-Verlag, Berlin, 1966
W. Günther, Zur Statik und Kinematik des Cosseratschen Kontinuums, Abh. Braunschweig. Wiss. Ges. 10, 195–213 (1958)
W. Nowacki, Couple-stresses in the theory of thermoelasticity. III, Bull. Acad. Polon. Sci. Sér. Sci. Tech. 14, 505–513 (1966)
E. Reissner and F. Y. M. Wan, A note on Günther’s analysis of couple stress, Proc. IUTAM Sympos. (1967), Mechanics of generalized continua, Springer-Verlag, Berlin, 1968
E. Reissner, A note on variational principles in elasticity, Internat. J. Solids Structures 1, 93 (1965)
J. L. Ericksen, Anisotropic fluids, Arch. Rat. Mech. Anal. 4, 231–237 (1960)
K. A. Kline, S. J. Allen and C. N. DeSilva, A continuum approach to blood flow, Biorheology 5, 111–118 (1968)
O. D. Kellogg, Foundations of potential theory, Dover, New York, 1953
S. J. Allen, C. N. DeSilva and K. A. Kline, Theory of simple deformable directed fluids, Phys. Fluids 10, 2551–2555 (1967)
A. E. Green and R. S. Rivlin, On Cauchy’s equations of motion, Z. Angew. Math. Phys. 15, 290–292 (1964)
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© Copyright 1970
American Mathematical Society