Variational-Lagrangian irreversible thermodynamics of nonlinear thermorheology
Author:
M. A. Biot
Journal:
Quart. Appl. Math. 34 (1976), 213-248
MSC:
Primary 80.49
DOI:
https://doi.org/10.1090/qam/462245
MathSciNet review:
462245
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A principle of virtual dissipation generalizing d’Alembert’s principle to nonlinear irreversible thermodynamics provides a unifying foundation which leads to an extremely general variational-Lagrangian analysis of dissipative phenomena. Thus a synthesis is achieved between thermodynamics and classical mechanics. The present paper applies this principle to the nonlinear thermomechanics of continua with dissipation and heat conduction. Field equations, constitutive equations and Lagrangian equations with generalized coordinates are derived for nonlinear thermoviscoelastcity, nonlinear thermoelasticity and heat conduction, plasticity, and compressible heat conducting fluids with Newtonian and non-Newtonian viscosity. The thermodynamics of instability is also analyzed from the same fundamental viewpoint.
M. A. Biot, Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena, J. Appl. Phys. 25, 1385–1391 (1954)
- M. A. Biot, Variational principles in irreversible thermodynamics with application to viscoelasticity, Phys. Rev. (2) 97 (1955), 1463–1469. MR 70514
- M. A. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27 (1956), 240–253. MR 77441
- M. A. Biot, Linear thermodynamics and the mechanics of solids, Proceedings of the Third U.S. National Congress of Applied Mechanics, Brown University, Providence, R.I., June 11-14, 1958, American Society of Mechanical Engineers, New York, 1958, pp. 1–18. MR 0102941
- M. A. Biot, Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Amer. 34 (1962), 1254–1264. MR 147041, DOI https://doi.org/10.1121/1.1918315
- M. A. Biot, Theory of stability and consolidation of a porous medium under initial stress, J. Math. Mech. 12 (1963), 521–541. MR 0153152
- Maurice A. Biot, Mechanics of incremental deformations. Theory of elasticity and viscoelasticity of initially stressed solids and fluids, including thermodynamic foundations and applications to finite strain, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0185873
M. A. Biot, Variational principles in heat transfer—a unified Lagrangian analysis of dissipative phenomena, Oxford University Press, 1970
- M. A. Biot, Theory of finite deformations of pourous solids, Indiana Univ. Math. J. 21 (1971/72), 597–620. MR 297184, DOI https://doi.org/10.1512/iumj.1972.21.21048
M. A. Biot, Non-linear thermoelasticity, irreversible thermodynamics and elastic instability, Indiana Univ. Math. J., 23, 309–335 (1973)
R. A. Schapery, A theory of non-linear thermoviscoelasticity based on irreversible thermodynamics, in Proc. Fifth U.S. Congress Appl. Mech., pp. 511–530, 1966
- M. A. Biot, A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics, Acad. Roy. Belg. Bull. Cl. Sci. (5) 61 (1975), 6–30. MR 371269
- M. A. Biot, Folding instability of a layered viscoelastic medium under compression, Proc. Roy. Soc. London Ser. A 242 (1957), 444–454. MR 92454, DOI https://doi.org/10.1098/rspa.1957.0187
- M. A. Biot, Thermoelastic buckling. An unstable thermodynamic equilibrium at minimum entropy, Acad. Roy. Belg. Bull. Cl. Sci. (5) 60 (1974), no. 2, 116–140. MR 386422
L. Onsager, Reciprocal relations in irreversible processes I, Phys. Rev. 37, 405–426 (1930)
L. Onsager, Reciprocal relations in irreversible processes II, Phys. Rev. 37, 2265–2279 (1931)
- L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2) 91 (1953), 1505–1512. MR 57765
- L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2) 91 (1953), 1505–1512. MR 57765
- Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI https://doi.org/10.1007/BF00283864
J. Meixner, Zur Thermodynamik der Theromodiffusion, Annalen der Physik 39, 333–356 (1941)
Y. C. Fung, Fundamentals of solid mechanics, Prentice Hall, N. J., 1965
O. W. Dillon, A non-linear thermoelastic theory, J. Mech. Phys. Solids 10, 128 (1962)
H. Parkus, Methods of solutions of thermoelastic boundary value problems, in High temperature structures and materials, pp. 317–347, Pergamon Press, London, 1964
- J. E. Adkins, Large elastic deformations, Progress in Solid Mechanics, Vol. II, North-Holland, Amsterdam, 1961, pp. 1–60. MR 0119550
M. Voisin, Sur une propriété de l’inertie, Annales Soc. Scientifique de Bruxelles 88, II, 219–226 (1974)
P. Lieber, O. Anderson and Koon-Sang-Wan, A principle of virtual displacements for real fluids, in Proc. 9th Int. Congress Appl. Mech., Brussels, 1956
M. A. Biot and H. Odé, Theory of gravity instability with variable overburden and compaction, Geophysics 30, 213–227 (1965)
M. A. Biot, Three-dimensional gravity instability derived from two-dimensional solutions, Geophysics 31, 153–166 (1966)
- I. Prigogine and P. Glansdorff, L’écart à l’équilibre interprété comme une source d’ordre. Structures dissipatives, Acad. Roy. Belg. Bull. Cl. Sci. (5) 59 (1973), 672–702 (French). MR 359863
- M. Reiner, A mathematical theory of dilatancy, Amer. J. Math. 67 (1945), 350–362. MR 12594, DOI https://doi.org/10.2307/2371950
- R. S. Rivlin, The hydrodynamics of non-Newtonian fluids. I, Proc. Roy. Soc. London Ser. A 193 (1948), 260–281. MR 25847, DOI https://doi.org/10.1098/rspa.1948.0044
H. Ziegler, Thermomechanics, Quart. Appl. Math. 30, 91–107 (1972)
J. C. Legros, Influence de la thermodiffusion sur la stabilité d’une lame liquide horizontal, Bull. Roy. Acad. Belgium (Classe des Sciences) 59, 382–391 (1973–74)
M. A. Biot, On a unified thermodynamic approach to a large class of instabilities of dissipative continua, in Advances in Chemical Physics 22, 13–16, Proc. Conf. on Instability and Dissipative Structures in Hydrodynamics, John Wiley & Sons, Inc., New York, 1975
M. A. Biot, Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena, J. Appl. Phys. 25, 1385–1391 (1954)
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. A. Biot, Linear thermodynamics and the mechanics of solids, in Proc. Third U.S. Nat. Congress Appl. Mech. (Amer. Soc. Mech. Eng.), pp. 1–18, New York, 1958
M. A. Biot, Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Am. 34, 1254–1264 (1962)
M. A. Biot, Theory of stability and consolidation of a porous medium under initial stress, J. Math. Mech. 12, 521–542 (1963)
M. A. Biot, Mechanics of incremental deformations, John Wiley & Sons Inc., New York, 1965
M. A. Biot, Variational principles in heat transfer—a unified Lagrangian analysis of dissipative phenomena, Oxford University Press, 1970
M. A. Biot, Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21, 597–620 (1972)
M. A. Biot, Non-linear thermoelasticity, irreversible thermodynamics and elastic instability, Indiana Univ. Math. J., 23, 309–335 (1973)
R. A. Schapery, A theory of non-linear thermoviscoelasticity based on irreversible thermodynamics, in Proc. Fifth U.S. Congress Appl. Mech., pp. 511–530, 1966
M. A. Biot, A virtual dissipation principle and Lagrangian equations in non-linear irreversible thermodynamics, Bull. Roy. Acad. Belgium (Classe des Sciences) 61, 6–30 (1975)
M. A. Biot, Folding instability of a layered viscoelastic medium in compression, Proc. Roy. Soc. Lond. A, 242, 444–454 (1957)
M. A. Biot, Thermoelastic buckling. An unstable thermodynamic equilibrium at minimum entropy, Bull. Roy. Acad. Belgium (Classe des Sciences) 60, 116–140 (1974)
L. Onsager, Reciprocal relations in irreversible processes I, Phys. Rev. 37, 405–426 (1930)
L. Onsager, Reciprocal relations in irreversible processes II, Phys. Rev. 37, 2265–2279 (1931)
L. Onsager and S. Machlup, Fluctuations and irreversible processes I, Phys. Rev. 91, 1505–1512 (1953)
S. Machlup and L. Onsager, Fluctuations and irreversible processes II, Phys. Rev. 91, 1512–1515 (1953)
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rat. Mech. Anal. 17, 1–46 (1964)
J. Meixner, Zur Thermodynamik der Theromodiffusion, Annalen der Physik 39, 333–356 (1941)
Y. C. Fung, Fundamentals of solid mechanics, Prentice Hall, N. J., 1965
O. W. Dillon, A non-linear thermoelastic theory, J. Mech. Phys. Solids 10, 128 (1962)
H. Parkus, Methods of solutions of thermoelastic boundary value problems, in High temperature structures and materials, pp. 317–347, Pergamon Press, London, 1964
J. E. Adkins, Large elastic deformations, Progress in Solid Mechanics II, Amsterdam, 1961
M. Voisin, Sur une propriété de l’inertie, Annales Soc. Scientifique de Bruxelles 88, II, 219–226 (1974)
P. Lieber, O. Anderson and Koon-Sang-Wan, A principle of virtual displacements for real fluids, in Proc. 9th Int. Congress Appl. Mech., Brussels, 1956
M. A. Biot and H. Odé, Theory of gravity instability with variable overburden and compaction, Geophysics 30, 213–227 (1965)
M. A. Biot, Three-dimensional gravity instability derived from two-dimensional solutions, Geophysics 31, 153–166 (1966)
I. Prigogine and P. Glansdorff, L’écart à l’équilibre interprété comme une source d’ordre. Structures dissipatives, Bull. Roy. Acad. Belgium (Classe des Sciences) 59, 672–701 (1973)
M. Reiner, A mathematical theory of dilatancy, Amer. J. Math. 67, 350 (1945)
R. S. Rivlin, The hydrodynamics of non-Newtonian fluids, Proc. Roy. Soc. Lond. A 193, 260 (1948)
H. Ziegler, Thermomechanics, Quart. Appl. Math. 30, 91–107 (1972)
J. C. Legros, Influence de la thermodiffusion sur la stabilité d’une lame liquide horizontal, Bull. Roy. Acad. Belgium (Classe des Sciences) 59, 382–391 (1973–74)
M. A. Biot, On a unified thermodynamic approach to a large class of instabilities of dissipative continua, in Advances in Chemical Physics 22, 13–16, Proc. Conf. on Instability and Dissipative Structures in Hydrodynamics, John Wiley & Sons, Inc., New York, 1975
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
80.49
Retrieve articles in all journals
with MSC:
80.49
Additional Information
Article copyright:
© Copyright 1976
American Mathematical Society
