Advertisement

Journal of Elasticity

, Volume 135, Issue 1–2, pp 3–72 | Cite as

Scientific Life and Works of Walter Noll

  • Paolo Podio-GuidugliEmail author
  • Epifanio G. Virga
Article

Abstract

Walter Noll (1925–2017) was an American mathematician of German birth who made lasting contributions to the foundations of continuum physics and the classical non-linear field theory. This essay is an attempt to put in a broader perspective Noll’s methods and achievements in the hope that young generations of researchers may find their inspiration in the talent and depth of the old. By no means should this be considered as a historical account on the development of continuum mechanics through the second half of the twentieth century. We are content to illuminate Noll’s precious legacy.

Keywords

Foundations of continuum Mechanics Balance equations Stress Contact interactions Constitutive equations Fading memory Thermodynamics Rheology Viscoelastic fluids Molecular mechanics Scale bridging Theory of elasticity Anisotropic media Geometric elasticity Media inhomogeneity 

Mathematics Subject Classification

74A10 74A15 74A20 74A25 74A50 74B20 74D10 74E05 74E10 76A02 76A05 76A10 76A15 82B21 82C21 

Notes

References

  1. 1.
    Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecule systems. J. Elast. 100(1), 63–143 (2010) zbMATHGoogle Scholar
  2. 2.
    Assis, A.K.T.: On Mach’s principle. Found. Phys. Lett. 2, 301–318 (1989) Google Scholar
  3. 3.
    Assis, A.K.T., Graneau, P.: Nonlocal forces of inertia in cosmology. Found. Phys. 26, 271–283 (1996) ADSzbMATHGoogle Scholar
  4. 4.
    Banfi, C., Fabrizio, M.: Sul concetto di sottocorpo nella meccanica dei continui. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. 66(2), 136–142 (1979). Available at http://www.bdim.eu/item?id=RLINA_1979_8_66_2_136_0 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Banfi, C., Fabrizio, M.: Global theory for thermodynamic behaviour of a continuous medium. Ann. Univ. Ferrara 27, 181–199 (1981) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bilby, B.A.: Continuous Distributions of Dislocations pp. 329–398. North-Holland, Amsterdam (1960) Google Scholar
  7. 7.
    Bragg, L.E.: On relativistic worldlines and motions, and on non-sentient response. Arch. Ration. Mech. Anal. 18, 127–166 (1965) zbMATHGoogle Scholar
  8. 8.
    Bunge, M. (ed.): Delaware Seminar in the Foundations of Physics. Studies in the Foundations Methodology and Philosophy of Science, vol. 1. Springer, Berlin (1967) zbMATHGoogle Scholar
  9. 9.
    Bunge, M.: Foundations of Physics. Springer Tracts in Natural Philosophy, vol. 10. Springer, New York (1967) zbMATHGoogle Scholar
  10. 10.
    Carathéodory, C.: Zur Axiomatik der Speziellen Relativitätstheorie. Sitz. Preuss. Akad. Wiss., Phys. Math. Kl. 5, 12–27 (1924) zbMATHGoogle Scholar
  11. 11.
    Di Carlo, A.: A major serendipitous contribution to continuum mechanics. Mech. Res. Commun. 93, 41–46 (2018) Google Scholar
  12. 12.
    Di Carlo, D.: Continuum mechanics as a computable coarse-grained picture of molecular dynamics. J. Elast. (2019, in press) Google Scholar
  13. 13.
    Coleman, B.D.: Kinematical concepts with applications in the mechanics and thermodynamics of incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. 9, 273–300 (1962) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Coleman, B.D.: Simple liquid crystals. Arch. Ration. Mech. Anal. 20, 41–58 (1965) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Coleman, B.D., Feinberg, M., Serrin, J. (eds.): Analysis and Thermomechanics: A Collection of Papers Dedicated to W. Noll on His Sixtieth Birthday. Springer, Berlin (1987) Google Scholar
  16. 16.
    Del Piero, G.: An axiomatic framework for the mechanics of generalized continua. Rend. Lincei Mat. Appl. 29, 31–61 (2018) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dunn, J.E., Fosdick, R.L.: The morphology and stability of material phases. Arch. Ration. Mech. Anal. 74, 1–99 (1980) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Einstein, A.: Prinzipielles zur allgemeinen Relativitätstheorie. Ann. Phys. 55, 241–244 (1918) zbMATHGoogle Scholar
  19. 19.
    Epstein, M., Maugin, G.A.: The energy-momentum tensor and material uniformity in finite elasticity. Acta Mech. 83, 127–133 (1990) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Epstein, M., Maugin, G.A.: Sur le tenseur de moment matériel d’Eshelby en é1asticité non linéaire. C. R. Acad. Sci. Paris 310, 675–678 (1990) zbMATHGoogle Scholar
  21. 21.
    Ericksen, J.L.: Anisotropic fluids. Arch. Ration. Mech. Anal. 4, 231–237 (1959) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ericksen, J.L.: Theory of anisotropic fluids. Trans. Soc. Rheol. 4, 29–39 (1960) MathSciNetGoogle Scholar
  23. 23.
    Ericksen, J.L.: Transversely isotropic fluids. Kolloid Z., 117–122 (1960) Google Scholar
  24. 24.
    Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961) MathSciNetGoogle Scholar
  25. 25.
    Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378 (1962) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. Lond. A 244, 87–112 (1951) ADSMathSciNetzbMATHGoogle Scholar
  27. 27.
    Feynman, R.P.: Forces in molecules. Phys. Rev. 56, 340–343 (1939) ADSzbMATHGoogle Scholar
  28. 28.
    Frank, F.C.: On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958) Google Scholar
  29. 29.
    Gallavotti, G.: Statistical Mechanics: a Short Treatise. Texts and Monographs in Physics. Springer, Berlin (1999) zbMATHGoogle Scholar
  30. 30.
    de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Clarendon, Oxford (1993) Google Scholar
  31. 31.
    Gibbs, J.W.: On the equilibrium of heterogeneous substances. In: The Collected Works, vol. I. Trans. Conn. Acad., vol. 3, pp. 108–248. Longmans, New York (1875–1878). 343–524, 55–353, 1928, Also available at https://gallica.bnf.fr/ark:/12148/bpt6k95192s/f82.image Google Scholar
  32. 32.
    De Giorgi, E., Colombini, F., Piccinini, L.C.: Frontiere Orientate di Misura Minima e Questioni Collegate. Quaderni Scuola Normale Superiore, vol. 1. Editrice Tecnico Scientifica, Pisa (1972) zbMATHGoogle Scholar
  33. 33.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Springer, New York (1984) zbMATHGoogle Scholar
  34. 34.
    Gurtin, M.: Configurational Forces as Basic Concepts of Continuum Physics. Applied Mathematical Sciences, vol. 137. Springer, Berlin (2000) zbMATHGoogle Scholar
  35. 35.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Contiuna. Cambridge University Press, Cambridge (2010) Google Scholar
  36. 36.
    Gurtin, M.E., Martins, L.C.: Cauchy’s theorem in classical physics. Arch. Ration. Mech. Anal. 60, 305–324 (1976) MathSciNetzbMATHGoogle Scholar
  37. 37.
    Gurtin, M.E., Mizel, V.J., Williams, W.O.: A note on Cauchy’s stress theorem. J. Math. Anal. Appl. 22, 398–401 (1968) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Gurtin, M.E., Williams, W.O.: An axiomatic foundation for continuum thermodynamics. Arch. Ration. Mech. Anal. 26, 83–117 (1967) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Gurtin, M.E., Williams, W.O., Ziemer, W.P.: Geometric measure theory and the axioms of continuum thermodynamics. Arch. Ration. Mech. Anal. 92, 1–22 (1986), reprinted in [15] MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hamel, G.: Über die Grundlagen der Mechanik. Math. Ann. 66, 350–397 (1908) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hand, G.L.: A theory of dilute suspensions. Arch. Ration. Mech. Anal. 7, 81–86 (1961) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Hand, G.L.: A theory of anisotropic fluids. J. Fluid Mech. 13, 33–46 (1962) ADSMathSciNetzbMATHGoogle Scholar
  43. 43.
    Hilbert, D.: Mathematische probleme. Arch. Math. Phys. 1, 213–217 (1901), reprinted in [44] zbMATHGoogle Scholar
  44. 44.
    Hilbert, D.: Gesammelte Abhandlungen, vol. 3. Springer, Berlin (1970) zbMATHGoogle Scholar
  45. 45.
    Ignatieff, Y.A.: The Mathematical World of Walter Noll. A Scientific Biography. Springer, Berlin (1996) zbMATHGoogle Scholar
  46. 46.
    Irving, J.H., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950) ADSMathSciNetGoogle Scholar
  47. 47.
    Jeffery, G.B.: The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179 (1922) ADSzbMATHGoogle Scholar
  48. 48.
    Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin (1929) zbMATHGoogle Scholar
  49. 49.
    Kröner, E.: Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin (1958) zbMATHGoogle Scholar
  50. 50.
    Lehoucq, R.B., Von Lilienfeld-Toal, A.: Translation of Walter Noll’s “Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics”. J. Elast. 100, 5–24 (2010) zbMATHGoogle Scholar
  51. 51.
    Leslie, F.M.: Some constitutive equations for anisotropic fluids. Q. J. Mech. Appl. Math. 19, 357–370 (1966) MathSciNetzbMATHGoogle Scholar
  52. 52.
    Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968) MathSciNetzbMATHGoogle Scholar
  53. 53.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge University Press, Cambridge (1927) zbMATHGoogle Scholar
  54. 54.
    Marsh Noll, M.: Ordinary Tasks. Madbooks, Pittsburgh (2016) Google Scholar
  55. 55.
    Maugin, G.A.: Continuum Mechanics through the Twentieth Century. A Concise Historical Perspective. Solid Mechanics and Its Applications, vol. 196. Springer, Dordrecht (2013) zbMATHGoogle Scholar
  56. 56.
    Maxwell, J.C.: On the dynamical theory of gases, IV. Philos. Trans. R. Soc. Lond. 157, 49–88 (1867). Reprinted in [57], pp. 26–78 ADSGoogle Scholar
  57. 57.
    Niven, W.D. (ed.): The Scientific Papers of James Clerk Maxwell, vol. 2. Dover, New York (1965) zbMATHGoogle Scholar
  58. 58.
    Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200(1063), 523–541 (1950) ADSMathSciNetzbMATHGoogle Scholar
  59. 59.
    Oseen, C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29(4), 883–899 (1933) zbMATHGoogle Scholar
  60. 60.
    Pinker, S.: How the Mind Works. Norton, New York (1997) Google Scholar
  61. 61.
    Pipkin, A.C., Rivlin, R.S.: The formulation of constitutive equations in continuum physics. Tech. Rep. DA 4531/4, Department of the Army, Ordnance Corps (1958) Google Scholar
  62. 62.
    Podio-Guidugli, P.: Inertia and invariance. Ann. Mat. Pura Appl. 172, 103–124 (1997) MathSciNetzbMATHGoogle Scholar
  63. 63.
    Podio-Guidugli, P.: Continuum Thermodynamics. SISSA Lecture Notes, vol. 1. Springer, Berlin (2019) zbMATHGoogle Scholar
  64. 64.
    Podio-Guidugli, P.: On the mechanical modeling of matter, molecular and continuum. J. Elast. (2019).  https://doi.org/10.1007/%2Fs10659-018-9709-y zbMATHGoogle Scholar
  65. 65.
    Prager, S.: Stress-strain relations in a suspension of dumbbells. Trans. Soc. Rheol. 1, 53–62 (1957) zbMATHGoogle Scholar
  66. 66.
    Ravi, R.: Comments on the culture of the force. Phys. Today 58(8), 15–16 (2005) Google Scholar
  67. 67.
    Rivlin, R.S.: Review of “The foundations of mechanics and thermodynamics: selected papers. W. Noll”. Am. Sci. 64, 100–101 (1976), the book reviewed is [B4] Google Scholar
  68. 68.
    Robb, A.A.: Geometry of Time and Space. A Theory of Time and Space. Cambridge University Press, Cambridge (1936). This is the second edition of a book published in 1914 with the title zbMATHGoogle Scholar
  69. 69.
    Schutz, J.W.: Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time. Lecture Notes in Mathematics, vol. 361. Springer, Berlin (1973) zbMATHGoogle Scholar
  70. 70.
    Seeger, A.: Recent advances in the theory of defects in crystals. Phys. Status Solidi 1, 669–698 (1961) Google Scholar
  71. 71.
    Sikorsky, R.: Boolean Albegras, 3rd edn. Springer, Berlin (1969) Google Scholar
  72. 72.
    Šilhavý, M.: The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch. Ration. Mech. Anal. 90, 195–212 (1985). Reprinted in [15] MathSciNetzbMATHGoogle Scholar
  73. 73.
    Šilhavý, M.: The scientific work of B.D. Coleman. J. Math. Mech. Solids (2019, in press) Google Scholar
  74. 74.
    Smith, H.E. (ed.): Autobiography of Mark Twain, vol. 1. University of California Press, Berkeley (2010) Google Scholar
  75. 75.
    Suppes, P.: Axioms for relativistic kinematics with or without parity. In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method, Studies in Logic and the Foundations of Mathematics, vol. 27, pp. 291–307. Elsevier, Amsterdam (1959) Google Scholar
  76. 76.
    Suppes, P.: The axiomatic method in the empirical sciences. In: Henkin, L. (ed.) Proceedings of the Turski Symposium. Proceedings of Symposia in Pure Mathematics, vol. 25, pp. 465–479. American Mathematical Society, Providence (1960) Google Scholar
  77. 77.
    Szekeres, G.: Kinematic geometry; an axiomatic system for Minkowski space-time: M.L. Urquhart in memoriam. J. Aust. Math. Soc. 8, 134–160 (1968) zbMATHGoogle Scholar
  78. 78.
    Tadmor, E.B., Miller, R.E.: Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press, Cambridge (2011) zbMATHGoogle Scholar
  79. 79.
    Tanner, R.I., Walters, K.: Rheology: An Historical Perspective. Rheology Series, vol. 7. Elsevier, Amsterdam (1998) zbMATHGoogle Scholar
  80. 80.
    Taylor, G.I.: Motion of solids in fluids when the flow is not irrotational. Proc. R. Soc. Lond. A 93, 99–113 (1917) ADSzbMATHGoogle Scholar
  81. 81.
    Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962) MathSciNetzbMATHGoogle Scholar
  82. 82.
    Truesdell, C.: The mechanical foundations of elasticity and fluid dynamics. J. Ration. Mech. Anal. 1, 125–300 (1952). Extensive additions and corrections appeared in [83] and further minor misprints were corrected in [84]. A corrected reprint of this essay incorporating all previous amendments and some other slips was published as a book in 1966 [87] MathSciNetzbMATHGoogle Scholar
  83. 83.
    Truesdell, C.: Corrections and additions to “The mechanical foundations of elasticity and fluid dynamics”. J. Ration. Mech. Anal. 2, 593–616 (1953) MathSciNetzbMATHGoogle Scholar
  84. 84.
    Truesdell, C.: Corrigenda. J. Ration. Mech. Anal. 3, 801–802 (1954) Google Scholar
  85. 85.
    Truesdell, C.: Hypo-elasticity J. Ration. Mech. Anal. 4, 83–133 (1955). 1019–1020 MathSciNetzbMATHGoogle Scholar
  86. 86.
    Truesdell, C.: Das ungelöste Hauptproblem der endlichen Elastizitätstheorie. Z. Angew. Math. Mech. 36, 97–103 (1956) MathSciNetzbMATHGoogle Scholar
  87. 87.
    Truesdell, C.: The Mechanical Foundations of Elasticity and Fluid Dynamics. International Science Review Series, vol. VIII, Part 1. Gordon & Breach, New York (1966) zbMATHGoogle Scholar
  88. 88.
    Truesdell, C.: Six Lectures on Modern Natutal Philosophy. Springer, Berlin (1966) zbMATHGoogle Scholar
  89. 89.
    Truesdell, C.: Essays in the History of Mechanics. Springer, Berlin (1968) zbMATHGoogle Scholar
  90. 90.
    Truesdell, C.: An Idiot’s Fugitive Essays on Science. Springer, New York (1984) zbMATHGoogle Scholar
  91. 91.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 2nd edn. Springer, Berlin (1992), see [92] for an annotated edition zbMATHGoogle Scholar
  92. 92.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004), edited by S.S. Antman zbMATHGoogle Scholar
  93. 93.
    Truesdell, C., Toupin, R.: The Classical Field Theories. In: Flügge, S. (ed.) Encyclopedia of Physics, vol. III/1. Springer, Berlin (1960) Google Scholar
  94. 94.
    Truesdell, C.A. (ed.): Continuum Mechanics II: The Rational Mechanics of Materials. International Science Review Series, vol. VIII, Part 2. Gordon & Breach, New York (1965) Google Scholar
  95. 95.
    Truesdell, C.A. (ed.): Continuum Mechanics III, Foundations of Elasticity Theory. International Science Review Series, vol. VIII, Part 3. Gordon & Breach, New York (1965) Google Scholar
  96. 96.
    Truesdell, C.A.: A First Course in Rational Continuum Mechanics, vol. 1, 2nd edn. Academic Press, San Diego (1991) zbMATHGoogle Scholar
  97. 97.
    Vol’pert, A.I., Hudjaev, S.I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Mechanics: Analysis, vol. 8. Nijhoff, Dordrecht (1985) Google Scholar
  98. 98.
    Wang, C.C.: A general theory of subfluids. Arch. Ration. Mech. Anal. 20, 1–40 (1965) MathSciNetzbMATHGoogle Scholar
  99. 99.
    Wang, C.C.: On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch. Ration. Mech. Anal. 27, 33–94 (1967) MathSciNetzbMATHGoogle Scholar
  100. 100.
    Wilczek, F.: Whence the force of \({F} = ma\)? I: Culture shock. Phys. Today 57(10), 11–12 (2004) Google Scholar
  101. 101.
    Wilczek, F.: Whence the force of \({F} = ma\)? II: Rationalizations. Phys. Today 57(12), 10–11 (2004) ADSGoogle Scholar
  102. 102.
    Wilczek, F.: Comments on the culture of the force. Phys. Today 58(8), 17 (2005) Google Scholar
  103. 103.
    Wilczek, F.: Whence the force of \({F} = ma\)? III: Cultural diversity. Phys. Today 58(7), 10–11 (2005) Google Scholar
  104. 104.
    Willems, J.C.: Dissipative dynamical systems, part I: general theory. Arch. Ration. Mech. Anal. 45, 321–351 (1972) zbMATHGoogle Scholar
  105. 105.
    Zaremba, S.: Le principe des mouvements relatifs et les équations de la mécanique physique. Bull. Int. Acad. Sci. Cracovie, 614–621 (1903). Available at https://www.biodiversitylibrary.org/item/47339#page/665/mode/1up
  106. 106.
    Zaremba, S.: Sur une forme perfectionée de la théorie de la relaxation. Bull. Int. Acad. Sci. Cracovie, 594–614 (1903). Available at https://www.biodiversitylibrary.org/item/47339#page/665/mode/1up
  107. 107.
    Zaremba, S.: Sur une conception nouvelle des forces intérieures dans un fluide en mouvement. Mém. Sci. Math. Gauthier-Villars, No. 82 (1937) Google Scholar
  108. 108.
    Ziemer, W.P.: Cauchy flux and sets of finite perimeter. Arch. Ration. Mech. Anal. 84, 189–201 (1983) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Accademia Nazionale dei LinceiRomaItaly
  2. 2.Department of MathematicsUniversity of Rome TorVergataRomaItaly
  3. 3.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

Personalised recommendations