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.
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Notes
In his learned account on the history of continuum mechanics in the twentieth century, Maugin writes about these treatises [55, p. 61]: “Impressive encyclopedic contributions by Truesdell, Toupin and Noll were the lighthouses that ‘illuminated’ the world community of mechanics”.
We refer here to Lincoln E. Bragg’s 1963 thesis “On Relativistic Worldlines and Motions, and on Non-Sentient Response”; to 1978 dissertation by Ray E. Artz, Jr., entitled “An Axiomatic Foundation for Finite-Dimensional Quantum Theory”; and to Vincent J. Matsko’s thesis, entitled “Mathematical Structures of Special Relativity”.
Our choice also reflects our taste; in this, we cannot but agree with Truesdell [88, p. 105]:“It is tasteless to recommend one’s own taste, but scarcely honest to recommend any other”.
A short personal account of his time in Bloomington is found in [C15] and in the autobiographical notes [WA1], which cover the years prior to 1956.
These and other biographical matters are found in [79]. Another valuable source is [45]. The six first chapters of [45] are devoted to a meticulous biography of Noll, private and professional, based not only on documents but also on interviews of, and letters from, coworkers and colleagues; the remaining seven chapters attempt an account by themes of Noll’s research work. In this book, appeared in print in 1996, Y.A. Ignatieff bravely took up an unusual endeavour for a historian of mathematics, namely, to write a scientific biography of a living mathematician. We are especially grateful to him for his bravery: his book has been one of our sources, to which we often turned.
Such a lack of virtues is perhaps the very reason why a good friend of mine recently said to me (with the generosity that only a true friend can afford) that “no Virga’s scientific obituary would be justified” after my death. I can only assure the reader that I’ll see that none will appear before.
This is also in David Owen’s recollection recorded in Sect. 3.5 below.
This is also the title of the book [54] that collects some of her recent, minimalistic poems, seeking beauty in everyday life.
A Service of Remembrance was held for Walter on the 21st of June 2017 at the Hunt Botanical Library of Carnegie Mellon University. The service was led by Pastor Chris Taylor of the Fox Chapel Presbyterian Church; the eulogy was read by Peter Noll, Walter’s son; remembrances were given, in the order, by William O. Williams and David R. Owen, both Carnegie Mellon’s Emeriti, and by William J. Hrusa, Professor at Carnegie Mellon University and Brian Seguin, Assistant Professor at the Loyola University of Chicago. The service was enriched by music: Marianne Cornetti, a world-renown mezzo-soprano (http://www.mariannecornetti.com/index.html), accompanied by pianist Mark Carver, sang Pie Jesu by Andrew Lloyd Webber, who was a composer very dear to Walter, and the Ode to Joy (from Beethoven’s 9th Symphony) by Henry J. van Dyke. Marianne, who had been a student of Resie, Walter’s late wife, has remained very close to Walter after Resie’s death and has been most kind to Marilyn as well.
M.F. is presently the president and CEO of the Houston Methodist Research Institute in Houston; among several other things, he also serves as executive vice president for Houston Methodist Hospital. https://en.wikipedia.org/wiki/Mauro_Ferrari.
For example, the expository paper [A29] is based on a series of lectures delivered at the Technion of Haifa, Israel, in the summer of 1972. It is typical of how Noll’s thoughts progressed, through a sequence of successive refinements involving higher levels of abstraction. His earlier views about a comprehensive mathematical infrastructure of continuum mechanics and thermodynamics were first presented in 1965 in the Bressanone Lectures [C6]; seven years later, they had already become obsolete for him.
A well-known exposition of this theory, including all developments up to 1965, is found in Chap. C of [B1], entitled “The general theory of material behavior”. Some seven years later, Noll conceived and published [A28], that is, in his words from [WB8], “a general framework for the formulation of constitutive laws, using the concept of a state of a material element”.
‘State’ is a loaded term. The information about aggregation these state equations embody is singled out by performing, even only mentally, creep and relaxation experiments; the impact of Noll’s and Coleman’s work on rheology will be discussed in Sect. 4.4. The fortune of the term ‘constitutive equation’, which to our knowledge was used in this paper for the first time, has been such as to be eventually registered in the Oxford English Dictionary (http://www.oed.com/view/Entry/39861?rskey=iEMOLh&result=1&isAdvanced=true#eid8396069), making his coiner proud.
“These two cases are also the extreme cases—pure fluidity and pure elasticity—in the landscape so beautifully described in his “continuity of states” by Walter Noll in 1955.” [55, p. 2].
This adjective, an artefact combining the Greek adjectives
for fluid and 
for solid, had some modest fortunes (e.g., references are made to ‘Noll’s theory of hygrosteric materials’), insufficient to make it to the Oxford English Dictionary. Neither they made it to the OED two terms, hereafter italicized, that became widely adopted though with acceptations slightly different from Noll’s original ones, that is, extra-stress for the stress diminished by the pressure stress possibly to be found in a hygrosteric body at ease (more about this term later).We read in Sect. 14 of [B1] that a material is simple if “…the collection of kinematical facts needed to determine the stress reduces to the history of the deformation gradient alone.”
Other names were proposed, in diminuendo: principle of objectivity of material properties, principle of material objectivity, principle of objectivity (cf. paragrah \(\zeta \) on p. 702 of [93] and Sect. 19A of [B1], where a history of the principle is told, from Hooke’s 1675 Philosophical-Scales up to 1965, the year when [B1] appeared in print). We shall have more to say about material frame-indifference in Sect. 4.5.2.
Curiously enough, in 1950 continuum mechanics was called rheology by Oldroyd and hydrodynamics by Irving and Kirkwood [46], as that discipline was probably referred to in their respective scientific communities.
The only somehow relevant quotation is found in footnote 4 on p. 138 of [82]:
Oldroyd bases his dynamical proposals upon an identification principle, in which the spatial and the material co-ordinates are made to coincide at a given instant and the numerical values of the components of a material tensor coincide with those of the corresponding components of an associated spatial tensor.
A more accurate reference to [58] is found both in footnote 3 on p. 703 of [93] and in Sect. 19A of [B1] (here [105], another relevant paper by Zaremba, is also cited; by the way, in [58] Zaremba is not mentioned at all), where the same quotation from [58] appears:
The form of the completely general equations must be restricted by the requirement that the equations describe properties independent of the frame of reference … Moreover, only those tensor quantities need be considered which have a significance for the material element independent of its motion as a whole in space [italics in the cited text].
Eventually, with characteristic intellectual honesty, Truesdell acknowledged in full the importance of Oldroyd’s contribution. We read on p. 44 of [90], apropos of the criteria he had chosen to decide coverage and focus of his own papers [82, 83]:
My failure to appreciate at its true value the fundamental memoir of Oldroyd [58] is a symptom of my failure to see then that what the field needed was generality in order to reach simplicity and clarity [italics in the cited text].
For a discussion of this interpretation and an alternative to it, we refer the reader to [64].
Similarly, it remains to be explored how reaching stresses can be related to both contactors and semi-contactors, which feature in the general theory of interactions, see Sect. 4.3.4.
A step in the proof of this result consists in an application of Fubini’s theorem, that is, of the Iterated Integration Theorem according to Walter, who used to advocate that theorems, laws of physics et sim. should be designated by alluding to their contents rather than be named after their discoverers.
In Truesdell’s translation [90, p. 507].
Again in Truesdell’s translation [90, p. 136].
We also learn from [89, p. 113] that D’Alembert said that “…we shall be content to remark that the principle [of accelerating force], be it true or be it dubious, be it clear or be it obscure, is useless to mechanics and ought therefore to be banished from it.”
On a more general note, we cannot say either whether he was the only one who solved it, or, for that matter, whether the sport (so dear to popular science) of naming those who supposedly solved any of Hilbert’s problems has any scientific significance. It is difficult to be in anyone else’s mind, but perhaps Hilbert had not meant to start a mind contest; his only purpose might have been to set the scene for a modern, pervasive role of mathematics in science. A century later, we may at least say that this program has been successful and that that shift in perspective for the role of mathematics has been beneficial for all applied sciences, which have become more mathematical, even beyond their desire; this is the only lasting legacy of Hilbert’s paper [43], and the merit is shared by the many mathematicians who have embraced the new century’s mentality.
A more wordy presentation, with many illustrative examples, can be found in Sects. I.2 and I.3 of [96].
For which a classical reference is the textbook [71].
As an example, the reader may consult pp. 24–26 of [96] for an application of this theory to mass-point dynamics.
As illustrated in [96, pp. 26–29], Rizzo later extended Noll’s theory of interactions to interpenetrating bodies.
As was typical in Noll’s evolving terminology, displacements, were later to be renamed transplacements (in [A33], for example). They are commonly called deformations in most contemporary textbooks, such as [35], for example. Noll felt always strongly about notation and terminology, to the point of apologizing, in a rather peculiar way, for his (frequent) changes of mind, as in the following footnote, taken from [A42]:
In 1958, Noll introduced the unfortunate terms “configuration” and “deformation” for what are called “placement” and “transplacement” here. He apologizes. Since these old terms were used in [91], they were widely accepted, and we are now in the ironic position to fight against a terminology that Noll introduced.
Noll’s interest for category theory [WC2] also permeated his views on the foundations of continuum mechanics [C12]. In the language of that theory, the class of all displacements is the class of morphisms of a category whose objects are pairs \((\mathscr{B}, \mathcal{E})\), where ℬ is a subset of ℰ. Likewise, a morphism from \((\mathscr{B},\mathcal{E})\) to \(( \mathscr{B}',\mathcal{E}')\) is an inverbible mapping from ℬ onto \(\mathscr{B}'\) that can be extended to a \(C^{2}\)-diffeomorphism from ℰ onto \(\mathcal{E}'\).
What came to be designated as a fit region had already been called a nice region [C12].
More precisely, the fit regions that are parts of a given fit region ℬ should be materially ordered by inclusion. They thus constitute the universe \(\varOmega _{\mathscr{B}}\), for which \(\varnothing \) is the empty set and \(\infty \) is ℬ itself.
As already remarked above, Noll advocated for the use of self-descriptive names to be given to theorems and principles. This theorem, in particular, is best known in the Western literature as the Gauss-Green Theorem.
A simple counterexample is constructed in [A33] as follows. Think of a rectangle deformed in such a way that one side becomes the graph of a smoothly oscillating function, such as \(f(x)=\exp (-1/x^{2})\sin (1/x)\), say for \(0< x<1\). Then take the intersection with an adjacent, undeformed rectangle, so as to chop off all protruding humps. The result is a countable union of chops, which cannot be regarded as a set with piecewise smooth boundary.
These are also known as Caccioppoli sets; see, for example, Chap. 1 of [33]. Perhaps, Banfi and Fabrizio [4] were the first to realize the role played by these sets in continuum physics (see also [5]). Other applications can be found in the papers [108], [72], and [39], all published not long before [A33]. However, the class of regions proposed in [4], besides being unnecessarily large, did not obey the requirement of constituting a material universe. All three requirements put forward in [A33] were instead satisfied by the class proposed by Gurtin, Williams, and Ziemer [39], but their development, being based on subtle measure-theoretic concepts, was unnecessarily sophisticated.
A regularly open set is an open set that coincides with the interior of its closure. The reason for this condition is to prevent the boundary of a fit region from being other than its outer delimiter. Similarly, a regularly closed set is a closed set that coincides with the closure of its interior. It makes no substantial difference whether one uses the class of regularly open or regularly closed sets as an ambient for fit regions, because the process of taking the closure is a natural one-to-one correspondence from the former class to the latter, its inverse being the process of taking the interior.
A subset of ℰ is negligible if for every \(\varepsilon >0\) it can be covered by a finite collection of balls, the sum of whose volumes does not exceed \(\varepsilon \). Thus a negligible set has zero volume-measure. Conversely, a compact set with zero volume-measure is negligible. Since the boundary of a bounded set is compact, condition (iv) could be replaced by “a fit region has boundary of zero volume-measure”.
Truesdell [96, p. 88] praises this book saying “Be it noted that this clear, excellent, and compact book is written by and for engineers.”
The reader should recall Bunge’s words in Sect. 3.6.
This quote also illuminates Noll’s unwearying attitude towards science since his youth.
This paper is based on a lecture that Walter gave in April 1993 at the meeting of the Society for Natural Philosophy that took place at Carnegie Mellon University on the occasion of his retirement from teaching.
Here, we adopt Noll’s notation \(\mathrm{Usph}\,\mathcal{V}\) for the unit sphere in the translation space \(\mathcal{V}\). It is admittedly rather unusual.
This specifically applies here to the outward unit normal field.
It is perhaps not completely useless to stress that a proto-contactor is indeed a special form of surface density for a contact flux.
In this context, surface tension arises from the balance equation obtained for a special constitutive choice of the tensor that plays the role of an edge stress. Traditionally, that very same balance equation is obtained by assuming that the boundary of the body behaves like a material skin. Such an assumption was not made in [A34], where the surface tension emerges, as it were, from within the body.
We also learn from [90, p. 546] that this postulate dates from 1959 but was only published in 1963.
Revisitations and extensions of Noll’s Axiom of Dynamics and its links with the evolution equations of various mechanical systems (classical and generalized alike) are too numerous to be mentioned here. We just refer the reader to [16], which we take as a representative for many others.
In a different, related perspective, the connection between inertia and invariance properties has been explored in [62].
B.D. Coleman, “an extraordinary figure of the 20th century continuum mechanics and thermodynamics[, who] made a number of deep and permanent contributions that changed the way these sciences are now understood, presented, and applied” [73], passed away on August 2nd, 2018.
This book features a limpid exposition of explicit solutions to motion problems that, in addition to being interesting per se, are of direct rheological interest, in that they point at a feasible experimental program to achieve the constitutive characterisation in terms of three viscometric functions of the shear rate of a class of incompressible Rivlin-Ericksen fluids whose extra-stress is sensitive to the past history of the deformation gradient. A preliminary presentation of the method of viscometric functions on which this book is based was given at the First International Symposium on Pulsatile Blood Flow, held in 1963 at the Presbyterian Hospital in Philadelphia [C4], see also [45, p. 128].
In particular, [A12] appeared in the fourth volume of this journal. In [89, p. 328], Truesdell tells an interesting story about the origin of this paper and its connection with a critical retrospective view on the history of mechanics:
This paper is directed explicitly toward a problem that originated in historical study. Noticing the thermodynamic method of Duhem for deriving the inequalities satisfied by the linear elasticities and linear viscosities in a lecture at Berlin in 1955 I had pointed out that corresponding considerations for the theory of finite elastic strain were not known, and I proposed the problem of finding the restrictions necessarily satisfied by the strain-energy function of an elastic material [86].
In the parlance of ‘rational’ continuum mechanicists, a physical concept is termed ‘basic’ when it is regarded as ‘primitive’ and, as such, it is defined by the “mathematical entities” chosen to represent and manipulate it. As is not infrequent and indeed healthy when a theory is in a phase of development and consolidation, both the list of basic concepts and the properties of their mathematical counterparts were to change, not always slightly, in later expositions of continuum theories, beginning with the publication of volume III/3 of the Handbuch der Physik [B1], coauthored by Noll with Tuesdell [C15].
Recall footnote 17.
As shown in Sect. 1.8.1 of [63], life gets much harder (but not impossible) if the proportionality presumption is renounced. Various later papers of Noll (e.g., [A41], [A42], and [WB8]) testify to his faithfulness to that presumption.
In fact, more can be deduced from (37), namely, that a thermodynamically compatible generalisation of Fourier law of heat conduction is
$$ \hat{\boldsymbol{q}}({\boldsymbol{F}},\eta ,\operatorname{grad}\vartheta )= \hat{\boldsymbol{K}}({\boldsymbol{F}},\eta ,\operatorname{grad}\vartheta ) \operatorname{grad}\vartheta , $$with the conductivity tensor\(\hat{\boldsymbol{K}}\) such that
$$ \operatorname{grad}\vartheta \cdot \hat{\boldsymbol{K}}({\boldsymbol{F}},\eta , \operatorname{grad}\vartheta )\operatorname{grad}\vartheta \geq 0 \quad \textrm{for all}\;({\boldsymbol{F}},\eta ) $$(see [63], Sect. 1.6).
This collection of works is dedicated to the memory of Clifford Truesdell, his mentor, who had revived the use of Natural Philosophy to designate the endeavour to understand nature by using conceptual mathematical tools. See also [A38], for a further discussion on the modern use of this old term.
See also [A45], which reproduces a lecture given at the Seminario Matematico e Fisico di Milano on October 4, 2012.
This is nothing but the neo-classical space-time introduced in [C7], the paper that Noll read at the Delaware Seminar in the Foundations of Physics organized by Bunge in 1965, about which we have said already in Sect. 3.6 above.
This booklet, written with Vincent J. Matsko who was Noll’s student, grew out of a course that Noll taught every two or three years since 1961. The manuscript was based on handwritten notes by V.J.M. who had taken the course as a graduate student in 1987. V.J.M. contributed many original ideas, especially for Chap. 3, provided details of proofs, and produced virtually all the exercises.
Ideas contained in this paper were further developed in two papers published in 1977 and 1978 with J.J. Schäffer [A31], [A32].
A fuller account, to which we refer the interested reader, is given in Chap. 16 of [45].
Thus, Maugin [55, p. 75] can say “…Toupin [and] Noll also tried their hands at the relativistic framework, but without much response and enthusiasm from relativists.”
We are indebted to M. Dafermos for this comment.
We have also learned this from M. Dafermos. Maybe Noll was not familiar with Robb’s work, but he must have known about it, because in Suppes’ paper [75], which was one of Noll’s motivations, that work is cited.
See also [C15] and [WA5].
Noll himself acknowledges that a similar notion of state, although based on different axioms, had been introduced by Willems [104].
This is a new term for what Noll used to call simply a body, a concept including a no longer required connectedness.
These assumptions are meant to replace the ones that are already common in continuum mechanics textbooks, namely, the principles of determinism, local action, and material frame-indifference (see, for example, Sect. IV.2 of [96], for a detailed account).
In this regard, the following quote from [A45] is appropriate:
In about 1970, at a conference in Germany, I was asked by a Physics professor whether General Relativity could be made coordinate-free. I said yes, but it took me decades to develop the necessary background. The reason was that a fully coordinate-free reformulation of the theory of differentiable manifolds had to be created first.
The unimodular group comprises all (invertible) tensors whose determinant has modulus 1. In [A22] Noll gave an original, alternative proof of the fact that the unimodular group is the maximal subgroup of the orthogonal group.
A dilatation group is the group of all unimodular tensors having three fixed linearly independent vectors as their eigenvectors.
Ericksen [21] explicitly renounces seeking inspiration for his equations in the molecular theories for suspensions of ellipsoidal [47] or dumbell particles [65], which had been derived by Jeffery and Prager, respectively. This task was undertaken by Hand [41, 42], who proposed a theory for anisotropic fluids with a symmetric stress tensor and an evolution equation for a supplementary symmetric tensor representing the alignment of suspended particles. Hand showed that his theory includes as special cases Jeffery’s, Prager’s, and Ericksen’s.
The history of the dynamics of liquid crystals has seen more than the attempts to represent them at all costs as simple fluids. Ericksen [25] reformulated in a continuum mechanics context the static theories of Oseen [59] and Frank [28]. Unfortunately, however, these Authors’ theory could not be obtained as the static limit of Ericksen’s theory of anisotropic fluids recalled above. Leslie [52] succeeded in extending an earlier constitutive theory of his [51] consistent with Ericksen’s theory of anisotropic fluids, thus arriving at what has become known as the Ericksen-Leslie theory for the dynamics of liquid crystals, perfectly compatible with Ericksen’s static theory, and blessed with an impressive number of experimental confirmations [30, Chap. 5]. In this theory, liquid crystals are no longer regarded as simple fluids. This confirms the suspicion that Coleman and Noll [A16] had already raised:
It is possible that the theory of simple fluids cannot be successfully applied to the physics of liquid crystals. My colleague and I, however, are not sufficiently familiar with the mechanical behavior of such substances to hazard a categorical statement on this.
The Gibbsian stability of phases was also to be later considered in [17], among other issues. The authors of this paper had, however, missed [A25], presumably on the account of the mathematical language it was clad in. In a note added in proof they say:
We believe that this interesting paper has not gained wide recognition because of the non-standard although elegant mathematical formalism it employs to examine the morphology of material phases. In fact, it was only after our work was completed that we came to appreciate Noll’s contribution.
In retrospect, it is perhaps fair to consider Noll’s paper [A25] as the precursor of the growth of interest in the variational approach to phase transformations that was to follow in the mathematical literature.
That is, the Reviews of Modern Physics.
The paper Noll refers to here is [A14], see Sect. 4.4.1.
This paper, which had been submitted in 1995, was an earlier version of the Contribution On Material Frame Indifference, which eventually appeared in [WB2], see Sect. 4.5.2.
We here quote verbatim a passage from a message by D.R. Owen to us, whose contents we incorporated in the above considerations.
His late writings posted on the web are a treasure trove of reflections and worth perusing conjectures.
for fluid and 
for solid, had some modest fortunes (e.g., references are made to ‘Noll’s theory of hygrosteric materials’), insufficient to make it to the Oxford English Dictionary. Neither they made it to the OED two terms, hereafter italicized, that became widely adopted though with acceptations slightly different from Noll’s original ones, that is, extra-stress for the stress diminished by the pressure stress possibly to be found in a hygrosteric body at ease (more about this term later).References
Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecule systems. J. Elast. 100(1), 63–143 (2010)
Assis, A.K.T.: On Mach’s principle. Found. Phys. Lett. 2, 301–318 (1989)
Assis, A.K.T., Graneau, P.: Nonlocal forces of inertia in cosmology. Found. Phys. 26, 271–283 (1996)
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
Banfi, C., Fabrizio, M.: Global theory for thermodynamic behaviour of a continuous medium. Ann. Univ. Ferrara 27, 181–199 (1981)
Bilby, B.A.: Continuous Distributions of Dislocations pp. 329–398. North-Holland, Amsterdam (1960)
Bragg, L.E.: On relativistic worldlines and motions, and on non-sentient response. Arch. Ration. Mech. Anal. 18, 127–166 (1965)
Bunge, M. (ed.): Delaware Seminar in the Foundations of Physics. Studies in the Foundations Methodology and Philosophy of Science, vol. 1. Springer, Berlin (1967)
Bunge, M.: Foundations of Physics. Springer Tracts in Natural Philosophy, vol. 10. Springer, New York (1967)
Carathéodory, C.: Zur Axiomatik der Speziellen Relativitätstheorie. Sitz. Preuss. Akad. Wiss., Phys. Math. Kl. 5, 12–27 (1924)
Di Carlo, A.: A major serendipitous contribution to continuum mechanics. Mech. Res. Commun. 93, 41–46 (2018)
Di Carlo, D.: Continuum mechanics as a computable coarse-grained picture of molecular dynamics. J. Elast. (2019, in press)
Coleman, B.D.: Kinematical concepts with applications in the mechanics and thermodynamics of incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. 9, 273–300 (1962)
Coleman, B.D.: Simple liquid crystals. Arch. Ration. Mech. Anal. 20, 41–58 (1965)
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)
Del Piero, G.: An axiomatic framework for the mechanics of generalized continua. Rend. Lincei Mat. Appl. 29, 31–61 (2018)
Dunn, J.E., Fosdick, R.L.: The morphology and stability of material phases. Arch. Ration. Mech. Anal. 74, 1–99 (1980)
Einstein, A.: Prinzipielles zur allgemeinen Relativitätstheorie. Ann. Phys. 55, 241–244 (1918)
Epstein, M., Maugin, G.A.: The energy-momentum tensor and material uniformity in finite elasticity. Acta Mech. 83, 127–133 (1990)
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)
Ericksen, J.L.: Anisotropic fluids. Arch. Ration. Mech. Anal. 4, 231–237 (1959)
Ericksen, J.L.: Theory of anisotropic fluids. Trans. Soc. Rheol. 4, 29–39 (1960)
Ericksen, J.L.: Transversely isotropic fluids. Kolloid Z., 117–122 (1960)
Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)
Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378 (1962)
Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. Lond. A 244, 87–112 (1951)
Feynman, R.P.: Forces in molecules. Phys. Rev. 56, 340–343 (1939)
Frank, F.C.: On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)
Gallavotti, G.: Statistical Mechanics: a Short Treatise. Texts and Monographs in Physics. Springer, Berlin (1999)
de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Clarendon, Oxford (1993)
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
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)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Springer, New York (1984)
Gurtin, M.: Configurational Forces as Basic Concepts of Continuum Physics. Applied Mathematical Sciences, vol. 137. Springer, Berlin (2000)
Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Contiuna. Cambridge University Press, Cambridge (2010)
Gurtin, M.E., Martins, L.C.: Cauchy’s theorem in classical physics. Arch. Ration. Mech. Anal. 60, 305–324 (1976)
Gurtin, M.E., Mizel, V.J., Williams, W.O.: A note on Cauchy’s stress theorem. J. Math. Anal. Appl. 22, 398–401 (1968)
Gurtin, M.E., Williams, W.O.: An axiomatic foundation for continuum thermodynamics. Arch. Ration. Mech. Anal. 26, 83–117 (1967)
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]
Hamel, G.: Über die Grundlagen der Mechanik. Math. Ann. 66, 350–397 (1908)
Hand, G.L.: A theory of dilute suspensions. Arch. Ration. Mech. Anal. 7, 81–86 (1961)
Hand, G.L.: A theory of anisotropic fluids. J. Fluid Mech. 13, 33–46 (1962)
Hilbert, D.: Mathematische probleme. Arch. Math. Phys. 1, 213–217 (1901), reprinted in [44]
Hilbert, D.: Gesammelte Abhandlungen, vol. 3. Springer, Berlin (1970)
Ignatieff, Y.A.: The Mathematical World of Walter Noll. A Scientific Biography. Springer, Berlin (1996)
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)
Jeffery, G.B.: The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179 (1922)
Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin (1929)
Kröner, E.: Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin (1958)
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)
Leslie, F.M.: Some constitutive equations for anisotropic fluids. Q. J. Mech. Appl. Math. 19, 357–370 (1966)
Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge University Press, Cambridge (1927)
Marsh Noll, M.: Ordinary Tasks. Madbooks, Pittsburgh (2016)
Maugin, G.A.: Continuum Mechanics through the Twentieth Century. A Concise Historical Perspective. Solid Mechanics and Its Applications, vol. 196. Springer, Dordrecht (2013)
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
Niven, W.D. (ed.): The Scientific Papers of James Clerk Maxwell, vol. 2. Dover, New York (1965)
Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200(1063), 523–541 (1950)
Oseen, C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29(4), 883–899 (1933)
Pinker, S.: How the Mind Works. Norton, New York (1997)
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)
Podio-Guidugli, P.: Inertia and invariance. Ann. Mat. Pura Appl. 172, 103–124 (1997)
Podio-Guidugli, P.: Continuum Thermodynamics. SISSA Lecture Notes, vol. 1. Springer, Berlin (2019)
Podio-Guidugli, P.: On the mechanical modeling of matter, molecular and continuum. J. Elast. (2019). https://doi.org/10.1007/%2Fs10659-018-9709-y
Prager, S.: Stress-strain relations in a suspension of dumbbells. Trans. Soc. Rheol. 1, 53–62 (1957)
Ravi, R.: Comments on the culture of the force. Phys. Today 58(8), 15–16 (2005)
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]
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
Schutz, J.W.: Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time. Lecture Notes in Mathematics, vol. 361. Springer, Berlin (1973)
Seeger, A.: Recent advances in the theory of defects in crystals. Phys. Status Solidi 1, 669–698 (1961)
Sikorsky, R.: Boolean Albegras, 3rd edn. Springer, Berlin (1969)
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]
Šilhavý, M.: The scientific work of B.D. Coleman. J. Math. Mech. Solids (2019, in press)
Smith, H.E. (ed.): Autobiography of Mark Twain, vol. 1. University of California Press, Berkeley (2010)
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)
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)
Szekeres, G.: Kinematic geometry; an axiomatic system for Minkowski space-time: M.L. Urquhart in memoriam. J. Aust. Math. Soc. 8, 134–160 (1968)
Tadmor, E.B., Miller, R.E.: Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press, Cambridge (2011)
Tanner, R.I., Walters, K.: Rheology: An Historical Perspective. Rheology Series, vol. 7. Elsevier, Amsterdam (1998)
Taylor, G.I.: Motion of solids in fluids when the flow is not irrotational. Proc. R. Soc. Lond. A 93, 99–113 (1917)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
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]
Truesdell, C.: Corrections and additions to “The mechanical foundations of elasticity and fluid dynamics”. J. Ration. Mech. Anal. 2, 593–616 (1953)
Truesdell, C.: Corrigenda. J. Ration. Mech. Anal. 3, 801–802 (1954)
Truesdell, C.: Hypo-elasticity J. Ration. Mech. Anal. 4, 83–133 (1955). 1019–1020
Truesdell, C.: Das ungelöste Hauptproblem der endlichen Elastizitätstheorie. Z. Angew. Math. Mech. 36, 97–103 (1956)
Truesdell, C.: The Mechanical Foundations of Elasticity and Fluid Dynamics. International Science Review Series, vol. VIII, Part 1. Gordon & Breach, New York (1966)
Truesdell, C.: Six Lectures on Modern Natutal Philosophy. Springer, Berlin (1966)
Truesdell, C.: Essays in the History of Mechanics. Springer, Berlin (1968)
Truesdell, C.: An Idiot’s Fugitive Essays on Science. Springer, New York (1984)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 2nd edn. Springer, Berlin (1992), see [92] for an annotated edition
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004), edited by S.S. Antman
Truesdell, C., Toupin, R.: The Classical Field Theories. In: Flügge, S. (ed.) Encyclopedia of Physics, vol. III/1. Springer, Berlin (1960)
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)
Truesdell, C.A. (ed.): Continuum Mechanics III, Foundations of Elasticity Theory. International Science Review Series, vol. VIII, Part 3. Gordon & Breach, New York (1965)
Truesdell, C.A.: A First Course in Rational Continuum Mechanics, vol. 1, 2nd edn. Academic Press, San Diego (1991)
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)
Wang, C.C.: A general theory of subfluids. Arch. Ration. Mech. Anal. 20, 1–40 (1965)
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)
Wilczek, F.: Whence the force of \({F} = ma\)? I: Culture shock. Phys. Today 57(10), 11–12 (2004)
Wilczek, F.: Whence the force of \({F} = ma\)? II: Rationalizations. Phys. Today 57(12), 10–11 (2004)
Wilczek, F.: Comments on the culture of the force. Phys. Today 58(8), 17 (2005)
Wilczek, F.: Whence the force of \({F} = ma\)? III: Cultural diversity. Phys. Today 58(7), 10–11 (2005)
Willems, J.C.: Dissipative dynamical systems, part I: general theory. Arch. Ration. Mech. Anal. 45, 321–351 (1972)
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
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
Zaremba, S.: Sur une conception nouvelle des forces intérieures dans un fluide en mouvement. Mém. Sci. Math. Gauthier-Villars, No. 82 (1937)
Ziemer, W.P.: Cauchy flux and sets of finite perimeter. Arch. Ration. Mech. Anal. 84, 189–201 (1983)
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Podio-Guidugli, P., Virga, E.G. Scientific Life and Works of Walter Noll. J Elast 135, 3–72 (2019). https://doi.org/10.1007/s10659-019-09728-w
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DOI: https://doi.org/10.1007/s10659-019-09728-w
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