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“Reality” and Representation in Mechanics: The Legacy of Walter Noll

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Abstract

The foundations of the mechanics of generalized continua are revisited in the light of the theoretical progress made in the last decades. The paper includes a summary of the scientific activity of W. Noll, to whom a large part of this progress is due.

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Change history

  • 20 September 2019

    <Emphasis Type="Bold">Correction to: J. Elast. (2019) 135: 117–148</Emphasis> <ExternalRef><RefSource><Emphasis Type="Bold">https://doi.org/10.1007/s10659-018-9697-y</Emphasis></RefSource><RefTarget Address="10.1007/s10659-018-9697-y" TargetType="DOI"/></ExternalRef>

    This erratum concerns a series of misprints, due to a human error occurred after the final proofreading. Due to this error, some capital <Emphasis Type="Italic">F</Emphasis> of the manuscript became lower-case <Emphasis Type="Italic">f</Emphasis> in the final version. Precisely, <Emphasis Type="Italic">all f contained in</Emphasis> (37) <Emphasis Type="Italic">to</Emphasis> (40) <Emphasis Type="Italic">and</Emphasis> (78)<Emphasis Type="Italic">,</Emphasis> (79)<Emphasis Type="Italic">,</Emphasis> (85)<Emphasis Type="Italic">,</Emphasis> (90) <Emphasis Type="Italic">should be read as F</Emphasis>.

Notes

  1. See, e.g., Laplace’s introduction to his “Essai philosophique sur les probabilités”.

  2. For instance, the line can be piecewise linear, or piecewise polynomial of different orders.

  3. In Newton’s words, “Absolute space, in its own nature and without anything external, always remains similar and immovable”. Cited in Mach [29], p. 226.

  4. The coincidence of the distances is possible only if the physical space has itself the structure of an Euclidean space. Otherwise the coincidence is only approximate, just as for the distances between points on the earth’s surface and between their images on the support.

  5. The “filling” operation may consist, for example, in joining the neighboring points of \(\frak{X}^{N}_{R}\) with a finite-element mesh, and taking as \(\Omega _{R}\) the region covered by the mesh, possibly with some regularization at the boundary.

  6. After receiving a finite number of distances between particles, it is up to the placer to decide between a discrete and a continuous representation. The origins of continuum mechanics go back to the deliberate choice of the continuous representation made by Cauchy [4].

  7. In many cases, it is not even clear what the physical boundary of a body is. The regularity to be attributed to the boundary was a major concern of Noll [39, 43]. The properties of what he called fit regions are in fact restrictions on the choices to be made by the placer.

  8. For example, for differentiable functions \(f\) the preservation of the orientation and the local injectivity at \(x_{R}\) are ensured by the analytical condition \(\det \nabla\! f(x _{R})>0\).

  9. For example, a second-order continuum need not represent a body with a twice differentiable structure. The body’s structure may even be discrete. But, by a decision of the placer, only twice differentiable deformations are considered admissible. An example is Ambrosio and Tortorelli’s model for fracture [2], in which cracked configurations are represented by differentiable functions with very large values of the deformation gradient inside the cracked zone.

  10. Noll [33].

  11. For example, as shown by (4), differentiability is preserved under composition if all deformations from the reference placement are differentiable and have a differentiable inverse.

  12. Modified by the scale factor, here taken equal to one for simplicity.

  13. The solution of this system is unique because the vectors \(e^{i}\) are linearly independent. I omit the proof that the solution is independent of \(i\).

  14. Just as the regularity in space, the regularity in time can be decided by the placer, possibly without any physical motivation.

  15. According to ([51], Sect. 17), a reference system is “a set of objects whose mutual distances change comparably little in time, like the walls of a laboratory, the fixed stars, or the wooden horses on a merry-go-round”. To assume that the reference system does not change with \(t\) does not mean that these objects “occupy a fixed position in the physical space”, since only an absolute spaces has “positions”, and we are not supposing that the physical space is an absolute space. We are only supposing that the mutual distances of the points of the reference system do not change during the evolution.

  16. The identification of observers with reference frames, see, e.g., Truesdell and Noll [51], Sect. 17, is an example of confusion between “reality” and representation, which may lead to incorrect conclusions. See the discussion on the law of gravitation in Sect. 8.5 below.

  17. Since \(f^{\alpha }\) is differentiable, the conditions (i), (ii) imply that the tensor \(\nabla\! f^{\alpha }(x ^{\alpha })\) is invertible and with positive determinant. Then the field \(F\) is made of invertible second-order tensors with positive determinant. However, being the gradient of a different function \(f^{\alpha }\) at each \(x_{R}^{\alpha }\), \(F\) is not expected to be itself a gradient.

  18. For an account of the literature on the subject see Deseri and Owen [15] and Owen [45]. In plasticity, the decompositions in (27) are called the Kröner-Lee and the Clifton decomposition, respectively. In the paper [11] they are compared with the additive decomposition (28).

  19. Its relevance is due to the form of the energy of a structured deformation, see Sect. 5.2 below.

  20. It has been said that “force has the same status in science as the notion of epicycles in astronomy; although the use of these terms may lead to correct results, it should not be supposed that they are part of nature itself” (attributed to G. Berkeley by Jammer [27], p. 204), and that forces should be rather considered as “methodological intermediates (between physical reality and representation) …comparable to the so-called middle term in the traditional syllogism” (ibid., preface to the Dover Edition, 1999).

  21. More in general, a dependence on additional state variables can be assumed, see Sect. 8 below.

  22. In fact there are energies, such as the frictional contact energy, which are only Gâteaux differentiable, see footnote 27. Here we assume differentiability just for simplicity. Note that \(b_{Rt}(x_{R})\) and \(s_{Rt}(x_{R})\) are vectors, while \(B_{Rt}(x_{R})\) and \(S_{Rt}(x_{R})\) are second-order tensors.

  23. Del Piero and Owen [14].

  24. Choksi and Fonseca [5], Owen [44], Del Piero [8].

  25. Barroso et al. [3], Theorem 3.2.

  26. This additional dependence is generally assumed, see, e.g., the theory of Gurtin and Anand discussed below. Perhaps, in the theorem proved in [3] this dependence is excluded by some restrictive hypothesis.

  27. We recall that a dissipation potential is a positive Gâteaux differentiable function, and that a function \(\phi \) is Gâteaux differentiable if the directional derivative

    $$ \breve{\nabla }\phi \bigl(F^{d}\bigr)\triangleright V =\lim _{\varepsilon \to 0^{+}} \frac{\phi (F^{d} +\varepsilon V) -\phi (F^{d}) }{\varepsilon } $$

    exists at all \(F^{d}\) and for all directions \(V\). For homogeneous functions \(\frak{h}\) of degree one, the notation \(\frak{h}(V) = \frak{h}\triangleright V\) was introduced in [13], Sect. 2.2, to which we refer for further details.

  28. See, e.g., Gurtin et al. [25], Sect. 29.

  29. Therefore, the external actions are indifferent, in the sense that they obey the transformation law (14) for vector fields.

  30. Note that the “moment of momentum” is evaluated at the points \(x_{t}\) of the region \(f_{t}(\Omega _{R})\) and not at the points \(x_{R t}\) of \(\Omega _{R}\).

  31. This deduction of the balance laws from indifference is due to Noll [35]. It answers the questions raised by Truesdell about the nature of this law [49].

  32. Since the property (47) is a direct consequence of the conservation principle, to call it principle is inappropriate. In Noll’s words, “the principle of material frame-indifference is not a law of physics… it is merely a prescription for avoiding nonsense” [42].

  33. Truesdell [50], p. 154. This assumption excludes, for example, the presence of different surface effects at the physical boundary and at the interior surfaces of the body.

  34. This was a conjecture of Cauchy, proved by Noll in [34].

  35. This is the famous tetrahedron theorem of Cauchy.

  36. In particular, for opposite directions \(n_{R}\) and \(-n_{R}\) this implies

    $$ s_{R}(x_{R},n_{R}) =-s_{R}(x_{R},-n_{R}), $$

    which is the local form of Newton’s law of action and reaction.

  37. We recall that \((\det F)^{-1}T_{R} \nabla\! f^{T}\) is the Cauchy stress tensor. Therefore, this condition states the symmetry of the Cauchy tensor.

  38. Germain [20, 21].

  39. See Choksi and Fonseca [5].

  40. It is remarkable that a single equation collects the information traditionally given by two separate conditions, the energy balance and the entropy imbalance, see, e.g., [25], Sects. 26, 27. Note that in the present analysis it is assumed that the dissipation potential is positively homogeneous of order one. In reality, as explained in the post-post scriptum at the end of the paper, alternative options are possible.

  41. As done, for example, in the gradient plasticity theory of Sect. 7.4.

  42. Coleman and Noll [7]. In particular, (60) is a mechanical counterpart of their relation (5.1), and the inequality on the right is a mechanical counterpart of the Clausius-Duhem inequality ([51], Sect. 79). However, while in [7] the stress tensor \(T_{R}\) is regarded as an internal action, and for it a constitutive equation is postulated, here \(T_{R}\) is an internal action, and the constitutive equation (58) is deduced from the conservation principle.

  43. Here and in the following for the “time” derivative I use the symbol \(\delta \), reserving the more familiar superimposed dot to the case in which “time” is the physical time.

  44. In the paper [13] it has been shown that \(\breve{\nabla }\phi ^{d}\) is the bounding map which determines the elastic region in which the tensor \((T_{R}-\nabla \varphi ^{d}(F^{d}))\) must lie, and that \(\nabla \varphi ^{d}(F^{d})\) is the backstress tensor which appears, for example, in the kinematical hardening model. If the boundary of the elastic region has corner points, the determination of \(\delta F^{d}\) at such points is not unique.

  45. Equation (67)2 implies the existence of the second directional derivative

    $$ \breve{\nabla }^{2}\phi ^{d}(A)\{H\}\triangleright K = \lim _{\varepsilon \to 0^{+}}\frac{\breve{\nabla }\phi ^{d}(A+ \varepsilon H)\triangleright K -\breve{\nabla }\phi ^{d}(A)\triangleright K }{\varepsilon }. $$

  46. That is, the volume density for the contact actions is the opposite of the volume density for the distance actions. The assumption (74) was made in 1973 by Noll, who called it area-volume continuity ([37], p. 78). The contact actions with this property were called weakly balanced Cauchy fluxes by Gurtin and Martins [23] and by Šilhavý [47], who used assumption (74) to weaken the regularity hypotheses made in the standard proof of the tetrahedron theorem.

  47. Del Piero [9]. Equation (74) suggests that the contact actions can be represented as distance actions and vice versa. The founders of continuum mechanics were well aware of this fact, see, e.g., Maxwell [30] and (Love [28], Note B of the Appendix).

  48. In components, \((\operatorname{div} \mathbb{T}_{R} F ^{dT})_{ih} =\mathbb{T}_{ijk,k}F^{d}_{hj}\) and \((\mathbb{T}_{R} \nabla\! f^{dT})_{ih} =\mathbb{T}_{ijk}F^{d}_{hj,k}\).

  49. Aifantis [1], Fleck and Hutchinson [18], Gudmundson [22], Fleck and Willis [19], etc.

  50. Note that it is incorrect to write \(\hat{\mathbb{T}} _{R}^{\mathit{diss}}\triangleright \nabla V^{p}\) as an inner product as done in (90.48) of [25], since \(\hat{\mathbb{T}}_{R}^{\mathit{diss}}\) is the map \(\breve{\nabla }_{2}\phi ^{d}\), which is not linear but only homogeneous of order one.

  51. Truesdell and Noll [51], Sect. 1.

  52. See Noll’s comments in [32], p. 8, [51], Sect. 15, [36], p. 2. For an intrinsic formulation of elasticity see [41], and for an extension to thermomechanics see Seguin’s thesis [46].

  53. The paper [36] in which the New Theory was formulated did not receive an adequate attention. This had the unfortunate consequence that most authors refer to Noll’s initial views rather than to their more mature developments. This is the case of the theory of fading memory which, as said in the next section, was criticized when it had already been set aside by the New Theory.

  54. For example, this is the case of microscopic deformations due to variations of temperature or of the electric or magnetic field.

  55. This is the case in the New Theory and in the theory of generalized standard materials of Halphen and Nguyen [26].

  56. Noll [33], Truesdell and Noll [51], Sect. 28.

  57. Noll [33], Truesdell and Noll [51], Sect. 38.

  58. Coleman and Noll [6].

  59. See [16, 17], and other papers cited therein.

  60. See my comments in Sects. 1 and 7 of [10] and in Sect. 4 of [12].

  61. Berkeley’s objections are summarized in Jammer [27], Chap. 11. For Mach see [29], Chapt. II.VI.8 and Appendix XX. For Noll see [35, 40].

  62. In a two-scale continuum, the kinetic energy can be augmented with a term depending on the microscopic velocity \(V\) and proportional to a microscopic equivalent of the mass. In a multi-scale continuum this leads to “generalized” conceptions of kinetic energy and mass, see e.g. [48].

  63. …“on regarde les forces d’inertie comme des forces véritables qui sont les interactons entre les corps dans notre syst‘eme solaire et la totalité des objets dans le reste de l’univers” [35].

  64. In general, the “fixed stars” are taken as the reference system representing the “rest of the universe”. For experiments in which the gravitational effect of the earth is considered predominant, the “rest of the universe” can be identified with the earth, and the walls of the laboratory can be taken as the reference system. “I have remained to the present day the only one who insists upon referring the law of inertia to the earth, and in the case of motions of great spatial and temporal extent, to the fixed stars”, Mach [29], p. 568.

  65. Usually the kinetic energy is placed, with the plus sign, in the integral representation (37) of the internal energy, though the inertia force (107) is generally regarded as an external action.

  66. The superimposed dot denotes the derivative with respect to the physical time.

  67. This criticism of apparent forces is mine. There is no trace of it in Noll’s papers. Perhaps the reason is that, according to his updated definition of placement, each placement of the body belongs to a different Euclidean space [36, 37]. Therefore, the idea of a “variable reference frame” makes no sense. On the contrary, this idea is meaningful when all placements belong to the same ℰ.

  68. For the revolutionary character of this change see a more detailed comment in Sect. 3.7 of my paper [12].

  69. This agrees with Noll’s belief that “the basic concepts of mechanics should not include items such as momentum, kinetic energy, and angular momentum, because they are relevant only when inertia is important”, [40], paper N2.

  70. [40], paper N1.

  71. Noll himself quotes a dictionary’s definition saying that the physical space is “the unlimited and indefinitely three-dimensional expanse in which all material objects are located and all events occur”. He seems to accept this definition, “vague and ambiguous as it may be”.

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    Gianpietro Del Piero

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    Gianpietro Del Piero

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Del Piero, G. “Reality” and Representation in Mechanics: The Legacy of Walter Noll. J Elast 135, 117–148 (2019). https://doi.org/10.1007/s10659-018-9697-y

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