2.1 Preamble
Most researchers will never have been fortunate enough to have heard Walter either teach or talk, and be directly acquainted with his work and ideas only via his papers or articles. The concise nature of these cannot always convey viewpoint, motivation or reasoning behind his particular approach.
3 While the value and depth of his papers were widely advertised by his former adviser Clifford Truesdell, this sometimes proved to be less helpful than intended. Truesdell’s writings were stimulating and greatly informative, but often their confrontational stance could alienate researchers with different scientific cultural backgrounds; here one only has to think of understandable responses to his designation of ‘rational’ mechanics in respect of his favoured approaches. Unfortunately, as a consequence of such alienation, and lack of acquaintance with contemporary mathematical concepts and notation, Noll’s work has not been fully appreciated. In several cases criticism has come from within the Continuum Mechanics fraternity.
2.2 Simple Fluids
Rivlin took issue with Noll’s definition of a simple fluid. Noll (cf. [
9,
10]) had shown that the stress response functional for any simple fluid can be expressed via material frame-indifference
4 (mfi) in terms of a reduced functional whose domain consists only of positive-definite symmetric tensor histories which at current time take the value
1, the identity tensor. The restriction imposed by mfi at this stage is that such reduced functional be invariant under the orthogonal group. Rivlin [
11] argued that the definition of simple fluid for the functional in question mandated invariance under the unimodular group, a stronger restriction than that of Noll, given the larger nature of the unimodular group. However, such restriction for Rivlin involved histories which at current time take the form
\(\mathbf{H}^{T}\mathbf{H}\) with
H unimodular. Since
\(\mathbf{H}^{T}\mathbf{H} = \mathbf{1}\) if and only if
H is orthogonal, this criticism is unfounded: such histories do not lie in the domain of the reduced response functional unless
H is orthogonal. This was pointed out by myself in [
12] and the source of the error identified. This error stemmed from a failure to distinguish between the domain of a functional and that which corresponds to a reduced form of this functional. Rivlin did not accept this technicality and replied with a repetition of his mistake [
13,
14]. This proved to be a salutary lesson in having to appreciate that, for whatever reason, one cannot always convince someone of their misconceptions or errors. Such a situation, of much greater significance, also arose concerning the status and interpretation of mfi.
2.3 Material Frame-Indifference
In a summer school held in 1982 at Noto, Sicily, I learned from Ingo Müller of his approximative constitutive relations for stress and heat flux fields in a rarefied gas derived on the basis of kinetic theory (cf. [
15]). These he claimed to violate mfi because these fields were dependent upon spin, contradicting the impossibility of such dependence as argued in most texts on the basis of mfi (cf., e.g., [
16], p. 156, [
17], p. 251). However, what such texts accomplish is to rule out dependence upon spin as computed relative to the frame of a general observer. Crucially, however, Müller’s spin is computed with respect to an (and hence any) inertial frame
5 and accordingly was a field upon which all observers could agree, since the notion of inertial frame is central to classical physics (in serving to establish the concept of force). However, from my understanding of mfi as expounded in Noll’s lectures, such dependence upon inertial spin was indeed possible, and this I indicated to Ingo. Since he strongly disagreed, I thought long and hard on the matter and wrote [
18] to explain my perspective. I also wrote to Walter to see if I had misunderstood the nature of mfi. In his reply of November, 1983, Walter wrote ‘As to the matter concerning frame-indifference, I fully agree with your viewpoint, and I told Ingo Müller so when I recently met him at a meeting in Providence, RI.’ Unfortunately Ingo never accepted this and the nature of mfi/objectivity continues to be controversial, particularly if it is identified with ‘invariance under superposed rigid motions’ (isrbm). However, the possibility that material response could be sensitive to spin relative to an inertial frame was accepted by both Noll and Müller and does not violate any fundamental tenet of Physics.
The essence of the notions of mfi and isrbm stems from recognition that constitutive relations which purport to describe material behaviour as manifest to different observers must be subject to some restriction in order to make sense. Said differently, any two observers should be able to agree upon the nature and behaviour of any given material of interest, and to recognise how this imposes restrictions upon, and relations between, response functions. How to label, frame and phrase discussion of this fundamental aspect of modelling has long been problematic. As detailed on pp. 28–29 of [
19], over the years Walter variously described the concept as ‘isotropy of space’ and ‘objectivity’ before finally settling on ‘frame-indifference’.
It clarifies matters to discard once and for all the isrbm interpretation. While at first sight it is ‘obvious’ that rotation of a body without distortion will merely rotate (
inter alia) the stress field therein in precisely the same way, a little thought suggests this might not be the case. If the body is initially at rest in an inertial frame, then subsequent rotation with respect to this frame will involve a motion which introduces non-uniform acceleration of points of the body. Accordingly, some form of external force system is required to preserve rigidity. Just how this might be achieved is problematic, not to say metaphysical. Of course, what isrbm immediately achieves by its very nature is to negate any constitutive dependence upon spin since no restriction is placed upon the rate at which rotation is performed. Since material behaviour rarely seems to exhibit spin dependence, its absence might be regarded as evidence of the veracity of isrbm. However, fundamental considerations indicate that spin dependence is to be expected quite generally: common experience merely indicates that sensitivity thereto is usually negligible. To appreciate this somewhat subtle point, one must recognise that continuum fields describe physical behaviour which is monitored by measuring devices. In order to interpret recorded data it is necessary to relate localised measurement values to continuum field values. However, local measurements involve associated scales of both length and time, since no measuring device can register physical information that is either instantaneous or localised at a geometrical point.
6 Rather, measurement values represent local space-time averages of material behaviour at specific scales. Thus any measurement made on a body must be expected to be sensitive to rotation of the body (which occurs during the time lapse required to register any given measurement) relative to the measuring device employed.
7 Accordingly, fields such as stress and heat flux must
a priori be expected to be sensitive
8 to rotation with respect to an (hence any) inertial frame. In [
15] Müller had exhibited such sensitivity via kinetic theory. The foregoing considerations should suffice to put paid to isrbm, but it has proved a hard beast to slay and continues to attract believers.
It was helpful to talk privately with Rivlin, at Nottingham in 1985, on the matter of isrbm. In a few words he disposed of its standard interpretation by saying that of course one should not envisage rigidly rotating a body of interest relative to the observer, but rather rotating the observer relative to the body. Although such motions would appear identical to the observer (consider films taken of the body with a camera held by the observer in which any ‘background’ has been erased: both films reveal the same relative rotation) there is a physical difference between the two relative rotations. The state of the body has not changed as a consequence of rotation of the observer (for whom field values will merely be perceived to be rotated versions of those which correspond to absence of relative rotation). Rivlin [
23] described this viewpoint as ‘invariance under superposed rotation’. In practice this is almost identical with ‘invariance under change of observer’ (cf. [
24], p. 206): here a change of observer, from
\(O\) to
\(O^{*}\) say, is envisaged in which the same response functions are employed by
\(O\) and
\(O^{*}\). This notion may be re-interpreted as a change in perspective of a single observer were he/she to have undergone a rotation. In so doing there can be no query (otherwise reasonable) as to why the same response functions should be involved: for example, in the case of elastic bodies, just why
\(O\) and
\(O^{*}\) should of necessity select the same reference configuration becomes immediately apparent.
Since isrbm continued to be regarded widely as an absolute statement, in 2002 I wrote [
25] to point out that it was helpful to emphasise the nature of objectivity in its broad sense as stemming from observer consensus. Here I was influenced by comments made by Jerry Ericksen, in a letter of November, 1980, when discussing a model of heat conduction which I felt illuminated the isrbm/mfi controversy (cf. (2) in [
26]): ‘As I interpret the word, objectivity refers to what it is that different observers can agree upon. First, different observers can and do make their own choice of reference configuration’. However, I was ignoring the advice of Truesdell who, in October, 1980, wrote ‘I hope you can avoid using “objectivity” because that term is often used (e.g. by Müller) to denote “invariance under superposed rigid motions”.’ In [
25] attention was drawn to observer consensus implicit in usage of mathematical ideas and results, to physical concepts such as inertial frames, events, and, for classical physics, to time lapses and distances between simultaneous events. The specific additional aspects of objectivity for continuum modelling of macroscopic behaviour were postulated to be: agreement upon the nature of any given ideal material, and upon all possible responses of an ideal material, no matter what the relative motion of the observers might be. Derivation of restrictions upon response functions involved precisely Rivlin’s notion of behaviour conceived in terms of two separate motions of a single observer relative to the body. The argumentation employed a second observer in order to stress that different observers may choose different response functions, echoing Ericksen’s above remark.
A more detailed discussion which includes microscopic considerations is to be found in Chap. 12 of [
27].