Karl Weissenberg - The 80th Birthday
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The Role of Similitude in Continuum Mechanics
One of Karl Weissenberg’s conceptions viewed after a quarter of a century
DR. P. U. A. GROSSMAN
Division of Building Research, CSIRO, Melbourne, Australia
1. PURPOSE OF THE ESSAY
Karl Weissenberg belongs to those pioneers in science who view a problem as part of the picture of nature as a whole rather than in isolation, who find a solution satisfying only if it fits into a general wider scheme, and who apply notions and methods originally developed for a different subject. He has made his mark in a variety of fields but I shall deal here with only one of the subjects with which his name is linked.
I had the good fortune of working closely with him on the background to what is known as the “Weissenberg effect”. When I addressed my first letter to “K.W.” in 1956 the essentials of his ideas had long been published, but there were details to be clarified and new questions considered which invariably arise after the first breakthrough. Close cooperation over two years offered me great and valuable stimulation.
Since that time we have both become active in different directions. Progress in rational mechanics has made a great impact on the interpretation of rheological effects and techniques of measurement have been refined. It will be of value to reassess Karl Weissenberg’s ideas, to place them into perspective and to emphasize those points that have not been generally appreciated. Perhaps he will disagree with some things that I have to say, perhaps I have misunderstood as so many have misunderstood him. I may also lay myself open to the charge of branching off into matters I am not competent to handle. However, close though brief cooperation gave me an advantage over those who know Karl Weissenberg only from his writings or from casual meetings.
Cannot the full story be gleaned from his own writings? Why a disciple’s secondhand account? Do I presume to have more ability in expressing his ideas when indeed he gave me valuable hints on written presentation?
Weissenberg is not a prolific writer and it is perhaps significant that many of his publications are contained in proceedings of conferences. Besides, the main purpose of this essay is not to restate his thoughts but to comment on their significance.
2. WEISSENBERG’S CONTRIBUTION AS COMMONLY VISUALIZED
Let us first recall the commonly accepted picture of Karl Weissenberg’s contribution to the field of “normal stresses”. He is credited with developing ingenious experiments to demonstrate striking effects shown by certain liquids under rotational shear. He is credited with postulating that in shear flow the second normal-stress difference is zero, a condition often regarded as theoretically unjustified and experimentally disproved. His name is also linked with an instrument which all too often is seen primarily to be a versatile precision viscometer with constant or periodic drive and with only optional facilities to measure the first normal stress difference.
It is true that Karl Weissenberg prepared striking experiments to demonstrate tension in the flow direction. It is true that his propositions lead to the equivalence of the other two normal stresses in the case of laminar shear. It is also true that part of the function of the rheogoniometer is the measurement of viscosity. But these statements do not place the emphasis correctly. A few specific practical consequences, significant as they are, have concealed the mainspring of Weissenberg’s considerations, an idea that is far from exhausted, outdated or superseded. The basis of his conclusions is what he called “similitude of anisotropy”.
3. REQUIREMENTS FOR SIMILITUDE
“Two systems are similar if their corresponding variables are proportional at corresponding locations and times”. (Skoglund1). Note that the constant of proportionality need not be stated and can vary from one location to another and from one time to another. Before we can decide whether two systems are similar or not, we must be able to determine which variables of one system correspond to those of the other.
Conventionally, strain is defined by a mapping of one configuration of points onto another while stress is defined by a mapping of forces onto surface elements or facets. There are no criteria for a correspondence between the set of surface forces at one location and time and the set of vectors that are obtained from the mapping and describe the deformed configuration.
Weissenberg 2 , 4 pointed to the need for new definitions of stress or strain such that a correspondence could be established. For example, the conventional stress tensor may be retained while a new strain tensor is introduced which maps onto the same set of facets a set of vectors indicative of the deformed state. This led to the concept of the spacing bar2 or the separation vector5 (not to be confused with the separation ratio6). The spacing bar is a contra-variant vector which in the undeformed reference state coincides with the covariant vector density representing a facet and is embedded in the material. In other words, in the undeformed configuration the spacing bar is normal to the facet and its length is a measure of the surface area. Throughout the deformation it links the same two material points and does not generally remain normal to the facet. The direction and length change of the spacing bar are indicative of the mutual displacement of parallel surfaces.
4. WEISSENBERG’S PROPOSITION
A brief restatement of the proposition is required here. For fuller discussions the reader is referred to the original papers 2,4 and to 5 and 6. Consider stress in an initially isotropic material as a function of configurational changes within the body. Because of the common experience that rheological bulk properties are independent of shear properties and, in particular, in incompressible materials the hydrostatic stress component is not determined by the deformation, a relationship need be established only between the deviatoric or anisotropic components of stress and strain.
While it is futile to search for a quantitative relationship of universal validity, Karl Weissenberg argued that a purely geometric restriction on the directional distribution of the vectors describing deformation and stress may be generally applicable and indeed plausible on physical grounds. He proposed a similitude between the anisotropic component of stress and the anisotropic component of strain, strain being defined as the mapping of spacing bars (separation vectors) onto surface elements. When strain history has an influence, the hypothesis is useful only if all relevant strains are mutually similar, a condition that is often fulfilled.
The proposition relates the direction of the stress vector and the separation vector (spacing bar) for any surface element. Because of the isotropic component, these vectors are not required by similitude of anisotropic parts to be colinear but they must lie in a plane containing also the normal to the element. Other forms of the proposition have been listed elsewhere 2,5 and their mutual equivalence shown5. They include notably the condition that differently oriented pairs of parallel surface elements are subject to the same normal forces if they have undergone the same normal separation.
5. JUSTIFICATION OF THE PROPOSITION
Let us first see that some directional restrictions on a constitutive equation have been widely accepted. When stress is an isotropic symmetric tensor function of a single variable it is expressible by a power expansion of that variable 7 (p. 32) and it must therefore have the same principal axes. This holds whether the independent tensor is strain8, strain rate9 or the gradient of any higher acceleration10. It would seem physically absurd in an isotropic body to expect principal stress directions other than the directions favoured by the variable on which the stress solely depends, because any other principal strain direction in an isotropic body would give rise to the question “Why this and no other ?”
By virtue of defining stress and strain in analogous forms Weissenberg can go one step further. He need not confine directional arguments to the directions characterizing the tensors as a whole but can discuss directions separately for vector belonging to each surface element. If his proposition were invalid, if the stress vector for a given surface element lay outside the plane defined by the separation vector and the normal, one would reasonably ask: “Why this direction and no other ?”
Having established the physical reasons for a proposition restricting the range of permissible constitutive equations, we must check its compatibility with other requirements governing constitutive equations. The need for preserving invariance to a wide range of transformations has been paramount in Weissenberg’s considerations but rather than follow his listing we shall check compatibility with the principles as laid down for the field theories 7, 11 of mechanics.
(a) Consistency with balance equations. Balance of mass or energy is not affected by a proposition that does not restrict scalars. The definition of stress used is conventional and leaves the balance of momentum undisturbed.
(b) Coordinate invariance. Stress and strain as defined by Weissenberg are both contravariant tensor densities, their transformation laws being identical. Force and separation vectors are also both contravariant. The proposition is thus invariant to transformations of cooordinates.
(c) Dimensionsal invariance. Similarity can be conceived as a transformation from one unit of measurement to another12. Similar systems remain similar under a change of units. This is one of the main attractions of Weissenberg’s proposition.
(d) Material indifference. The proposition is based on a set of embedded vectors representing the surface elements through a point in the material. Any relation between vectors that are linear vector functions of the same embedded vector must be independent of an observer’s frame of reference.
(e) Equipresence. Stress can depend on many variables other than the motion. Any variables that are scalars, e.g. temperature, in no way influence Weissenberg’s directional relation. The same holds for vector or tensor variables that affect stress only through their invariant values. Should stress be a function of two or more variables, each of which is expressible as a mapping onto the same surface elements, then the proposition gives useful information only when there is a similitude between all independent variables. An example was mentioned earlier, namely the case where more than one previous configuration determine the stress.
(f) Local action. The postulate applies for the close vicinity of any one point and does not say anything about changes from point to point.
The hypothesis rests on several premises and cannot be expected to hold when these premises are not valid.
We are dealing with a continuum that is isotropic in a stress-free ground-state. Consideration has been given to extending the rule to certain more restricted isotropy groups.
We are dealing with stress as a function of a single variable, strain. Weissenberg’s proposition does remain valid also for more than one independent variable of tensor character, requiring that the space in which the force on a given surface be contained is the space containing the independent vectors. However, it becomes useless information unless there are similitude relations between the independent variables.
We are dealing with a relationship that leaves magnitudes unspecified and allows these magnitudes to vary from one location and one instant to another. Similitude of anisotropy of stress and strain does not provide the whole constitutive equation but must be incorporated into a wider theoretical framework and supplemented by empirical measurement.
7. EXPERIMENTAL EVIDENCE
One consequence of the Weissenberg hypothesis is the requirement in simple shear flow, or indeed in any viscometric flow, that normal stresses in the direction of the flow gradient and in the neutral direction be equal, i.e. that what is known as the second normal-stress difference be zero. Methods used to measure these stress differences have been reviewed by Lodge6 and more recently by Ginn and Metzner13 and by Olabisi and Williams14. Although it is agreed that the second normal stress difference is usually much smaller in absolute value than the first, a number of researchers report values significantly different from zero, some positive and some negative. The second normal-stress difference could even exceed the first15. The existence of secondary flow phenomena16 and the surface shape in channel flow17 have also been linked with a non-zero second normal-stress difference. All this throws doubt on the universal validity of the hypothesis and has led to the introduction of a class of “Weissenberg fluids”18 for which alone this difference is zero.
Experimental evidence may have established non-zero secondary differences, favouring at present negative values but this need not necessarily prove the Weissenberg hypothesis false. Assumed conditions and prerequisites for deriving the equality of certain normal stresses from the hypothesis may not be strictly applicable, e.g. the flow may not everywhere be viscometric, or the stress may depend on tensor variables other than strain which are not codirectional or similar. The latter applies in Giesekus’19 model in which the introduction of a mobility tensor can cause the second normal-stress difference to become non-zero.
The demonstration of effects such as the climb of elastic liquids up stirring rods; the condition that the second normal stress difference be zero in simple shear flow; the development of an instrument capable of providing all the measurements during flow to derive stress at any angle, a “flow-angle measurer” or rheogoniometer; all these developments were based on the cardinal idea of similitude of anisotropy of stress and strain.
By introducing the idea of similitude into tensor functions and pointing to the prerequisite of basing the systems on a common set of geometric entities, Karl Weissenberg disclosed a powerful theoretical weapon that could profitably be applied also to fields other than mechanics.
The author gratefully acknowledges the help received in discussions with Dr. P. F. Lesse and Dr. A. V. Ramamurthy.
1. Skoglund, V. J., “Similitude”. (International Textbook Co., Scranton, Pa. 1967).
2. Weissenberg, K., Report of the Gen. Conf. British Rheologists’ Club (London 1946).
3. Weissenberg, K., Nature 1959, 310 (147).
4. Weissenberg, K., Proc. 1st International Congress on Rheology (North Holland Publ. Co., Amsterdam 1949).
5. Grossman, P.U.A., Kolloidzeitschrift, 174, 97 (1961).
6. Lodge, A. S., “Elastic Liquids” (Academic Press, London, 1964).
7. Truesdell, G. and Noll, W., The Non-linear Field Theories of Mechanics. In “Encyclopedia of Physics” III/3. ed. Flügge. (Springer-Verlag, Berlin, 1965).
8. Managhan, F. D., Am. J. Math., 59, 235 (1937).
9. Reiner, M., Am. J. Math., 67, 350 (1945).
10. Rivlin, R. S. and Ericksen, J. L., J. rat. Mech. Analysis, 4, 323 (1955).
11. Truesdell, C. and Toupin, R. A., Principles of Classical Mechanics and Field Theory. In “Encyclopedia of Physics” III/1 ed. Flügge (Springer-Verlag, Berlin, 1960).
12. Sedov, L. I., “Similarity and Dimensional Methods in Mechanics” (transl.), (Infosearch, London, 1959).
13. Ginn, R. F., and Metzner, A. B., Trans. Soc. Rheol., 13, 429 (1969).
14. Olabisi, O. and Williams, M. C., Trans. Soc. Rheol., 16, 727 (1972).
15. Han, C. D. and Charles, M., Trans. Soc. Rheol., 14, (1970).
16. Criminale, W. O. Jr., Ericksen, J. L. and Filby, G. L. Jr., Arch. rat. Mech. Analysis, 1, 410 (1958).
17. Wineman, A. S. and Pipkin, A. L., Acta Mech., 2, 104 (1966).
18. Giesekus, H., Rheol. Acta., 2, 50 (1962).
19. Giesekus, H., Rheol. Acta., 5, 29 (1966).
Preface / Acknowledgements / Biographical Notes
Weissenberg’s Influence on Crystallography
Karl Weissenberg and the Development of X-Ray Crystallography
The Isolation of, and the Initial Measurements of the Weissenberg Effect
The Role of Similitude in Continuum Mechanics
The Effect of Molecular Weight and Concentration of Polymers in Solutions on the Normal Stress Coefficient
Elasticity in Incompressible Liquids
The Physical Meaning of Weissenberg's Hypothesis with Regard to the Second Normal-Stress Difference
A Study of Weissenberg's Holistic Approach to Biorheology
The Weissenberg Rheogoniometer Adapted for Biorheological Studies
Dr. Karl Weissenberg, 1922-28
Weissenberg’s Contributions to Rheology
The Early Development of the Rheogoniometer
Some of Weissenberg's More Important Contributions to Rheology: An Appreciation
Publications of Karl Weissenberg and Collaborators / List of Contributors
© Copyright John Harris