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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”. (Skoglund^{1}). 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 bar^{2} or
the separation vector^{5} (not to be confused with the separation ratio^{6}).
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 shown^{5}. 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 strain^{8},
strain rate^{9} or the gradient of any higher acceleration^{10}.
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
another^{12}. 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.

6. LIMITATIONS

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 Lodge^{6} and more recently by Ginn
and Metzner^{13} and by Olabisi and Williams^{14}. 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 first^{15}. The existence
of secondary flow phenomena^{16} and the surface shape in channel flow^{17}
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.

8. CONCLUSION

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.

9. ACKNOWLEDGEMENTS

The author gratefully acknowledges the help received in discussions with Dr. P. F. Lesse and Dr. A. V. Ramamurthy.

REFERENCES

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**

**
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**

**
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**