*Karl Weissenberg - The 80th Birthday
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**Elasticity in
Incompressible Liquids**

**PROFESSOR A. S. LODGE**

*University of Wisconsin-Madison, Rheology
Research Centre,U.S.A.*

Weissenberg said, “In liquids, confess

That large strains sometimes govern the stress.”

Remarked Rivlin, “No drivelling!

In flows steady, though swivelling,

The elastic part must evanesce.”

Then along came young Oldroyd, J. G.

“Things aren’t quite as they seem, now,” said he;

“With components convected

(As H. Hencky selected),

One obtains a new D by Dt.”

**1. INTRODUCTION**

In this essay, I propose to discuss Weissenberg’s hypothesis that the stress tensor in a flowing viscoelastic liquid is an isotropic function of the recoverable strain tensor. This hypothesis does not appear to have received adequate attention in the literature hitherto and is of considerable significance in relation to many contemporary studies of the rheological properties of concentrated polymer solutions and molten polymers. The hypothesis seems to have received less attention than another hypothesis of Weissenberg’s which led to the prediction that the second normal stress difference in shear flow should be zero.

In addition to making detailed references to Weissenberg’s publications, I shall draw on my recollections of conversations which I had with Weissenberg, particularly during the period 1948-1950 when I was privileged to begin my own studies of rheology under Weissenberg’s direction. I take this opportunity to acknowledge the great help which I received from Weissenberg, whose ideas have contributed much to my own subsequent research. In particular, it was an attempt to furnish at least one complete set of constitutive equations to illustrate his idea that the stress tensor should be isotropically related to a finite-strain tensor in flowing viscoelastic liquids that helped me to formulate the ‘rubber-like liquid’ equations on the basis of a molecular network theory for polymeric liquids; and it was Weissenberg’s pioneering appreciation of the importance of normal stress differences in shear flow which encouraged me to persist in a rather lengthy experimental programme for measuring them.

Although Weissenberg’s ideas on the effect of finite-strain elasticity encompass solids and compressible materials as well as incompressible liquids, I am restricting the present discussion to incompressible liquids partly for brevity and partly because this application is particularly important and controversial. I consider, therefore, viscoelastic liquids subjected to flows and deformations at constant volume and constant temperature.

**2. ELASTIC RECOVERY AND WEISSENBERG EFFECTS**

Weissenberg (1947, fig. 1) and Freeman and Weissenberg (1948) described experimental observations involving the flow of a variety of viscoelastic liquids (including saponified oils and polymer solutions) when sheared between rigid walls in relative rotation. The behaviour is very different from that of Newtonian liquids (pure oils and liquids of low molecular weight). The main observations with viscoelastic liquids may be summarized as follows:

(2.1) The liquids climb the inner cylinder in a wide-gap concentric cylinder apparatus; generate a non-uniform pressure distribution on the plate of a parallel-plate apparatus, with the pressure increasing from near zero (i.e. atmospheric) at the rim to a maximum at the centre (on the axis of rotation); and generate a positive axial thrust in the cone-and-plate apparatus. For brevity, I shall refer to these effects collectively as ‘Weissenberg Effects’ (see §7 below).

(2.2) When various liquids are tested, it is found that the Weissenberg Effects are more pronounced in those liquids which (in separate measurements) exhibit the larger recoverable shear strain.

(2.3) Weissenberg Effects are attributed to the generation of stress components representing a ‘pull along the lines of flow’. With those liquids which possess sufficiently long relaxation times, the shear flow can be suddenly stopped and cuts made in the liquid perpendicular to the previous lines of flow; the observed opening-up of such cuts is interpreted as direct evidence of the existence of a pull along the lines of flow.

The magnitude of the recoverable shear strain was presumably measured from the recoil observed in one or more of the rotational apparatuses when the shear stress was suddenly removed. Such techniques had been standard for some time, but are now believed to yield ‘constrained recovery’ (in which the normal stress differences take time to vanish) rather than ‘free recovery’ (in which the stress instantaneously becomes isotropic and remains isotropic during recovery) (Lodge, 1964, Ch. 7). This distinction was not made by Weissenberg (1947, para. 2) whose term ‘recoverable strain’ refers to the change of shape involved when a material element deforms from an arbitrary state at a time following some flow history to a ‘ground-state’ defined in the following sentence: “….we suddenly release all external forces and allow the material to drift back into some ground-state (which may or may not have been the one from which it was started)”. The use of the word ‘drift’ implies that ‘recoverable strain’ is to be interpreted as that involving ultimate (rather than instantaneous or delayed) recovery.

It thus appears that Weissenberg’s ‘recoverable strain’ should be regarded as the strain involved in ultimate free recovery in the terminology used, for example, by Lodge (1964, Ch. 7).

The explanation of the effects (2.1) in terms of a ‘pull along the lines of flow’ was proposed by Weissenberg (1947, para. 5) as follows: “If, as in our experiments, the lines of flow are closed circles, the pull along these lines strangulates the liquid and forces it inwards against the centrifugal forces and upwards against the forces of gravity.” Equations based on this proposal were given by Weissenberg and his collaborators (Freeman and Weissenberg, 1948, para. 5: Russell, 1946; Roberts, 1952). The main points involved are as follows (c.f. Lodge, 1964, Ch. 9).

We first consider a homogeneous, rectilinear,
shear flow, of shear rate ś (t). Referred to a rectangular Cartesian coordinate
system Ox_{1}x_{2}x_{3}, fixed in space, the velocity
components v_{i} are given by the equations

**(2.4)**

The shearing surfaces x_{2} = const.
move without stretching; lines parallel to Ox_{1} are lines of flow or
lines of shear; the volume is constant; t denotes time. The shear flow is
‘unidirectional’ in the sense that the same family of material lines (parallel
to Ox_{1}) are lines of shear throughout; if, while retaining the same
shearing surfaces, we changed the lines of shear at some instant, the shear flow
would be multidirectional.

Let p_{ij} (i, j = 1, 2, 3) denote
components of the Cartesian stress tensor, referred to Ox_{1}x_{2}x_{3},
with the sign convention that p_{11} > 0 for a tensile normal component
of stress. It is assumed that body couples and contact couples are absent, so
that p_{ij} = p_{ji}. It is assumed that the material is
“mechanically determined” in the sense that the stress p_{ij} is
determined to within an additive isotropic stress by the flow history (cf.
Weissenberg, 1931, p. 7, para. 3). It is also assumed that the material is
isotropic in the sense that its constitutive equation can be expressed in a form
in which all the material constants are scalars (cf. Weissenberg, 1935, p. 151,
10). It then follows that the stress tensor must possess the same symmetry
property as that possessed by the velocity field (2.4), namely, symmetry with
respect to 180^{o} rotation about Ox_{3} hence

**
(2.5)**

Thus there are at most four non-zero independent stress components, of which three combinations, e.g.

**
(2.6)**

are of direct rheological significance for
incompressible liquids. N_{1} and N_{2} are, respectively, the
primary and secondary normal stress differences. When ś and p_{21} are
independent of time, p_{21}/ś is the viscosity, η. For an
incompressible, Newtonian liquid, N_{1} = N_{2} = 0 and η is
independent of ś.

A positive value for p_{11} could be
described as a pull along the lines of flow; for incompressible liquids,
however, it is to be expected that such a pull would not affect the flow in the
presence of components p_{22} and p_{33} each equal to p_{11}.
It is therefore reasonable to consider instead the value of p_{11} taken
with reference to some convenient local reference level of normal tensile
stress. Using p_{22} as such a reference level, a pull along the lines
of flow then corresponds to a positive value of N_{1}.

One could, alternatively, use p_{11} -
p_{33} or P_{11} - (p_{11} + P_{22} + p_{33})/3
as a measure of the pull along the lines of flow. Both of thesequantities are
positive when N_{1} is positive and N_{2} is small or zero.
Weissenberg’s explanation of the effects (2.1) can be expressed in the form

**
(2.7) (2.8)**

In addition to emphasizing the importance of N_{1}
and N_{2} in shear flow of viscoelastic liquids, Weissenberg showed how
curvilinear shear flows could be used to measure N_{1} and N_{2}.
For the theory of these methods, it is convenient to use orthogonal coordinate
systems: cylindrical polar, for Couette flow between coaxial cylinders and for
torsional flow between circular parallel plates; spherical polar, for shear flow
between cone and plate with small gap angle. p_{ij} can then be used to
denote physical components of the stress tensor associated with the appropriate
orthogonal coordinate system; p_{ij} can therefore be taken to be equal
to Cartesian components of the stress tensor referred to a local rectangular
Cartesian coordinate system Qx_{1}x_{2}x_{3}, where Q is
the place occupied by a typical particle P at time t, Qx_{1} is
tangential to the line of shear through Q, and Qx_{2} is normal to the
shearing surface at Q.

Using (2.5) and the stress equations of motion
with neglect of inertial forces, equations can be obtained from which, in
principle, values of N_{1} and N_{2} can be derived from
pressures and forces measured in the different types of apparatus. For the
cone-and-plate apparatus (which plays a crucial role in the task of eliminating
a certain class of inapplicable constitutive equations), the basic equations may
be written in the following form:

**
(2.9) (2.10) (2.11)**

p_{a} denotes atmospheric pressure. - p_{22}(r)
denotes the pressure which the liquid exerts on the plate at a distance r from
the axis of rotation (which coincides with the cone axis). R is the radius of
the cone. In (2.10) and (2.11), it is assumed that the liquid is confined to the
gap between cone and plate, and that the shear flow persists up to the free
liquid boundary (which approximates in shape to part of the surface of a sphere
of radius R). F denotes the total thrust normal to the cone exerted by the
liquid, minus the contribution due to an isotropic stress of magnitude -p_{a}.

The observation by Freeman and Weissenberg (1948) that F, the axial thrust in the cone-and-plate apparatus, is positive for a variety of viscoelastic liquids is thus by itself sufficient to establish the existence of a pull along the lines of flow. As far as I am aware, all published measurements on liquids exhibiting effects of the kind (2.1) for which F has been measured yield positive values of F.

Early evidence for the smallness of N_{2}
was provided in two ways:

(i) observed values of the rim pressure -p_{22}(R)
in the cone-and-plate and parallel-plate systems (for which (2.10) is
applicable) were close to atmospheric pressure p_{a} (Weissenberg, 1947,
fig. 1; Roberts, 1954; Russell, 1946);

(ii) the wall pressure -p_{22} near the
upper free liquid surface in Couette flow in a narrow-gap concentric cylinder
apparatus was approximately equal to the atmospheric pressure (Russell, 1946).

There is now reason to believe that such
measurements of N_{2} may be unreliable owing to the use of
liquid-filled holes for pressure measurement (Broadbent, Kaye, Lodge, and Vale,
1968). Recent measurements do, however, show that, for some polymer solutions at
least, the magnitude of N_{2} is appreciably smaller than that of N_{1}
(Kaye, Lodge, and Vale, 1968; Broadbent and Lodge, 1971; Christiansen and
Miller, 1971; van Es, 1972).

The evidence at present available thus confirms (2.7) for all liquids exhibiting Weissenberg effects, and confIrms (2.8) for some of these liquids.

It should be emphasized that (2.9) - (2.11) can
be derived without making any assumption about the value of N_{2}. The
form in which Freeman and Weissenberg (1948, para. 5) give equations of this
type appears to involve the assumption that N_{2} = 0, and that
atmospheric pressure is taken as a reference level from which stress components
are measured (so that our p_{11} equals their Pφφ - p_{a}); they
use a cylindrical (instead of spherical) polar coordinate system for the
cone-and-plate system.

**3. ORIENTATION OF THE PRINCIPAL AXES OF
STRESS **

Weissenberg (1947, para. 6) states that, “As a
first approximation, a theory was proposed^{4}, containing an
arbitrarily assumed law of elasticity, and from this theory predictions were
made for a pull along the lines of flow and for a great variety of rheological
phenomena, including such as are reproduced in fig. 1.….”. Weissenberg’s
reference 4 (which should be Arch. Sci. Phys. et Nat. (5), **140**, 44
(1935), not (5) 17, 1 (1934) as given by Weissenberg (1947)) contains no
explicit reference to a pull along the lines of flow in shear flow. The link
between the 1935 and 1947 papers, however, involves a very simple and well-known
equation, namely

**
(3.1)**

where *X’*
denotes the angle between Ox_{1} and either of the two principal axes of
stress which lie in the Ox_{1}x_{2} - plane. (2.5) implies that
one principal axis of stress is parallel to Ox_{3} and hence that the
other two principal axes lie in the Ox_{1}x_{2} - plane. Without
loss of generality, we shall, for brevity, confine the discussion (except where
otherwise stated) to that principal axis of stress for which

**
(3.2)**

It is interesting to note that (3.1), which is
a standard equation in plane stress analysis, was mentioned briefly by Lawrence
(1938) in connection with the shear flow of birefringent solutions but was
discarded with a statement (p. 42) that the tension N_{1} (= T, in
Lawrence’s notation) could not be measured.

It follows at once from (3.1) that *X’*
= 45^{o} if, and only if, N_{1}/p_{21} = 0, and that, as
N_{1}/p_{21} increases from zero, *X’ *
decreases from 45^{o}. Thus the
generation of a pull along the lines of flow (in the sense (2.7)) is associated
with a change in the orientation of the principal axes of stress from the
orientation which one would obtain (for example) with an incompressible
Newtonian liquid, namely *X’=* 45^{o}.

Weissenberg (1935, p. 163) considered a more general ‘perfectly viscous’ material for which the deviatoric stress tensor P is an isotropic function of the rate-of-strain tensor Ś:

**
(3.3)**

P is defined by the equation

**
(3.4)**

where Tr denotes trace and 1 denotes the unit
tensor (1935, p. 151, 3^{o}). Ś^{3} = Ś. Ś. Ś, etc., where a dot
between tensors denotes a single contraction. The coefficients η_{i} are
scalars. The relation between Weissenberg’s definition of Ś and the more
familiar definition

**
(3.5)**

is given in the Appendix to this essay.

The equation

**
(3.6)**

for an incompressible Newtonian liquid is evidently a particular case of (3.3).

Weissenberg noted that, according to (3.3), P
and Ś necessarily have the same principal directions (1935, p. 163, c)) and that
accordingly, in shear flow, *X’* =
45^{o} (1935, p. 167). It follows that constitutive equations of the
form (3.3) cannot give a valid description of the Weissenberg Effects (2.1) as
they have been observed with viscoelastic liquids to date because (3.3) predicts
zero axial thrust in the cone-and-plate system, according to (2.11) and (3.1).

The same statement is obviously valid for the closely related constitutive equation

**
(3.7)**

where p is an arbitrary scalar and A and B are scalar functions of the invariants of Ś. This equation was considered by Reiner (1945) and Rivlin (1948) without reference to Weissenberg’s earlier use of (3.3). Application of the Cayley-Hamilton theorem to (3.3) and replacement of the deviatoric stress tensor P by the extra stress tensor p + p1 immediately leads to an equation of the form (3.7). The conflict between the predictions of (3.3) or (3.7) with the observed data on the Weissenberg Effects has been emphasized by Roberts (1954) and Markovitz (1957).

From (2.4), (3.5), and (3.7), it follows that η = A and that

**
(3.8)**

It follows that the purely viscous liquid (3.7)
can exhibit certain Weissenberg Effects (e.g. non-uniform pressure distributions
in the cone-and-plate and parallel-plate systems) associated with a non-zero
secondary normal stress difference N_{2}, but it seems that such a
possibility is of rather academic interest only. The prediction of zero total
thrust in the cone-and-plate apparatus justifies my earlier statement in §2
above that this apparatus is crucial.

Weissenberg justifies the omission of
even-powered terms in (3.3) by appeal to an unspecified symmetry argument; I
believe that this would involve the tacit assumption that the coefficients η_{i}
in (3.3) are constants. This would be consistent with the definition of a
function of tensor which Weissenberg was using (1935, pp. 97, 98). On p. 163,
line 9, it is clear that the word ‘linéare’ should be replaced by ‘analytique’,
which is evidently required and which is used in a similar context on p. 159,
line 9.

Even apart from the use of p (instead of - ⅓Tr p) in (13.7), Reiner’s equation (3.7) is formally more general than Weissenberg’s equation (3.3) because the former involves two arbitrary functions A, B of two independent scalar invariants of Ś while the latter involves only one function F(x) of a scalar variable x (Weissenberg, 1935, pp. 97, 98). The difference in ‘degree of generality’ is somewhat reduced by the fact that, for real materials, A and B should be derivable from a single function of the two invariants of Ś (the ‘dissipation function’).

In referring to Reiner’s equation (3.7), Truesdell (1952, p. 225) states, “This formula is given by Reiner, to whom we owe the introduction of tensorial methods in general fluid dynamics.” I have remarked that Reiner’s equation (3.7) is only a little more general than Weissenberg’s equation (3.3), published in terms of tensor analysis by Weissenberg 11 years before Reiner92s paper. Moreover, Weissenberg’s 1931 and 1935 papers use tensor analysis throughout and make substantial use of tensor methods in a field of fluid dynamics which is significantly wider than the field encompassed by Reiner’s 1945 paper. Weissenberg’s papers deal with liquids which have elastic and viscous properties which are described by constitutive equations which may involve stress and strain tensors and their time derivatives of all orders. Truesdell’s statement should surely read, “It is to Weissenberg that we owe the introduction of tensorial methods in general fluid dynamics.”

Weissenberg’s use of the deviatoric stress
tensor instead of the extra-stress tensor is somewhat puzzling; it does not
affect the points at issue in the present essay, because N_{1} and N_{2}
are not affected, but it might be of importance in connection with certain
scalar equations such as the energy balance equations which form a main
background for Weissenberg’s two early papers (1931, 1935). The extra-stress
tensor p + p1 could conceivably occur in the arguments of the free energy
function and the dissipation function in eq. (10) of (1931), p. 15, and it is
not obvious how the isotropic pressure variable p would be determined.
Furthermore, the isotropic pressure variable plays an important role in a
certain calculation of free recovery for a rubberlike liquid (Lodge, 1964, Ch.
7), being equal to the reciprocal of a certain modulus for the liquid.

**4. ELASTIC MATERIALS**

It is clear from the above that, in order to furnish a correct description of the Weissenberg Effects actually observed, some mechanism must be postulated whereby the orientation of the principal axes of stress in shear flow can be made to differ from the orientation of the principal axes of Ś. We know today that such a difference of orientations is predicted by very many constitutive equations, but the importance of having such a difference was first emphasized by Weissenberg (1931; 1935, pp. 149, 150, 171) and was rather slow in achieving widespread recognition, if one judges by published criticisms (§5 below) and by the number of papers published on the Reiner equation (3.7).

Weissenberg. (1935, p. 159, b)) considers a
material for which the deviatoric stress tensor P(t) in a state t is an
isotropic function of a finite-strain tensor S_{p}(t_{o},t):

**
(4.1)**

S_{p}, is a ‘post-rotational’ strain
tensor describing strain for the deformation t_{o} → t, and is defined
by the equation

**
(4.2)**

where ψ is the deformation-gradient tensor whose components are given by the equation

**
(4.3)**

x_{i} and x^{o}_{i}
denote coordinates (referred to Ox_{1}x_{2}x_{3}) of the
places which a typical particle occupies in states t and t_{o},
respectively (see Appendix). The coefficients
ץ_{i}
are scalars.

Since (4.1) represents an isotropic relation of some generality, it does not matter what function is used in (4.2), provided that the strain tensor

_{~} _{~}

Ψ.Ψ (not Ψ.Ψ) is used. Weissenberg establishes this point by using rectangular axes which follow the rotational part of the deformation Ψ. From (4.3), we have

**
(4.4)**

_{
~}

so that B_{ij} ≡ Ψ.Ψ is just the Finger
strain tensor for t_{o} → t. The states t_{o}, t are labelled
(0), (1) respectively by Weissenberg.

A basic question concerns the definition of the
state t_{o}. In the first instance, t_{o} is taken to label a
stress-free state in which the material is at rest (relative to axes fixed in
space) (Weissenberg, 1935, pp. 69, 70, 147, 149). After an arbitrary homogeneous
flow history, a state t is reached, in which the stress is, in general,
non-zero.

Let us suppose that the stress is again made
and kept zero, and that the material comes to rest in some state which we label
t’_{o}. The strain t → t’_{o} is thus the ultimate free
recoverable strain.

If, in all cases, the states t_{o} and
t’_{o} are identical, the material is an elastic solid. Weissenberg is
careful not to impose this restriction, however, in order that his formalism may
encompass both solids and liquids. The application to liquids is made in two
distinct ways, as follows.

I. First, in the 1935 paper (p. 166), it is
assumed that (a) the liquid can be subjected to a deformation (starting from t_{o})
which is so rapid that the stress tensor is determined by an ‘elasticity law’ of
the form (4.1). It is further assumed that (b) the liquid can then be made to
flow in such a manner that the stress tensor does not change with time. If the
rapid strain in (a) is a simple shear of the form x^{o}_{i} → x_{i},
with

**
(4.5)**

so that s is the magnitude of the shear strain,
and if the flow in (b) is a shear flow given by (2.4), then it is a
straight-forward matter to show that *X’*
is less than 45^{o}.

I think that, in certain aspects, these
assumptions are open to question: if one adjusted the shear rate ś(t) in the
flow (b) so as to keep the shear stress p_{21} constant, there is no
reason to suppose that N_{1} (generated in (a)) would remain constant.
Provided, however, that N_{1} does not go to zero, one would still get
*X’* < 45^{o}.

II. The second application to liquids (1947,
para. 6) is made by (c) replacing t_{o }in (4.1) by t’_{o}, so
that P(t) is assumed to be an isotropic funcfion of the recoverable strain
tensor S_{p}, (t’_{o}, t) or equivalently of B (t’_{o},
t) (for the ultimate free recovery when the stress is zero from time t on). It
is also assumed (d) that, for a shear flow of type (2.4) up to time t, the
recovery t → t’_{o} is a simple shear of type (4.5). Weissenberg also
states that agreement with available data of type (2.1) can be secured if the
following particular form of (4.1) is chosen (see Appendix):

**
(4.6)**

G(P,R) here denotes a scalar function of the
invariants of P and R, which presumably varies from one liquid to another. It is
a straightforward matter to show from (4.4), (4.5), and (4.6) that *X’*
in the steady shear flow is given by the equation

**
(4.7)**

where s denotes the magnitude of the recoverable strain.

Thus a finite recoverable shear strain is
associated with a value of *X’*
less than 45^{o} and with a positive value of N_{1}, since p_{21}
is positive. Thus Weissenberg’s ‘Recoverable Strain Hypotheses’ (II) (c), (d)
give a possible correlation of the experimental facts (2.1), (2.2) and (2.3).

It is evident that an essential feature of
Weissenberg’s approach involves the replacement of an isotropic relation (3.3)
between P and Ś by an isotropic relation (4.1) between P and a finite-strain
tensor. Reiner (1948) subsequently showed that the relation between *X’*
and s, given in (4.7), does not depend on the choice of the particular
stress/strain relation (4.6) and is valid for any isotropic relation between p
and B of the general form

**
(4.8)**

where p is an arbitrary scalar and a, b are
scalar functions of the invariants of B. The Cayley-Hamilton Theorem enables one
to express (4.1) in the compact form (4.8). p thus has the same principal
directions as B; by making a plausible assumption about B (namely, (II) (d)),
Weissenberg gets *X’* < 45^{o}
for shear flow. Again, for consistency, I think that one needs to interpret P in
(4.1) and (4.6) as the extra-stress tensor because incompressible liquids are
being considered; (4.6) is in fact incompatible with the assumption that P is
deviatoric (i.e. that Tr P = 0).

For the shear (4.5), it follows from (4.4) that

**
(4.9)**

It can thus be seen that the statement (II) (d)
above is equivalent to the statement given by Weissenberg (1947, para. 6): “the
recoverable strain has components R_{mn} which are all zero except R_{11},
R_{12}, and R_{21}, with R_{11} = (R_{12})^{2}
= (R_{21})^{2}”.

**5. CRITICISMS OF WEISSENBERG’S
RECOVERABLE-STRAIN HYPOTHESES**

Rivlin (1948, P. 280) stated, “any viscoelastic material, which is fluid-like in its ability to be continuously deformed, cannot be distinguished from a visco-inelastic fluid solely by means of steady-state experiments, in which the flow is laminar. For example, suppose the fundamental stress-strain-velocity relations involve time derivatives of the stress components. For a steady state of flow, these vanish and the stress-strain-velocity equations are indistinguishable from those describing a true fluid ... Effects similar to those predicted from the theory of §§15 and 16” (i.e. from the Reiner equation (3.7) above) “have been described in a number of fluids by Weissenberg (1947). They are interpreted by him as arising from the fact that the fluids concerned have highly-elastic as well as viscous properties. It does not, however, appear that this explanation can be correct, from a phenomenological point-of-view, since, in the experiments, the fluids are in a steady-state of laminar flow.”

Oldroyd (1950) pointed out that Rivlin’s criticism is invalid because rheologically-admissible time derivatives of tensors (such as the stress tensor) must involve the vorticity tensor and need not vanish in steady laminar shear flow. Rivlin subsequently withdrew his criticism. Weissenberg was evidently aware of the complications involved in defining rheologically-admissible time derivatives of stress and strain tensors when the flow history is not rotation-free (1931, pp. 8, 16: 1935, pp. 79, 147978).

Truesdell (1952, p. 243, footnote), in discussing the climbing of liquids up the inner cylinder in Couette flow (2.1), stated, “Weissenberg himself attempts an explanation by his theory of elasticity (§51), and Reiner (1952) proposes experiments to decide whether the effect arises from elastic or viscous properties. To me the issue seems not to exist: the theory of elasticity is applicable only to substances in which the stress does not subside until recovery of strain from some specified reference configuration, but the physical fluids used in the experiments show no such tendency whatever, and an attempt to represent phenomena appearing in fluids by a theory of elasticity seems unnecessary, irrelevant, and inappropriate.”

The statement that the “fluids ... show no such tendency whatever” is incorrect because the experiments showed that the extent of rod-climbing (and the other effects (2.1)) increased as the measured elastic recovery increased (2.2). The issue whether elastic or viscous properties govern the effects certainly does exist; in the cone-and-plate experiment, elastic properties give an axial thrust, whereas viscous properties alone give zero thrust. It is true that the theory of elasticity as developed before Weissenberg’s papers were published in 1931 and 1935 applied to materials which were solids in the sense that they possessed a unique stress-free state; but it is surely very reasonable to attempt (as Weissenberg did) to modify the theory so as to apply to liquids which experiments showed possessed both elastic and viscous properties. Truesdell’s criticism of this attempt appears to be no more than a statement of opinion unaccompanied by any detailed arguments.

Elsewhere, Truesdell (1952, pp. 199, 210) does make a more detailed criticism which is equivalent to the statement that Weissenberg’s hypothesis (II) (c) above (i.e. that the stress tensor p is an isotropic function of the recoverable strain tensor R) lacks useful content unless additional information is given which is sufficient to enable one to calculate R in terms of the flow history. Truesdell links Weissenberg92s theory with a theory of Reiner’s (p. 210) which “will not permit the solution of a single boundary problem, for it is altogether indeterminate” (p. 199).

It is true that Weissenberg did not give equations sufficient to enable one to calculate R in terms of the flow history or to “solve boundary problems”. Weissenberg was fully aware of the need to discover constitutive equations (which could be used, in particular, to solve boundary problems); his 1931 paper made it clear that such equations could be very complicated and would, in general, involve stress and strain tensors and their appropriate time derivatives of various orders. Weissenberg’s aims in his 1947 and 1948 papers were more limited than the discovery of complete constitutive equations but were nonetheless important and valuable, and could furnish a possible guide to the choice of constitutive equations: his aim was to explain at least qualitatively the effects (2.1), (2.2), and (2.3) in terms of a few basic plausible postulates. His postulates (II) (c,d) above achieve this aim; in particular, the tensor R is not left completely undetermined, as Truesdell implies, because Weissenberg postulates three important properties for R, namely, (i) that R is a finite-strain tensor, (ii) that R describes the ultimate free recovery after an arbitrary flow history, and (iii) that, for recovery following a unidirectional shear flow, R describes a simple shear with the same shearing surfaces and lines of shear.

The formulation of complete constitutive
equations and the solution of boundary-value problems are matters of importance
but they do not by themselves exhaust the useful subject matter of theoretical
rheology. Viscoelastic materials commonly exhibit a great variety of striking
properties, and it is surely a matter of importance to attempt to correlate some
of these properties on the basis of one or two physically-plausible postulates.
A recent example of such an attempt (Lodge and Meissner, 1972) involves a
correlation of values of N_{1}, N_{2}, and P_{21}
(associated with the superposition of a rapid shear strain on a shear flow) on
the basis of a single postulate (that the extra stress in a flowing polymeric
liquid is determined by the configurational entropy of a Gaussian molecular
network); this attempt is closely connected with two of Weissenberg’s ideas (the
use of a finite strain which is so quick that there is a change in free energy
without dissipation; and the instantaneous free recovery); complete constitutive
equations are neither known nor used.

**6. RECOVERABLE STRAIN IN THE “RUBBERLIKE
LIQUID”.**

During the twenty-five years which have elapsed since Weissenberg’s 1947 paper, many constitutive equations have been published which describe elastic and viscous properties in idealized incompressible liquids. It is instructive to consider whether Weissenberg’s Recoverable Strain Hypotheses are valid for these liquids. As far as I am aware, however, calculations of elastic recovery in terms of flow history have been published for two, and only two, constitutive equations, namely, the ‘rubberlike liquid’ equation

**
(6.1)**

and the formally similar ‘covariant’ integral equation

**
(6.2)**

The memory function μ is a continuous,
decreasing function of the time interval t-t’; the Cauchy strain tensor C is the
reciprocal of the Finger strain tensor B. The constant volume condition is det B
= 1 = det C. The ‘contravariant’ equation (6.1) has been derived from a
molecular network theory (Lodge, 1954, 1968). It has been shown (Lodge, 1964, P.
125, (7.1)) that the rubberlike liquid (6.1) satisfies a modified form of
Weissenberg’s Recoverable Strain Hypothesis (II) (c) in which the ultimate free
recovery t → t’_{o} is replaced by the instantaneous free recovery t →
t*; in place of Weissenbergs equation (4.6), we have the very similar equation

**
(6.3)**

(obtained by transfer of (7.5) of Lodge (1964)
to space at time t). This also differs from Weissenberg’s (4.6) in that the
modulus μ_{o} is determined by the flow history (instead of by P and R)
and the extra stress tensor p + p1 is used (instead of the deviatoric stress
tensor P). When μ(t)
= a_{1} exp (-t/t_{1}),
however, the instantaneous and ultimate free recoveries are equal (Lodge, 1964,
(7.6)).

For a flow history which is a steady shear flow
of the form (2.4) up to time t, the instantaneous free recovery is not a simple
shear (as postulated by Weissenberg, (II) (d) above) but is a simple shear
superposed on lateral expansions (parallel to Ox_{2} and Ox_{3})
and a longitudinal contraction (parallel to Ox_{1}) (Lodge, 1964, p.
134, and 1958). Thus the recoverable strain tensor B(t*,t), though different in
detail from Weissenberg’s recoverable strain tensor B(t_{o}’, t) given
by (4.9), does involve a finite shear component and in consequence does give a
positive value of N_{1}(t) (and a value of *X’*
less than 45^{o}). There is thus a significant measure of agreement
between the predictions of the rubberlike-liquid equation (6.1) and the
Recoverable Strain Hypotheses (III) (c,d) of Weissenberg (1947).

The method used by Lodge (1964, p. 125-127) to calculate the instantaneous free recovery for the rubberlike liquid (6.1) should be immediately applicable to a variety of other constitutive equations when expressed in convected components or in terms of body tensors (Lodge, 1972). For (6.2), one obtains an equation similar to (6.3) but with B (t*, t) replaced by C (t*, t). Other constitutive equations would give other instantaneous recoverable strain tensors. For recovery following shear flow, (6.2) gives longitudinal and lateral contractions and a lateral expansion superposed on a finite shear.

**7. TERMINOLOGY: THE ‘WEISSENBERG EFFECTS’**

Garner and Nissan (1946) and Garner, Nissan,
and Wood (1950) described experiments in which the rod-climbing effect was
demonstrated (1946) together with the non-uniform pressure distribution in the
parallel-plate system (1950); the latter was demonstrated by means of a series
of capillary tubes situated on one of the plates. The explanation proposed by
Garner et al. is that the stress tensor has equal normal components p_{11},
p_{22}, and p_{33} whose magnitude is a function of the local
shear rate ś. In accordance with this explanation, they used a concentric system
with a non-uniform annulus in their rod-climbing demonstration, and they found a
uniform distribution of pressure in their cone-and-plate apparatus. The latter
is in disagreement with all subsequent published measurements of which I am
aware.

The work of Weissenberg and co-workers and the work of Garner and co-workers was carried out during wartime in England and was not allowed to be published until after the war. There were discussions between the two groups during the course of the work, and there have been questions raised as to the correct attribution of priority for the various contributions.

There seems to be no doubt, however, that Weissenberg should be accorded priority for the following contributions: that curvilinear shear flows can be used to measure normal stress differences, and that one or more non-zero normal stress differences are responsible for the effects (2.1); in particular, that in shear flow between cone and plate, non-zero normal stress differences give rise to a non-uniform pressure distribution and a positive axial thrust.

In a letter dated September 8, 1959,
Weissenberg wrote to me as follows: “The whole set-up of experiments with
torsional shear was devised by me in order to derive from the pressure
distribution of P_{22} the angular deviation of the stress tensor from
the strain velocity tensor, and the pull along the lines of flow (which was then
already calculated from the formulae of the French paper). The pressure
distribution was at the time measured by Russell and myself by varying the
diameter of the plates, and using an annulus. While this was going on, Garner
and Nissan checked on my experiments and used for this purpose capillary gauges.
They were thus the first to observe the pressure distribution directly, as
Russell said in his thesis, since my observations at the time required a series
of experiments….. Before Garner and Nissan started on these experiments they had
been shown my experiments and were introduced to my theory in the Committee of
the Flame Warfare Department……”

Truesdell (1952, pp. 243, 245) refers to effects of the type (2.1) as ‘Poynting Effects’ because of Poynting’s early demonstration (1913) that a rubber cylinder increases in length when subjected to torsional shear stresses. Poynting’s experiments were made with a solid; Weissenberg’s experiments were made with liquids, and, moreover, involved pressure and axial force measurements not made by Poynting. Although today there is reason to believe that the Poynting Effects and the Weissenberg Effects would be produced in materials (whether solid or liquid) for which the stress tensor is an isotropic function of a finite-strain tensor, the idea of extending elastic properties from solids to liquids in this connection is due to Weissenberg, as described above. It was, previous to Weissenberg’s work, by no means obvious that effects, such as (2.1), observed with liquids could be in any way analogous to effects in solids. Truesdell himself in the same monograph (1952, p. 243, footnote) criticized Weissenberg for his proposal to extend the theory of elasticity from solids to liquids.

**APPENDIX**

In this appendix, relations will be derived between equations used above and equations given by Weissenberg (1935). Page numbers will refer to the latter paper. A very clear exposition of Weissenberg’s ideas applied to shear flow has been given by Burgers (1948).

From pp. 55, 65, 71, and 72, the rate-of-strain tensor Ś is given by the equation

**
(A.1)**

where

**
(A.2)**

r and ŕ denote relative position vectors of two
particles P_{M}**,** P_{N} at times t**, **t + dt
respectively (in states labelled (1), (I’) by Weissenberg). All deformations are
considered to be homogeneous, so that it makes no difference whether the two
particles are neighbouring or not. It also follows that v (**R**, t**),**
the velocity vector of particle P_{M} at time t, is given by the
equation

**
(A.3)**

where R = (OP_{M})_{t}, the
position vector of P_{M} at time t drawn from an origin 0 fixed in space
(p. 54).

It follows that

**
(A.4)**

and hence, from (A.2), that

**
(A.5)**

From (A.5), it follows that, to the first order in dt,

**
(A.6)**

where the tilde denotes the transpose.

Corresponding to a function f(x) = a_{0}
+ a_{1} x + a_{2} x^{2} + …….represented by a convergent
power series, Weissenberg (p. 97) uses a definition f(T) = a_{0} **1**
+ a_{1 }**T** + a_{2} **T**^{2} + ….. for the
corresponding function of a second rank tensor **T**. It follows, in
particular, that log_{e} (**1 + A**) = A - ½ **A**^{2} +
….., and hence, from (A.6), that

**
(A.7)**

From (A.1), we thus obtain the required result that

**
(A.8)**

which is the equation whose component form is
(3.5), where x_{i} denote the components of R referred to the
rectangular Cartesian coordinate system Ox_{1}x_{2}x_{3},
fixed in space.

To prove (4.3), we have

**
(A.9)**

where r^{o} = (P_{M}P_{N})
t_{o} (p. 70), and hence

**
**

**
(A.10)**

**
**

where b_{i} are independent of position.
(4.3) follows at once on differentiation of (A. 10).

Finally, we show that (4.6) is equivalent to the equation

**
(A.11)**

where a, b, c and u, v, w are permutations of 1,
2, 3, (Weissenberg, 1947, (1)). Q_{au} are evidently the components of a
symmetric tensor Q = P - G R referred to an arbitrary rectangular Cartesian
coordinate system. From (A.11), we have

**
(A.12)**

Since Q is symmetric, there exists a rectangular
Cartesian coordinate system whose axes are parallel to the principal axes of Q;
in this system, Q_{23} = Q_{31} = Q12 = 0. Hence Q_{ij}=
0 (i,j = 1, 2, 3), and therefore Q = 0, which proves (4.6).

**
REFERENCES**

Broadbent, J. M., A. Kaye, A. S. Lodge, and D.
G. Vale (1968), *Nature (London)* 217, 55.

Broadhent, J. M., and A. S. Lodge (1971) *
Rheol. **Acta.*
10, 557.

Burgers, J. M. (1948), *Proc. Kon.
Ned. **Akad. v. Wet.*,
51, no. 7, 787.

Christiansen E. B., and M. Miller (1971), *
Trans. Soc. Rheol.*, 15, 189.

Freeman, S. M., and K. Weissenberg (1948), *
Nature (London)*, 162, 320.

Garner, F. H., and A. H. Nissan (1946), *
Nature (London)*, 158, 634.

Gamer, F. H., A. H. Nissan, and G. F. Wood
(1950), *Phil. Trans*., A243, 37.

Kaye, A., A. S. Lodge, and D. G. Vale (1968),
*Rheol. Acta* 7, 368.

Lawrence, A. S. C. (1938), *J. Roy. Microscop.
Soc*., 58, 30.

Lodge, A. S. (1954), Proc. 2nd. Int. Congr. Rheol. (Butterworth, Ed. Harrison).

Lodge, A. S. (1958), in “Rheology of Elastomers” (Pergamon, London, Ed. P. Mason and N. Wookey).

Lodge, A. S. (1964), “Elastic Liquids” (Academic Press, London and New York).

Lodge, A. S. (1968), *Rheol. Acta*, 7,
379.

Lodge, A. S. (1972), *Rheol. Acta*, 11,
106.

Lodge, A. S., and J. Meiβner, (1972), *Rheol.
Acta* 11, 351.

Markovitz, H. (1957), *Trans. Soc. Rheol*.,
1, 37.

Oldroyd, J. G. (1950), *Proc. Roy. Soc*.,
A200, 523.

Poynting, J. H. (1913), *India-Rubber J*.,
Oct. 4, p. 6.

Reiner, M. (1945), *Amer. J. Math*., 67,
350.

Reiner, M. (1948), *Amer. J. Math*., 70,
433.

Roberts, J. E. (1952), U. K. Ministry of Supply Report ADE 13/52.

Roberts, J. E. (1954), Proc. 2nd. Int. Congr. Rheol. (Butterwoth, Ed. Harrison), p. 91.

Russell, R. J. (1946), Ph.D. Thesis, Imperial College, University of London.

van Es, H. (1972), 6th Int. Congr. Rheol., and private communication.

Weissenberg, K. (1931), Abh. Preuss. Akad. Wiss. Phys.-Math. K1., Nr. 2.

Weissenberg, K. (1935), *Arch. Sci. Phys. et
Nat*. (5), 140, 44, 130.

Weissenberg, K. (1947), *Nature (London)*,
159, 310.

**
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**