thixotropic elasto-viscoplastic model for structured fluids

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Thixotropic elasto-viscoplastic model for structured fluids

Paulo R. de Souza Mendes*

Received 19th September 2010, Accepted 20th December 2010

DOI: 10.1039/c0sm01021a

A constitutive model for structured fluids is presented. Its predictive capability includes thixotropy,

viscoelasticity and yielding behavior. It is composed by two differential equations, one for the stress and

the other for the structure parameter—a scalar quantity that represents the structuring level of the fluid.

The equation for stress is obtained in accordance with a simple mechanical analog composed by

a structuring-level-dependent Maxwell element in parallel with a Newtonian element, leading to an

equation of the same form as the Jeffreys (or Oldroyd-B) equation. The relaxation and retardation

times that arise are functions of the structure parameter. The ideas found in de Souza Mendes, J. Non-

Newtonian Fluid Mech., 2009, 164, 66 are employed for the structure parameter equation as well as for

the dependencies on the structure parameter of the structural viscosity and structural shear modulus.

The model is employed in constant-rate, constant-stress, and oscillatory shear flows, and its predictive

capability is shown to be excellent for all cases.

1. Introduction

Structured fluids are present in everyday life. Most suspensions,

emulsions, and foams are structured fluids. Many foods,

personal care products, paints, inks, cements, adhesives, greases,

natural muds, drilling muds, crude oils, gels, and various slurries

are some examples of structured fluids.

The mechanical behavior of most structured fluids is highly

non-Newtonian. At small stress levels, their behavior is visco-

elastic. Beyond a certain stress threshold, usually called yield

stress, a major microstructure collapse occurs which causes

dramatic drops in viscosity and elasticity.

The microstructure of a structured fluid usually acquires

a stable configuration if exposed for a long time to a constant

stress or shear rate. This steady state is the result of the equi-

librium between the microstructure buildup and breakdown

rates. If a new equilibrium is not achieved instantaneously after

a step change to a new stress or rate, then the structured fluid is

said to be time-dependent. Note that yield-stress as well as other

structured fluids can in principle be either time-dependent or

time-independent, but in most cases this classification itself is

a function of the combination and range of parameters under

study.

A time-dependent fluid is said to be thixotropic if its steady-

state viscosity decreases with the shear rate (i.e. if it is shear-

thinning), and if in addition the viscosity changes are reversible.

On the other hand, a time-dependent fluid is said to be antith-

ixotropic if its steady-state viscosity increases with the shear rate

Department of Mechanical Engineering, Pontif�ıcia UniversidadeCat�olica-RJ, Rua Marques de Sao Vicente 225, Rio de Janeiro, RJ22453-900, Brazil. E-mail: [email protected]

This journal is ª The Royal Society of Chemistry 2011

(i.e. if it is shear-thickening), and if in addition the viscosity

changes are reversible.

Time dependency is observed in most structured fluids, espe-

cially in the small stress range.1 Irreversibility of the micro-

structure changes is also common, which render rather difficult

the tasks of modeling and characterizing such fluids.2,3

A number of useful reviews on thixotropy are available in the

literature, among which the ones published by Barnes4 and, more

recently, Mewis and Wagner.3 Mujumdar et al.5 also gives

a thorough discussion of the thixotropy literature, including

a quite complete comparison between the various structural

kinetics models then found in the literature, and de Souza

Mendes6 discusses the main aspects of these literature reviews.

The more recent thixotropy models found in the literature

include elasticity effects.7,8,9,10 Some models consider elasticity

and yielding, but no thixotropy effects e.g. ref. 11,12. One

shortcoming of these thixotropy models that include elasticity is

their incapability of predicting what experimentally distinguishes

thixotropy from viscoelasticity, namely an instantaneous drop in

shear stress when the shear rate is suddenly decreased.3

In summary, so far most of the available thixotropy models

rely heavily on ad hoc assumptions that most often lack justifi-

cations other than simplicity’s sake. As a result, their predictive

capability is often inadequate and, in addition, restricted to

a narrow range of applications. It is not unusual that the

obtained predictions are qualitatively wrong even for the steady-

state shear viscosity.

More recently, de Souza Mendes6 proposed a model for elasto-

viscoplastic thixotropic fluids that introduced a number of novel

ideas. The stress equation was developed strictly in accordance

with a Maxwell-like mechanical analog and well founded phys-

ical arguments. The proposed evolution equation for the

Soft Matter, 2011, 7, 2471–2483 | 2471

http://dx.doi.org/10.1039/c0sm01021a






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structure parameter was also developed in a conceptually

different manner, viz., the instantaneous stress and the steady-

state viscosity data were included in the breakdown term

according to a clear physical basis. The resulting formulation is

more straightforward than the previous ones found in the liter-

ature, and the drawbacks presented by the previously published

viscoelasticity-based thixotropy models were removed alto-

gether. Consequently, the predictions of the model proposed by

de Souza Mendes6 were shown to be faithful to experimental

observation for a much wider range of rheological flows.

The present paper follows the lines of ref. 6 to propose a new

model for structured fluids whose differential equation for stress

is developed consistently with a mechanical analog composed by

a strucuring-level-dependent Maxwell-like element in parallel

with a Newtonian element. The presence of the Newtonian

element is shown to enhance significantly the quality of the

predictions. Results obtained with the model for different rheo-

logical test flows demonstrate a far superior predictive capability.

2. The model

This section describes the assumptions and gives the equations

that compose the proposed constitutive model for thixotropic

materials. One of the key assumptions is the existence of

a microstructure whose state can be described by a single scalar

parameter. Let l be this parameter that expresses the state of the

structure. By definition, it ranges from 0 to 1, 0 corresponding to

a completely unstructured state and 1 corresponding to

a completely structured state.

2.1. Equation for stress

The differential equation for the shear stress s used in the present

model is now derived with basis on the mechanical analog shown

in Fig. 1. In this figure, Gs(l) is the shear modulus of the

microstructure; hs(l) is the structural viscosity, a function that

describes the purely viscous response of the microstructure; hN is

the viscosity corresponding to the completely unstructured state

(i.e. to l ¼ 0); ge is the elastic shear strain of the microstructure

when it is submitted to the shear stress s; gv is the viscous shear

strain; and g is the total shear strain. This analog corresponds to

the Jeffreys (or Oldroyd-B) viscoelastic constitutive model,

except that here both Gs and hs are assumed to be functions of the

structure parameter l. It is clear that, according to this analog,

a very large value of hs combined with a finite value of Gs implies

Fig. 1 The mechanical analog.

2472 | Soft Matter, 2011, 7, 2471–2483

the behavior of a viscoelastic solid (with a retardation time equal

to hN/Gs); conversely, a very large value of Gs combined with

a finite value of hs implies the behavior of a purely viscous fluid

(whose viscosity is hv ^ hs + hN).

It is easy to see from the model represented in Fig. 1 that

ge + gv ¼ g 0 _ge + _gv ¼ _g (1)

s ¼ s1 þ s20_s ¼ _s1 þ _s2 (2)

where the dot on top of the variables denotes differentiation with

respect to time t. s1 is the structural stress and s2 is the structure-

independent stress. It is clear that

s2 ¼ hN _g (3)

Thus,

s1 ¼ s� hN _g0_s1 ¼ _s� hN€g (4)

Moreover, the following two expressions also hold for s1:

s1 ¼ hs _gv (5)

s1 ¼ Gs(ge � ge,n) (6)

where ge and ge,n are deformations measured from an arbitrary

fixed reference configuration of the microstructure. ge is the

elastic deformation corresponding to the current configuration,

and ge,n is the deformation corresponding to the natural or

neutral configuration, i.e. the configuration assumed when s ¼ 0.

The neutral configuration is a characteristic of the microstruc-

ture, and hence it is expected to change if (and only if) the

microstructure changes. Consequently, ge,n is expected to be

a sole function of the structure parameter l. Differentiation of

eqn (6) with respect to time gives

_s1 ¼ _Gs

�ge � ge;n

�þ Gs

�_ge � _ge;n

�(7)

The hypothesis that changes in ge,n are solely due to changes in

the microstructure (i.e. k_¼ 0 0 _Gs ¼ 0, _ge,n ¼ 0) leads to the

conclusion that changes in ge are solely due to changes in s1, i.e.

_ge ¼ 05_s1 ¼ 0.

In other words, eqn (7) can be decomposed into two inde-

pendent expressions:

0 ¼ _Gs

�ge � ge;n

�� Gs _ge;n or _ge;n ¼

_Gs

Gs

�ge � ge;n

�(8)

and

_s1 ¼ Gs _ge (9)

The above considerations grant the desired behavior of the

model. For example, it is worth examining the case in which the

material microstructure is initially completely destroyed due to

a long exposure to high shear stresses, and then at a given instant

of time the stress is set to zero so that the microstructure starts

building up. In this case, the model predicts that ge,n ¼ ge (see

eqn (6)), _ge ¼ 0 (see eqn (9)), and thus _ge,n ¼ 0, despite the fact

that _Gs s 0 (see eqn (8)). In other words, in this case the model

predicts that the building up of the microstructure will render the

material elastic while keeping it in its relaxed configuration ge,n,

as it should.



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Now eqn (1) is multiplied by hs and then eqn (5) and (9) are

plugged into it. The result is

s1 þhs

Gs

_s1 ¼ hs _g or s1 ¼ �q1 _s1 þ hs _g (10)

where q1 ¼hs

Gs.

Now eqn (2), (4), and (10) are combined, yielding

s ¼ s1 þ s2 ¼ �q1 _s1 þ ðhs þ hNÞ _g¼ �q1

�_s� hN€g

�þ ðhs þ hNÞ _g (11)

or

sþ q1 _s ¼ ðhs þ hNÞ _gþ q1hN€g (12)

Rearranging, a Jeffreys-like equation for stress is finally

obtained:

sþ q1 _s ¼ hv

�_gþ q2€g

�(13)

In this equation,

hv ¼ hs þ hN and q2 ¼hshN

Gsðhs þ hNÞ¼ hshN

Gshv

¼ hN

hv

q1 (14)

Thus, q2 < q1.

Because the characteristic times q1 and q2 vary continuously

with the structuring level, eqn (13) actually represents a family of

Jeffreys materials parametrized by the structuring level. This

feature lends a remarkable predictive capability to eqn (13), as it

will be illustrated in Sec. 3 below.

2.2. Relaxation and retardation times

It is clear that q1(l) is a relaxation time, while q2(l) is a retarda-

tion time. These quantities can be written in the following form:

q1 ¼�

1� hN

hvðlÞ

�hvðlÞGsðlÞ

(15)

q2 ¼�

1� hN

hvðlÞ

�hN

GsðlÞ(16)

When the material is highly structured (l / 1, and hence

hv [ hN), then q1zhvðlÞGsðlÞ

, and q2zhN

GsðlÞ. In this case s/hv is

very small, and eqn (13) approaches the one pertaining to the

Kelvin–Voigt viscoelastic solid model, namely

_gþ hN

Gs

€g z_s

Gs

(17)

Thus, when the material is highly structured, the behavior as

predicted by eqn (13) is essentially independent of the structural

viscosity hs.

On the other extreme, when the material becomes unstructured

(l / 0) both q1 and q2 go to zero, and eqn (13) reduces to the

Newtonian fluid model (s ¼ hN _g).

2.3. Viscosity function h

The viscosity function is defined as


hhs_g

(18)

It can be written as a product of two functions, one carrying

the elasticity information and the other the purely viscous

character. To this end, eqn (1) is rewritten as:

1� _ge

_g¼ _gv

_g¼ hv _gv

hv _g¼ hs _gv þ hN _gv

hv _g

¼ hs _gv þ hN _g� hN _ge

hv _g

z}|{¼ s

¼ s_ghv

� hN

hv

z}|{¼q2=q1

_ge

_g¼ h

hv

� q2

q1

_ge

_g(19)

Thus,

h ¼�

1��

1� q2

q1

�_ge

_g

�hv (20)

where the term between brackets is the elastic contribution to the

viscosity function h.

2.4. Generalization of the stress equation

A tridimensional, frame-indifferent version of eqn (13) for

general flows is readily obtained by replacing (i) the shear stress sby the extra-stress tensor s ^ T + p1 (T is the total stress tensor

field, p is the pressure field, and 1 is the unit tensor), (ii) the shear

rate _g by the rate-of-deformation tensor _g ^ Vv + VvT (v is the

velocity vector field), and (iii) the time derivative by the upper-

convected time derivative (for example):

sþ q1ðlÞsV ¼ hvðlÞ

�_gþ q2ðlÞ _g

V �(21)

This paper restricts the discussion to homogeneous shear

flows, for which eqn (21) yields13 s22 ¼ s23 ¼ s32 ¼ s33 ¼ 0, and�sþ q1ðlÞ _s ¼ hvðlÞð _gþ q2ðlÞ€gÞ ð21� componentÞ

s11 þ q1ðlÞ _s11 ¼ 2sq1ðlÞ _g� 2hvðlÞq2ðlÞ _g2 ð11� componentÞ(22)

where 1 is the flow direction, 2 is the gradient direction, and 3 is

the neutral direction. Eqn (22) show that the 21–component of

eqn (21) coincides with eqn (13) for homogeneous shear flows.

Moreover, a non-zero first normal stress difference s11 � s22 is

predicted by eqn (21). This predicted normal stress difference

depends of course on the structure parameter l, and it tends to

zero as l / 0, as expected for an unstructured fluid.

Eqn (22) also show that the shear stress s as predicted by the

present model is independent of the normal stresses for homo-

geneous shear flows.

2.5. The structural shear modulus function Gs(l)

A physically reasonable function Gs(l) should ensure that both q1

and q2 are monotonically decreasing functions of l (see eqn (15)

and (16)). This criterion is satisfied, for example, if (but not only

if) (i) Gs(l)¼ constant, and (ii) Gs(l) is smallest when the material

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is fully structured (l ¼ 1), and increases monotonically as l is

decreased.

One possible choice for Gs(l) that is consistent with the above

discussion is6

Gs ¼Go

lm (23)

where Go is the structural shear modulus of the completely

structured material and m is a dimensionless positive constant.

2.6. The structural viscosity function hs(l)

Eqn (20) shows that, when _ge ¼ 0, the viscosity function reduces

to h ¼ hv ¼ hs + hN. In particular, for steady-state flows,

hss( _g) ¼ hv(lss( _g)) (24)

where hss is the steady-state viscosity function and lss the steady-

state structure parameter. Therefore, the steady-state viscosity

function carries useful information regarding the dependence of

h and hv on the structure parameter l.6 For example, it is clear

that both hss and hv vary from hN to ho, where ho is the steady-

state viscosity of the completely structured material (l ¼ 1).

Thus, the function hv(l) should map the range [0,1] into the range

[hN, ho].

The above considerations suggest the following form for

hv(l):6

hvðlÞ ¼�

ho

hN

�l

hN (25)

Eqn (25) can be seen as the definition of the structure

parameter. It can be solved for l to yield

l�

_g; t�¼�

ln hvð _g; tÞ � ln hN

ln ho � ln hN

�(26)

In particular, the above equation reduces to the following

expression for the steady-state structure parameter lss:

lss

�_g�¼�

ln hssð _gÞ � ln hN

ln ho � ln hN

�(27)

Therefore, once the flow curve is determined experimentally,

the steady-state structure parameter is also determined from eqn

(27)6.

2.7. The evolution equation for the structure parameter

The structure parameter l is now assumed to obey the following

evolution equation:6

dl

dt¼ 1

teq

�ð1� lÞa�f ðsÞlb

�(28)

where teq is a characteristic time of change (of l), and a and b are

positive dimensionless constants. In this equation, the first term

on the right-hand side is a structure buildup term, while the

second one is a breakdown term.

As discussed in ref. 6, in the literature the function f appearing

in the breakdown term of eqn (28) is always taken to depend on

the shear rate _g, such that it is zero at _g ¼ 0 and increases

monotonically as _g is increased e.g. see ref. 4,5,14. However, it

2474 | Soft Matter, 2011, 7, 2471–2483

seems more adequate that the breakdown term be a function of

the shear stress s instead, such that it is zero at s¼ 0 and increases

monotonically as s is increased. This is so because what breaks

the microstructure is the level of stress, and not the level of

deformation rate. For example, if a completely structured

material, initially at rest, is suddenly submitted to a constant

shear rate, the shear stress, initially null, will increase linearly

with time. It is reasonable to expect that (i) the breakdown rate is

small at early times, when the microstructure is nearly unde-

formed and under small stresses, and (ii) as time elapses and the

stress builds up, the breakdown rate increases. However, note

that if a dependence of f on _g is assumed, a non-physical response

is obtained. In this situation, f does not change with time. As

a consequence, the breakdown rate, lbf, is maximum at the onset

of the flow (when the stress is zero), and decreases as the

microstructure breaks and l decreases.

In steady state (dl/dt ¼ 0), the function f(s) should reduce to

f ðsssÞ ¼ f�hv

�lss

�_g��

_g�¼ ½1� lssð _gÞ�a

lssð _gÞbin steady state (29)

In view of the above discussion, the following form for f(s) has

the adequate properties:

f ðsÞ ¼ ½1� lssð _gÞ�a

lssð _gÞb�

shvðlÞ _g

�c

(30)

where c is a dimensionless positive constant. Note that, in steady

state, eqn (30) reduces to eqn (29) as it should. The terms

hvðlÞ _gis

of course equal to the term between brackets of eqn (20), and it

introduces elasticity and stress level effects in the breakdown

term.

Thus the model uses the steady-state experimental information

as an input for the evolution equation, because the function

lss( _g) used is determined from the observed material behavior

(see eqn (27)).

Combination of eqn (28) and (30) gives

dl

dt¼ 1

teq

�ð1� lÞa�ð1� lssÞa

�l

lss

�b� shvðlÞ _g

�c�(31)

In the related literature, the function f appearing in eqn (28) is

often assumed to be of the form k _gd, where k is a fitting constant

and d is in most cases taken to be equal to 1 for simplicity (e.g.

ref. 5,14,15. This assumption defines the form of the steady-state

structure parameter lss( _g) (see eqn (29)), which is not necessarily

compatible with the experimentally observed steady-state

viscosity function hss( _g) e.g. see ref. 5,15.

In this work a reverse approach is adopted,6 viz. a representative

form for hss( _g) is chosen, and then it is used in eqn (27) to obtain

lss. This approach yields a more elaborate function f, which is

consistent with the observed behavior of the material while

flowing in steady state. An immediate benefit of this procedure is

the certainty of excellent predictions of steady shear flow.

Eqn (31) can be easily adapted for usage in complex,

non-homogeneous flows, by interpreting (i)dl

dtas the material

derivative of l, i.e.dl

dt¼ vl

vtþ v,Vl, (ii) _g as the second invariant



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of the rate of deformation tensor, i.e. _g ¼ffiffiffiffiffiffiffiffiffiffi1

2tr _g

r2, and (iii) s as

the second invariant of the extra-stress tensor, i.e. s ¼ffiffiffiffiffiffiffiffiffiffiffi1

2trs2

r.

2.8. The steady-state viscosity function

Many of the steady-state viscosity functions available in the

literature can be used in this model, provided it fits well to the

steady-state data. In this work the form proposed by de Souza

Mendes6 is chosen:

hss

�_g�¼�

1� exp

�� ho _g

so

��so � sod

_ge� _g= _god þ sod

_gþ K _gn�1

þ hN

(32)

In this equation, so is the static yield stress, sod the dynamic

yield stress, _god a shear rate that marks the transition in stress

from so to sod, K the consistency index, and n the power-law

index. Eqn (32) is capable of predicting all the features observed

in steady-state data for viscoplastic materials.

2.8.1. Features of the steady-state viscosity function. For

completeness, the main features of the steady-state viscosity

function given by eqn (32) are now discussed, with the aid of

Fig. 2. It is clear that, consistently with the assumptions and

definitions above, eqn (32) obeys the limiting values ho and hN,

as _g / 0 and _g / N respectively.

In Fig. 2 it can be seen that the stress changes from the

static yield stress so to the dynamic yield stress sod in the

vicinity of _g ¼ _god. As discussed in ref. 6, the reason for

including the static yield stress in the flow curve stems from

the fact that the static yield stress must eventually be attained

in steady state, provided the shear rate is low enough,

Fig. 2 The steady-state shear stress, viscosity, and structure parameter

as a function of the shear rate.


otherwise the structuring/unstructuring process would not be

reversible as required by the definition of a thixotropic

material. In the viscosity function given by eqn (32), this

sufficiently low shear rate is roughly _god.

This figure illustrates that, in addition to _god, three other shear

rates mark important transitions in the flow curve, namely _g0, _g1,

and _g2. These shear rates are given by:6

_go ¼so

ho

; _g1 ¼�sod

K

�1=n

; _g2 ¼�hN

K

�1=n�1

(33)

The steady-state viscosity of the completely structured mate-

rial (l ¼ 1), ho, corresponds to the Newtonian plateau observed

in Fig. 2 in the shear rate range _g < _go. Therefore, _go is roughly

the maximum shear rate at which the material structure is

unaffected.

Another transition shear rate is _g1, which marks the beginning

of the power-law region that follows the sharp viscosity decrease

observed at s ¼ sod. _g1 can be conveniently used as a character-

istic shear rate to non-dimensionalize viscoplastic fluid

mechanics problems.16

The highest transition shear rate, _g2, is the one at which the

transition from power-law behavior to Newtonian behavior

occurs, in the high shear rate region (see Fig. 2) where the

material structure is completely destroyed.

Fig. 2 illustrates that the flow curve predicted by eqn (32) is

non-monotonic within the stress range sod < s < so, as observed

in flow curves of many thixotropic materials. This feature is

known to be directly related to interesting phenomena such as

shear banding.17

2.9. Parameters of the model

In summary, eqn (13), (23), (25), (15), (16), (27), (31), and (32)

compose the thixotropy model proposed in this paper. These

equations are gathered below:

sþ q1 _s ¼ hv

�_gþ q2€g

�(13)

Gs ¼Go

lm (23)

hvðlÞ ¼�

ho

hN

�l

hN (25)

q1 ¼�

1� hN

hvðlÞ

�hvðlÞGsðlÞ

(15)

q2 ¼�

1� hN

hvðlÞ

�hN

GsðlÞ(16)

lss

�_g�¼�

lnhssð _gÞ � lnhN

lnho � lnhN

�(27)

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dl 1�

a a

�l�b� s

�c�
dt¼
teq

ð1� lÞ �ð1� lssÞlss hvðlÞ _g

(31)

hss

�_g�¼�

1� exp

�� ho _g

so

��so � sod

_ge� _g= _god þ sod

_gþ K _gn�1

þ hN

(32)

Thus, comparing the present model with the one proposed by

de Souza Mendes,6 it is seen that the parameters that appear in

the two models are exactly the same, i.e. no additional parameter

was needed for the inclusion of the retardation time in the present

model.

Specifically, the parameters are: ho, hN, so, sod, _god, K, n, Go,

m, teq, a, b, and c.

The flow curve parameters sod, K, and n can be determined

from a least-squares fitting to the flow curve, while the remaining

parameters can in principle be obtained via fittings to data per-

taining to transient flows. Some transient flows are discussed in

Sec. 3, where possible procedures for determination of parame-

ters are suggested. However, a detailed discussion on the meth-

odology to determine the model parameters is beyond the

intended scope of this text.

3. Results and discussion

To demonstrate the predictive capability of the model, this

section presents solutions of eqn (13), (23), (25), (15), (16), (27),

(31), and (32) for some selected flows. Simultaneous integration

of the differential equations involved was performed numerically

with a fourth-order Runge–Kutta discretization in time.

The results are presented in dimensionless form, following the

ideas described elsewhere.16 The characteristic shear rate

employed in the scaling is _g1 (defined in eqn (33), see also Fig. 2),

whereas the characteristic shear stress is sod.

All the results below pertain to the following set of parameter

values: ho _g1/sod ¼ 107; n ¼ 0.5; so/sod ¼ 2; _god/ _g1 ¼ 10�4; hN _g1/

sod ¼ 10�2; Go/sod ¼ 1; m ¼ 0.1; _g1teq ¼ 10; a ¼ 1; b ¼ 1; c ¼ 0.1.

3.1. Steady-state predictions

As discussed earlier, the steady-state viscosity function predicted

by the present model is, by construction, exactly the same as the

one used to compose the evolution equation for the structure

parameter, eqn (31) (see eqn (27) and (30)). Thus, in the present

case, the predicted steady-state viscosity function is given by eqn

(32) and plotted in Fig. 2.

Fig. 3 Stress evolution for breakdown or rejuvenation experiments: the

material is at rest for a long time and, at t ¼ 0, a constant shear rate is

imposed.

3.2. Constant shear rate flows

The predictions of the model for the flows characterized by a step

change in shear rate are now examined. The flow is initially

steady at a given shear rate value, _gi, and, at time t¼ 0, the shear

rate is abruptly changed to another value, _gf. That is,

_g(t) ¼ _gi + ( _gf � _gi)H(t) (34)

where H(t) is the Heaviside step function.

2476 | Soft Matter, 2011, 7, 2471–2483

For the simple case in which both Gs and hv are constant,

eqn (13) can be integrated to give the following solution for s:

sðtÞ ¼ _gf

�hv � ðhv � hNÞ

�1� _gi

_gf

�exp

�� Gs

hv � hN

t

�(35)

This equation is useful in the analysis of the results for

constant shear rate flows presented below.

3.2.1. Breakdown or rejuvenation experiments. In this flow,

the material is initially at rest for a long time, and hence

completely structured ( _gi¼ 0; li¼ 1). At time t¼ 0, the shear rate

is suddenly changed from _gi ¼ 0 to _gf, and kept constant until

steady state is attained.

The evolution of the shear stress s and structure parameter l

are given in Fig. 3 and 4, respectively, for different values of _gf. In

this flow, it is easy to see from eqn (35) and also from the

mechanical analog in Fig. 1 that s ¼ hN _gf at time t ¼ 0, whereas

s / hss( _gf) _gf as t / N (steady state), for all values of _gf.

However, Fig. 3 shows that the transient response of s depends

strongly on _gf.

The time needed for steady state to be reached increases

dramatically as _gf is decreased. When _gf is very small so that

lss z 1 ( _gf < _go), then the solution given by eqn (35) is valid. In

this case, using the fact that Gs x Go and hv x ho [ hN, it

reduces to

s ¼ _gf

�hN þ ho

�1� exp

�� Go

ho

t

��(36)

Therefore, in this case very large times (of the order of ho/Go)

are needed for the stress to reach its steady-state value, namely

s ¼ ho _gf. Moreover, while t � ho/Go the material behaves elas-

tically, i.e. the stress increases linearly with time:



Fig. 4 Structure parameter evolution for breakdown or rejuvenation

experiments: the material is at rest for a long time and, at t¼ 0, a constant

shear rate is imposed.

Fig. 5 Stress evolution for buildup or aging experiments: the material is

previously pre-sheared and, at t ¼ 0, a different constant shear rate is

imposed.

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s z hN _gf + Go _gft (37)

Eqn (37) is just the asymptote of eqn (36) for small values of

Go

ho

t, and it could of course be inferred directly from the expected

behavior of the mechanical analog (Fig. 1) for this case.

Even in the range _go < _gf� _g1, when _gf is still small enough so

that q1(l) _gf (a Deborah number) is very large at all times during

the transient flow (see curves for _gf/ _g1 ¼ 10�3 and 10�5 in Fig. 3),

the stress increases linearly with time and then levels off as steady

state is approached. In these cases, the microstructure parameter

is always large and changes mildly during the flow. The transient

flow is thus dominated by elastic strain, and steady state occurs

at t � O(q1(lss)), when the microstructure ceases to deform

elastically and hence the stress levels off at so or sod, depending

on the value of _gf (see Fig. 2).

For larger values of _gf, the solution given in eqn (36) is valid

only up to the time when no appreciable breakdown has

occurred. Large values of _gf imply a fast stress growth, and thus

an early onset of microstructure breakdown. Because micro-

structure breakdown is not instantaneous (Fig. 4), the stress

keeps growing during the earlier stages of the breakdown

process, until the sharp breakdown related to yielding takes

place. This is the cause of the stress overshoot observed in Fig. 3,

in the curves for _gf/ _g1 ¼ 10�1 and larger. It is thus clear that the

stress overshoot is of elastic nature, and the maximum transient

stress attained at a given shear rate is sometimes referred to in the

literature as the ‘‘elastic yield stress.’’ In contrast to the dynamic

and static yield stresses, which are clearly steady-state properties,

the elastic yield stress is a shear-rate dependent transient quan-

tity.

Finally it is seen that, as the shear rate is increased, the ratio of

the maximum (elastic yield) stress to the steady-state stress


initially increases and then decreases. This trend is related to the

fact that the breakdown rate of the microstructure increases as

the stress is increased, which constitutes a regulating mechanism

that controls stress growth.

All the trends observed in Fig. 3 are plentifully corroborated

by data found in the literature e.g. see ref. 14,18.

3.2.2. Aging or buildup experiments. In this flow, the material

is initially flowing in steady state at a relatively high shear rate

( _gi/ _g1 ¼ 100 in the present example), and hence its structuring

level is low (li ¼ 0.12 in the present example). At time t ¼ 0, the

shear rate is suddenly changed from _gi ¼ 100 _g1 to _gf, and kept

constant until steady state is attained.

Fig. 5 and 6 show the model predictions for the buildup

experiments. The evolution of the shear stress s and structure

parameter l are given in these figures for different values of _gf.

For the buildup experiments, from eqn (35) it is clear that the

shear stress at t ¼ 0 is

s(0) ¼ hN _gf + (hss( _gi) � hN) _gi (38)

where the first term on the right hand side is the hN—dashpot

contribution, while the second term is the contribution of the

Maxwell element (see Fig. 1). Eqn (35) also gives that s /

hss( _gf) _gf as t / N, for all values of _gf.

It is first noted in Fig. 5 and 6 that the curves for _gf/ _g1 ¼ 103

and 105 actually correspond to breakdown experiments, inas-

much as _gf > _gi in these cases. They appear in these figures just to

illustrate the case of further micrustructure breakdown of a pre-

sheared material. It is seen that mild stress overshoots occur in

these cases despite the low structuring level at t ¼ 0, because

q1(li) _gf is still quite large in these cases (q1(li) _gf¼ 89.17and 8917,

respectively).

Soft Matter, 2011, 7, 2471–2483 | 2477


Fig. 6 Structure parameter evolution for buildup or aging experiments:

the material is previously pre-sheared and, at t ¼ 0, a different constant

shear rate is imposed.

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For the microstructure buildup experiments ( _gf < _gi) relative to

the curves for _gf/ _g1¼ 10�5, 10�3, 10�1, 1, and 10, it is observed that

a steep l increase occurs in the time interval from _g1t � 0.1 up to

t� O(teq) (Fig. 6). Subsequently, either its steady-state value lss is

achieved ( _gf/ _g1¼ 1 and 10) or it mildly overshoots and then levels

off to lss ( _gf/ _g1¼ 10�5, 10�3, and 10�1). The l overshoot occurs at

lower shear rates because in these cases, during the elastic defor-

mation the stress increases very slowly and there is plenty of time

for microstructure buildup before the stress becomes large enough

to start causing appreciable breakdown.

Because the structure parameter does not change significantly

in the early stages of the flow ( _g1t ( 0.5), all the curves for the

buildup experiments nearly coincide in this range, displaying

a stress relaxation that obeys eqn (35):

sðtÞ ¼ hss

�_gi

�_gf þ

�hss

�_gi

�� hN

��_gi � _gf

�exp

�� GsðlssÞ

hssð _giÞ � hN

t

�(39)

Thus, this relaxation at early times occurs within times of the

order of (hss( _gi) � hN)/Gs(lss). It is clear that this relaxation time

would be zero if the material were initially completely unstruc-

tured (lss ¼ 0).

The steep l increase observed in the time interval from _g1t �0.1 up to t � O(teq) (Fig. 6) causes the stress increase observed in

Fig. 5. Due to the fast structuring, important changes in struc-

tural shear modulus (decrease) and structural viscosity (increase)

occur in this interval, causing the interesting transition observed

in the stress curves in this time interval.

Fig. 7 Shear rate evolution for viscosity bifurcation experiments: the

material is at rest for a long time and, at t ¼ 0, a constant shear stress is

imposed.

3.3. Constant shear stress flows

The predictions of the model for two different constant shear

stress flows are now examined.

2478 | Soft Matter, 2011, 7, 2471–2483

The first flow is characterized by a step change in shear stress,

i.e. it is initially steady at a given shear stress value, si ¼ hss( _gi) _gi,

and, at time t¼ 0, the shear stress is abruptly changed to another

value, sf. That is,

s(t) ¼ si + (sf � si)H(t) (40)

For the simple case in which both Gs and hv are constant, eqn

(13) can be integrated to give the following solution for _g:

_gðtÞ ¼ sf

�1

hv

þ�

1

hN

� 1

hv

��1� si

sf

�exp

�� hv

hv � hN

Gs

hN

t

�(41)

This equation is useful in the analysis of the results presented

in Sec. 3.3.1, where an example of this flow is discussed.

The second constant shear stress flow examined is the one

followed by pre-shear. The material is in steady-state flow at

a relatively high shear rate value, and then it is brought to rest

( _g ¼ 0) for some period of time after which a constant stress is

imposed until steady state is achieved. This flow is discussed in

Sec. 3.3.2.

3.3.1. Viscosity bifurcation when the material is initially at

rest. Fig. 7, 8, and 9 show results for the so-called viscosity

bifurcation experiments. The material is at rest and thus

completely structured (li¼ 1), and at time t¼ 0 the shear stress is

suddenly changed from zero to sf, and kept constant until steady

state is attained. The evolution of the viscosity h, shear rate _g,

structure parameter l are given in Fig. 7, 8, and 9, respectively,

for different values of sf.

When sf is very small so that lss z 1, then Gs x Go and hv xho [ hN, and hence eqn (41) reduces to



Fig. 8 Viscosity evolution for viscosity bifurcation experiments: the

material is at rest for a long time and, at t ¼ 0, a constant shear stress is

imposed.

Fig. 9 Structure parameter evolution for viscosity bifurcation experi-

ments: the material is at rest for a long time and, at t¼ 0, a constant shear

stress is imposed.

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_gðtÞ ¼ sf

�1

ho

þ�

1

hN

� 1

ho

�exp

�� Go

hN

t

�(42)

Therefore, when sf is not large enough to cause notable

microstructure breakdown, the shear rate falls due to the

increasing elastic resistance of the deformed microstructure, and

a time of about 30hN/Go is needed for the shear rate to reach its

steady-state value, namely _g ¼ sf

ho

(thus very small). Moreover,


when the microstructure equilibrium time teq is significantly

larger than 30hN/Go, it happens that eqn (42) is always the

solution of eqn (13) for t < teq, even for large values of sf, because

significant microstructure breakdown starts at t � O(teq) only.

Thus, a pseudo-steady state is observed where _gxsf

ho

, roughly in

the time range 30hN/Go < t < teq (provided 30hN/Go < teq). Fig. 7,

8, and 9 illustrate this interesting behavior.

These figures also illustrate that, in the cases pertaining to sf/

sod ¼ 0.01, 0.5, 1, 1.5, and 1.98, steady state is achieved without

much breaking of the microstructure. However, the same is not

true for the curves pertaining to sf/sod¼ 1.99, 2, 3, 5, and 7, where

the steady-state viscosities are all several orders of magnitude

lower than ho, indicating a dramatic breakdown of the micro-

structure in these cases.

The viscosity bifurcation that occurs between sf/sod¼ 1.98 and

1.99 is related to the local maximum slmax (1.98 < slmax/sod <

1.99) that exists in the flow curve in the range of small shear rates

(see Fig. 2). Indeed, from eqn (32) it is easy to see that this local

maximum is slmax/sod ¼ 1.98547, and occurs at _glmax/ _g1 ¼1.3921 � 10�6. This bifurcation obviously occurs as a conse-

quence of the non-monotonic nature of the flow curve. When the

material is initially at rest and the stress is set to a value below

slmax, no important microstructure breakdown occurs, and the

steady state is achieved at a shear rate value smaller than _glmax.

However, when the stress is set to a value above slmax, no matter

how close to it, the microstructure eventually collapses, a shear

rate jump occurs, and the steady state is achieved at a viscosity

several orders of magnitude lower than ho. It is worth noting that

slmax is essentially equal to the static yield stress so, and so in

principle the flow corresponding to Fig. 7, 8, and 9 can be used to

determine experimentally the static yield stress.

It is especially remarkable that, for stress levels around the

yield stress (e.g. curve for sf/sod ¼ 1.99 in Fig. 7), the shear rate

seems to be leveling off to a negligibly low value, and, after a very

large delay time (O(100teq) in this example), it suddenly increases

steeply and then levels off to a much higher shear rate value. In

addition, this delay time decreases as the shear stress is increased.

This predicted behavior agrees with experimental evidence

available in the literature.3,17,19

3.3.2. Viscosity bifurcation when the material is pre-sheared.

Another type of flow at constant stress is now examined. The

material is pre-sheared to bring about a low structuring level in

steady state (li ¼ 0.12 in this example). Subsequently, the shear

rate is set to zero, and the stress is allowed to start relaxing. At

the moment t ¼ 0 when the relaxing stress reaches sf, it is kept

constant until steady state is achieved. It is assumed that no

significant changes in the structuring level occur during the

resting period (time interval between the cessation of

pre-shearing and t ¼ 0). Therefore, the initial shear rate is null

( _g(0) ¼ 0).

It is worth noting that negative values of _g(0) (recoil) would

occur if the flow started immediately after pre-shearing and the

pre-shearing stress were larger thanhvðliÞ

hvðliÞ � hN

sf (see eqn (41)).

On the other hand, if the experiment started after the stress

generated during pre-shearing had relaxed to zero, it is possible

that significant structuring would have occurred, which implies

Soft Matter, 2011, 7, 2471–2483 | 2479


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an uncertain initial condition. For these reasons, it seems more

representative of laboratory practice to impose the _g(0) ¼0 initial condition described in the previous paragraph. It is clear

that each value of sf implies a different resting period. If instead

a fixed resting period were imposed, each imposed value of sf

would cause a different shear rate jump at t ¼ 0.

Fig. 10, 11, and 12 show the evolution of the shear rate _g,

viscosity h, and structure parameter l for this flow, for different

Fig. 10 Shear rate evolution for viscosity bifurcation experiments: the

material is pre-sheared, then brought to rest, and the stress is allowed to

relax up to s ¼ sf. From this moment (t ¼ 0) on, it is kept constant at sf.

Fig. 11 Viscosity evolution for viscosity bifurcation experiments: the

material is pre-sheared, then brought to rest, and the stress is allowed to

relax up to s ¼ sf. From this moment (t ¼ 0) on, it is kept constant at sf.

Fig. 12 Structure parameter evolution for viscosity bifurcation experi-

ments: the material is pre-sheared, then brought to rest, and the stress is

allowed to relax up to s ¼ sf. From this moment (t ¼ 0) on, it is kept

constant at sf.

2480 | Soft Matter, 2011, 7, 2471–2483

values of sf. In these figures, all the imposed stresses up to sf/sod¼10, correspond to steady-state structure parameter values lss

which are higher than li, which means that some microstructure

buildup is expected for these cases. The remaining two cases

shown, namely, sf/sod ¼ 50 and 100, correspond to breakdown

experiments, inasmuch as lss < li for these cases, and were

included for completeness to illustrate cases of further micro-

structure breakdown. The following discussion, however, focuses

on the cases pertaining to lss > li, which carry more interesting

features.

Fig. 12 shows that the structure parameter does not change

significantly until t� O(0.01teq), when it starts increasing to reach

a steady state at t � O(teq) or later, depending upon the level of

the stress sf. Thus it is not difficult to see that, while t < O(0.01teq)

the shear rate should obey

_gðtÞ ¼ sf

hvðliÞ

�1� exp

�� hvðliÞ

hvðliÞ � hN

GsðliÞhN

t

�(43)

Thus, for any shear stress value, the shear rate departs from

zero, increases monotonically and, at t � O((hv(li) � hN)hN/

(hv(li)Gs(li))), reaches a pseudo-steady state value, namely

_g ¼ sf

hssðliÞ, which lasts up to t� O(0.01teq) (Fig. 10). In this same

range of early times, for all shear stress values the viscosity

function departs from infinity at t ¼ 0, then decreases mono-

tonically and, at t � O((hv(li) � hN)hN/(hv(li)Gs(li))), levels off

to its pseudo-steady state value h ¼ hss(li), and remains

unchanged up to t � O(0.01teq) (Fig. 11).

This counterintuitive initial rise of the shear rate—even when

the imposed shear stress is much lower than the yield stress—has

been observed experimentally (e.g. see ref. 17 Fig. 3 therein), and

is explained through the present model as follows. At t ¼ 0 the



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material is at rest but the microstructure is elastically deformed,

and the stress at the Maxwell element is sf. As time elapses, the

relaxing process continues at the Maxwell element, and thus the

stress at the Newtonian element (which was zero at t ¼ 0) must

increase, meaning that _g also must increase. It is thus clear that

this behavior is of elastic nature and is a consequence of a non-

zero retardation time. For this reason, the Maxwell-like model

proposed by de Souza Mendes6 is not able to predict such

behavior.

Similarly to what was observed with no imposed pre-shearing,

a viscosity bifurcation is also observed in Fig. 10, 11, and 12. In

this case the bifurcation indicates that a dramatic buildup of the

microstructure occurs when the stress is set to a value below

a local minimum slmin of the flow curve which exists in the range 1

< slmin/sod < 1.1. Actually this local minimum is slmin/sod ¼1.02694, and occurs at _glmin/ _g1 ¼ 6.2111 � 10�4 (see Fig. 2). In

this case it is clear that slmin is essentially equal to the dynamic

yield stress sod.

It is clear that the flows corresponding to Fig. 8 and 11 provide

respectively a lower and upper bound for the parameter _god that

appears in eqn (32), namely, _glmax and _glmin.

The above described trends predicted for startup flows at

constant shear stress by the present model are in qualitative

agreement with the experimental observations reported in the

literature e.g. see ref. 15,17.

Before proceeding to the next section, a final comment is in

order. The results presented in Secs. 3.2 and 3.3 were obtained via

simultaneous numerical integration of eqn (13) and (31).

However, the analytical expression given in eqn (35) turned out

to be an excellent approximation for s(t) in constant shear rate

flows, the same being true regarding eqn (41) for _g(t) in constant

shear stress flows, provided hv and Gs in these expressions are

seen as functions of l as given by integration of eqn (31).

In narrow time ranges containing stress overshoots and

sudden changes in time derivative, however, the just mentioned

approximation becomes poor. Nevertheless, this fact is certainly

potentially useful in the determination of some of the model

parameters.

3.4. Oscillatory flows

The model predictions for small-amplitude oscillatory flows are

now presented. As the results below will illustrate, the intro-

duction of a non-zero retardation time rendered the present

Jeffreys-like model far superior to the Maxwell-like model

recently proposed by de Souza Mendes,6 as far as the predictive

capability of oscillatory flows is concerned.

A sinusoidal shear g(t) ¼ gasin(ut) (ga is the shear amplitude,

and u is the frequency of oscillation) is imposed, resulting in

a shear rate _g(t) ¼ _gacos(ut), where _ga ¼ uga is the shear rate

amplitude. As usual in small-amplitude oscillatory flows, the

shear stress is assumed to be a linear function of both g(t) and

_g(t):

s(t) ¼ ga{G0sin(ut) + G00cos(ut)} (44)

where G0 and G00 are respectively the storage and loss moduli.

Differentiation with respect to time yields


_sðtÞ ¼ ga

�_G0 � uG00

�sinðutÞ þ

�uG0 þ _G0 0

�cosðutÞ (45)

The combination of eqn (44) and (45) with eqn (13), yields the

following differential equations for G0 and G0 0:

_G0 þ GsðlÞhvðlÞ � hN

G0 ¼ uG00 � hNu2 (46)

_G00 þ GsðlÞhvðlÞ � hN

G00 ¼ u

�hvðlÞGsðlÞhvðlÞ � hN

� G0�

(47)

The above equations are to be solved in conjunction with the

evolution equation for l, eqn (31). The expected solution for l(t)

is an initial transient—at the end of which the l level has adjusted

to the imposed shear stress level—followed by a periodic

response. However, when the typical times of change of l are

much larger than 1/u, then the amplitude of this periodic solu-

tion for l is negligibly small, and hence nearly time-independent

values of l, G0, and G0 0 are eventually reached. In this pseudo-

steady state, _G0 x 0, _G0 0 x 0, and the above differential equa-

tions reduce to the following expressions for G0 and G0 0:

G0ðlÞ ¼u2hvðlÞ

2GsðlÞ

�1� hN

hvðlÞ

�2

GsðlÞ2þu2hvðlÞ2

�1� hN

hvðlÞ

�2(48)

and

G0 0ðlÞ ¼uhvðlÞGsðlÞ2

1þ u 2

hNhvðlÞGsðlÞ2

�1� hN

hvðlÞ

�2!

GsðlÞ2þu2hvðlÞ2�

1� hN

hvðlÞ

�2(49)

Eqn (48) and (49) are equivalent to the well known expressions

for G0 and G00 as predicted by the classical Jeffreys model (usually

written in terms of the relaxation and retardation times, q1 and

q2), except that in the present model G0 and G0 0 depend on the

structure parameter l.

If the shear stress amplitude is sufficiently small to keep the

material structure intact, then hv ¼ ho [ hN and Gs ¼ Go �uho, and hence the above expressions reduce to

G0ohG0ð1Þ ¼u2h2

oGo

�1� hN

ho

�2

G2o þ u2h2

o

�1� hN

ho

�2x Go (50)

and

G00ohG00ð1Þ ¼uhoG2

o

�1þ u2

hNho

G2o

�1� hN

ho

�2!

G2o þ u2h2

o

�1� hN

ho

�2xuhN (51)

Actually, if the frequency u is not too low, then hv(l) [ hN

and Gs(l) � uhv(l) up to l values much lower than unity, and

hence, it is still true in this range that G0(l) x Gs(l) and that

G0 0(l) x uhN.

Therefore, the model parameters Go, m, and hN can in prin-

ciple be obtained directly from shear amplitude sweep data (see

Soft Matter, 2011, 7, 2471–2483 | 2481


Fig. 13 Structure parameter, storage, and loss moduli as a function of

the shear amplitude.

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Fig. 13). This way of obtaining hN is particularly convenient due

to inertia problems that occur in steady-state flow measurements

in the high shear rate range.

The model predictions for a shear amplitude sweep test are

illustrated in Fig. 13. It is firstly observed that both G0 and G00 are

ga-independent (and equal to Go and uhN respectively) for

a wide range of ga, as discussed above. When ga reaches O(1), the

weak dependency of the structural shear modulus Gs on l

(because m ¼ 0.1) becomes visible in the G0 curve, i.e. it is clear

that in this range G0 increases mildly with ga, but subsequently it

drops sharply. In this same range, G0 0starts increasing steeply,

reaches a maximum, and from there on decreases with ga,

although less steeply than G0. The two curves cross (G0 ¼ G0 0) at

the same ga value where the G0 0 maximum occurs.

Fig. 14 Structure parameter, storage, and loss moduli as a function of

the frequency.

2482 | Soft Matter, 2011, 7, 2471–2483

The curve shapes of the storage and loss moduli obtained

experimentally for thixotropic fluids and reported in the recent

literature are typically very close to the ones seen in Fig. 13 (e.g.

see refs. 18,20,21). The observed trends are faithfully described

by eqn (48) and eqn (49). It is interesting to notice that the abrupt

changes in the G0 and G0 0 curves occur when the structural

viscosity is orders of magnitude lower than the one observed

when the material is at rest. This fact suggests that there is no

justification for the often employed method of yield stress

measurement based on stress amplitude sweep data.

Fig. 14 gives the model predictions for frequency sweep tests at

a fixed shear amplitude value. Both the moduli and the structure

parameter are given as functions of the frequency. The trends

observed in this figure are in full agreement with the ones typi-

cally observed experimentally for structured liquids (e.g., see ref.

22 p.92, Fig.12 therein). All the classical frequency sweep regions

are predicted: the viscous region, the transition-to-flow region,

the rubbery region, the leathery crossover region, and the glassy

region. The deepness of the G0 0 ‘valley’ and the slope of G0 on the

‘plateau’ and subsequent regions depend on the structuring level,

which varies with the imposed shear amplitude ga.

4. Final remarks

This paper describes a model for elasto-viscoplastic thixotropic

fluids that is a followup of the model recently proposed by de

Souza Mendes.6 It is composed of two differential equations, one

for stress and the other for the structure parameter. The equation

for stress is identical to the Jeffreys fluid viscoelastic model,

except that the relaxation and retardation times depend on the

structure parameter.

The model proposed here preserves all the advantages of the

former model, while achieving a significantly better predictive

capability. No additional parameter is needed.

The most remarkable features of the model are:

� The formulation is simple and all the assumptions are

justified by physical arguments;

� The concept of a neutral configuration that changes with the

microstructure introduced in ref. 6 is employed, which yields

a differential equation for stress that obeys a well defined

mechanical analog;

� A retardation time arises in the stress equation, yielding

a Jeffreys-like differential equation whose relaxation and retar-

dation times are structuring-level dependent. As a consequence,

the model is able to predict quite accurately even the most

complex trends related to viscoelasticity that are observed in the

rheological data for structured fluids found in the literature;

� As shown in ref. 6 the breakdown term of the evolution

equation for the structure parameter is assumed to depend on the

stress rather than on the shear rate;

� The evolution equation for the structure parameter is con-

structed such as to be consistent with the steady-state flow curve,

as proposed by de Souza Mendes.6 This is certainly one of the

main reasons for the excellent model performance;

� The non-monotonic steady-state viscosity function proposed

by de Souza Mendes6 is employed. This function allows the model

to accommodate two yield stresses, a static and a dynamic one.

Because the physics involved in the model are quite clear and

its predictions are in excellent qualitative agreement with



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experimental observation for all flows investigated, then the

proposed model also constitutes an important tool to better

understand the fundamentals of the mechanical behavior of

structured fluids.

5.Acknowledgments

The author is indebted to Petrobras S.A., MCT/CNPq, CAPES,

FAPERJ, and FINEP for the financial support to the Group of

Rheology at PUC-RIO.

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thixotropic elasto-viscoplastic model for structured fluids

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