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A new constitutive equation for elastoviscoplastic uid
ows
Pierre Saramito
To cite this version:
Pierre Saramito. A new constitutive equation for elastoviscoplastic uid ows. Journal of Non-
Newtonian Fluid Mechanics, Elsevier, 2007, 145 (1), pp.1-14. �10.1016/j.jnnfm.2007.04.004�. �hal-
00109101v4�
A new constitutive equation for elastoviscoplastic fluid flows
Pierre Saramito
a
a
CNRS – LJK, B.P. 53, 38041 Grenoble cedex 9, France
Abstract – From thermodynamic theory, a new three-dimensional model for ela stoviscoplastic fluid
flows is presented. It e xtends both the Bingham viscoplastic and the Oldroyd viscoelastic models.
Fundamental flows are studied: simple shea r flow, uniaxial elongation and large amplitude oscillatory
shear. The complex moduli (G
′
, G
′′
) are fo und to be in qualitative agreement with experimental data
for materials that present microscopic network structures and large scale rearrangements. Various fluids
of practica l interest, such as liquid foams, droplet emulsions or bloo d, present such elastoviscoplastic
behavior : a t low stress, the material behave s as a viscoelastic solid, whereas at stresses above a yie ld
stress, the material behaves as a fluid.
Keywords – non-Newtonian fluid; vis coelasticity; viscoplasticity; constitutive equation.
1. Introductio n
1.1. Historical background
The development of viscoplastic rheological model based on yield stress started in 1900 when Schwed-
off [20], s tudying a gelatin suspension, presented a one-dimensional plas tic viscoelastic version of the
Maxwell model:
˙ε = 0, when τ ≤ τ
0
,
λ
dτ
dt
+ (τ − τ
0
) = η
m
˙ε, when τ > τ
0
,
(1)
where τ is the stress, ˙ε the rate of deformation, η
m
> 0 the visc osity, τ
0
≥ 0 the yield stress and λ ≥ 0
a r e laxation time. In steady shear flow this reduces to τ = τ
0
+ η
m
˙ε when τ > τ
0
. In 1922, Bingham [1]
proposed the one-dimensional stress-deformation r ate equation for a viscous fluid with a yield stress:
max
0,
|τ| − τ
0
|τ|
τ = η
m
˙ε ⇐⇒
|τ| ≤ τ
0
when |˙ε| = 0,
τ = η
m
˙ε + τ
0
˙ε
|˙ε|
otherwise.
(2)
Notice that this model is equivalent – up to the sign of ˙ε, assumed positive – to the steady case of the
model proposed by Schwedoff. Numerous attempts have been made to modify this simple equation to
Email address: Pierre.Saramito@imag.fr (Pierre Saramito).
Preprint submitted to Elsevier Science April 2, 2007
account for more complex behavior of such materials. In 1926, Hers chel and Bulkley [6] proposed to
model the observed shear str e ss dependence of the viscosity on the shear ra te ˙ε after yielding by explicitly
defining the viscosity η
m
as a power-law function of |˙ε|.
In 1932, Prage r [14], using the von Mises [24] y ie lding criterion, proposed to extend the Bingham model
to the three-dimensional c ase and Oldroyd, in 1947, in a collection of papers (see e.g. [11]) studied the
Bingham three- dimensional model and its Herschel-Bulkley extension coupled with the Navier-Stokes
equations for the motion of the fluid. Oldroyd also proposed a three-dimensional constitutive equation
which combines the yielding criterion together with a linear Hookean elastic b e havior before yielding and
a viscous behavior after yielding . In the one-dimensional case, the model can be written as:
τ = µε when τ ≤ τ
0
,
|τ| − τ
0
|τ|
τ = η
m
˙ε when τ > τ
0
.
(3)
When compared to (1), this model is an improvement, since the material is no longer rigid before yielding.
Here, the yield stress τ
0
is related to the c ritical strain ε
0
= τ
0
/µ. Since the first equation in (3 ) describes
stresses in term of strain ε and the s e c ond equation in (3) in term of strain rate ˙ε, the stress-strain curve
predicted by this model must exhibit a disco ntinuity at the critical s train ˙ε = ˙ε
0
at which the stress
jumps from τ = τ
0
to τ = τ
0
+ η ˙ε. T his is an approximation of the true behavior of materials: the real
deformation at the transition is expected to be smooth, at least continuous.
In 1950, Oldroyd [12] developed a theory for the invariant forms of rheological equations of state and
proposed a three-dimensional viscoelastic model, that can be expressed in its one-dimensional version as:
λ
dτ
dt
+ τ = η
m
˙ε (4)
where the total stress σ = η ˙ε + τ. In this approach the stress τ is the elastic part of the total stress, from
which the elastic deformation can ea sily be obtained. The constant η > 0 is another viscosity, often c alled
the solvent viscosity in the context of polymer solutions.
1.2. One-dimensional presentation of the proposed model
Let us introduce an one-dimensional version of our model in order to combine the two previous models (2)
and (4):
λ
dτ
dt
+ max
0,
|τ| − τ
0
|τ|
τ = η
m
˙ε. (5)
where the total stress is σ = η ˙ε + τ . When λ = η = 0 we obtain (2) while when τ
0
= 0 our model reduces
to (4). Observe that (5) differs both from (1) and (3). Schwedoff proposed a rigid behavior ˙ε = 0 when
|τ| ≤ τ
0
and Oldroyd proposed a brutal change of model when rea ching the yield value. Our proposition
assures a continuous change from a solid to a fluid behavior of the material.
The mechanical model is represented in Fig. 1.c. A friction τ
0
has bee n inserted in the Oldroyd viscoelastic
model (Fig. 1.b). At stre sses below the yield stress, the friction element remains rigid. The level of the
elastic strain ener gy required to break the frictio n element is determined by the von Mises yielding crite-
rion. Consequently, before yielding, the whole system predicts only recoverable Kelvin-Voigt viscoelastic
deformation due to the spring and the viscous element η: the Kelvin-Voigt viscoelastic model is described
by a s pring and a viscous body in pa rallel (Fig. 1.a). The elastic behavior τ = µε is expressed in (5) in
differential form where µ = η
m
/λ is the elasticity of the spring in Fig. 1.c. Before yielding, the total stress
2
(a) Kelvin-Voigt (b) Oldroyd
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
µ
η
σ
ε
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
η
σ
ε
ε
e
ε
m
µη
m
(c) present model (d) creeping σ(t) = σ for the present model
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
η
σ
τ
0
ε
ε
e
η
m
ε
m
µ
σ < τ
0
σ > τ
0
t
ε(t)
0
0
Figure 1. The proposed elastoviscoplastic model.
is σ = µε + η ˙ε + τ. As soon as the strain energy exceeds the level required by the von Mise s criterion,
the stress in the fr ic tion element attains the yield value and the element breaks allowing deformation
of all the other elements. After yielding, the deformation of these elements describes the Oldroyd-type
viscoelastic behavior.
3
The evolution in time of elongation ε(t) for a fixed imposed traction σ (creeping) is repre sented on Fig 1.d.
When σ ≤ τ
0
, the elongation for a fixed imposed traction is bounded in time, which mea ns that such
material behaves as a solid. Otherwise, σ > τ
0
, the elongation is unbounded in time, which means that
the material behaves as a fluid. Notice the continuous behavior of the solution: the change of regime
occurs a t t = t
0
for the solution
σ > τ
0
and there is neither jump nor loss of differentiability in the curve.
The solution on Fig 1.d is computed explicitly and details are reported in appendix B .
1.3. Comparison with other recent and closely related models
A numb e r of other closely related models have appeared in the literature.
In 1991, Beris et al.. [3], in order to recover a continuous approximation of the solution of the elastic-
viscoplastic model (3) proposed by Oldroyd, introduced an ad-hoc recovery procedure. Before yielding, the
material behaves as an elastic solid while after yielding it behaves as a power-law viscous non-Newtonian
fluid. Despite the la ck of a thermodynamical analysis of their model, these authors have been able to
propose, based on their computational r e sults, a use ful Cox-Merz rule extension that was found to be in
good agreement with experimental data on a suspension o f silicon particles in polyethylene.
In 2003, Puzrin and Houlsby [15, p. 254] inserted friction and a spring into the viscoelastic Kelvin-Voigt
model and the resulting model is represented in Fig. 2.a. Before y ielding, the friction is rigid and the
material behaves a s an elastic solid, thanks to the spring µ. After yielding, the deformation is governed
by the so-called standard mo del: a K e lv in-Voigt element plus a spring in series (see e.g. [10, p. 42]). The
predicted elongation under a constant traction is presented on Fig. 2.b. The elongation jumps immediately
at t = 0 from ε = 0 to ε =
σ/µ, since the spring element has no time scale. Next, when t > 0, if σ ≤ τ
0
, then
the elongation remains constant, since the spring is fully extended. Otherwise, if σ > τ
0
, the elo ngation
grows and tends to a b ounded value: the mater ial be haves as a solid. By this way Puzrin and Houlsby were
able to capture some releva nt aspects of the behavior of saturated c lays. These authors have prop osed in
a collectio n of papers [7,16] many variants of their model. Nevertheless, it is not applicable to material
such as human blood or liquid foams that deform under a low stress and flow under a sufficient stress.
In 1990 Isayev and Fan [9] proposed a viscoelastic plastic constitutive equation for flow of particles
filled polymers. The main idea of these authors is to insert a friction and a spring into the viscoelas tic
Oldroyd model and the resulted model is represented on Fig. 2.c. Before yielding, the friction is rigid and
the material behaves as a simple elas tic solid, thanks to the spring µ. After yielding, the deformatio n
is described by the Oldroyd viscoelastic model. These authors have added in parallel other Max well
elements in order to replace the Oldroyd v iscoelastic element by a multi-mode Leonov viscoelastic model.
The predicted e longation under a constant trac tion is presented on Fig. 2.d. As for the Puzrin-Houlsby
model, the e longation jumps immediately at t = 0 from ε = 0 to ε =
σ/µ. Next, when t > 0 , if σ ≤ τ
0
,
then the elongation remains constant. Other wise, if σ > τ
0
, the elongation grows and is not bounded: in
that case, the material behaves as a fluid. The frictio n element was able to describe the stress generated
in the disperse phase of the filled polymer melts and this contribution has represented an impo rtant
conceptual advance in this domain. Since the effect of added solid particles in a polymer melt is usually
to reduce the viscoelasticity, the need of this kind of for mulation arises primarily in the case of highly
viscoelastic polymers such as rubbers. Nevertheless, the model presented in Fig. 1.c and. 1.d is a definite
improvement: the instanta neous jump at t = 0 followed by a constant elongation under traction of both
the Puzrin-Houlsby and the one-mode simplified Isayev-Fan models is replaced by a smoother b e havior.
During the last two decades, the thermodynamic framework developed and furnished some robust theo-
retical tools for managing efficiently rheolo gical models: see e.g. [22, p. 26] or [10] for concise presentations
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