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1. Ministry of Higher Education and Scientific Research
University of Tikrit
College of Engineering
Department of Chemical Engineering
Elasticity and Linear Viscoelastic Models
by
Mohammed Hassan Mohammed
SUPERVISED BY
Asst. Prof. Dr. Safaa M. Rasheed

2.  𝜏𝑖𝑗= 𝑘 𝑖𝜇𝑖𝑗𝑘𝑙
𝑑𝑥𝑘
𝑑𝑥𝑙
 where i, j, k, and 1 may be 1,2,3
 𝜇𝑖𝑗𝑘𝑙 "viscosity coefficients."
 The quantities x1, x2, x3 in the derivatives denote the Cartesian coordinates x, y, z,
 vx, vy, vz, are the same as vx, vy, vz.
 In the discussion about generalizing Newton's "law of viscosity," we specifically
excluded time derivatives and time integrals in the construction of a linear
expression for the stress tensor in terms of the velocity gradients. In this, we allow
for the inclusion of time derivatives or time integrals, but still require a linear
relation between 𝜏 and 𝛾 This leads to linear viscoelastic models.
Elasticity and Linear Viscoelastic Models
Newton's "law of
viscosity

3.  Viscoelasticity is the property of materials that exhibit both
viscous and elastic characteristics when undergoing deformation.
 Viscous materials, like honey, resist shear flow and strain
linearly with time when a stress is applied.
 Elastic materials strain when stretched and quickly return to their
original state once the stress is removed.
 Viscoelastic materials have elements of both of these properties
and, as such, exhibit time-dependent strain. Whereas elasticity is
usually the result of bond stretching along crystallographic planes
in an ordered solid, viscosity is the result of the diffusion of
atoms or molecules inside an amorphous material.
Viscoelasticity

5.  In the nineteenth century, physicists such as Maxwell, Boltzmann, and Kelvin researched
and experimented with creep and recovery of glasses, metals, and rubbers.
 Viscoelasticity was further examined in the late twentieth century when synthetic
polymers were engineered and used in a variety of applications. Depending on the change
of strain rate versus stress inside a material the viscosity can be categorized as having a
linear, non-linear, or plastic response.
 When a material exhibits a linear response it is categorized as a Newtonian material. In
this case the stress is linearly proportional to the strain rate. If the material exhibits a non-
linear response to the strain rate, it is categorized as Non-Newtonian fluid.
Background

7. Types
Linear viscoelasticity is when the function is separable in both creep response
and load. All linear viscoelastic models can be represented by a Volterra equation
connecting stress and strain:
Linear viscoelasticity is usually applicable only for small deformations.
 Nonlinear viscoelasticity is when the function is not separable. It usually happens
when the deformations are large or if the material changes its properties under
deformations.
 anelastic material is a special case of a viscoelastic material: an anelastic material
will fully recover to its original state on the removal of load.
Types

8. Viscoelasticity is studied using dynamic mechanical analysis,
applying a small oscillatory stress and measuring the resulting
strain.
Purely elastic materials have stress and strain in phase, so that the
response of one caused by the other is immediate.
In purely viscous materials, strain lags stress by a 90 degree phase
lag.
Viscoelastic materials exhibit behavior somewhere in the middle of
these two types of material, exhibiting some lag in strain.
Dynamic modulus

9. Constitutive models of linear viscoelasticity
Constitutive
models of
linear
viscoelasticity
Maxwell
model
Standard linear
solid model
Kelvin-
Voigt model

10.  The model can be represented by the following equation:
 The Maxwell model can be represented by a purely viscous damper and a purely
elastic spring connected in series, as shown in the diagram.
Maxwell model

11.  The Kelvin–Voigt model, also known as the Voigt model, consists of a Newtonian
damper and Hookean elastic spring connected in parallel, as shown in the picture. It
is used to explain the creep behaviour of polymers.
 The constitutive relation is expressed as a linear first-order differential equation:
Kelvin–Voigt model

12.  For this model, the governing constitutive relation is:
 The standard linear solid model effectively combines the Maxwell model and a
Hookean spring in parallel. A viscous material is modeled as a spring and a dashpot in
series with each other, both of which are in parallel with a lone spring.
Standard linear solid model

13.  The Generalized Maxwell model also known as the Maxwell–Wiechert model is the
most general form of the linear model for viscoelasticity. It takes into account that the
relaxation does not occur at a single time, but at a distribution of times.
 The figure on the right shows the generalised Wiechert model.
 Applications: metals and alloys at temperatures lower than one quarter of their
absolute melting temperature
Generalized Maxwell model

14.  H. T. BANKS, J. B. HOOD, N. G. MEDHIN AND J. R. SAMUELS, A stick-slip/Rouse hybrid
model for viscoelasticity in polymers, Technical Report CRSC-TR06-26, NCSU, November,
2006, Nonlinear. Anal. Real., 9 (2008), pp. 2128–2149.
 Transport Phenomena 2nd Edition by R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot 2002.
 Carisey, A.; Tsang, R.; Greiner, A. M.; Nijenhuis, N.; Heath, N.;Nazgiewicz, A.; Kemkemer, R.; Derby,
Spatz, J.; Ballestrem, C.Vinculin regulates the recruitment and release of core focal adhesion
 H. T. BANKS, N. G. MEDHIN AND G. A. PINTER, Modeling of viscoelastic shear: a nonlinear
stick-slip formulation, CRSC-TR06-07, February, 2006, Dyn. Sys. Appl., 17 (2008), pp.383–406.
 H. T. BANKS, G. A. PINTER, L. K. POTTER, M. J. GAITENS AND L. C. YANYO, Modeling of quasistatic
dynamic load responses of filled viscoelastic materials, CRSC-TR98-48, NCSU,December, 1998;
11 in Mathematical Modeling: Case Studies from Industry (E .Cumberbatch and A. Fitt, eds.),
University Press, 2001, pp. 229–252.
REFERENCES