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## Lesson 8. VISCOELASTIC MODELS

*Module 1. Rheology of foods*

**Lesson 8**

**VISCOELASTIC MODELS **

** 8.1 Introduction **

The fluids can be classified into following categories depending on the response to the applied shear force. Visco-elastic models are developed for Newtonian and non-newtonian fluids by different scientist considering different elements like dashpot and spring in series, in parallel and in combination. Widely used rheological models are Kelvin model, Maxwell Model and Burgers Models, which are described here.

Newtonian fluids are fluids which exhibit a linear increase in the shear stress with the rate of shear. These fluids exhibit a linear relationship between the shear stress and the rate of shear.

The equation for characterizing Newtonian fluid is

*Τ* = µ (-dv/dx) ----- (Eq-1)

Where, *Τ = *shear stress, µ = dynamic viscosity (η = µ/ρ) , -dv/dx = velocity gradient

A non-Newtonian fluid is broadly defined as one for which the relationship between shear stress and shear rate is not a constant. When the shear rate is varied, the shear stress doesn't vary in the same proportion. These fluids exibit either shear thinning or shear thickening behaviour and some exhibit a yield stress.

The two most commonly used equations for characterizing non-Newtonian fluids are the power law model (Eq-2) and Herschel-Bulkley model (Eq-3) for fluids.

*Τ* = K ( γ )^{n} -------(Eq-2)

* Τ * = * Τ ** _{ 0 }* + K ( γ )

^{n}-------(Eq-2)

Where, *Τ =*shear stress, K = consistency constant, γ = shear rate, n = flow behaviour index,

* Τ _{0 = } * yield stress

** 8.2 Rheological Models **

Several models have been developed to describe the viscoelastic behaviour of materials. There are two basic viscoelastic models viz Kelvin and Maxwell. Other complex viscoelastic behaviors are described by using combinations of these basic models (Fig. 8.1 & Fig. 8.2).

** (i) Kelvin model **

The Kelvin model employs the spring (elastic component) and dashpot (viscous component) in parallel. In this stress is the sum of two components of which one is proportional to the strain and the other is proportional to the rate of shear. Since the elements are in parallel they are forced to move together at constant rate. When a constant load is applied to Kelvin model, initially a retarded deformation is obtained followed by a final steady state deformation. When the load is removed the Kelvin model recovers completely but not instantaneously. The model is expressed mathematically as:

ε_{t} = σ_{0} / E [ 1 - e ^{(-t/ Tret)}] …………..(Eq. 5)

where ε_{t} is strain at time t, σ_{0} is applied stress, E is elastic modulus and T_{ret} is retardation time.

** (ii) Maxwell model **

The Maxwell model employs a spring and dashpot in series. In this model the deformation is composed of two parts, one purely viscous and the other purely elastic, When a constant load is applied to Maxwell body, instantaneous elastic deformation will take place followed by continuing viscous flow, which will continue indefinitely as it is not limited by the spring component. When load is removed, the Maxwell body recover instantly but completely. The Maxwell body shows stress relaxation but Kelvin body does not. stress-strain-time relationship in Maxwell model can be given as:

ε _{ t } = σ _{ 0 } / E_{d} [ 1 - e ^{(-t/ Tret)}] + E_{0}…………..(Eq. 6)

where, σ_{t} is stress at time t, σ_{o} is fixed strain, E_{d} is elastic decay modulus and T_{red} is relaxation time and E_{0} is equilibrium modulus.

** (iii) Burgers model **

This 4-element model is one of the best known rheological model which has been used to predict the creep behaviour in a number of materials. The model is composed of spring and dashpot in series with another spring and dashpot in parallel. When a burger's body is subjected to constant load, there is instantaneous deformation (E_{0}) is followed by retarded flow. When the load is removed there is instantaneous recovery followed by incomplete and slow recovery. The stress-strain time relationship can be given as:

ε_{t} = σ_{0} / E_{o} + σ_{0} / E_{t}( 1 - e^{(-t/ Tret)} ) + σ_{0 }t / nv…………..(Eq. 7)

In terms of compliance function J_{t} which is reciprocal of Young’s modulus (E) the above equation can be given as:

J_{t} = J_{0} + J_{t} ( 1 - e^{(-t/ Tret)} ) + t / nv ………………………… (Eq. 8)

Where, J_{0} is (l/E_{0}) initial compliance, J_{t} is (l/E_{t}) retarded compliance and t / nv is Newtonian compliance.

** (iv) Generalized Maxwell model **

A generalised Maxwell model is composed of n Maxwell elements with a spring in parallel with nth element. The elastic modulus E_{0} of last spring corresponds to the equilibrium modulus in the stress relaxation test. The stress-strain time relationship is given by :

ε_{t} = σ_{0} ( E_{d1} + e^{(-t/T1)} + E_{d2} e^{(-t/T2)} + …………………..+ E_{dn} e^{(-t/Tn)} + E_{o})…………..(Eq. 9)

where, T_{1}, T_{2}……………………. T_{n} are relaxation times.

** (v) Generalised Kelvin model **

Experimental data on may viscoelastic materials including biological materials have shown more than one relaxation time or retardation time. For these materials, complete behaviour cannot be represented by a singly Maxwell or single Kelvin model or elements model. Each or these models have only one time constant. To represent viscoelastic behaviour more realistically a chain of Kelvin models, each with its own time retardation is assumed and the model is called a generalized Kelvin model. It consists of Kelvin elements connected in series with an initial spring and final ‘viscous element. The equation for generalised kelvin model is:

ε_{t} = { 1 / E_{0} + 1/Et_{1} ( 1 - e ^{-t/T1}) + 1 / E_{t2} (1-e ^{-t/T2}) + ------+ 1 / E _{tn} ( 1- e ^{-t / Tn}) + t / nv ---------------------(Eq. 10)

where, T1, T2……………….. T_{n} are relaxation times.

** (vi) Plasto - visco - elastic or Bingham model **

A more common type of body is the plasto-visco-elastic or Bingham body. When the stress is applied which is below the yield stress the Bingham body reacts as an elastic body. At stress values beyond the yield stress there are two components. One is constant and is represented by the friction element and the other is proportional to the shear rate and represents the viscous flow element. In a creep ,experiment with stress not exceeding yield value, the creep curve would be similar to the one for a elastic body. When the shear tress is greater than the yield stress, the strain increases with time similar to the behaviour of a Maxwell body. Upon removal of stress at time the strain decreases instaneously and remains constant thereafter. The decrease represents the elastic components and the plastic deformation is permanent.

** (vii) Psychorheological Models **

Psycho rheological models consist of a mathematical expression relating sensory rheological data to the corresponding mechanical data. These two sets of data are usually considered as output and input respectively. Associations between subjective and objective texture measurement may be expressed by grap1iieal or mathematic / statistical terms. Various con-elation coefficients quantify the relation between variables.

Using regression analysis, one can ascertain the relation it seIf beyond developing a measure of relatedness of two variables with the assumption of unilateral casuality. Regression analysis helps the experimentor to : (a) select a variables, and (b) to estimate the parameters of that equation by statistical analysis.

** 8.5 Viscoelastic Charaterization of Materials **

There are a number of tests which may be used to study viscoelastic materials and determine the relation among stress-strain-time for a given type of deformation and a given type of loading pattern. The most important tests include stress relaxation, creep and dynamic tests.

** (i) Creep measurement (Fig. 8.1) **

In this test the stress is suddenly applied and held constant, and strain ( 'γ' or 'ε') is measured as a function of time. For a viscoelastic material the slope (dγ/ dt = γ) gives (from σ/ γ) an apparent Viscoelasticity. The deformation γ_{0} is a measure of the elastic part. From the instantaneous shear modulus G_{0} may be calculated (σ/y_{0}) or the instantaneous compliance J_{0}_{o} / σ). The whole curve gives J_{t}, which, in principle, can be calculated to yield G_{t}. The rheological model to represent the creep behaviour is the Kelvin model and 4 elements Burgers model. Creep measurement are very useful for studying stand up properties of foods. (γ

** (ii) Stress relaxation (Fig. 8.2) **

_{t}in tension or compression, G

_{t}in shear or K

_{t}in bulk compression. The rheological models representing stress relaxation are Maxwell model and generalised Maxwell model. One of the most important viscoelastic parameters which can be obtained from stress relaxation test is the relaxation time. It is the time at which the stress in the body resembling a Maxwell model decay to 1/e of initial stress. It is the measure of the rate at which a material dissipates stress after receiving a sudden force. There are a number of methods for treating experimental data on stress relaxation and finding the relaxation time.

** (iii) Dynamic Measurement **