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Viscoelastic Characterization Dr. Muanmai Apintanapong

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Elastic deformation - Flow behavior

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Elastic behavior

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Newtonian behavior

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Newtonian liquid

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Introduction to Viscoelasticity All viscous liquids deform continuously under the influence of an applied stress – They exhibit viscous behavior. Solids deform under an applied stress, but soon reach a position of equilibrium, in which further deformation ceases. If the stress is removed they recover their original shape – They exhibit elastic behavior. Viscoelastic fluids can exhibit both viscosity and elasticity, depending on the conditions. Viscous fluid Viscoelastic fluid Elastic solid

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Shear Stress

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Shear Rate

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Practical shear rate values

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Viscosity =resistance to flow

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Viscosity of fluids at 20 C Go to stress relaxation

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Viscosity: temperature dependence

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Flow curve and Viscosity curve

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Flow behavior: flow curve

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Flow behavior: viscosity curve

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Stress Relaxation Universal Testing Machine Instron, TA XT2 Force sensor

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Stress Relaxation Test Time, t Strain Stress Elastic Viscoelastic Viscous fluid 0 Stress Viscous fluid

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Ü Strain is applied to sample instantaneously (in principle) and held constant with time. Stress is monitored as a function of time (t). Stress Relaxation Experiment Strain 0 time

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Stress Relaxation Experiment Stress decreases with time starting at some high value and decreasing to zero. Response of Material Stress time 0 Response of Classical Extremes time 0 stress for t>0 is constant time 0 stress for t>0 is 0 Stress Hookean Solid Newtonian Fluid

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Stress is applied to sample instantaneously, t 1, and held constant for a specific period of time. The strain is monitored as a function of time ( (t) or (t)). The stress is reduced to zero, t 2, and the strain is monitored as a function of time t or t Creep Recovery Experiment Stress time t1t1 t2t2

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Creep Recovery Experiment Response of Classical Extremes – Stain for t>t1 is constant – Strain for t >t2 is 0 time Strain time Strain time – Stain rate for t>t1 is constant – Strain for t>t1 increase with time – Strain rate for t >t2 is 0 t2t2 Stress t1t1 t1t1 t2t2 t2t2 t1t1

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Reference: Mark, J., et.al.,Physical Properties of Polymers,American Chemical Society, 1984, p Creep Recovery Experiment: Response of Viscoelastic Material Creep > 0 time t 1 t 2 Recoverable Strain Recovery = 0 (after steady state) Strain Strain rate decreases with time in the creep zone, until finally reaching a steady state. In the recovery zone, the viscoelastic fluid recoils, eventually reaching a equilibrium at some small total strain relative to the strain at unloading.

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time Recovery ZoneCreep Zone Less Elastic More Elastic Creep 0 Recovery = 0 (after steady state) // Strain t1t1 t2t2 Creep Recovery Experiment

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Rheological Models Mechanical components or elements

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Elastic (Solid-like) Response A material is perfectly elastic, if the equilibrium shape is attained instantaneously when a stress is applied. Upon imposing a step input in strain, the stresses do not relax. The simplest elastic solid model is the Hookean model, which we can represent by the “spring” mechanical analog.

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Elasticity deals with mechanical properties of elastic solids (Hooke’s Law) Stress, LL Strain, = L/L L E= /

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Strain, Stress,

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Elastic (Solid-like) Response Stress Relaxation experiment strain) time stress) time t o =0 Creep Experiment stress) time t o =0 strain) time t o =0 tsts tsts oo o /E oo

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Viscous (Liquid-like) Response A material is purely viscous (or inelastic) if following any flow or deformation history, the stresses in the material become instantaneously zero, as soon as the flow is stopped; or the deformation rate becomes instantaneously zero when the stresses are set equal to zero. Upon imposing a step input in strain, the stresses relax as soon as the strain is constant. The liquid behavior can be simply represented by the Newtonian model. We can represent the Newtonian behavior by using a “dashpot” mechanical analog:

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Theory of Hydrodynamics In Newtonian Fluids, Stress is proportional to rate of strain but independent of strain itself Newton’s Law

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Strain, , Stress,

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Viscous (Liquid-like) Response Stress Relaxation experiment (suddenly applying a strain to the sample and following the stress as a function of time as the strain is held constant). strain) time stress) time t o =0 Creep Experiment (a constant stress is instantaneously applied to the material and the resulting strain is followed as a function of time) stress) time t o =0 strain) time t o =0 tsts tsts oo oo

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Energy Storage/Dissipation Elastic materials store energy (capacitance) Viscous materials dissipate energy (resistance) t Energy t E Viscoelastic materials store and Viscoelastic materials store and dissipate a part of the energy dissipate a part of the energy t

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What causes viscoelastic behavior? Long polymer chains at the molecular scale, make polymeric matrix viscoelastic at the microscale Reference: Dynamics of Polymeric Liquids (1977). Bird, Armstrong and Hassager. John Wiley and Sons. pp: 63. Energy Storage +Dissipation

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Specifically, viscoelasticity is a molecular rearrangement. When a stress is applied to a viscoelastic material such as a polymer, parts of the long polymer chain change position. This movement or rearrangement is called Creep. Polymers remain a solid material even when these parts of their chains are rearranging in order to accompany the stress, and as this occurs, it creates a back stress in the material. When the back stress is the same magnitude as the applied stress, the material no longer creeps. When the original stress is taken away, the accumulated back stresses will cause the polymer to return to its original form. The material creeps, which gives the prefix visco-, and the material fully recovers, which gives the suffix –elasticity.

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Examples of viscoelastic foods: Food starch, gums, gels Grains Most solid foods (fruits, vegetables, tubers) Cheese Pasta, cookies, breakfast cereals Almost all solid foods and fluid foods containing long chain biopolymers

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Viscoelasticity Experiments Static Tests –Stress Relaxation test –Creep test Dynamic Tests –Controlled strain –Controlled stress (When we apply a small oscillatory strain and measure the resulting stress)

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Why we want to fit models to viscoelastic test data? To quantify the data – mathematical representation For use with other food processing applications -Some food drying models require viscoelastic properties -Design of pipelines, mixing vessels etc., using viscoelastic fluid foods To obtain information at different test conditions –Example: Extrusion To obtain an estimate of elastic properties and relaxation times –Helps to quantify glass transition

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Viscoelastic Models Maxwell Model Kelvin-Voigt Model Used for stress relaxation tests Used for creep tests

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Viscoelastic Response – Maxwell Element A viscoelastic material (liquid or solid) will not respond instantaneously when stresses are applied, or the stresses will not respond instantaneously to any imposed deformation. Upon imposing a step input in strain the viscoelastic liquid or solid will show stress relaxation over a significant time. At least two components are needed, one to characterize elastic and the other viscous behavior. One such model is the Maxwell model:

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Viscoelastic Response Let’s try to deform the Maxwell element E Strain, Stress,

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Maxwell Model Response The Maxwell model can describe successfully the phenomena of elastic strain, creep recovery, permanent set and stress relaxation observed with real materials Moreover the model exhibits relaxation of stresses after a step strain deformation and continuous deformation as long as the stress is maintained. These are characteristics of liquid-like behaviour Therefore the Maxwell element represents a VISCOELASTIC FLUID.

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Maxwell Model-when is applied 1. will be same in each element 2. Total = sum of individual

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Maxwell Model Response 1) Creep Experiment: If a sudden stress is imposed (step loading), an instantaneous stretching of the spring will occur, followed by an extension of the dashpot. Deformation after removal of the stress is known as creep recovery:. stress) time t o =0 time t o =0 tsts tsts oo o /E Or by defining the “creep compliance”: Elastic Recovery Permanent Set o /E dashpot spring

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Maxwell Model Response 2) Stress Relaxation Experiment: If the mechanical model is suddenly extended to a position and held there ( =const., =0): Exponential decay strain) time t o =0. oo Also recall the definition of the “relaxation” modulus: and stress) time t o =0 o=Gooo=Goo = /E = Relaxation time = the time required by biopolymers to relax the stresses

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Generalized Maxwell Model The Maxwell model is qualitatively reasonable, but does not fit real data very well. Instead, we can use the generalized Maxwell model 11 2 2 3 3 n n E1E1 E2E2 E3E3 EnEn

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n Generalized Maxwell Model Applied for stress relaxation test

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Determination of parameters for Generalized Maxwell Model There are 4 methods. –Method of Instantaneous Slope –Method of Central Limit Theorem –Point of Inflection Method –Method of Successive Residuals direct method and more popular Optional

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Method of Successive Residuals First plot-semilog plot: if it is linear, use single Maxwell Model If it is not linear, use Generalized Maxwell Model

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Divided into many parts and plot of each part until the curvature disappears. time t o =0 ln Second plot Slope of straight line = 1/ 2 Plot until it is straight time t o =0 ln First plot Slope of straight line = -1/ 1 ln 1 ln 2

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Example: Genealized maxwell model for stress relaxation test Test sample has 2 cm diameter and 4 cm long Area = X m 2 t (min)F (kg)

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first plot ln 1 = =y-intercept 1=1= slope= 1=1= second plot ln 2 = =y-intercept 2=2= slope= 2=2=

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first plotsecond plot t(s)t(s)stress Pa

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Voigt-Kelvin Model Response The Voigt-Kelvin element does not continue to deform as long as stress is applied, rather it reaches an equilibrium deformation. It does not exhibit any permanent set. These resemble the response of cross- linked rubbers and are characteristics of solid-like behaviour Therefore the Voigt-Kelvin element represents a VISCOELASTIC SOLID. The Voigt-Kelvin element cannot describe stress relaxation. Both Maxwell and Voigt-Kelvin elements can provide only a qualitative description of the response Various other spring/dashpot combinations have been proposed.

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Viscoelastic Reponse Voigt-Kelvin Element The Voigt-Kelvin element consists of a spring and a dashpot connected in parallel. E

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Creep Recovery Experiment: applied 0 (step loading) strain) time t o =0 oo + strain) time t o =0 = /E = characteristic time = time of retardation time t 0 t Slope= Strain (t) 0 /E

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Generalized Voigt-Kelvin Model E1E1 E2E2 E3E3 EnEn

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Three element Model Standard linear solid

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Four element Model E1E1 E2E2 22 11 strain) time t o =0 oo strain) time t o =0 strain) time t o =0 spring Kevin dashpot

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CB A. Creep test: use 4-element model time t 0 Strain (t) 0 /E 2 = r 0 /E 1 = 0 a = 2 /E 2 = ret.. Dashpot, 1 Kelvin, 2 /E 2 Spring, E 1 Slope = 0 / 1

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Generalized four-element model Combination of four-element model in series

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Example Analyze the given experimental creep curve in terms of the parameters of a 4-element model. Cylindrical specimen (2 cm in diameter and 5 cm long) Applied step load is 10 kg.

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r a ret 0 =0.2 slope = = dashpot kelvin spring

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Length =0.05m Diameter =0.02m Area = m2m2 Load =10kg Pa slope =Deformation/time =0.0266cm/s slope = per sec per sec Pa s 0.125cm =0.025m/mm/m Pa a = 0.9 = Pa s 0 /E 1 = 0.2cm =0.04m/mm/m Pa

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