8 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 fluidViscoelastic fluidElastic solid
26 Stress Relaxation Test StrainElasticStressStressViscoelasticStressViscous fluidViscous fluidViscous fluidTime, t
27 Stress Relaxation Experiment Strain is applied to sample instantaneously(in principle) and held constant with time.Stress is monitored as a function of time (t).Straintime
28 Stress Relaxation Experiment Response of Classical ExtremesElasticViscousHookean SolidNewtonian FluidStressStressstress for t>0is 0stress for t>0is constanttimetimeStress decreases with timestarting at some high valueand decreasing to zero.Response of MaterialViscoelasticStresstime
29 Creep Recovery Experiment Stress is applied to sample instantaneously, t1, 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, t2, and the strain is monitored as a function of timetortStresst1t2time
30 Creep Recovery Experiment DeformationStresst1timet2Response of Classical ExtremesElasticViscousStain rate for t>t1 is constantStrain for t>t1 increase with timeStrain rate for t >t2 is 0Stain for t>t1 is constantStrain for t >t2 is 0StrainStraint1timet2t1timet2
31 Creep Recovery Experiment: Response of Viscoelastic Material timet12RecoverableStrainRecovery = 0 (after steady state)/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.Reference: Mark, J., et.al.,Physical Properties of Polymers,American Chemical Society, 1984, p. 102.
33 Rheological ModelsMechanical components or elements
34 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.
35 Elasticity deals with mechanical properties of elastic solids (Hooke’s Law) Stress, LStrain, = L/LLE=/
38 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:
39 Theory of Hydrodynamics Newton’s LawIn Newtonian Fluids, Stress is proportional to rate of strain but independent of strain itself
41 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).o (strain) (stress)to=0to=0timetimeCreep Experiment (a constant stress is instantaneously applied to the material and the resulting strain is followed as a function of time)t (stress) (strain)totststo=0timeto=0time
42 Energy Storage/Dissipation Elastic materials store energy (capacitance)Viscous materials dissipate energy (resistance)EnergytEnergytEViscoelastic materials store anddissipate a part of the energyt
43 What causes viscoelastic behavior? Energy Storage +DissipationFood Chemists – More on nature of these polymersReference: Dynamics of Polymeric Liquids (1977). Bird, Armstrong and Hassager. John Wiley and Sons. pp: 63.Long polymer chains at the molecular scale, make polymeric matrix viscoelastic at the microscale
44 Specifically, viscoelasticity is a molecular rearrangement 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.
45 Examples of viscoelastic foods: Almost all solid foods and fluid foods containing long chain biopolymersFood starch, gums, gelsGrainsMost solid foods (fruits, vegetables, tubers)CheesePasta, cookies, breakfast cereals
46 Viscoelasticity Experiments Static TestsStress Relaxation testCreep testDynamic TestsControlled strainControlled stress(When we apply a small oscillatory strain and measure the resulting stress)
47 Why we want to fit models to viscoelastic test data? To quantify the data – mathematical representationFor use with other food processing applicationsSome food drying models require viscoelastic propertiesDesign of pipelines, mixing vessels etc., using viscoelastic fluid foodsTo obtain information at different test conditionsExample: ExtrusionTo obtain an estimate of elastic properties and relaxation timesHelps to quantify glass transition
48 Viscoelastic Models Maxwell Model Kelvin-Voigt Model Used for stress relaxation testsUsed for creep tests
49 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:
50 Viscoelastic Response Strain,Stress,Let’s try to deform the Maxwell element
51 Maxwell Model Response The Maxwell model can describe successfully the phenomena of elastic strain, creep recovery, permanent set and stress relaxation observed with real materialsMoreover 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 behaviourTherefore the Maxwell element represents a VISCOELASTIC FLUID.
52 Maxwell Model-when is applied 1. will be same in each element2. Total = sum of individual
53 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:.Or by defining the “creep compliance”:Elastic Recovery (stress)o/EPermanentSetdashpotoo/Espringto=0tsto=0tstimetime
54 Maxwell Model Response 2) Stress Relaxation Experiment: If the mechanical model is suddenly extended to a position and held there (o=const., =0):.Exponential decayAlso recall the definition of the “relaxation” modulus:and (stress)timeto=0o=Gooo(strain)to=0time = /E = Relaxation time = the time required by biopolymers to relax the stresses
55 Generalized Maxwell Model The Maxwell model is qualitatively reasonable, but does not fit real data very well.Instead, we can use the generalized Maxwell model1 2 3 nE1E2E3En
56 Generalized Maxwell Model Applied for stress relaxation test
57 Determination of parameters for Generalized Maxwell Model There are 4 methods.Method of Instantaneous SlopeMethod of Central Limit TheoremPoint of Inflection MethodMethod of Successive Residuals direct method and more popularOptional
58 Method of Successive Residuals First plot-semilog plot: if it is linear, use single Maxwell ModelIf it is not linear, use Generalized Maxwell Model
59 Plot until it is straight timeto=0ln First plotSlope of straight line = -1/1ln 1Divided into many parts and plot of each part until the curvature disappears.ln Second plotSlope of straight line = 1/2ln 2to=0timePlot until it is straight
60 Example: Genealized maxwell model for stress relaxation test t (min)F (kg)1000.574166.51.5612572.554.53533.551.54514.550549648.5747.5847946104511441243Test sample has 2 cm diameter and 4 cm longArea = X 10-4 m2
61 first plotln 1==y-intercept1=slope=1=second plotln 2==y-intercept2=slope=2=
62 first plotsecond plott(s)stress Pa306090120150180210240270300360420480540600660720
63 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 behaviourTherefore 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 responseVarious other spring/dashpot combinations have been proposed.
64 Viscoelastic Reponse Voigt-Kelvin Element The Voigt-Kelvin element consists of a spring and a dashpot connected in parallel.E
65 Creep Recovery Experiment: applied 0 (step loading) (strain)(strain)+to=0to=0timetimetimetSlope=/Strain (t)0/E = /E = characteristic time = time of retardation