In strainsine waves, the complex modulus can be

In order to investigate the viscoelasticity behavior of each material an ad hocmeasuring technique called Dynamical mechanical analysis (DMA) is used.The DMA applies an oscillatory force, which causes a sinusoidal stress to thesample, generating a sinusoidal strain. From the amplitude of the deformationat the peak of the sine wave and the time delay between the stress and strainsine waves, the complex modulus can be calculated.(insert picture from Dynamic mechanical analysis : a practical introduc-tion)Complex modulusThe complex modulus is calculated from the ratio of the sinusoidal stress tostrain: E(asterisk)=sigma(t)/epsilon(t).  as a function of time.  The strainwhich is applied on the material under investigation during the experimentis kept at low levels to maintain the material’s response to stresses linearviscoelastic. Therefore the stress and strain change linearly in relation to eachother and the complex modulus can characterize the material’s deformationbehaviour.It is assumed that the temperature is constant and thus it does not appearas a variable. There is no well-established theoretical model on what is theexact influence level of temperature on material’s viscoelasticity.  Howeverthere is an empirical model which relates the time and temperature effecton viscoelasticity and it’s called the time – temperature superposition (TTS)principle and it will be discussed later in detail.The complex modulus is comprised of real and imaginary components:E(asterisk)=Eâ2+iEâ2â29 10CHAPTER 3.  DYNAMICAL MECHANICAL ANALYSISEâ2isknownasthestoragemodulusanditmeasurestheenergythatisstoredintothematerialandrepresentstheelasticsegmentofit;Eâ3isknownasthelossmodulusanditmeasurestheenergytransformedtoheatanditrepresentstheviscoussegmentofthematerial.If the solid is fully elastic the generated strain and stress waves will beabsolutely in phase.  In contrast, for a purely viscous fluid, the generatedstrain wave will have a pi/2 phase delay in relation to the stress one.For better understanding of the physical meaning of the storage modulus,it is described as the level of the stiffness of the material. Whereas the lossmodulus shows the level of the material’s damping capacity.(insert picture from Dynamic mechanical analysis : a practical introduc-tion)Additionally, another property which is well-used for DMA analysis is thetangent of the phase angle (tan delta) and it is the ratio of the loss modulus tothe storage modulus and therefore is independent of the part’s geometry . It isdefined as Tan Î ? = Eâ3/Eâ2Thetangentdisalsoanindexofhowmuchthemateriallosesenergytomolecularmovementsandinternalfrictionanditisknownasthemateriallossfactororlosstangent.Tan Î ? ranges from zero for an ideal elastic substance to infinity for anideal viscous one.If the type of deformation is shear, the shear storage modulus, ( G â2),shearlossmodulus(Gâ3),andtanÎ?=Gâ3/Gâ2areused.Analogicallytheirdefinitionsaredefined,usingshearstressandshearstraininsteadofthetensileonesTime – temperature superpositionIn order to describe the viscoelastic behaviour of a linear polymer for a broadrange of frequencies TTS is used. The latter is a tool with which it is possibleto shift values acquired from various temperatures to a chosen reference one.The three variables, frequency (Ï), time (t) and temperature (T), havea closely interrelated effect on polymers.  By decreasing the time length ofan experiment is equivalent to decreasing the temperature, so decreasing thefrequency of oscillation imposed on the material is equivalent to increasing theexperiment’s time length or temperature.To apply the the TTS technique temperature – dependent shift factorsare used on the measured stress, time or frequency values. The vertical shiftfactor is multiplied with a stress measured at a temperature T to calculate astress that is the equivalent value at the reference temperature. Likewise, thehorizontal shift factor multiplies a frequency or divides a time to calculate areduced value equal to the equivalent value at a specific temperature.Thus, a curve can be constructed for a broad frequency range using temper-atures values over smaller frequency range or time and by moving the curveson the log time axis so that they fit each other and align as a continuouscurve. Hence TTS, is based on the observation that the curves representingthe viscoelastic properties of a single material, determined at several differenttemperatures, are similar in shape when plotted against log t or log Ï .The single curve formed by superposition of the various curves shifted onthe time axis to the given reference temperature is referred to as a â mastercurve”. The latter shows viscoelastic behavior over a much larger range oftimes or frequencies than could be studied using a single rheometer at onetemperature. 3.1.  DYNAMICAL MECHANICAL ANALYSIS11Moreover, master curve clearly displays the five different characteristicviscoelastic regions or physical states with respect to the time or frequencyaxis. (write them)Transitions shift to higher temperatures at higher frequencies. The shift tohigher temperatures is a direct consequence of time â temperature equivalence.As the frequency is increased, only shorter timescale motions are possible; thusthe polymer responds more as if it were at a lower temperature than a samplerun at a lower frequency but the same temperature.  Consequently, highertemperatures are required for a sample to achieve an equivalent mechanicalstate at higher frequencies, and the transitions shift to higher temperatures.While time â temperature superposition is very useful, it will not workin all cases. Predictions based on TTS conform well to the observed beha-vior of many polymers, but others exhibit behavior inconsistent with TTS. Anumber of assumptions inherent in the principle of time â temperature equi-valence are incorrect for many polymers. For example, implicit in TTS is theassumption that the effect of temperature on the relaxation time spectrum,is consistent for the entire spectrum, but this is frequently in error. First,we have noted that relaxations in different temperature regions have differentactivation energies, so they will not superpose by a single set of a T values.Also, the polymer must be homogeneous. An example in which this clearlydoes not hold is the case of multiphase blends or block copolymers. Thesematerials will have multiple relaxations that are unrelated, and therefore maybe expected to have differing shift factors for any given reference temperature,even when each phase in isolation conforms to the WLF equation. In addition,TTS assumes that the chemical structure and morphology of the polymer isconstant.  Any polymer that undergoes a chemical change with time, suchas oxidation, UV degradation, additional polymerization, or crosslinking, willbe expected to undergo a change in properties in a manner inconsistent withTTS. Similarly, if a polymer undergoes a change in morphology as a resultof crystallization, partial melting, or crystal refi nement during annealing, itsrelaxation spectrum will be altered and it will not follow TTS.Thus, caution is warranted when using TTS to predict properties for timesand temperatures outside the range over which data were taken and for sys-tems that are nonhomogeneous.However, as a general rule, good alignment (i.e., a smooth fi t of thecurves), when shifting the data taken at different temperatures, is a goodindicator that the assumptions inherent in TTS are valid for the system ofinterest.The procedure can be applied in a fast and easy way for most of thestandard polymer melts or âthermo-rheologicallyâ simple materials. Polymerswith a high level of crystalline structures and high molar mass such as HDPEdo not show significant temperature dependenc