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For usual damping materials, the complex modulus depends not only of frequency but also of other environmental factors such as temperature, prestress, etc. The following sections discuss these factors and the representation in constitutive laws.
Temperature is the environmental factor that has the most influence on viscoelastic material characteristics []. At various temperatures, these materials typically have four different regions shown in figure ??: glassy, transition, rubberlike and fluid. Depending on the considered material, the operating temperature can be in any of the four regions. For polymer blends, each polymer can be in a different region.
In the first region, associated with low temperatures, the material in its glassy state is characterized by a storage modulus that reaches its maximum value and has low variations with temperature. The loss factor is very small and diminishes with temperature. Material deformations are then small.
The transition region is characterized by a modulus decreasing with temperature and a loss factor peaking in the middle of the region. Typically the maximum corresponds to the point of maximum slope for the storage modulus. The associated temperature is called transition temperature. Note that this can be confusing since the transition temperature depends on the frequency used to generate figure ??.
In the rubberlike region storage modulus and loss factor are both characterized by relatively small values and low temperature dependence. The fourth region corresponds to a fluid state. This region is rarely considered because of its inherent instability.
For damping applications, one typically uses viscoelastic materials in the transition region. This choice is motivated by the fact that the loss factors presents a maximum in this area, thus allowing an efficient use of the material damping properties.
In the frequency / temperature domain, figure ?? illustrates the existing inverse relation of the effects of temperature and frequency. Experimentally, one often finds that by shifting isotherm curves along the frequency axis by a given factor αT, one often has good superposition.
This property motivates the introduction of the reduced frequency α(T)ω and a description of the complex modulus under the form
| (1.18) |
The validity of this representation is called the frequency temperature superposition principle [, ] and the curve Ê is called a master curve.
Various authors have given thermodynamic justifications to the frequency temperature superposition principle. These are limited to unique polymers. For polymer blends, which have significant advantages, it is not justified.
The superposition principle is used to build a standard representation called nomogram which simplifies the analysis of properties as a function of temperature T and frequency ω. The product ωαT corresponds to an addition on a logarithmic scale. One thus defines true frequencies on the right vertical axis, and isotherm lines allowing to read the reduced frequency graphically on the horizontal axis. For a frequency ωj and a temperature Tk, one reads the nomogram in three step shown in figure ??
Various parametric expression have been proposed to model the temperature shift factor αT. The empirical equation of Williams-Landel-Ferry [], called WLF equation,
| (1.19) |
is often used. Various papers state that C1 = 17.4 et C2 = 51.6 are realistic values for many materials with changing T0 but this claims seems mostly unfounded.
On should also cite the αT model based on Arrhenius equation used in thermodynamics to quantify the relation between the rate of a chemical reaction and it's temperature
| (1.20) |
where T is the temperature in degree Kelvin, R=8.314× 10−3 kJ mol−1K−1 is constant of perfect gas and Ea corresponds to the activation energy of the reaction. This relation is less used than the WLF equation or other models uniquely based on Δ T, but the reason is probably only linked to easiness in the determination of parameters.
Figure ?? shows typical αT curves and their expression as a function of Δ T (difference between T and a given reference temperature T0). One clearly sees that these curves mostly differ in their low temperature behavior. In practical applications, one can usually adjust parameters of any law to be appropriate. The solution preferred here is to simply interpolate between points of an αT table.
Between the other environmental factors influencing the behavior, one essentially distinguishes non linear effects (static and dynamic) and history effects (exposition to oil, high temperatures, vacuum, ...).
Non linear dynamic effects are very hard to characterize since high amplitude variations of the induced strain are typically correlated with significant energy dissipation and thus temperature changes. The effects of level and temperature changes are then coupled for materials of interest which are typically in the transition region. Experimentally, such non linear studies are thus limited to the rubber like region. The effects are similar to those of temperature although of smaller magnitude.
Non linear static effects, that is effects linked to a static prestress assumed to be constant, are significant and easier to characterize. It is known to be essential when considering machinery suspensions or constrained viscoelastic sandwiches where press-forming induced significant pre-stress [].
History effects are generally associated with extreme solicitation that one seeks to avoid but whose probability of occurrence is non zero.
As for temperature, one generally represent the effect of other environmental factors as shift factors [], although the superposition hypothesis may not be as well verified [].
v_mat_test.tex
