The Rheology Handbook. Thomas Mezger

= Pa-1], with the re-formation γr [%]. The following holds for the equilibrium (shear) compliance Je:

      Equation 6.10

      Je = limt → ∞ Jr(t) (= J∞)

      Je [1/Pa = Pa-1] is the limiting value of the Jr(t)-function after an “infinitely long” period of time, i. e. when t = ∞. The following holds: Je = J0

      In the equilibrium state, the recovery compliance shows the same value if determined on the one hand via J(t) in an “infinitely short” period of time as above (in terms of J0), or, on the other hand via Jr(t) after an “infinitely long” period (in terms of Je or J∞), i. e. at the very end of the test when finally reaching again steady-state behavior or equilibrium of forces, respectively. However, when using the function of J(t) or Jr(t), the same value for J0 and Je may only be achieved if indeed the limiting value of the LVE deformation range has never been exceeded.

      6.3.4.2.3c) Determination of the limiting value of the LVE range

      Evaluating creep curves, many users prefer the J(t)-function to the γ(t)-function.

      Since J(t) = γ(t) / τ0, all J(t)-curves essentially overlap with one another, independent of the preset stress τ0 as long as the limit of the LVE range is not exceeded. Since the elasticity law (τ / γ = const) applies to each measuring point, the preset stress results in the corresponding proportional deformation value. Therefore, the corresponding ratio value of J(t) is independent of the preset stress value when still measuring in the LVE range.

      It is easy to check whether the limit of the LVE range has been exceeded by presenting in the same diagram all of the individual J(t)-curves resulting from several creep tests which are performed by presetting a different constant τ0-value for each individual test. Those J(t)-curves, which are not overlapping with one another but deviate significantly upwards, have obviously been measured under conditions outside the LVE range (see curve b in Figure 6.9). In this case, the internal structure has been deformed already too much by the preset stress, which was too high in this case. As a consequence, the partially destroyed structure of the material is yielding more, i. e. it will be deformed to a greater extent as corresponding to the elasticity law. In other words: The compliance of the structure increases, and then, higher J-values are obtained.

      Therefore, tests which are nevertheless carried out under these conditions reveal information on non-linear behavior outside the LVE range. For the above reasons, J(t)-diagrams are well suited to check whether the LVE range has been exceeded or not. For detailed information on the LVE range, see Chapters 8.3.2 and 8.3.3 (oscillatory tests), and about non-linear behavior, see Chapter 8.3.6 (LAOS tests).

      Note: Master curve of J(t)-functions via time/temperature shift (WLF method)

      For all thermo-rheologically simple materials tested in the LVE deformation range, a temperature-invariant master curve of the J(t)-function can be determined from several individual J(t)-functions of which each one was measured at a different temperature. By the way, this is also possible for all other parameters of rheology, if data are measured within the LVE range. The corresponding master curve is generated using the WLF relation and the time/temperature shift (TTS) method (see Chapter 8.7.1). This method is useful to produce J(t)-values also in a time-frame for which no measuring data are available, e. g. in the short-term range.

       6.3.4.3Retardation time

      Ideal-elastic materials are showing as well immediate deformation after applying a step-like stress, as well as immediate re-formation after removing the stress afterwards in the step-like form again. For all VE samples, this elastic behavior occurs with a certain time delay. To evaluate this time-

      dependent deformation behavior, two parameters have been defined:

      the relaxation time λ [s] and the retardation time Λ [s], both are pronounced lambda.

      Typically, the term relaxation time is used in combination with controlled shear strain tests (or controlled shear deformation, CSD, resp.), and controlled strain rate tests (or controlled shear rate, CSR, resp.), e. g. when performing relaxation tests.

      Relaxation is a process in the state at rest after a forced deflection or strain and can be described as “delayed elasticity” in the sense of “delayed stress decrease” (see Chapter 7.3.3.2). On the other hand, the term retardation time is used when performing controlled force tests or controlled shear stress tests (CSS), such as creep tests. Retardation is a delayed response to an applied force or stress and can be described as “delayed elasticity” in the sense of “delayed re-formation”.

      6.3.4.3.1a) Retardation time Λ in the Kelvin/Voigt model

      Behavior of VE solids becomes clear when using the differential equation according to Kelvin/Voigt (see Chapter 5.2.2.1b): τ = η ⋅ γ ̇ + G ⋅ γ

      UsingΛ = η/Gorη = Λ ⋅ Gthen:

      Equation 6.11

      τ = (Λ ⋅ G) ⋅ γ ̇ + G ⋅ γorτ/G = Λ ⋅ γ ̇ + γ

      Here, the symbol Λ is taken for the retardation time. Some authors choose the symbol λK (or τκ) to show the correlation between the Kelvin/Voigt model – which is used to characterize the rheological behavior of VE solids – and this specific time [6.1]. The retardation time determines the time-dependent deformation and re-formation behavior of the parallel connected components spring and dashpot of the Kelvin/Voigt model for both intervals, as well for the stress phase as well as for the rest phase. The solution of the differential equation leads to the following time-­dependent exponential function:

      Equation 6.12

      γ = (τ / G) ⋅ [1 – exp(-t/Λ)]

      1 At the time point t = Λ, the following applies to the creep phase:

      γ(Λ) = (τ / G) ⋅ [1 – (1/e)] = 63.2 % ⋅ (τ / G) = 63.2 % ⋅ γmax

      Therefore counts for the creep phase: The retardation time Λ of the Kelvin/Voigt model is reached if the γ-value has increased to 63.2 % of the maximum deformation γmax which will finally occur at the end of the stress interval (see also Chapter 6.3.3a).

      1 At the time point t = Λ, the following applies to the creep recovery phase:

      γ(Λ) = γmax – (τ / G) ⋅ [1 – (1/e)] = γmax – 63.2 % ⋅ (τ / G) = (100 % – 63.2 %) ⋅ γmax

      and thus: γ(Λ) = 36.8 % ⋅ γmax

      Therefore counts for the creep recovery phase: The retardation time Λ of the Kelvin/Voigt model is reached if the γ-value has decreased to 36.8 % of γmax (see also Chapter 6.3.3b).

      6.3.4.3.2b) Retardation time Λ in the Burgers model

      Using the Burgers model, the calculation seems to be even more complex since here, the immediate re-formation of the first spring S1 in the recovery phase also has to be taken into account for the determination of the γ-values. However, since the spring recoils immediately, i. e. idealized

      “in zero-time”, there is no influence on the value of Λ. Therefore, it has no effect on the time-

      dependent behavior at all.

       6.3.4.4Retardation time spectrum

      When describing the behavior of real polymers, it is important to take into account that the molecules do not have a single molar mass only since they may have a more or less wide molar mass distribution (MMD). Therefore a retardation time spectrum should be given preference, since here, a single retardation time only is not sufficient for an appropriate analysis.