The Rheology Handbook. Thomas Mezger
model)
3 Continuously increasing deformation of dashpot D3 with a constant rate of deformation (change in deformation within a period of time, or shear rate)
After a sufficiently long test period, all springs and dashpots are deflected to a certain extent dependent on the test conditions, i. e. on the constant stress value and on the period of time of the stress applied.
1 Creep recovery phase
When removing the force F, the following re-formation behavior occurs:
1 Immediate, step-like elastic re-formation of spring S1
2 Delayed re-formation of spring S2 and dashpot D2 (as with the Kelvin/Voigt model)
3 Dashpot D3 remains completely deflected
If the sample completely returns to its initial position, then it is a VE material showing the character of a viscoelastic solid (see Figure 6.5, no. 2). In this case, the dashpot D3 is without function in the Burgers model, and can therefore be ignored for analysis.
Figure 6.8: Creep curve and creep recovery curve γ(t)
6.3.3Curve discussion
In order to explain the analysis in detail, the creep curve (from t0 to t2) and the creep recovery curve (from t2 to t4) are divided into the following sectors, see Figure 6.8:
1 Δγ1 = (γ1 – 0) = γ1 as step-like, purely elastic change in deformation occurring immediately after the beginning of the test, idealized: “without any time delay, in zero-time” (behavior of spring S1)
2 Δγ2 = (γ2 – γ1) as delayed viscoelastic change in deformation (spring S2 together with dashpot D2)
3 Δγ3 = (γ3 – γ2) as purely viscous change in deformation (dashpot D3); after reaching steady-state behavior, i. e. a constant rate of deformation (or shear rate), the γ-curve is showing a constant slope angle β then
4 γmax = Δγ1 +Δγ2 + Δγ3 = γ3 is the maximum deformation, at the end of the stress phase
5 Δγe = (γmax – γv) is the change in deformation as re-formation in the creep recovery phase (representing the elastic proportion of the viscoelastic behavior)
6 γv is the extent of remaining deformation after the creep recovery phase (viscous proportion)
6.1.2.1.1a) Creep curve
Various denominations are used for the first test interval: creep curve, deformation curve, load(ing) phase, or stress phase.
Transient flow behavior with a non-constant rate of deformation (shear rate) γ ̇ occurs between the time points t0 and t1, then: γ = f(τ0, t). Here, the slope value of the time-dependent deformation curve depends as well on the applied shear stress τ0 as well as on the passing time. Steady-state behavior with γ ̇ = Δγ / Δt = const is reached at time point t1 when γ = f(τ0). Here counts: γ ̇ = (γ3 – γ2) / (t2 – t1). From now on the γ-curve shows a constant slope, now depending on the applied stress τ0 only but no longer on the time. Steady-state creep is reached when the curve displays a constant slope angle β finally.
The creep function, describing the time-dependent deformation behavior during the stress phase, can be formulated as follows:
Equation 6.1
γ(t) = Δγ1 + Δγ2(t) + Δγ3(t) = (τ0 / G1) + (τ0 / G2) ⋅ [1 – exp(-t/Λ)] + (τ0 ⋅ t) / η0
with the shear modulus G1 [Pa] = τ0 / γ1 (corresponding to the spring constant of S1 and visible in the creep curve as an immediate deformation step due to the purely elastic behavior); the retardation time Λ [s] = η2 / G2 (pronounced: “lambda”, see Chapter 6.3.4.3) with the shear modulus G2 [Pa], (the spring constant of S2) and the shear viscosity η2 [Pas], (the dashpot constant of D2); and the zero-shear viscosity η0 [Pas], (the dashpot constant of D3). The medium term of the formula is obtained from the differential equation according to Kelvin/Voigt (see Chapter 5.2.2.1b):
τ = G2 ⋅ γ + η2 ⋅ γ ̇
The following applies for t = 0:
γ(0) = (τ0 / G1) + (τ0 / G2) ⋅ [1 – e0)] + (τ0 ⋅ 0) / η0 = (τ0 / G1) + (τ0 / G2) ⋅ (1 – 1) + 0
thus: γ(0) = (τ0 / G1)
Summary: At the very beginning of the test the only element which is deflected is the spring S1, which is deformed immediately, without any delay.
The following applies for t = ∞ (infinity), or for practical users, after a “very long” time:
γ(∞) = (τ0 / G1) + (τ0 / G2) ⋅ [1 – (1/e∞)] + C = (τ0 / G1) + (τ0 / G2) ⋅ (1 - 0) + C
thus: γ(∞) = (τ0 / G1) + (τ0 / G2) + C
Summary: S1 and S2 are fully deflected and therefore also D2. The deformation of D3 would be, strictly speaking, “infinitely” large (γ3 = ∞), this is formulated here in terms of a C for a correspondingly large value which is reached in the very end of the creep phase.
The following applies for t = Λ, i. e., when reaching the retardation time:
γ(Λ) = (τ0 / G1) + (τ0 / G2) ⋅ [1 – (1 / e)] + (τ0 ⋅ Λ) / η0
thus: γ(Λ) = (τ0 / G1) + 0.632 ⋅ (τ0 / G2) + (τ0 ⋅ Λ) / η0
Summary: S1 is fully deflected, and up to this time point, S2 and therefore also D2 are deflected partially to an extent of 63.2 %. D3 however, is deflected to a very small and not significant extent only, since the third term has merely reached a relatively low value up to this point.
6.1.2.1.2b) Creep recovery curve
Also for the second test interval various denominations are used: creep recovery curve, re-formation curve, retardation curve, or rest phase.
Transient flow behavior during the re-formation occurs between the time points t2 and t3. Finally, the deformation value reached is Δγe. The latter represents the elastic proportion of the viscoelastic behavior. For a test being sufficiently long, a VE liquid will show a certain, permanently remaining deformation value then (which is γv = const), and therefore steady-state deformation behavior . For a VE solid, steady-state (equilibrium state) is reached when the material displays complete re-formation after all.
The creep recovery function, describing the time-dependent re-formation behavior during the rest phase, can be formulated as follows:
Equation 6.2
γ(t) = γmax – Δγ1 – Δγ2(t) = γmax – (τ0 / G1) – (τ0 / G2) ⋅ [1 – exp(-t/Λ)]
The following applies for t = 0:
γ(0) = γmax – (τ0 / G1) – (τ0 / G2) ⋅ [1 – e0] = γmax – (τ0 / G1) – (τ0 / G2) ⋅ (1 – 1)
thus: γ(0) = γmax – (τ0 / G1)
i. e., immediately after releasing the load, only the spring S1 recoils without any delay.
The following applies for t = ∞ (infinity), or for practical users, after a “very long” time:
γ(∞) = γmax – (τ0 / G1) – (τ0 / G2) ⋅ [1 – (1/e∞)] = γmax – (τ0 / G1) – (τ0 / G2) ⋅ (1 – 0)
thus: γ(∞) = γmax – (τ0 / G1) – (τ0 / G2)
i. e., S1 and S2, and therefore also D2 are fully reset; D3 however, remains deflected. At the very end of the creep recovery phase it is merely the deformation value γv which still occurs.
The following applies