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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.