Читать книгу The Rheology Handbook - Thomas Mezger - Страница 246

The following components are combined in series to analyze creep and creep recovery behavior: spring S1, spring S2 and dashpot D2 (both in parallel); and dashpot D3 (see Figure 6.7). Indeed, this is a combination of the Maxwell model (S1 and D3) and the Kelvin/Voigt model (S2 and D2). This combined model is called the Burgers model, since in 1935 it was presented by J. M. Burgers (1895 to 1981) [6.2].

Figure 6.7: The Burgers model to simulate the behavior of viscoelastic materials when performing a creep and creep recovery test

In order to analyze deformation behavior on the basis of the Burgers model a differential equation of the second order is used containing the following parameters [6.3] [6.4] [6.5]:

 τ, shear stress

  τ ̇ , the first time derivative of shear stress τ, or time-dependent rate of change in shear stress (stress rate); meaning: How fast is the shear stress changing with time?

  τ ̇ , the second time derivative of shear stress τ, or time-dependent rate of change in stress rate τ ̇ ; meaning: How fast is the stress rate changing with time?

 γ, shear deformation (or shear strain)

  γ ̇ , the first time derivative of deformation γ (or strain), or time-dependent rate of change in deformation (or deformation rate, strain rate, shear rate, “shear velocity”); meaning: How fast is deformation changing with time?

  γ ̇ , the second time derivative of deformation γ, or time-dependent rate of change in deformation rate γ ̇ (change in shear rate, “acceleration in shearing”); meaning: How fast is the deformation rate (shear rate) changing with time?

Viscoelastic behavior, illustrated by use of the Burgers model:

1 Creep phase

When applying the force F, the following deformation behavior occurs:

1 Immediate, step-like deformation of spring S1 at the beginning of the test

2 Delayed deformation of spring S2 and dashpot D2 (as with the Kelvin/Voigt 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)