The Rheology Handbook. Thomas Mezger

(inverted commas, since in the stress phase there is no creep at all), and creep recovery curve (in the rest phase there is an immediate recovery step in deformation, back to zero)

       Figure 6.4: Creep curve and “recovery curve” of an ideal-viscous fluid (inverted commas, since in the rest phase there is no recovery of deformation at all)

       Figure 6.5: Creep and creep recovery curves of two different viscoelastic materials, in the rest phase both are showing delayed re-formation:

      (1) only partially for the VE liquid, and

      (2) finally completely for the VE solid

6.2.4Viscoelastic

      behavior

      For VE materials under a constant stress, the resulting deformation can be imagined to be composed of proportions: the first one occurs immediately and the second one delayed. After removing the stress, delayed re-formation takes place either partially or completely. The γ(t)-diagram shows for both creep curve and creep recovery curve, the shape of an increasing or decreasing exponential function, respectively (e-function; see Figure 6.5).

      The degree of re-formation depends on the elastic proportion of the VE behavior of the sample.

      For VE liquids (1) after releasing the stress and even after a longer rest phase, a certain extent of deformation is remaining permanently. This value represents the viscous proportion of the VE behavior of the sample. This behavior is exhibited by concentrated polymer solutions and polymer melts. Here, a load-and-removal cycle is an irreversible process, since the shape of the sample remains permanently changed after the experiment is finished.

      For VE solids (2), delayed but complete re-formation occurs if the period of testing is sufficiently long. This behavior is displayed by chemically crosslinked materials, gels and concentrated dispersions with a gel-like structure at rest. Here, a load-and removal cycle is a reversible process since the shape of the test material will be the same again finally when compared to the initial shape.

6.3Analysis

       6.3.1Behavior of the molecules

      Behavior of a polymer sample can be illustrated by observing the process taking place in a conglomeration of polymer molecules, when regarding a single macromolecule only (see Figure 6.6).

      6.1.2.1.1a) At rest, before the step in stress

      Still in a stress-less state, the chain of an individual polymer molecule occurs as a spherical molecule coil which may have many entanglements with neighboring chains. This is the state at rest requiring minimum expense of energy.

      6.1.2.1.2b) Under constant stress, in the creep phase

      The observed macromolecule reacts to the constant shear stress showing slow, creeping motion. Forced by the constant load, the spherical coil begins to leave its state of rest exhibiting increasing deformation. It changes more and more to the shape of an ellipsoid whose axis is oriented lengthways between the shear direction and the direction of the shear gradient. More illustrative: An ellipsoid is shaped “like an American football”. As a consequence, deformation is increasing now as well of the individual molecule, as well as of the superstructure of the whole polymer sample due to the entanglements between the molecule chains.

      6.1.2.1.3c) After releasing the stress, in the creep recovery phase

      Immediately after the step-like removal of the stress, each one of the molecules is trying to return to a rest position bare of any deformation. Continuously, the molecules are recovering more and more from the previously occurred deformation, reducing it continuously by slow and compensating, creep motions within the superstructure. This is a delayed process which is also called “retarded”. For uncrosslinked polymers, the extent of re-formation may tend to zero if the creep phase was long enough, i. e., if a correspondingly great number of disentanglements occurred in the stress interval. For samples showing a chemical or physical network, the extent of partial re-formation corresponds to the elastic proportion. Fully crosslinked polymers are re-forming completely as shown in Figure 6.5 (no. 2) if the LVE range was not exceeded. Finally they will achieve the same shape as they displayed initially, before the stress was applied (as shown in Figure 6.6, no. 1).

       Figure 6.6: Deformation process of a polymer molecule when performing a creep test (Figure from [6.1]); (1) at rest, before applying the stress (2) under stress after a certain period of time

      In rheology, a delayed deformation or re-formation process after applying or removing a stress, respectively, is referred to as retardation . As a comparison: The term “relaxation” is used to describe the behavior at rest after applying strain (deformation), e. g. when performing relaxation tests (see Chapter 7).

      For “Mr. and Ms. Cleverly”

6.3.2The Burgers model

      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