The Rheology Handbook. Thomas Mezger

are trying permanently to recoil against the external forces, and this counter action leads to three-dimensional effects producing normal forces (see Chapter 5.3).

      Note 2: Tack and stringiness

      Stringiness and tack can be tested and scientifically analyzed, adjusted and controlled via the damping factor tanδ obtained when performing oscillatory tests. See also the Notes and Examples in Chapters 8.2.4a (Note 2) and 8.3.4.5 (Example 2 about construction adhesives in automotive industry). A further testing mode are tack tests, see Chapter 10.8.4.2.

      Note 3: Normal stresses

      Extrudate swell, the Weissenberg effect, as well as all other viscoelastic effects are resulting from normal forces or normal stresses, respectively (see Chapter 5.3).

      End of the Cleverly section

      Summary: Behavior of viscoelastic liquids

      In order to get an optimal behavior for practical use, the viscous and elastic portions of viscoelastic materials should show a well-balanced ratio.

5.2.2Viscoelastic solids according to Kelvin/Voigt

      5.1.2.1.1Experiment 5.4: Breaking behavior of a silicone rubber with a low degree of crosslinking

      This VE mass breaks into two pieces when applying a high tensile stress. Afterwards, the edges are moving slowly just a bit backwards. Finally, the edges of the two pieces remain stable in their dimension without exhibiting any further reformation.

      5.1.2.1.2Experiment 5.5: Comparison of two rubber balls

      Two rubber balls having the same weight, but consisting of different base materials, are hit on a stone floor. The balls are bouncing back to different heights, clearly indicating different elastic behavior. One of the balls does not bounce back as high since obviously, this rubber consists of a viscoelastic material showing a comparably smaller elastic portion. See also Experiment 8.2 in Chapter 8.4.3a (oscillatory tests): bouncing rubber balls, and damping factor tanδ.

      For “Mr. and Ms. Cleverly“

      5.1.2.1.35.2.2.1 The Kelvin/Voigt model

      Behavior of a VE solid can be illustrated using the combination of a spring and a dashpot in parallel connection (see Figure 5.6). Both components are connected by a rigid frame. The mathematical fundamentals, in the form of a differential equation, were first presented by O. E. Meyer (1874 [5.10] [5.2]). However, due to later works of W. Thomson (Kelvin, 1878 [5.11]) and W. Voigt (1892 [5.12]), usually nowadays, the model is called the Kelvin/Voigt model. Sometimes it is also named Voigt/Kelvin model (as in DIN 1342-1), or Kelvin model [5.13], or Voigt model [5.14].

       Figure 5.6: The Kelvin/Voigt model to simulate the deformation behavior of viscoelastic solids

       Figure 5.7: Simulation of the deformation

      behavior of a viscoelastic solid using the

      Kelvin/Voigt model

      5.1.2.1.4a) Viscoelastic behavior, illustrated by use of the Kelvin/Voigt model (see Figure 5.7):

      5.1.2.1.51) Before applying a load

      Both components of the model – as well the spring as well as the dashpot – exhibit no deformation.

      5.1.2.1.62) When under load

      Deformation is increasing continuously as long as a constant loading force is applied. The two components can only be deformed together, which means simultaneously and to the same extent, because they are connected by a rigid frame. Since its motion is slowed down by the presence of the dashpot, the spring cannot carry out an immediate, step-like deformation as it would do if it were independent of the dashpot.

      As a result of the load phase, the deformation process occurs in the γ(t)-diagram as a curved, time-dependent exponential function (e-function) showing moderately increasing deformation values until a certain maximum value γmax is reached finally.

      5.1.2.1.73) When removing the load

      The spring immediately aims to elastically step back to its initial shape, and this driving force is causing both components to reach their initial positions finally. However, this step occurs only after a certain period of time. Therefore, due to the presence of the dashpot there is a delayed process going on. But indeed, in the very end the model will show no longer any remaining deformation.

      As a result of the phase after removing the load, the re-formation process occurs again as a curved, time-dependent e-function in the γ(t)-diagram, but now of course showing moderately decreasing deformation values. After a sufficiently long period of time the reformation will be completed, i. e. displaying γ = 0 again, like at the beginning of the test.

      Summary: Behavior of the Kelvin/Voigt model

      After a load-and-removal cycle, these kinds of samples show indeed delayed but complete recovery (re-formation) which fully compensates for the previously occurred deformation. There is a reversible deformation process taking place since such a sample occurs in an unchanged shape in the very end of the test. These kinds of materials behave essentially like a solid, and therefore, they are referred to as viscoelastic solids or Kelvin/Voigt solids.

      5.1.2.1.8b) Differential equation according to the Kelvin/Voigt model

      In order to analyze Kelvin/Voigt behavior during a load cycle, the following differential equation is used (again with “v” for the viscous portion and “e” for the elastic one):

      Assumption 1: The total shear stress applied will be distributed on the two model components.

      τ = τv + τe

      Assumption 2: Deformation of both components occurs to the same extent, and this applies also to the shear rate.

      γ = γv = γe or γ ̇ = γ ̇ v = γ ̇ e

      with γ ̇ = dγ/dt (as explained in Chapter 4.2.1)

      The viscosity law applies to the viscous element: τv = η ⋅ γ ̇ v

      The elasticity law applies to the elastic element: τe = G ⋅ γe

      The sum of the shear stresses results in the differential equation according to Kelvin/Voigt:

      Equation 5.2

      τ = τv + τe = η ⋅ γ ̇ v + G ⋅ γe = η ⋅ γ ̇ + G ⋅ γ

      The solution and use of this differential equation are described in Chapters 6.3.3 a/b and 6.3.4.3 (creep tests).

      End of the Cleverly section

      5.1.2.1.95.2.2.2Examples of the behavior of VE solids in practice

      The