[4.13]Hooke, R., Micrographia, 1665; De potentia restitutiva, John Marty, London, 1678
[4.14]Euler, L., Mechanica, 1736; Theoria motus corporum (“second mechanica”), 1765
[4.15]Cauchy, A. L., De la pression ou tension dans un corps solide, 1827; Sur les équations differentielles ou de mouvement pour les points matériels, 1829
[4.16]DIN SPEC 143-1 (pre-norm): Modern rheological test methods, part 1: Determination of the yield point (fundamentals and comparative test methods), Beuth, Berlin, 2005/2019 (in German: Moderne rheologische Prüfverfahren, Teil 1: Bestimmung der Fließgrenze, Grundlagen und Ringversuch)
[4.17]Mezger, T. G., Applied Rheology – with Joe Flow on Rheology Road, Anton Paar, Graz, 2020, 7th ed. (up to now also available in the following languages: Chinese, French, German, Italian, Japanese, Korean, Portuguese, Russian, Spanish)
5Viscoelastic behavior
In this chapter are explained the following terms given in bold:
| Liquids | Solids | ||
| (ideal-) viscousflow behaviorviscosity law(according to Newton) | viscoelastic flow behavior Maxwell model | viscoelastic deformation behavior Kelvin/Voigt model | (ideal-) elasticdeformation behaviorelasticity law(according to Hooke) |
| flow/viscosity curves | creep tests, relaxation tests, oscillatory tests | ||
5.1Introduction
Viscoelastic (VE) materials are always showing viscous and elastic behavior simultaneously. The viscous portion behaves according to the viscosity law which was presented in Chapter 2, and the elastic portion behaves according to the elasticity law which was explained in Chapter 4. Depending on their rheological behavior, VE liquids can be distinguished from VE solids. VE materials display a time-dependent, delayed response as well when a stress or a strain is applied as well as when it is removed.
5.2Basic principles
5.2.1Viscoelastic liquids according to Maxwell
5.1.2.1.1Experiment 5.1: The thickened liquid in a glass beaker
After swirling the filled glass a few times and then stopping the motion immediately, the VE liquid still continues to swirl. When the liquid comes to rest, the surface is leveling again after a short period of time.
5.1.2.1.25.2.1.1The Maxwell model
Behavior of a VE liquid can be illustrated using the combination of a spring and a dashpot in serial connection (see Figure 5.1). Both components can be deflected independently of each other. The spring model shows Hookean behavior as described in Chapter 4.3.1b, and the dashpot model displays Newtonian behavior as explained in Chapter 2.3.1b. The model was named in honor to James C. Maxwell (1831 to 1879), who first presented the corresponding mathematical fundamentals (in 1867/1868, in the form of a differential equation; see also Chapter 14.2: 1855) [5.1] [5.2]. H. Freundlich used the (German) term Fließelastizität (literally: flow elasticity), in order to indicate to anomalities in the flow behavior, and others spoke of elastoviscous liquids [5.23] [5.24].
Figure 5.1: The Maxwell model to simulate the behavior of viscoelastic liquids
5.1.2.1.3a) Viscoelastic flow behavior , illustrated by use of the Maxwell model (see Figure 5.2):
5.1.2.1.41) Before applying a load
Both components of the model – as well the spring as well as the dashpot – exhibit no deformation.
5.1.2.1.52) When under load
2a. When applying a constant force, only the spring displays an immediate, step-like deformation until it reaches a constant deflection value which is proportional to the value of the loading force. Therefore, immediately after the beginning of the load phase it is only the spring which is deformed.
2b. Afterwards, when still under the acting constant force, also the piston of the dashpot is beginning to move, and it is now continuously moving on as long as the force is applied. After a certain period of time under load, both components are showing a certain extent of deformation which corresponds as well to the degree of the force as well as to the time of loading.
As a result of the load phase, the time-dependent deformation function occurs in a γ(t)-diagram as an immediate deformation step, i. e. a step which is independent of time, followed by a straight line sloping upwards.
5.1.2.1.63) When removing the load
The spring recoils elastically, i. e. it moves back immediately and completely. The distance travelled by the dashpot, however, remains unchanged.
As a result of the phase after removing the load, the γ(t)-function occurs as an immediate reformation step, i. e. a step which is independent of time. Afterwards, the γ-value remains unchanged on a constant value.
Figure 5.2: Simulation of the deformation behavior of viscoelastic liquids using the Maxwell model
Summary: Behavior of the Maxwell model
After a load-and-removal cycle, these kinds of samples remain partially deformed. The extent of the reformation represents the elastic portion, and the extent of the permanently remaining deformation corresponds to the viscous portion. There is an irreversible deformation process taking place since such a sample occurs in a changed shape at the end of the process because its reformation is not complete, even after a long period of time at rest. These kinds of materials behave essentially like a liquid, and therefore, they are referred to as viscoelastic liquids or Maxwell fluids.
5.1.2.1.7b) Differential equation according to the Maxwell model
In order to analyze Maxwellian behavior during a load-and-removal cycle, the following differential equation is used (with the index “v” for the viscous portion and “e” for the elastic one):
Assumption 1: The total deformation is the sum of the individual deformations applied to the two model components.
γ = γv + γe
This applies also to the shear rates: γ ̇ = γ ̇ v + γ ̇ e
since γ ̇ = dγ/dt (as explained in Chapter 4.2.1)
Assumption 2: The same shear stress is acting on each one of the two components.
τ = τv = τe
The viscosity law applies to the viscous element: η = τv / γ ̇ vor γ ̇ v = τv /η
The elasticity law applies to the elastic element:G = τe /γeorγe = τe /G
and γ ̇ e = τ ̇ e /G respectively, with the change of the shear stress over time as τ ̇ = dτ/dt [Pa/s], this is