The Rheology Handbook. Thomas Mezger

kinds of materials often display effects like wall-slip, sliding and plug flow, when forced to move through capillaries, tubes and pipes. Similar effects may also occur in the gap of a rheometer measuring system. For these kinds of samples, rheological behavior is not constant throughout the whole volume of the test material: a part of the sample is flowing, and another one does not [3.34]. Inhomogeneous shear effects should always be expected when testing dispersions showing a high concentration of solid matter since here, phase separation of the sample may occur. Below a critical shear rate value shear banding effects may occur (see Figure 2.9, no. 6 and 7) [3.26].

      3.3.4.2.4Example 1: Plastic behavior of metals for cold forging processes

      Plastic deformation occurs in cold forging processes of metals, or with other crystal-forming materials, if lattice dislocation takes place between the atomic levels or crystals [3.35] [3.36]. Below the yield point, there is elastic and reversible deformation behavior. Above the yield point, however, behavior is irreversible and inelastic and then, deformation is no more homogeneously distributed throughout the entire shear gap [3.22].

      3.3.4.2.5Example 2: Plastic deformation of the landscape and soil flow (solifluction )

      In the Ice Age (which, for example, came to an end in Northern Germany around 14,000 years ago), the shear force of glaciers moved loose layers of soil masses and pieces of solid rock, e. g. boulders, weighing several tons. The so-called “solifluction” of the partially thawed permafrost soil resulted in plastic deformation of the landscape.

      3.3.4.2.6Example 3: Plastic land subsidence caused by mining or dike construction

      As a result of mining, land subsidence may occur when loosened sediment layers meet ground water. Here, and also in the construction of water protection dikes, the soil-mechanical, rheologically plastic behavior is a crucial point (creep, sliding, flow).

      3.3.4.2.7Example 4: Plastic flow of debris and mud avalanches in the mountains

      Debris and mud avalanches are an inhomogeneous mixture of water, fine sediment (max. particle size up to d = 0.1 mm), mud (clay, silt, sand, with 0.1 mm < d < 20 mm), and larger “particles” (granules, gravel, stones, boulders, up to d = 20 cm or even 1 m, – which ideed was a little bit too large for a normal rheometer …). They may show mean velocities of v = 1 to 30 m/s, thus, more than 100 km/h. Considering a longitudinal section, two zones of the velocity distribution occur: In the lower range close to the bottom a shear zone showing clearly increasing velocity values from the bottom upwards (i. e. flowing layers showing a velocity gradient), and above a zone close to the surface of almost constant velocity (i. e. v = const, showing “plug flow”) [3.37]. For corresponding rheological tests of dipersions showing particel size up to 10 mm, a ball measuring system can be used (see Chapter 10.6.5). Shear experiments on bulk solids and powders are described in Chapter 13.

      3.3.4.2.8Illustration, using the Two-Plates model (see Figure 2.9: no. 4)

      In order to illustrate plastic behavior: Not all flowing layers in the Two-Plates model are shifting along one another to the same extent. Therefore across the shear gap, the resulting shear rate (or “velocity gradient“) and shear deformation (or deflection gradient),respectively, are not constant. Only some layers are in motion and others remain still at rest. After removing the load, any re- formation of the layers depends on the elastic portion. Related to the entire gap this re-formation is also inhomogeneously. In comparison to this, Figures 2.1, 2.9 (no.2) and 4.1 exhibit the behavior desired by rheologists at homogeneous deformation and flow conditions.

      3.3.4.2.9Examples of plastic materials

      Solids or dispersions with a high concentration of solids showing high interactive and cohesive forces; i. e. plasticine (see Experiment 1.1b of Chapter 1.2), wax, a piece of soap, sealants, sludge, loam, mocha coffee grounds, plasters, surfactant, systems showing shear-banding (see Chapter 9.2.2).

      When performing rheological experiments in a scientific sense, however, it is assumed that there are homogeneous shear conditions across the entire shear gap. Only then, a constant shear rate or shear deformation can be expected to occur during the whole shearing process and after removing the load, and viscous, viscoelastic or elastic behavior. And only under these preconditions, the behavior can be described formally and mathematically by the relations according to Newton, Hooke, Maxwell, Kelvin/Voigt and Burgers.

      3.3.4.2.10Summary

      Plastic behavior cannot be described unequivocally using the usual scientific fundamentals of mathematics and physics since it is an inhomogeneous behavior. Therefore, it can only be presented in terms of relative values obtained from empirical tests.

      Unfortunately, in many industrial laboratories the terms plastic, ideal-plastic, viscoplastic or elastoplastic still are used to mean a lot of different things. It is useful to understand what these terms might mean, but their use should be avoided when performing and analyzing scientific rheological tests. Within a limited deformation range, in most cases, samples can be characterized as viscoelastic (DIN 13343). However, if a material cannot be sheared homogeneously it is often necessary to use special relative measuring systems (see Chapter 10.6). In this case, it is better to work only with the measured raw data such as torque, rotational speed and deflection angle, instead of any rheological parameter such as shear stress, shear rate, shear deformation, viscosity, and shear modulus.

      3.3.4.2.11Note by the way

      Both, the plastic surgeon and the sculptor (working with wood, stone or plaster to create plastic sculptures), are performing usually inhomogeneous and irreversible deformation processes, related to the structure of the material as a whole. Hopefully, the patients, operated beauties and lovers of fine arts are pleased with the end-products of these kinds of plastic deformations.

      3.3.4.2.12d) Practical example: Yield point and wet layer thickness of a coating

      Using the yield point value, it is possible to make a simple, rough estimation of the wet layer thickness on a vertical wall (see Figure 3.23).

      The following holds: τ = F/A, with the area to be coated: A = b ⋅ c,

      and the weight force of the layer due to gravity: F = FG = m ⋅ g = V ⋅ ρ ⋅ g

      with the mass m [kg] and the volume

      V [m3] = a ⋅ b ⋅ c of the volume element,

      the gravitation constant g = 9.81 m/s2, and the density ρ [kg/m3] of the coating,

      it follows that: FG = a ⋅ b ⋅ c ⋅ ρ ⋅ g

      Result: τ = τ0 = FG/A = (a ⋅ b ⋅ c ⋅ ρ ⋅ g)/(b ⋅ c) = a ⋅ ρ ⋅ g

      3.3.4.2.13Summary

      A coating layer with the layer thickness “a” remains on the wall only then, if the limiting value of the shear stress between the states of rest and flow is not exceeded. This τ-value is the yield point τ0 [Pa]. As the calculation shows, the yield point is independent of the layer width and length. Thus, the wet layer thickness of a coating which remains on a vertical wall can be calculated as:

      Equation 3.5

      a = τ0 /(ρ ⋅ g)

      3.3.4.2.14Examples