Читать книгу The Rheology Handbook - Thomas Mezger - Страница 232

This chapter is intended for giving merely in a nutshell some basic information on normal stress tests and on the corresponding terminology in this special field of rheology.

If a viscoelastic material is deformed, there are not only one-dimensional forces or stresses acting in the main direction of shear deformation. Due to its elastic proportion, a material will permanently show a certain ability of molecular or structural re-formation or recovery, already during the deformation process. Besides the shear stress τ = F / A (see Two-Plates model, Figures 2.1 and 4.1), so-called normal stresses are occurring, which have an effect in perpendicular direction to the shear stress. The result is a state of three-dimensional deformation. This can be illustrated using a (3 × 3) tensor (see also Chapter 14.2: A. L. Cauchy, 1827 [5.19], and J. C. Maxwell, 1855 [5.1]). It contains nine values, which can be displayed in a three-dimensional Cartesian coordinate system. In order to explain this, we can imagine the behavior of an infinitely (infinitesimal) small cube-shaped volume element of our test material being within the shear gap (Figure 5.8).

Taking the shear direction or deformation direction as x-direction; in the Two-Plates model the x-axis is pointing to the right (see Chapter 4.2, Figure 4.1). The y-direction is the direction of the shear gradient; in the Two-Plates model the y-axis is pointing upwards. The z-direction is called the indifferent or neutral direction; in the Two-Plates model the z-axis is pointing out of the writing plane, towards the reader.

Figure 5.8: An infinitesimal small volume element of the sample, and the effective directions in the Cartesian coordinate system:

x as shear direction, y as direction of the shear gradient, and z as neutral direction (with the z-axis pointing perpendicular out of the page)

Figure 5.9: An infinitesimal small volume element of the sample, shear stress τ and the three

normal stresses τxx, τyy and τzz

(the latter works vertically out of the picture

surface)

The stress tensor indicates the following stress values:

1 Into x-direction:τxx, τyx, τzx

2 Into y-direction:τxy, τyy, τzy

3 Into z-direction:τxz, τyz, τzz

The first index of the stress tensor values indicates the position of the area of the cube-shaped volume element on which the stress is acting. The term normal direction is used in mathematics and physics to determine the position of an area. For example here, the area with the x-direction as the normal direction means, it is the area on which the x-coordinate is standing in a right-angled position. The normal force or normal stress, respectively, acts in the direction of the surface-normal standing vertically on the regarded area, and it is presented in the form of a vector (arrow). The second index of the stress sensor values indicates the direction of the force or stress acting on the area which is described by the first index. Thus, for the x-direction counts here, it is pointing to the right side.

The tensor contains the three normal stresses τxx, τyy and τzz (see Figure 5.9). For most of the viscoelastic liquids, the τxx component clearly shows the highest value of the three normal stresses. The other six tensor stresses are shear stresses. If a liquid is flowing, i. e. if viscous behavior clearly dominates, these six stress values can be reduced to the τyx component only, and the other five stress values can be ignored. This is the reason why in this textbook, beside of the chapter at hand, there is mentioned this one τ-value only, and therefore here, it can be written without any index.

The shear stress τyx = F / A = τ is acting in x-direction on an area which is parallel to the plates of the Two-Plates model, and the normal direction of the corresponding area is the y-direction. We can imagine this area as an individual flowing layer. An illustrative example is a stack of beer mats (see Experiment 2.1 and Figure 2.2 of Chapter 2.2). Here, each individual beer mat represents an area with the y-direction as the normal direction, and each beer mat is moving into x-direction when applying a shear force F.

Usually, the following curve functions are presented in a diagram, typically on a double-

logarithmic scale.

1 The 1 st normal stress difference N1( γ ̇ ), as a function of the shear rate

Equation 5.3

N1 [Pa] = τxx – τyy

1 The 2 nd normal stress difference, as N2( γ ̇ )

Equation 5.4

N2 [Pa] = τyy – τzz

1 The 1 st normal stress coefficient, as ψ1( γ ̇ ); psi, pronounced: “psee” or “sy”

Equation 5.5

ψ1 [Pa ⋅ s2] = N1 / γ ̇ 2

showing the 1st zero-normal stress coefficient as a plateau value in the low-shear range

Equation 5.6

ψ1,0 [Pa ⋅ s2] = lim γ ̇ → 0 ψ1( γ ̇ ) = const

1 The 2 nd normal stress coefficient, as ψ2( γ ̇ )

Equation 5.7

ψ2 [Pa ⋅ s2] = N2 / γ ̇ 2

showing the 2nd zero-normal stress coefficient as a plateau value in the low-shear range

Equation 5.8

 ψ2,0 [Pa ⋅ s2] = lim γ ̇ → 0 ψ2( γ ̇ ) = const

In order to determine the first normal stress difference, the raw data measured by a rheometer are the values of the normal force FN in [N] in axial direction (y-direction). Normal forces of samples are forces acting into the direction of the shaft of the measuring bob, trying to push the upper plate or the cone upwards or the lower plate downwards, respectively (when using a parallel-plate or cone-and-plate measuring geometry). For Information on tests using the normal force control (NFC) option, see Chapters 10.4.6 and 10.7b.

Effects of normal forces occur in different forms. They may cause large problems in several application fields in industry. Examples (see Figures 5.3, 5.4, 5.10 and 5.11):

1 When performing rotational tests with viscoelastic samples, the corresponding effects may occur in the form of streaks and similar defects on the surface of cylinder measuring geo­metry or on the edges of cone-and-plate and parallel-plate systems. For stiff samples, these effects are also influenced by the stiffness of as well the measuring geometry as well as of the measuring instrument used.

2 Post-extrusion swell or die swell effects and as melt fracture when extruding polymer melts

3 The Weissenberg effect or rod-climbing effect when stirring

Figure 5.10: Cone-and-plate (CP) measuring geo­metry in side view during a rotational test, with a certain constriction of the sample at the edge of the cone. In the direction along the cone axis (axial direction), the normal force tries to push apart cone and plate, and with a stiff bottom plate system available, it may be measured on top as FN

Figure 5.11: CP or PP geometry in top view during a rotational test on a polymer sample: (1) with coiled macromolecules when at rest or under low shear conditions, and (2) with stretched molecules under the shear force FS working in x-direction causing a high shear deformation or shear rate, respectively. Due to the resetting elasticity of the molecules, as a consequence, normal forces are resulting, as well in y-direction (axial, along the axis of the measuring geometry, see Figure 5.10) as well as in z-direction (towards the axis)

Further information on normal stresses can be ­found e. g. in DIN 13316 [5.4] [5.13].

Note: Lodge/Meissner relation

In 1976, Arthur S. Lodge (1942 to 2007) and Joachim Meissner (1929 to 2011) presented the following relation [5.20] [5.21]:

Equation 5.9

N1,LM = γ ⋅ τ

Using this LM-relation, with the values of shear strain γ [1] and shear stress τ [Pa] which were preset or determined via relaxation tests (see Chapter 7), the 1st normal stress difference N1 [Pa] can be calculated. Numerous tests with many standard polymers have confirmed this empirical rule. Comparisons have shown good correlation between

1 N1 values which are measured directly by the normal stress sensor of a rheometer, and

2 N1,LM values, which are calculated by use of the LM relation from data which are measured by stress relaxation tests [5.22]

End of the Cleverly section

In the following Chapters 6 to 8, the most important types of tests are presented which can be performed to measure viscoelastic behavior: creep tests, relaxation tests, and oscillatory tests.