The Rheology Handbook. Thomas Mezger

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4Elastic behavior and

      shear modulus

      In this chapter are explained the following terms given in bold:

Liquids		Solids
(ideal-) viscousflow behaviorviscosity law(according to Newton)	viscoelasticflow behaviorMaxwell model	viscoelasticdeformation behaviorKelvin/Voigt model	(ideal-) elastic deformation behavior elasticity law(according to Hooke)
flow/viscosity curves	creep tests, relaxation tests, oscillatory tests

4.1Introduction

      Before 1990, there were only few users of torsional and oscillatory rheometers performing tests on solid materials under scientific measuring conditions. Meanwhile however, the progress in technology of the corresponding instruments has enabled many users to characterize the elastic behavior of solid samples also in a range of very low deformations, and therefore, in a non-

      destructive range.

4.2Definition of terms

      The following section uses the Two-Plates model in order to define further rheological parameters (see also Chapter 2.2). The lower plate is fixed (deflection s = 0). The upper plate with the (shear) area A is deflected by the (shear) force F and the resulting deflection s is measured (see Figure 4.1). Between the plates there is the constant distance h, and the sample is sheared in this shear gap. It is assumed that the following shear conditions are given:

      1 The sample shows adhesion to both plates without any wall-slip effects.

      2 The sample is deformed homogeneously throughout the entire shear gap, i. e., no inhomogeneous “plastic deformation” is occurring (see also Chapter 3.3.4.2c and Figure 2.9, no. 4).

      Accurate calculation of the rheological parameters is only possible if both conditions are met.

      The real geometric conditions in rheometer measuring systems are not as simple as in the Two-Plates model. However, if a shear gap is narrow enough, the necessary requirements are largely met and the definitions of the following rheological parameters can be used.

       Figure 4.1: Two-Plates model for shear tests to illustrate deformation of a material in the shear gap

4.2.1Deformation and strain

      Definition of the shear deformation, also termed shear strain:

      Equation 4.1

      γ = s/h

      γ (pronounced: gamma), with the deflection path s [m] and the distance h [m] between the plates, see Figure 4.1. The following holds: s/h = tanφ, with the deflection angle φ [°], (phi, pronounced: fee or fi).

      The unit of the shear deformation γ is [1], it is therefore dimensionless.

      Most samples have to be measured at very low γ-values in order to remain in that limited part of the deformation range which can be analyzed scientifically. Therefore, in most cases, it is useful to specify the γ-values in % (= 0.01 = 10-2).

      Example: A deformation of γ = s/h = 0.1 = 10 % occurs in a shear gap of h = 1 mm if one of the plates is deflected by s = 0.1 mm.

      Note: Use of the terms deformation and strain

      Sometimes, the terms “deformation” and “strain” are used as synonyms. In order to use a clear language, strain should be chosen if a controlled shear strain test is performed. And “deformation” should be selected to outline the consequences for the passively reacting sample if a controlled shear stress test is carried out. However, in many industrial laboratories people use both terms without making a difference. In most other languages besides English there are existing no different terms for γ when performing these two different test modes, and therefore then, in both cases the term deformation is used.

      For “Mr. and Ms. Cleverly”

      The relation between deformation γ and shear rate γ ̇

      The symbol γ ̇ for the shear rate is derived from γ. The following holds:

      Equation 4.2

      Δγ / Δt = (γ1 – γ0) / (t1 – t0)

      with the deformation γ0 [%] at the beginning, and γ1 [%] at the end of the test, with the time points t0 [s] at the beginning and t1 [s] at its end, with the change in deformation Δγ [%] and the test period as time interval Δt [s]. Using the scientific notation for infinitesimal parameters:

      Equation 4.3

      dγ / dt = γ ̇

      Therefore, a shear rate γ ̇ is an infinitely small change in deformation (dγ) which takes place in an infinitely short time period (dt). In other words (according to ASTM D4092): The shear rate γ ̇ is “the time rate of change of shear strain”. Expressed mathematically: γ ̇ with the unit [s-1] is the time derivative of γ. In other words: The shear rate is the time-dependent rate of deformation, or briefly, strain rate.

      End of the Cleverly section

4.2.2Shear modulus

      When measuring ideal-elastic solids at a constant temperature, the ratio of the shear stress τ and corresponding deformation γ is a material constant if testing is performed within the rever­sible-elastic deformation range, the so-called linear-elastic range. This material specific value is referred to as the shear modulus G and reveals information about the rigidity of a material. Materials showing comparably stronger intermolecular or crystalline cohesive forces exhibit higher internal rigidity, and therefore, also a higher G-value. Sometimes, the shear modulus is also called modulus of elasticity in shear or rigidity modulus. Definition of the shear modulus:

      Equation 4.4

      G = τ / γ

      Note: The plural of shear modulus is shear moduli.

      The unit of the shear modulus is [Pa], (pascal), and 1 Pa = 1 N/m2

      For rigid solids, the following units are also used:

      1 kPa (kilopascal) = 1000 Pa = 103 Pa

      1 MPa (megapascal) = 1000 kPa