Dynamic modulus

The complex shear modulus describes the behavior of viscoelastic body at a oscillating shear stress in the vibration rheometry. It links the shear stress acting on the sample with the resulting shear deformation.

The complex shear modulus can be measured technically relatively easy with a rheometer ( for liquids) and by dynamic mechanical analysis are determined (for solids ). The change of the complex shear modulus with variation of amplitude, frequency, or other parameters such as temperature provides information on the properties of the material. Thus, for example, determine the linear range, closed on the molecular structure or a cross-linking process to be examined.

Storage and loss modulus

Generally, the complex shear modulus is in the form of a complex number:

With

  • The memory module (real part ), which represents the elastic component. It is proportional to the percentage of the strain energy, which is stored in the material and can be recovered again upon release from the material.
  • The loss modulus (imaginary part ) which is the viscous component. It corresponds to the loss of the energy, which is converted into heat by internal friction.

The quotient of the loss factor and is:

He takes for an ideal elastic body to the value 0 for an ideal viscous body he goes to infinity

Basics

Experienced by a body shearing

So is this the shear stress

Necessary

With

  • - Shear angle
  • - deformation
  • - Thickness of the body under consideration
  • - Deformation force
  • - Surface.

The behavior of a viscoelastic material, ie the relationship between shear stress and shear can be modeled by rheological model body, which consist of

  • Springs ( Hooke elements) that represent the purely elastic component. Here, the shear stress on the shear modulus of the shear associated with:
  • Damping cylinders ( Newton - elements with a Newtonian fluid ) for the ideal viscous component. Here, the shear stress on the viscosity is linked to the shear rate, that is, with the first derivative of shear:

Simple models for the description of a viscoelastic solid are eg

  • The Maxwell model, that is, the series connection of a spring with a damping element, or
  • The Kelvin model, that is, the parallel combination of a spring with a damping cylinder.

Derivation of the Kelvin body

In a Kelvin body is due to the parallel shear the same in both branches, the total shear stress is now composed of the shear stresses in Hooke and Newton element together:

Formally comply with the relations of the parallel connection of an ohmic resistor and an inductor in the theory of electricity. As there, the equation can be transformed with a vector diagram to:

With

And

The shear stress is thus shifted by the phase angle, which may assume a value between 0 ° and 90 °, with respect to the shear.

In analogy to the complex AC circuit analysis, the variables can also be described with complex functions:

Then the complex shear modulus of the quotient of complex shear stress and shear complex:

The reciprocal of

Is called the complex compliance.

Complex viscosity

On passing from, we obtain the complex shear rate:

This allows the complex viscosity calculated:

Your sum is:

Swell

  • Georg Meichsner, Thomas Mezger, Jörg Schröder: paint properties measure and control. Vincentz Network GmbH & Co KG, Hannover 2003, ISBN 978-3-8787-0739-4, chap. 4.3.4. The oscillatory testing - vibration rheometry, pp. 73-80 ( limited preview on Google Book Search ).
  • Material property
  • Rheology
Zero crossing
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