Complex shear modulus

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In vibrational rheometry, the complex shear modulus describes the behavior of viscoelastic bodies under an oscillating shear load . It links the shear stress acting on the specimen with the resulting shear deformation .

The complex shear modulus can be measured relatively easily with a rheometer (for liquids) and by dynamic mechanical analysis (for solids). The change in the complex shear modulus with variation of amplitude , frequency or other parameters such as temperature provides information about the properties of the material. So z. B. the linear range can be determined, the molecular structure can be concluded or a crosslinking process can be examined.

Storage and loss module

In general, the complex shear modulus has the form of a complex number :

With

  • the storage modulus ( real part ), which stands for the elastic part. It is proportional to the portion of the deformation energy that is stored in the material and can be recovered from the material after the load is removed.
  • the loss module ( imaginary part ), which stands for the viscous part. It corresponds to the loss of energy, which is converted into heat through internal friction .

The quotient from and is the loss factor :

For an ideally elastic body it assumes the value 0, for an ideally viscous body it approaches infinity

Basics

Principle of shear
Maxwell body

A body experiences a shear

,

so is the shear stress

necessary

With

  • - shear angle
  • - deformation path
  • - Thickness of the body under consideration
  • - deformation force
  • - Surface.

The behavior of a viscoelastic material, i.e. H. The relationship between shear stress and shear can be simulated by rheological model bodies, which are composed of

  • Feathers ( Hooke elements), which represent the purely elastic part. Here, the shear stress on the shear modulus with the shear connected:
If the body is exposed to a sinusoidal shear with amplitude and angular frequency :
so in the elastic branch:
In the case of an ideally elastic body, the shear stress also has a sinusoidal curve, namely in phase with the shear: when the oscillation passes through zero , the body does not experience any deformation and therefore no shear stress is necessary to overcome the restoring force . In contrast, the body is maximally deformed in terms of amplitude, and then the restoring force is also greatest.
An ideally viscous body has a 90 ° phase shift in the shear stress: at the zero crossing of the deformation, the change in the deformation is greatest, which is why the resistance of the liquid that has to be overcome is greatest. In terms of amplitude, the direction of movement is reversed, the shear rate and thus also the shear stress become zero for a moment.

Simple models for describing a viscoelastic solid are e.g. B.

Derivation for the Kelvin body

Kelvin body

In the case of a Kelvin body, due to the parallel connection, the shear is the same in both branches, whereas the total shear stress is made up of the shear stresses in the Hooke and Newton elements:

In purely formal terms, the relationships correspond to the parallel connection of an ohmic resistor with an inductance in electricity theory. As there the equation can be transformed with a vector diagram to:

With

and

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

In analogy to the complex AC calculation , the quantities can also be described with complex functions:

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

The reciprocal

is known as complex compliance .

Complex viscosity

If one derives , one obtains the complex shear rate :

This allows the complex viscosity to be calculated:

Your amount is:

The dynamic viscosity is calculated from the real part of :

swell

  • Georg Meichsner, Thomas Mezger, Jörg Schröder: Measuring and controlling paint properties . Vincentz Network GmbH & Co KG, Hanover 2003, ISBN 3-87870-739-8 , chap. 4.3.4. The oscillation experiment - vibrational rheometry , p. 73–80 ( limited preview in Google Book search).