Dynamic-mechanical Analysis (DMA) – General Principles

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Dynamic-mechanical analysis (DMA) – General principles

Fundamentals

In dynamic-mechanical analysis (see also: elastic modulus), a test specimen with a defined geometry is subjected to periodically alternating loading. By varying the frequency, it is possible to characterise the time dependence of the material behaviour at a constant temperature. If these tests are carried out in a temperature control chamber, the test method is referred to as DMTA and the temperature dependence of the materials in question is characterised. The relationship between the stress time t and the frequency f or angular frequency ω is given by Eq. (1).

$t={\frac {1}{2\pi f}}={\frac {1}{\omega }}$

(1)

The DMA is characterised by the fact that only relatively short test times are required to determine viscoelastic characteristic values in a wide frequency range. In addition, it is relatively easy to investigate the material behaviour as a function of temperature using dynamic-mechanical thermal analysis (DMTA), although longer test times are required here due to the necessary temperature stability [1].

Performing the DMA

There are different variants for performing DMA, which differ in terms of the achievable frequency range, the type of mechanical loading and the material parameter determined. Another classification is based on the type of vibration excitation in the methods

• with forced vibrations,
• with free damped vibrations and
• with resonance vibrations.

In the very high frequency range, the propagation of sound and ultrasonic waves or dielectric spectroscopy is also used to determine characteristic values. The different methods of DMA or DMTA are standardised in ISO 6721-1 [2].

Method with force-induced vibrations

To characterise the viscoelastic properties of plastics using forced vibrations, the test specimen is subjected to a sinusoidal mechanical loading with a constant frequency and constant amplitude (Fig. 1). In the case of linear-viscoelastic behaviour, the temporal changes in stress and strain in the oscillating state have the same frequency but different phase positions.

Fig. 1:

Temporal change in stress and strain during dynamic-mechanical analysis using forced vibrations

In the case of normal stress, Eqs. (2) and (3) apply, and in the case of shear stress, Eqs. (4) and (5) apply.

$\varepsilon \left(t\right)=\varepsilon _{0}\cdot \sin \omega t$

(2)

$\sigma \left(t\right)=\sigma _{0}\cdot \sin \left(\omega t+\delta \right)$

(3)

$\gamma \left(t\right)=\gamma _{0}\cdot \sin \omega t$

(4)

$\tau \left(t\right)=\tau _{0}\cdot \sin \left(\omega t+\delta \right)$

(5)

As a result of the phase shift δ between stress (tension) and deformation (strain or shear), the modulus must be introduced as a complex quantity E* or G* according to Eqs. (6) and (7) in order to describe the stress–strain (shear) relationship.

$E^{\ast }=E^{\prime }+iE^{\prime \prime }$

(6)

$G^{\ast }=G^{\prime }+iG^{\prime \prime }$

(7)

The complex module can be viewed as a vector in the complex number plane (Fig. 2), whose direction is given by the phase angle δ and whose magnitude is given by the ratio of the amplitude values of voltage and strain.

Fig. 2:

Representation of the moduli E* and G* in the complex number plane

The absolute value of the respective module is calculated from the ratio of the initial loading to the initial deformation according to Eqs. (8) and (9).

$\left|E^{\ast }\right|={\frac {\sigma _{0}}{\varepsilon _{0}}}$

(8)

$\left|G^{\ast }\right|={\frac {\tau _{0}}{\gamma _{0}}}$

(9)

Using simple trigonometric relationships, it is possible to divide the energy into a real part Eʼ or Gʼ and an imaginary part Eʼʼ or Gʼʼ, which is done using Eqs. (10) to (13). The real part Eʼ or Gʼ is referred to as the storage modulus and is a measure of the reversible energy W_rev stored during one oscillation period. The imaginary part E‘‘ or G‘‘ records the energy dissipated during the period W_irrev and is referred to as the loss module. The ratio of the loss module to the storage module gives the loss factor d = tan δ, which characterises the damping behaviour of the material according to Eqs. (14) and (15).

$E^{\prime }=E^{\ast }\cdot \cos \delta ={\frac {\sigma _{0}}{\varepsilon _{0}}}\cdot \cos \delta$

(10)

$G^{\prime }=G^{\ast }\cdot \cos \delta ={\frac {\tau _{0}}{\gamma _{0}}}\cdot \cos \delta$

(11)

$E^{\prime \prime }=E^{\ast }\cdot \sin \delta ={\frac {\sigma _{0}}{\varepsilon _{0}}}\cdot \sin \delta$

(12)

$G^{\prime \prime }=G^{\ast }\cdot \sin \delta ={\frac {\tau _{0}}{\gamma _{0}}}\cdot \sin \delta$

(13)

$\tan \delta ={\frac {E^{\prime \prime }}{E^{\prime }}}={\frac {1}{2\pi }}\cdot {\frac {W_{irrev}}{W_{rev}}}$

(14)

$\tan \delta ={\frac {G^{\prime \prime }}{G^{\prime }}}={\frac {1}{2\pi }}\cdot {\frac {W_{irrev}}{W_{rev}}}$

(15)

The forced vibration method is limited to frequencies below the resonance frequency of the test specimen. Commercial devices operate in the range from approx. 10^-2 Hz to 10² Hz, with the power consumption of the drive motor serving as the measured variable. The measure can be performed in both strain- and stress-controlled modes, which enables the determination of the complex modulus E* or G* and the complex compliance C* = 1 / E*. Axial and torsional testing devices can be used to simulate a wide range of different stress cases (tension, compression, bending, shear and torsion) using a suitable test specimen holder. This allows complex elasticity and shear moduli to be determined in a wide stiffness range from 10^-3 MPa to 10⁶ MPa. However, the biggest disadvantage of this method is its low sensitivity when measuring small damping values (tan δ < 0.01), i.e. very stiff or high-modulus materials. Due to their wide range of applications, forced vibration methods now play a dominant role in the dynamic-mechanical analysis of polymer materials.

Method with free damped vibrations

Free damped vibrations are actually only used in measurements with the torsion pendulum, whereby significantly lower measurement frequencies are possible here. If a test specimen is deflected from its resting position by a single impulse-like loading, it returns to its equilibrium state in free damped vibrations. The eigenfrequency of the vibration and the temporal decrease in the vibration amplitudes depend on the viscoelastic properties of the material and the test temperature (Fig. 3). Free damped vibrations are used at frequencies in the range from 0.1 to 10 Hz, whereby the investigation of materials with low damping of tan δ ≤ 0.1 is preferred.

Fig. 3:

Freely decaying damped vibration

Since investigations dependent on temperature cause a change in the eigenfrequency of the system due to the change in modulus, modulus–temperature curves are therefore usually measured at a sliding frequency. However, it is possible in principle to compensate for the frequency changes by varying the moment of inertia of the flywheel mass. The main advantages of the torsion pendulum are its simple design and measurement acquisition, as well as its high sensitivity.

Resonance method

If forced vibrations are generated at a frequency whose wavelength reaches the size of the test specimen dimensions, resonance phenomena occur. If the test specimen is excited in the resonance range with a constant force amplitude, the amplitude of the deflection reaches a maximum, whereby the respective resonance frequency f_i and half-value width Δf_i are determined, which are related to the viscoelastic properties of the material under investigation. Forced resonance vibrations are preferably used in bending or tensile vibration tests to determine the complex modulus. Excitation can be servo-hydraulic, capacitive or electromagnetic, and the vibrations are usually measured contact-free using electromagnetic transducers.

Determining the glass transition temperature T_g

Figure 4 shows schematic modulus–temperature diagrams under tensile vibration loading for various types of plastics, which are also preferred for determining the glass transition temperature T_g . In amorphous plastics (Fig. 4a), the high mobility of the chains and chain segments usually results in a clearly defined transition range in which the glass temperature can be determined relatively easily. Due to the crystalline content, this transition is less pronounced depending on the degree of crystallinity (Fig. 4b). Due to the high degree of cross-linking in duromers, such as EP or UP resins, the glass transition is often difficult to detect in DMTA (Fig. 4c), whereas in elastomers, the glass temperature is easier to determine due to the much lower degree of cross-linking (Fig. 4d).

Fig. 4:	Schematic storage module temperature curves for (a) amorphous, (b) semi-crystalline thermoplastics, (c) thermosets and (d) elastomeric plastics
	∎	–	Glass state
	∎	–	Glass transition range
	∎	–	Rubber elasticity
	∎	–	Processing range

See also

• Dynamic-mechanical analysis (DMA) – Bend loading
• Dynamic-mechanical analysis (DMA) – Tensile stress
• Dynamic-mechanical analysis (DMA) – Torsional stress
• Elastic modulus
• Elastic Modulus – Examples and material values

References

[1]	Lüpke, T.: Fundamental Principles of Mechanical Behavior. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 87–100 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-8807-5; see AMK-Library under A 22)
[2]	ISO 6721-1 (2019-04): Plastics – Determination of Dynamic Mechanical Properties – Part 1: General Principles