Dynamic-mechanical Analysis (DMA) – Torsional Stress

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Dynamic-mechanical Analysis (DMA) – Torsional stress

Fundamentals

Dynamic-mechanical analysis (DMA) or dynamic-mechanical-thermal analysis (DMTA) using torsional loading can basically be carried out using two test methods:

• forced vibrations and
• free damped vibrations.

In dynamic-mechanical analysis, a test specimen with a specified geometry is subjected to a periodically alternating stress. By varying the frequency, it is possible to characterise the time dependence of the material behaviour at a constant temperature. When these tests are carried out in a temperature-controlled chamber, the test method is referred to as DMTA and the temperature dependence of the materials under investigation is characterised. 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 comparatively 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 [1−3].

Methods with forced oscillations

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

Fig. 1:

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

The value δ is the phase angle, which lies in the range between 0 and π/2. In the case of shear stress, Eqs. (1) and (2) apply

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

(1)

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

(2)

As a result of the phase shift δ between loading (stress) and deformation (shear), the modulus must be introduced as a complex quantity G* according to Eq. (3) in order to describe the stress – shear relationship.

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

(3)

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 stress and strain.

Fig. 2:

Graphical representation of the module G* in the complex number plane

The absolute value of the respective module is calculated from the ratio of the initial stress to the initial deformation according to Eq. (4).

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

(4)

Using simple trigonometric relationships, it is possible to divide the energy into a real part G‘ and an imaginary part G‘‘, which is done using Eqs. (5) and (6). The real part G‘ 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 G‘‘ corresponds to the energy W_irrev dissipated during the period and is called the loss modulus. The ratio of the loss modulus to the storage modulus gives the loss factor d = tan δ, which describes the damping behaviour of the material according to Eq. (7).

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

(5)

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

(6)

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

(7)

The forced torsional 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. If, for example, the glass transition (see: glass transition temperature) is reached as a result of an increase in the test temperature, the damping (Fig. 3) increases significantly and the power of the motor must be increased to maintain the test frequency and stress or deformation amplitude.

Fig. 3:

Storage modulus G‘ and loss factor tan δ of polypropylene (abbreviation: PP)

The measurement can be performed in both strain-controlled and stress-controlled modes, which allows the determination of the complex modulus G* and the complex compliance C* = 1 / G*. This enables the determination of complex shear moduli in a wide stiffness range from approx. 10^-3 MPa to 10⁶ MPa. However, the biggest disadvantage of this method is its low sensitivity when measuring plastics with very low damping (tan δ < 0.01), i.e. very stiff or high-modulus materials (see: elastic modulus – Examples and material value). Due to their wide range of applications, test methods with forced vibrations now play a dominant role in the dynamic-mechanical analysis of polymeric materials.

In rare cases, special torsion or hybrid testing machines for high forces are used in dynamic-mechanical analysis or spectroscopy using torsional stress. In most applications, however, table testing systems (stand-alone systems) are used for smaller test forces. What all methods have in common is that the deformation of the test specimen is very small and should not exceed the linear-viscoelastic range. As a result of these small deformations, high test frequencies of up to 100 Hz can be achieved with DMA or DMTA in the temperature range from approx. –180 °C to 400 °C with mechanical excitation.

In polymer testing using DMA or DMTA, the tests are normally performed in a force- or strain-controlled manner (see: dynamic-mechanical analysis (DMA) – tensile stress). The primary control loop is used to maintain a constant stress or deformation amplitude, while the second control loop monitors the constant mean stress or strain in order to compensate for stress relaxation or creep of the test specimen. In some cases, a third control loop is used to compensate for losses in stiffness during the measurement.

Fig. 4:

Schematic test setup (a) and commercially available table-top testing machine with temperature control chamber (b)

Depending on the test force and equipment, there is also a wide range of commercial testing systems available for table testing systems, which are usually equipped with additional devices such as strain sensors and temperature control chambers (Fig. 4).

Methods with free damped oscillations

Free damped oscillations 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 oscillations. The deflection should not exceed the range of linear-viscoelastic deformation. The eigenfrequency of the vibration and the temporal decrease in vibration amplitudes (damping) depend on the viscoelastic properties of the material under investigation and the test temperature (Fig. 5).

The free damped oscillations are used at frequencies in the range from 0.1 to 10 Hz, whereby the examination of materials with low damping of tan δ ≤ 0.1 is preferred. Since tests dependent on temperature cause a change in the natural frequency of the system due to the change in modulus, modulus – temperature curves are therefore usually measured at a sliding frequency, whereby compensation for the frequency changes is possible in principle by varying the moment of inertia of the oscillating mass.

Fig. 5:

Freely decaying damped vibration

The principle of free damped oscillations is applied in technical applications in the form of the torsion pendulum method and is standardised in ISO 6721-2 [3]. The basic structure of a torsion pendulum is shown schematically in Fig. 6. A test specimen, preferably prismatic in shape, is clamped firmly at one end, while the other end is connected to a flywheel mass that influences the moment of inertia and thus the eigenfrequency of the entire system. At the same time, however, this mass also causes expansion as a result of mechanical and thermal stress (Fig. 6a). To avoid these superimposed normal stresses in the longitudinal direction of the test specimen, a weight compensation can be used in the form of a compensation mass (Fig. 6b). An impulse-like initial deflection φ of the flywheel mass excites the test specimen to freely decaying torsional vibrations, as shown schematically in Fig. 6c.

Fig. 6:

Schematic representation of the torsion pendulum setup (a) without weight compensation, (b) with weight compensation, and (c) test arrangement with initial deflection φ and test specimen

The storage module G‘ can be determined from the eigenfrequency of the vibration according to Eq. (8) [3].

$G^{\prime }=4\pi l_{p}\cdot \left(f_{d}^{2}F_{d}-f_{0}^{2}\right)F_{g}$

(8)

Here, f_d is the eigenfrequency of the pendulum with the test specimen and f₀ is the eigenfrequency of the pendulum without the test specimen (when working without weight compensation, f₀ = 0). Other influencing factors to be taken into account are the moment of inertia I_p of the flywheel mass with clamping, a damping correction factor F_d and the geometry factor F_g. When using prismatic test specimens with a clamping length L, width b, thickness h and an h/b ratio ≤ 6, the geometric correction factor is F_c = 1 − 0.63 h/b, whereby G‘ is then calculated according to Eq. (9).

$G^{\prime }=12\pi ^{2}f_{d}^{2}l_{p}\cdot \left(1-\left({\frac {\Lambda }{2\pi }}\right)^{2}-\left({\frac {f_{0}}{f_{d}}}\right)^{2}\right){\frac {L}{bh^{3}F_{c}}}$

(9)

The logarithmic decrement Λ characterises the damping of the system. It is determined from the ratio of the amplitudes or elongations of successive vibrations according to Eqs. (10) or (11) (Fig. 7).

Fig. 7:

Schematic representation of the measurement of the decrement from (a) the elongation and (b) the amplitude

$\Lambda =\ln {\frac {A_{n-1}}{A_{n}}}=\ln {\frac {t_{n}}{t_{n-1}}}$

(10)

$\Lambda =2\ln {\frac {A_{n-1}}{A_{n}}}=2\ln {\frac {t_{n}}{t_{n-1}}}$

(11)

The loss modulus G‘‘ can be calculated using the logarithmic decrement according to Eq. (12).

$G^{\prime \prime }=4\pi f_{d}l_{p}\left(\Lambda -\Lambda _{0}\right)F_{g}$

(12)

If no weight compensation is used, the logarithmic decrement of the pendulum without test specimen Λ₀ is equal to 0. With weight compensation, low intrinsic damping of the pendulum results in Λ₀ << Λ for test specimens with a rectangular cross-section and a small h/b ratio, the loss modulus is given by Eq. (13).

$G^{\prime \prime }={\frac {2\pi f_{d}^{2}l_{p}\Lambda L}{bh^{3}F_{c}}}$

(13)

The loss factor tan δ can then be determined from the storage and loss module according to Eq. (14). The main advantages of the torsion pendulum are its simple design and measurement acquisition, as well as its high sensitivity.

$\tan \delta ={\frac {G^{\prime \prime }}{G^{\prime }}}$

(14)

See also

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

References

[1]	Lüpke, T.: Mechanical Spectroscopy. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 91–93 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	ISO 6721-1 (2019-04): Plastics – Determination of Dynamic Mechanical Properties – Part 1: General Principles
[3]	ISO 6721-2 (2019-04): Plastics – Determination of Dynamic Mechanical Properties – Part 2: Torsion-pendulum Method