Dynamic-mechanical Analysis (DMA) – Bend Loading

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Dynamic-mechanical analysis (DMA) – Bend loading

Fundamentals

In dynamic-mechanical analysis under bend loading, the test specimen is subjected to periodically alternate loading, whereby the time dependence of the material behaviour can be characterised by varying the frequency of the forced resonant vibration (DMA). If a temperature control chamber is also used, the temperature dependence of the properties being investigated for the plastics in question can be recorded and displayed at the same time (DMTA).

Performing DMA under bend loading

DMA or DMTA under flexural loading is a dynamic-mechanical testing method using resonant vibrations, which is used to characterise viscoelastic properties and glass transition temperature.

For this purpose, the test specimen is subjected to a sinusoidal mechanical load of variable frequency, with an increase in amplitude occurring near the resonances. In the case of linear-viscoelastic material behaviour, the temporal changes in stress and strain have the same frequency but different phase positions according to Eqs. (1) and (2) (Fig. 1).

Fig. 1:

Temporal change in bend loading and peripheral fibre strain during dynamic-mechanical analysis using forced vibrations

$\varepsilon _{f}\left(t\right)=\varepsilon _{f0}\cdot \sin \omega t$

(1)

$\sigma _{f}\left(t\right)=\sigma _{f0}\cdot \sin \left(\omega t+\delta \right)$

(2)

As a result of the phase shift δ between stress (tension) and deformation (strain or shear), the modulus as a complex quantity E_f* according to Eq. (3) is valid for describing the stress–strain relationship.

$E_{f}^{\ast }=E_{f}^{\prime }+iE_{f}^{\prime \prime }$

(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 the applied forced stress and strain.

Fig. 2:

Representation of the flexural modulus Ef* 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 Eq. (4).

$\left|E_{f}^{\ast }\right|={\frac {\sigma _{f0}}{\varepsilon _{f0}}}$

(4)

Using simple trigonometric relationships, a division into the real part E_fʼ and the imaginary part E_fʼʼ is then made using Eqs. (5) and (6). The real part E_fʼ is referred to as the storage modulus and is a measure of the reversible energy W_rev stored during an oscillation period. The imaginary part E_fʼʼ records the energy W_irrev dissipated in the respective period and is referred to as 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). The value δ is the so-called phase angle, which can take values between 0 and π/2 [1].

$E_{f}^{\prime }=E_{f}^{\ast }\cdot \cos \delta ={\frac {\sigma _{f0}}{\varepsilon _{f0}}}\cdot \cos \delta$

(5)

$E_{f}^{\prime \prime }=E_{f}^{\ast }\cdot \sin \delta ={\frac {\sigma _{f0}}{\varepsilon _{f0}}}\cdot \sin \delta$

(6)

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

(7)

If a test specimen is excited in a flexural vibration test according to ISO 6721-3 [2] with forced resonance vibrations, typical test specimen resonances corresponding to the wavelength occur, depending on the clamping used and the test specimen geometry. If the excitation takes place in the resonance range with a constant force amplitude, the amplitude of the deflection passes through a maximum (Fig. 3), whereby the respective resonance frequency f_i and its half-value width Δf_i are related to the viscoelastic properties of the material under investigation.

Fig. 3:

Schematic resonance curve of a viscoelastic material

Prismatic rods are used as test specimens, which are either clamped on one side (method A) or suspended from textile threads in the vibration nodes (method B) [2] (Fig. 4). The flexural vibrations are excited and recorded without contact using inductive or capacitive transducers, which are connected to thin metal plates glued to the surface of the plastic test specimen.

Fig. 4:

Test setup for determining viscoelastic properties with forced vibrations in the resonance range (a) with fixed clamping and (b) free suspension of the test specimen

The differential equation Eq. (8) of the bending vibration can be solved using the BERNOULLI approach according to Eq. (9), resulting in the spatial and temporal functions of the vibration in Eqs. (10) and (11).

$E_{f}l_{y}\cdot {\frac {\partial ^{4}w}{\partial t^{4}}}-F{\frac {\partial ^{2}w}{\partial x^{2}}}+\rho \cdot A{\frac {\partial ^{2}w}{\partial t^{2}}}+k{\frac {\partial w}{\partial t}}=p\left(x,t\right)$

(8)

$w(x,t)=X(x)T(t)\!$

(9)

$X^{(4)}(x)-{\frac {F}{E_{f}l_{y}}}){X}^{\prime \prime }(x)-{\frac {\beta _{i}^{4}}{L^{4}}}X(x)=0$

(10)

${\ddot {T}}(t)+{\frac {k}{\rho A}}{\dot {T}}(t)+\omega ^{2}T(t)=p(x,t)$

(11)

with:	Ef ly	–	bending stiffness
	L	–	test specimen length
	A	–	test specimen cross-section
	F	–	test force
	ρ	–	test specimen density
	k	–	test specimen damping
	w	–	test specimen deflection
	x	–	amplitude
	ω	–	circular frequency
	βi	–	i-th eigenvalue

Solving the eigenvalue problem according to Eq. (10) yields the eigenvalue equation of the differential equation for the resonance frequencies according to Eq. (12) for the different mounting cases of the test specimen β_i², which are shown in Table 1.

${\frac {\rho A}{E_{f}l_{y}}}\cdot \omega ^{2}={\frac {\beta _{i}^{4}}{L^{4}}}$

(12)

Table 1: Eigenvalues for determining the storage module

Order number i	clamped on one side (method A)	freely suspended (method B)
1	3.52	22.4
2	22.0	61.7
>2	(i - 1/2)·π2	(i + 1/2)·π2

With the aid of a frequency generator, the excitation frequency f can be continuously varied within a range of approximately 10¹ Hz to 10³ Hz. When scanning through this frequency range, several maxima of the vibration amplitude are registered at the detector, which correspond to the resonance points of different orders i (i = 1, 2, 3, ...), whereby the resonance amplitude becomes smaller and smaller with increasing order number as the damping increases. The real part (storage modulus of the complex elastic modulus) can be determined according to Eq. (13) from the resonance frequency f_i of the i-th resonance point, the density ρ of the material under investigation and the dimensions (free length L and thickness h).

$E^{\prime }=48\pi ^{2}\cdot \rho {\frac {L^{4}}{h^{2}}}\left[{\frac {f_{i}}{\beta _{i}^{2}}}\right]^{2}$

(13)

From the half-value width of the resonance curve Δf_i and the resonance frequency f_{i</dub>, the loss factor tan δ can be calculated as a further parameter according to Eq. (14) and the loss modulus according to Eq. (15), provided that the loss factor of the material is ≤ 0.1.}

$\tan \delta ={\frac {\Delta f_{i}}{f_{i}}}$

(14)

$E^{\prime \prime }=E^{\prime }\cdot \tan \delta$

(15)

For materials with low intrinsic damping (tan δ < 0.01), the analysis of freely decaying vibrations after switching off the exciter at resonance frequency should be used. This results in a decrease in the amplitudes of successive vibrations, from which the loss factor can also be determined using the logarithmic decrement Λ (see Eq. (16)).

$\tan \delta ={\frac {\Lambda }{\pi }}={\frac {1}{\pi }}\ln {\frac {A_{n-1}}{A_{n}}}$

(16)

A decisive disadvantage of the method is that often only relatively few resonance points are available for evaluation and that the position of the resonance points can only be influenced by changing the dimensions of the test specimen. Temperature-dependent measurements cause changes in the resonance frequencies, making it impossible to determine characteristic values at a constant frequency.

Device systems for performing DMA

In dynamic-mechanical analysis or spectroscopy using bend loading, tabletop testing systems (stand-alone systems) are usually used for smaller test forces. What all devices have in common is that the deformation of the test specimen is very small and should not exceed the linear-viscoelastic range. The frequency generator can be used to generate an automatic frequency sweep (Fig. 5), which excites the test specimen in the measuring stand to vibrate. The amplitude increases that occur at defined frequencies can be recorded and evaluated by a vibration analyser, which also provides information on damping and vibration modes.

Fig. 5:

Schematic diagram of a DMA system from Brüel & Kjaer, Denmark

See also

• Dynamic-mechanical analysis (DMA) – General principles
• 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 6721-3 (2021-02): Plastics – Determination of Dynamic Mechanical Properties – Part 3: Flexural Vibration – Resonance-Curve Method