View source for Dynamic-mechanical Analysis (DMA) – Tensile Stress

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<span style="font-size:1.2em;font-weight:bold;">Dynamic-mechanical analysis (DMA) – Tensile stress</span>
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==General information==

In dynamic-mechanical analysis under tensile stress, the [[Specimen|test specimen]] used is subjected to periodically alternating [[Stress|stress]], whereby the characterisation of the time dependence of the [[Material & Werkstoff|material]] behaviour is possible by varying the frequency of the forced vibration (DMA). If a temperature control chamber is also applied to the measuring system, the temperature dependence of the [[Plastics|plastics]] in question can also be recorded and displayed at a constant frequency (DMTA).

==Performing DMA under tensile stress==

[[Elastic Modulus#Dynamic-mechanical analysis (DMA)|DMA or DMTA]] under tensile stress is one of the dynamic-mechanical testing methods with [[Dynamic-mechanical Analysis (DMA) – General Principles#Method with force-induced vibrations|forced vibrations]], which is used to characterise [[Viscoelastic Material Behaviour|viscoelastic properties]] and the [[Glass Transition Temperature|glass transition temperature]]. For this purpose, the [[Specimen|test specimen]] is subjected to a sinusoidal mechanical stress of constant frequency and constant amplitude. In the case of [[Linear-viscoelastic Behaviour|linear-viscoelastic]] [[Material & Werkstoff|material]] behaviour, the temporal changes in stress and strain in the steady state have the same frequency but different phase positions according to '''Eqs. (1)''' and '''(2)''' ('''Fig. 1''').

[[File:DMA Torsion-Fig1.jpg|500px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 1''': 
|width="600px"|Temporal change in stress and strain during dynamic mechanical analysis using forced vibrations
|}


{| 
|- valign="top"
|width="350px"|<math>\varepsilon \left ( t \right )=\varepsilon_{0}\cdot \sin \omega t</math>
|width="50px"|(1)
|}

{| 
|- valign="top"
|width="350px"|<math>\sigma \left ( t \right )=\sigma_{0}\cdot \sin \left ( \omega t+\delta \right )</math>
|width="50px"|(2)
|}

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

{| 
|- valign="top"
|width="350px"|<math>E^{\ast }=E^{\prime}+iE^{\prime\prime}</math>
|width="50px"|(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 induced forced stress and strain.

[[File:DMA_Zugbeanspruchung-2.jpg|400px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 2''': 
|width="600px"|Representation of the '''E*''' module 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)'''.

{| 
|- valign="top"
|width="350px"|<math>\left | E^{\ast } \right |=\frac{\sigma_{0}}{\varepsilon_{0}}</math>
|width="50px"|(4)
|}

Using simple trigonometric relationships, the energy is then divided into the real part ''Eʼ'' and the imaginary part ''Eˮ'', which is done using '''Eqs. (5)''' and '''(6)'''. The real part ''Eʼ'' is referred to as the storage modulus and is a measure of the reversible energy ''W''<sub>rev</sub> stored during one vibration period. The imaginary part ''Eˮ'' records the energy dissipated in the respective period ''W''<sub>irrev</sub> 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 & Werkstoff|material]] according to '''Eq. (7)'''. The value ''δ'' is the so-called phase angle, which can take values between 0 and π/2.

{| 
|- valign="top"
|width="350px"|<math>E^{\prime}=E^{\ast }\cdot \cos \delta=\frac{\sigma_{0}}{\varepsilon_{0}}\cdot \cos \delta</math>
|width="50px"|(5)
|}

{| 
|- valign="top"
|width="350px"|<math>E^{\prime\prime}=E^{\ast }\cdot \sin \delta=\frac{\sigma_{0}}{\varepsilon_{0}}\cdot \sin \delta</math>
|width="50px"|(6)
|}

{| 
|- valign="top"
|width="350px"|<math>\tan \delta=\frac{E^{\prime\prime}}{E^{\prime}}=\frac{1}{2\pi }\cdot \frac{W_{irrev}}{W_{rev}}</math>
|width="50px"|(7)
|}

The forced vibration method is limited to frequencies below the resonance frequency of the test specimen. Commercially available devices operate in the range from approx. 10<sup>-2</sup> Hz to 10<sup>2</sup> Hz, with the power consumption of the [[Drives Materials Testing Machines|drive motor]] serving as the [[Measured Variable|measured variable]]. The [[Measure|measurement]] can be strain- or stress-controlled, which allows the complex modulus ''E*'' and the complex compliance with ''C*'' = 1 / ''E*'' to be determined. These devices allow complex [[Elastic Modulus|elastic moduli]] to be determined in a wide [[Stiffness|stiffness]] range from 10<sup>-3</sup> MPa to 10<sup>6</sup> 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 [[Material & Werkstoff|materials]]. Due to their wide range of applications, however, forced vibration methods now play a dominant role in the dynamic-mechanical analysis of [[Polymer|polymer]] [[Material & Werkstoff|materials]].

==Device systems for performing DMA and DMTA==

[[Servo-hydraulic Testing Machine|Servo-hydraulic universal testing machines]] for high forces or tabletop testing systems (stand-alone systems) for lower test forces can be used for dynamic mechanical analysis or spectroscopy using tensile stress ('''Fig. 3'''). What all methods have in common is that the [[Deformation|deformation]] of the [[Specimen|test specimen]] is very small and should not exceed the [[Linear-viscoelastic Behaviour|linear-viscoelastic range]]. As a result of these small deformations, high test frequencies of up to 200 Hz can be achieved with [[Dynamic-mechanical Analysis (DMA) – General Principles|DMA or DMTA]] in the temperature range from approx. –180 °C to 400 °C with mechanical excitation [1, 2]. For [[Polymer Testing|polymer testing]] using DMA or DMTA under [[Tensile Test|tensile stress]] [3], small hydraulic, pneumatic or electrodynamic testing machines are mainly used to generate sinusoidal strains or stresses, whereby the tests can be carried out in a force- or strain-controlled manner (see: [[Tensile Test Control|tensile test control]]).

[[File:DMA Tension-Fig3.jpg]]
{| 
|- valign="top"
|width="50px"|'''Fig. 3''': 
|width="600px"|DMTA system (a) Schematic layout, (b) [https://www.mts.com/de MTS 858 servo-hydraulic system, Berlin], and (c) ZWICK LTS 5 kN system, [https://www.zwickroell.com/de/ZwickRoell GmbH & Co. KG, Ulm]
|}

Depending on the test force and equipment, there is also a wide range of commercial testing systems available for table testing systems (see also: [[Load Framework|load frames]]), which are usually equipped with additional devices such as path sensors and temperature control chambers ('''Fig. 4''').

[[File:DMA_Zugbeanspruchung-4.jpg]]
{| 
|- valign="top"
|width="50px"|'''Fig. 4''': 
|width="600px"|DMTA systems (a) [https://www.mt.com/de/de/home.html Mettler Toledo, Greifensee, Switzerland] (b) MPAS from [https://analyzing-testing.netzsch.com/de Gabo Qualimeter GmbH], Ahlden
|}

The test system then applies a constant sinusoidal strain or stress amplitude with a defined and constant frequency to the test specimen, whereby in [[Polymer Testing|polymer testing]], the tensile threshold range is preferred due to the geometry of the test specimen. The amplitude of the stress or strain, as well as the middle stress or strain and the temperature, are kept at a constant level by means of PID control in order to compensate for [[Creep Plastics|creep]] or [[Relaxation Plastics|relaxation effects]] during the test.

==See also==

* [[Dynamic-mechanical Analysis (DMA) – General Principles|Dynamic-mechanical analysis (DMA) – General principles]]
* [[Dynamic-mechanical Analysis (DMA) – Torsional Stress|Dynamic-mechanical analysis (DMA) – Torsional stress]]
* [[Dynamic-mechanical Analysis (DMA) – Bend Loading|Dynamic-mechanical analysis (DMA) – Bend loading]]
* [[Elastic Modulus|Elastic modulus]]
* [[Elastic Modulus – Examples and Material Values|Elastic modulus – Examples and material values]]
* [[Shear Modulus|Shear modulus]]

==References==

{|
|-valign="top"
|[1]
|Lüpke, T.: Dynamic-Mechanical Analysis (DMA). In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 88–96 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-8807-5; see [[AMK-Library]] under A 22)
|-valign="top"
|[2]
|ISO 6721-1 (2019-04): Plastics – Determination of Dynamic Mechanical Properties – Part 1: General Principles 
|-valign="top"
|[3]
|ISO 6721-4 (2019-05): Plastics – Determination of Dynamic Mechanical Properties – Part 4: Tensile Vibration – Non-resonance Method 
|}

[[Category:Deformation]]
[[Category:Thermoanalytical Methods]]