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{{Language_sel|LANG=ger|ARTIKEL=Dauerschwingversuch}}
{{PSM_Infobox}}
Vibration test, continuous vibration test or cyclic loading test
__FORCETOC__

==Thermal failure==

The work to be applied per unit volume during a sinusoidal [[Stress|stress]] in a fatigue test consists of two components: the elastically stored work ''W''', which is recovered as mechanical work during unloading, and the loss work ''W'''', which is converted into heat.

{|
|-
|width="20px"|
|width="500px" | <math>W\,=\,W'+W'' </math>
|}

The loss work, relative to 1 cycle, is calculated as

{|
|-
|width="20px"|
|width="500px" | <math>W''\,=\,\sigma_a\,\varepsilon_a\,\pi\,\sin\delta</math>
|}

or, per unit of time, as

{|
|-
|width="20px"|
|width="500px" | <math>W''\,=\,\sigma_a\,\varepsilon_a\,N\,\pi\,\sin\delta</math>
|}

The loss angle ''δ'' is generally less than 15° for [[Polymer|polymers]]. Therefore, sin ''δ'' can be set equal to the loss factor ''d'' = tan ''δ'', which is also temperature-dependent. For the stress-controlled fatigue test (''σ''m = const.), the following then applies:

{|
|-
|width="20px"|
|width="500px" | <math>W''\,=\,\sigma_a^2\,\varepsilon_a\,N\,E'\,\pi\,\tan\delta</math>
|}

and for the strain-controlled fatigue test (''ε''m = const.), the following applies:

{|
|-
|width="20px"|
|width="500px" | <math>W''\,=\,\varepsilon_a^2\,\varepsilon_a\,N\,E'\,\pi\,\tan\delta</math>
|}

with

{|
|-
|''E'''
|width="15px" |
|real part of the [[Elastic Modulus|modulus of elasticity]] of the fatigue test
|}

The test [[Specimen|specimen]] will then fail thermally under oscillating [[Stress|stress]] when the point under the highest stress reaches the softening temperature. This is the case when the heat ''W'''' generated by the internal damping of the material is greater than the amount of heat ''Q'' dissipated.

{|
|-
|width="20px"|
|width="500px" | <math>Q\,=\,\alpha\,f\,(\vartheta-\vartheta_1)</math>
|}

{|
|- valign="top"
|width="30px" rowspan="5"|with:
|width="30px"|''α''
|width="30px"|–
|width="540px"|heat transfer coefficient
|-
|width="30px"|''f''
|width="30px"|–
|width="540px"|shape factor surface/cross-section
|-
|width="30px"|''&thetasym;''
|width="30px"|–
|width="540px"|test specimen temperature
|-
|width="30px"|''&thetasym;''1
|width="30px"|–
|width="540px"|ambient temperature
|-
|width="30px"|''k''
|width="30px"|–
|width="540px"|correction factor internal temperature
|}

This results in the following conditions for thermal failure with ''W'''' > ''Q'' for ''σ''m = const.:

{|
|-
|width="20px"|
|width="500px" | <math>\frac{k^2\,\sigma_a^2\,N}{a\,f}\,>\,\frac{E'\,(\vartheta-\vartheta_1)}{\tan\delta}</math>
|}

and for ''ε''m = const.

{|
|-
|width="20px"|
|width="500px" | <math>\frac{k^2\,\varepsilon_a^2\,N}{a\,f}\,>\,\frac{\vartheta-\vartheta_1}{E'\,\tan\delta}</math>
|}

The factor ''k'' takes into account the temperature in the test specimen cross-section.

{|
|- valign="top"
|width="60px"|''k'' = 1:
|width="540px"|temperature is the same across the entire cross-section
|-
|width="60px"|''k'' = 0.5:
|width="540px"| good heat conduction
|-
|width="60px"|''k'' > 1:
|width="540px"|heat accumulation
|}

The heating behaviour varies greatly between individual [[Plastics|plastics]], as shown in '''Fig. 1''' below.

[[File:dsv_term1.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Heating in continuous vibration testing using polycarbonate ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PC) and polyamide ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PA) as examples [1]
|}

The specified relationships can be used to calculate the magnitude of the [[Stress|stress]] at which no additional heating occurs or to determine the equilibrium temperature that occurs at a given stress magnitude.

If the heat transfer coefficient ''α'' and the factor ''k'' cannot be derived from empirical values, ''k''²/''α'' is determined experimentally in a continuous vibration test with a constant load cycle number ''N'', constant form factor ''f'' and constant ambient temperature.

==Influences==

In [[Plastics|plastics]], the WÖHLER curve (S–N curve) shifts to lower load limits due to notches, weld true, increased frequencies, larger cross-sections and medial stresses. Increased load-bearing capacity can be expected with increasing molecular orientation, in composites with increased [[Fibre Orientation|fibre orientation]], with favourable heat dissipation and during load breaks ('''Fig. 2''').

[[File:Continuous Vibration Test-2.jpg|600px]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Hysteresis and factors influencing the S–N curve [2]
|}

In general, the fatigue strength of [[Plastics|plastics]] is only set at 20 % to 30 % of the [[Tensile Strength|tensile strength]], whereas for structural steels this value is 60 % to 80 %.

When using the test method described to determine the [[Fatigue Strength|fatigue strength]], the deformation behaviour of the materials to be tested must be taken into account. In [[Plastics|plastics]], the deformation behaviour is at least time-dependent, i.e. the [[Stress|stress]] applied does not immediately produce a corresponding deformation, but the [[Deformation|deformation]] that occurs is delayed. Under cyclic loading, this results in a hysteresis loop in the [[Tensile Test|stress–strain diagram]] (hysteresis test method).

This means that the deformation energy stored by the test [[Specimen|specimen]] is not completely recovered during the return deformation, so additional energy must be applied for the return deformation. A certain amount of deformation work is absorbed during each load cycle. This is proportional to the area enclosed by the hysteresis loop and is converted into heat. The low [[Thermal Conductivity|thermal conductivity]] of plastics plays a significant role here. Continuous vibration tests on plastics therefore lead to heating of the test [[Specimen|specimens]], which causes a decrease in the [[Elastic Modulus|modulus of elasticity]] and an increase in [[Creep Plastics|creep]] and [[Relaxation Plastics|relaxation effects]].

In order to ensure constant test conditions, the test parameters must therefore be monitored and deviations adjusted. In addition, depending on the type, [[Stress|stress]] and cooling options, the frequency for [[Plastics|plastics]] is limited to 500 to 3,000 load cycles per minute (50 Hz).

==Regulation==

The control of continuous vibration tests is complicated in that, on the one hand, the amplitude (stress or strain variation) must be controlled, while on the other hand, in the case of [[Plastics|plastics]], the mean stress decreases or the mean strain increases as a result of [[Relaxation Plastics|relaxation]] or [[Creep Plastics|retardation]]. The control of the amplitude therefore corresponds to control loop 1 and that of the respective mean value to control loop 2. Since the modulus of elasticity or compliance does not remain constant due to material damage, but serves as a basic setting parameter, the control behaviour may also change. In modern [[Material Testing Machine|testing systems]], a third control loop is therefore implemented to monitor the [[Stiffness|stiffness]] of the test specimen (see: [[Specimen Compliance|specimen compliance]]) and continuously adjust the control parameters in the event of changes. Regardless of the display of the signal behaviour on the monitor of the connected computer, monitoring of the analogue or incremental output signal using an oscilloscope should not be dispensed with, as signal processing such as filtering or smoothing can lead to a changed signal curve.

The following control variables can be set for oscillating loads:

* proportional amplification ''P'
* integral amplification ''I''
* derivative amplification ''D''

The P-part in particular has a major influence on the stability of the control system:

{|
|-
|width="20px"|
|width="500px" | <math>A_1-A_0\,=\,\left[1-\frac{A_f}{100}\right]P</math>
|}

{|
|- valign="top"
|width="30px" rowspan="4"|with:
|width="30px"|''A''0
|width="30px"|–
|width="540px"|initial amplitude
|-
|width="30px"|''A''1
|width="30px"|–
|width="540px"|new amplitude
|-
|width="30px"|''A''f
|width="30px"|–
|width="540px"|control deviation of the amplitude
|-
|width="30px"|''P''
|width="30px"|–
|width="540px"|proportional amplification
|}

The quality of the control depends largely on the time response of the control loop. The difference between the setpoint and actual signal (control deviation) is amplified, integrated and differentiated depending on the PID setting. Too high a P part always leads to control loop instability.

==Example==

In the case of [[Bend Loading|bend loading]], heat is generated at the point of greatest stress, i.e. the [[Surface|surface]], while heat derivation is also easier. In addition, the stresses relax more quickly there, so that higher-frequency dynamic loads are possible overall. If stresses are reduced at the [[Surface|surface]] or if [[Crack|cracks]] even occur, the cross-section and thus the moment of resistance or the stresses will decrease with the 2nd power. Due to the reduction in stress, the flexural fatigue test (strain-controlled) is then continued at a significantly lower flexural stress level. Flexural fatigue tests therefore often result in higher, fictitious strengths at higher load cycles than tensile fatigue tests, as the level of stress remains constant in stress-controlled tests.

The following '''Fig. 3''' shows the results of the vibration test on polymethyl methacrylate ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PMMA) under [[Bend Loading|bend loading]] in the alternating range and constant mean strain.

[[File:Continuous Vibration Test-3.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |Bending vibration test on PMMA; stress amplitude (a), temperature at the test specimen surface (b) [3]
|}

When εm and ''ε''a = const., the stress amplitude at the start of the test decreases with increasing number of load cycles. The same applies to the [[Elastic Modulus|modulus of elasticity]]. At the same time, the level of the stress amplitude at the start of the test causes an increase in the test specimen temperature, which can lead to thermal failure.

In the example shown, this occurs at stress amplitudes of 45 MPa and 54 MPa, as test specimen temperatures of approx. 90 °C are reached. At a stress amplitude of 43 MPa, the heat generated by internal damping and the heat dissipated are in equilibrium, resulting in fatigue failure.

The following '''Fig. 4''' shows the results of continuous vibration tests in the tensile threshold range on glass fibre-reinforced polyamide ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PA/GF). The load cycle frequency was 7 Hz, the stress amplitude 36 MPa and the mean stress 38 MPa.

[[File:Continuous Vibration Test-4.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px" |Continuous vibration test in the tensile stress range on PA/GF; stress amplitude (a), temperature on the test specimen surface (b) [3]
|}

==See also==

* [[Fatigue]]
* [[Test Specimen for Fatigue Tests|Test specimen for fatigue tests]]
* [[Vibration Strength|Vibration strength]]
* [[Vibration-induced Creep Fracture|Vibration-induced creep fracture]]
* [[Vibration Fracture|Vibration fracture]]

'''References'''

{|
|-valign="top"
|[1]
|Oberbach, K.: Das Verhalten von Kunststoffen bei kurzzeitiger und langzeitiger Beanspruchung, Kennwerte und Kennfunktionen. Materialwissenschaft und Werkstofftechnik 2 (2004) 65, pp. 281–291 DOI: https://doi.org/10.1002/mawe.19710020602
|-valign="top"
|[2]
|Oberbach, K.: Schwingfestigkeit von Thermoplasten – ein Bemessungskennwert ?. Kunststoffe 77 (1987) 4 pp. 409–414, see [https://www.dnb.de/EN/Home/home_node.html Deutsche Nationalbibliothek]
|-valign="top"
|[3]
|Dallner, C., [[Ehrenstein, Gottfried W.|Ehrenstein, G. W.]]: Thermische Einsatzgrenzen von Kunststoffen, Part II: Dynamisch-Mechanische Analyse unter Last. Zeitschrift Kunststofftechnik 2 (2006) 4 pp. 1–33 [https://res.cloudinary.com/sternwald-systems/raw/upload/v1/hugoprd/ARTIKEL_ATTACH/0023F281_3E3D467AA9D5/e8bf11b9b644f4a867234a6ac6e8266f6122815e/WAK_2006_04_Thermische-Einsatzgrenzen-von-Kunststoffen.pdf Download als pdf]
|}

'''Further References'''

* Baur, E., Brinkmann, S., Schmachtenberg, E.: Saechtling Kunststoff-Taschenbuch. Carl Hanser, Munich Vienna (2008) 30th Edition (ISBN 978-3-446-40352-9)
* Oberbach, K.: Das Verhalten von Kunststoffen bei kurzzeitiger und langzeitiger Beanspruchung, Kennwerte und Kennfunktionen. Materialwissenschaft und Werkstofftechnik 2 (2004) 65, pp. 281–291 DOI: https://doi.org/10.1002/mawe.19710020602
* Hellrich, W., Harsch, G., Haenle, S.: Werkstoff-Führer Kunststoffe, Eigenschaften – Prüfungen – Kennwerte. Carl Hanser, Munich Vienna (2004) 9th Edition (ISBN 3-446-22559-5)
* Becker, G. W., Braun, D., Carlowitz, B.: Die Kunststoffe. Chemie, Physik, Technologie, Kunststoff-Handbuch Volume 1. Carl Hanser, Munich Vienna (1990) 1st Edition (ISBN 978-3-4461-4416-1)
* Ehrenstein, G. W.: Faserverbund-Kunststoffe, Werkstoffe – Verarbeitung – Eigenschaften. Carl Hanser, Munich Vienna (2006), 2nd Edition (ISBN 978-3-446-22716-3; see [[AMK-Büchersammlung|AMK-Library]] under G 6-2)
* Bargel, H.-J., Schulze, G.: Werkstoffkunde. Springer Berlin Heidelberg 11th Edition (2012) (ISBN 978-3-642-17716-3; see AMK-Library under L 43)
* [[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]] (Eds.): Kunststoffprüfung. Carl Hanser, Munich (2025) 4th Edition, p. 165 and p. 170 (ISBN 978-3-446-44718-9; E-Book: ISBN 978-3-446-48105-3; see [[AMK-Büchersammlung|AMK-Library]] under A 23)
* Erhard, G.: Konstruieren mit Kunststoffen. Carl Hanser, Munich (2008) 4th Edition, (ISBN 978-3-446-41646-8; see [[AMK-Büchersammlung|AMK-Library]] under G 59)
* Renz, R., [[Altstädt, Volker|Altstädt, V.]], Ehrenstein, G. W.: Schwingfestigkeitsverhalten von faserverstärkten Kunststoffen (SMC); Faserverbundwerkstoffe. Volume III, Springer, Berlin (1986), pp. 441–518 (ISBN 978-3-642-82624-5)
* Oberbach, K.: In: Henkhaus, R.: Schwingfestigkeit von Kunststoffen. DVM, 1. Sitzung des AK „Polymerwerkstoffe“ Frankfurt/M., 21 and 22 October 1986, Proceedings pp. 25–40 (see [[AMK-Büchersammlung|AMK-Library]] under C 15)

[[Category:Fatigue]]

Continuous Vibration Test - Revision history