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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Plastic-Hinge-Modell}} {{PSM_Infobox}} Plastic hinge model (Türangelmodell) __FORCETOC__ ==Fundamentals== The plastic hinge model is a widely used model for determining the critical crack opening displacement for three-point bending specimens (SENB-specimens) [1]. While under quasi-static loading the critical..."

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{{Language_sel|LANG=ger|ARTIKEL=Plastic-Hinge-Modell}}
{{PSM_Infobox}}
Plastic hinge model (Türangelmodell)
__FORCETOC__

==Fundamentals==

The plastic hinge model is a widely used model for determining the critical [[Extended CTOD Concept|crack opening displacement]] for [[SENB-Specimen|three-point bending specimens]] (SENB-specimens) [1].

While under [[Quasi-static Test Methods|quasi-static loading]] the critical [[Crack Opening|crack opening]] can be determined by optical methods or by [[Measure|measuring]] the notch expansion ''v'' and extrapolating to the [[Crack|crack tip]], under dynamic loading (see: [[Impact Loading Plastics|impact loading plastics]]) only an indirect determination via electronically measured deflection is possible. In [2], Zeislmair and Dahl compile numerous methods for determining crack opening values under static stress, most of which were developed empirically, and compare the results. A prerequisite for determining the [[Crack Opening|critical crack opening]] under dynamic loading is that a quasi-static stress state (see: [[Plane Stress and Strain State|plane stress and strain state]]) develops in the test [[Specimen|specimen]], the formation of which is ensured by compliance with the condition of '''Eq. (1)'''.

{|
|-
|width="20px"|
|width="500px" | <math> t_{B}\geq 2,3 \ ... \ 3 \tau </math>
|(1)
|}

mit
{|
|-
|''t''B
|width="15px" |
|[[ICIT – Experimental Conditions|fracture time]]
|-
|''τ''
|
|period of characteristic inertial oscillation (see also: [[Inertial Load|inertial load]])
|}

is controlled.

==Principle of the plastic hinge model==

Comparative studies of various models for [[Bend Loading|bend loading]] on a [[SENB-Specimen|SENB-specimen]] have led to the conclusion that the critical crack opening should be determined on the basis of the ‘plastic hinge’ model (door hinge model) [2−5]. In this model, the [[Crack|crack flanks]] open under increasing test load, like a hinge, around a point designated as the centre of rotation in front of the [[Crack|crack tip]], whereby only one half of the specimen was considered due to the symmetry of the test [[Specimen|specimen]] ('''Fig. 1''').

[[File:plastichingemodell1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Principle of the plastic hinge model for SENB- specimens (1-centre of rotation, 2 sharp [[Notch|notch]], 3-[[Support Distance|support]]) [6]
|}

The experimentally determined deflection ''f''max is composed of a part caused by the bending of the unnotched part ''f''B and a part caused by the [[Deformation|deformation]] in the area of the notch ''f''K according to '''Eq. (2)''',

{|
|-
|width="20px"|
|width="500px" | <math> f_{max}=f_{B}+f_{K} \! </math>
|(2)
|}

where the bending part is calculated using the relationship

{|
|-
|width="20px"|
|width="500px" | <math> f_{B}=\frac{F_{max} \ s^3}{4 \ E \ B \ W^3}</math>
|(3)
|}

mit
{|
|-
|''E''
|width="15px" |
|[[Elastic Modulus|modulus of elasticity]]
|-
|''s''
|
|[[Support Distance|support distance]]
|-
|''B''
|
|test specimen thickness
|-
|''W''
|
|specimen width
|}

is calculated.

==Determination of the critical crack opening==

For the [[SENB-Specimen|SENB-specimen]] stressed by bending, the following equation applies based on the plastic hinge model ('''Fig. 1''')

{|
|-
|width="20px"|
|width="500px" | <math> \delta_{k} = \frac{1}{n}(W-a)\frac{4 \ f_{K}}{s}</math>
|(4)
|}

This reduces the calculation of the critical crack opening (see: [[Extended CTOD Concept|extended CTOD concept]]) to the area at the notch tip by subtracting the part of the deflection of an unnotched test specimen from the maximum deflection ''f''max of the notched test specimen (according to '''Eq. (2)''').

As can be seen from the literature [2, 7‒9], the rotation factor ''n'' depends on the load and moves towards the [[Crack|crack tip]] as the load increases.

==The rotation factor ''n''==

[[File:plastichingemodell2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Dependence of the rotation factor n on the load for PP 1 a) (1 to 5 selected heat treatment conditions) and PP 2 b) (6 to 9 selected degrees of orientation) [7, 8]
|}

For [[CT-Specimen|CT-specimens]] loaded under [[Quasi-static Test Methods|quasi-static]] conditions, simultaneous registration of notch expansion ''v''c and load-point displacement ''v''L for various [[Polymer|polymer]] [[Material & Werkstoff|materials]] showed that the rotation factor assumes the limit value ''n'' = 4 at the moment of [[Fracture|fracture]] (see '''Fig. 2''') [6, 10].

==Influence of the a/W-ratio==

'''Figure 3''' shows the [[Extended CTOD Concept|critical crack opening]] ''δ''dk calculated for a polyamide material ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PA) with [[Deformation#Elastic deformation|elastic material behaviour]] according to '''Eq. (4)''' with a rotation factor ''n'' = 4 in comparison to ''δ''d and ''δ''dB as a function of the ''a''/''W''-ratio.

[[File:plastichingemodell3.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |Influence of the ''a''/''W''-ratio on the critical crack openings ''δ''d, ''δ''dB and ''δ''dK for unstable [[Crack Propagation|crack propagation]] in polyamide
|}

To calculate ''δ''d and ''δ''dB, ''f''K is replaced by ''f''max and ''f''B respectively in '''Eq. (4)'''. Taking into account the results presented in [1] for PE-HD+Hp and another PP material, PP 3, it can be seen that

# the bending component is dominant in determining critical crack opening values for small ''a''/''W'', with ''δ''dB becoming smaller as ''a''/''W'' increases. For polyamide, ''δ''dB is 75 % at ''a''/''W'' = 0.1 and only 25 % at ''a''/''W'' = 0.7,
# the proportion ''δ''dB for PP 3 at ''s''/''W'' = 7 becomes negligibly small for high ''a''/''W'',
# the crack openings ''δ''dK determined from the notch proportion ''f''K according to '''Eq. 4''' are independent of the ''a''/''W''-ratio.

The independence of ''δ''dK from the ''a''/''W''-ratio is shown in '''Fig. 4''' for selected [[Polymer|polymer]] [[Material & Werkstoff|materials]].

[[File:plastichingemodell4.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px" |Dependence of critical crack openings ''δ''dK on the ''a''/''W''-ratio for PP 3 (''v''H = 1.5 ms-1; ''s''/''W'' = 7), PE-HD+Bw (''v''H = 1.5 ms-1; ''s''/''W'' = 4), PVCC (''v''H = 1 ms-1; ''s''/''W'' = 4), PE-HD+Hp (''v''H = 1.5 ms-1; ''s''/''W'' = 4) and PA (''v''H = 1 ms-1; ''s''/''W'' = 4)
|}

The increase in ''δ''dK values apparent for ''a''/''W'' < 0.2 is due to the very high energy absorption of the test [[Specimen|specimens]] during the impact process for the lowest ''a''/''W''-ratios (see: [[Impact Loading Plastics|impact loading plastics]]), so that the empirical condition for controlling the energy absorption during the impact process (see: [[ICIT – Experimental Conditions|ICIT ‒ Experimental conditions]]), according to which the impact energy ''A''H specified by the pendulum chamber for the [[Fracture|fracture process]] must be greater than 3 times the deformation energy ''A''G consumed by the test [[Specimen|specimen]], cannot be fulfilled with sufficient certainty for all materials and for PVCC only from ''a''/''W'' ≤ 0.17, which is why the curve in '''Fig. 4''' is only marked with a dashed line.

The caption for '''Fig. 4''' clearly shows the [[Stress|stress conditions]] selected for the various polymer materials in terms of compliance with the experimental conditions of the [[Instrumented Charpy Impact Test|instrumented Charpy impact test]], which lead to different [[Deformation Rate|deformation rates]] at the [[Crack|crack tip]]. The crack opening velocity according to '''Eq. (5)'''

{|
|-
|width="20px"|
|width="500px" | <math> \frac{d\delta_{dk}}{dt}=\frac{\delta_{dk}}{t_{B}}</math>
|(5)
|}

can be used as a measure to describe them (see also: [[Levels of Knowledge in Fracture Mechanics|levels of knowledge in fracture mechanics]]).

The fracture time ''t''B is calculated according to '''Eq. (6)'''.

{|
|-
|width="20px"|
|width="500px" | <math> t_{B}=\frac{f_{max}}{v_{H}-\frac{1}{3}\Delta v}</math>
|(6)
|}

Δ''v'' refers to the change in pendulum hammer speed during the deformation process, which, according to '''Eq. (7)''', is expected to be greater the higher the deformation energy ''A''G of the [[Specimen|test specimen]] and the pendulum hammer speed ''v''H before the impact:

{|
|-
|width="20px"|
|width="500px" | <math> \Delta v = v_{H} \left ( v_{H}^2-\frac{2}{m_{H}}A_{G} \right ) ^{\frac{1}{2}}</math>
|(7)
|}

''m''H = mass of the pendulum hammer, determined by weighing in a horizontal position [11].

==Crack opening velocity of selected plastics==

'''Figure 5''' shows the crack opening velocities determined from '''Eq. (5)''' as a function of the ''a''/''W''-ratio for selected [[Plastics|plastics]]. With the exception of PP 3, the increases in d''δ''dK/d''t'' up to approximately ''a''/''W'' = 0.5 result primarily from the decrease in ''f''max with increasing ''a''/''W'', and the decrease in d''δ''dK/d''t'' for ''a''/''W'' > 0.5 results from the fact that fmax increases again with a simultaneous continuous reduction in Δ''v''.

[[File:plastichingemodell5.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 5''':
|width="600px" |Crack opening velocities d''δ''dK/d''t'' for various plastics
|}

The dependence of d''δ''dK/d''t'' (see: [[Levels of Knowledge in Fracture Mechanics|levels of knowledge in fracture mechanics]]) on the ''a''/''W'' ratio for PP 3 indicates that for ''s''/''W'' = 7, the [[Stress|stress]] on the material in the [[Crack|crack tip area]] is comparable to that of other [[Plastics|plastics]], whereby, in addition to the high fracture times (here, the influence of the test [[Specimen|specimens]] pulling through the [[Support Distance|supports]] must be taken into account), the low bending component ''δ''dB has an effect, meaning that the notch component at ''s''/''W'' = 7 is already of decisive importance even for small ''a''/''W'' values.

When the pulling of the [[Specimen|test specimens]] through the abutments is taken into account (see: [[Support Distance|support distance]]), the d''δ''dK/d''t'' curve is shifted to higher values.

The crack opening velocity d''δ''/d''t'' (see: [[Levels of Knowledge in Fracture Mechanics|levels of knowledge in fracture mechanics]]) also offers the possibility of converting the different stress conditions arising for different test specimen geometries to a test specimen-invariant [[Deformation Rate|deformation rate]] at the [[Crack|crack tip]] and using this as a parameter in material-specific investigations.

==See also==

* [[Extended CTOD Concept|Extended CTOD concept]]
* [[Crack Tip Opening Displacement Concept (CTOD)|Crack tip opening displacement concept]]
* [[Crack Model according to DUGDALE|Crack model according to DUGDALE]]
* [[Geometry Criterion#Geometry criterion, crack opening displacement|Geometry criterion, crack opening displacement]]

==References==

{|
|-valign="top"
|[1]
|[[Grellmann,_Wolfgang|Grellmann, W.]]: Beurteilung der Zähigkeitseigenschaften von Polymerwerkstoffen durch bruchmechanische Kennwerte. Habilitation (1986), [https://de.wikipedia.org/wiki/Technische_Hochschule_Leuna-Merseburg Technische Hochschule Leuna-Merseburg], Wiss. Zeitschrift TH Merseburg 28 (1986), No. 6, pp. 787‒788 ([https://www.polymerservice-merseburg.de/fileadmin/inhalte/psm/veroeffentlichungen/Habil_Grellmann_Inhaltsverzeichnis.pdf Content], [https://www.polymerservice-merseburg.de/fileadmin/inhalte/psm/veroeffentlichungen/Habil_Grellmann_Kurzfassung.pdf summary])
|-valign="top"
|[2]
|Schwalbe, K. H.: Bruchverhalten keramischer Werkstoffe: Methoden und Ergebnisse. Fortschritts-Berichte VDI-Z. Reihe 18, Nr. 10, VDI-Verlag, 1981 (ISBN 978-3-1814-1118-6)
|-valign="top"
|[3]
|[[Blumenauer, Horst|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig (1982), (ISBN: VLN 152-915/62/73; see [[AMK-Library]] under E 29-1)
|-valign="top"
|[4]
|Kobayashi, T.: On the information about fracture characteristics obtained from instrumented impact test of A533 steel for reactor pressure vessel. Eng. Fracture Mechanics [https://www.sciencedirect.com/science/article/abs/pii/0013794484900699 19 (1984) 67]; [https://doi.org/10.1016/0013-7944(84)90069-9 DOI]
|-valign="top"
|[5]
|Schwalbe, K.-H.: Bruchmechanik metallischer Werkstoffe. Carl Hanser, Munich Vienna (1980), (ISBN 3-446-12983-9; see [[AMK-Library]] under E 15)
|-valign="top"
|[6]
|[https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 236–239 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see [[AMK-Library]] under A 22)
|-valign="top"
|[7]
|Hille, E.: Untersuchungen zum Bruchverhalten des orientierten isotaktischen Polypropylen. Dissertation, [https://de.wikipedia.org/wiki/Technische_Hochschule_Leuna-Merseburg Technische Hochschule Leuna-Merseburg (1983)] (1983)
|-valign="top"
|[8]
|Newe, R.: Untersuchungen zum Bruchverhalten des nichtorientierten isotaktischen PP. Dissertation, Technische Hochschule Leuna-Merseburg (1980)
|-valign="top"
|[9]
|Hollstein, T., Blauel, J. G., Wenk, K.: 12. Sitzung des Arbeitskreises Bruchvorgänge im DVM Freiburg, 7–8 October 1980
|-valign="top"
|[10]
|Jungbluth, M.: Untersuchungen zum Verformungs- und Bruchverhalten von PVC-Werkstoffen. Dissertation, Technische Hochschule Leuna-Merseburg (1987) (see AMK-Library under B 1-1) ([https://www.polymerservice-merseburg.de/fileadmin/inhalte/psm/veroeffentlichungen/Jungbluth_Untersuchungen_zum_Verformungs-_und_Bruchverhalten_von_PVCC-Werkstoffen.pdf Short bibliographic summary /Content])
|-valign="top"
|[11]
|ISO 13802 (2025-08): Plastics ‒ Verification of Pendulum Impact-testing Machines – Charpy-, Izod- and Tensile Impact Testing
|}

[[Category:Fracture Mechanics]]

Plastic Hinge Model - Revision history