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Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Bruchmechanik}} {{PSM_Infobox}} Fracture mechanics FORCETOC ==Linear-elastic fracture mechanics== Fracture mechanics assumes that the fracture of a component and thus of the material occurs as a result of the propagation of cracks. It investigates the conditions for Crack Propagation|crack propaga..."

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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Bruchmechanik}} {{PSM_Infobox}} Fracture mechanics __FORCETOC__ ==Linear-elastic fracture mechanics== Fracture mechanics assumes that the fracture of a component and thus of the material occurs as a result of the propagation of cracks. It investigates the conditions for Crack Propagation|crack propaga..."

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{{Language_sel|LANG=ger|ARTIKEL=Bruchmechanik}}
{{PSM_Infobox}}
Fracture mechanics
__FORCETOC__

==Linear-elastic fracture mechanics==

Fracture mechanics assumes that the [[Fracture|fracture]] of a [[Plastic Component|component]] and thus of the [[Material & Werkstoff|material]] occurs as a result of the [[Crack Propagation|propagation of cracks]]. It investigates the conditions for [[Crack Propagation|crack propagation]] and allows quantitative relationships to be established between the external [[Stress|stress]], i.e. the nominal stress acting on the component or [[Specimen|test specimen]], the size and shape of the cracks, and the resistance of the material to [[Crack Propagation|crack propagation]].

The LEFM concept describes the stress state in the vicinity of the [[Crack|crack tip]] using the stress intensity factor ''K'' ('''Fig. 1'''):

{|
|-
|width="20px"|
|width="500px" | <math>\sigma_{ij}\,=\,\frac{K}{\left( 2\, \pi \, r \right)^\frac {1}{2}} \cdot g_{ij} \cdot \left( \Theta \right)</math>
|(1)
|}

with
{|
|-
|<math>\sigma</math>ij
|width="15px" |
|normal and/or shear stress
|-valign="top"
|''r'', <math>\Theta</math>
|
|polar coordinates with [[Crack|crack tip]] as point of origin
|-valign="top"
|''g''ij
|
|dimensional function.
|}

[[file:LEBM_1.jpg|300px]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Coordinate system for describing stress state at crack tip
|}

The stress intensity factor introduced by IRWIN [1] is given by

{|
|-
|width="20px"|
|width="500px" | <math>K\,=\,\sigma_N\, \left( \pi\, a \right)^\frac{1}{2}</math>
|(2)
|}

with
{|
|-
|<math>\sigma</math>N
|width="15px" |
| nominal stress
|-valign="top"
|''a''
|
|[[Crack]] length
|}

The finite geometry of each component and [[Specimen|test specimen]], as well as the crack geometry, are taken into account by introducing ''a'' [[Geometry Function|geometry function]] ''f'' (''a''/''W''), whereby '''Eq. 2''' takes the form

{|
|-
|width="20px"|
|width="500px" | <math>K\,=\,\sigma_N\, \left( \pi\, a \right)^\frac{1}{2}\, f \left( \frac{a}{W}\right)</math>
|(3)
|}

can be written. The functions ''f'' (''a''/''W'') have been calculated for a large number of [[Fracture Mechanics Test Specimens|fracture mechanics test specimens]] [2, 3]. '''Figs. 2''' and '''3''' show the dimensions of test [[Specimen|specimens]] preferably used for [[Plastics|plastics]]. For an infinitely extended test [[Specimen|specimen]] and the limiting case of a [[Crack|crack]] with a notch radius ''ρ'' ~ 0, the [[Geometry Function|geometry function]] ''f'' (''a''/''W'') = 1.

At the start of unstable [[Crack Propagation|crack propagation]], the stress intensity factor reaches a critical value ''K''Ic, which is referred to as fracture or [[Crack Toughness|crack toughness]] and is measured in MPa mm1/2. The index I indicates [[Fracture Modes|mode I]] loading, in which the load is applied perpendicular to the crack surface.

[[file:Frakture Mechanics Fig2.jpg|550px]] 
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Test specimen shape [[SENB-Specimen|SENB]] with dimensions, the corresponding equations for calculating the fracture toughness (stress intensity factor) and the [[Geometry Function|geometric functions]]
|}

[[file:LEBM_Tabelle_2.jpg]] 
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |Specimen shapes [[SENT-Specimen|SENT]] and [[Compact Tension (CT) Specimen|CT]] with their dimensions, the corresponding equations for calculating fracture toughness and the [[Geometry Function|geometric functions]]
|}

For the most important technical case of [[Stress|stress]], the [[Fracture Safety Criterion|fracture safety criterion]] is

{|
|-
|width="20px"|
|width="500px" | <math>K_I\,\le\, K_{Ic}</math>
|(4)
|}

according to which the fracture safety of a [[Plastic Component|component]] is guaranteed as long as the critical value is not exceeded.

In addition to the simple [[Crack Opening|crack opening]] according to mode I, '''Figure 1''' also shows [[Fracture Modes|fracture modes]] II and III, which occur under shear or torsional stress.

Depending on the test specimen geometry, different [[Multiaxial Stress State|multiaxial stress states]] form in front of the [[Crack|crack]] tip.

'''Figure 4''' shows the influence of the test specimen thickness on the [[Fracture Behaviour|fracture behaviour]] using the example of post-chlorinated PVC ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PVCC) and polypropylene ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PP), whereby a macroscopic increase in normal stress fracture is observed as a result of the transition from the [[Plane Stress and Strain State|plane stress state to the plane strain state]].

If the plane strain state is present at the crack tip, the fracture toughness becomes independent of the test specimen geometry. It reflects the influence of the material structure, the [[Velocity|velocity]] and the ambient temperature on the [[Toughness|toughness]].

[[file:Frakture Mechanics Fig4.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px" |Dependence of fracture toughness Kc, ''K''Ic at room temperature on the test specimen thickness under [[Quasi-static Test Methods|quasi-static]] [[Stress|stress]] for PVCC with with ''K''Ic = 110 MPamm1/2 (a) and for PP with ''K''Ic = 139 MPamm1/2 (b) bei einer [[Crosshead Speed|crosshead speed]] of ''v''T = 8,3 • 10-4 ms-1
|}

Using a linear-elastic approach, the geometric parameters ''B'', ''a'' and the ligament extension (''W'' – ''a'') are estimated using the empirically determined relationship [2, 4–6]

{|
|-
|width="20px"|
|width="500px" | <math>B, a, \left(W-a\right)\, \ge \, \beta \left( \frac{K}{\sigma_y}\right)^2</math>
|(5)
|}

with
{|
|-
|<math>\sigma</math>y
|width="15px" |
|[[Yield Stress|yield stress]] (yield point)
|}

The geometric constant ''β'' is material-dependent [4, 7–10].

'''References'''

{|
|-valign="top"
|[1]
|Irwin, G. R.: Analysis of Stress and Strain Near the End of a Crack Traversing a Plate. J. Appl. Mech. 24 (1957) 361
|-valign="top"
|[2]
|Anderson, T. L.: Fracture Mechanics. Fundamentals and Applications. 2nd Ed., CRC Press, Boca Raton (1995) (ISBN 978-0849342608; see [[AMK-Büchersammlung|AMK-Library]] under E 8-1)
|-valign="top"
|[3]
|Tada, H., Paris, P. C.; Irwin, G. R.: The Stress Analysis of Cracks Handbook. 3th Ed., ASME Press, New York (2000) (ISBN 0791801535)
|-valign="top"
|[4]
|[[Blumenauer, Horst|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (1993) (ISBN 3-342-00659-5; see [[AMK-Büchersammlung|AMK-Library]] under E 29-3)
|-valign="top"
|[5]
|Francois, D., Pineau, A. (Eds.): From Charpy to Present Impact Testing. ESIS Publication 30, Elsevier Science Ldt. Oxford (2002) (ISBN 9780080439709)
|-valign="top"
|[6]
|Akay, M.: Fracture Mechanics Properties. In: Brown, R. (Ed.): Handbook of Polymer Testing. Marcel Dekker Inc., New York (1999) (ISBN 1-85957-324-X) pp. 533–588
|-valign="top"
|[7]
|[[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]], Hesse, W.: Prozedur zur Ermittlung des Risswiderstandsverhaltens mit dem instrumentierten Kerbschlagbiegeversuch. In: [https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.], Seidler, S.: Deformation und Bruchverhalten von Kunststoffen. Springer, Berlin Heidelberg (1998) S. 75–90, (ISBN 3-540-63671-4; e-Book (2014): ISBN 978-3-642-58766-5; see [[AMK-Büchersammlung|AMK-Library]] under A 6)
|-valign="top"
|[8]
|[https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.], Seidler, S. (Eds.): Deformation and Fracture Behaviour of Polymers. Springer, Berlin Heidelberg (2001) (ISBN 978-3540412472; see [[AMK-Büchersammlung|AMK-Library]] under A 7)
|-valign="top"
|[9]
|[https://de.wikipedia.org/wiki/Wolfgang_Grellmann Grellmann, W.], Seidler, S., [https://researchgate.net/profile/Ralf-Lach Lach, R.]: Geometrieunabhängige bruchmechanische Werkstoffkenngrößen – Voraussetzung für die Zähigkeitscharakterisierung von Kunststoffen. Material und Werkstofftechnik 32 (2001) 552–561
|-valign="top"
|[10]
|Grellmann, W., Seidler, S.: Determination of Geometry Independent Fracture Mechanics Values of Polymers. Int. J. of Fracture 68 (1994) R19–R22
|}

==Linear-elastic fracture mechanics with small-scale yielding==

By taking into account the [[Fracture Mirror|fracture mirror]] ''a''BS when describing [[Toughness|toughness]] ('''Fig. 5'''), whereby the [[Initial Crack Length|initial crack length]] ''a'' must be extended by the microscopically measured length of stable crack growth,

{|
|-
|width="20px"|
|width="500px" | <math>a_{eff}\,=\,a + a_{BS}</math>
|(6)
|}

the transition to LEFM with small-scale yielding is formally completed. The length of the [[Fracture Mirror|fracture mirror]], which is predominantly determined experimentally using light microscopy, is the [[Measured Variable|measured variable]] for the radius of the plastic zone, which is introduced into fracture mechanics in the [[Crack Model according to IRWIN and Mc CLINTOCK|crack model according to IRWIN and Mc CLINTOCK]]. The sum of the [[Initial Crack Length|length of the initial crack]] (notch length) and the [[Fracture Mirror|fracture mirror]] is referred to as the [[Effective Crack Length|effective crack length]]. In very brittle structures (gross spherulitic) and at high [[Test Speed|loading speed]] or low temperatures, the [[Fracture Mirror|fracture mirror]] is negligible.

[[file:Frakture Mechanics Fig5.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 5''':
|width="600px" |Fracture surface of an ethylene-propylene random copolymer with 4 mol.-% etyhylene (a) and schematic diagram of characteristic areas (b)
|}

==Elastic–plastic fracture mechanics (EPFM)==

In the case of macroscopically brittle [[Fracture|fracture]] of a [[Plastic Component|component]], the critical [[Error Limit|error size]] is often caused by stable crack growth of existing incipient cracks. Of particular technical significance here are crack enlargement due to mechanical [[Stress|stress]] (static, dynamic, oscillating) and medial stress ([[Environmental Stress Cracking Resistance|stress cracking corrosion]]).

If the expansion of the [[Plastic Zone|plastic zone]] (see also: ‘[[Effective Crack Length|effective crack length]]’) is not small in relation to the component or specimen dimensions, the [[Fracture|fracture]] is preceded by plastic flow in larger areas of the material in front of the [[Crack|crack tip]]. Since this case occurs in most structural materials under normal operating conditions, linear-elastic fracture mechanics has been further developed into yield fracture mechanics, i.e. fracture mechanics with general plastic deformation.

The theoretical basis is provided by the [[Crack Model according to DUGDALE|DUGDALE crack model]] derived by Wells in 1961, which is based on the assumption that the [[Fracture Formation|fracture process]] is deformation-determined. The formation of a microstructurally induced [[Plastic Zone|plastic zone]] is permitted. In addition to the [[Crack Tip Opening Displacement Concept (CTOD)|Cack Tip Opening Displacement (CTOD) concept]] based on this assumption, the [[J-Integral Concept|J-integral concept]] and the [[Crack Resistance (R) Curve|crack resistance (R) curve]] concept are established as further concepts of elastic–plastic fracture mechanics (EPFM).

In the case of elastic--plastic material behaviour, the fracture process is characterised by the stages of crack blunting (see: [[Crack Resistance Curve – Experimental Methods|crack resistance curve]]), [[Crack Initiation|crack initiation]], stable [[Crack Propagation|crack propagation]] and, subsequently, possibly unstable crack propagation. This entire process can be described by the [[Crack Resistance (R) Curve|crack resistance curve]] (R curve) of yield fracture mechanics.

In recent years, considerable progress has been made in determining material science [[Material Value|characteristic values]] using the concepts of yield fracture mechanics, with particular attention being paid to the specific deformation and [[Fracture Behaviour|fracture behaviour]] of [[Plastics|plastics]]. Methods for structural analysis and methods for explaining deformation mechanisms have made an essential contribution here.

With regard to the applicability of fracture mechanical [[Material Parameter|material parameters]] in plastics development, particular attention is paid to the quantification of energy-dissipative processes using generalised integral criteria of fracture mechanics. This includes the [[JTJ-Concept|JTJ-concept]] developed by Michel and Will in 1986, which enables the quantification of energy-dissipative processes during stable crack growth. The suitability of this concept for establishing quantitative morphology–toughness correlations was demonstrated by [[Seidler, Sabine|Seidler]] in 1996. Phase distributions, sizes and interactions in [[Polymer|polymeric]] multiphase systems are the focus of interest as morphological variables.

By combining fracture mechanics investigation methods and morphological investigations, taking into account the test temperature, correlations between morphology and crack initiation and propagation behaviour are clarified, which form the basis for quantitative morphology–toughness correlations.

==See also==

* [[Fracture Mechanical Testing|Fracture mechanical testing]]
* [[Fracture]]
* [[Levels of Knowledge in Fracture Mechanics|Levels of knowledge in fracture mechanics]]
* [[Crack]]
* [[Notch]]
* [[Toughness]]
* [[Crack Toughness|Crack toughness]]

'''Refereces'''

* [[Blumenauer, Horst|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (2003), 3rd Edition, (ISBN 3-342-00659-5; see [[AMK-Büchersammlung|AMK-Library]] under E 29-3)
* [[Seidler, Sabine|Seidler, S.]]: Anwendung des Risswiderstandskonzeptes zur Ermittlung strukturbezogener bruchmechanischer Werkstoffkenngrößen bei dynamischer Beanspruchung, Habilitation (1997), Martin-Luther-Universität Halle-Wittenberg, VDI-Reihe 18: Mechanik/Bruchmechanik Nr. 231, VDI Verlag Düsseldorf (1998) (ISBN 978-3-1832-3118-8; see [[AMK-Büchersammlung|AMK-Library]] under B 2-1)
* Anderson, T. L.: Fracture Mechanics; Fundamental and Applications. CRC Press, Boca Raton (2005) (ISBN 978-0849342608; see [[AMK-Büchersammlung|AMK-Library]] under E 8-2)
* Krüger, L., Trubitz, P., Hentschel, S.: Bruchmechanisches Verhalten unter quasistatischer und dynamischer Beanspruchung. In: Biermann, H., Krüger, L.: Moderne Methoden der Werkstoffprüfung. Wiley-VCH Verlag GmbH, Weinheim (2014) pp. 1–52; ISBN 978-3-527-33413-1 (see [[AMK-Büchersammlung|AMK-Library]] under M 35)

[[category:Fracture Mechanics]]
[[category:Scientific Disciplines]]

Fracture Mechanics - Revision history

Oluschinski at 06:32, 15 December 2025