GRIFFITH's Criteria - Revision history

Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=GRIFFITH-Kriterium}} {{PSM_Infobox}} GRIFFTH`s criteria FORCETOC ==GFRIFFITH`s fracture criterion== The energy balance in the case of crack propagation in an infinitely extended plate with finite thickness was first discussed by Griffith [1]. According to GRIFFITH, crack propagation occurs when the elastic distortion energy ''W''
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Created page with "{{Language_sel|LANG=ger|ARTIKEL=GRIFFITH-Kriterium}} {{PSM_Infobox}} <span style="font-size:1.2em;font-weight:bold;">GRIFFTH`s criteria</span> FORCETOC ==GFRIFFITH`s fracture criterion== The energy balance in the case of crack propagation in an infinitely extended plate with finite thickness was first discussed by Griffith [1]. According to GRIFFITH, crack propagation occurs when the elastic distortion energy ''W''<su..."

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{{Language_sel|LANG=ger|ARTIKEL=GRIFFITH-Kriterium}}
{{PSM_Infobox}}
<span style="font-size:1.2em;font-weight:bold;">GRIFFTH`s criteria</span>
FORCETOC

==GFRIFFITH`s fracture criterion==

The energy balance in the case of [[Crack Propagation|crack propagation]] in an infinitely extended plate with finite thickness was first discussed by [[Griffith, Alan Arnold|Griffith]] [1]. According to GRIFFITH, crack propagation occurs when the elastic distortion energy ''W''<sub>e</sub> released by the elongation of the [[Initial Crack Length|initial crack]] is equal to or greater than the [[Surface Energy|surface energy]] ''W''<sub>o</sub> required to form the [[Fracture Surface|fracture surfaces]], i.e.

{|
|-
|width="20px"|
|width="500px" | <math>W_e\, \ge \, W_o</math>
|(1)
|}

The criterion for [[Crack Propagation|unstable crack propagation]] is

{|
|-
|width="20px"|
|width="500px" | <math>\frac{dW_e}{da}\, \ge \, \frac{dW_o}{da}</math>
|(2)
|}

Elastic distortion energy ''W''<sub>e</sub> is assumed,

{|
|-
|width="20px"|
|width="500px" | <math>W_e\,=\,\frac{\pi \sigma^2 a^2}{E}</math>
|(3)
|}

with
{|
|-
|a
|width="15px" |
|[[Initial Crack Length|Crack length]]
|-
|E
|
|[[Elastic Modulus|Elastic modulus]]
|}

while the [[Surface Energy|surface energy]] ''W''<sub>o</sub> is assumed for both fracture surfaces.

{|
|-
|width="20px"|
|width="500px" | <math>W_o\,=\,4\, a\, \gamma_o</math>
|(4)
|}

with
{|
|-
|<math>\gamma</math><sub>o</sub>
|width="15px" |
|[[Surface Tension and Interfacial Tension|Surface tension]]
|}

amounts to.

The criterion of [[Crack Propagation|crack propagation]] then leads to

{|
|-
|width="20px"|
|width="500px" | <math>\frac{2\, a\, \pi\, \sigma^2}{E} \, \ge \, 4\, \gamma_o</math>
|(5)
|}

==Energy balance of unstable crack propagation==

Under the conditions of '''Eq. (5)''', an existing [[Crack|crack]] propagates in an unstable manner, i.e. without external energy input ('''Figure''').

[[File:Griffith's Criteria Fig.jpg|300px]]
{|
|- valign="top"
|width="50px"|'''Figure''':
|width="600px" |Energy balance during unstable crack propagation [2]<br>a) [[Surface Energy|Surface energy]]<br>b) Elastic distortion energy<br>c) Total energy
|}

The stress required for crack propagation is obtained as

{|
|-
|width="20px"|
|width="500px" | <math>\sigma_c\,=\,\sqrt{\frac{2 E \gamma_o}{\pi a}}</math>
|(6)
|}

while the corresponding [[Initial Crack Length|critical crack length]] (Griffith length) is ''a''<sub>o</sub>

{|
|-
|width="20px"|
|width="500px" | <math>a_o\,=\,\frac{2 E \gamma_o}{\pi \sigma^2}</math>
|(7)
|}

Experimental investigations have confirmed the usefulness of the relationship for critical stress and crack length in extremely brittle material behaviour [3].

The two basic hypotheses of the GRIFFITH criterion are [4]:

{|
|width="92px" height="25px"|'''Hypothese 1:'''
|The [[Crack|crack]] propagates in the direction of the crack.
|- valign ="top"
|width="92px" height="25px"|'''Hypothese 2:'''
|Crack growth occurs if ''J''<sub>e</sub> ≥ 2 ''γ''<sub>o</sub><br>
with J<sub>e</sub> – elastic component.<br><br>
The [[Crack Propagation|crack propagation]] is unstable if the following applies:
|- valign ="top"
|
|<math>\frac{\delta K_I}{\delta a}\, \ge \, 0</math>
|}

==See also==

* [[Crack Model according to GRIFFITH|Crack model according to GRIFFITH]]
* [[Griffith, Alan Arnold]]
* [[Crack]]

==References==

{|
|-valign="top"
|[1]
|[[Griffith, Alan Arnold|Griffith, A. A.]]: The Phenomena of Rupture and Flow in Solids. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 221 (1921) pp. 163 – 198. JSTOR; DOI: https://www.jstor.org/stable/91192
|-valign="top"
|[2]
|[[Blumenauer, Horst|Blumenauer, H.]]: Bruchmechanik – Grundlagen, Prüfmethoden, Anwendungsgebiete. Deutscher Verlag für Grundstoffindustrie, Leipzig (1973) pp. 37–38 (see [[AMK-Library]] under E 28)
|-valign="top"
|[3]
|[[Blumenauer, Horst|Blumenauer, H.]]: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig (1987) 2nd Edition p. 22 (ISBN 3-342-00096-1; see [[AMK-Library]] under E 29-2)
|-valign="top"
|[4]
|Sähn, S., Göldner, H.: Bruch- und Beurteilungskriterien in der Festigkeitslehre. Fachbuchverlag, Leipzig Köln (1993) 2nd Edition, p. 139 (ISBN 3-343-00854-0; see [[AMK-Library]] under E 26)
|}

==Weblinks==

* Wikipedia– The Free Encyclopedia: Griffith, Alan Arnold https://en.wikipedia.org/wiki/Alan_Arnold_Griffith
* 100 Jahre Bruchmechanik: Alan Arnold Griffith. DVM-Nachrichten N 71 • Frühjahr 2020 p. 9 [https://dvm-berlin.de/sites/default/files/files/publications/field-teaser-pdfdownloads/1224620288.pdf Download als PDF]

[[Category:Fracture Mechanics]]