Geometry Criterion - Revision history

Oluschinski at 06:39, 4 September 2026

2026-09-04T06:39:02Z

← Older revision		Revision as of 08:39, 4 September 2026
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	The geometry constant <math>\beta</math> is material-dependent.		The geometry constant <math>\beta</math> is material-dependent.

	Experimental results regarding the influence of the test specimen thickness ''B'' on the fracture mechanical properties (see: [[Fracture Mechanical Testing \| fracture mechanical testing]]) for plastics are available in the literature. '''Figure 1''' shows the dependence of the coefficient according to the above equation on the fracture toughness determined under quasi-static and impact loading (see: [[Impact Loading of Plastics \| impact loading of plastics]]) for various [[Plastics \| plastics]]. The relationship shown was established on the basis of experimentally determined thickness and ''a''/''W'' dependencies and has a high degree of generalisation, as a common relationship results regardless of the type of stress (quasi-static, impact) and the material failure (stable, unstable) (see: [[Crack Propagation \| Crack propagation]]).		Experimental results regarding the influence of the test specimen thickness ''B'' on the fracture mechanical properties (see: [[Fracture Mechanical Testing\|fracture mechanical testing]]) for plastics are available in the literature. '''Figure 1''' shows the dependence of the coefficient according to the above equation on the fracture toughness determined under quasi-static and impact loading (see: [[Impact Loading Plastics\|impact loading of plastics]]) for various [[Plastics\|plastics]]. The relationship shown was established on the basis of experimentally determined thickness and ''a''/''W'' dependencies and has a high degree of generalisation, as a common relationship results regardless of the type of stress (quasi-static, impact) and the material failure (stable, unstable) (see: [[Crack Propagation\|Crack propagation]]) [1–5].

	[[file:Bild-Geometrie-K-Lexikon.jpg\|400px]]		[[file:Bild-Geometrie-K-Lexikon.jpg\|400px]]
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	\|width="600px" \|Dependence of the coefficient <math>\beta</math> on the fracture toughness ''K''<sub>Ic</sub>, ''K''<sub>Id</sub> for different plastics		\|width="600px" \|Dependence of the coefficient <math>\beta</math> on the fracture toughness ''K''<sub>Ic</sub>, ''K''<sub>Id</sub> for different plastics
	\|}		\|}


	~~'''References'''~~

	* [[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)
	* Anderson, T. L.: Fracture Mechanics. Fundamentals and Applications. 3rd Ed., CRC Press, Boca Raton (2005) (ISBN 978-0849342608; see [[AMK-Büchersammlung \| AMK-Library]] under E 8-2), DOI: [https://doi.org/10.1201/9781315370293 https://doi.org/10.1201/9781315370293]
	* [[Grellmann,_Wolfgang\|Grellmann, W.]], [[Seidler,_Sabine\|Seidler, S.]], [https://researchgate.net/profile/Ralf-Lach Lach, R.]: Geometrieunabhängige bruchmechanische Werkstoffkenngrößen – Voraussetzung für die Zähigkeitscharakterisierung von Kunststoffen. Materialwissenschaften und Werkstofftechnik 32 (2001) 552–561, https://doi.org/10.1002/1521-4052(200106)32:6%3C552::AID-MAWE552%3E3.0.CO;2-O
	* Akay, M.: Fracture Mechanics Properties. In: Brown, R. P. (Ed.): Handbook of Polymer Testing. Marcel Dekker Inc., New York (1999) 533–588 (ISBN 978-0824701710; see [[AMK-Büchersammlung \| AMK-Library]] under C 5)

	==Geometry criterion, J-integral concept==		==Geometry criterion, J-integral concept==
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	\|}		\|}

	Knowledge of the general <math>\varepsilon</math>-J relationship allows the required test [[Specimen \| specimen]] thicknesses to be estimated. The advantage of determining fracture mechanics values under impact loading lies in the possibility of obtaining geometry-independent values even at low test specimen thicknesses.		Knowledge of the general <math>\varepsilon</math>-J relationship allows the required test [[Specimen \| specimen]] thicknesses to be estimated. The advantage of determining fracture mechanics values under impact loading lies in the possibility of obtaining geometry-independent values even at low test specimen thicknesses [3, 6].


	~~'''References'''~~

	* [~~[Grellmann~~,~~_Wolfgang\|Grellmann, W.]], [[Seidler,_Sabine\|Seidler, S.]], [https://researchgate.net/profile/Ralf-Lach Lach, R.~~]~~: Geometrieunabhängige bruchmechanische Werkstoffkenngrößen – Voraussetzung für die Zähigkeitscharakterisierung von Kunststoffen. Materialwissenschaften und Werkstofftechnik 32 (2001) 552–561~~
	* [https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.]: New Developments in Toughness Evaluation of Polymers and Compounds by Fracture Mechanics. In: Grellmann, W., [[Seidler,_Sabine\|Seidler, S.]]: Deformation and Fracture Behaviour of Polymers. Springer Berlin Heidelberg (2001) p. 3–26, (ISBN 3-540-41247-6; see [[AMK-Büchersammlung \| AMK-Library]] under A 7)

	==Geometry criterion, crack opening displacement==		==Geometry criterion, crack opening displacement==
Line 101:		Line 87:
	\|}		\|}

	In addition to the [[J-Integral Concept \| J-integral concept]], the [[Crack Tip Opening Displacement Concept (CTOD) \| CTOD concept]] is used in particular to describe deformation-determined fracture processes. The prerequisite for determining critical crack openings is the formation of a quasi-static stress state. On the basis of the “Plastic Hinge Model”, the critical crack opening is determined for impact-type loading, which is independent of the ''a''/''W'' ratio at ''B'' = 4 mm for ''a''/''W'' > 0.2. '''Figure 3''' shows that <math>\xi</math>-values between 10 and 90 can be assumed and that a considerable overestimation of the required minimum test specimen dimensions is possible if the necessary notch depth or test specimen thickness is still unknown.		In addition to the [[J-Integral Concept \| J-integral concept]], the [[Crack Tip Opening Displacement Concept (CTOD) \| CTOD concept]] is used in particular to describe deformation-determined fracture processes. The prerequisite for determining critical crack openings is the formation of a quasi-static stress state. On the basis of the “Plastic Hinge Model”, the critical crack opening is determined for impact-type loading, which is independent of the ''a''/''W'' ratio at ''B'' = 4 mm for ''a''/''W'' > 0.2. '''Figure 3''' shows that <math>\xi</math>-values between 10 and 90 can be assumed and that a considerable overestimation of the required minimum test specimen dimensions is possible if the necessary notch depth or test specimen thickness is still unknown [7, 8].

	==See also==		==See also==
Line 109:		Line 95:
	*[[Crack Tip Opening Displacement Concept (CTOD) \| Crack tip opening displacement concept (CTOD)]]		*[[Crack Tip Opening Displacement Concept (CTOD) \| Crack tip opening displacement concept (CTOD)]]

			==References==

	~~'''References'''~~		{\|
			\|-valign="top"
	* [[Grellmann,_Wolfgang\|Grellmann, W.]], [[Seidler,_Sabine\|Seidler, S.]]: Determination of Geometry Independent Fracture Mechanics Values of Polymers. Int. J. of Fracture 68 (1994) R19–R22, https://doi.org/10.1007/BF00032333		\|[1]
	* Grellmann, W., [[Seidler,_Sabine\|Seidler, S.]], Hesse, W.: Procedure for Determining the Crack Resistance Behaviour Using the Instrumented Charpy Impact Test. In: Grellmann, W., Seidler, S.: Deformation and Fracture Behaviour of Polymers. Springer Berlin Heidelberg (2001) S. 71–86, (ISBN 3-540-41247-6; [[~~AMK-Büchersammlung \|~~ AMK-Library]] under A 7)		\|[[Blumenauer, Horst\|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (1993) (ISBN 3-342-00659-5; see [[AMK-Library]] under E 29-3)
			\|-valign="top"
			\|[2]
			\|Anderson, T. L.: Fracture Mechanics. Fundamentals and Applications. 3rd Ed., CRC Press, Boca Raton (2005) (ISBN 978-0849342608; see [[AMK-Library]] under E 8-2), DOI: [https://doi.org/10.1201/9781315370293 https://doi.org/10.1201/9781315370293]
			\|-valign="top"
			\|[3]
			\|[[Grellmann,_Wolfgang\|Grellmann, W.]], [[Seidler,_Sabine\|Seidler, S.]], [https://researchgate.net/profile/Ralf-Lach Lach, R.]: Geometrieunabhängige bruchmechanische Werkstoffkenngrößen – Voraussetzung für die Zähigkeitscharakterisierung von Kunststoffen. Materialwissenschaften und Werkstofftechnik 32 (2001) 552–561, https://doi.org/10.1002/1521-4052(200106)32:6%3C552::AID-MAWE552%3E3.0.CO;2-O
			\|-valign="top"
			\|[4]
			\|Akay, M.: Fracture Mechanics Properties. In: Brown, R. P. (Ed.): Handbook of Polymer Testing. Marcel Dekker Inc., New York (1999) 533–588 (ISBN 978-0824701710; see [[AMK-Library]] under C 5)
			\|-valign="top"
			\|[5]
			\|[[Williams,_James_Gordon\|Williams, J. G.]]: Fracture Mechanics of Polymers. Ellis Horwood Ltd., Publisher (1984) (ISBN 978-0470-20013-1; see [[AMK-Library]] under E 42)
			\|-valign="top"
			\|[6]
			\|[https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.]: New Developments in Toughness Evaluation of Polymers and Compounds by Fracture Mechanics. In: Grellmann, W., [[Seidler,_Sabine\|Seidler, S.]]: Deformation and Fracture Behaviour of Polymers. Springer Berlin Heidelberg (2001) p. 3–26, (ISBN 3-540-41247-6; see [[AMK-Library]] under A 7)
			\|-valign="top"
			\|[7]
			\|[[Grellmann,_Wolfgang\|Grellmann, W.]], [[Seidler,_Sabine\|Seidler, S.]]: Determination of Geometry Independent Fracture Mechanics Values of Polymers. Int. J. of Fracture 68 (1994) R19–R22, https://doi.org/10.1007/BF00032333
			\|-valign="top"
			\|[8]
			\|Grellmann, W., [[Seidler,_Sabine\|Seidler, S.]], Hesse, W.: Procedure for Determining the Crack Resistance Behaviour Using the Instrumented Charpy Impact Test. In: Grellmann, W., Seidler, S.: Deformation and Fracture Behaviour of Polymers. Springer Berlin Heidelberg (2001) S. 71–86, (ISBN 3-540-41247-6; [[AMK-Library]] under A 7)
			\|}

	[[Category:Fracture Mechanics]]		[[Category:Fracture Mechanics]]

Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Geometriekriterium}} {{PSM_Infobox}} Geometry criterion FORCETOC ==Geometry criterion, fracture tougness== In the linear-elastic approach, the geometric variables ''B'', ''a'' and the ligament expansion (''W''–''a'') are estimated using the empirically determined relationship {| |- |width="20px"| |width="500px" | $B{,}\ a{,}\ \left( W-a\right)\,\ge\,\beta \left( \frac{K}{\..."$

2025-12-02T08:39:57Z

Created page with "{{Language_sel|LANG=ger|ARTIKEL=Geometriekriterium}} {{PSM_Infobox}} Geometry criterion __FORCETOC__ ==Geometry criterion, fracture tougness== In the linear-elastic approach, the geometric variables ''B'', ''a'' and the ligament expansion (''W''–''a'') are estimated using the empirically determined relationship {| |- |width="20px"| |width="500px" | <math>B{,}\ a{,}\ \left( W-a\right)\,\ge\,\beta \left( \frac{K}{\..."

New page

{{Language_sel|LANG=ger|ARTIKEL=Geometriekriterium}}
{{PSM_Infobox}}
Geometry criterion
__FORCETOC__

==Geometry criterion, fracture tougness==

In the linear-elastic approach, the geometric variables ''B'', ''a'' and the ligament expansion (''W''–''a'') are estimated using the empirically determined relationship
{|
|-
|width="20px"|
|width="500px" | <math>B{,}\ a{,}\ \left( W-a\right)\,\ge\,\beta \left( \frac{K}{\sigma_y}\right)^2</math>
|}

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

The geometry constant <math>\beta</math> is material-dependent.

Experimental results regarding the influence of the test specimen thickness ''B'' on the fracture mechanical properties (see: [[Fracture Mechanical Testing | fracture mechanical testing]]) for plastics are available in the literature. '''Figure 1''' shows the dependence of the coefficient according to the above equation on the fracture toughness determined under quasi-static and impact loading (see: [[Impact Loading of Plastics | impact loading of plastics]]) for various [[Plastics | plastics]]. The relationship shown was established on the basis of experimentally determined thickness and ''a''/''W'' dependencies and has a high degree of generalisation, as a common relationship results regardless of the type of stress (quasi-static, impact) and the material failure (stable, unstable) (see: [[Crack Propagation | Crack propagation]]).

[[file:Bild-Geometrie-K-Lexikon.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Dependence of the coefficient <math>\beta</math> on the fracture toughness ''K''Ic, ''K''Id for different plastics
|}

'''References'''

* [[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)
* Anderson, T. L.: Fracture Mechanics. Fundamentals and Applications. 3rd Ed., CRC Press, Boca Raton (2005) (ISBN 978-0849342608; see [[AMK-Büchersammlung | AMK-Library]] under E 8-2), DOI: [https://doi.org/10.1201/9781315370293 https://doi.org/10.1201/9781315370293]
* [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]], [https://researchgate.net/profile/Ralf-Lach Lach, R.]: Geometrieunabhängige bruchmechanische Werkstoffkenngrößen – Voraussetzung für die Zähigkeitscharakterisierung von Kunststoffen. Materialwissenschaften und Werkstofftechnik 32 (2001) 552–561, https://doi.org/10.1002/1521-4052(200106)32:6%3C552::AID-MAWE552%3E3.0.CO;2-O
* Akay, M.: Fracture Mechanics Properties. In: Brown, R. P. (Ed.): Handbook of Polymer Testing. Marcel Dekker Inc., New York (1999) 533–588 (ISBN 978-0824701710; see [[AMK-Büchersammlung | AMK-Library]] under C 5)

==Geometry criterion, J-integral concept==

Due to the elastic-plastic material behaviour typical of plastics, especially with decreasing test specimen thickness, decreasing stress velocity and increasing temperature, and the limits for the applicability of linear-elastic [[Fracture Mechanics | fracture mechanics]], it is necessary to use the [[J-Integral Concept | J-integral concept]] to describe the geometry dependence. The critical ''J''-values are geometry-independent if the criterion

{|
|-
|width="20px"|
|width="500px" | <math>B{,}\ a{,}\ \left( W-a\right)\,\ge\,\varepsilon \frac{J}{\sigma_y}</math>
|}

with
{|
|-
|<math>\varepsilon</math>
|width="15px" |
|material-dependent constant of the geometry criterion of the ''J''-integral concept
|}

is fulfilled.

For the geometry constant from this criterion, '''Figure 2''' shows a tendency to decrease with increasing toughness, which <math>\varepsilon</math>, like the geometry constant <math>\beta</math>, must be regarded as a material-dependent variable and can assume values between 5 and 1220, which represent maximum values for impact loading.

[[file:Bild-Geometrie-d-Lexikon.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Dependence of the coefficients <math>\varepsilon</math> on the ''J''-value for different plastics
|}

Knowledge of the general <math>\varepsilon</math>-J relationship allows the required test [[Specimen | specimen]] thicknesses to be estimated. The advantage of determining fracture mechanics values under impact loading lies in the possibility of obtaining geometry-independent values even at low test specimen thicknesses.

'''References'''

* [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]], [https://researchgate.net/profile/Ralf-Lach Lach, R.]: Geometrieunabhängige bruchmechanische Werkstoffkenngrößen – Voraussetzung für die Zähigkeitscharakterisierung von Kunststoffen. Materialwissenschaften und Werkstofftechnik 32 (2001) 552–561
* [https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.]: New Developments in Toughness Evaluation of Polymers and Compounds by Fracture Mechanics. In: Grellmann, W., [[Seidler,_Sabine|Seidler, S.]]: Deformation and Fracture Behaviour of Polymers. Springer Berlin Heidelberg (2001) p. 3–26, (ISBN 3-540-41247-6; see [[AMK-Büchersammlung | AMK-Library]] under A 7)

==Geometry criterion, crack opening displacement==

The requirements for the test specimen geometry are estimated using the [[Crack Tip Opening Displacement Concept (CTOD) | Crack tip opening displacement concept]]
{|
|-
|width="20px"|
|width="500px" | <math>B{,}\ a{,}\ \left( W-a\right)\,\ge\,\xi \cdot \delta</math>.
|}

with
{|
|-
|<math>\xi</math>
|width="15px" |
|material-dependent constant of the geometry criterion of the [[Crack Tip Opening Displacement Concept (CTOD) | CTOD concept]]
|}

[[file:Bild-Geometrie-J-Lexikon.jpg|400px]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |Dependence of the coefficient <math>\xi</math> on the critical crack opening <math>\delta</math>Idk
|}

In addition to the [[J-Integral Concept | J-integral concept]], the [[Crack Tip Opening Displacement Concept (CTOD) | CTOD concept]] is used in particular to describe deformation-determined fracture processes. The prerequisite for determining critical crack openings is the formation of a quasi-static stress state. On the basis of the “Plastic Hinge Model”, the critical crack opening is determined for impact-type loading, which is independent of the ''a''/''W'' ratio at ''B'' = 4 mm for ''a''/''W'' > 0.2. '''Figure 3''' shows that <math>\xi</math>-values between 10 and 90 can be assumed and that a considerable overestimation of the required minimum test specimen dimensions is possible if the necessary notch depth or test specimen thickness is still unknown.

==See also==
*[[Fracture Mechanics | Fracture mechanics]]
*[[Crack Models | Crack models]]
*[[J-Integral Concept | J-integral concept]]
*[[Crack Tip Opening Displacement Concept (CTOD) | Crack tip opening displacement concept (CTOD)]]

'''References'''

* [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]]: Determination of Geometry Independent Fracture Mechanics Values of Polymers. Int. J. of Fracture 68 (1994) R19–R22, https://doi.org/10.1007/BF00032333
* Grellmann, W., [[Seidler,_Sabine|Seidler, S.]], Hesse, W.: Procedure for Determining the Crack Resistance Behaviour Using the Instrumented Charpy Impact Test. In: Grellmann, W., Seidler, S.: Deformation and Fracture Behaviour of Polymers. Springer Berlin Heidelberg (2001) S. 71–86, (ISBN 3-540-41247-6; [[AMK-Büchersammlung | AMK-Library]] under A 7)

[[Category:Fracture Mechanics]]