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{{Language_sel|LANG=ger|ARTIKEL=Auswertemethode nach Kanazawa}}
{{PSM_Infobox}}
''J''-integral estimation method according to KANAZAWA (K)
__FORCETOC__

==Basic assumption of the estimation method==

[[J-Integral Evaluation Method (Overview) | ''J''-integral estimation methods]] are used for the determination of fracture mechanics [[Material Value | values]] according to the [[J-Integral Concept | ''J''-integral concept]] [1].

In the ''J''-integral evaluation method according to Kanazawa [2–4], a complementary deformation energy ''A''K is introduced to determine ''J''IK values. He modified the calculation approach according to RICE, since RICE obtained too small ''J'' values for small crack lengths. KANAZAWA derived a correction function for this.

{|
|-
|width="20px"|
|width="500px" | <math> J_{I}^{K}=\frac{c_{1} A_{G}+c_{2} A_{K}-c_{3} A_{0}}{B(W-a)}</math>
|}

{|
|-
|width="20px"|
|width="500px" | with <math> c_{1}=2; \ c_{2}=\alpha (\frac{W-a}{W}); \ c_{3}=2+\alpha (\frac{W-a}{W})</math>
|}

Thus, the ''J'' value generally results in:

{|
|-
|width="20px"|
|width="500px" | <math> J_{I}^{K}=\frac{2}{B(W-a)}(A_{G}-A_{0})+\frac{\alpha }{BW}(F_{max}f_{max}-A_{G}-A_{0})</math>
|}

[[file:Evaluation-Method K1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Determination of ''J'' integral according to KANAZAWA
|}

==Determination equation for Single-Edge-Notched Bend (SENB) specimen==

For the specific case of the [[SENB-Specimen | SENB test specimen]], the following then applies:

{|
|-
|width="20px"|
|width="500px" | <math> J_{Id}^{K}=\frac{1}{B}\left [ (\frac{2}{W-a}-\frac{\alpha}{W})A_{G}+\frac{\alpha}{W}(F_{max}f_{max})-(\frac{2}{W-a}+\frac{\alpha}{W})A_{0} \right ]</math>
|}

with: ''A''K = ''F''max ''f''max − ''A''G as complementary deformation energy

for 0 < ''a''/''W'' < 1 and

{|
|-
|width="20px"|
|width="500px" | <math> \alpha=\frac{f^2(\frac{a}{W})}{\int_{0}^{\frac{a}{W}}f^2(\frac{a}{W})d(\frac{a}{W})}-\frac{2}{1-\frac{a}{W}}</math>
|}

The significance of ''α'' for the determination of fracture-mechanical parameters with the aid of three-point bending test specimens can be derived from the graphical representation in '''Fig. 2''' using the corresponding geometry function ''f''(''a''/''W'') from Tada [6].

{|
|-
|width="20px"|
|width="500px" | <math> f(\frac{a}{W})=2.9(\frac{a}{W})^\frac{1}{2}-4.6(\frac{a}{W})^\frac{3}{2}+21.8(\frac{a}{W})^\frac{5}{2}-37.6(\frac{a}{W})^\frac{7}{2}+38.7(\frac{a}{W})^\frac{9}{2}</math>
|}

[[file:Evaluation-Method K2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Geometry function of ''J''-integral evaluation procedure according to method of KANAZAWA in dependence on ''a''/''W'' ratio for three-point bend loading and ''s''/''W'' = 4
|}

==Determination equation for Compact Tension (CT) specimen==

The following determination equations apply to the [[Compact Tension (CT) Specimen | CT specimen]]:

{|
|-
|width="20px"|
|width="500px" | <math> J_{Ic}^{K}=\frac{1}{B}\left [ \frac{1}{(W-a)}A_{G}+(\frac{\alpha}{W}-\frac{1}{(W-a)})A_{K}-\frac{\alpha}{W}A_{0} \right ]</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math> J_{Ic}^{K}=\frac{1}{B}\left [ \frac{1}{(W-a)}A_{G}+(\frac{\alpha}{W}-\frac{1}{(W-a)})(F_{max}f_{max}-A_{G})-\frac{\alpha}{W}A_{0} \right ]</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math> J_{Ic}^{K}=\frac{1}{B}\left [ \frac{A_{G}}{(W-a)}\frac{\alpha}{W}F_{max}f_{max}-\frac{\alpha}{W}A_{G}-\frac{F_{max}f_{max}}{(W-a)}+\frac{A_{G}}{(W-a)}-\frac{\alpha}{W}A_{0} \right ]</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math> J_{Ic}^{K}=\frac{1}{B}\left [ \frac{2A_{G}-F_{max}f_{max}}{(W-a)}+\frac{\alpha}{W}(F_{max}f_{max}-A_{G}-A_{0}) \right ]</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math> J_{Ic}^{K}=\frac{1}{B}\left [ \frac{1}{(W-a)}(2A_{G}-F_{max}f_{max})+\frac{\alpha}{W}(F_{max}f_{max}-A_{G}-A_{0}) \right ]</math>
|}

{|
|-
|width="20px"|
|width="500px" | with <math> \alpha=\frac{f^2(\frac{a}{W})}{\int_{0}^{\frac{a}{W}}f^2(\frac{a}{W})d(\frac{a}{W})}</math>
|}

As a result of extensive investigations on the crack length dependence of the ''J''-integral, it was proven in [1, 5] that the ''J'' evaluation methods of KANAZAWA and [[RICE, PARIS and MERKLE – J-Integral Estimation Method | RICE, PARIS and MERKLE]] provide too high fracture-mechanical [[Material Value | characteristic values]] for small crack lengths.

==See also==

*[[J-Integral Concept | J-integral concept]]
*[[J-Integral Evaluation Method (Overview) | J-Integral evaluation method (overview)]]
*J-integral estimation methods of
::– [[BEGLEY and LANDES – J-Integral Estimation Method | BEGLEY and LANDES]]
::– [[RICE, PARIS and MERKLE – J-Integral Estimation Method | RICE, PARIS and MERKLE]]
::– [[SUMPTER and TURNER – J-Integral Estimation Method | SUMPTER and TURNER]]
::– [[MERKLE and CORTEN – J-Integral Estimation Method | MERKLE and CORTEN (MC)]]

''''References'''

{|
|-valign="top"
|[1]
|[[Grellmann,_Wolfgang|Grellmann, W.]]: Beurteilung der Zähigkeitseigenschaften von Polymerwerkstoffen durch bruchmechanische Kennwerte. Habilitation (1986), Technische Hochschule „Carl Schorlemmer“ Leuna-Merseburg und Wiss. Zeitschrift TH Merseburg 28 (1986), H 6, p. 787–788
|-valign="top"
|[2]
|Schwalbe, K.-H.: Bruchmechanik metallischer Werkstoffe. Carl Hanser Munich Vienna (1980), (ISBN 3-446-12983-9; see [[AMK-Büchersammlung | AMK-Library]] under E 15)
|-valign="top"
|[3]
|Kanazawa, T., Machida, D., Onozuka, M., Kaned, S.: Report of the University of Tokyo HWx-779-75 in [4]
|-valign="top"
|[4]
|Kromp, K., Pabst, R. F.: Über die Ermittlung von J-Integralwerten bei keramischen Werkstoffen im Hochtemperaturbereich. Materialprüfung 22 (1980) 6, p. 241–245
|-valign="top"
|[5]
|[[Grellmann,_Wolfgang|Grellmann, W.]], Sommer, J.-P.: Beschreibung der Zähigkeitseigenschaften von Polymerwerkstoffen mit dem J-Integralkonzept. Institut für Mechanik, Berlin und Karl-Marx-Stadt, Fracture Mechanics, Micromechanics and Coupled Fields – (FMC)-Series (1985) 17, p. 48–72
|-valign="top"
|[6]
|Tada, H., Paris, P. C., Irwin, G. R.: The Stress Analysis of Cracks Handbook. Hellertown Pennsylvania, Del. Res. Corp. (1973)
|}

[[Category:Fracture Mechanics]]
[[Category:Instrumented Impact Test]]

KANAZAWA – J-Integral Estimation Method - Revision history