KANAZAWA – J-Integral Estimation Method

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J-integral estimation method according to KANAZAWA (K)

Basic assumption of the estimation method

J-integral estimation methods are used for the determination of fracture mechanics values according to the 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_I^K 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.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J_{I}^{K}={\frac {c_{1}A_{G}+c_{2}A_{K}-c_{3}A_{0}}{B(W-a)}}}

with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1}=2;\ c_{2}=\alpha ({\frac {W-a}{W}});\ c_{3}=2+\alpha ({\frac {W-a}{W}})}

Thus, the J value generally results in:

$J_{I}^{K}={\frac {2}{B(W-a)}}(A_{G}-A_{0})+{\frac {\alpha }{BW}}(F_{max}f_{max}-A_{G}-A_{0})$

Fig. 1:

Determination of J integral according to KANAZAWA

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

For the specific case of the SENB test specimen, the following then applies:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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]}

with: A_K = F_max f_max − A_G as complementary deformation energy

for 0 < a/W < 1 and

$\alpha ={\frac {f^{2}({\frac {a}{W}})}{\int _{0}^{\frac {a}{W}}f^{2}({\frac {a}{W}})d({\frac {a}{W}})}}-{\frac {2}{1-{\frac {a}{W}}}}$

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].

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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}}}

Fig. 2:

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 CT specimen:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ]}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ]}

with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \alpha ={\frac {f^{2}({\frac {a}{W}})}{\int _{0}^{\frac {a}{W}}f^{2}({\frac {a}{W}})d({\frac {a}{W}})}}}

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 provide too high fracture-mechanical characteristic values for small crack lengths.

See also

• J-integral concept
• J-Integral evaluation method (overview)
• J-integral estimation methods of

– BEGLEY and LANDES

– RICE, PARIS and MERKLE

– SUMPTER and TURNER

– MERKLE and CORTEN (MC)

'References

[1]	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
[2]	Schwalbe, K.-H.: Bruchmechanik metallischer Werkstoffe. Carl Hanser Munich Vienna (1980), (ISBN 3-446-12983-9; see AMK-Library under E 15)
[3]	Kanazawa, T., Machida, D., Onozuka, M., Kaned, S.: Report of the University of Tokyo HWx-779-75 in [4]
[4]	Kromp, K., Pabst, R. F.: Über die Ermittlung von J-Integralwerten bei keramischen Werkstoffen im Hochtemperaturbereich. Materialprüfung 22 (1980) 6, p. 241–245
[5]	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
[6]	Tada, H., Paris, P. C., Irwin, G. R.: The Stress Analysis of Cracks Handbook. Hellertown Pennsylvania, Del. Res. Corp. (1973)