View source for J-Integral Concept

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<span style="font-size:1.2em;font-weight:bold;">''J''-integral concept</span>
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==Energetic consideration of the fracture process==

The ''J''-integral introduced by Cherepanov [1] and Rice [2] has gained the greatest importance for [[Plastics | plastics]] due to the energetic consideration of the fracture process (see: [[Fracture Mechanics | Fracture mechanics]] and [[Fracture Types | types of fracture]]). The path-independent contour integral encloses the plastically deformed region and runs in the elastic deformed region with a closed integration path around the crack tip ('''Figure 1''').

[[file:J-Integral.jpg|500px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 1''': 
|width="600px" |Determination of the J-integral: path-independent contour integral with 1 – plastically deformed area (energy-dissipative zone) and 2 – elastically deformed area (a), experimentally determined load vs. load-line displacement curves of different crack lengths (b), energy determined by planimetrating the dependency ''F'' = ''F''(''v'', ''f'') dependence, related to the specimen thickness as a function of the crack length (c) and ''J''-integral (d) determined by differentiating the curves (c) [3].
|}

The x- und y-components are defined by

{|
|-
|width="20px"|
|width="500px" | <math>J_x\,=\, \int_{R} \left( W \,dy-T_{ij} \cdot n_j \frac{\partial u}{\partial x}dR\right)</math> und 
|}

{|
|-
|width="20px"|
|width="500px" | <math>J_y\,=\, \int_{R} \left( -W \,dx-T_{ij} \cdot n_j \frac{\partial u}{\partial x}dR\right)</math>.
|}

with
{|
|-
|''W''
|width="15px" |
|elastic energy density
|-
|''T''
|
|stress tensor
|-
|''n''
|
|components oft the unit vector to R around the crack tip
|-
|''u''
|
|displacement vector components
|}

==Experimental determination of J-values==

The experimental determination is carried out according to '''Fig. 1 b to d''' by determining the deformation energy ''A''<sub>G</sub> from the registered load vs. load line displacement curves with different notch depths by planimetry and displaying the ratio ''A''<sub>G</sub>/''B'' as a function of ''a''. The deformation energy ''A''<sub>G</sub> is then determined by the graphical differentiation.

Using graphical differentiation, the following results

{|
|-
|width="20px"|
|width="500px" | <math>J\,=\,\frac{1}{B} \frac{\partial A_G}{\partial a}</math>
|}

as function of the load-line displacement resp. deflection.

Since the effort to determine ''J''-values according to this procedure is too high for practical [[Material Value | characteristic value]] determination, approximation formulas have been developed. The best-known procedures are:

* Approximation method according to [[BEGLEY and LANDES – J-Integral Estimation Method | BEGLEY and LANDES]],
* Approximation method according to [[RICE, PARIS and MERKLE – J-Integral Estimation Method | RICE, PARIS and MERKLE]],
* Approximation method according to [[KANAZAWA – J-Integral Estimation Method | KANAZAWA ]],
* Approximation methods according to [[SUMPTER and TURNER – J-Integral Estimation Method | SUMPTER and TURNER]] and
* Approximation methods according to [[MERKLE and CORTEN – J-Integral Estimation Method | MERKLE and CORTEN]].

==Correlations of the J-integral to the stress intensity factor and the crack opening displacement==

For elastic material behaviour, the ''J''-integral is identical to the [[Energy Release Rate | energy release rate ''G'']]:

{|
|-
|width="20px"|
|width="250px" | <math>J_I\,=\,G_I\,=\,\frac{{K_I}^2}{E}</math>
|for ESZ resp.
|-
|
|<math>J_I\,=\,G_I\,=\,\frac{{K_I}^2}{E} \left(1- {\nu}^2 \right)</math>
|for EDZ.
|}

These equations are to be used for the conversion of ''J''<sub>Ic</sub> values into K<sub>Ic</sub><sup>J</sup> values.

The relationship between ''J''-integral and [[Crack Tip Opening Displacement Concept (CTOD) | Crack tip opening displacement (CTOD)]] concept provides

{|
|-
|width="20px"|
|width="500px" | <math>J\,=\,m \cdot \sigma_y \cdot \delta_{Ic}</math>,
|}

where ''m'' is called the constraint factor according to [4, 5]. The critical ''J''-values are geometry-independent, i.e. real [[Material Value | material values]], 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.

In [6], the relationship between the fracture mechanical [[Material Parameter | parameters]] determined according to the ''J''-integral and the [[Crack Tip Opening Displacement Concept (CTOD) | CTOD concept]] is considered using the example of the temperature dependence of the toughness of unoriented and cold-rolled oriented polypropylene ([[Plastics – Symbols and Abbreviated Terms | abbreviation]]: PP). For the constraint factor, ''m'' = 0.7 is given for the examined PP material [7].

==See also==

*[[Fracture Mechanics | Fracture mechanics]]
*[[Fracture Types | Fracture types]]
*[[Crack Tip Opening Displacement Concept (CTOD) | Crack tip opening displacement concept (CTOD)]]


'''References'''

{|
|-valign="Top"
|[1]
|Cherepanov, G. P.: On Crack Propagation in Continuous Media. Applied Mechanics and Mathematics 31 (1967) 503
|-valign="Top"
|[2]
|Rice, J. R.: A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks. J. Appl. Mech. (1968) 379–386
|-valign="Top"
|[3]
|[[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.):Polymer Testing. Carl Hanser Verlag, Munich (2022) 3. Edition, p. 239–241 (ISBN 978-1-56990-806-8; E-Book ISBN: 878-1-56990-807-5; see [[AMK-Büchersammlung | AMK-Library]] under A 23)
|-valign="Top"
|[4]
|[[Blumenauer, Horst|Blumenauer, H.]], Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (1993) 3. Auflage, (ISBN 3-342-00659-5; siehe [[AMK-Büchersammlung | AMK-Library]] under E 29-3)
|-valign="Top"
|[5]
|Anderson, T. L.: Fracture Mechanics. Fundamentals and Applications. 2nd Ed., CRC Press, Boca Raton (1995) 2. Auflage, (ISBN 978-0849342608; siehe [[AMK-Büchersammlung | AMK-Library]] under E 8-1), DOI: [https://doi.org/10.1201/9781315370293]
|-valign="Top"
|[6]
|[https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.], Che, M.: Assessment of Temperature-dependent Fracture Behaviour with Different Fracture Mechanics Concepts on Example of Unoriented and Cold-rolled Polypropylene. J. Applied Polymer Science 66 (1997) 1237–1249; https://doi.org/10.1002/(SICI)1097-4628(19971114)66:7%3C1237::AID-APP4%3E3.0.CO;2-H
|-valign="Top"
|[7]
|Hille, E.: Untersuchungen zum Bruchverhalten des orientierten isotaktischen Polypropylen. Ph.D. Dissertation, [https://de.wikipedia.org/wiki/Technische_Hochschule_Leuna-Merseburg Technische Hochschule Leuna-Merseburg (1983)]
|}

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