View source for Crack Resistance (R) Curve

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<span style="font-size:1.2em;font-weight:bold;">Crack resistance (R) curve concept</span>
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==Construction of an R curve==

When applying the [[J-Integral Concept|J-integral concept]], it should be noted that, in most cases, [[Fracture|fracture]] is initiated by stable [[Crack Propagation|crack propagation]]. The assessment of [[Crack Toughness|crack toughness]] is based on crack resistance (R) curves. To construct the R curve, the J-value is selected as the loading parameter and plotted as a function of the stable crack growth Δ''a'' ('''Fig. 1''').

[[File:Crack_Resistance_(R)_Curve_Fig1.jpg|500px]]
{| 
|- valign="top"
|width="50px"|'''Fig. 1''': 
|width="600px" |Crack resistance curve of the elastic-plastic fracture mechanics
|}

==Phases of crack propagation==

The ''J''-Δ''a'' curve, also known as the JR-curve, consists of two regions that describe the phases of crack blunting and [[Crack Propagation|crack propagation]]. The blunting line marks the region where blunting occurs at the [[Crack|crack]] tip in the form of a bulge in the crack front (see: [[Crack Opening|crack opening]]) and the [[Stretch Zone|stretch zone]] forms before stable crack growth begins. The following equation applies to the blunting line

{|
|-
|width="20px"|
|width="500px" | <math>J\,=\,q \cdot \sigma_F \cdot \Delta a,</math>
|(1)
|}

where the yield strength (see: [[Yield Stress|yield stress]]) is determined on the basis of the applied load. The factor ''q'' depends on the [[Material & Werkstoff|material]]’s hardening behaviour and is generally given as ''q'' = 2.

The crack propagation behaviour is described by a power law of the form,

{|
|-
|width="20px"|
|width="500px" | <math>J\,=\, c_1 \, \Delta a^{c_2}</math>
|(2)
|}

where
{|
|-
|''c''<sub>1</sub>, ''c''<sub>2</sub>
|...
|represent material constants.
|}

==Physical crack initiation==

The [[Crack Initiation|crack initiation]] phase is quantified by the actual physical crack initiation value and, approximately, by technical crack initiation values. The physical crack initiation value is determined at the site of the original crack termination by measuring the width of the [[Stretch Zone|stretch zone]] ('''Fig. 1'''). This procedure requires SEM images of the [[Fracture Surface|fracture surface]].

Technical crack initiation values ''J''<sub>0.2</sub> are determined at a crack extension Δ''a'' = 0.2 mm onset or from the intersection of the crack propagation curve ('''Eq. 2''') and the blunting line shifted parallel by 0.2 mm ('''Fig. 1''').

For plastics, in contrast to metallic [[Material & Werkstoff|materials]], the determination of ''J''<sub>0.2</sub> at Δ''a'' = 0.2 mm onset in accordance with ESIS TC 4 has become the established practice. When determining crack initiation values, it must be taken into account that the evaluation may only be carried out within specific validity ranges.

The [[Tearing Modulus|tearing modulus]] is derived from the rise of the ''J''-Δ''a'' curve as an additional [[Material Parameter|material parameter]],

{|
|-
|width="20px"|
|width="500px" | <math>T_J\,=\, \frac {dJ}{d \Delta a} \frac{E}{\sigma_F^2}</math>
|(3)
|}

which quantifies the resistance to stable crack propagation.

==The ''J''<sub>TJ</sub>-concept by WILL and MICHEL==

WILL and MICHEL propose a model according to which stable [[Crack Propagation|crack growth]] occurs when the energy dissipated in the [[Plastic Zone|plastic zone]], which varies depending on the material, compensates for the excess available energy caused by the crack propagation. According to this model, stable crack growth should be understood as JTJ -controlled crack growth.

In the same way as the crack field parameter ''J'', the [[Crack Opening|crack opening]] displacement ''δ'' can be used to construct the R-curve. From the ''δ''-Δa curves, the physical crack initiation value ''δ''<sub>i</sub>, the technical crack initiation value ''δ''<sub>0.2</sub> and the parameter ''δT''<sub>δ</sub> can be derived. The parameters based on the CTOD concept differ in their informative value from those determined from J-Δa curves, due to the evaluation being based on plastic deformation.

==See also==

* [[Crack Resistance Curve – Experimental Methods|Crack resistance curve – Experimental method]]
* [[Crack Resistance Curve – Examples|Crack resistance curve – Examples]]
* [[Crack resistance curve – Elastomers Quasistatic|Crack resistance curve – Elastomers quasistatic]]
* [[ICIT – Stop Block Method|ICIT – Stop block method]]
* [[ICIT – Extended Stop-Block Method|ICIT – Extended stop-block method]]
* [[Stretch Zone|Stretch zone]]
* [[J-Integral Concept|J-integral concept]]
* [[JTJ-Concept|JTJ-concept]]

==References==

* Will, P., Michel, B., Zerbst, U.: JTJ-gesteuertes Risswachstum und die Energiebilanz am duktilen Riss. Technische Mechanik 7 (1986) 58–60; [https://journals.ub.ovgu.de/index.php/techmech/article/view/1640 https://journals.ub.ovgu.de/index.php/techmech/article/view/1640] (Access: 17.04.2026)
* [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Kunststoffprüfung. Carl Hanser, Munich (2025) 4th Edition, pp. 250/251 (ISBN 978-3-446-44718-9; E-Book: ISBN 978-3-446-48105-3; see [[AMK-Library]] under A 23)
* ESIS TC 4 (2001): A Testing Protocol for Conducting J-Crack Resistance Curve Test on Plastics
* PD ISO/TS 28660 (2023-03): Plastics – Determination of J-R-Curves – Fracture Toughness
* ASTM D 6068 (2010; reapproved 2018): Standard Test Method for Determining J-R Curves of Plastic Materials