Levels of Knowledge in Fracture Mechanics

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Levels of knowledge in fracture mechanics
Information content of fracture mechanical material parameters

General remarks

When applying fracture mechanics working methods (see: fracture mechanics and fracture mechanical testing) to plastics,

• on the one hand, a number of fundamental methodological findings and experimental procedures can be adopted that were gained during the evaluation of the toughness of metallic materials, and,
• on the other hand, the pronounced time and temperature dependencies of materials with a polymer matrix must lead to plastic-specific methodological developments (see, for example: toughness temperature dependence) [1, 2].

Levels of knowledge in fracture mechanics

From a methodological point of view, the fracture mechanical material parameters or fracture toughness parameters can be divided into three levels of knowledge [3‒5]:

1. Crack toughness as resistance against unstable crack initiation
2. Crack toughness as resistance against stable crack initiation and crack propagation
3. Crack toughness as resistance against the rate of change of loading variables

Fracture toughness as resistance against crack initiation

Level I contains all fracture mechanics material parameters that enable the determination of crack toughness as resistance against unstable crack initiation (see Table 1).

Column 1 lists the respective fracture mechanics concept used to analyse the material behaviour. The second column lists the essential fracture mechanics parameters for characterising the stress and deformation field near the crack tip, and the last two columns list the respective crack failure criterion (see: fracture safety criterion) and describe the informative value of the material parameters.

The concept of linear-elastic fracture mechanics (LEFM), LEFM with small-scale yielding and the equivalent energy concept enable quantitative recording of failure on the basis of force- or stress-determined fracture toughness, as only the force or stress measurements are included in the determination equations. Thus, based on their informative value, LEFM with small-scale yielding and the equivalent energy concept can be classified analogously to LEFM.

Table 1: Level of knowledge I – Fracture mechanics material parameters as resistance against unstable crack initiation

Fracture mechanics concept	Parameters for characterising the stress and deformation field near the crack tip			Fracture safety criterion	Practical relevance
Crack toughness as resistance against unstable crack initiation
linear-elastic fracture mechanics (LEFM)	stress intensity factors			static crack initiation	force- or stress-determined
LEFM LEFM with small scale yielding equivalent energy concept	$K_{I}(a_{0})$ $K_{I}(a_{eff})$ $K_{I}(F_{Q}^{*},\,a_{eff})$	${\Biggr \rbrace }$	critical values: $K_{Ic},\,K_{Id}$	$K_{I}\geq K_{Ic}$
				dynamic crack initiation	fracture toughness (crack toughness)
			Mode I, II, III: $K_{I},\,K_{II},\,K_{III}$	$K_{I}\geq K_{Id}$
elastic-plastic fracture mechanics (EPFM)
CTOD-concept	crack opening displacement $\delta _{Ic},\delta _{Id}$ Mode I, II, III, Mixed Mode			$\delta _{I}\geq \delta _{Ic}$ $\delta _{I}\geq \delta _{Id}$	deformation-determined crack opening displacement
J-integral concept	J-integral $J_{Ic},J_{Id}$ Mode I, II, III, Mixed Mode			$J_{I}\geq J_{Ic}$ $J_{I}\geq J_{Id}$	energy determined J-integral

The parameters of crack opening displacement δ_Ic and δ_Id under static and dynamic loading, determined according to the Crack Tip Opening Displacement Concept (CTOD) of elastic-plastic fracture mechanics (EPFM), prove to be deformation-determined fracture mechanical material parameters, since only deformation values (crack expansion, crack opening and maximum deflection) are taken into account.

In connection with the testing of composite materials, i.e. the toughness assessment of fibre composites but also adhesive bonds, the experimental methods and evaluation formalisms for Mode II, Mode III and mixed-mode stress (see: fracture modes and crack opening modes) have become increasingly important.

While the CTOD concept is deformation-determined and the LEBM concept is fracture force-determined, the J-integral concept enables an energetic interpretation of the fracture behaviour, as both force and deformation measurements are taken into account. The J-integral concept thus plays a central role in the evaluation of fracture behaviour.

Fracture toughness as resistance against stable crack initiation and propagation

When applying the J-integral concept to determine fracture mechanical material parameters, it should be noted that in most cases of mechanical stress, the resulting fracture is initiated by stable crack propagation. The evaluation of fracture toughness as resistance to stable crack initiation and propagation (see Table 2) is based on the crack resistance (R) concept.

Table 2: Level of knowledge II – Fracture mechanics material parameters as resistance against stable crack initiation and propagation

Fracture mechanics concept	Parameters for characterising the stress and deformation field near the crack tip			Fracture safety criterion	Practical relevance
Crack toughness as resistance against stable crack initiation and propagation
crack resistance (R) concept	crack initiation	crack propagation	generalised crack propagation	crack initiation
				$\delta ;J_{component}\geq \delta _{i};J_{i\,resp.\,0.2}$	$J\cdot T_{J},\delta \cdot T_{\delta }$-controlled stable crack propagation
CTOD-resistance curve	$\delta _{/phys},\delta _{0.2}$	$T_{\delta }={\frac {d\,\delta }{d(\Delta a)}}\,{\frac {E}{R_{e}}}$	$\delta \cdot T_{\delta }$
				crack propagation
J-R-curve	$J_{/phys},J_{0.2}$	$T_{J}={\frac {d\,J}{d(\Delta a)}}\,{\frac {E}{R_{e}^{2}}}$	$J\cdot T_{J}$	${\frac {d\,J_{component}}{d(\Delta a)}}\geq {\frac {d\,J_{material}}{d(\Delta a)}}$
J-T-stability diagram	instability value $J_{50}$			Instability caused by advanced crack propagation	energy determininated plastic instability
				$T_{component}\geq T_{material}$

For the construction of crack resistance (R) curves, the crack opening δ and the J-integral value are preferred as loading parameters over the stress intensity factor K_I of the LEFM concept.

The resistance to actual crack initiation is denoted by J_iphys or δ_iphys, and the resistance to stable crack propagation by T_δ or T_J, which represent the rise of the R curve multiplied by E/R_e or E/R_e², respectively.

Based on the energy balance at the crack, Will and Michel [6, 7] introduced a practical model for evaluating stable crack growth (see: JT_J-concept).

After that, stable crack growth occurs when the energy dissipated in the plastic zone specific to the material compensates for the excess available energy caused by the crack growth.

Crack growth is then controlled by the product JT_J or δT_δ and is referred to as JT_J- or δT_δ-controlled stable crack growth. Numerous examples of model systems provide experimental evidence for the existence of JT_J-controlled crack growth in the literature [3‒5].

Paris and Johnson [8] introduced the instability parameter J₅₀ as a further toughness criterion. A graphical method [9] is used to determine this experimentally, transforming J-Δa- into J-T-stability diagrams.

Such diagrams make it possible to draw conclusions about the growth instability of cracks in components without using the method for determining crack toughness as resistance to unstable crack propagation.

Fracture toughness as resistance against the rate of change of parameters

For the determination of crack resistance as resistance to the rate of change of parameters based on knowledge level III, on the one hand, only limited experimental results are available, and on the other hand, decisive progress can be expected here in material-related toughness assessment [10‒13] (see Table 3).

Table 3: Level of knowledge III – Fracture mechanics material parameters as resistance against the rate of change of parameters by taking into account the temporal change in deformation

Fracture mechanics concept	Parameters for characterising the stress and deformation field near the crack tip	Fracture safety criterion	Praktical relavance
Crack toughness as resistace against rate of chance of parameters
	consideration of time-dependent changing of deformation energy
modified CTOD- and J-concept	${\dot {\delta }}_{Ic}={\frac {\delta _{Ic}}{t_{B}}};\;{\dot {\delta }}_{Id}={\frac {\delta _{Id}}{t_{B}}}$ t B – time to fracture ${\dot {J}}_{Ic}={\frac {J_{Id}}{t_{B}}};\;{\dot {J}}_{Id}={\frac {J_{Id}}{t_{B}}}$	${\dot {\delta }}_{I}\geq {\dot {\delta }}_{Ic};\;{\dot {\delta }}_{I}\geq {\dot {\delta }}_{Id}$ ${\dot {J}}_{I}\geq {\dot {J}}_{Ic};\;{\dot {J}}_{I}\geq {\dot {J}}_{Id}$	${\dot {\delta }}$- bzw. ${\dot {J}}$-controlled crack propagation (energy-rate-determined plastic instability)
${\dot {R}}$-concept
${\dot {\delta }}-\Delta a$ ${\dot {J}}-\Delta a$	${\dot {\delta }}_{i};\;{\dot {T}}_{\delta }\;\;\;{\dot {\delta }}_{i}\cdot {\dot {T}}_{\delta }$ ${\dot {J}}_{i};\;{\dot {T}}_{J}\;\;\;{\dot {J}}_{i}\cdot {\dot {T}}_{J}$	${\dot {T}}_{\delta _{component}}\geq {\dot {T}}_{\delta _{material}}$ ${\dot {T}}_{J_{component}}\geq {\dot {T}}_{J_{material}}$

The loading speed is known to have a significant influence on the fracture behaviour of polymeric materials. When conducting fracture mechanics tests, the crosshead speed of the materials testing machine, the impact velocity of a pendulum or drop hammer, or the impact velocity of projectiles in arrest tests are usually specified. However, the data on the impressed velocity are not comparable, as different test specimen types cause different conversions of the load-line displacement into the deformation of the crack tip area (see also: fracture process zone).

The crack opening displacement velocity appears to be a suitable measure for describing toughness as resistance to the rate of change, as it is a comparable parameter that compensates for the influence of different test specimen geometries. Analogous to this description of crack growth, the J-integral and the R-curve concept, as well as the LEFM concept, can be used to assess the material based on the change in loading over time. The parameter J can thus be understood as a change in energy rate, and this is referred to as J-controlled crack growth.

Assessment of the state of development of fracture mechanics values determination and the automation of stable crack growth detection

With the classification of fracture mechanics material parameters, the transition to the next higher level of knowledge increases the theoretical demands regarding knowledge of fracture mechanics concepts, the degree of difficulty in experimental methodology, and also the effort required for automation, e.g., in determining the length of stable crack growth (Table 4).

Table 4: State of development in fracture mechanics material testing, problems in determining characteristic values and objectives

	Characteristic values as resistance against crack initiation and propagation
Level	I	II	III
State	for unstable crack growth	for stable crack growth	as function of the rate of change
effort required to determine the values	relatively low, easy to automate	relatively high, very difficult to automate to date	very high
current distribution of toughness values	widespread use in industrial practice	limited distribution to specialised testing institutes and knowledge carriers	still a current research topic
target objectives	fully automated value determination	automated crack length determination	distribution in practice, development of evaluation procedures, standardisation
information content	single-parameter description of fracture behaviour	multi-parameter description of fracture behaviour	comprehensive assessment of crack propagation kinetics

See also

• Fracture mechanics
• Fracture mechanical testing
• Fracture safety criterion
• J-integral concept
• Crack Tip opening displacement concept
• JTJ-concept
• Fracture behaviour of plastic components

References

[1]	Blumenauer, H., Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig Stuttgart (2003), 3rd Edition, (ISBN 3-342-00659-5; see AMK-Library under E 29-3)
[2]	Anderson, T. L.: Fracture Mechanics ‒ Fundamental and Applications. CRC Press, Boca Raton (2005) (ISBN 978-0849342608; see AMK-Library under E 8-2), DOI: https://doi.org/10.1201/9781315370293
[3]	Grellmann, W., Seidler, S.: Anwendung des instrumentierten Kerbschlagbiegeversuches in der Werkstoffentwicklung von Kunststoffen. DVM-Tagung „Werkstoffprüfung 1990“, December 6 and 7, 1990, Bad Nauheim Proceedings. pp. 79‒88
[4]	Grellmann, W.: Aussagefähigkeit bruchmechanischer Werkstoffkenngrößen bei der Werkstoffentwicklung von Polymerblends. Proceedings 3. Erlanger Kunststoff-Tage, Erlangen, April 21‒23, 1993, Neue polymere Werkstoffe, Zahradnik, F., Kaschta, J. (Eds.), Self-Publishing (1993), Erlangen, Proceedings pp. 175‒195
[5]	Grellmann, W.: Neue Entwicklungen bei der bruchmechanischen Zähigkeitsbewertung von Kunststoffen und Verbunden. In: Grellmann, W., Seidler, S. (Eds.): Deformation und Bruchverhalten von Kunststoffen. Springer Berlin Heidelberg (1998) pp. 3‒26 (ISBN 3-540-63671-4, see AMK-Library under A 6), DOI: https://link.springer.com/chapter/10.1007/978-3-642-58766-5_1#citeas
[6]	Will, P.; Michel. B., Zerbst, U.: JTJ-gesteuertes Risswachstum und die Energiebilanz am duktilen Riss. Technische Mechanik 7 (1986) 58‒60
[7]	Will, P.: JTJ-Konzept und dissipative Energien am Riss. In: Grellmann, W., Seidler, S. (Eds.) Deformation und Bruchverhalten von Kunststoffen. Springer Berlin Heidelberg (1998) pp. 27‒34 (ISBN 3-540-63671-4; see AMK-Library under A 6)
[8]	Paris P. C., Johnson R. E.: Fracture Resistance Curves and Engineering Applications. ASTM STP 803 Vol II: 5 (1983)
[9]	Lach, R., Grellmann, W.: JTJ- und δTδ -Stabilitätsdiagramme als Grundlage einer alternativen Methode zur Ermittlung von Instabilitätswerten aus Risswiderstandskurven. In: Grellmann, W., Seidler, S. (Eds.) Deformation und Bruchverhalten von Kunststoffen. Springer Berlin Heidelberg (1998) 145‒154 (ISBN 3-540-63671-4; see AMK-Library under A 6), DOI: https://link.springer.com/chapter/10.1007/978-3-642-58766-5_11#citeas
[10]	Seidler, S. (1998): Anwendung des Risswiderstandskonzeptes zur Ermittlung strukturbezogener bruchmechanischer Werkstoffkenngrößen bei dynamischer Beanspruchung. Fortschritt-Berichte, VDI-Reihe 18: Mechanik/Bruchmechanik No. 231, VDI-Publishing Düsseldorf (ISBN 3-18-323118-2; see AMK-Library under B 2-1)
[11]	Lach, R., Grellmann, W.: Time-dependent Fracture Behaviour of Polymers at Impact and Quasi-Static Loading Conditions. In: Grellmann, W., Langer, B. (Eds.): Deformation and Fracture Behaviour of Polymer Materials. Springer, Berlin (2017) 3‒21 (ISBN 978-3-319-41877-3; see AMK-Library under A 19), DOI: https://link.springer.com/chapter/10.1007/978-3-319-41879-7_1#citeas
[12]	Lach, R., Seidler, S., Grellmann, W.: Resistance Against the Intrinsic Rate of Fracture Mechanics Parameters for Polymeric Materials under Moderate Impact Loading. Mechanics of Time-Dependent Materials 9 (2005) 103‒119, DOI: https://doi.org/10.1007/s11043-005-1084-y
[13]	Lach, R., Grellmann, W.: Time- and Temperature-dependent Fracture Mechanics of Polymers: General Aspects at Monotonic Quasistatic and Impact Loading Conditions. Macromolecular Materials and Engineering 273 (2008) 555‒567, DOI: https://doi.org/10.1002/mame.200700417

Additional literature on assessing the toughness of polymers using the JTJ concept

• Grellmann, W., Seidler, S.: Risszähigkeit von Kunststoff-Messungen bei dynamischer Beanspruchung. Materialprüfung 33 (1991) 7-8, pp. 213–218

• Grellmann, W., Seidler, S., Oberbach, K.: Ermittlung dynamischer Risswiderstandskurven von Polymerblendes mit Hilfe des instrumentierten Kerbschlagbiegeversuches, 23. Vortragsveranstaltung, DVM Arbeitskreis "Bruchvorgänge", Berlin, Februar 26‒ 27, 1991, Proceedings, pp. 401‒412

• Seidler, S., Grellmann, W.: Bruchverhalten und Morphologie von PC/ABS Blends ‒ Anwendung moderner Konzepte der Fließbruchmechanik zur Optimierung der Zähigkeit. 2nd Erlanger Kunststoff Tage, Erlangen, April 17‒19, 1991, Werkstoffcharakterisierung und Qualitätssicherung, Zahradnik, F., Kaschta, J. (Eds.), Self-Publishing (1991), Erlangen, pp. 125‒145

• Grellmann, W., Seidler, S., Oberbach, K.: Dynamische Risswiderstandskurven von Polymerblends. 14. Gesa Symposium "Experimentelle Mechanik in Forschung und Praxis", Berlin, April 25‒ 26, 1991, VDI Berichte Nr. 882 (1991), pp. 433‒443

• Seidler, S., Grellmann, W.: Charakterisierung des Risswiderstandsverhaltens von Kunststoffen mit dem JTJ-Konzept. "Werkstoffprüfung 1992", Bad-Nauheim, December 3 and 4, 1992, Proceedings pp. 387‒393

• Seidler, S., Grellmann, W.: Anwendung bruchmechanischer Werkstoffkenngrößen in der Kunststoffentwicklung. Plaste und Kautschuk 40 (1993) 8., pp. 263‒269 Download as pdf

• Grellmann, W., Seidler, S., Langer, B.: J-Integral-Analyse von Kurzfaser-Verbundwerkstoffen. "Werkstoffprüfung 1993", Bad Nauheim, December 2 and 3, 1993 Proceedings pp. 317‒325

• Grellmann, W., Seidler, S., Jung, K.: Stand und Entwicklungstendenzen bei der Anwendung des Risswiderstandskonzeptes in der Kunststoffprüfung. Werkstoffprüfung 1994, Bad-Nauheim, December 1 and 2, 1994, Proceedings pp. 273‒281

• Seidler, S., Grellmann, W.: Application of the Instrumented Impact test to the Toughness Characterization of High Impact Thermoplastics. Impact and Dynamic Fracture of Polymers and Composites, ESIS Publication 19 (Edited by J. G. Williams and A. Pavan), Mechanical Engineering Publications, London, 1995 pp. 171‒178

• Seidler, S., Grellmann, W., Langer, B.: Anwendbarkeit des Risswiderstandskonzeptes zur Zähigkeitsbewertung von kurzfaserverstärktem Polyamid. 27. Vortragsveranstaltung, DVM Arbeitskreis "Bruchvorgänge" Köln, Februar 14 and 15, 1995, Proceedings pp. 63‒72

• Seidler, S., Grellmann, W.: Application of the Instrumented Impact Test to the Toughness Characterization of High Impact Thermoplastics. Polymer Testing 14 (1995) 453‒469; https://doi.org/10.1016/0142-9418(95)00003-B

• Grellmann, W.: Morphologie-Zähigkeits-Korrelation polymerer Mehrphasenwerkstoffe ‒ Aussagefähigkeit und strukturelle Empfindlichkeit bruchmechanischer Werkstoffkenngrößen. Tagung "Gefüge und Bruch", Leoben, March 20‒22, 1996, Proceedings pp. 1‒8

• Grellmann, W., Seidler, S., Jung, K., Gahleitner, M., Fiebig, J.: Bruchverhalten und Morphologie von PP-Reaktorblends. Werkstoffwoche`96, Stuttgart, May 28‒31, 1996, Symposium 7 "Materialwissenschaftliche Grundlagen", Proceedings pp. 933‒938

• Seidler, S., Grellmann, W.: Fracture Behaviour and Morphology of Polymers. Ninth International Conference on Fracture, ICF 9 Sydney, April 1‒5, 1997, Proceedings Volume 2, pp. 1021‒1027

• Seidler, S., Grellmann, W.: Application of the Instrumented Impact Test to the Toughness Characterization of High Impact Thermoplastics. Polymer Testing 14 (1995) 453‒469 DOI: https://doi.org/10.1016/0142-9418(95)00003-B