Essential Work of Fracture (EWF)-Concept

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Essential Work of Fracture (EWF)-Konzept

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Essential work of Fracture (EWF)-concept

Basic assumption

The Essential Work of Fracture (EWF) concept was first mentioned by Broberg [1] in 1968 and subsequently developed further [2, 3].

The concept is based on the assumption that a wide range of materials, particularly plastics and their modifications, are capable of transmitting forces even under large deformations. It assumes that the inelastic region at the tip of a crack can be divided into an inner region, where the actual fracture process (process zone) takes place, and an outer region, in which energy dissipation occurs through plastic deformation (see also: plastic zone).

The EWF concept is part of the Post-Yield Fracture Mechanics (PYFM) concept, which is primarily applied under conditions of plane stress.

Specific fracture energy

The total work of fracture required to cause a notched specimen to fracture is composed of two components, W_e and W_p, corresponding to the two zones in front of the crack tip.

We is referred to as the essential work of fracture and describes crack growth and the creation of new surfaces (see: fracture surface). W_p is referred to as the non-essential work of fracture, which encompasses the dissipation of energy resulting from plastic deformation in the fracture process zone.

W_{f}=W_{e}+W_{p}

(1)

Based on the initial cross-sectional area (the product of the ligament length l and the specimen thickness B), the following equation is obtained

W_{f}=w_{f}lB=w_{e}lB+\beta w_{p}l^{2}B

(2)

where β is the shape factor of the plastic zone. Dividing the equation by the cross-sectional area of the test specimen results in the specific fracture energy:

w_{f}=w_{e}+\beta w_{p}l

.

(3)

The essential work of fracture we is a measure of resistance to crack initiation. The product of w_p and the plastic zone form factor β is a measure of resistance to stable crack propagation [4, 5].

A critical analysis of the applicability of the EWF concept to ductile plastics under conditions of plane strain at quasi-static and impact loading rate in a three-point bending setup (SENB-specimens) and a tensile testing setup (SENT- and DENT-specimens) was presented by Kotter in [6].

Application examples

Dependence of specific fracture energy on ligament length for a PE pipe material

In [7], Langer and Berthold demonstrate the application of the EWF concept to a polyethylene (PE) pipe material. For the investigations, double edge-notched tension DENT-specimens with a geometry of 10 x 1.5 mm² and ligament lengths of 2, 4, 6 and 8 mm were used. The investigations were carried out at room temperature and at 80 °C, with a loading speed of 500 mm/min. Figure 1 shows the fracture energy determined as a function of ligament length, using the pipe material as an example.

Fig. 1:

Application of the EWF concept: Dependence of the specific fracture work w_f on the ligament length l, and determination of w_e and β w_p using the example of a PE pipe material [7]

Dependence of the specific fracture work on ligament length for an isotactic PP material

In [9], Karger-Kocsis uses the EWF concept to demonstrate the differences in the toughness behaviour of α- and β-nucleated isotactic polypropylene (see Fig. 2).

Fig. 2:

Specific fracture work as a function of ligament length for α- and β-nucleated isotactic polypropylene [9]

Evaluation of the artificial ageing of a biopolymer film using the EWF concept

In [10], Monami demonstrates the sensitivity of the EWF method in detecting age-related changes in the toughness properties of biodegradable mulch films (see also: bio-plastics – impact modified), whose functionality is achieved through controlled ageing behaviour. For the quasi-static tests, double-edge-notched (DENT)-specimens with a geometry of 100 mm in length, 25 mm in width and a ligament length of 3 mm, 6 mm, 9 mm, 12 mm, 15 mm and 18 mm were used. The test speed was 10 mm/min. The films were 15 µm thick and were subjected to artificial ageing in water and in air. The test specimens were cut in the machine direction. Figure 3 illustrates, using the example of a commercially available biopolymer film with the trade name Mater-Bi®, a starch blend based on maize starch, the percentage change in the fracture mechanical properties essential (w_e) and non-essential (W_p or βw_p) fracture work relative to the initial values.

Fig. 3:

Changes in the essential (we) and non-essential (βwp) fracture work of the mulch film in water and air at 80 °C in each case

The non-essential fracture work shows a strong increase after 24 hours; with further artificial ageing in air, this characteristic value remains virtually constant over the period studied, but decreases for films stored in water as the ageing period increases. The essential fracture work of the biopolymer film decreases significantly after just 24 hours for both films stored in water and in air, and remains virtually constant as the ageing duration increases. Upon failure of the aged film, the energy dissipated in the fracture process zone to form new fracture surfaces decreases, as described by we, whilst more work is consumed in the outer plastic zone, as described by βw_p. Thus, the application of the EWF method not only allows the detection of changes in mechanical properties caused by artificial ageing in water or air, but also the detection of changes in fracture mechanisms resulting from a shift in energy absorption from the formation of new surfaces towards the dissipation of work in the outer plastic zone. The results demonstrate the sensitivity of the EWF method in detecting age-related changes in properties [10].

Numerous publications demonstrate the successful application of the EWF concept to various plastics for assessing the toughness of thin specimens or films (under conditions of plane stress) [8‒15].

References

[1]	Broberg, K. B.: Critical review of some theories in fracture mechanics. International Journal of Fracture Mechanics 4 (1968) 11‒19; DOI: https://link.springer.com/article/10.1007/BF00189139#citeas9
[2]	Broberg, K. B.: Crack growth criteria and non-linear fracture mechanics. Journal of Mechanics and Physics of Solids 19 (1971) 407‒418; DOI: https://doi.org/10.1016/0022-5096(71)90008-1
[3]	Broberg, K. B.: On stable crack growth. Journal of Mechanics and Physics of Solids 23 (1975) 215‒237; DOI: https://doi.org/10.1016/0022-5096(75)90017-4
[4]	Mai, Y.-W., Cotterell, B.: On the essential work of ductile fracture in polymers. International Journal of Fracture 32 (1986) 105‒125; DOI: https://link.springer.com/article/10.1007/BF00019787#citeas
[5]	Mai, Y.-W., Powell, P.: Essential work of fracture and J-integral measurements for ductile polymers. Journal of Polymers Science: Part B: Polymer Physics 29 (1991) 758‒793; DOI: https://doi.org/10.1002/polb.1991.090290702
[6]	Kotter, I.: Morphologie-Zähigkeits-Korrelationen von EPR-modifizierten Polypropylenwerkstoffen. Martin-Luther-Universität Halle-Wittenberg, Dissertation, 2003, ISBN 978-3-898206440, Mensch & Buch Verlag Berlin, 2003 (see AMK-Library under B 1-11)
[7]	Langer, B., Berthold, A., Grellmann, W., Enderle, H.-F.: Mechanische Kurzzeitprüfung zur Bewertung des Verhaltens von PE-Rohrwerkstoffen beim langsamen Risswachstum. Materialprüfung 54 (2012) 9, pp. 580‒585; DOI: https://doi.org/10.3139/120.110364
[8]	Mouzakis, D. E.: Application of the Essential Work of Fracture Method for Ductile Polymer Systems. Mensch & Buch Verlag, Berlin, 1999
[9]	Karger-Kocsis, J.: How does „Phase transformation toughening“ work in semicrystalline polymers?. Polymer Engineering and Science 36 (1996) 203‒210; DOI: https://doi.org/10.1002/pen.10403
[10]	Monami, A., Langer, B., Grellmann, W.: Moderne Methoden der Kunststoffprüfung zur Werkstoffentwicklung und Bauteilprüfung. Werkstoffprüfung 2016, Fortschritte in der Werkstoffprüfung für Forschung und Praxis. December 1 and 2, 2016, Neu-Ulm, Proceedings pp. 219–224 (ISBN 978-3-514-00830-4; see AMK-Library under M 61)
[11]	Karger-Kocsis, J.: For what a kind of polymer is the toughness assessment by the essential work concept straightforward?. Polymer Bulletin 37 (1996) 119‒126; DOI: https://link.springer.com/article/10.1007/BF00313827#citeas
[12]	Marchal, Y., Oldenhove, B., Daoust, D., Legras, R., Delannay, F.: Characterization of the fracture toughness of rubber-toughened polypropylene thin plates. Polymer Engineering and Science 38 (1998) 2063‒2071
[13]	Ferrer-Balas, D., Maspoch, M. L., Martinez, A. B., Santana, O. O.: On the essential work of fracture method: Energy partitioning of the fracture process in iPP films. Polymer Bulletin 42 (1999) 101‒108
[14]	Maspoch, M. L., Ferrer, D., Gordillo, A., Santana, O. O., Martinez, A. B.: Effect of the specimen dimensions and the test speed on the fracture toughness of iPP by the essential work of fracture method. Journal of Applied Polymer Science 73 (1999) 177‒187
[15]	Lach, R., Celevics, S., Jahn, I., John, M., Teuscher, N., Tillner, B., Langer, B., Grellmann, W.: Mechanical and fracture mechanics investigations of uni-directionally fibre-reinforced thermoplastic polymer tapes. Structural Integrity Procedia (2025) 1337‒1342; DOI: https://doi.org/10.1016/j.prostr.2025.06.208

Standard reference

ISO 23524 (2022-10): Plastics – Determination of Fracture Toughness of Films and Thin Sheets ‒ The Essential Work of Fracture (EWF) Method