Plastic Hinge Model

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Plastic hinge model (Türangelmodell)

Fundamentals

The plastic hinge model is a widely used model for determining the critical crack opening displacement for three-point bending specimens (SENB-specimens) [1].

While under quasi-static loading the critical crack opening can be determined by optical methods or by measuring the notch expansion v and extrapolating to the crack tip, under dynamic loading (see: impact loading plastics) only an indirect determination via electronically measured deflection is possible. In [2], Zeislmair and Dahl compile numerous methods for determining crack opening values under static stress, most of which were developed empirically, and compare the results. A prerequisite for determining the critical crack opening under dynamic loading is that a quasi-static stress state (see: plane stress and strain state) develops in the test specimen, the formation of which is ensured by compliance with the condition of Eq. (1).

t_{B}\geq 2,3\ ...\ 3\tau

(1)

mit

t_B		fracture time
τ		period of characteristic inertial oscillation (see also: inertial load)

is controlled.

Principle of the plastic hinge model

Comparative studies of various models for bend loading on a SENB-specimen have led to the conclusion that the critical crack opening should be determined on the basis of the ‘plastic hinge’ model (door hinge model) [2−5]. In this model, the crack flanks open under increasing test load, like a hinge, around a point designated as the centre of rotation in front of the crack tip, whereby only one half of the specimen was considered due to the symmetry of the test specimen (Fig. 1).

Fig. 1:

Principle of the plastic hinge model for SENB- specimens (1-centre of rotation, 2 sharp notch, 3-support) [6]

The experimentally determined deflection f_max is composed of a part caused by the bending of the unnotched part f_B and a part caused by the deformation in the area of the notch f_K according to Eq. (2),

f_{max}=f_{B}+f_{K}\!

(2)

where the bending part is calculated using the relationship

f_{B}={\frac {F_{max}\ s^{3}}{4\ E\ B\ W^{3}}}

(3)

mit

E		modulus of elasticity
s		support distance
B		test specimen thickness
W		specimen width

is calculated.

Determination of the critical crack opening

For the SENB-specimen stressed by bending, the following equation applies based on the plastic hinge model (Fig. 1)

\delta _{k}={\frac {1}{n}}(W-a){\frac {4\ f_{K}}{s}}

(4)

This reduces the calculation of the critical crack opening (see: extended CTOD concept) to the area at the notch tip by subtracting the part of the deflection of an unnotched test specimen from the maximum deflection f_max of the notched test specimen (according to Eq. (2)).

As can be seen from the literature [2, 7‒9], the rotation factor n depends on the load and moves towards the crack tip as the load increases.

The rotation factor n

Fig. 2:

Dependence of the rotation factor n on the load for PP 1 a) (1 to 5 selected heat treatment conditions) and PP 2 b) (6 to 9 selected degrees of orientation) [7, 8]

For CT-specimens loaded under quasi-static conditions, simultaneous registration of notch expansion v_c and load-point displacement v_L for various polymer materials showed that the rotation factor assumes the limit value n = 4 at the moment of fracture (see Fig. 2) [6, 10].

Influence of the a/W-ratio

Figure 3 shows the critical crack opening δ_dk calculated for a polyamide material (abbreviation: PA) with elastic material behaviour according to Eq. (4) with a rotation factor n = 4 in comparison to δ_d and δ_dB as a function of the a/W-ratio.

Fig. 3:

Influence of the a/W-ratio on the critical crack openings δ_d, δ_dB and δ_dK for unstable crack propagation in polyamide

To calculate δ_d and δ_dB, f_K is replaced by f_max and f_B respectively in Eq. (4). Taking into account the results presented in [1] for PE-HD+Hp and another PP material, PP 3, it can be seen that

the bending component is dominant in determining critical crack opening values for small a/W, with δ_dB becoming smaller as a/W increases. For polyamide, δ_dB is 75 % at a/W = 0.1 and only 25 % at a/W = 0.7,
the proportion δ_dB for PP 3 at s/W = 7 becomes negligibly small for high a/W,
the crack openings δ_dK determined from the notch proportion f_K according to Eq. 4 are independent of the a/W-ratio.

The independence of δ_dK from the a/W-ratio is shown in Fig. 4 for selected polymer materials.

Fig. 4:

Dependence of critical crack openings δ_dK on the a/W-ratio for PP 3 (v_H = 1.5 ms^-1; s/W = 7), PE-HD+Bw (v_H = 1.5 ms^-1; s/W = 4), PVCC (v_H = 1 ms^-1; s/W = 4), PE-HD+Hp (v_H = 1.5 ms^-1; s/W = 4) and PA (v_H = 1 ms^-1; s/W = 4)

The increase in δ_dK values apparent for a/W < 0.2 is due to the very high energy absorption of the test specimens during the impact process for the lowest a/W-ratios (see: impact loading plastics), so that the empirical condition for controlling the energy absorption during the impact process (see: ICIT ‒ Experimental conditions), according to which the impact energy A_H specified by the pendulum chamber for the fracture process must be greater than 3 times the deformation energy A_G consumed by the test specimen, cannot be fulfilled with sufficient certainty for all materials and for PVCC only from a/W ≤ 0.17, which is why the curve in Fig. 4 is only marked with a dashed line.

The caption for Fig. 4 clearly shows the stress conditions selected for the various polymer materials in terms of compliance with the experimental conditions of the instrumented Charpy impact test, which lead to different deformation rates at the crack tip. The crack opening velocity according to Eq. (5)

{\frac {d\delta _{dk}}{dt}}={\frac {\delta _{dk}}{t_{B}}}

(5)

can be used as a measure to describe them (see also: levels of knowledge in fracture mechanics).

The fracture time t_B is calculated according to Eq. (6).

t_{B}={\frac {f_{max}}{v_{H}-{\frac {1}{3}}\Delta v}}

(6)

Δv refers to the change in pendulum hammer speed during the deformation process, which, according to Eq. (7), is expected to be greater the higher the deformation energy A_G of the test specimen and the pendulum hammer speed v_H before the impact:

\Delta v=v_{H}\left(v_{H}^{2}-{\frac {2}{m_{H}}}A_{G}\right)^{\frac {1}{2}}

(7)

m_H = mass of the pendulum hammer, determined by weighing in a horizontal position [11].

Crack opening velocity of selected plastics

Figure 5 shows the crack opening velocities determined from Eq. (5) as a function of the a/W-ratio for selected plastics. With the exception of PP 3, the increases in dδ_dK/dt up to approximately a/W = 0.5 result primarily from the decrease in f_max with increasing a/W, and the decrease in dδ_dK/dt for a/W > 0.5 results from the fact that fmax increases again with a simultaneous continuous reduction in Δv.

Fig. 5:

Crack opening velocities dδ_dK/dt for various plastics

The dependence of dδ_dK/dt (see: levels of knowledge in fracture mechanics) on the a/W ratio for PP 3 indicates that for s/W = 7, the stress on the material in the crack tip area is comparable to that of other plastics, whereby, in addition to the high fracture times (here, the influence of the test specimens pulling through the supports must be taken into account), the low bending component δ_dB has an effect, meaning that the notch component at s/W = 7 is already of decisive importance even for small a/W values.

When the pulling of the test specimens through the abutments is taken into account (see: support distance), the dδ_dK/dt curve is shifted to higher values.

The crack opening velocity dδ/dt (see: levels of knowledge in fracture mechanics) also offers the possibility of converting the different stress conditions arising for different test specimen geometries to a test specimen-invariant deformation rate at the crack tip and using this as a parameter in material-specific investigations.

References

[1]	Grellmann, W.: Beurteilung der Zähigkeitseigenschaften von Polymerwerkstoffen durch bruchmechanische Kennwerte. Habilitation (1986), Technische Hochschule Leuna-Merseburg, Wiss. Zeitschrift TH Merseburg 28 (1986), No. 6, pp. 787‒788 (Content, summary)
[2]	Schwalbe, K. H.: Bruchverhalten keramischer Werkstoffe: Methoden und Ergebnisse. Fortschritts-Berichte VDI-Z. Reihe 18, Nr. 10, VDI-Verlag, 1981 (ISBN 978-3-1814-1118-6)
[3]	Blumenauer, H., Pusch, G.: Technische Bruchmechanik. Deutscher Verlag für Grundstoffindustrie, Leipzig (1982), (ISBN: VLN 152-915/62/73; see AMK-Library under E 29-1)
[4]	Kobayashi, T.: On the information about fracture characteristics obtained from instrumented impact test of A533 steel for reactor pressure vessel. Eng. Fracture Mechanics 19 (1984) 67; DOI
[5]	Schwalbe, K.-H.: Bruchmechanik metallischer Werkstoffe. Carl Hanser, Munich Vienna (1980), (ISBN 3-446-12983-9; see AMK-Library under E 15)
[6]	Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 236–239 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[7]	Hille, E.: Untersuchungen zum Bruchverhalten des orientierten isotaktischen Polypropylen. Dissertation, Technische Hochschule Leuna-Merseburg (1983) (1983)
[8]	Newe, R.: Untersuchungen zum Bruchverhalten des nichtorientierten isotaktischen PP. Dissertation, Technische Hochschule Leuna-Merseburg (1980)
[9]	Hollstein, T., Blauel, J. G., Wenk, K.: 12. Sitzung des Arbeitskreises Bruchvorgänge im DVM Freiburg, 7–8 October 1980
[10]	Jungbluth, M.: Untersuchungen zum Verformungs- und Bruchverhalten von PVC-Werkstoffen. Dissertation, Technische Hochschule Leuna-Merseburg (1987) (see AMK-Library under B 1-1) (Short bibliographic summary /Content)
[11]	ISO 13802 (2025-08): Plastics ‒ Verification of Pendulum Impact-testing Machines – Charpy-, Izod- and Tensile Impact Testing