Equivalent Energy Concept – Application Limits

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Equivalent energy concept – Application limits

General

The equivalent energy concept is based on the assumption that the load-load-line displacement curves of test specimens that are geometrically similar but have different thicknesses lie on a common curve [1, 2]. To illustrate the significance of the equivalent energy concept and thus the limits of its applicability (see also: levels of knowledge in fracture mechanics), two specific examples are considered [3].

Example 1: Glass fibre-reinforced polyethylene (PE-HD+GF)

material:	PE-HD + GF (E-glass)
matrix:	ρ = 0.960 g cm-3; MW = 87,300 g mol-1
fibre volume content:	0.09, 0.14 and 0.28
fibre length:	l = 200 µm
fibre diametre:	d = 10 µm
l/d-ratio:	20
experimental Method:	Instrumented Charpy Impact Test (ICIT) and recording of load (F)-deflection (f) diagrams
conditions:	support span/specimen width (s/W) = 4; hammer velocity = 1 m s-1; notch depth/specimen width (a/W) = 0.2
measured variable:	Fmax – maximum load at the point of unstable crack propagation fmax – maximum deflection FQ* – pseudo-elastic load

Experimental results:

Fig. 1:

Dependence of the maximum load Fmax and the pseudo-elastic load FQ* (a) and the maximum deflection fmax on the fibre volume content for PE-HD + GF composites (b)

For determining fracture mechanical characteristic values (see also: fracture mechanical testing) of the critical stress intensity factors K_I

$K_{I}=\sigma (\pi \cdot a)^{1/2}\,f\left({\frac {a_{eff}}{W}}\right)$

the following determination equations are valid (SENB-specimens):

Linear-elastic fracture mechanics (LEFM):

$K_{I}^{LEBM}={\frac {F_{max}\,\cdot \,s}{B\,\cdot \,W^{3/2}}}f\left({\frac {a}{W}}\right)$

Linear-elastic fracture mechanics with small-scale yielding:

$K_{I}^{LEBM}={\frac {F_{max}\,\cdot \,s}{B\,\cdot \,W^{3/2}}}f\left({\frac {a_{eff}}{W}}\right)$

Equivalent energy concept:

$K_{I}^{E}={\frac {F_{Q}^{*}\,\cdot \,s}{B\,\cdot \,W^{3/2}}}f\left({\frac {a_{eff}}{W}}\right)$

(equation for the geometry function, see above)

Experimental results:

Fig. 2:

Dependence of various critical stress intensity factors on fibre volume content for PE-HD + GF composites

A comparative analysis of Fig. 1 and Fig. 2 clearly shows that the critical stress intensity factor is a load- or stress-determined parameter. According to the equations of determination, only the maximum impact load or the pseudo-elastic load is included in the calculation as a direct measured variable.

In the case of larger plastic deformations, the stress intensity factor is only a formal calculation variable that is insufficient for describing toughness either qualitatively or quantitatively, as the deformation behaviour is not taken into account.

The fracture mirror determined by light microscopy on the fracture surface shows the dependence depicted in Fig. 3.

Fig. 3:

Dependence of the fracture mirror as on the fibre volume content for PE-HD+GF composites

Taking into account the amount of stable crack growth in the form of the fracture mirror leads to an increase in the effective crack length a_eff. The initial crack length in these materials was 2 mm, i.e. it can be assumed that the condition of a small plastic zone compared to the initial crack length is not met. In the case of dependencies such as those shown in this example, more information about the material behaviour can be obtained by considering the measured variables than with the fracture mechanical material parameter K_Id. However, since the measured variables are geometry-dependent quantities, another way must be found to quantitatively describe the toughness behaviour of these composites.

The critical crack opening displacement, determined according to DUGDALE's crack model, is suitable for describing the deformability of the materials.

$\delta _{Id}={\frac {1}{n}}(W-a)\,{\frac {4\,f_{max}}{s}}$

Fig. 4:

Dependence of the critical crack opening displacement δId on the fibre volume content φv for PE-HD+GF composites

The parameter ‘critical crack opening displacement’ is always advantageous when the material user or developer is looking for a parameter that clearly describes the increasing embrittlement and represents a geometry-independent variable (see: geometry criterion).

From the above descriptions, it can be deduced that the limit of the suitability of the equivalent energy concept is reached when deformation is impeded, which manifests itself in a decrease in the critical crack opening displacement. If a force-determined and a deformation-determined toughness assessment lead to different conclusions, a parameter must be found for the energy-determined assessment of the fracture behaviour which takes the measured variables force and deflection into account in the evaluation equations.

The J-integral concept can be used for such a fracture mechanical evaluation. The J-integral evaluation method according to SUMPTER and TURNER [4] has proven to be suitable for polymer materials.

Fig. 5:

Dependence of JId-values on fibre volume content for PE-HD+GF composites

The decrease in initial deformability by approx. 40 % is of crucial importance for the evaluation of the failure process of this PE-HD+GF composite system. This influence is reflected in the course of the J-integral parameter, with a maximum in the J values (see Fig. 5) occurring for φ_v ≈ 0.1.

The failure process of short fibre-reinforced plastics (see: short fibre-reinforced composites) is characterised by various micromechanical fracture modes, such as the tearing of the bonds at the fibre end and along the fibre/matrix interface (see also: fibre – matrix adhesion), the onset of sliding processes between the fibre and matrix along a material-specific slip length, stable plastic matrix flow without fibre pull-out, and local brittle fracture (see: types of fracture) of the matrix with fibre pull-out [5].

Example 2: Unoriented and highly oriented polypropylene (PP)

Material system: unoriented and highly oriented PP [6] (cold rolling; degree of orientation fx = 80 %)

Fig. 6:

Arrangement of test specimens and notches in three-point bending specimens and CT-specimens in relation to the rolling direction

Experimental results:

• static loading

• dynamic loading

Fig. 7:

Dependence of the fracture strengths KICLEBM and KICE under static loading KIdLEBM and KIdE under dynamic loading on temperature

See also

• Equivalent energy concept – Basics
• Levels of knowledge in fracture mechanics
• Toughness temperature dependence
• Quasi-static test methods
• Impact loading plastics

References

[1]	Witt, F. J., Mager, T. R.: Fracture Toughness KIcd Values at Temperatures up to 550 °F for ASTM A 533 Grade B, Class 1 Steel. Nucl. Eng. Des. 17 (1971) 91–102 DOI: https://doi.org/10.1016/0029-5493(71)90042-2
[2]	Witt, F. J.: Fracture Behavior of Reactor Pressure Vessel Steel in the Frangible, Transitional and Tough Regimes. Nucl. Eng. Des. 20 (1972) 237–249 DOI: https://doi.org/10.1016/0029-5493(72)90029-5
[3]	Grellmann, W.: Zähigkeitsbewertung mit bruchmechanischen Methoden. In: Schmiedel, H. (Ed.): Handbuch der Kunststoffprüfung. Carl Hanser, Munich Vienna (1992), pp. 145–146 and 175 (ISBN 3-446-16336-0; see AMK-Library under A 3)
[4]	Sumpter, J. G. D., Turner, C. E.: Cracks and Fracture. ASTM STP 601 (1976) 3–18
[5]	Grellmann, W.: Zähigkeitsbewertung mit bruchmechanischen Methoden. In: Grellmann, W., Seidler, S.(Ed.): Kunststoffprüfung. Carl Hanser, Munich (2024) 4th Edition, p. 273 (ISBN 978-3-446-44718-9; E-Book: ISBN 978-3-446-48105-3; see AMK-Library under A 23)
[6]	Hille, E.: Untersuchungen zum Bruchverhalten des orientierten isotaktischen Polypropylen. Ph.D. Dissertation, Technische Hochschule Leuna-Merseburg (1983)