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{{Language_sel|LANG=ger|ARTIKEL=Äquivalentenergiekonzept – Anwendungsgrenzen}}
{{PSM_Infobox}}
Equivalent energy concept – Application limits
__FORCETOC__

==General==

The [[Equivalent Energy Concept – Basics|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|levels of knowledge in fracture mechanics]]), two specific examples are considered [3].

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

{|
! width="200px" |
! width="450px" |
! valign="top" |
|-
|[[Material & Werkstoff|material]]:
|PE-HD + GF (E-glass)
|-
|matrix:
|''ρ'' = 0.960 g cm-3; ''M''W = 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:
|[[ICIT – Support Span Method|support span/specimen width]] (''s''/''W'') = 4; [[ICIT – Influence of Pendulum Hammer Velocity|hammer velocity]] = 1 m s-1; notch depth/specimen width (''a''/''W'') = 0.2
|-
|[[Measured Variable|measured variable]]:
|<ul><li>''F''max – maximum load at the point of unstable [[Crack|crack]] propagation</li><li>''f''max – maximum deflection</li>
<li>''F''Q* – pseudo-elastic load</li></ul>
|}

''Experimental results:''

[[File:equiv_anwend_bild2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Dependence of the maximum load ''F''max and the pseudo-elastic load ''F''Q* (a) and the maximum deflection ''f''max on the fibre volume content for PE-HD + GF composites (b)
|}

For determining fracture mechanical characteristic [[Material Value|values]] (see also: [[Fracture Mechanical Testing|fracture mechanical testing]]) of the critical stress intensity factors ''K''I

{|
|-
|width="20px"|
|width="500px" | <math>K_I=\sigma(\pi\cdot a)^{1/2}\,f\left(\frac{a_{eff}}{W}\right)</math>
|}

the following determination equations are valid ([[SENB-Specimen|SENB-specimens]]):

'''[[Fracture Mechanics#Linear-elastic fracture mechanics|Linear-elastic fracture mechanics]] (LEFM):'''

{|
|-
|width="20px"|
|width="500px" | <math>K^{LEBM}_I=\frac{F_{max} \, \cdot\,s} {B \, \cdot\, W^{3/2}}f\left(\frac{a}{W}\right)</math>
|}

'''[[Fracture Mechanics#Linear-elastic fracture mechanics with small-scale yielding|Linear-elastic fracture mechanics with small-scale yielding]]:'''

{|
|-
|width="20px"|
|width="500px" | <math>K^{LEBM}_I=\frac{F_{max} \, \cdot\,s} {B \, \cdot\, W^{3/2}}f\left(\frac{a_{eff}}{W}\right)</math>
|}

'''Equivalent energy concept:'''

{|
|-
|width="20px"|
|width="500px" | <math>K^E_I=\frac{F^*_Q \, \cdot\,s} {B \, \cdot\, W^{3/2}}f\left(\frac{a_{eff}}{W}\right)</math>
|}

(equation for the [[Geometry Function|geometry function]], see above)

''Experimental results:''

[[File:Aquivalent-ApplLimits-Fig2.jpg]] 
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |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|measured variable]].

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

The [[Fracture Mirror|fracture mirror]] determined by light microscopy on the [[Fracture Surface|fracture surface]] shows the dependence depicted in '''Fig. 3'''.

[[File:Aquivalent-ApplLimits-Fig3.jpg]] 
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |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|fracture mirror]] leads to an increase in the [[Effective Crack Length|effective crack length]] ''a''eff. The [[Initial Crack Length|initial crack length]] in these materials was 2 mm, i.e. it can be assumed that the condition of a small [[Plastic Zone|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 Variable|measured variables]] are geometry-dependent quantities, another way must be found to quantitatively describe the [[Toughness|toughness behaviour]] of these composites.

The critical [[Extended CTOD Concept|crack opening displacement]], determined according to [[Crack Model according to DUGDALE|DUGDALE's]] [[Crack Models|crack model]], is suitable for describing the deformability of the [[Material & Werkstoff|materials]].

{|
|-
|width="20px"|
|width="500px" | <math>\delta_{Id} = \frac{1}{n}(W-a)\, \frac{4\,f_{max}}{s}</math>
|}

[[File:Aquivalent-ApplLimits-Fig4.jpg]] 
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px" |Dependence of the critical [[Crack Tip Opening Displacement Concept (CTOD)|crack opening displacement]] ''δ''Id on the fibre volume content ''φ''v for PE-HD+GF composites
|}

The parameter ‘critical crack opening displacement’ is always advantageous when the [[Material & Werkstoff|material]] user or developer is looking for a [[Material Parameter|parameter]] that clearly describes the increasing embrittlement and represents a geometry-independent variable (see: [[Geometry Criterion|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 [[Material Parameter|parameter]] must be found for the energy-determined assessment of the fracture behaviour which takes the [[Measured Variable|measured variables]] force and deflection into account in the evaluation equations.

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

[[File:Aquivalent-ApplLimits-Fig5.jpg]] 
{|
|- valign="top"
|width="50px"|'''Fig. 5''':
|width="600px" |Dependence of ''J''Id-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 Plastics|short fibre-reinforced composites]]) is characterised by various [[Micromechanics & Nanomechanics|micromechanical]] [[Fracture Modes|fracture modes]], such as the tearing of the bonds at the fibre end and along the fibre/matrix interface (see also: [[Fibre – Matrix Adhesion|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: [[Fracture Types|types of fracture]]) of the matrix with fibre pull-out [5].

==Example 2: Unoriented and highly oriented polypropylene (PP)==

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

[[File:Aquivalent-ApplLimits-Fig6.jpg]] 
{|
|- valign="top"
|width="50px"|'''Fig. 6''':
|width="600px" |Arrangement of test specimens and notches in [[SENB-Specimen|three-point bending specimens]] and [[CT-Specimen|CT-specimens]] in relation to the rolling direction
|}

''Experimental results:''

* [[Quasi-static Test Methods|static loading]]

[[File:Aquivalent-ApplLimits-Fig7.jpg]]

* [[Impact Loading Plastics|dynamic loading]]

[[File:Aquivalent-ApplLimits-Fig7-1 (dynamic loading).jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 7''':
|width="600px" |Dependence of the fracture strengths ''K''ICLEBM and ''K''ICE under static loading ''K''IdLEBM and ''K''IdE under dynamic loading on temperature
|}

==See also==

* [[Equivalent Energy Concept – Basics|Equivalent energy concept – Basics]]
* [[Levels of Knowledge in Fracture Mechanics|Levels of knowledge in fracture mechanics]]
* [[Toughness Temperature Dependence|Toughness temperature dependence]]
* [[Quasi-static Test Methods|Quasi-static test methods]]
* [[Impact Loading Plastics|Impact loading plastics]]

==References==

{|
|-valign="top"
|[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 https://doi.org/10.1016/0029-5493(71)90042-2]
|-valign="top"
|[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 https://doi.org/10.1016/0029-5493(72)90029-5]
|-valign="top"
|[3]
|[[Grellmann,_Wolfgang|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)
|-valign="top"
|[4]
|Sumpter, J. G. D., Turner, C. E.: Cracks and Fracture. ASTM STP 601 (1976) 3–18
|-valign="top"
|[5]
|[https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.]: Zähigkeitsbewertung mit bruchmechanischen Methoden. In: [https://de.wikipedia.org/wiki/Wolfgang_Grellmann Grellmann, W.], [[Seidler,_Sabine|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)
|-valign="top"
|[6]
|Hille, E.: Untersuchungen zum Bruchverhalten des orientierten isotaktischen Polypropylen. Ph.D. Dissertation, [https://de.wikipedia.org/wiki/Technische_Hochschule_Leuna-Merseburg Technische Hochschule Leuna-Merseburg (1983)]
|}

[[Category:Fracture Mechanics]]
[[Category:Instrumented Impact Test]]

Equivalent Energy Concept – Application Limits - Revision history