View source for Equivalent Energy Concept – Basics

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<span style="font-size:1.2em;font-weight:bold;">Equivalent energy concept – Basics</span>
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==Fundamentals==

The equivalent energy concept developed by Witt and Mager [1, 2] can be classified in terms of its predictive capabilities within the methods of linear elastic [[Fracture Mechanics|fracture mechanics]] [3]. The development and application of the [[J-Integral Concept|J-integral concept]] has caused the equivalent energy method to fade into the background, to the extent that it is no longer mentioned in most recent publications [4].

The concept developed by Witt and Mager on the basis of an energy comparison in the elastic-plastic stress state is based on the investigation of the deformation behaviour of geometrically similar test specimens with [[Crack|cracks]] but different thicknesses.

Load-load contact attack point–displacement curves (or shorter load-line displacement curves) are recorded on these [[Specimen for Fracture Mechanics Tests|fracture mechanics test specimens]], and the loads ''F'' required to [[Fracture|fracture]] the test specimens and the resulting deflections ''f'' or general displacements ''v'' are recorded and plotted against each other in a diagram.

If a representation in standard coordinates (relative coordinates) is used, the deformation behaviour of all [[Specimen|test specimens]] can be represented in a single dependency.

[[File:equiv_fig1.jpg]]<br>
{| 
|- valign="top"
|width="50px"|'''Fig. 1''': 
|width="600px" |Load-line displacement curve in related coordinates ''F''/''B''<sup>2</sup> and ''v''/''B'' for geometrically similar test specimens A, C and D with different thicknesses ''B''
|}

===Basic assumption===

''The ratio of volume-related fracture energy is equal to the reciprocal ratio of thickness dependence.''

===Requirements===

# A, C and D are geometrically similar [[Specimen|specimens]]
# BD > BC > BA; The corresponding test specimens break at points A, C and D
# While the fracture of the test specimen with thickness ''B''<sub>D</sub> is still purely elastic, the ratio of the energies at fracture relative to test specimen D increases as the thickness decreases.

The energy required to fracture a small test specimen with elastic-plastic [[Deformation|deformation]] can now be used to determine the energy required to fracture a large test specimen with elastic deformation. The area under this curve has the dimension of energy relative to volume, which is why it is also referred to as volumetric energy. For this reason, this statement could also be applied to volumetric energy. The experimental determination of the [[Fracture Mechanics|stress intensity factor]] (also known as the ''K'' factor) is carried out by determining a pseudo-elastic load ''F''<sub>Q</sub><sup>*</sup>.

[[File:Equiv_fig2.jpg]]<br>
{| 
|- valign="top"
|width="50px"|'''Fig. 2''': 
|width="600px" |Determination of the pseudo-elastic load ''F''<sub>Q</sub><sup>*</sup> according to the equivalent energy concept
|}

First, the area ''A''<sub>1</sub> is determined under the load-load attack point--displacement curve – in the case of the [[Instrumented Charpy Impact Test|instrumented Charpy impact test]], this would be the impact load--deflection curve. Then, by applying the tangent to the load-line displacement curve, taking into account the area equality ''A''<sub>1</sub> ≡ ''A''<sub>2</sub>, the pseudo-elastic load is determined according to the equation

{|
|-
|width="20px"|
|width="500px" | <math>F^*_Q = \sqrt{2 A_1 tan\,\alpha}</math>
|}

According to '''Fig. 2''', tan α is the slope of the load-line displacement curve for [[CT-Specimen|CT]] and [[SENT-Specimen|SENT]] test specimens or the load–deflection curves for three-point bending test specimens.

Neglecting the derivation, the equation for the stress intensity factors for the individual test specimens is obtained as follows:

====A. Three-point bend test specimen ([[SENB-Specimen|SENB-specimen]] – Single Edge Notched Bend)====

{|
|-
|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>
|}
 
{|
|-
|width="20px"|
|width="500px" | <math>with\qquad f\left(\frac{a}{W}\right)\, = \,2.9\,\left(\frac{a}{W}\right)^{1/2} - 4.6\,\left(\frac{a}{W}\right)^{3/2} \,+ \,21.8\left(\frac{a}{W}\right)^{5/2}\,- \,37.6 \left(\frac{a}{W}\right)^{7/2} \,+ \,38.7\left(\frac{a}{W}\right)^{9/2}</math>
|}
 
{|
|-
|width="20px"|
|width="500px" | <math>and\qquad a\, =\, a_{eff}</math>
|}


====B. Compact tension test specimen ([[CT-Specimen|CT-specimen]] – Compact Tension)====

{|
|-
|width="20px"|
|width="500px" | <math>K^E_I=\frac{F^*_Q}{B \, \cdot\, W^{1/2}}f\left(\frac{a_{eff}}{W}\right)</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math>with\qquad f\left(\frac{a}{W}\right)\, = \,29{.}6\,\left(\frac{a}{W}\right)^{1/2}\,-\, 1855\left(\frac{a}{W}\right)^{3/2}\,+ \,655{.}7\left(\frac{a}{W}\right)^{5/2}\,- \,1017\left(\frac{a}{W}\right)^{7/2}\,+\, 638{.}9\left(\frac{a}{W}\right)^{9/2}</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math>and\qquad a\, =\, a_{eff}</math>
|}

====C. Single-edge notched tension specimen ([[SENT-Specimen|SENT-specimen]])====

{|
|-
|width="20px"|
|width="500px" | <math>K^E_I=\frac{F^*_Q \, \cdot\,a^{1/2}} {B \, \cdot\, W}f\left(\frac{a_{eff}}{W}\right)</math> 
|}

{|
|-
|width="20px"|
|width="500px" | <math>with\qquad f\left(\frac{a}{W}\right)\, = \,1{.}99\,-\, 0{.}41\left(\frac{a}{W}\right)\,+\, 18{.}7\left(\frac{a}{W}\right)^{2}\,-\, 38{.}48\left(\frac{a}{W}\right)^{3}\,+ \,53{.}85\left(\frac{a}{W}\right)^{4}\,</math>
|}

{|
|-
|width="20px"|
|width="500px" | <math>and\qquad a\, =\, a_{eff}</math>
|}

The verification of geometry independence is performed using the relationships

{|
|-
|width="20px"|
|width="500px" | <math>a_{eff}, B_{min},(W-a)\ge\beta\left(\frac{K^E_Q}{R_e}\right)^2</math>
|}

with β as material-dependent constant: 0.6 …. 8.3 [5] or

{|
|-
|width="20px"|
|width="500px" | <math>a_{eff}, B,(W-a)\ge\beta_1\frac{(K^E_Q)^2}{R_e\cdot E}</math>
|}

with β<sub>1</sub> = 15 …..125  [5, 6], which are also referred to as [[Geometry Criterion|geometry criteria]] or, more simply, thickness criteria.

==Application limits and examples==

see: [[Equivalent Energy Concept – Application Limits|Equivalent energy concept – Application limits]]

==See also==

* [[Levels of Knowledge in Fracture Mechanics|Levels of knowledge in fracture mechanics]]
* [[Toughness Temperature Dependence|Toughness temperature dependence]]
* [[SENB-Specimen|SENB-specimen]]
* [[CT-Specimen|CT-specimen]]
* [[SENT-Specimen|SENT-specimen]]

==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 
|-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 
|-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]
|[https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22) 
|-valign="top"
|[5]
|Own results, unpublished  
|-valign="top"
|[6]
|[https://de.wikipedia.org/wiki/Wolfgang_Grellmann Grellmann, W.], Che, M.: Assessment of Temperature-dependent Fracture Behaviour with Different Fracture Mechanic Concepts on Example of Unoriented and Cold-rolled Polypropylene. J. Appl. Polym. Sci. 66 (1997) 1237–1249 ; https://doi.org/10.1002/(SICI)1097-4628(19971114)66:7%3C1237::AID-APP4%3E3.0.CO;2-H
|}

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