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	==Introduction==		==Introduction==

	In the context of [[Fracture Mechanics \| fracture mechanics]] [[Toughness \| toughness]] evaluation, three crack opening modes must be distinguished according to the three spatial directions, which either act individually or are superimposed in different ways, whereby this is referred to as mixed-mode loading. In crack opening mode I, crack opening occurs under [[Tensile Test \| tensile]] stress, while crack opening modes II and III each describe shear stress acting orthogonally to each other in relation to the [[Crack Propagation \| crack propagation]] direction. The crack opens either as a result of a [[Stress \| stress]] parallel to the crack propagation direction (crack opening mode II) or a stress orthogonal to it (crack opening mode III). Of all the possible more complex mixed-mode stresses in the form of superposition states of the individual modes, the superpositions of Mode I and Mode II and of Mode I and Mode III are of particular practical importance for [[Plastics \| plastics]]. The stress intensity known from linear-elastic [[Fracture ~~Machanics~~ \| fracture mechanics]] (LEBM) as a measure of the intensity of the stress field in the vicinity of the tip of a crack can be divided into the components ''K''<sub>I</sub>, ''K''<sub>II</sub> and ''K''<sub>III</sub> associated with the respective crack opening modes.		In the context of [[Fracture Mechanics\|fracture mechanics]] [[Toughness\|toughness]] evaluation, three crack opening modes must be distinguished according to the three spatial directions, which either act individually or are superimposed in different ways, whereby this is referred to as mixed-mode loading. In crack opening mode I, crack opening occurs under [[Tensile Test\|tensile]] stress, while crack opening modes II and III each describe shear stress acting orthogonally to each other in relation to the [[Crack Propagation\|crack propagation]] direction. The crack opens either as a result of a [[Stress\|stress]] parallel to the crack propagation direction (crack opening mode II) or a stress orthogonal to it (crack opening mode III). Of all the possible more complex mixed-mode stresses in the form of superposition states of the individual modes, the superpositions of Mode I and Mode II and of Mode I and Mode III are of particular practical importance for [[Plastics\|plastics]]. The stress intensity known from linear-elastic [[Fracture Mechanics\|fracture mechanics]] (LEBM) as a measure of the intensity of the stress field in the vicinity of the tip of a crack can be divided into the components ''K''<sub>I</sub>, ''K''<sub>II</sub> and ''K''<sub>III</sub> associated with the respective crack opening modes.

	The fracture modes are also presented in the explanation of the concept of fracture mechanics in '''Fig. 1'''.		The fracture modes are also presented in the explanation of the concept of fracture mechanics in '''Fig. 1'''.

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{{Language_sel|LANG=ger|ARTIKEL=Mixed-Mode-Rissausbreitung}}
{{PSM_Infobox}}
Mixed-mode crack propagation in brittle thermoplastics
__FORCETOC__

==Introduction==

In the context of [[Fracture Mechanics | fracture mechanics]] [[Toughness | toughness]] evaluation, three crack opening modes must be distinguished according to the three spatial directions, which either act individually or are superimposed in different ways, whereby this is referred to as mixed-mode loading. In crack opening mode I, crack opening occurs under [[Tensile Test | tensile]] stress, while crack opening modes II and III each describe shear stress acting orthogonally to each other in relation to the [[Crack Propagation | crack propagation]] direction. The crack opens either as a result of a [[Stress | stress]] parallel to the crack propagation direction (crack opening mode II) or a stress orthogonal to it (crack opening mode III). Of all the possible more complex mixed-mode stresses in the form of superposition states of the individual modes, the superpositions of Mode I and Mode II and of Mode I and Mode III are of particular practical importance for [[Plastics | plastics]]. The stress intensity known from linear-elastic [[Fracture Machanics | fracture mechanics]] (LEBM) as a measure of the intensity of the stress field in the vicinity of the tip of a crack can be divided into the components ''K''I, ''K''II and ''K''III associated with the respective crack opening modes.

The fracture modes are also presented in the explanation of the concept of fracture mechanics in '''Fig. 1'''.

==Application example poly(methyl methacrylate) (PMMA)==

The device in '''Fig. 1''' with a [[Compact Tension Shear (CTS) Specimen | CTS (Compact Tension Shear) test specimen]] made of poly(methyl methacrylate) ([[Plastics – Symbols and Abbreviated Terms | abbreviation]]: PMMA) subjected to external tension enables the realization of pure modes I and II as well as mixed-mode I/II conditions by varying the angle ''φ'' between the notch and the direction of loading in the range from 0° to 90°.

[[file:Mixed-Mode_1.jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px"|Mixed-mode loading device with clamped [[Compact Tension Shear (CTS) Specimen | CTS-specimen]] (''φ'' = 45°)
|}

The crack initiation angle ''β'' determined according to '''Fig. 2''' depends on the direction of loading ''φ''.

[[file:Mixed-Mode_2.jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px"|Crack formation depending on the loading angle ''φ''
|}

The nearly linear-elastic material behavior typical for PMMA allows the use of the LEBM to determine the stress intensity factor Kc according to the equation

{|
|-
|width="20px"|
|width="500px" | <math> K_{c} = \frac {F_{max}} { B \sqrt{W}} \cdot f \left(\frac {a} {W} \right)</math>
|(1)
|}
with the geometry function ''f''(''a''/''W'')
{|
|-
|width="20px"|
|width="500px" | <math> f \left(\frac {a} {W}\right) = \frac {\sqrt {2\,\text{tan}{\frac{\pi\,a}{2\,W}}}} {\text {cos} \frac {\pi\,a} {2\,W}} \; \left[0.752 + 2.02 \left(\frac {a} {W} \right) + 0.37 \left(1 - \text {sin} \frac {\pi\,a} {2\,W} \right)^3 \right] </math>
|(2)
|}

and the maximum load ''F''max by evaluating the force–displacement diagrams recorded during the test. Here it means: ''B'' and ''W'' – test specimen thickness and width, ''a'' – notch depth.

By determining the Mode I and Mode II parts ''K''I and ''K''II of the [[Fracture Mechanics | stress intensity factor]] according to

{|
|-
|width="20px"|
|width="500px" | <math> K_{I} = K_{Ic} \cdot \text {cos}^2 \varphi </math>
|(3)
|}
and
{|
|-
|width="20px"|
|width="500px" | <math> K_{II} = K_{IIc} \cdot \text {sin}^2 \varphi </math>
|(4)
|}

By means of the experimentally determined fracture toughness under pure Mode I or pure Mode II loading (''K''Ic and ''K''IIc ), a comparative value ''K''V of the stress intensity factor with

{|
|-
|width="20px"|
|width="500px" | <math> K_{V} = \sqrt {{K_I}^2 + {K_II}^2} </math>
|(5)
|}

and compare it with the experimental data ''K''c of the stress intensity factor according to equation (1), as shown in '''Fig. 3'''. The good agreement between the experimentally determined ''K''c values and the ''K''V values according to equation (5) can be explained by the assumption of a normal stress fracture hypothesis. The initial crack propagation angle ''β'' and the loading angle ''φ'' are identical up to about ''φ'' = 60°, i.e. the crack propagates perpendicular to the external stress according to this hypothesis ('''Fig. 4''').

[[file:Mixed-Mode_3.jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px"|Stress intensity factor as a function of the loading angle ''φ''
|}

According to the maximum tensile stress criterion, referred to below as the MTS criterion (maximum tensile stress), the initial crack propagation angle ''β'' after

{|
|-
|width="20px"|
|width="500px" | <math> \beta = - \text {arcsin} \left( \frac {M-3 \cdot \sqrt {M^2 + 8}} {M^2 + 9} \right) \text {with} \; M = \frac {K_I} {K_{II}} - \frac {T \cdot \sqrt {\pi \, a}} {K_c} </math>
|(6)
|}

where the influence of the T stresses compared to the conventional MTS criterion is taken into account by an additional term (biaxiality ratio) to the ratio ''K''I/''K''II of the stress intensity factors (mixed-mode ratio) (modified MTS criterion).

[[file:Mixed-Mode_4.jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px"|Crack initiation angle  as a function of the loading angle ''φ''
|}

In comparison to the singular stress parts of the stress field in cracked solids, which can be quantified by the stress intensity factor, the T-stresses are non-singular stress parts of constant magnitude parallel to the crack, which for [[SENT-Specimen | Single-Edge-Notched Tension (SENT) test specimens]] under mixed-mode loading according to

{|
|-
|width="20px"|
|width="500px" | <math> T = \frac {F_{max}} {B \cdot (W-a)} \cdot (\text {cos}^2 \varphi - \text {sin}^2 \varphi) </math>
|(7)
|}
can be estimated.

This means that, compared to the conventional MTS criterion, they lead to a further reduction in the crack propagation angle  for  > 45° according to equation (4).

The crack initiation angles  calculated according to the modified MTS criterion tend to agree with the experimentally determined values, with  increasing up to  = 60° and changing less for  > 60° ('''Fig. 4''').

==Mixed-mode I/II stress of thermoplastics==

Compared with long-fiber-reinforced plastics and bonded joints, [[Crack Propagation | crack propagation]] in brittle amorphous [[Thermoplastic Material | thermoplastics]] under mixed-mode stress has been investigated relatively rarely to date. However, a number of studies on crack propagation behavior under mixed-mode I/II loading have already been carried out specifically on PMMA as a model [[Material & Werkstoff | material]] for brittle [[Fracture Behaviour | fracture behaviour]] (see also: [[Fracture Types | fracture types]]) for a compilation of the literature, see [1–4]). Either internally notched tensile test specimens with variation of the angle between the notch and the loading direction in the range of 0° to 90° were used without realizing a pure Mode II loading, or other [[Specimen | test specimen]] shapes, such as asymmetrically loaded four-point bending test specimens, semicircular bending test specimens with variation of the notch angle, one-sided obliquely notched tensile and bending test specimens, asymmetrically loaded test specimens with internal crack or compact tensile shear test specimens, were used. Compact tensile shear test specimens are used (see: [[Specimen for Fracture Mechanics Tests | specimen for fracture mechanics test]]).

Different fracture criteria are described in the literature for linear-elastic material behavior under combined tensile and shear loading (e.g. under Mode I/II), with the MTS criterion, the maximum [[Energy Release Rate | energy release rate]] criterion and the minimum distortion energy density criterion being the best known. Of these criteria, the MTS criterion in particular has been successfully applied to PMMA, whereby the agreement between the experimental results and the theoretical description could be further improved by taking into account non-singular stress components (T-stresses).

==See also==

* [[Fracture Mechanical Testing | Fracture mechanical testing]]
* [[Specimen for Fracture Mechanics Tests | Specimen for fracture mechanics tests]]
* [[Fracture Types | Fracture types]]
* [[Fracture Behaviour | Fracture behaviour]]
* [[Fracture Mechanics | Fracture mechanics]]
* [[Crack Opening | Crack opening]]
* [[Composite Materials Testing | Composite materials testing]]

'''References'''

{|
|-valign="top"
|[1]
|[https://researchgate.net/profile/Ralf-Lach Lach, R.], [[Grellmann,_Wolfgang|Grellmann, W.]]: Mixed mode fracture mechanics behaviour of PMMA. Macromolecular Symposia 373 (2017) 1600108 (6 pages)
|-valign="top"
|[2]
|Lach, R., [https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.]: Mixed Mode-Rissausbreitung in spröden Thermoplasten am Beispiel von Polymethylmethacrylat. In: [https://de.wikipedia.org/wiki/Michael_Pohl_(Metallurg) M. Pohl] (Hrsg.): Konstruktion, Qualitätssicherung und Schadensanalyse. Tagungsband Werkstoffprüfung 2007, 29./30.11.2007, Neu-Ulm, Verlag Stahleisen, Düsseldorf (2007), 189–194
|-valign="top"
|[3]
|Lach, R., [https://de.wikipedia.org/wiki/Wolfgang_Grellmann Grellmann, W.]: Mixed Mode-Rissausbreitung in spröden Thermoplasten am Beispiel von Polymethylmethacrylat. 12. Problemseminar: Deformation und Bruchverhalten von Kunststoffen, 24.–26.06.2009, Merseburg (2009), Tagungsband (CD-ROM), 364–370
|-valign="top"
|[4]
|Borreck, S.: Bewertung des Rissausbreitungsverhalten spröder Kunststoffe unter Mixed Mode-Beanspruchung. Unveröffentlichte Studienarbeit, Martin-Luther-Universität Halle-Wittenberg, Halle (2010)
|}

[[Category:Hybrid Methods]]
[[Category:Fracture Mechanics]]
[[Category:Morphology and Micromechanics]]

Mixed-Mode Crack Propagation - Revision history

Oluschinski at 05:22, 15 December 2025