Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics
The main goal of this research is to propose a failure criterion based on the linear elastic notch fracture mechanics (LENFM) for predicting tensile crack initiation from a blunt V-notch, encountering large plasticity at the notch vicinity. First, some most recently published experimental results on...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1736232020-12-14T01:26:06Z Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics Torabi, A.R. Campagnolo, A. Berto, F. Научно-технический раздел The main goal of this research is to propose a failure criterion based on the linear elastic notch fracture mechanics (LENFM) for predicting tensile crack initiation from a blunt V-notch, encountering large plasticity at the notch vicinity. First, some most recently published experimental results on tensile failure of V-notched ductile aluminum plates are briefly described. Then, with the aim to avoid complex and time-consuming elastic-plastic analyses, the equivalent material concept (EMC) is employed together with a LENFM-based fracture criterion, namely the averaged strain energy density (ASED) criterion, for predicting the load-carrying capacity of the V-notched aluminum plates. A very good agreement is shown to exist between the experimental results and theoretical predictions of the EMC-ASED criterion. Предложен критерий разрушения, основанный на линейной упругой механике разрушения, для прогнозирования инициирования трещины растяжения от затупленного V-образного надреза при большой пластичности вблизи надреза. Описаны некоторые современные экспериментальные результаты по разрушению при растяжении пластичных алюминиевых пластин с V-образным надрезом. С целью избежания сложного и трудоемкого упругопластического анализа для прогнозирования несущей способности алюминиевых пластин с V-образным надрезом используются концепция эквивалентного материала и критерий усредненной плотности энергии деформации, базирующийся на линейной упругой механике разрушения. Показано хорошее соответствие между экспериментальными данными и результатами прогнозирования на основе концепции эквивалентного материала и критерия усредненной плотности энергии деформации. Запропоновано критерій руйнування, що базується на лінійній пружній механіці руйнування, для прогнозування ініціювання тріщини розтягу від затупленого V-подібного надрізу при значній пластичності поблизу надрізу. Описано деякі сучасні експериментальні результати щодо руйнування при розтязі пластичних алюмінієвих пластин із V-подібним надрізом. Із метою запобігання складного і трудомісткого аналізу для прогнозування несівної здатності алюмінієвих пластин із V-подібним надрізом використовуються концепція еквівалентного матеріалу і критерій усереднено ї щільності енергії деформації, що базується на лінійній пружній механіці руйнування. Показано хороший збіг експериментальних даних із результатами прогнозування на основі концепції еквівалентного матеріалу і критерію усередненої щільності енергії деформації. 2017 Article Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics / A.R. Torabi, A. Campagnolo, F. Berto // Проблемы прочности. — 2017. — № 2. — С. 18-29. — Бібліогр.: 39 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173623 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Torabi, A.R. Campagnolo, A. Berto, F. Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics Проблемы прочности |
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The main goal of this research is to propose a failure criterion based on the linear elastic notch fracture mechanics (LENFM) for predicting tensile crack initiation from a blunt V-notch, encountering large plasticity at the notch vicinity. First, some most recently published experimental results on tensile failure of V-notched ductile aluminum plates are briefly described. Then, with the aim to avoid complex and time-consuming elastic-plastic analyses, the equivalent material concept (EMC) is employed together with a LENFM-based fracture criterion, namely the averaged strain energy density (ASED) criterion, for predicting the load-carrying capacity of the V-notched aluminum plates. A very good agreement is shown to exist between the experimental results and theoretical predictions of the EMC-ASED criterion. |
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Article |
| author |
Torabi, A.R. Campagnolo, A. Berto, F. |
| author_facet |
Torabi, A.R. Campagnolo, A. Berto, F. |
| author_sort |
Torabi, A.R. |
| title |
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics |
| title_short |
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics |
| title_full |
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics |
| title_fullStr |
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics |
| title_full_unstemmed |
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics |
| title_sort |
large-scale yielding failure prediction of notched ductile plates by means of the linear elastic notch fracture mechanics |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2017 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/173623 |
| citation_txt |
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means of the Linear Elastic Notch Fracture Mechanics / A.R. Torabi, A. Campagnolo, F. Berto // Проблемы прочности. — 2017. — № 2. — С. 18-29. — Бібліогр.: 39 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-15T10:22:09Z |
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2025-07-15T10:22:09Z |
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1837708010121592832 |
| fulltext |
UDC 539.4
Large-Scale Yielding Failure Prediction of Notched Ductile Plates by Means
of the Linear Elastic Notch Fracture Mechanics
A. R. Torabi,
a
A. Campagnolo,
b
and F. Berto
c,d,1
a Fracture Research Laboratory, Faculty of New Sciences & Technologies, University of Tehran,
Tehran, Iran
b Department of Industrial Engineering, University of Padova, Padova, Italy
c Department of Management and Engineering, University of Padova, Vicenza, Italy
d Department of Mechanical and Industrial Engineering, Norwegian University of Science and
Technology (NTNU), Trondheim, Norway
1 filippo.berto@ntnu.no
ÓÄÊ 539.4
Ïðîãíîçèðîâàíèå ðàçðóøåíèÿ ïðè ïîëíîìàñøòàáíîì òå÷åíèè â ïëàñòè-
íàõ ñ íàäðåçîì íà îñíîâå ëèíåéíîé óïðóãîé ìåõàíèêè ðàçðóøåíèÿ
À. Ð. Òîðàáè
à
, À. Êàìïàíüîëî
á
, Ô. Áåðòî
â,ã
à Ëàáîðàòîðèÿ èññëåäîâàíèÿ ðàçðóøåíèÿ, Ôàêóëüòåò èííîâàöèîííûõ òåõíîëîãèé, Òåãåðàíñêèé
óíèâåðñèòåò, Òåãåðàí, Èðàí
á Ôàêóëüòåò îðãàíèçàöèè ïðîèçâîäñòâà, Ïàäóàíñêèé óíèâåðñèòåò, Ïàäóÿ, Èòàëèÿ
â Ôàêóëüòåò ìåíåäæìåíòà è èíæèíèðèíãà, Ïàäóàíñêèé óíèâåðñèòåò, Âè÷åíöà, Èòàëèÿ
ã Ìåõàíèêî-ìàøèíîñòðîèòåëüíûé ôàêóëüòåò, Íîðâåæñêèé óíèâåðñèòåò åñòåñòâåííûõ è òåõíè-
÷åñêèõ íàóê, Òðîíõåéì, Íîðâåãèÿ
Ïðåäëîæåí êðèòåðèé ðàçðóøåíèÿ, îñíîâàííûé íà ëèíåéíîé óïðóãîé ìåõàíèêå ðàçðóøåíèÿ, äëÿ
ïðîãíîçèðîâàíèÿ èíèöèèðîâàíèÿ òðåùèíû ðàñòÿæåíèÿ îò çàòóïëåííîãî V-îáðàçíîãî íàäðåçà
ïðè áîëüøîé ïëàñòè÷íîñòè âáëèçè íàäðåçà. Îïèñàíû íåêîòîðûå ñîâðåìåííûå ýêñïåðèìåí-
òàëüíûå ðåçóëüòàòû ïî ðàçðóøåíèþ ïðè ðàñòÿæåíèè ïëàñòè÷íûõ àëþìèíèåâûõ ïëàñòèí ñ
V-îáðàçíûì íàäðåçîì. Ñ öåëüþ èçáåæàíèÿ ñëîæíîãî è òðóäîåìêîãî óïðóãîïëàñòè÷åñêîãî
àíàëèçà äëÿ ïðîãíîçèðîâàíèÿ íåñóùåé ñïîñîáíîñòè àëþìèíèåâûõ ïëàñòèí ñ V-îáðàçíûì
íàäðåçîì èñïîëüçóþòñÿ êîíöåïöèÿ ýêâèâàëåíòíîãî ìàòåðèàëà è êðèòåðèé óñðåäíåííîé ïëîò-
íîñòè ýíåðãèè äåôîðìàöèè, áàçèðóþùèéñÿ íà ëèíåéíîé óïðóãîé ìåõàíèêå ðàçðóøåíèÿ. Ïîêà-
çàíî õîðîøåå ñîîòâåòñòâèå ìåæäó ýêñïåðèìåíòàëüíûìè äàííûìè è ðåçóëüòàòàìè ïðîãíîçè-
ðîâàíèÿ íà îñíîâå êîíöåïöèè ýêâèâàëåíòíîãî ìàòåðèàëà è êðèòåðèÿ óñðåäíåííîé ïëîòíîñòè
ýíåðãèè äåôîðìàöèè.
Êëþ÷åâûå ñëîâà: íàäðåç, ïëàñòè÷íûé ìàòåðèàë, çàðîæäåíèå òðåùèíû, ïîíÿòèå ýêâè-
âàëåíòíîãî ìàòåðèàëà, êðèòåðèé óñðåäíåííîé ïëîòíîñòè ýíåðãèè äåôîðìàöèè, ïîëíî-
ìàñøòàáíîå òå÷åíèå.
Introduction. In general, ductile metallic materials are divided into two main
categories. The first and second categories involve those materials exhibiting negligible and
considerable strain-hardening in the plastic zone, respectively. An accurate factor to realize
the category is normally the difference between the yield and tensile strengths of the
material. The fracture mechanics analyses for the two categories are really different. The
© A. R. TORABI, A. CAMPAGNOLO, F. BERTO, 2017
18 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2
materials inside the first category have usually valid fracture toughness value in the context
of the linear elastic fracture mechanics (LEFM); i.e., K-based fracture toughness K cI or
K c , where K cI is utilized in plane-strain conditions, which is usually satisfied for
relatively large thicknesses, and K c in plane-stress conditions, relating to small thicknesses.
For ductile metallic materials inside the second category, a LEFM-based fracture toughness,
K cI or K c , is not obtained according to the standard code ASTM E399-12e3 [1], but
instead, some other fracture toughness values in the context of the elastic-plastic fracture
mechanics (EPFM), e.g., the crack tip opening displacement (CTOD), crack tip opening
angle (CTOA), resistance curve (R-curve), critical J-integral, etc., are utilized to characterize
the resistance of material against crack propagation [2, 3]. Despite valid K cI or K c
values for the first category, if the size of the plastic zone ahead of the crack tip at the
propagation instance becomes considerable, particularly for small thicknesses, the effects of
the plastic zone on the prediction of crack propagation should be considered in the fracture
models.
Although the physical meaning of the notch fracture toughness (NFT) is completely
different from that of the fracture toughness (FT); the NFT means the resistance of a notch
against initiating crack(s) from its border, while the FT means that of a crack against
propagation; the failure process in notched members is very similar to that in cracked ones.
For example, in a V-notched ductile member, a plastic region initiates from the notch
border during loading and grows till the onset of crack initiation from the notch border at
which the notched member shows its load-carrying capacity (LCC). For predicting the LCC
of a V-notched ductile component, various criteria exist depending on the existence of valid
or invalid K-based fracture toughness value. For ductile materials inside the first category,
the thickness of the notched component plays an important role in selecting appropriate
failure criteria. If the thickness becomes larger than a specified value, which is usually
computable, no considerable plastic zone will form around the notch at the crack initiation
instance and hence, the failure criteria in the context of the linear elastic notch fracture
mechanics (LENFM) can accurately be utilized for the LCC prediction. For small thicknesses
(e.g., thin plates), however, the plastic zone size is significant; so considering the effects of
plastic deformations around the notch in the LCC prediction is necessary, and using directly
the LENFM-based criteria is not allowable.
Aluminum alloys are extensively utilized in engineering structures, particularly
aerospace structures, due to their good strength/weight ratio, high toughness and ductility,
good fatigue resistance and fracture toughness etc. Most of such alloys locate inside the
first category of ductile materials and hence, they have valid values of K-based fracture
toughness. Consequently, selecting the failure criterion for predicting the NFT for such
alloys strongly depends on thickness. Recently, two papers have been published on the NFT
of aluminum alloys [4, 5]. The mode I NFT of a commercial aluminum alloy containing
U-notches of various tip radii has been evaluated by Vratnica et al. [4]. They have carried
out the fracture experiments on 10 mm thick single-edge-notched-bend (SENB) specimens;
measured the NFT, and predicted successfully the experimentally obtained NFT values by
means of a stress-based criterion that employs the linear-elastic stress distributions around a
blunt crack-like notch [4]. Madrazo et al. [5] have also performed a research work on mode
I fracture of notched Al 7075-T651 specimens. They have conducted tensile fracture tests
on the notched compact-tension (CT) specimens of 20 mm thick and experimentally
measured the mode I NFT values [5]. Such values have then been well predicted by means
of the linear elastic theory of critical distances (TCD) [5]. It is worth noting that the TCD
has two well-known approaches, namely the point method and the line method [6–9] which
both have been employed in [5] for theoretical fracture assessments. According to the
requirements reported in [1] for a valid K cI test and considering the materials tensile and
fracture toughness properties reported in [4, 5]; the minimum thicknesses for valid fracture
toughness tests can simply be calculated to be equal to about 40 and 5 mm, respectively.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2 19
Large-Scale Yielding Failure Prediction of Notched Ductile Plates ...
Comparing theses values with the notched specimen thicknesses reported in [4, 5] indicates
that the notched specimens of [4] do not satisfy plane-strain conditions, while those of [5]
do so. Accordingly, considerable plastic zone has been formed around notches of [4] at the
crack initiation instance and hence, ignoring such a zone in the theoretical failure
predictions is questionable.
Most recently, due to wide application of thin aluminum plates in aero-structures, a
few research works have been carried out on ductile failure analysis of thin notched
aluminum plates under various loading conditions [10–15]. In [10–15], Torabi and
co-researchers have performed numerous failure tests on thin rectangular plates weakened
by blunt V- and U-notches, and made of the aluminum alloys Al 7075-T6 and Al 6061-T6
under pure mode I and mixed mode I/II loading conditions from which the cracking
behavior of the plates has been studied; the failure regime, e.g., the moderate-scale yielding
(MSY), large-scale yielding (LSY) etc., has been determined; and the LCC of the notched
plates has been experimentally measured. As observed during the experiments, for such
thin plates, the plastic zone around the notch at the crack initiation instance is considerable
and hence, the failure criteria in the context of the LENFM could not be used for predicting
the LCC of the notched plates. In order to avoid time-consuming and complex
elastic-plastic failure analyses and to make the LENFM-based failure criteria usable for
such ductile problems, the equivalent material concept (EMC), proposed originally by the
first author [16–19], has been employed. The experimentally obtained LCCs have been
successfully predicted by using some well-known stress-based brittle fracture criteria,
namely the point-stress (PS), mean-stress (MS) and maximum tangential stress (MTS)
criteria together with the EMC [10–15].
The most recent research on ductile failure of notched components has been
performed by Torabi et al. [20]. In [20], the experimentally recorded tensile LCCs of the
V-notched Al 7075-T6 plates reported in [10] have been well predicted by means of the
EMC in conjunction the well-known energy-based brittle fracture criterion, namely the
averaged strain energy density (ASED) criterion [21–27]. It has been found in [20] that the
EMC-ASED criterion could work well on mode I ductile failure of notched aluminum
plates under the MSY failure regime. In order to examine the success of the EMC-ASED
criterion also under the LSY failure regime, it is attempted in the present research to predict
the tensile LCCs of the V-notched Al 6061-T6 plates reported most recently in [14]. A very
good agreement is shown to exist between the theoretical and experimental results,
demonstrating that the combination of the EMC and the ASED criterion is successful,
regardless of the size of plastic zone around the notch at crack initiation instance.
1. Ductile Fracture Test Results Reported in the Literature. Most recently, Torabi
and Keshavarzian [14] have published a set of ductile fracture test results on blunt
V-notched thin plates under pure mode I loading. The material investigated has been the
aluminum alloy Al 6061-T6 with the chemical composition and mechanical properties
presented in Tables 1 and 2, respectively [14]. Figure 1 shows the tensile engineering and
true stress–strain curves of the material [14]. As seen in Fig. 1, Al 6061-T6 exhibits ductile
behavior in tension with negligible strain-hardening in the plastic region.
T a b l e 1
Al 6061-T6 Chemical Composition [14] (in wt.%)
Si Fe Cu Mn Mg Zn Ni Cr Pb Sn Ti
0.61 0.48 0.17 0.05 0.86 0.02 0.003 0.2 0.001 0.001 0.08
B Cd Bi Ca P Sb V Zr Co Li Al
0.004 0.001 0 0.001 0.002 0.008 0.025 0.0002 0.003 0.001 97.6
20 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
The specimen utilized in [14] to perform the fracture experiments has been a
rectangular plate of 4 mm thick containing a centrally located rhombic slit with four
round-tip V-notches; two of which are subjected to pure mode I loading (i.e., the opening
mode) as a result of a tensile load remotely applied to the specimen. A scheme of the
specimen is depicted in Fig. 2, including the geometrical and loading parameters [14].
The parameters 2�, �, 2a, L, W , and P in Fig. 2 are the notch angle, the notch
radius, twice the notch length (i.e., the horizontal diameter of the rhombic slit), the
specimen length, the specimen width, and the applied tensile load, respectively. The
specimen dimensions have been considered in the fracture tests to be as follows: 2 30�� ,
60, and 90�, �� 1, 2, and 4 mm, 2 25a� mm, L� 160 mm, and W � 50 mm [14]. Totally,
twenty seven displacement-controlled monotonic fracture tests have been conducted in
[14].
Three experimentally obtained evidences have been reported in [14], demonstrating
that the V-notched Al 6061-T6 plates have failed by the LSY regime. Note that the failure
has been defined in [14] to be the crack initiation from the notch tip. The evidences are:
(i) permanent deformations of the specimen ligament in large amount at the crack initiation
instance (it could be captured by naked eye); (ii) considerable opening of the notch mouth
T a b l e 2
Some of the Mechanical Properties of Al 6061-T6 [14]
Mechanical property Value
Elastic modulus E, GPa
Poisson’s ratio
Tensile yield strength (MPa)
Ultimate tensile strength (MPa)
Elongation at break (%)
Engineering strain at maximum load
True fracture stress (MPa)
Fracture toughness Kc , MPa m
Strain-hardening coefficient (MPa)
Strain-hardening exponent
67
0.33
276
292
11
0.034
299
38
314
0.021
Fig. 1. Tensile engineering and true stress–strain curves of Al 6061-T6 [14].
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2 21
Large-Scale Yielding Failure Prediction of Notched Ductile Plates ...
before failure, and (iii) large nonlinear portion in the load–displacement curve before the
peak. To numerically confirm the experimentally observed LSY failure of the V-notched Al
6061-T6 plates, some elastic-plastic finite element (FE) analyses have been performed on
the plates, and it has been shown that a large portion of the ligament (more than 50%)
experiences plastic deformations at the onset of crack initiation from the notch tip, proving
well the LSY failure regime [14].
The experimental values of the LCC of the tested specimens reported in [14] are
summarized in Table 3 for various notch angles and notch radii. In Table 3, the parameter
Pi (i� 1, 2, 3) denotes the LCC in the three repeated tests and Pav is the average of the
three LCCs.
T a b l e 3
Experimental LCCs of the V-Notched Al 6061-T6 Specimens
for Various Notch Angles and Notch Radii [14]
2�, deg �, mm P1 , N P2 , N P3 , N Pav , N
30 1
2
4
30,567
31,674
31,862
30,350
31,884
31,615
29,997
31,913
31,652
30,305
31,824
31,710
60 1
2
4
31,325
31,668
32,119
31,685
31,538
31,869
31,876
31,711
31,946
31,629
31,639
31,978
90 1
2
4
29,334
31,501
31,516
30,626
31,468
31,398
30,114
31,485
31,450
30,025
31,485
31,455
Fig. 2. Scheme of the test specimen including the geometrical and loading parameters [14].
22 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
In the next section, the EMC is briefly described, capable of equating the real ductile
material having elastic-plastic behavior with a virtual brittle material exhibiting linear-
elastic behavior.
2. Equivalent Material Concept. The EMC has been originally proposed by Torabi
[16] with the aim to take a permission to analyze ductile failure of notched members by
means of the LENFM principles. Two main mechanical properties, namely the fracture
toughness and the tensile strength are necessary for failure analysis of notched components
in the context of the LENFM. For the first property, it is assumed that the ductile material
showing considerable plastic zone in the tensile stress-strain curve has valid K-based
fracture toughness (K cI or K c ) value. For the second property, the ultimate tensile
strength of the ductile material cannot directly be used in the LENFM-based fracture
criteria, since the material exhibits considerable nonlinear behavior in the plastic region. To
overcome this restriction, the EMC is utilized in which the ductile material is equated with
a virtual brittle material with linear elastic behavior; both absorb the same amount of the
tensile strain energy density (SED) for the crack initiation to take place. By computing the
SED for ductile material till the ultimate point and setting it equal to that for the virtual
brittle material till final fracture, the tensile strength of the equivalent material can be
computed.
Considering the power-law relationship between the stress and the plastic strain in the
plastic region of the ductile material, an expression has been reported in a few recent
references for computing the tensile strength of the equivalent material, � f
* . It is [10–15]:
� � �f Y u true
n nEK
n
*
,[ ( . ) ],� �
�
� �2 1 12
1
0002 (1)
where the parameters � f
* , �Y , E, K , n, and �u true, are the tensile strength of the
equivalent material, yield strength of the real ductile material (e.g., Al 6061-T6 in [14]),
elastic modulus, strain-hardening coefficient, strain-hardening exponent, and the true
plastic strain at maximum load, respectively. Simply, �u true, can be calculated by using
Eq. (2), in which the parameter �u denotes the engineering plastic strain at maximum load:
� �u true u, ln( ).� �1 (2)
Now, the K cI (or K c ) and � f
* , which are the two main fracture properties of the
equivalent material, can simply be utilized in various fracture criteria in the field of the
LENFM, e.g., the ASED criterion, for predicting the onset of crack initiation from the
notch tip in ductile materials. In the forthcoming sections, the ASED criterion is briefly
described and by linking it to the EMC [i.e., using K cI (or K c ) and � f
* values in the
criterion], the tensile LCCs of the V-notched Al 6061-T6 plates (see Table 3) that fail by
LSY regime are theoretically predicted.
3. Brief Description of the ASED Criterion. The most important point for designers
is certainly the existence of appropriate failure models to predict the load-carrying capacity
of components weakened by notches. With the aim to provide such models, a strain energy
density based criterion has been proposed by Lazzarin and co-authors [21, 22], by which
the experimental fracture loads of notched specimens can properly be estimated.
The strain energy density factor S was defined for sharp cracks by Sih [28] as the
product of the strain energy density by a specified critical distance measured from the crack
tip. Fracture was thought of as controlled by a critical value S c , whereas the crack growth
direction was determined by imposing a minimum condition on the factor S .
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The method proposed by Sih is a point-wise criterion, while the ASED approach as
presented in [21, 22] suggests that brittle fracture takes place when the strain energy density
averaged over a known control volume is equal to a critical value Wc . This value varies
from material to material but it is independent of the notch geometry. The control volume is
thought of as dependent on the ultimate tensile strength and on the fracture toughness K cI
in the case of brittle or quasi-brittle materials subjected to static and monotonic loads. Such
a method was formalized and applied first to sharp V-notches under mode I and mixed
mode I/II loadings [21] and later extended to blunt U- and V-notches [22, 29], also under
multiaxial I/III static loadings [24, 25]. Some recent developments and applications are
reported in [22, 30–32], with some considerations also to three-dimensional effects
[33–35], which have been widely discussed in [36].
For sharp cracks, the control volume is a circle of radius Rc centered at the crack tip
(Fig. 3a). Under plane-strain conditions, the critical length, Rc , can be evaluated by the
following expression [37]:
R
K
c
c
t
�
�
�
�
�
�
�
�
( )( )
.
1 5 8
4
2
� �
� �
I
(3)
In Eq. (3), K cI is the fracture toughness, � is Poisson’s ratio, and � t is the ultimate
tensile strength of the material.
For a sharp V-notch, the control volume becomes a circular sector of radius Rc
centered at the notch tip (Fig. 3b), while for a blunt V-notch under mode I loading, the
volume assumes the crescent shape, as shown in Fig. 3c, where Rc is the depth measured
along the notch bisector line. The outer radius of the crescent shape is equal to R rc � 0,
being r0 the distance between the notch tip and the origin of the local coordinate system
(see Fig. 3c). Such a distance depends on the V-notch opening angle 2�, according to the
expression r0 2 2 2� � � � � �( ) ( ). For more details see also [22].
4. Application of EMC in Combination with ASED Criterion. The ASED approach
is applied here by considering the mechanical properties of the equivalent material. The
critical strain energy density, Wc EMC, , is evaluated by using Eq. (4) and considering
� f
*
� 1066 MPa, as obtained by applying the EMC according to Eq. (1):
W
E
c EMC
f
,
*( )
.�
�
2
2
(4)
The critical strain energy density results to be equal to 8.48 MJ/m3. The control radius
Rc is evaluated by using Eq. (3) with � �t f� �
* 1066 MPa. It has been found Rc �
� 0.317 mm.
a b c
Fig. 3. Control volume (area) for sharp crack (a), sharp V-notch (b) and blunt V-notch (c) under
mode I loading. Distance r0 2 2 2� � � � � �( ) ( ).
24 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
The ASED occurring inside the control volume embracing the edges of V-notches, W ,
is calculated numerically by using the FE code ANSYS®. For each geometry, a model is
created which requires an accurate definition of the control volume where the ASED
should be averaged (see Fig. 3c). All FE analyses are performed by using second order
eight-node plane elements (PLANE 183 of the ANSYS® element library) under plane-strain
conditions. A detail of the FE mesh adopted inside the control volume is reported in Fig. 4.
In particular, typical refined and coarse FE meshes are shown in Fig. 4a and 4b,
respectively. As it is well-known from [30–32, 38, 39], the ASED can correctly be
evaluated also with very coarse FE meshes and, for this reason, all numerical models are
performed here using the FE mesh shown in Fig. 4b.
5. Results and Discussion. Table 4 summarizes the outlines of the experimental,
numerical and theoretical findings for V-notched specimens with three different notch radii
(�� 1, 2, and 4 mm) and opening angles (2 30�� , 60, and 90�), analyzed by means of the
ASED approach. In particular, Table 4 reports the experimental loads to failure (P) for all
notch radii � and opening angles 2� compared with the theoretical values (PASED ),
obtained on the basis of the ASED evaluation. PASED is the theoretical load obtained by
keeping a constant averaged strain energy density (W ) equal to 8.48 MJ/m3, that is the
material critical value, over the control volume.
The last columns of the table present the deviations between the values of the
experimental failure loads and the theoretical ones evaluated by means of ASED criterion.
� is defined as the ratio between the experimental load and the theoretical one for each
case.
It is clearly seen in Table 4 that the great majority of predictions are well inside the
scatter of �20%, with some results inside the scatter �10%. Only few exceptions are
characterized by a deviation between 20 and 25%.
A synthesis in terms of the square root value of the local strain energy averaged over
the control volume of radius Rc (namely, W ), normalized with respect to the critical
energy of the equivalent material, Wc EMC, , as a function of the notch opening angle 2� is
shown in Fig. 5. The plotted parameter is proportional to the failure load. The aim is to
investigate the influence of the notch geometry on the fracture prediction based on the
ASED approach. Also from the graphical point of view, it is obvious that most of values
fall inside a scatter ranging from 0.80 to 1.20 with the majority of the data inside 0.90 to
1.10. Only few exceptions are outside the scatter band, having a normalized ASED between
a b
Fig. 4. Example of the refined FE mesh employed in the control volume (a). The same results can be
achieved by using a coarse FE mesh, as that shown in (b).
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Large-Scale Yielding Failure Prediction of Notched Ductile Plates ...
0.75 and 0.80. The synthesis confirms also the choice of the control volume, which seems
to be suitable to characterize the material behavior under pure mode I loading. The scatter
of the experimental data presented here is in very good agreement with the recent database
in terms of ASED reported in a recent review of the approach, which dealt with brittle and
quasi-brittle materials [22].
Conclusions. The present research is aimed to predict tensile crack initiation in Al
6061-T6 thin plates weakened by blunt V-notches. Some experimental results relevant to
tensile failure of V-notched ductile aluminum plates, published in a recent contribution,
were briefly described. In that work, on the basis of experimental observations, a
large-scale yielding failure regime was demonstrated for the considered aluminum plates.
With the aim of theoretically predicting the experimental results, the equivalent material
concept was adopted in conjunction with the averaged strain energy density evaluated over
a control volume. Without requiring elastic-plastic finite element analyses, it was shown
T a b l e 4
Critical Loads Predicted by Means of ASED Criterion in Combination with EMC
2�, deg �, mm P1 , N P2 , N P3 , N PASED , N �1 �2 �3
30 1
2
4
30,567
31,674
31,862
30,350
31,884
31,615
29,997
31,913
31,652
31,711
35,951
42,518
0.96
0.88
0.75
0.96
0.89
0.74
0.95
0.89
0.74
60 1
2
4
31,325
31,668
32,119
31,685
31,538
31,869
31,876
31,711
31,946
29,943
34,349
41,092
1.05
0.92
0.78
1.06
0.92
0.78
1.06
0.92
0.78
90 1
2
4
29,334
31,501
31,516
30,626
31,468
31,398
30,114
31,485
31,450
28,559
32,465
39,009
1.03
0.97
0.81
1.07
0.97
0.80
1.05
0.97
0.81
Fig. 5. Synthesis of fracture data in terms of normalized ASED.
26 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2017, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
that the combination of the averaged strain energy density approach and the equivalent
material concept can successfully predict the load-carrying capacity of the V-notched Al
6061-T6 plates, characterized by large plastic deformations ahead of the notch tip at the
onset of crack initiation.
Ð å ç þ ì å
Çàïðîïîíîâàíî êðèòåð³é ðóéíóâàííÿ, ùî áàçóºòüñÿ íà ë³í³éí³é ïðóæí³é ìåõàí³ö³
ðóéíóâàííÿ, äëÿ ïðîãíîçóâàííÿ ³í³ö³þâàííÿ òð³ùèíè ðîçòÿãó â³ä çàòóïëåíîãî V-ïîä³á-
íîãî íàäð³çó ïðè çíà÷í³é ïëàñòè÷íîñò³ ïîáëèçó íàäð³çó. Îïèñàíî äåÿê³ ñó÷àñí³
åêñïåðèìåíòàëüí³ ðåçóëüòàòè ùîäî ðóéíóâàííÿ ïðè ðîçòÿç³ ïëàñòè÷íèõ àëþì³í³ºâèõ
ïëàñòèí ³ç V-ïîä³áíèì íàäð³çîì. ²ç ìåòîþ çàïîá³ãàííÿ ñêëàäíîãî ³ òðóäîì³ñòêîãî
àíàë³çó äëÿ ïðîãíîçóâàííÿ íåñ³âíî¿ çäàòíîñò³ àëþì³í³ºâèõ ïëàñòèí ³ç V-ïîä³áíèì
íàäð³çîì âèêîðèñòîâóþòüñÿ êîíöåïö³ÿ åêâ³âàëåíòíîãî ìàòåð³àëó ³ êðèòåð³é óñåðåä-
íåíî¿ ù³ëüíîñò³ åíåð㳿 äåôîðìàö³¿, ùî áàçóºòüñÿ íà ë³í³éí³é ïðóæí³é ìåõàí³ö³
ðóéíóâàííÿ. Ïîêàçàíî õîðîøèé çá³ã åêñïåðèìåíòàëüíèõ äàíèõ ³ç ðåçóëüòàòàìè ïðî-
ãíîçóâàííÿ íà îñíîâ³ êîíöåïö³¿ åêâ³âàëåíòíîãî ìàòåð³àëó ³ êðèòåð³þ óñåðåäíåíî¿
ù³ëüíîñò³ åíåð㳿 äåôîðìàö³¿.
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Received 12. 12. 2016
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