Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density
Получены новые экспериментальные данные о хрупком разрушении по механизму отрыва в концентраторах напряжения типа замочной скважины и проверена возможность их прогнозирования с помощью критерия плотности локальной энергии деформации. Выполнено 21 испытание на разрушение образцов из полиметилметакрил...
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Torabi, A.R. Campagnolo, A. Berto, F. 2020-12-04T19:51:27Z 2020-12-04T19:51:27Z 2016 Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density / A.R. Torabi, A. Campagnolo F. Berto // Проблемы прочности. — 2016. — № 2. — С. 80-92. — Бібліогр.: 50 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173457 539.4 Получены новые экспериментальные данные о хрупком разрушении по механизму отрыва в концентраторах напряжения типа замочной скважины и проверена возможность их прогнозирования с помощью критерия плотности локальной энергии деформации. Выполнено 21 испытание на разрушение образцов из полиметилметакрилата в виде модифицированных бразильских дисков с центральным вырезом гантелеобразной формы (типа замочной скважины) при комнатной температуре. Экспериментальные величины разрушающих нагрузок достаточно точно рассчитаны для различных значений радиусов гантелей согласно критерию плотности локальной энергии деформации, усредненному по некоторому контрольному объему, охватывающему вершину концентратора. Показано, что экспериментальные данные хорошо согласуются с расчетными. Отримано нові експериментальні дані щодо крихкого руйнування за механізмом відриву в концентраторах напружень типу замкової свердловини і перевірено можливості їх прогнозування за допомогою критерію щільності локальної енергії деформації. Проведено 21 дослідження на руйнування зразків із поліметилметакрилату у вигляді модифікованих бразильских дисків із центральним вирізом у вигляді гантелі (типу замкової свердловини) за кімнатної температури. Експериментальні величини руйнівних напружень достатньо точно розраховані для різних значень радіусів гантелей згідно з критерієм щільності локальної енергії деформації, усередненим по деякому контрольному об єму, що охоплює вершину концентратора. Показано, що експериментальні дані добре узгоджуються з розрахунковими. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density Анализ хрупкого разрушения по механизму отрыва в концентраторах напряжения типа "замочной скважины" с помощью критерия плотности энергии деформации (на англ. яз.) Article published earlier |
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Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density |
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Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density Torabi, A.R. Campagnolo, A. Berto, F. Научно-технический раздел |
| title_short |
Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density |
| title_full |
Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density |
| title_fullStr |
Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density |
| title_full_unstemmed |
Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density |
| title_sort |
tensile fracture analysis of key-hole notches by means of the strain energy density |
| author |
Torabi, A.R. Campagnolo, A. Berto, F. |
| author_facet |
Torabi, A.R. Campagnolo, A. Berto, F. |
| topic |
Научно-технический раздел |
| topic_facet |
Научно-технический раздел |
| publishDate |
2016 |
| language |
English |
| container_title |
Проблемы прочности |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| format |
Article |
| title_alt |
Анализ хрупкого разрушения по механизму отрыва в концентраторах напряжения типа "замочной скважины" с помощью критерия плотности энергии деформации (на англ. яз.) |
| description |
Получены новые экспериментальные данные о хрупком разрушении по механизму отрыва в концентраторах напряжения типа замочной скважины и проверена возможность их прогнозирования с помощью критерия плотности локальной энергии деформации. Выполнено 21 испытание на разрушение образцов из полиметилметакрилата в виде модифицированных бразильских дисков с центральным вырезом гантелеобразной формы (типа замочной скважины) при комнатной температуре. Экспериментальные величины разрушающих нагрузок достаточно точно рассчитаны для различных значений радиусов гантелей согласно критерию плотности локальной энергии деформации, усредненному по некоторому контрольному объему, охватывающему вершину концентратора. Показано, что экспериментальные данные хорошо согласуются с расчетными.
Отримано нові експериментальні дані щодо крихкого руйнування за механізмом відриву в концентраторах напружень типу замкової свердловини і перевірено можливості їх прогнозування за допомогою критерію щільності локальної енергії деформації. Проведено 21 дослідження на руйнування зразків із поліметилметакрилату у вигляді модифікованих бразильских дисків із центральним вирізом у вигляді гантелі (типу замкової свердловини) за кімнатної температури. Експериментальні величини руйнівних напружень достатньо точно розраховані для різних значень радіусів гантелей згідно з критерієм щільності локальної енергії деформації, усередненим по деякому контрольному об єму, що охоплює вершину концентратора. Показано, що експериментальні дані добре узгоджуються з розрахунковими.
|
| issn |
0556-171X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/173457 |
| citation_txt |
Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy Density / A.R. Torabi, A. Campagnolo F. Berto // Проблемы прочности. — 2016. — № 2. — С. 80-92. — Бібліогр.: 50 назв. — англ. |
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2025-11-25T23:55:34Z |
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2025-11-25T23:55:34Z |
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1850584377122095104 |
| fulltext |
UDC 539.4
Tensile Fracture Analysis of Key-Hole Notches by Means of the Strain Energy
Density
A. R. Torabi,
a
A. Campagnolo,
b
and F. Berto
b,1
a Fracture Research Laboratory, Faculty of New Science and Technologies, University of Tehran,
Tehran, Iran
b Department of Management and Engineering, University of Padova, Vicenza, Italy
1 berto@gest.unipd.it
ÓÄÊ 539.4
Àíàëèç õðóïêîãî ðàçðóøåíèÿ ïî ìåõàíèçìó îòðûâà â êîíöåíòðàòîðàõ
íàïðÿæåíèÿ òèïà “çàìî÷íîé ñêâàæèíû” ñ ïîìîùüþ êðèòåðèÿ ïëîòíîñòè
ýíåðãèè äåôîðìàöèè
À. Ð. Òîðàáè
à
, À. Êîìïàíüîëî
á
, Ô. Áåðòî
á
à Ëàáîðàòîðèÿ èññëåäîâàíèÿ ðàçðóøåíèÿ, Ôàêóëüòåò èííîâàöèîííûõ òåõíîëîãèé, Óíèâåðñèòåò
ã. Òåãåðàí, Èðàí
á Ôàêóëüòåò ìåíåäæìåíòà è èíæèíèðèíãà, Óíèâåðñèòåò ã. Ïàäóÿ, Âè÷åíöà, Èòàëèÿ
Ïîëó÷åíû íîâûå ýêñïåðèìåíòàëüíûå äàííûå î õðóïêîì ðàçðóøåíèè ïî ìåõàíèçìó îòðûâà â
êîíöåíòðàòîðàõ íàïðÿæåíèÿ òèïà “çàìî÷íîé ñêâàæèíû” è ïðîâåðåíà âîçìîæíîñòü èõ
ïðîãíîçèðîâàíèÿ ñ ïîìîùüþ êðèòåðèÿ ïëîòíîñòè ëîêàëüíîé ýíåðãèè äåôîðìàöèè. Âûïîëíåíî
21 èñïûòàíèå íà ðàçðóøåíèå îáðàçöîâ èç ïîëèìåòèëìåòàêðèëàòà â âèäå ìîäèôèöèðîâàííûõ
“áðàçèëüñêèõ äèñêîâ” ñ öåíòðàëüíûì âûðåçîì ãàíòåëåîáðàçíîé ôîðìû (òèïà çàìî÷íîé ñêâà-
æèíû) ïðè êîìíàòíîé òåìïåðàòóðå. Ýêñïåðèìåíòàëüíûå âåëè÷èíû ðàçðóøàþùèõ íàãðóçîê
äîñòàòî÷íî òî÷íî ðàññ÷èòàíû äëÿ ðàçëè÷íûõ çíà÷åíèé ðàäèóñîâ “ãàíòåëåé” ñîãëàñíî êðèòå-
ðèþ ïëîòíîñòè ëîêàëüíîé ýíåðãèè äåôîðìàöèè, óñðåäíåííîìó ïî íåêîòîðîìó êîíòðîëüíîìó
îáúåìó, îõâàòûâàþùåìó âåðøèíó êîíöåíòðàòîðà. Ïîêàçàíî, ÷òî ýêñïåðèìåíòàëüíûå äàííûå
õîðîøî ñîãëàñóþòñÿ ñ ðàñ÷åòíûìè.
Êëþ÷åâûå ñëîâà: êðèòåðèé ïëîòíîñòè ýíåðãèè äåôîðìàöèè, êîíöåíòðàòîð íàïðÿ-
æåíèé òèïà “çàìî÷íîé ñêâàæèíû”, îáðàçåö òèïà “áðàçèëüñêîãî äèñêà”, õðóïêîå
ðàçðóøåíèå.
Introduction. Depending on the crack length emanating from a notch border, different
repairing methods are utilized to remove or stop the crack. For very large cracks, the most
common method is to drill a stop hole leading to surround the crack tip and prevent it from
extension. For notched members damaged by small cracks emanating from the notch
border, a common repairing method is to remove the cracks by drilling a hole with the
radius actually equal to the crack length. For U-shaped notches damaged by cracks, a
key-hole or a nipple-shaped notch can be obtained from such repairing method depending
on the ratio U-notch tip radius/crack length. As seen in Fig. 1, key-hole and nipple-shaped
notches are resulted for the ratios smaller and larger than one, respectively. Since the level
of stress concentration around the notch depends seriously on the notch feature, the fracture
toughness of repaired notches should be theoretically and/or experimentally evaluated.
© A. R. TORABI, A. CAMPAGNOLO, F. BERTO, 2016
80 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2
When one deals with brittle or quasi-brittle materials like graphite, ceramics, high-
strength metals and alloys etc., much more attention is required to be paid to the design
process since fracture in such materials happens suddenly with usually no precaution. If a
brittle component or structure is weakened by notches, as man-made stress raisers, brittle
fracture becomes much more serious because of the stress concentration. Thus, fracture
toughness of notched brittle members should be carefully evaluated, either for original
(e.g., U-shaped) or repaired (e.g., key-hole or nipple-shaped) notches.
Brittle fracture in notched components has been commonly studied in the past based
on the linear elastic behavior of materials. Based on this particular behavior, a new field,
namely the linear elastic notch fracture mechanics (LENFM), was born inside the fracture
mechanics in which brittle fracture of notched members are analyzed by means of the
well-known failure concepts of the fracture mechanics like the maximum tangential stress
(MTS), the mean stress (MS), the cohesive zone crack model (CZM) and the strain energy
density (SED), etc. Among various notch features, U and V-shaped notches have attracted
great interest to be analyzed because of their extensive engineering applications (e.g., to
provide a path of motion for reciprocating components, to fabricate bolts with V- and
U-shaped threads, to provide grooves on shafts etc.). Moreover, in-plane loading conditions,
namely, mode I, mode II and mixed mode I/II loadings have also been much more
interested in comparison with the other types of loading such as mode III, again because of
very common applications. A brief literature survey on the notch fracture mechanics is
presented herein.
Gogotsi [1] investigated the fracture toughness of ceramics and ceramic-based
composites considering different geometries of specimens under three and four point
bending. Knesl [2] suggested a criterion for V-notch stability. Stress distribution near a
large blunt V-shaped notch has been utilized by Nui et al. [3] in order to study the fracture
toughness of brittle materials by means of the critical notch stress intensity factor. Seweryn
[4] proposed a brittle fracture criterion for sharp V-notches and verified its validity by
means of some experimental results. Considering plasticity at the notch tip vicinity,
Strandberg [5] investigated successfully the fracture in V-notched specimens made of a
type of soft annealed tool steel at � �50 C. Gomez and Elices [6] proposed and utilized the
CZM for estimating brittle fracture of sharp V-notches under opening mode. They verified
the CZM predictions by means of mode I fracture test results obtained from V-notched
samples made of PMMA [6]. In [7], a similar approach on the basis of the CZM has been
utilized to predict mode I brittle fracture in U-notched specimens made of PMMA and
Fig. 1. Key-hole and nipple-shaped notches obtained from removing the cracks from U-notch border.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2 81
Tensile Fracture Analysis of Key-Hole Notches ...
subjected to three-point bending. Some recent investigations have been performed on
notched members both theoretically and experimentally [8–14]. In the theoretical part, a
successful use of the point-stress (PS) and the mean stress (MS) fracture criteria for
predicting the mode I apparent notch fracture toughness (NFT) for V- and U-shaped
notches is carried out. The theoretical NFTs have been verified by means of different brittle
and quasi-brittle materials and various test specimens.
One of the most important theories in the context of brittle fracture of engineering
components is certainly the strain energy density (SED) criterion. For cracked members,
Sih [15] proposed the strain energy density factor (SEDF) by taking into account
simultaneously both the strain energy density (SED) and the critical distance measured
from the crack tip. According to the Sih criterion, brittle fracture takes place in a cracked
brittle member when the SEDF attains its critical value [15]. The crack growth direction
could also be determined by setting a minimum condition on the SEDF [15]. The Sih
criterion is a point-wise criterion.
After that pioneering work [15], other researchers extended the SED concept to sharp
and blunt notches (sharp and blunt V-notches, U-notches, etc.) by averaging SED over a
specified control volume which embraces the notch edge with the aim to predict brittle
fracture in notched components under different loading conditions [16–22]. The predictions
of SED have been frequently verified by using the experimental results obtained from
testing different specimens, materials and notches under various types of loading (mode I,
mixed mode I/II etc.) [16–22]. Some recent works on the fracture analysis of blunt notches
by means of SED are those published in [23–26]. The SED has been employed to assess the
load-carrying capacity of U-notched Brazilian disk (UNBD) specimens made of
polycrystalline graphite under pure mode I [23], pure mode II [24] and mixed mode I/II
[25] loading conditions. It has been also analyzed mode I fracture of rounded-tip V-notches
introduced in three different test samples made of graphite by means of SED [26]. Some
recent studies deal with torsion [27] and compression loading [28] of graphite samples and
the extension of the SED criterion to these cases.
Regarding brittle fracture of key-hole notches, only few papers have been published
in the literature [29–31]. Kullmer and Richard [29] made satisfactorily use of a stress-based
criterion to predict the fracture loads of the compact-tension-shear-notch (CTSN)
specimens made of PMMA and subjected to mixed mode I/II loading. Lazzarin et al. [30]
employed the local SED criterion to estimate the fracture loads of isostatic graphite plates
weakened by horizontal (for mode I loading) and inclined (for mixed mode I/II loading)
key-hole notches. The mode I test results reported in [30] have been re-analyzed by Torabi
[31] using the two stress-based criteria, namely the PS and the MS criteria.
In the present work, tensile fracture of key-hole notches was studied both
experimentally and theoretically. In the experimental part, 21 fracture tests were carried out
on a new version of the well-known Brazilian disk specimen containing a central
dumbbell-shaped slit with two key-ends (Key-BD) made of PMMA. The experimentally
recorded fracture loads of the Key-BD specimens were theoretically estimated by means of
the SED criterion for various notch radii. It was found that the SED criterion could predict
the experimental results successfully with a scatter band of about �10%.
1. Experiments. New fracture tests were conducted at room temperature on key-hole
notched specimens made of PMMA. Details of the experiments are presented in this
section.
1.1. Material. The most common material utilized in brittle fracture experiments is
certainly the plexiglass or polymethyl-methacrylate (PMMA). Most of PMMAs exhibit
quasi-brittle behavior at room temperature and brittle behavior at low temperatures (usually
below � �60 C). The PMMA utilized in the present experiments was the same is used in
[32–34]. The mechanical properties of this material at room temperature are presented in
Table 1 [32–34].
82 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
1.2. Specimen. The Brazilian disk specimen containing a central dumbbell-shaped slit
with two key-ends (Key-BD specimen) was utilized to perform the mode I fracture
experiments at room temperature (see Fig. 2). As seen in Fig. 2, the remote compressive
load is applied to the specimen along the slit bisector line and hence, the key-ends
experience pure opening mode (i.e., pure mode I loading).
The disk diameter (D) and the thickness t were equal to 80 mm and 10 mm,
respectively. Two various slit lengths (d) of 24 and 40 mm were considered in the tests.
With these values of d , however, the values of the relative notch length ratio (RNL) d D
were equal to 0.3 and 0.5. To fabricate the Key-BD specimens, first, a PMMA plate of 10 mm
in thickness was provided. Then, the precise sketch of each specimen was given to a
high-precision 2-D CNC water jet cutting machine for fabrication. Finally, the cut surfaces
were precisely polished by means of appropriate abrasive papers. Three notch radii of 1, 2,
and 4 mm and four radii of 1, 2, 4, and 6 mm were considered for the specimens with
d D� 0.3 and 0.5, respectively. The test speed was set to be equal to 0.3 mm/min
providing monotonic loading conditions. Each test was repeated three times with the aim to
check the repeatability of the fracture tests. 21 tests were totally carried out in this
investigation. A Key-BD specimen is represented in Fig. 3 during mode I fracture test.
Also, Fig. 4 shows several Key-BD specimens after fracture.
The experimentally recorded fracture loads are presented in Table 2 for various notch
radii and different values of the RNL ratio.
As seen in Fig. 5, the load-displacement plots obtained from the fracture tests were
linear up to final fracture. Also, a sudden fall of the curve from the maximum load to zero
proves that fracture occurred in a brittle matter.
T a b l e 1
Properties of the Tested PMMA at Room Temperature [32–34]
Material property Value
Elastic modulus E, GPa 2.96
Poisson’s ratio � 0.38
Ultimate tensile strength �t , MPa 70.5
Plane-strain fracture toughness K cI , MPa m 1.96
Fig. 2. The Key-BD specimen under mode I loading.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2 83
Tensile Fracture Analysis of Key-Hole Notches ...
In the forthcoming sections, the experimental fracture loads are theoretically predicted
by means of the local SED criterion.
T a b l e 2
The Experimentally Recorded Fracture Loads of the Key-BD Specimens
for Various Notch Radii and Different Values of the RNL Ratio
RNL
(d D)
�,
mm
P1 ,
N
P2 ,
N
P3 ,
N
P ,
N
0.3 1 9391 9080 9302 9258
0.3 2 8664 9361 9691 9239
0.3 4 10891 10635 10768 10765
0.5 1 6928 6859 6748 6845
0.5 2 7015 6991 7040 7015
0.5 4 6889 5992 6441 6441
0.5 6 7173 6608 5957 6579
Fig. 3. A Key-BD specimen during fracture test.
Fig. 4. Key-BD specimens after fracture.
Fig. 5. A load–displacement plot for a Key-BD PMMA specimen.
Fig. 3 Fig. 4
84 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
2. Fracture Criterion Based on the Strain Energy Density Averaged Over a
Control Volume. In order to estimate the fracture load of notched PMMA components,
designers need a suitable fracture criterion based on the mechanical behavior of material
around the notch tip. A strain-energy-density based criterion is described in this section, by
which the fracture loads obtained from the experiments can be estimated with a reasonable
accuracy.
As described in the introduction, dealing with cracked components, the strain energy
density factor S was defined first by Sih [15] as the product of the strain energy density by
a critical distance from the point of singularity. Failure was thought of as controlled by a
critical value S c , whereas the direction of crack propagation was determined by imposing
a minimum condition on S . Furthermore, this theory was used to study three problems of
structural failure, namely the problem of slow stable growth of an inclined crack in a plate
subjected to uniaxial tension, the problem of fracture instability of a plate with a central
crack and two notches, and the problem of unstable crack growth in a circular disc
subjected to two equal and opposite forces. The results of stress analysis were combined
with the strain energy density theory to obtain the whole history of crack growth from
initiation to instability. A length parameter was introduced to define the fracture instability
of a mechanical system. Fracture trajectories were obtained for fast unstable crack
propagation. The study of crack initiation and propagation is still an active research topic as
demonstrated by some recent papers in the field [35–39]. This is particularly true when
complex loading modes are taken into account [40–42].
Different from the Sih criterion, which is a point-wise one, the averaged strain energy
density criterion (SED) as presented in [16–18], states that brittle failure occurs when the
mean value of the strain energy density over a given control volume is equal to a critical
value Wc . This critical value varies from material to material but it does not depend on the
notch geometry and sharpness. The control volume, reminiscent of Neuber’s concept of
elementary structural volume [43], is thought of as dependent on the ultimate tensile
strength and the fracture toughness K cI , in the case of brittle or quasi-brittle materials
subjected to static loads.
Such a method was formalized and applied first to sharp (zero radius), V-notches
under mode I and mixed I/II loading [16] and later extended to blunt U- and V-notches
[17]. Some recent developments and applications are summarized in [44–46] with some
considerations of the three-dimensional effects [47–50].
When dealing with cracks, the critical volume is a circle of radius Rc centered at the
tip [16]. Under plane strain conditions, the critical length, Rc , can be evaluated according
to the following expression [16, 46]:
R
K
c
c�
�
�
��
�
��
( )( )
,
1 5 8
4
� �
� �
I
t
(1)
where K cI is the fracture toughness, � is Poisson’s ratio, and � t is the ultimate tensile
strength of a plain specimen that obeys a linear elastic behavior.
For a key-hole, the volume assumes the crescent shape shown in Fig. 6 and defined in
[28], where Rc is the depth measured along the notch bisector line. The outer radius of the
crescent shape is equal to Rc � 2 (Fig. 6).
Under mixed mode loading, the critical volume is no longer centered on the notch tip,
but rather on the point where the principal stress reaches its maximum value along the
edge of the notch. It was assumed that the crescent shape volume rotates rigidly under
mixed mode, with no change in shape and size. This is the governing idea of the
‘equivalent local mode I’ approach, as proposed and applied to U- and V-notches [18, 19,
46].
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Tensile Fracture Analysis of Key-Hole Notches ...
Avoiding any simplified assumption, the SED values can be directly derived from
finite element (FE) models. The advantage of the direct evaluation of the SED from a FE
model is that the value of this parameter is mesh-independent as described in [50]. A very
coarse mesh can be adopted for the SED evaluation contrary to the mesh required to
evaluate the notch stress intensity factors or other stress-based parameters. An example is
shown in Figure 7 comparing a typical fine mesh (Fig. 7a) with a coarse one (Fig. 7b) as
employed in the present analyses.
3. SED Approach in Fracture Analysis of the Tested PMMA Specimens. The
fracture criterion described in the previous section is employed here to estimate the fracture
loads obtained from the experiments conducted on the PMMA specimens weakened by
key-hole notches. In order to determine the SED values, a finite element model of the
specimen was generated. A typical mesh used in the numerical analyzes is shown in Fig. 7a
and 7b for all the shapes of the specimens considered in the present investigation. The
averaged strain energy density criterion (SED) states that failure occurs when the mean
value of the strain energy density over a control volume, W , is equal to a critical value Wc ,
which depends on the material but not on the notch geometry [16]. This critical value can
be determined from the ultimate tensile strength � t according to the Beltrami expression:
W
E
c
t�
� 2
2
. (2)
Fig. 6. Control volume under mixed mode loading for key-hole notched specimens.
a b c
Fig. 7. Typical mesh used to evaluate the SED: (a) fine mesh; (b) coarse mesh; (c) SED contour lines.
86 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
In parallel, the control volume definition via the control radius Rc needs the
knowledge of the fracture toughness K cI and Poisson’s ratio � [see Eq. (1)]. The critical
load that is sustainable by a notched component can be estimated by imposing W to be
equal to the critical value Wc . This value is considered here constant under mode I, mode II
and in-plane mixed-mode conditions. This assumption has been extensively verified for a
number of different brittle and quasi-brittle materials [18, 46].
As mentioned earlier, the properties of the PMMA considered in the present
investigation are: � t � 70.5 MPa, K cI � 1.96 MPa m, and Poisson’s ratio � � 0.38. As a
result, the critical SED for the tested PMMA is Wc � 0.839 MJ/m3, whereas the radius of
the control volume is Rc � 0.166 mm considering trial plane strain conditions.
The SED occurring inside the control volume embracing the edges of the notches has
been calculated numerically by using the FE code ANSYS. For each geometry, a model
was created defining the control volume where the strain energy density should be averaged
(see Fig. 7c). All the analyses have been carried out by using eight-node elements under the
hypothesis of plane-strain conditions. Only negligible differences (less than 1%) occur by
using fine or coarse meshes.
Table 3 summarizes the outlines of the experimental, numerical and theoretical
findings for the tested PMMA specimens, analyzed here by means of SED. In particular, the
table summarizes the experimental loads to failure P for every notch radius � compared
with the theoretical values Pth based on the SED evaluation. The tables also give the SED
value as obtained directly from the FE models of the PMMA specimens by applying the
average experimental load P to the model.
The last columns of the tables report the relative deviations between experimental and
theoretical loads. As widely discussed in [18, 46], acceptable engineering values range
from 0.8 to 1.2. As visible from the table, this range is satisfied for all the summarized test
data.
The results are given also in graphical form in Fig. 8 where the experimental values of
the critical loads (open dots) have been compared with the theoretical predictions based on
the constancy of the SED in the control volume (solid line). The plots are given for the
notched PMMA specimens as a function of the notch tip radius �. The trend of the
theoretically predicted loads is in a good agreement with the experimental ones.
T a b l e 3
Experimental Fracture Loads Compared with Theoretical Ones Evaluated
by Means of SED
RNL
(d D)
�,
mm
P ,
N
SED,
MJ/m3
Pth ,
N
P1 ,
N
P2 ,
N
P3 ,
N
�1 ,
%
�2 ,
%
�3 ,
%
0.3 1 9258 0.873 9076 9391 9080 9302 �3.5 0.0 �2.5
0.3 2 9239 0.718 9989 8664 9361 9691 13.3 6.3 3.0
0.3 4 10765 0.842 10744 10891 10635 10768 �1.4 1.0 �0.2
0.5 1 6845 1.055 6103 6928 6859 6748 �13.5 �12.4 �10.6
0.5 2 7015 0.895 6794 7015 6991 7040 �3.3 �2.9 �3.6
0.5 4 6441 0.660 7262 6889 5992 6441 5.1 17.5 11.3
0.5 6 6579 0.709 7155 7173 6608 5957 �0.3 7.6 16.7
Note. The SED in the table has been evaluated applying the numerical model P P� .
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2 87
Tensile Fracture Analysis of Key-Hole Notches ...
A synthesis in terms of the square root value of the local energy averaged over the
control volume (of radius Rc ), normalized with respect to the critical energy of the material
as a function of the notch tip radius is shown in Fig. 9. The plotted parameter is
proportional to the fracture load. The new data are plotted together independent of the
notch geometries and specimens shape. The aim is to investigate the influence of the notch
tip radius on the fracture assessment based on SED. From the figure, it is clear that the
scatter of the data is very limited and almost independent of the notch radius. All the values
fall inside a scatter ranging from 0.80 to 1.20 with the majority of the data inside 0.90 to
1.10. The synthesis confirms also the right 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 a good agreement with the recent database in terms
of SED reported in [16].
a
b
Fig. 8. Comparison between theoretical fracture loads obtained by SED and experimental data for
d D � 0.3 (a) and d D � 0.5 (b).
88 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2
A. R. Torabi, A. Campagnolo, and F. Berto
Conclusions. A new set of experimental results regarding tensile brittle fracture in
key-hole notches is provided in the paper. The fracture loads of PMMA specimens
weakened by key-hole notches were theoretically predicted by means of the well-
established brittle fracture criterion, namely the strain energy density (SED) over a
specified control volume, which embraces the notch edge. Different notched test specimens
with two different d D ratios were considered in the predictions. All the theoretical results
fall inside a scatter band of �20%, the majority of which being inside a scatter band of
�10% demonstrating the effectiveness and the repeatability of the SED criterion. The
choice of the control volume size seems to be suitable for the considered material.
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ìåíòàëüí³ âåëè÷èíè ðóéí³âíèõ íàïðóæåíü äîñòàòíüî òî÷íî ðîçðàõîâàí³ äëÿ ð³çíèõ
çíà÷åíü ðàä³óñ³â “ãàíòåëåé” çã³äíî ç êðèòåð³ºì ù³ëüíîñò³ ëîêàëüíî¿ åíåð㳿 äåôîð-
ìàö³¿, óñåðåäíåíèì ïî äåÿêîìó êîíòðîëüíîìó îá’ºìó, ùî îõîïëþº âåðøèíó êîí-
öåíòðàòîðà. Ïîêàçàíî, ùî åêñïåðèìåíòàëüí³ äàí³ äîáðå óçãîäæóþòüñÿ ç ðîçðàõóí-
êîâèìè.
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Received 21. 09. 2015
92 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 2
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|