Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density
The main aim of the present work is to check the suitability of the brittle fracture model, namely the local strain energy density (SED), in predicting the experimental results on mode I fracture of blunt V-notched graphite components. For this purpose, a wide range of test results reported in...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2013
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irk-123456789-1126782020-12-17T17:32:34Z Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density Torabi, A.R. Bertoh, F. Научно-технический раздел The main aim of the present work is to check the suitability of the brittle fracture model, namely the local strain energy density (SED), in predicting the experimental results on mode I fracture of blunt V-notched graphite components. For this purpose, a wide range of test results reported in the recent literature on brittle fracture of V-notched test specimens characterized by different geometries is considered. The specimens are made of the same type of coarsegrained polycrystalline graphite. The fracture assessment is carried out predicting theoretically the fracture loads by means of the SED criterion. The SED parameter is evaluated by averaging the local energy over a well-defined control volume which embraces the notch edge. It is found that the SED criterion allows assessing the fracture behavior of graphite specimens characterized by different notch angles and tip radii. Проводится контроль пригодности модели хрупкого разрушения, а именно: локальной плотности энергии деформации, при прогнозировании результатов экспериментальных исследований по разрушению нормальным отрывом графитовых образцов с тупым V-образным надрезом. Рассмотрены результаты испытаний на хрупкое разрушение образцов с V-образным надрезом с разной геометрией, представленные в литературных источниках. Образцы изготовляли из однотипного крупнозернистого поликристаллического графита. Оценку разрушения проводили посредством теоретического прогнозирования разрушающей нагрузки с помощью критерия плотности энергии деформации. Параметр плотности энергии деформации вычислен путем усреднения значения локальной энергии по определенному контрольному объему, который охватывает кромку надреза. Обнаружено, что данный критерий позволяет оценивать поведение графитовых образцов с разными углами надреза и радиусами у его вершины при разрушении. Проводиться контроль придатності моделі крихкого руйнування, а саме: локальної густини енергії деформації, при прогнозуванні результатів експериментальних досліджень щодо руйнування нормальним відривом графітових зразків із тупим V-подібним надрізом. Розглянуто результати випробувань на крихке руйнування зразків із V-подібним надрізом із різною геометрією, відомі з літературних джерел. Зразки виготовляли з однотипного крупнозернистого полікристалічного графіту. При оцінці руйнування використовували теоретичне прогнозування руйнівного навантаження за допомогою критерію густини енергії деформації. Параметр густини енергії деформації обчислено шляхом усереднення значення локальної енергії за визначеним контрольним об’ємом, що охоплює кромку надрізу. Установлено, що даний критерій дозволяє оцінити поведінку графітових зразків із різними кутами надрізу і радіусами у його вершині при руйнуванні. 2013 Article Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density / A.R. Torabi, F. Bertoh // Проблемы прочности. — 2013. — № 6. — С. 5-21. — Бібліогр.: 42 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112678 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Torabi, A.R. Bertoh, F. Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density Проблемы прочности |
| description |
The main aim of the present work is to check
the suitability of the brittle fracture model,
namely the local strain energy density (SED), in
predicting the experimental results on mode I
fracture of blunt V-notched graphite components.
For this purpose, a wide range of test results
reported in the recent literature on brittle
fracture of V-notched test specimens characterized
by different geometries is considered. The
specimens are made of the same type of coarsegrained
polycrystalline graphite. The fracture assessment
is carried out predicting theoretically
the fracture loads by means of the SED criterion.
The SED parameter is evaluated by averaging
the local energy over a well-defined control
volume which embraces the notch edge. It is
found that the SED criterion allows assessing
the fracture behavior of graphite specimens
characterized by different notch angles and tip
radii. |
| format |
Article |
| author |
Torabi, A.R. Bertoh, F. |
| author_facet |
Torabi, A.R. Bertoh, F. |
| author_sort |
Torabi, A.R. |
| title |
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density |
| title_short |
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density |
| title_full |
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density |
| title_fullStr |
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density |
| title_full_unstemmed |
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density |
| title_sort |
fracture assessment of blunt v-notched graphite specimens by means of the strain energy density |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2013 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112678 |
| citation_txt |
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means of the Strain Energy Density / A.R. Torabi, F. Bertoh // Проблемы прочности. — 2013. — № 6. — С. 5-21. — Бібліогр.: 42 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT torabiar fractureassessmentofbluntvnotchedgraphitespecimensbymeansofthestrainenergydensity AT bertohf fractureassessmentofbluntvnotchedgraphitespecimensbymeansofthestrainenergydensity |
| first_indexed |
2025-07-08T04:23:43Z |
| last_indexed |
2025-07-08T04:23:43Z |
| _version_ |
1837051281857839104 |
| fulltext |
ÍÀÓ×ÍÎ-ÒÅÕÍÈ×ÅÑÊÈÉ
ÐÀÇÄÅË
UDC 539.4
Fracture Assessment of Blunt V-Notched Graphite Specimens by Means
of the Strain Energy Density
A. R. Torabia and F. Bertob,1
a University of Tehran, Tehran, Iran
b University of Padova, Vicenza, Italy
1 Berto@gest.unipd.it
ÓÄÊ 539.4
Îöåíêà ðàçðóøåíèÿ ãðàôèòîâûõ îáðàçöîâ ñ òóïûì V-îáðàçíûì
íàäðåçîì ñ ïîìîùüþ êðèòåðèÿ ïëîòíîñòè ýíåðãèè äåôîðìàöèè
À. Ð. Òîðàáèà, Ô. Áåðòîá
à Òåãåðàíñêèé óíèâåðñèòåò, Òåãåðàí, Èðàí
á Ïàäóàíñêèé óíèâåðñèòåò, Âè÷åíöà, Èòàëèÿ
Ïðîâîäèòñÿ êîíòðîëü ïðèãîäíîñòè ìîäåëè õðóïêîãî ðàçðóøåíèÿ, à èìåííî: ëîêàëüíîé ïëîò-
íîñòè ýíåðãèè äåôîðìàöèè, ïðè ïðîãíîçèðîâàíèè ðåçóëüòàòîâ ýêñïåðèìåíòàëüíûõ èññëåäî-
âàíèé ïî ðàçðóøåíèþ íîðìàëüíûì îòðûâîì ãðàôèòîâûõ îáðàçöîâ ñ òóïûì V-îáðàçíûì
íàäðåçîì. Ðàññìîòðåíû ðåçóëüòàòû èñïûòàíèé íà õðóïêîå ðàçðóøåíèå îáðàçöîâ ñ V-îáðàç-
íûì íàäðåçîì ñ ðàçíîé ãåîìåòðèåé, ïðåäñòàâëåííûå â ëèòåðàòóðíûõ èñòî÷íèêàõ. Îáðàçöû
èçãîòîâëÿëè èç îäíîòèïíîãî êðóïíîçåðíèñòîãî ïîëèêðèñòàëëè÷åñêîãî ãðàôèòà. Îöåíêó ðàçðó-
øåíèÿ ïðîâîäèëè ïîñðåäñòâîì òåîðåòè÷åñêîãî ïðîãíîçèðîâàíèÿ ðàçðóøàþùåé íàãðóçêè ñ
ïîìîùüþ êðèòåðèÿ ïëîòíîñòè ýíåðãèè äåôîðìàöèè. Ïàðàìåòð ïëîòíîñòè ýíåðãèè äåôîð-
ìàöèè âû÷èñëåí ïóòåì óñðåäíåíèÿ çíà÷åíèÿ ëîêàëüíîé ýíåðãèè ïî îïðåäåëåííîìó êîíòðîëü-
íîìó îáúåìó, êîòîðûé îõâàòûâàåò êðîìêó íàäðåçà. Îáíàðóæåíî, ÷òî äàííûé êðèòåðèé
ïîçâîëÿåò îöåíèâàòü ïîâåäåíèå ãðàôèòîâûõ îáðàçöîâ ñ ðàçíûìè óãëàìè íàäðåçà è ðàäèóñàìè
ó åãî âåðøèíû ïðè ðàçðóøåíèè.
Êëþ÷åâûå ñëîâà: V-îáðàçíûé íàäðåç, õðóïêîå ðàçðóøåíèå, ãðàôèò, ïëîò-
íîñòü ýíåðãèè äåôîðìàöèè, íàãðóæåíèå íîðìàëüíûì îòðûâîì.
Introduction. Graphite materials are normally utilized as heat shields in
various industries like aerospace, steel making and nuclear industries etc. Such
materials are vulnerable to abrupt fracture because of weak mechanical properties
and high degree of brittleness. The fracture problem would be quite serious if stress
© A. R. TORABI, F. BERTO, 2013
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 5
raisers such as defects, scratches, cracks and notches exist in the graphite
component. The stress raisers concentrate stresses around themselves and make a
hazardous region with high risk of initiating crack(s) and provoking sudden
fracture. Therefore, the fracture resistance of graphite parts should be essentially
investigated under mechanical loads, particularly in the presence of stress
concentrators [1–5] or sharp cracks [6–8].
In general, crack problems have been more extensively investigated by the
researchers than notch problems in the context of fracture mechanics, since the
origin of fracture mechanics was on those components containing pre-existing
cracks. A comprehensive literature review showed that crack propagation and
fracture have been widely assessed by several researchers on different types of
graphite materials. Under mode I loading conditions where the crack faces open
with respect to each other, two main contributions were found. The first one deals
with evaluating the fracture toughness of a type of nuclear graphite performed by
Shi et al. [9] using a three-point bending specimen. The second investigation has
been carried out by Etter et al. [10] on fracture toughness of a type of poly-
crystalline graphite using a single-edge cracked beam specimen. Dealing with
mixed mode fracture in graphite materials, however, more works were found in
literature survey. Awaji and Sato [11] were probably the first researchers who
investigated experimentally the fracture toughness of two various graphite materials
by means of the cracked Brazilian disk (CBD) specimen. The fracture toughness of
different graphite materials has been experimentally evaluated by Yamauchi et al.
[12, 13] by means of the CBD and the semi circular bend (SCB) specimens under
asymmetric three-point bending. Moreover, Li et al. [14] utilized a single-edge
cracked sample to measure the mixed mode fracture toughness of a type of
poly-granular graphite material. Other investigations on fracture toughness of
graphite materials are also worth of mentioning (see [15–22] and references
reported therein).
Nowadays, the fracture problems in which a brittle component is weakened by
a notch are investigated by means of the principles of the notch fracture mechanics
(NFM). Specific needs, like to connect various parts of structures and machines,
force designers to introduce notches of different shapes, particularly V- and
U-shaped notches, into the engineering components and structures. As mentioned
above, notches concentrate stresses in the proximity of their tip making the
component more prone to crack initiation. If the component is made of a brittle or
quasi-brittle material (such as most of industrial graphite materials), the crack
nucleates from the notch border and propagates quickly causing a sudden fracture
which usually takes place immediately after a rapid crack growth. Thus, the
fracture behavior of brittle components containing notches should be essentially
studied by using theoretical and/or experimental methods.
Few studies have been previously carried out on the fracture of graphite
materials weakened by notches. At the best of authors’ knowledge, the pioneering
researchers in this field were Bazaj and Cox [23] and Kawakami [24] who first
investigated the notch sensitivity on different graphite materials. In the past five
years, brittle fracture assessment of notched graphite components has been
conducted by taking into account mainly the notch fracture mechanics approaches.
Dealing with mode I loading, some recent results of research in this field have been
A. R. Torabi and F. Berto
6 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
published in [25] where the brittle fracture in V-notched graphite components has
been investigated both theoretically and experimentally. Some mode I fracture tests
on three different shapes of V-notched graphite specimens, which were completely
different in overall geometry, were carried out and the fracture loads were assessed
by means of a stress-based fracture model named in [25] the mean stress (MS)
fracture model. In [26], some experimental results were conducted on brittle
fracture of V-notched Brazilian disk (V-BD) specimens made of a type of
polycrystalline graphite under mixed mode loading. A stress-based brittle fracture
model, namely the V-notched maximum tangential stress (V-MTS) criterion, was
utilized for predicting the test results [26].
Beside the works performed in [25, 26] where commonly stress-based fracture
models have been employed for the fracture assessment, the strain energy density
(SED) approach, as first suggested in [27], has also been successfully used in
[28–30] with the aim to assess a large bulk of static mixed mode fracture tests from
various brittle and quasi-brittle materials. One of the most important advantages of
the mean SED approach is the mesh independency. In fact, contrary to some
parameters integrated in the local criteria (e.g., maximum principal stress,
hydrostatic stress, deviatoric stress), which are mesh-dependent, the SED averaged
over a control volume is substantially insensitive to the mesh refinement. As
widely documented in [31, 32], refined meshes are not necessary, because the mean
value of the SED on control volume can be directly determined via the nodal
displacements, without involving their derivatives. As soon as the average SED is
known, the notch stress intensity factors (NSIFs) can be calculated a posteriori on
the basis of very simple expressions linking the local SED and the relevant NSIFs.
This holds true also for the stress concentration factors (SCFs), at least when the
local stress distributions ahead of the blunt notch are available for the plane
problem. The extension of the SED method to three-dimensional cases is also
possible as well as its extension to notched geometries exhibiting small-scale
yielding.
Dealing with mixed mode I/II fracture of graphite specimens containing
rounded-tip notches of various shapes, the fracture load of notches has been
recently predicted successfully by using the SED model (see [33–35]).
In a more recent work, a large bulk of out-of-plane fracture tests have been
conducted on laboratory-scaled V-notched graphite bars and predicted well the
maximum torque that each notched bar can sustain by means of the SED criterion
[36]. As a non-conventional fracture study, brittle fracture of isostatic graphite
subjected to compression has been also studied experimentally by means of
prismatic specimens weakened by sharp and rounded-tip V-notches [37]. The SED
model has been utilized for the fracture assessment of notched graphite specimens
under compressive loading, as an extension of what has been suggested in previous
papers dealing with the cases of in-plane tension/shear and torsion loading in
notched graphite specimens.
Some new fracture test results on U-notched graphite specimens have been
reported in [38] dealing with mode I loading conditions. The specimen tested was a
disc-type sample containing a central bean-shaped slit with two U-shaped ends,
called U-notched Brazilian disk (UNBD). The experimental results of mode I notch
fracture toughness have been assessed by means of some stress-based criteria [38].
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 7
In this research, the main target was to verify the suitability of the well-known
SED criterion in predicting tensile fracture load of several blunt V-notched graphite
specimens reported previously in literature [25]. The results revealed that the SED
criterion could predict the experimental results very well for various notch angles
and different notch tip radii.
1. Experimental Results Reported in Literature.
1.1. Material. The material utilized in [25] for fabricating the test samples was
a type of commercial coarse-grained polycrystalline graphite with the properties
presented in Table 1.
1.2. Specimen. Three blunt V-notched specimens; completely different in
overall geometry; have been utilized in [25] for mode I fracture experiments. They
have been the rounded-tip V-notched three-point bend (RV-TPB), the rounded-tip
V-notched semi-circular bend (RV-SCB) and the rounded-tip V-notched Brazilian
disk (RV-BD) specimens [25]. Figure 1 displays such specimens schematically. The
dimensions of the three test specimens are presented in Tables 2, 3, and 4.
8 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. R. Torabi and F. Berto
T a b l e 1
Properties of the Graphite Material [25]
Material property Value
Elastic modulus E, GPa 8.05
Poisson’s ratio � 0.2
Ultimate tensile strength �t , MPa 27.5
Plane-strain fracture toughness K cI , MPa m 1.0
Bulk density (kg/m3) 1710
Mean grain size (�m) 320
Porosity (%) 9
T a b l e 2
Dimensions of the RV-TPB Specimens [25]
Specimen �, mm a, mm L, mm S , mm W , mm 2�, deg
RV-TPB 1.2 10 100 60 20 30, 60, 90
4.0 16 160 96 32 30, 60, 90
T a b l e 3
Dimensions of the RV-SCB Specimens [25]
Specimen a, mm D , mm S , mm �, mm 2�, deg
RV-SCB 15 60 45 1, 2, 4 30, 60, 90
T a b l e 4
Dimensions of the RV-BD Specimens [25]
Specimen a, mm D , mm �, mm 2�, deg
RV-BD 15 60 1, 2, 4 30, 60, 90
The parameters a, �, and 2� denote the notch length, the notch tip radius,
and the notch angle, respectively. The thickness for the entire specimens has been
equal to 8 mm [25]. For each type of specimens represented in Fig. 1, three notch
angles of 30, 60, and 90� and also three notch tip radii of 1, 2, and 4 mm have been
considered. To check the repeatability of the experimental results, three samples
have been tested for any of twenty seven geometries providing totally eighty one
test results [25]. The fracture loads of the V-notched graphite specimens are
presented in Table 5 [25].
Since the load–displacement curves recorded during the experiments have
been linear up to final fracture [25], using the brittle fracture criteria in the context
of linear elastic notch fracture mechanics (LENFM) such as the SED criterion is
allowable for predicting the experimental results.
2. Fracture Criterion Based on the Strain Energy Density Averaged Over
a Control Volume. In order to estimate the fracture load of notched graphite
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.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 9
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
a b
c
Fig. 1. The blunt V-notched test specimens used in [25] for fracture experiments: RV-TPB specimen
(a), RV-SCB specimen (b), and RV-BD specimen (c).
Dealing with cracked components, the strain energy density factor S [39]
was defined first by Sih 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 [40].
Different from Sih’s criterion, which is a point-wise criterion, the averaged
strain energy density criterion (SED) as presented in [27, 28] 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 [41], 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 [27] and later extended to blunt U-
and V-notches [28–30].
When dealing with cracks, the critical volume is a circle of radius Rc centered
at the tip (Fig. 2). Under plane strain conditions, the critical length, Rc , can be
evaluated according to the following expression [30]:
R
K
c
c
t
�
�
�
�
�
��
( )( )
,
1 5 8
4
2
� �
� �
I
(1)
10 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. R. Torabi and F. Berto
T a b l e 5
Fracture Loads (N) of the V-Notched Graphite Specimens [25]
Specimen 2 30� � � 2 60� � � 2 90� � �
� �1
(mm)
� � 2
(mm)
� � 4
(mm)
� �1
(mm)
� � 2
(mm)
� � 4
(mm)
� �1
(mm)
� � 2
(mm)
� � 4
(mm)
RV-TPB 153
163
157
181
195
189
261
291
317
162
174
170
196
209
216
292
327
345
162
170
166
190
200
212
305
315
326
RV-SCB 512
542
568
587
633
620
638
685
718
471
490
508
558
577
602
667
726
801
469
532
550
549
572
630
591
614
694
RV-BD 1893
1939
1875
1890
2060
2119
2057
2204
2023
1379
1452
1485
1758
1574
1606
1688
1932
1545
787
1004
938
960
1080
940
1110
1250
1060
where K cI is the fracture toughness, � Poisson’s ratio, and � t the ultimate
tensile strength of a plain specimen that obeys a linear elastic behavior.
For a sharp V-notch, the critical volume becomes a circular sector of radius Rc
centered at the notch tip (Fig. 2) while for a blunt V-notch under mode I loading,
the volume assumes the crescent shape shown in Fig. 2c, 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 (Fig. 2). Such a distance depends on the V-notch opening
angle 2�, according to the expression r0 2 2 2�
� � � � �( ) ( ) [28].
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 [29, 30].
When the area embraces the semicircular edge of the notch (and not its
rectilinear flanks), the mean value of SED can be expressed in the following form
[28]:
W F H R
Ec
tip
1
2
2 2� ( ) ( , ) ,� � �
�
(2)
where F( )2� and H Rc( , )2� � depend on previously defined parameters and are
listed in a previous reference [28]. By simply using the definition of the mode I
NSIF for blunt V-notches [42], a simple relationship between � tip and K1� can
be obtained as follows:
K F tip1
1
2 1
�
�� � ��
( ) . (3)
Then, it is possible to rewrite Eq. (3) in a more compact form:
W H R
K
E R
c
c
1
1
2
2 1
2
1
1
�
( , ) .
( )
� �
�
� (4)
Equation (4) can be used to evaluate the SED under mode I loading once K1�
is known.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 11
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
a b c
Fig. 2. Control volume (area) for sharp V-notch (a), crack (b), and blunt V-notch (c) under mode I
loading. (Distance r0 2 2 2�
� � � � �( ) ( ), for a U-notch r0 2� � .)
Alternatively 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 [31, 32]. 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.
3. SED Approach in Fracture Analysis of the Tested Graphite Specimens.
The fracture criterion described in the previous section is employed here to
estimate the fracture loads obtained from the experiments conducted on the
graphite specimens and taken from [25]. In order to determine the SED values, first
a finite element model of the graphite specimens was generated. A typical mesh
used in the numerical analyzes is shown in Fig. 3 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 notch geometry [27, 28]. This critical value can
be determined from the ultimate tensile strength � t according to Beltrami’s
expression:
W
Ec
t
�
� 2
2
. (5)
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 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
[27–30].
As mentioned earlier, the properties of the graphite material used in the
present investigation are: � t � 27.5 MPa, K cI �1 MPa m, and Poisson’s ratio
� � 0.2. As a result, the critical SED for the tested graphite is Wc � 0.0469 MJ/m3
whereas the radius of the control volume is Rc � 0.429 mm considering realistic
plane strain conditions. The material properties are the same of that reported in [25,
33].
The SED occurring inside the control volume embracing the edges of
V-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. 3). All the analyses have been carried out by
using eight-node elements under the hypothesis of plane-strain conditions.
Figure 4, which refers to the case ��1 mm and 2 30�� � , reports the strain
energy density contour lines inside the control volume for the three different
shapes of specimens considered herein. Note that the SED is symmetric with
respect to the notch bisector line.
Tables 6–8 summarize the outlines of the experimental, numerical and
theoretical findings for the tested graphite specimens with three different notch tip
radii and notch opening angles investigated in the present research and re-analyzed
12 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. R. Torabi and F. Berto
by means of SED. In particular, each 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 graphite specimens.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 13
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
a
b
Fig. 3. Typical mesh used to evaluate the SED in RV-TPB (a), RV-SCB (b), and RV-BD (c)
specimens.
c
T a b l e 6
Synthesis of the Results from RV-TPB Specimens
2�, deg �, mm W , MJ/m3 P, N Pth , N W Wc
30 1 0.0415
0.0470
0.0436
153
163
157
162
162
162
0.940
1.001
0.965
2 0.0473
0.0549
0.0516
181
195
189
180
180
180
1.005
1.082
1.049
4 0.0412
0.0512
0.0608
261
291
317
278
278
278
0.938
1.045
1.139
60 1 0.0455
0.0525
0.0501
162
174
170
164
164
164
0.985
1.058
1.034
2 0.0548
0.0623
0.0665
196
209
216
181
181
181
1.081
1.153
1.192
4 0.0517
0.0648
0.0722
292
327
345
278
278
278
1.050
1.176
1.241
14 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. R. Torabi and F. Berto
Fig. 4. SED contour lines.
90 1 0.0446
0.0491
0.0468
162
170
166
166
166
166
0.976
1.024
0.999
2 0.0514
0.0569
0.0640
190
200
212
181
181
181
1.047
1.102
1.168
4 0.0563
0.0601
0.0643
305
315
326
278
278
278
1.096
1.132
1.172
T a b l e 7
Synthesis of the Results from RV-SCB Specimens
2�, deg �, mm W , MJ/m3 P, N Pth , N W Wc
30 1 0.0615
0.0689
0.0757
512
542
568
447
447
447
1.145
1.212
1.270
2 0.0636
0.0740
0.0710
587
633
620
504
504
504
1.165
1.256
1.231
4 0.0546
0.0630
0.0692
638
685
718
591
591
591
1.079
1.159
1.215
60 1 0.0510
0.0552
0.0593
471
490
508
452
452
452
1.043
1.085
1.125
2 0.0575
0.0615
0.0669
558
577
602
504
504
504
1.107
1.145
1.195
4 0.0598
0.0709
0.0863
667
726
801
591
591
591
1.129
1.229
1.356
90 1 0.0488
0.0628
0.0671
469
532
550
460
460
460
1.020
1.157
1.196
2 0.0555
0.0602
0.0731
549
572
630
505
505
505
1.088
1.133
1.248
4 0.0471
0.0508
0.0650
591
614
694
589
589
589
1.002
1.041
1.177
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 15
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
Continued Table 6
T a b l e 8
Synthesis of the Results from RV-BD Specimens
2�, deg �, mm W , MJ/m3 P, N Pth , N W Wc
30 1 0.0445
0.0467
0.0437
1893
1939
1875
1942
1942
1942
0.974
0.998
0.965
2 0.0486
0.0577
0.0610
1890
2060
2119
1856
1856
1856
1.018
1.109
1.141
4 0.0652
0.0748
0.0630
2057
2204
2023
1745
1745
1745
1.179
1.263
1.159
60 1 0.399
0.0443
0.0463
1379
1452
1485
1493
1493
1493
0.923
0.972
0.994
2 0.0643
0.0516
0.0537
1758
1574
1606
1500
1500
1500
1.172
1.049
1.070
4 0.0599
0.0785
0.0502
1688
1932
1545
1493
1493
1493
1.131
1.294
1.035
90 1 0.0216
0.0352
0.0307
787
1004
938
1158
1158
1158
0.679
0.866
0.809
2 0.0302
0.0381
0.0289
960
1080
940
1197
1197
1197
0.802
0.902
0.785
4 0.0371
0.0471
0.0338
1110
1250
1060
1247
1247
1247
0.889
1.002
0.849
The last column of the tables reports the square root of the SED normalized by
the critical value for the material. This non-dimensional parameter is proportional to
a load and quantifies the accuracy in the fracture assessment. It should be equal to
1.00 for a 0% of relative deviation between predicted and experimental values. As
widely discussed in [30], acceptable engineering values range between 0.8 and 1.2.
As visible from the tables, this range is satisfied for the majority of the summarized
test data with only few exceptions.
Dealing with TPB specimens and notch opening angles equal to 30 and 90�,
the results are given also in graphical form in Fig. 5 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 graphite specimens as a function of the notch tip radius �.
16 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. R. Torabi and F. Berto
The trend of the theoretically predicted loads is in good agreement with the
experimental ones. The same comparison is shown in Fig. 6 for SCB specimens.
Also in this case, it is evident that the SED is able to assess the fracture loads with
a good accuracy being the predicted values also in the safe direction. Figure 7
reports the comparison between theoretical and experimental fracture loads for the
BD specimens characterized by a notch opening angle 2 60�� � .
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 normalized notch tip radius is shown in Fig. 8. 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 and only few
data outside the range 0.8–1.2. The synthesis confirms also the choice of the
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 17
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
a b
Fig. 5. Comparison between theoretical fracture loads obtained by SED and experimental data for
RV-TPB specimens with a notch opening angle 2 30� � � (a) and 2 90� � � (b).
a b
Fig. 6. Comparison between theoretical fracture loads obtained by SED and experimental data for
RV-SCB specimens with a notch opening angle 2 30� � � (a) and 2 90� � � (b).
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 good agreement with the recent database in terms of SED reported in [29, 30].
Conclusions. The fracture loads of extensive V-notched graphite specimens
reported in recent literature were theoretically predicted very well 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. Three V-notched
test specimens with completely various overall geometries (disk, semi-disk and
rectangle) were considered in the predictions. All of the theoretical results fall
inside a scatter band of �20% with the majority of which inside a scatter band of
�10% demonstrating the effectiveness and the repeatability of the SED criterion.
Only very few results fall outside the scatter band of �20% that may or may not
be attributed to possible inaccuracy in the experiments.
Ð å ç þ ì å
Ïðîâîäèòüñÿ êîíòðîëü ïðèäàòíîñò³ ìîäåë³ êðèõêîãî ðóéíóâàííÿ, à ñàìå: ëî-
êàëüíî¿ ãóñòèíè åíåð㳿 äåôîðìàö³¿, ïðè ïðîãíîçóâàíí³ ðåçóëüòàò³â åêñïåðè-
ìåíòàëüíèõ äîñë³äæåíü ùîäî ðóéíóâàííÿ íîðìàëüíèì â³äðèâîì ãðàô³òîâèõ
çðàçê³â ³ç òóïèì V-ïîä³áíèì íàäð³çîì. Ðîçãëÿíóòî ðåçóëüòàòè âèïðîáóâàíü íà
18 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
A. R. Torabi and F. Berto
Fig. 7. Comparison between theoretical fracture loads obtained by SED and experimental data for
RV-BD specimens with a notch opening angle 2 60� � �.
Fig. 8. Synthesis of brittle failure data from graphite specimens.
êðèõêå ðóéíóâàííÿ çðàçê³â ³ç V-ïîä³áíèì íàäð³çîì ³ç ð³çíîþ ãåîìåòð³ºþ,
â³äîì³ ç ë³òåðàòóðíèõ äæåðåë. Çðàçêè âèãîòîâëÿëè ç îäíîòèïíîãî êðóïíî-
çåðíèñòîãî ïîë³êðèñòàë³÷íîãî ãðàô³òó. Ïðè îö³íö³ ðóéíóâàííÿ âèêîðèñòî-
âóâàëè òåîðåòè÷íå ïðîãíîçóâàííÿ ðóéí³âíîãî íàâàíòàæåííÿ çà äîïîìîãîþ
êðèòåð³þ ãóñòèíè åíåð㳿 äåôîðìàö³¿. Ïàðàìåòð ãóñòèíè åíåð㳿 äåôîðìàö³¿
îá÷èñëåíî øëÿõîì óñåðåäíåííÿ çíà÷åííÿ ëîêàëüíî¿ åíåð㳿 çà âèçíà÷åíèì
êîíòðîëüíèì îá’ºìîì, ùî îõîïëþº êðîìêó íàäð³çó. Óñòàíîâëåíî, ùî äàíèé
êðèòåð³é äîçâîëÿº îö³íèòè ïîâåä³íêó ãðàô³òîâèõ çðàçê³â ³ç ð³çíèìè êóòàìè
íàäð³çó ³ ðàä³óñàìè ó éîãî âåðøèí³ ïðè ðóéíóâàíí³.
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theory,” Arch. Appl. Mech., 82, No. 10-11, 1361–1376 (2012).
41. H. Neuber, Kerbspannungslehre, 2nd edition, Springer-Verlag, Berlin (1958).
42. P. Lazzarin and S. Filippi, “A generalised stress intensity factor to be applied
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(2006).
Received 03. 09. 2013
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 21
Fracture Assessment of Blunt V-Notched Graphite Specimens ...
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