Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches
In the present paper, crack initiation in mixed-mode (I+II) fracture has been studied using notched circular ring specimens. A new criterion of brittle mixed-mode (I+II) fracture based on the notch tangential stress and the volumetric approach has been developed. The critical value of the equi...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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| Цитувати: | Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches / H.El Minor, M. Louah, Z. Azari, G. Pluvinage, A. Kifani // Проблемы прочности. — 2002. — № 6. — С. 61-71. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-469292013-07-07T23:13:26Z Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches Minor, H. El Louah, M. Azari, Z. Pluvinage, G. Kifani, A. Научно-технический раздел In the present paper, crack initiation in mixed-mode (I+II) fracture has been studied using notched circular ring specimens. A new criterion of brittle mixed-mode (I+II) fracture based on the notch tangential stress and the volumetric approach has been developed. The critical value of the equivalent notch stressintensity factor has been considered as fracture toughness in mixed-mode (I+II) fracture. Исследуется зарождение трещины по смешанному механизму разрушения (типа I+II) в образцах кольцевого типа с внутренним надрезом. Предложен новый критерий для описания хрупкого разрушения смешанного типа I+II, в основу которого положен объемный подход, а базовым параметром служит касательное напряжение в надрезе. Предлагается в качестве параметра вязкости разрушения для смешанного механизма разрушения по типу I+II использовать предельное значение эквивалентного коэффициента интенсивности напряжений в надрезе. Досліджується зародження тріщини за змішаним механізмом руйнування (типу І+ІІ) в зразках кільцевого типу з внутрішнім надрізом. Запропоновано новий критерій для описання крихкого руйнування змішаного типу І+ІІ, в основу якого покладено об’ємний підхід, а базовим параметром є дотичне напруження у надрізі. За параметр в ’язкості руйнування для змішаного механізму руйнування за типом І+ІІ пропонується використовувати граничне значення еквівалентного коефіцієнта інтенсивності напружень у надрізі. 2002 Article Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches / H.El Minor, M. Louah, Z. Azari, G. Pluvinage, A. Kifani // Проблемы прочности. — 2002. — № 6. — С. 61-71. — Бібліогр.: 15 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46929 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
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 Minor, H. El Louah, M. Azari, Z. Pluvinage, G. Kifani, A. Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches Проблемы прочности |
| description |
In the present paper, crack initiation in
mixed-mode (I+II) fracture has been studied using
notched circular ring specimens. A new criterion
of brittle mixed-mode (I+II) fracture
based on the notch tangential stress and the volumetric
approach has been developed. The critical
value of the equivalent notch stressintensity
factor has been considered as fracture
toughness in mixed-mode (I+II) fracture. |
| format |
Article |
| author |
Minor, H. El Louah, M. Azari, Z. Pluvinage, G. Kifani, A. |
| author_facet |
Minor, H. El Louah, M. Azari, Z. Pluvinage, G. Kifani, A. |
| author_sort |
Minor, H. El |
| title |
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches |
| title_short |
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches |
| title_full |
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches |
| title_fullStr |
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches |
| title_full_unstemmed |
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent Notch Stress Intensity Factor to the Cracks Emanating From Notches |
| title_sort |
brittle mixed-mode (i+ii) fracture: application of the equivalent notch stress intensity factor to the cracks emanating from notches |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2002 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46929 |
| citation_txt |
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent
Notch Stress Intensity Factor to the Cracks Emanating From Notches / H.El Minor, M. Louah, Z. Azari, G. Pluvinage, A. Kifani // Проблемы прочности. — 2002. — № 6. — С. 61-71. — Бібліогр.: 15 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-04T06:28:18Z |
| last_indexed |
2025-07-04T06:28:18Z |
| _version_ |
1836696731213889536 |
| fulltext |
UDC 539.4
Brittle Mixed-Mode (I+II) Fracture: Application of the Equivalent
Notch Stress Intensity Factor to the Cracks Emanating From Notches
H. El M inor,a M. Louah,a Z. A zari,b G. Pluvinage,b and A. K ifanic
a Laboratory of Applied Mechanics and Applied Technology, Rabat, Morocco
b L.F.M. Faculty of Sciences, University of Metz, Metz, France
c Laboratory of Mechanics and Physics of Materials, Rabat’s Faculty of Sciences,
University of Rabat, Rabat, Morocco
У Д К 539.4
Хрупкий механизм разрушения смешанного типа I+II в трещинах,
растущ их из концентраторов напряжения: эквивалентный
коэффициент интенсивности напряжений в концентраторе
напряжения
Х. Эль М инора, М. Луаха, 3. Азари6, Г. П лю винаж 6, А. К иф анив
а Лаборатория прикладной механики и технологии, Рабат, Марокко
6 Лаборатория механической надежности, Метц, Франция
в Лаборатория механики и физики материалов, Рабат, Марокко
Исследуется зарождение трещины по смешанному механизму разрушения (типа I+II) в
образцах кольцевого типа с внутренним надрезом. Предложен новый критерий для описания
хрупкого разрушения смешанного типа I+II, в основу которого положен объемный подход, а
базовым параметром служит касательное напряжение в надрезе. Предлагается в качестве
параметра вязкости разрушения для смешанного механизма разрушения по типу I+II
использовать предельное значение эквивалентного коэффициента интенсивности напря
жений в надрезе.
К лю чевы е слова : хрупкий механизм разрушения смешанного типа I+II,
влияние надреза, проблема угловой трещины, критерий максимального ка
сательного напряжения, эффективное касательное напряжение, эффективная
касательная протяженность, градиент относительного касательного напря
жения, весовая функция, эквивалентный коэффициент интенсивности на
пряжений, вязкость разрушения.
Introduction. First studies of fracture mechanics were performed on
precracked specimens loaded according to an elementary fracture mode I, because
it is considered to be the most dangerous fracture mode. In reality, some situations
result in a simultaneous presence of two modes of fracture: mode I (opening mode)
and mode II (shear mode). In the latter case, no standardized test procedure exists.
In recent years, a diversity of test conditions and types of specimens have been used
in the fracture toughness investigations, which resulted in the development of
© H. EL MINOR, M. LOUAH, Z. AZARI, G. PLUVINAGE, A. KIFANI, 2002
ISSN 0556-171X. Проблемы прочности, 2002, № 6 61
H. El Minor, M. Louah, Z. Azari, et al.
various mixed-mode fracture criteria, techniques of measuring threshold values of
fatigue and determining the crack growth laws applicable to multiaxial stress
situations. Some of these types of specimens have been presented and critically
analyzed by Richard [1]. Recent investigations [2, 3] have shown that precracked
circular ring specimens are well suited for the mixed-mode tests.
However, the above mentioned type of precracked specimens presents two
inconveniences:
(i) manufacturing of these specimens is regarded to be time-consuming and
expensive;
(ii) for very brittle materials, such as ceramics and high-strength steels, it is
practically impossible to precrack specimens and, therefore, the use of a notched
specimen is preferred instead.
At present, Creager’s solution [4] is used for analytical representation of the
stress intensity factor at the notch tip. This solution has been constructed by
adding a geometrical correction factor to Irwin’s solution [5]. The respective
method is based on the assumption that the characteristic distance (or the process
volume diameter) is equal to p / 2 (where p is the notch radius). However, this
procedure has a limited applicability.
Recently it has been proposed [6, 7] to characterize fracture conditions for a
notched specimen by using the actual stress gradient at the notch root. This stress
gradient can be characterized by a relationship different from the crack-tip stress
gradient. This method has been used in the present work to determine fracture
resistance in the applied mixed-mode (I+II) fracture using notched circular ring
specimens. The toughness of high-strength steel 45SCD6 was defined by a critical
equivalent notch stress-intensity factor. This approach is based on the generic
stress of the “notch fracture mechanics.”
M aterial and Specimens. Here we propose to check the validity of this
fracture criterion using experimental results obtained from notched high-strength
steel specimens subjected to compression loading. Stress distribution in the
specimen has been determined using the finite element method.
The material studied is high-strength steel 45CDS6 according to the French
standard. Its mechanical properties are listed in Table 1. Microanalysis of the
material gives the following chemical composition: 0.45%C, 1.6%Si, 0.6%Mn,
0.6%Cr, and 0.25%Mo.
T a b l e 1
Mechanical Properties of 45CDS6 Steel
Density V A, % E , Rp0.2, R ,Rm , K Ic,
(kg/m3) MPa MPa MPa MP^Vm
7800 0.28 2.8 210,0б5 1463 1662 97
Tests were performed using U-notched circular ring specimens (Fig. 1) with
the external radius R e = 20 mm, internal radius R t = 10 mm, thickness B = 7 mm,
and the notch length a = 4 mm. Different notch radii were obtained using a
wire-cutting electrical discharge machine (EDM) and wires of different diameter.
The notch-root radius was measured using a profile projector.
62 ISSN G556-Î7ÎX. Проблемыг прочности, 2GG2, № б
Brittle Mixed-Mode (I+II) Fracture: Application o f the
Fig. 1. U-notched circular ring specimen. Fig. 2. Mechanical testing.
Definition o f D ifferent Fracture M odes:
a) 0 = 0° - mode I [8];
b) 0 = 33° - mode II [2] and [3];
c) 0°< 0 < 33° - mixed mode (I+II) fracture.
M echanical Testing. Figure 2 shows a scheme of mechanical testing of
notched specimens subjected to compression loading.
Experim ental Procedure and Results. Critical Load Pc. Figure 3 shows
evolution of the experimental critical load Pc versus the notch radius p and
inclination angle 0 (0 = 0° in mode I, and 0 = 33° in mode II). It is evident from
Fig. 3 that the critical load Pc increases linearly with the notch radius p.
Fig. 3. Critical load Pc versus notch radius p for various inclination angles 0.
Bifurcation Angle. The experimental values of the bifurcation angle 0 o
measured by an optical microscope (Fig. 4) are compared to the respective
numerical values calculated using the maximum tangential stress criterion.
M echanisms o f Crack Initiation: “Volumetric A p p r o a c h According to an
engineering approach, crack initiation occurs under whatever high stress
concentration conditions and is defined as an appearance of a short crack
detectable with a magnification of X 50.
Figure 5 shows that blunted notches are characterized by the short crack
mechanism, which means increasing of the crack growth rate after a decreasing
period.
ISSN 0556-171X. npoôëeubi npounocmu, 2002, № 6 63
H. El Minor, M. Louah, Z Azari, et al.
c
Fig. 5. Fracture micromechanism: (a) short crack mechanism; (b) evolution of microcracks; (c) crack
growth.
The mechanisms of crack initiation have been described widely. These
involve various intrusion and extrusion mechanisms in pure ductile metals or
dislocation pile-up on inclusions, decohesion of the matrix and, finally, crack
initiation.
The following two considerations support the notion that a certain physical
volume is required for the crack initiation mechanisms to take place:
- The probability of crack initiation is proportional to the process volume,
where the probability to find an initiation site (inclusion) is assumed to be
uniform.
- Crack resistance is influenced by the specimen dimensions and relative
stress gradient (i.e., the stress gradient divided by the stress value), which are
dimensional parameters.
It has been proved that crack initiation mechanisms at the notch root cannot
be explained by the spot approach, i.e., one of the key factors is the maximum
local stress. We consider that the effective stress range acting in the crack
initiation process volume plays an important role in this respect.
64 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N 6
Brittle Mixed-Mode (I+II) Fracture: Application o f the
Within this volume, the average stress is high enough to promote crack
initiation, while the relative stress gradient is not too high, which makes all points
within this volume to be sufficiently stressed (by the so-called “effective
tangential stress”). The role of the relative stress gradient in the crack-initiation
process has been mentioned previously by Buch [9].
Proposed Criterion for the M ixed-M ode F ractu re Initiated from
Notches: the Equivalent Notch Stress Intensity Factor. This criterion assumes
that in mixed-mode (I+II) fracture crack initiation from notches is governed by
the tangential stress.
For various notch radii and inclination angles we have determined the
maximum tangential stress points over the notch contour. We have analyzed the
distribution of tangential stresses at the notch tip according to this approach. This
analysis shows a “pseudo singularity” stress distribution governed by the
equivalent notch stress-intensity factor K eq. We will show below that the critical
values of this parameter can be used to determine the fracture toughness in
mixed-mode (I+II) fracture.
Direction o f the M axim um Tangential Stress at the Notch Tip. Finite element
calculations (Castem 2000) have been used to determine the maximum tangential
stress in all points of the notch contour. It is noteworthy that the respective
direction varies linearly with the notch radius. This variation is given by the
following correlations:
Q0 = A (P ) ̂ + B (P ) (in degrees), (1)
where
A( p ) = -0 .0222P 2 + 1.4983P, (2)
B( P ) = -0.0456P 2 + 3.6178P. (3)
In mode II (3 = 33°) for p = 0: Q0 = B(33°) = 70.39°^ 70.5° (Sih and
Erdogan [10]) or « 70.33° (Stroh [11]).
The numerical values obtained according to the maximum tangential stress
criterion [Eq. (1)] are compared to the experimental values measured by an
optical microscope (Fig. 6).
3 (degree)
Fig. 6. Direction of the maximum tangential stress в0 versus the inclination angle 3
ISSN 0556-Î7ÎX. Проблемы прочности, 2002, № 6 65
H. El Minor, M. Louah, Z. Azari, et al.
Figure 6 shows that the propagation of the angled crack 6 0 described
according to the maximum tangential stress criterion correlates closely with the
experimental data, and the above criterion can be used to predict the bifurcation
angle for cracks emanating from notches.
Tangential Stress Distribution at the Notch Tip. In Fig. 7, we plot the
evolution of the stress distribution at the notch tip versus the distance p according
to the direction of the maximum tangential stress 6 o (for p = 0.5 mm and
fl = 18°).
Figure 7 shows that the tangential stress a 66 is appreciably higher than a rr,
a zz, and r r6 . This important difference is observed for various notch radii p and
inclination angles fl.
Fig. 7. Distribution of stresses aee, arr, azz, and ггв at the notch tip.
We assume that the mixed-mode (I+II) notch-initiated fracture is governed
by the tangential stress (while a 66 plays the major role in the cracking process).
Further, we have studied this stress distribution according to the maximum
tangential stress direction.
We have plotted the tangential stress distribution in a bilogarithmic graph
(Fig. 8) according to the procedure described elsewhere [6, 7, 12].
log(r)
66
Fig. 8. Tangential stress distribution at the notch tip.
ISSN 0556-171X. Проблемы прочности, 2002, № 6
Brittle Mixed-Mode (I+II) Fracture: Application o f the
Zone I : “High-stress” region.
Zone II: “Pseudo singularity” stress distribution governed by the equivalent
notch stress-intensity factor.
K
o - = O r ■ <4)
where o 00 is the tangential stress (MPa), r is the distance from the notch tip
(mm), and K eqp is the equivalent notch stress intensity factor (MPaVm).
Zone III : “distant” region.
Effective Tangential D istance r0f . The upper limit of the “pseudo
singularity” stress distribution has the following coordinates (r0ef , o 0f ), where
r0f is the effective tangential distance and o 0ef is the effective tangential stress.
By definition, the effective tangential distance is the diameter of the process
volume assuming it has a cylindrical shape. A typical example of the tangential
stress distribution is presented in Fig. 9. The relative tangential stress has also
been plotted versus distance r .
to
Fig. 9. Tangential stress distribution at the notch tip. Relative tangential stress gradient versus the
distance. Determination of the effective tangential distance.
The relative tangential stress is defined as
1 d(°00 ( r )
X o 00( r) dr ' (5)
The effective tangential distance can be determined using the following
considerations:
1. According to [13], the effective tangential distance exceeds the plastic
zone diameter [Rp (0 0 )].
2. The effective tangential distance is located in the stressed region, where
the stress gradient is not too high.
Weight Function. All the stressed points in the process volume have a certain
role in crack initiation from notches. This role is different for each point and
depends both on the distance between this point and the notch, and on the stress
gradient. We can define the weighted stress, which takes into account these roles,
as follows:
ISSN 0556-171X. npoôëeubi npounocmu, 2002, № 6 67
H. El Minor, M. Louah, Z. Azari, et al.
° ij = ° j p (r, xX (6)
where p ( r , x ) is the weight function. According to Weixing [14], this function is
defined by the following relationship:
P ( r , X ) = 1_ r x. (7)
Effective Tangential Stress a f . The effective tangential stress is the stress,
which corresponds to the effective distance of stress distribution. It is defined as
1 ref
— f ° e e p ( r , X)d r.a f = — J o ee<p(r, x )d r. (8)
ef 0
This is the average value of the weighted tangential stress.
Critical Equivalent Notch Stress-Intensity Factor K ceqp and Fracture
Toughness K ic. In this section, we have determined the critical equivalent notch
stress-intensity factor K ceqp in mode I, mode II, and mixed mode (I+II) fracture.
We compared its value to the fracture toughness K Ic according to Jones [8], who
considered a notch to be equivalent to a crack.
For circular ring specimens, Jones [8] expressed the fracture toughness K Ic
by the following relationship:
2.61Pc
K Ic B r F ~ , (9)
where 0.625< ( a + R i ) /R 0 < 0.845 and R i / R 0 = 0.5; a > 0.4b (b = R 0 — R i),
while 2.61 corresponds to the value of the geometrical correction and the critical
load Pc.
According to [8], the fracture toughness K lc and the critical equivalent
notch stress intensity factor K ceqp have been determined from experimental and
numerical results and presented, respectively, in Figs. 10 and 11 versus the notch
radius p.
It is noteworthy that if the notch radius is less than p c, the critical equivalent
notch stress-intensity factor K ^ is practically constant and independent of both
the notch radius and the inclination angle. The critical value of this parameter is
nearly equal to the notch stress intensity factor K IP in mode I fracture:
K cqp ~ K Ip « K Ic (for p < p c = 0.75 mm) « 97 MPaVm. (10)
However, according to [8], for p > p c = 0.75 mm, the fracture toughness
K ic exhibits a linear relationship with the notch radius p and is not an intrinsic
material characteristic.
68 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 6
Brittle Mixed-Mode (I+II) Fracture: Application o f the
p (mm)
Fig. 10. The influence of the notch radius p on the fracture toughness K lc.
p (mm)
Fig. 11. The influence of the notch radius p on the critical equivalent-notch stress intensity factor
K c .Keqp•
M ixed-M ode (I+II) Criterion Based on the Equivalent Notch Stress- Intensity
Factor KCqp • The tangential stress distribution at the notch tip can be described
by two types of fracture criteria: global and local. In the case of a notch, there is
no stress singularity at the crack tip, but the maximum stress is followed by a
“pseudo singularity,” where the stress distribution is governed by the equivalent
notch stress-intensity factor.
Global Fracture Criterion. It is assumed that crack initiation occurs:
(i) in the direction perpendicular to the tangential stress when it reaches its
maximum value;
(ii) when the equivalent notch stress-intensity factor K c reaches its critical
value:
K<eqp= ° Ce f ^ = K Ip . (11)
Local Fracture Criterion (Volumetric Approach). The local fracture criterion
is based on the following considerations: for physical reasons, a certain volume
(called “effective volume”) is required for the fracture process to occur. Within
ISSN 0556-171X. npoôëeubi npounocmu, 2002, № 6 69
H. El Minor, M. Louah, Z. Azari, et al.
this volume, the effective tangential stress can be considered as an average stress
tangential weight, which takes into account the tangential stress distribution. This
process volume can be described by the distance r f , or the so-called “effective
tangential distance” considering that the specimen thickness is constant and the
process volume is cylindrical.
The crack initiation is assumed to occur when both the effective tangential
stress o f and the effective tangential distance reach their critical values.
Conclusions. For brittle mixed-mode (I+II) fracture emanating from notches,
the maximum tangential stress criterion, which is based on the effective tangential
stress and the equivalent notch stress-intensity factor, agrees well with the
experimental data and can be used to predict the bifurcation angle for cracks
emanating from notches.
The volumetric approach envisages that the cracking process requires a
physical volume to occur. The effective tangential stress acting in this volume
(called the effective volume) is given by the value of the minimum relative
tangential stress.
The results of this work and some other ones [6, 12, 15] indicate that this
approach provides a relatively good description of the notch effect.
Р е з ю м е
Досліджується зародження тріщини за змішаним механізмом руйнування
(типу І+ІІ) в зразках кільцевого типу з внутрішнім надрізом. Запропоновано
новий критерій для описання крихкого руйнування змішаного типу І+ІІ, в
основу якого покладено об’ємний підхід, а базовим параметром є дотичне
напруження у надрізі. За параметр в ’язкості руйнування для змішаного
механізму руйнування за типом І+ІІ пропонується використовувати гра
ничне значення еквівалентного коефіцієнта інтенсивності напружень у над
різі.
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Received 19. 03. 2002
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N 6 71
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