Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis
In order to evaluate the mechanical behavior around small-scale yielding crack tip for both plane strain and plane stress, the asymptotic governing equations and their boundary conditions by considering fracture mechanisms are formulated. A total deformation theory of plasticity with a power-law ha...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2005
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| Цитувати: | Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis / V.N. Shlyannikov, Zh.M. Sakhabutdinov // Проблемы прочности. — 2005. — № 1. — С. 60-75. — Бібліогр.: 29 назв. — англ. |
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irk-123456789-473242013-07-11T21:08:36Z Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis Shlyannikov, V.N. Sakhabutdinov, Zh.M. Научно-технический раздел In order to evaluate the mechanical behavior around small-scale yielding crack tip for both plane strain and plane stress, the asymptotic governing equations and their boundary conditions by considering fracture mechanisms are formulated. A total deformation theory of plasticity with a power-law hardening is used. The analysis of the near-tip fields is carried out for both the maximum tensile and shear stress crack growth direction criteria, as well as for the complete range of mixity parameters and various strain-hardening levels. The new scheme of mixed-mode problem solution is proposed. Realationships between elastic and plastic mixity parameters are given as functions of the crack growth direction criterion and the strain-hardening exponent. На основі домінуючого механізму руйнування для оцінки поведінки матеріалу за маломасштабної текучості в області вістря тріщини сформульовано асимптотичні керуючі рівняння та їх граничні умови для випадків плоскої деформації і плоского напруженого стану. Рівняння базуються на дефор маційній теорії пластичності зі степеневим зміцненням. Аналіз близько лежачих полів напружень виконано для критеріїв максимальних нормальних і дотичних напружень, повного діапазону змішаних форм деформування та різних рівнів показника зміцнення. Запропоновано нову схему розв’язку задач змішаних форм руйнування. Установлені співвідношення між визначальними пружними і пластичними параметрами змішаності пред ставлено як функції виду критерію ростутріщини і показника деформаційного зміцнення матеріалу. Исходя из рассмотрения доминирующего механизма разрушения для оценки поведения материала при маломасштабной текучести в области вершины трещины сформулированы асимптотические управляющие уравнения и их граничные условия для случаев плоской деформации и плоского напряженного состояния. Уравнения основаны на деформационной теории пластичности со степенным упрочнением. Анализ близлежащих полей напряжений выполнен для критериев максимальных нормальных и касательных напряжений, полного диапазона смешанных форм деформирования и различных уровней показателя упрочнения. Предложена новая схема решения задач смешанных форм разрушения. Установленные соотношения между определяющими упругими и пластическими параметрами смешанности представлены как функции вида критерия роста трещины и показателя деформационного упрочнения материала. 2005 Article Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis / V.N. Shlyannikov, Zh.M. Sakhabutdinov // Проблемы прочности. — 2005. — № 1. — С. 60-75. — Бібліогр.: 29 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/47324 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Shlyannikov, V.N. Sakhabutdinov, Zh.M. Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis Проблемы прочности |
| description |
In order to evaluate the mechanical behavior around small-scale yielding crack tip for both plane strain and plane stress, the asymptotic governing equations and their boundary conditions by considering fracture mechanisms are formulated. A total deformation theory of plasticity with a power-law hardening is used. The analysis of the near-tip fields is carried out for both the maximum tensile and shear stress crack growth direction criteria, as well as for the complete range of mixity parameters and various strain-hardening levels. The new scheme of mixed-mode problem solution is proposed. Realationships between elastic and plastic mixity parameters are given as functions of the crack growth direction criterion and the strain-hardening exponent. |
| format |
Article |
| author |
Shlyannikov, V.N. Sakhabutdinov, Zh.M. |
| author_facet |
Shlyannikov, V.N. Sakhabutdinov, Zh.M. |
| author_sort |
Shlyannikov, V.N. |
| title |
Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis |
| title_short |
Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis |
| title_full |
Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis |
| title_fullStr |
Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis |
| title_full_unstemmed |
Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis |
| title_sort |
evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. part 1. preliminary analysis |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2005 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/47324 |
| citation_txt |
Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis / V.N. Shlyannikov, Zh.M. Sakhabutdinov // Проблемы прочности. — 2005. — № 1. — С. 60-75. — Бібліогр.: 29 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-04T07:06:12Z |
| last_indexed |
2025-07-04T07:06:12Z |
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1836699115392598016 |
| fulltext |
UDK 539.4
Evaluation of the Elastic-Plastic Mixity Parameters on the Base of
Different Crack Propagation Criteria. Part 1. Preliminary Analysis
V. N. Shlyannikov and Zh. M. Sakhabutdinov
Kazan State Power Engineering University, Kazan, Russia
УДК 539.4
Оценка упругопластического параметра смешанности на основе
различных критериев роста трещины. Сообщение 1. Предвари
тельный анализ
В. Н. Ш лянников, Ж . М. Сахабутдинов
Казанский государственный энергетический университет, Казань, Россия
Исходя из рассмотрения доминирующего механизма разрушения для оценки поведения мате
риала при маломасштабной текучести в области вершины трещины сформулированы
асимптотические управляющие уравнения и их граничные условия для случаев плоской
деформации и плоского напряженного состояния. Уравнения основаны на деформационной
теории пластичности со степенным упрочнением. Анализ близлежащих полей напряжений
выполнен для критериев максимальных нормальных и касательных напряжений, полного
диапазона смешанных форм деформирования и различных уровней показателя упрочнения.
Предложена новая схема решения задач смешанных форм разрушения. Установленные
соотношения между определяющими упругими и пластическими параметрами смешан
ности представлены как функции вида критерия роста трещины и показателя деформа
ционного упрочнения материала.
Ключевые слова : поведение трещин при смешанном типе деформирования,
маломасштабная текучесть, смена механизма разрушения, рост трещины,
параметры смешанности.
Introduction. Fatigue crack growth can be controlled by a mixture of
processes (ductile and brittle), mechanisms (static, fatigue, creep) and loading
modes (tension, torsion, biaxial/multiaxial) [1]. Additionally, mixed-mode crack
extension can be affected by many other considerations such as artefact geometry
(thin plates, thick shells, and the size, shape and orientation of the defect),
environmental effects (temperature, gaseous and liquid environment), material
state (crystallographic structure, heat treatment and route of manufacture), and
stress conditions (out-of-phase and random loading effects).
One of the important issues is that, for a large number of known mixed-mode
crack growth problems there are two fundamentally distinct classes of growth:
maximum principal stress-dominated and shear-dominated ones. This is true
regardless of whether we consider static or cyclic loading conditions. Another
issue is the intimate connection of the crack tip displacement concept to
mixed-mode elastic-plastic fracture and fatigue processes. Several elastic-plastic
finite element analyses [2-13] showed non-uniform deformation and damage
fields near an initially smooth notch tip under mixed mode loading.
© V. N. SHLYANNIKOV, Zh. M. SAKHABUTDINOV, 2005
60 ISSN 0556-171X. Проблемы прочности, 2005, № 1
Evaluation o f the Elastic-Plastic Mixity Parameters
Aoki et al. [2] predicted that two competing process zones may be associated
with the tip of crack; one process zone dominated by tensile stress and the other
one dominated by shear stress. The side of the notch, dominated by tensile stress,
blunts, while the other side, dominated by shear strains, sharpens. It was shown in
[10] and [14] that the stable crack under mixed-mode loading conditions
propagates either as a Mode I crack approximately in the direction normal to the
maximum tangential stresses (that is referred as “tensile crack”) or as a Mode II
crack in the maximum shear strain direction (that is referred as “shear crack”). It
is clear from the preceding discussion that there are two competing fracture
mechanisms that are operative near the sharpened and blunted part of the notch,
respectively, in a ductile material under mixed-mode loading. The dominant
mechanism (between the two considered) establishes the stable crack growth
direction. Furthermore, under mixed-mode loading crack growth would no longer
take place in self similar manner, that is, it will grow along a curvilinear path. The
principal feature of such crack growth is that the stable crack propagates either as
dominating by tensile crack fracture mechanism approximately in the direction
normal to the maximum tangential stresses or as dominated by shear crack
fracture mechanism in the maximum strain direction.
Several analytical and numerical studies have been undertaken to analyze
stress-strain state at the tip in order to understand elastic-plastic mixed-mode
crack behavior. For example, Shih [15] has examined the line crack subjected to
combined Mode I and Mode II loading using a “small-scale yielding” analysis of
an elastic-plastic body under plane strain conditions (i.e., extending the HRR-
solution [16, 17] on Mode I fracture to the mixed mode case). Shih [15] has
shown that two parameters, the /-integral and the mixity parameter M p ,
completely define the near-tip asymptotic stress field. The analysis was only
related to the plane strain tensile crack fracture mechanism (boundary conditions)
for small scale yielding. In [15], the plane stress crack behavior and the fracture
mechanism associated with shear crack were not taken into consideration.
All the above analytical and numerical analyses on the effect of the dominant
fracture mechanism under mixed-mode loading are focused on Mode I boundary
conditions. A similar investigation for both tensile and shear cracks in elastic-
plastic solids has not been carried out in the past. In the present paper, a
steady-state crack in elastic-plastic solids is simulated using a dominant
singularity solution governing the asymptotic behavior at the crack tip. Our
investigation is carried out within the framework of mixed-mode (combining
Mode I and Mode II), both plane strain and plane stress, and small scale yielding
conditions.
Most of the developed theories concerning the mechanisms of fracture are
based naturally on microscopic structure of materials. The theories dealing
primarily with the mechanism of ductile fracture implicate the influence of the
voids, creating at the core region around the crack tip, on the mode of initiation
and propagation of a crack. The macroscopic effect of this mechanism of
nucleation, growth and coalescence of voids is an increment of crack length in
*
direction of initial crack propagation 0 , which is apparent in various degrees in
all materials from the quasi-brittle to the quasi-plastic. The simultaneous presence
of yielding and fracture implies that several failure mechanisms are interfering in
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N 1 61
V. N. Shlyannikov and Zh. M. Sakhabutdinov
fracture of ductile materials, and the zone of influence of these mechanisms is
quite extended. Since it includes a plastic zone around the crack tip and an elastic
layer adjacent to it containing the voids. These zones may be large or insignificant,
depending on the particular properties of the material.
In brittle or quasi-brittle materials, the plastic zone developed before the
final fracture remains rather small with respect to the crack length and therefore
linear elastic fracture mechanics theory yields satisfactory results. Ductile
materials under plane strain and plane stress conditions also respond in small
scale yielding fracture as those for example, which exhibit a discontinuous mode
of fracture in which some of the grains cleave ahead of the main crack, and the
bridging regions between them then break in a ductile manner, and for which the
extent of the plastic zones may also be very limited. Then for this type of fracture,
the singular elastic solution may also be considered as a true representation of the
stress and strain fields outside the plastic zone.
Elastic Stress-Strain State. Let us consider a crack in a general mixed
mode stress field and examine the extension of the crack from the tip. All the
geometric configurations, characterizing the loaded cracked plate as well as the
stress field, are shown in Fig. 1.
a
■fr 'O' fr D
T\G<p
y a
A S '
A
X
r|CT
4 - £ 4}
CT
*Fig. 1. The biaxially loaded inclined crack and the direction 9 of propagation.
For an elastic and isotropic plate, under conditions of plane strain, containing
an inclined internal crack of length 2a and obliqueness y3, which is submitted at
infinity to a biaxial stressed state, defined by the stress o yn = o and o xn =
along two adjacent sides of the plate, the components of stresses at the tip crack
are given by relations [18-20]
K 1 9 ( 9 39) K 2 9 ( 9 39)
o xx = Æ F cosï l ’ — sm ï sm T j — Æ F sm ï l 2 + “ s ï “ s T j + T ■
K , 9 ( 9 39) K 2 9 9 39
o yy = ,----- cos- 11 + sin — sin---- I + ,------sin — cos —cos----, (1)
yy V2^ r 2 \ 2 2 ) V2^T 2 2 2 ’ w
K 1 9 9 39 K 2 9 ( 9 39)
o xy = ,-----sin — cos —cos------+ I-----cos —11 — sin — sin----I,
xy V2^ r 2 2 2 2 \ 2 2 y
62 ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N9 1
Evaluation o f the Elastic-Plastic Mixity Parameters
where the stress intensity factors K 1 and K 2 as well as the nonsingular term T
are given by
o 4 n a „ o^lna „
K 1 = 2 [1 + V — (1— ^)cos2y3], K 2 = 2 ( l - ^ ) sm 2f (2)
T = o(1— v)cos2f.
In these relations, V is the biaxiality factor of the stresses o yn and o xn at
infinity, v = o xnj o yn, and f is the angle subtended by the axis of loading and
the crack plane, a is the half of the crack length. The crack orientation angle f
(Fig. 1) is measured positive clockwise and is reckoned from vertical direction or
the direction of major tension. For the case of the oblique crack, the crack-field
mode mixity parameter M E expressed by relation (1) and (2), is given by
1+ V — (1— V)cos2f
(1— v)sin2f
(3)
A number of criteria is available for the prediction of direction of the initial
*
crack extension 0 . All these are defined using either some aspects of the
stress-strain field existing prior to the start of propagation or some modification
thereof occurring as a consequence of the extension. The criteria belonging to the
first category are the prior stress field criteria. The local symmetry criterion [21,
*
22] belongs to the second group and the predictions of 0 are based on changes
in the crack tip stress-strain field arising out of an infinitesimal extension of the
main crack. This criterion predicts the extension in the direction given by either
F2 = 0 (where F 2 is the crack extension force at the tip normal to its own plane),
or K 2 (where K 2 is the Mode II stress intensity factor at the tip of a propagating
crack). Note that for an inclined crack under remote uniform biaxial tension, the
prediction based on the local symmetry criterion is close to those due to some of
the prior field criteria [23, 24]). The present paper deals with an application of the
maximum tangential stress and the maximum shear stress criteria, because, as it is
clear from above discussion, they are corresponding to the two main dominant
fracture mechanisms. The criteria of maximum tangential stress and the maximum
shear stress are the examples of the prior stress field criteria. They directly use
certain functions of stress or strain components in an element ahead of the crack*
tip to specify the basis for the determination of fracture angle 0 . In all cases, the
crack is considered to extend in the radial direction for which the basis function
has an extremum, and the propagation begins when the function reaches a critical
value.
In the case of the maximum tangential stress (MTS) criterion [23], the crack
is considered to extend from the crack tip in the radial direction given by the
point of maximum tangential stress on a circle of finite radius from the point of
fracture initiation. The crack propagation angle 0 * is measured negative
clockwise from the crack axis and passing through the point of fracture initiation.
-1 ( 0 = 0)
r-»oo O rg (0 = 0)
2
— tan
n
-1 2
= — tan
n
-1
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N9 1 63
V. N. Shlyannikov and Zh. M. Sakhabutdinov
In the case of the maximum shear stress (MSS) criterion [24, 25], the initial
extension occurs along the plane of the maximum shear stress around the crack
tip. Both MTS and MSS criteria were further used by Williams and Ewing [26],
Eftis et al. [20] and Chrysakis [24, 25] who incorporated in the analysis the
second (non-singular) term in the series expansion of the crack tip stress field.
This idea improved considerably the fit of the theoretical predictions to the
experimental data.
The mathematical formulation of the above criteria is expressed as follows.
The M axim um Tangential Stress C riterion. Let r and 0 denote the polar
coordinates centered at the crack tip. Then, adopting the notation
Fki = 1 + t f - (1- ^ ) c o s 2 3,
Fk 2 = ( 1 - ^ )sin23,
0 / 0 30,
Fyi = cos -11 + sin - sin— \,
0 0 30
F y2 = sin2 cos2 cosy ’
0 30
F 01 = 3 cos — + cos — ,
0 30.
Fx1 = cos —11- sm —sm —- 1,
2 2
30
F 2 = sin—12+ cos —cos — I,
2 2
0 0 30
Fxy1 = sin 2 cos 2 c o s y ,
0 / 0 30
Fxy2 = cos 2 1 1 - sin 2 ^ T
0 30
F 02 = sm — + s m - - ,
(4)
it is for the biaxially loaded crack
K 1 / 0 30\ K 2 / 0 30\
7 00 = — ; ^ = l 3cos — + cos — I - — ; ^ = l 3sin— + 3sin — | +
00 4^2*7 V 2 2 ) 4 a & V 2 2
+ 7 yn ( 1 - ^ )cos2y3 sin 0,
or in dimensionless form
7 00 1
7 yn 8a/2 y R 7 [Fk1F 01 - 3Fk2F 02] + Tsin2 0.
(5)
(6)
The derivative of 700 = 7 0 0 /7 yn is
d 7 00 1
8-J l'
+ 2T sin 0 cos ( (7)
Now the maxima of 700 are determined by certain solutions of the equation
4 0 7 [FmF 01 - 3Fk 2 F02] + 1^V2 T sin 0 cos 0 = 0. (8)
where ()' = d/d0.
64 ISSN 0556-171X. npo6n.eMH npounocmu, 2005, N 1
Evaluation o f the Elastic-Plastic Mixity Parameters
The M axim um Shear Stress Criterion. Similarly, from [19, 20]
T max _ 4 ( 0 yy xx )2 + 0 2y (9)
and taking into account Eqs. (1)-(4):
111 a
o yn
[Fki(Fyi — Fxi) + Fk2(Fy2 + Fx2 )] — T \ +
+ 8r [Fk1F xy1 + Fk2 Fxy2 ]2
1/2
(10)
The equation dx dQ (r rQ = x rdj o yn), after some algebra, takes the form
P,
1 F k 1(F y1 — FX1) + Fk2(F y2 + FX2)]+ P2 [Fk1FXy1 + Fk2 FXy2] = 0, (11)
where
and
P1 V 8r [Fk1(F y1 F x1 ) + Fk2 (F y2 + Fx2 )] T
P2 = [Ffc1 Fxy1 + Fk 2 Fxy2 ].
It is expedient to study the positions of the maxima for both criteria for
different inclinations of the crack axis. These positions are defined by their angle* _
Q subtended between the direction of the respective o ee - and x rd -maximum
*
and the crack axis. In order to define angle of crack propagation Q we zero the
first derivative [Eqs. (8) and (11), respectively], examining also the sign of its
second derivative, that is
d 2 o ,
< 0 and < 0. (12)
Both criteria (5) and (9) are based on singular solutions, the point of view
being that the stress-strain characteristics are not determined at the crack tip itself,
but at some distance r from it. Sih [15] considered that this radius must be
constant and greater than the radius of a so-called core region defined by the
limits of continuum mechanics, or by the curvature of the crack-tip. Theocaris
and Andrianopoulos [27] consider that the radius of the curve, which is suitable
for each criterion, must be, at least, equal to the elastic-plastic boundary, which,
in general, is not circular. This is valid also for brittle materials because all
2
2 2
ISSN 0556-171X. npoôëeMbi npounocmu, 2005, N9 1 65
V. N. Shlyannikov and Zh. M. Sakhabutdinov
materials present some kind of non-linearity in their stress-strain curves before
fracture. Thus, we must define the elastic-plastic boundary before applying any
fracture criteria. The need to introduce the elastic-plastic boundary as the limiting
curve when evaluating each criterion can also be seen, if one accepts that crack
propagation is the consequence of the transition of a generic elementary volume
in the vicinity of the crack-tip, from the elastic to the fracture state by passing
obligatorily through a non-linear or plastic state. Since, stresses in each
elementary volume depend strongly on the distance from the crack-tip, it is
obvious that this distance is extremely important.
Hence, taking into account the singular character of stress components
included in Eqs. (5) and (9) it is necessary to obtain the elastic-plastic boundary
contour. For isotropic power-law hardening materials the plane-strain yield
condition can be written [28] as
0 2
t - 2 -
/ o xx - o yy + o .y (13)2
while for plane stress it is
0 e 0 XX + ° yy 0 xx0 yy 30xy , (14)
where 0 e is the generalized effective tensile stress, r e is the generalized
effective shear stress. Replacing 0 e in Eqs. (13) and (14) by the yield stress 0 0
and substituting Eqs. (1) into it, one can determine the elastic-plastic boundary
around the crack tip.
Calculations for the max 0 00 and the max r rd criteria, modified so that the
influence of the non-singular (second) term in the expression of stresses be taken
into account. Behavior of the non-singular term T as a function of the crack
angle 0 and the elastic mixity parameter M E are shown in Fig. 2 for different
biaxial ratio ^ and fixed value of normalized nominal stress 0 — 0 ynj 0 0 — 0.5. It
is clear that the same value of the T-term can be obtained by various combinations
of the crack angle and the biaxial ratio. Our numerical results (Fig. 3) for each
criterion are given for three types of the stressed state: uniaxial tension (^ — 0),
biaxial tension (^ — 0.5) and equibiaxial tension-compression (^ — - 1).
It is interesting first to point out in Fig. 3 the differences between the values
of 0 corresponding to each criterion under pure shear (pure Mode II) loading
conditions, i.e., ^ — - 1 and 0 — 45°. Indeed, the maximum tangential stress
. . . 5̂ _ t sje
criterion gives the value of 0 — 76 for plane strain and 0 — 73 for plane
. . . *stress, while the maximum shear stress criterion predicts 0 — 0. These differences
%
in the 0 -values relate to the different dominant fracture mechanisms, lying in
the basis of each criterion. Furthermore, it should be kept in mind that influence
of orientation of the crack 0 and the mode of biaxiality in loading of the plate ^
%
on the direction 0 of crack growth can be directly specified by the elastic mode
66 ISSN 0556-171X. npo6n.eMH npounocmu, 2005, N9 1
Evaluation o f the Elastic-Plastic Mixity Parameters
mixity parameter M E with help of Eq. (3). Thus, using any fracture criterion, in
particular, the maximum tangential stress and the maximum shear stress criteria in
*
the form of Eqs. (6) and (10), the fracture initiation angle 0 can be transformed
into a function of the elastic mode mixity parameter M e for given values of
both the biaxiality factor ] and the crack angle p (Fig. 4). In other words, the
*
crack propagation angle 0 is defined in terms of the tensile and shear stresses at
the crack tip vicinity as a function of the elastic far-field mixity parameter M e ,
♦ ̂ * ♦ /ii.e., 0 = f ( M e , criterion). Note that, at the same values of ] and p, different
criteria will predict different values of 0 * . Besides, as it can be seen in Figs. 3
and 4 there is strong dependence for both criteria on the non-singular term. The
results which are shown in Figs. 3 and 4 by solid lines are related to constant
value of the core region radius r/a = 0.01 in singular solutions (6) and (10) while
the dashed lines represent the crack growth direction criteria when the crack
distance is defined as elastic-plastic boundary contour rp ja . From the point of
view of the elastic-plastic analysis the use of the plastic zone boundary is
preferable.
Fig. 2. Non-singular term T versus crack angle p and elastic mixity parameter Me under biaxial
loading.
Small-Scale Yielding. Here we briefly discuss the boundary conditions for
symmetric pure Mode I, anti-symmetric pure Mode II and mixed mode stress
fields.
Mode I C rack-Tip Field (Tensile Crack). We first examine the Mode I
crack-tip stress field for power-law hardening materials. The stress distribution of
the Mode I plane-strain near-tip field under small-scale yielding conditions was
presented in [16, 17]. The stress-free boundary on the upper and lower faces of
the crack requires that o ro = <^oo= 0 at 0 = —180° and 180°. For the symmetric
Mode I stress field, we consider the case where o rr > 0 on the stress-free crack
faces. In front of the crack tip (the macrocrack growth direction), the shear stress
o ro at 0 = 0 must vanish, while the normal stresses o rr and o 00 have a local
minimum and maximum, respectively for the symmetric Mode I stress field.
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V. N. Shlyannikov and Zh. M. Sakhabutdinov
Fig. 3. Directions of initial crack extension 0 according to the maximum tangential stress and the
maximum shear stress criteria.
M ode II C rack-T ip Field (Shear Crack). Now we examine the anti
symmetric Mode II stress field. The stress distribution of the Mode II plane-strain
small-scale yielding conditions for strain-hardening materials was presented in
[15, 16].
As for Mode I solution, the stress-free boundary conditions on the crack
faces require that o ro = ooo = 0 at 0 = —180° and 180°. We consider the loading
conditions which produce the anti-symmetric Mode II stress field near the crack
tip with o rr > 0 at 0 = —180° and o rr < 0 at 0 = 180°. Directly ahead of the
crack tip (0 = 0), the normal stresses o rr and o 00 should vanish, while the shear
stress o r0 has a local maximum for the anti-symmetric Mode II stress field.
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Evaluation o f the Elastic-Plastic Mixity Parameters
Fig. 4. Relations between Q and Me based on the MTS and MSS criteria in the cases of constant
core region radius r/a = 0.01 and elastic-plastic boundary rpja under uniaxial tension = 0) and
equibiaxial tension-compression ( ̂= — 1).
The following analytical results of the solution of the nonlinear compatibility
equation attract our attention. From our analytical solutions for plane strain and
plane stress conditions to be discussed later we find the Airy crack-tip stress
function and their derivatives which are plotted in Fig. 5 for both the predominant
shear crack (Fig. 5a and 5d) and tensile crack (Fig. 5b and 5e) loading conditions,
respectively. As shown in Fig. 5a and 5d the boundary conditions corresponding
to the shear crack is given at point 0 — 0, that is pure Mode II. On the other
hand, the predominant loading related to the tensile crack is given at point
0 — -7 6 .7 ° (plane strain) and 0 — -73.3° (plane stress) for n — 9, as it shown
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V. N. Shlyannikov and Zh. M. Sakhabutdinov
Fig. 5. Angular variation (a, b, d, e) of the Airy stress function and their derivatives (c, f) of the
crack-tip stress fields for pure Mode II, n = 9, (a, b, c) - plane strain, (d, e, f) - plane stress.
%
in Fig. 5b and 5e. It is interesting to note that in this case when Q = —76.7° the
Q-distributions of the Airy stress function and their derivatives in Fig. 5b with
respect to Q is similar to that of the plane strain pure Mode II crack-tip fields in
Fig. 5a. The characteristics of the Airy stress function fields for each of the
corresponding shear crack solution (Fig. 5a) and the tensile crack solution (Fig. 5b)
are the same and both lead to the stress state that corresponds to the pure Mode II
(Fig. 5c). Similar results for plane stress crack behavior are represented in Fig. 5d-
5f. The results shown in Fig. 5c are the crack-tip stresses for pure Mode II and
agree well with the results [15, 16]. As shown in Fig. 5c for plane strain, the
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Evaluation o f the Elastic-Plastic Mixity Parameters
general features of the crack-tip stress field are two equally possible crack growth
directions, that are 0 = 0 and 0 = —76.7 , while for plane stress they are
0 = 0 and 0 = —73.3°.
Mixed-Mode C rack-Tip Field. On the both lower and upper crack faces,
the stress-free condition requires that o ^ = o 00 = 0 at 0 = ±180°. We consider
the loading condition which produces o rr > 0 at 0 = —180°. At this moment we
will use, for instance, the Mode I stress field boundary conditions as the basis to
construct the mixed Mode I and II stress field. We consider the loading conditions
which alone would produce a Mode I crack-tip field. To generate a possible plane
strain mixed mode solution we displace point 0 = 0 (which specifies the
boundary conditions and corresponds to the crack growth direction) on the 0-axis% %
to the left to a position of points 0 = —17.1° (Fig. 6b), 0 = —39.9° (Fig. 6c),
0* = —57.2° (Fig. 6d), 0* = —68.7° (Fig. 6e), and 0* = —76.7° (Fig. 6f). The
displacement value depends upon the ratio of the Mode II load to the Mode I
load. In Fig. 6 are shown our numerical results determined from singularity
analysis, to be given later, concerning the stress fields for intermediate mixed
mode state for strain-hardening exponent n = 9. The point, which corresponds
to the crack growth direction at the crack tip, consequently has to be moved from
* * the position 0 = 0 (pure Mode I) to the position 0 = —76.7 (pure Mode II), in
order to cover the complete mixed mode range. Hence, the intermediate point
. . . .
0 = 0 of setting the boundary conditions governs of the stress fields under mixed
mode loading. For each point, which is the crack growth direction and defined by
* *
angle 0 = 0 (Fig. 6a -6f), the dimensionless stress component ~ 00(0 ) has the
* * local maximum, ~ rr (0 ) has the local minimum, while ~ ̂ (0 ) is equal to
zero, that corresponds to main postulates of both the maximum tangential stress
. . . ̂
and the local symmetry criteria. Since the stressed state at 0 = 0 has to
correspond to the pure Mode I (the tensile crack), while at 0 = —76.7° we have
the pure Mode II (the shear crack) stressed state, we therefore expect that a
change of the dominant fracture mechanism would occur. When point 0 = 0
moves to the position of 0 = —76.7 where the normal stresses o rr and o 0g
are anti-symmetric with respect to the vertical line, the general crack-tip stress
state reach the anti-symmetric pure Mode II crack-tip field. Our analytical results
concerning the dimensionless stresses distributions under plane strain, which are
presented in Fig. 6, in general, confirm the study of Shih [15].
. . . . .
In summary, by moving point (Fig. 6a -6f) from the position of 0 = 0 to the
position 0 = —76.7 for n = 9, we can generate a complete range of mixed
mode crack-tip field. The solutions for the stress field can be obtained for the
corresponding crack growth direction criterion (or the dominant fracture
. . . H*
mechanism). The study of this case suggests that, when point 0 = 0 of
application of the predominant tensile (shear) crack boundary conditions goes
from pure Mode I (Mode II) to the mixed mode loading, there is indeed an
intermediate stressed state where the crack tip field changes from the tensile
(shear) crack dominant fracture mechanism to the shear (tensile) one.
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V. N. Shlyannikov and Zh. M. Sakhabutdinov
Fig. 6. The solutions of the mixed mode crack-tip stress field for tensile crack predominant loading
conditions with n = 9, (1) orr; (2) ~00; (5) dr0; (4) oe.
*Then, taking into account the equation M E = f (0 , criterion, dominant
fracture mechanism), which obtained earlier, we can formulate a new scheme of
mixed mode problem solution:
criterion 0 = f (p , rç, dominant fracture mechanism)
\ ~ (15)
( p ^ ^ M E ^ 0 * ij ^ M P .
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Evaluation o f the Elastic-Plastic Mixity Parameters
The relationship between M E and M p is now known in terms of the
. . /-% . . . . . . .
inclined crack angle p, biaxiality factor and the fracture initiation angle 0 .
The elastic far-field and the plastic near-field can be regarded as completely
identified for mixed-mode plane strain small-scale yielding. Thus, we found new
method, which has enabled us to directly connect the near-field to the far-field
throughout the fracture initiation angle.
Unlike the investigation [15], where the finite-element approach [29] was
used to calculate the relationship between M E and M p , in the present paper,
the near-field mixity parameter M p was obtained directly from the compatibility
equation solution as 0-distribution of the dimensionless stresses ~ 00 and ~ ro
ranging from pure Mode I to pure Mode II and including an intermediate mixed
mode fields. In particular, M p was obtained at 0 = 0 when additional boundary
*condition in form of the crack growth direction was specified at 0 = 0 . This
addition boundary condition, when we are solving the non-linear compatibility
equation, is given a definite meaning by setting the maximum value of the
0-variation of the tangential (shear) stress in an intermediate point, which is
. . . . ̂ . .
defined by the fracture initiation angle value 0 . Hence complete specification of
the boundary conditions for the non-linear mixed mode problem includes both the
stress-free boundary condition on the upper (0 = 180°) and lower (0 = -180°)
crack faces and the condition maximum tangential (shear) stress in the crack
*propagation direction (0 = 0 ) depending on the fracture criterion.
Conclusions. The results of both numerical and analytical investigations
show that each type of boundary conditions correspond to a definite crack growth
direction criterion, in particular, in present paper, these are the maximum
tangential and shear stress criteria. The satisfaction of one or another crack
. . . . . . . . .
propagation criterion at 0 = 0 implies that in this direction one or another
dominant fracture mechanism will take place. Consequently, our approach makes
it possible to obtain stress fields corresponding to one or another dominant
fracture mechanisms. In the present work, these are the mechanisms that are
discussed above and referred to as tensile and shear crack dominant fracture
mechanisms.
Acknowledgements. Funding for this study was provided by the Russian
Foundation for Basic Researches through grant 03-01-96233 and the Foundation
of Academy of Sciences of Tatarstan Republic through grant 05-5.3-218/2003(f).
Р е з ю м е
На основі домінуючого механізму руйнування для оцінки поведінки мате
ріалу за маломасштабної текучості в області вістря тріщини сформульовано
асимптотичні керуючі рівняння та їх граничні умови для випадків плоскої
деформації і плоского напруженого стану. Рівняння базуються на дефор
маційній теорії пластичності зі степеневим зміцненням. Аналіз близько
лежачих полів напружень виконано для критеріїв максимальних нормаль
них і дотичних напружень, повного діапазону змішаних форм деформу
вання та різних рівнів показника зміцнення. Запропоновано нову схему
розв’язку задач змішаних форм руйнування. Установлені співвідношення
ISSN 0556-171X. Проблеми прочности, 2005, № 1 73
V. N. Shlyannikov and Zh. M. Sakhabutdinov
між визначальними пружними і пластичними параметрами змішаності пред
ставлено як функції виду критерію росту тріщини і показника деформа
ційного зміцнення матеріалу.
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Received 06. 09. 2004
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