Analytical solutions of the tensile strength and preponderant crack angle for the I-II mixed crack in brittle material/Trapiojo atsparumo tempiant ir atitinkamo leistinojo plysio kampo, esant vienaasiam apkrovimui bei I-II tipo misriam plysiui, teorinis sprendimas, pagristas maksimaliu ziediniu itempiu kriterijumi.
Yan, Zhang ; Hongguang, Ji ; Jianhong, Ye 等
1. Introduction
Generally, the cracks in real structural materials are loaded by
the complicated combined stress field due to the asymmetry of structures
and loads, anisotropy of materials or other reasons. It results in that
the stress fields around the cracks tip are significantly different from
that of pure mode I, II, III cracks. The stress field should be affected
by the pure mode I and II, even III cracks simultaneously. The cracks
different from pure mode I, II and III cracks are referred to as mixed
cracks. In the real structural materials, the mixed cracks exist
abundantly, and the I-II mixed cracks is one of the most common forms.
The mechanical behaviours of the I-II mixed cracks are always paid a
great number of attentions by engineers [1-3].
In the calculation analysis of fracture mechanics, linear-elastic
model is a common one and used widely. The linear-elastic model is not
only simple and easy to be applied, but also could avoid the problem to
some extent that the development of elastoplastic model is not perfect
and mature. For those structure materials with large brittleness, the
plastic deformation is very small when the brittle fracture occurring
under tensile stress loading. Therefore, the linear elastic model is
applicable for most brittle materials in solving fracture mechanical
problems.
Based on the linear elastic constitutive model, a series of
fracture criterions are proposed by some researchers which are
applicable for the mixed cracks, such as, the maximum circumferential stress criterion proposed by Erdogan and Sih [4]; the maximum energy
release rate criterion developed by Hussain et al. [5]. The maximum
tensile strain criterion [6]; the maximum strain energy density factor
criterion proposed by Sih [7]; equivalent stress intensity factors
criterion [8]; expansion/torsion strain energy density factor criterion
[9, 10]; J-integration criterion [11]. Among these fracture criterions,
the maximum circumferential stress criterion is the simplest one with
excellent applicability, which is frequently used by researchers and
engineers [12, 13]. It is indicated by the mixed cracks test using the
concretes that fracture angle determined by the maximum circumferential
stress criterion agrees well with experimental data [3]. It is shown
that the applicability of the maximum circumferential stress criterion
is very well for the brittle materials.
In practical engineering, it is important for us to know the
critical limited load and the initial fracture angle for propagating
once the stress intensity factors [K.sub.I], [K.sub.II] of structural
materials and fracture toughness [K.sub.IC] of model I crack are
calibrated through experimental methods. Recently, a great number of
attentions have been pain to investigate the ultimate strength of
material with crack [14-16]. In this study, the tensile strength and the
corresponding preponderant fracture angle for a I-II mixed crack
contained in infinite plate under uniaxial tensile stress are
investigated based on the linear-elastic maximum circumferential stress
criterion. Namely, it will be demonstrated that under how much the
tensile stress applied on the infinite plate making the I-II mixed crack
begin to propagate, and what the preponderant crack angle is making the
I-II mixed crack most easily to propagate in the infinite plate under
tensile stress.
2. Maximum circumferential stress criterion
Maximum circumferential stress criterion is proposed by Erdogan and
Sih [4] based on the experimental results that the mixed cracks
propagate along the direction which is perpendicular with the maximum
circumferential tensile stress. The basic statement is:
--cracks unstably propagate along the direction perpendicular with
the maximum circumferential stress [[sigma].sub.[theta][theta]] near the
tips of mixed cracks;
--the condition for cracks beginning to unstably propagate is that
the maximum [[sigma].sub.[theta][theta]] reaches a certain critical
value of the materials (tensile strength).
[FIGURE 1 OMITTED]
In the two dimensional model shown in Fig. 1 ([K.sub.III] = 0), the
stress fields near the tips of the I-II mixed cracks are expressed as
(in the form of polar coordinate; and the endpoints of crack is the
origin) [17].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [K.sub.I], [K.sub.II] are the stress intensity factors of
pure mode I and mode II cracks; [theta]([theta][-[pi],[pi]]) is positive
for counterclockwise situations. Otherwise, it is negative.
o([r.sup.-1/2]) is the high order small value in Eq. (1), which is
ignored in the following derivation. Additionally, the tensile stress is
taken as positive value in this study.
2.1. Direction of crack propagating
According to the maximum circumferential stress criterion, cracks
should propagate along the direction perpendicular to the maximum
[[sigma].sub.[theta][theta]] near the tips of I-II mixed crack. The
following conditions have to be satisfied
[partial derivative][[sigma].sub.[theta][theta]]/[partial
derivative][theta] = 0 and [[partial
derivative].sup.2][[sigma].sub.[theta][theta]]/[partial
derivative][[theta].sup.2] = 0 (2)
Differentiating at both sides of Eq. (1), we can obtain
[partial derivative][[sigma].sub.[theta][theta]]/[partial
derivative][theta] = -1/[square root of (2[pi]r)] 3/2 cos [theta]/2
[[K.sub.I]sin[theta] + [K.sub.II](3cos[theta] - 1)] (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
From [partial derivative][[sigma].sub.[theta][theta]]/[partial
derivative][theta] = 0, we get
cos [theta]/2 [[K.sub.I]sin[theta] + [K.sub.II](3cos[theta]-1)] = 0
(5)
A solution of Eq. (5) is that cos([theta]/2) = 0 ([theta] = [+ or
-] [pi], [[sigma].sub.[theta][theta]] = 0). However, if substituting
them into Eq. (4), it is found that [partial
derivative][[sigma].sub.[theta][theta]]/[partial derivative][theta] = 0.
It can not meet the condition of [[partial
derivative].sup.2][[sigma].sub.[theta][theta]]/[partial
derivative][[theta].sup.2] < 0. In addition, the fracture surface
described by this solution is the same with the surface of mixed cracks.
Actually, there is no physical meaning. Finally, the initial fracture
angle [[theta].sub.0] is determined by the following equation
[K.sub.I]sin[theta] + [K.sub.II](3cos[theta] - 1) = 0 (6)
When both [K.sub.I] and [K.sub.II] are not equal to 0 in Eq. (6),
we obtain
[[theta].sub.0] = 2arc tan [(1 + [square root of (1 +
8[([K.sub.II]/[K.sub.I]).sup.2])]]/4([K.sub.II]/[K.sub.I])
Substituting [[theta].sub.0] = 2arc tan [(1 + [square root of (1 +
8[([K.sub.II]/[K.sub.I]).sup.2])]]/4([K.sub.II]/[K.sub.I]) into Eq.(4),
we can obtain [[partial
derivative].sup.2][[sigma].sub.[theta][theta]]/[partial
derivative][[theta].sup.2] > 0. It means that the
[[sigma].sub.[theta][theta]] reaches its minimum value, namely the
maximum compressive stress. Therefore, the initial fracture angle
[[theta].sub.0] can be determined only by the following equation
[[theta].sub.0] = 2arc tan [(1 + [square root of (1 +
8[([K.sub.II]/[K.sub.I]).sup.2])]]/4([K.sub.II]/[K.sub.I]) (7)
The variable curve of [[theta].sub.0] is shown in Fig. 2.
[FIGURE 2 OMITTED]
From Eq. (6), we know that
[K.sub.II]/[K.sub.I] = -sin[[theta].sub.0]/[3cos[[theta].sub.0] -
1] (8)
From Eq. (8), it is found that 3cos[[theta].sub.0] - 1 > 0 and
[[theta].sub.0] < 70[degrees]32' due to that
[K.sub.II]/[K.sub.I] > 0 and sin[[theta].sub.0] < 0. Therefore,
substituting Eq. (6) or Eq. (8) into Eq. (4), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
It is indicated that the solution of Eq. (6) [theta] =
[[theta].sub.0] meets the condition letting the
[[sigma].sub.[theta][theta]] reaches its maximum value.
Comparing Eq. (6) and the third expression in Eq. (1), it is found
that [[tau].sub.r[theta]] = 0 on the plane where the
[[sigma].sub.[theta][theta]] reaches its maximum value. That is to say
the plane is the principle stress plane. Therefore, the maximum
circumferential stress is the maximum tensile stress near the tips of
the mixed cracks if only the singular term is retained in the all stress
components.
In Eq. (6), when [K.sub.I] = 0, [K.sub.II] [not equal to] 0 (pure
model II crack), namely, [K.sub.II]/[K.sub.I] [right arrow] [infinity],
it is derived that
[[theta].sub.0] = arccos 1/3 = -70[degrees]32'. (10)
2.2. Stress condition for cracks beginning to propagate
According to maximum circumferential stress criterion, the cracks
begin to unstably propagate when [[sigma].sub.[theta][theta]max] reaches
a certain critical value [[sigma].sub.[theta][theta]c] (tensile
strength). Generally, this critical value is determined through some
experimental methods of pure mode I crack. For a pure mode I cracks,
[K.sub.II] = 0 . From Eq. (7), we know that the initial facture angle
[[theta].sub.0] equals to 0 for a pure mode I crack which is known as
self-similar propagation. When the pure mode I crack begins to
propagate, the [[sigma].sub.[theta][theta]max] exactly reaches the
critical value [[sigma].sub.[theta][theta]c] of materials. From this
point of view and Eq. (1), it is obtained that
[[sigma].sub.[theta][theta]c] = 1/[square root or 2[pi]r]
[K.sub.IC] (11)
where [K.sub.IC] is the fracture toughness of pure model I crack.
Once this critical value of materials is determined, and combined with
Eq. (1), we know that the instability criterion of I-II mixed cracks is
1/2 cos [[theta].sub.0]/2 [[K.sub.I] (1 + cos[[theta].sub.0]) -
3[K.sub.II]sin[[theta].sub.0]] = [K.sub.IC] (12)
In a sense, the I-II mixed cracks can be considered as a kind of
equivalent mode I cracks. The equivalent stress intensity factor could
be formulated as
[K.sub.eff] =1/2 cos [[theta].sub.0]/2 [[K.sub.I](1 +
cos[[theta].sub.0]) - 3[K.sub.II]sin[[theta].sub.0]]
Then the instability criterion is
[K.sub.eff] = [K.sub.IC]
For pure mode II cracks, the initial fracture angle when the cracks
begin to propagate is [[theta].sub.0] = [cos.sup.-1] (1/3) which is
determined by Eq. (7). At this moment, [K.sub.II] = [K.sub.IIC.] If
substituted into Eq. (12), it is obtained
[K.sub.IIC]/[K.sub.IC] = [square root of 3]/2. (13)
3. Uniaxial brittle tensile strength and the corresponding crack
angle
As it is shown in Fig. 1 there is an inclined crack in infinite
plate. In this infinite plate, the far-field stresses can be determined
as following expressions according to the coordinate transformation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where [beta] is the sharp angle between the crack and the tensile
stress. From the definition of stress intensity factor, we know that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Substituting Eq. (15) into (6), the relationship between the
fracture angle [[theta].sub.0] and the crack angle [beta] could be
written as follow
tan[beta] = 1 - 3cos[[theta].sub.0]/sin[[theta].sub.0] (16)
From the viewpoint of causal relationship, the crack angle [beta]
should be a independent variable, and the fracture angle [[theta].sub.0]
should be the dependent variable. Therefore, equation (16) is an
implicit function between [beta] and [[theta].sub.0.] The explicit
function between [beta] and [[theta].sub.0] can be obtained based on the
Eqs. (7) and (15)
[[theta].sub.0] = 2arc tan [(1 - [square root of 1 +
8[cot.sup.2][beta]])]/4cot[beta] (17)
Similar with the definition of fracture toughness, we define the
tensile brittle capacity of materials as
[K.sub.j] = [[sigma].sub.c][square root of [pi]a] (18)
where the [[sigma].sub.c] is the far-field tensile stress when the
I-II mixed crack begin to propagate, [K.sub.J] is the equivalent
fracture toughness of the I-II mixed crack. When the crack begins to
propagate, the far-field tensile stress [sigma] applied to infinite
plate reaches [[sigma].sub.c] (Fig. 1), and Eqs. (15) and (12) are
satisfied simultaneously. Combining Eqs. (15) and (12), and letting
[sigma] = [[sigma].sub.c], we obtain
[K.sub.J] = [K.sub.IC]/1/4 cos [[theta].sub.0]/2 [1 +
cos[[theta].sub.0])(1 - cos2[beta]) - 3sin[[theta].sub.0]sin2[beta]]
(19)
Rewriting the above Eq. (19) in another form
[K.sub.J] = [K.sub.IC]/F([[theta].sub.0],[beta]) (20)
in which
F ([theta], [beta]) = 1/4 cos [theta]/2 x
[(1+cos[theta])(1-cos2[beta])-3sin[theta]sin2[beta]] (21)
where [theta], [beta] are treated as independent variables.
Comparing Eqs. (21) and (1), it is obtained that
[[sigma].sub.[theta][theta]] = [sigma][square root of
[pi]a]/[square root of 2[pi]r] F([theta],[beta]) (22)
According to the process of obtaining the fracture angle
[[theta].sub.0], we know
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From the Eq. (22), we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
It is assumed that the equivalent fracture toughness of the I-II
mixed crack [K.sub.J] reaches its minimum value when [beta] =
[[beta].sub.m,] namely
d[K.sub.J]/d[beta] = 0 and [d.sup.2][K.sub.J]/d[[beta].sup.2] = 0
(24)
Based on the equation (20), it is obtained that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)
In Eq. (25), [partial derivative]F([[theta].sub.0],[beta])/[partial
derivative][[theta].sub.0] = f ([[theta].sub.0],[beta]) = 0, and if
d[K.sub.J]/d[beta] = 0 according to the extremum principle, the Eq. (25)
can be written as
dF([[theta].sub.0],[beta]/d[beta] = 0 (27)
Equivalently, it is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)
Combining Eqs. (21) and (28), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)
As mentioned in above section, the roots of cos[theta]/2 = 0 can
not satisfy the condition of maximum value for
[[sigma].sub.[theta][theta]]. Therefore, only the following expression
could be obtained from Eq. (29)
(1 + cos[[theta].sub.0])sin2[beta] - 3sin[[theta].sub.0]cos2[beta]
= 0 (30)
Namely,
tan2[beta] = 3sin[[theta].sub.0]/[1 + cos[[theta].sub.0]] (31)
Combining Eqs. (16) and (31), it can be obtained that
sin[[theta].sub.0](12[cos.sup.2][[theta].sub.0] -
11cos[[theta].sub.0] + 1) = 0 (32)
The equations to determine the roots of Eq. (32) are
sin[[theta].sub.0] = 0 (33)
sin[[theta].sub.0] (12[cos.sup.2][[theta].sub.0] -
11cos[[theta].sub.0] +1) = 0 (34)
The first root can be obtained from the Eq. (33), it is
[[theta].sub.01] = 0[degrees]. Another two roots can be determined from
Eq. (34), they are [[theta].sub.02] = -35.48[degrees] and
[[theta].sub.03] = -84.13[degrees]. From Fig. 2 we know that [absolute
value of [[theta].sub.0]] < 70[degrees]32'. Therefore, the
[[theta].sub.03] can't satisfy the condition of maximum value for
[[sigma].sub.[theta][theta].] It should be rejected. Following, the
roots of [[theta].sub.01] = 0[degrees] and [[theta].sub.02] =
-35.48[degrees] will be verified.
Differentiating on the both sides of Eq. (16), it is obtained that
[sec.sup.2][beta] = 3 -
cos[[theta].sub.0]/[sin.sup.2][[theta].sub.0] d[[theta].sub.0]/d[beta]
(35)
Substituting Eq. (16) into Eq. (35)
d[[theta].sub.0]/d[beta] = [sin.sup.2][[theta].sub.0][(1 +
[tan.sup.2][beta])]/[3 - cos[[theta].sub.0]] = [2(1 -
3cos[[theta].sub.0] + 4[cos.sup.2] [[theta].sub.0]]/[3 -
cos[[theta].sub.0]] (36)
From Eq. (29), we can determine the terms [[partial
derivative].sup.2]F/[partial derivative][beta][partial
derivative][[theta].sub.0] and [[partial derivative].sup.2]F/[partial
derivative][[beta].sup.2] in Eq. (26)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)
[[partial derivative].sup.2]F/[partial derivative][[beta].sup.2] =
cos[theta]/2 [(1 + cos[theta])cos2[beta] + 3sin[theta]sin2[beta]] (38)
Similarly, from Eq. (21), the term [[partial
derivative].sup.2]F/[partial derivative][[theta].sub.0.sup.2] in Eq.
(26) can be determined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)
Substituting Eqs. (37), (38) and (39) into Eq. (26), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)
Following, the two roots of [[theta].sub.0] = 0[degrees] and
[[theta].sub.02] = 35.48[degrees] is verified respectively.
(1) [[theta].sub.0] = 0[degrees]
Substituting [[theta].sub.0] = 0[degrees] into Eq. (16), the [beta]
is determined as 90[degrees]. Substituting [[theta].sub.0] = 0[degrees]
and [beta] = [pi]/2 into Eq. (15), we know that [K.sub.II] = 0 (pure
mode I crack) under such conditions. And substituting [[theta].sub.0] =
0[degrees] and [beta] = [pi]/2 into Eq. (40), it is obtained
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Obviously, the above expression can not satisfy the condition of
[d.sup.2][K.sub.J]/d[[beta].sup.2] > 0. Therefore, [beta] = [pi]/2,
[[theta].sub.0] = 0[degrees] are not the solution which making the
[K.sub.J] reach its minimum value. Substituting [beta] = [pi]/2,
[[theta].sub.0] = 0[degrees] into Eq. (19), we get
[K.sub.J1] = [[sigma].sub.c][square root of [pi]a] = [K.sub.IC]
([beta] = [pi]/2,[[theta].sub.0] = 0) (41)
If the design tensile strength of materials [sigma] =
[[sigma].sub.c] is given, the maximum critical length of crack in the
direction of [beta] = [pi]/2 is
[a.sub.0] = [K.sub.IC]/[pi][[sigma].sup.2.sub.c] (42)
(2) [[theta].sub.02] = -35.48[degrees]
Substituting [[theta].sub.02] = -35.48[degrees] into Eq. (16), the
[beta] is determined as 68.09[degrees]. Substituting [[theta].sub.02] =
-35.48[degrees] and [beta] = 68.09[degrees] into Eq. (40), it is
obtained
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Obviously, the solution of [[theta].sub.02] = -35.48[degrees] and
[beta] = 68.09[degrees] satisfy the condition to let the [K.sub.J] reach
its minimum value. Substituting [[theta].sub.02] = -35.48[degrees] and
[beta] = 68.09[degrees] into Eq. (19), obtaining
[K.sub.J2] = [[sigma].sub.c][square root of [pi]a] = 0.97[K.sub.IC]
(43)
Comparing Eqs. (43) and (41), we find that [K.sub.J2] <
[K.sub.J1]. Therefore, this solution indeed can make [K.sub.J] reach its
minimum value. Here, we define the [[beta].sub.m] = 68.09[degrees] is
the preponderant crack angle of material under uniaxial tensile stress.
From Eq. (43), if the length of crack is a, the critical uniaxial
tensile strength of materials along the direction of preponderant crack
angle is
[[sigma].sub.c] = 0.97[K.sub.IC]/[square root of [pi]a] (44)
Comparing Eqs. (43) and (41), it is found that the uniaxil tensile
strength of materials if the direction of crack is [beta]m =
68.09[degrees] is smaller about 3% than that of the direction of crack
is [beta] = [pi]/2.
From Eq. (43), if the design tensile strength of materials [sigma]
= [[sigma].sub.c] is given, the critical crack length along the
direction [beta]m = 68.09[degrees] is
[a.sub.0] = 0.94[K.sup.2.sub.IC]/[pi][[sigma].sup.2.sub.c] (45)
In engineering, if the direction of force applied is variable or
uncertain, and the design tensile strength of materials [sigma] =
[[sigma].sub.c] is given, then the results determined by Eq. (45) could
be considered as the permitted maximum crack length at arbitrary
direction in engineering structural materials. This permitted maximum
crack length in structural materials could provide reliable theoretical
basis for the detecting and limiting the crack length in structural
design. It is noted that this result is obtained based on the brittle
fracture instability. The subcritical crack propagation and fatigue
fracture and other factors have not been considered.
The Eq. (19) can be rewritten as following form
[K.sub.J] = [K.sub.IC]/[1/2 sin[beta]cos [[theta].sub.0]/2 [(1 +
cos[[theta].sub.0])sin[beta] - 3sin[[theta].sub.0]cos[beta]]] (46)
In above equation, the values of [beta] making [K.sub.J] [right
arrow] [infinity] are determined by
sin[beta] [(l + cos[[theta].sub.0])sin[beta] -
3sin[[theta].sub.0]cos[beta]] = 0 (47)
One of the solutions of Eq. (47) is sin[beta] = 0 ([beta] = 0). The
crack is vertical and parallel with the far-field tensile stress. Under
such condition, [K.sub.I] = [K.sub.II] = 0; and the crack has no any
influence on the stress field near the crack.
Another solution of Eq. (47) is
(1 + cos[[theta].sub.0]) sin[beta] - 3sin[[theta].sub.0]cos[beta] =
0
Namely
tan[beta] = 3sin[[theta].sub.0]/[1 + cos[[theta].sub.0]]
Obviously, both the [beta] and [[theta].sub.0] are greater than 0.
This result completely can not satisfy the assumption of problem and the
experimental data; because if the [beta] is greater than 0 in the
problem, then the [[theta].sub.0] should be smaller than 0 according to
the stresses analysis and experiment. Therefore, this solution only
exists in mathematics. Actually, there is no any physical meaning.
4. Conclusion and discussion
1. In this study, the brittle tensile capacity and the tensile
strength of engineering materials for the I-II mixed crack is derived
under uniaxis tensile stress based on the linear elastic maximum
circumferential stress theory, see Eqs. (17), (18) and (19).
2. The preponderant fracture angle for the propagation of unstable
crack is [[beta].sub.m] = 68.09[degrees] for the I-II mixed crack under
uniaxis tensile stress; and the corresponding brittle tensile capacity
is [K.sub.J] = 0.97[K.sub.IC], tensile strength is [[sigma].sub.c] =
0.97[K.sub.IC]/[square root of [pi]a]. From this conclusion, it is
proposed that the maximum crack length in engineering materials
shouldn't be greater than [a.sub.0] =
0.94[K.sup.2.sub.IC]/[pi][[sigma].sup.2.sub.c], if the design tensile
strength [[sigma].sub.c] is given. This limitation for the crack length
in engineering materials provide reliable theoretical basis for
detecting the crack length in materials.
3. The theoretical results proposed in this study are different
from that of flat elliptical crack model (Jaeger and Cook, 1979), and
that of S criterion, in which the preponderant crack angle is determined
as [[beta].sub.m] = 90[degrees]. As stated in this context, the
difference of the brittle tensile capacity and the tensile strength for
the two types of preponderant crack angle [[beta].sub.m] (90[degrees]
and 68.09[degrees]) is not greater than 3%. It is relatively difficult
to check the theoretical results using experiment methods. However, the
great effort will be performed in the further study to verify the
theoretic solution proposed in this study.
4. The evolutionary point of the maximum value of [K.sub.J] is
[[beta].sub.[infinity]] = 0[degrees]. At this moment, the crack is
vertical, and parallel with the far-field tensile stress. The crack has
no any influence on the stress field in infinite plate.
http://dx.doi.org/ 10.5755/j01.mech.18.1.1275
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Zhang Yan, Department of Civil and Environmental Engineering,
University of Science and Technology Beijing, Beijing, 100083 China,
E-mail: yanzhang2009@gmail.com
Ji Hongguang, Department of Civil and Environmental Engineering,
University of Science and Technology Beijing, Beijing, 100083 China,
E-mail: jihongguang@ces.ustb.edu.cn
Ye Jianhong, Division of Civil Engineering, University of Dundee,
Dundee, DD1 4HN UK, E-mail: jzye@dundee.ac.uk
Ye Jianhong, Key Lab. of Eng. & Geomech., Institute of Geology
and Geophysics, Chinese Academy of Sci., Beijing, 100029 China, E-mail:
yejianhongcas@gmail.com
Li Shiyu, Institute of Geophysics, China Earthquake Administrate,
Beijing, 100081, China
Received February 10, 2011
Accepted February 09, 2112