Received: 1 May 2023
Revised: 8 June 2023
Accepted: 10 June 2023
DOI: 10.1111/ffe.14090
ORIGINAL ARTICLE
Mixed-mode I/II stress intensity factors of a biaxially loaded
plate with a central slant frictional crack: An analytical
comprehensive solution
Mohammad M. I. Hammouda
Department of Mechanical Engineering,
Al-Azhar University, Cairo, Egypt
Correspondence
Mohammad M. I. Hammouda
Email: mohammad.hammouda@yahoo.
com
Abstract
This work presents a comprehensive analytical solution for the mixed-mode
I/II stress intensity factor of a biaxially loaded plate that has a centrally
straight inclined crack with frictional surfaces. The solution addresses every
possible combination of coefficient of friction, biaxial ratio, crack length, and
angle. The present results agree with the available corresponding finite element solutions. For the tension-compression and compression-tension biaxial
loading, crack angles are recognized to achieve pure mode I, pure mode II,
and mixed-mode crack tip deformation in terms of stress biaxiality and friction
coefficient. In the compression-compression biaxial loading, only mode II
crack tip deformation is operative. The ranges of crack angles necessary to produce dormant cracks are determined for the biaxial loading patterns, where a
compressive stress is involved, as a function of the biaxial ratio and friction
coefficient. It is possible to apply the present solution to cracked plates with
two-dimensional stresses that include a shear stress.
KEYWORDS
biaxial loading, frictional crack surfaces, inclined cracks, mixed mode I/II stress intensity
factor, two-dimensional compound stresses
Highlights
• This work evaluates modes I/II stress intensity factor of central inclined frictional cracks in plates under biaxial loading.
• The solution addresses every possible combination of friction, biaxiality, and
crack angle and length.
• The solution covers the three biaxial loading patterns that include a compressive stress.
• The solution can be applied to cracks in plates under complex 2D stresses
that include a shear stress.
Fatigue Fract Eng Mater Struct. 2023;1–13.
wileyonlinelibrary.com/journal/ffe
© 2023 John Wiley & Sons Ltd.
1
1 | INTRODUCTION
Engineering materials and structures fail in service due
to the existence of either internal or external cracks,
which grow as a result of operating loads.1 The present
cracks are generally oriented in space with respect to the
applied loads, and this may develop different modes of
crack tip deformation (CTD).2–4 For example, the tips of
a through-thickness inclined crack located centrally in a
disc, which is subject to in-plane forces, experience
either (1) a mixed-mode I/II or (2) the pure mode II
deformation.5,6 Numerous similar crack-load configurations that exhibit that mode mixing can be
mentioned.7–9
Of interest in the fracture mechanics field and hence
to some researchers has been a slant through-thickness
crack centrally located in a plate subject to a biaxial
loading,10–12 which refers to a crack that is inclined at an
angle other than 90 to the loading direction. This type of
crack is typically found in materials that are subjected to
complex loading conditions such as turbine blades or jet
engine components.13,14
The presence of a slant crack in a plate can significantly reduce its load-carrying capacity as it can lead to
stress concentrations and, ultimately, failure of the
material. Such cases require careful considerations and
analysis to ensure the safety and structural integrity of
the material. The stress intensity factor (SIF) is the
most important parameter in linear elastic fracture
mechanics15,16 that has been invoked to deal with such
problems.12,17–27
Depending on the crack-load configuration, the system of a slant crack may exhibit (1) the pure opening
mode I, (2) the pure shear mode II, or (3) a mixed-mode
I/II CTD.28–30 The SIFs associated with mode I and mode
II CTD are respectively KI and KII. The two letters I and
II are not used in the present article as subscripts, as usually invoked in standard literature to give space for other
subscripts. Tension-tension biaxial loading (TTBL) leads
to an open crack where KI ≥ 0 and KII ≥ 0. When at
least one of the two involved stresses is compressive, that
is, the cases of tension-compression biaxial loading
(TCBL) and compression-tension biaxial loading (CTBL),
the two crack surfaces come into contact at some crack
angles, and KI becomes ≤ 0 with KII ≥ 0 (pure mode II
CTD). A crack in compression-compression biaxial
loading (CCBL) is closed, whatever its angle.
numerical,7,11,17–22,28–30,32,33
and
Experimental,2,9,31
10
analytical works in the literature determined the values
of KI and KII for an inclined crack in a plate under
biaxial loading.
Fatigue cracks grow in the opening mode I and in a
mixed mode,3,4,12,23,34,35 and they can also grow in a
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shear mode.24–27 For simplicity, when numerically or
experimentally dealing with cracked plates in biaxial
loading, the majority of researchers either (1) adjusted
the crack-load system to produce an open crack with
mixed-mode I/II CTD or (2) assumed smooth crack surfaces in the presence of closed cracks with the pure mode
II CTD. However, cracked surfaces are rough due to the
existence of asperities.36,37 Thus, closed cracks involve
additional features compared to open cracks because of
the normal and tangential friction forces arising on the
crack surfaces in contact.6,28–30,36–38
The role of crack surface friction has been examined experimentally in some cases.37 Experimental
quantification of the friction contribution to mode II
fatigue crack growth is difficult.37 Hammouda et al.28–30
used the finite element method (FEM) to compute the
SIFs of a slanting frictional crack centrally located in a
plate subjected to TTBL, TCBL, CTBL, and CCBL. The
analyzed controlling parameters were crack-length to
plate width ratio, crack angle, and friction coefficient.
However, numerical techniques such as FEM, by
definition, give an exact solution to an approximate
problem. For time and effort savings, more accurate
solutions, and ease of use, researchers prefer analytical
solutions.
In spite of its importance, the problem of sliding slant
cracks with friction surfaces in contact has not yet been
analytically addressed in relevant literature. Thus, the
main objective of the present work is to analytically compute the SIFs of a central straight slant frictional crack in
a plate that is biaxially loaded. The present solution is
comprehensive since it manipulates the entire range of
the controlling parameters of the problem, that is, crack
angle and length, friction coefficient, and load biaxiality.
Further, it is possible to apply the present solution to
cracked plates with two-dimensional stresses that include
a shear stress.
2 | P R E S E N T AN A L Y S I S
Figure 1 presents a square plate made of a homogeneous
isotropic material with a central through-thickness
straight slant crack. The ratio of the crack length 2a to
the plate width 2w is ϕ. The crack makes an angle θ with
the vertical Cartesian y-axis in the clockwise direction.
The plate is subject remotely to biaxial stresses,
σ y ¼ λy σ along the y-axis and σx = λxσ along the horizontal Cartesian x-axis. Here, σ is a normal positive reference
stress. The crack is first assumed to have frictionless surfaces. The mode II SIF of that crack, KIIS , is given by the
applied loads and the crack/plate geometry. The subscript
S refers to the frictionless crack.
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3
F I G U R E 1 A square plate with a central
through-thickness slant crack and biaxial
stresses, σ y and σ x , applied remotely.
The stresses generated within the uncracked plate
under the above remote load are as follows. An infinitesimal, two-dimensional material element located radially
at a distance r measured along the crack line from its
center O has the following three stresses:
ð1Þ
2. tangential stress σθθ normal to the crack line
σθθ ¼ λy s2 þ λx c2 σ
ð2Þ
3. radial normal stress σrr along the crack line
σrr ¼ λy c2 þ λx s2 σ
ð3Þ
Here, s ¼ sinθ, c ¼ cos θ, respectively. The mode II SIF
of the current crack in an infinite plate, KIIS∞ , is given
by σrθðπaÞ0:5 .15,16 Using Equation (1),
pffiffiffiffiffi
KIIS∞ ¼ λy λx sc σ πa
pffiffiffiffiffi
YIIS
σrθ πa
λy λx sc
ð6Þ
Invariably, KIIS ¼ 0 if λy ¼ λx . Equation (2)
indicates that the surfaces of the crack are open
over its entire length in the case of TTBL. For the
other three loading possibilities, TCBL, CTBL, and
CCBL, the two crack surfaces are expected to be in
contact at some crack angles that meet the requirement
of σθθ < 0. The range of those angles depends on ϕ, λy ,
and λx .
Friction between the crack surfaces reduces the
corresponding mode II SIF. Coulomb friction with a
coefficient μ is now assumed between the crack surfaces,
with the assumption of contact along the entire crack
line. Thus, the normalized mode II SIF of the present system with frictional crack surfaces KIIF is given by replacing the shear stress σrθ in Equation (6) by ðσrθ þ μσθθ Þ,
that is,
ð4Þ
KIIF ¼ For a frictionless crack in a finite plate,
ð5Þ
The normalized mode II SIF of the considered crack
system is YIIS . From Equation (1)
KIIS ¼ 1. shear stress σrθ acting along the crack line
σrθ ¼ λy λx sc σ
pffiffiffiffiffi
KIIS ¼ YIIS σ πa
pffiffiffiffiffi
YIIS
ðσrθ þ μσθθ Þ πa
λy λx sc
ð7Þ
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HAMMOUDA
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In the case of contact crack surfaces, σθθ has a negative sign. Equations (1) and (2) give σrθ and σθθ . From
Equations (2), (6), and (7)
(
λy s2 þ λx c2
1þ μ
λy λx sc
KIIF ¼ KIIS
The term μ
λy s2 þλx c2
ðλy λx Þsc
on the pattern of the applied loads; and (3) in outof-phase biaxial loading, the load mix continuously
changes during load cycling.
)
ð8Þ
is the effect of frictional contact
between the two crack surfaces in reducing
the corresponding mode II SIF. The normalized mode II
SIF of the considered crack with frictional surfaces,
YIIF , is
(
YIIF ¼ YIIS
λy s2 þ λx c2
1 þ μ
λy λx sc
)
ð9Þ
3 | R ES U L T S A N D D I S C U S S I O N
The present results are relevant to fatigue crack growth
experiments under biaxial loading, where the monitored
crack has a zigzag path with rough surfaces having
irregularly distributed asperities. This induces, at every
instant during load cycling, irregular mixing of CTD at
the front of the tested crack,33 whatever the pattern of
the applied loads is. An analysis of such a case is
extremely difficult, if not impossible for the time being.
Even when the existence of that actual configuration of
crack surfaces is ignored and instead the crack surfaces
are assumed to be either frictional or frictionless, (1) an
existing crack can change direction depending on its tip
load mix, which is thus continuously changing during
the test; (2) the load mix in a biaxially loaded component
can be different during loading and unloading depending
F I G U R E 3 Finite element data points of YI29 for open inclined
central cracks (θ = 0–90 ) in finite plates (ϕ = 0.3 and 0.5) subjected
to biaxial stress with λy = 1 and 1 and λx ranging from 1 to 1.
F I G U R E 2 Crack angle-loading configuration required to operate (1) a pure sliding mode II CTD with frictional crack surfaces (regions
(a)), (2) a mixed mode I/II CTD (regions (b)), (3) a pure mode I CTD, that is, θ = 0 or θ = 90 . Figure 3 gives the circular data points which
correspond to YI = 0. [Colour figure can be viewed at wileyonlinelibrary.com]
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5
With the present analyzed crack, see Figure 1, (1) dormant cracks, (2) pure opening mode I loading, (3) pure
mode II loading, and (4) mixed
mode I and II loading are
possible. By considering λy ¼ 1 and jλx j ≤ 1, the present
analysis covers the entire range of biaxial loading. The
main control variables for a given ϕ are λy , λx , θ, and μ.
In Figure 2, the two solid lines plot θ against λx for
σθθ = 0 in the cases of λy = 1, Figure 2A, and λy = 1,
Figure 2B (see Equation 2). The regions (b) in Figure 2
refer to the situations (σθθ > 0) where a mixed mode I/II
CTD operates. The regions (a) in Figure 2 define the situations where σθθ is less than 0, frictional sliding contact
occurs between the two crack surfaces, and the outcome
is a pure mode II crack tip deformation (CTD), with the
possibility of a dormant crack if KIIF = 0. Regardless of
the load biaxiality, cracks that have an angle of 0 or 90
have a pure mode I CTD.
The mode I SIF of the current crack in an infinite
plate,15,16 KI∞ , is
pffiffiffiffiffi
KI∞ ¼ λy s2 þ λx c2 σ πa
ð10Þ
Here, λy s2 þ λx c2 is the normalized mode I SIF,
YI∞ ¼ σσθθ . In a finite plate, the corresponding mode I SIF
and its normalized factor are KI and YI, respectively.
Friction between crack surfaces has no influence on YI.
Figure 3 presents published finite element (FE) data
points reported by Hammouda et al.29 of YI for open
inclined central cracks (θ = 0–90 ) in finite plates (ϕ = 0.3
and 0.5) subjected to biaxial stress with λy = 1 and 1
and λx ranging from 1 to 1.
Equation
(10) suggests plotting YI against
λy s2 þ λx c2 as shown in Figure 4, which concludes that
YI is fairly fitted to a linear relation in the form:
YI ¼ A λy s2 þ λx c2
ð11Þ
For the crack-loading systems shown in Figure 3,
Hammouda et al.29 used the FE method to numerically
compute the corresponding normalized mode II SIF with
frictionless crack surfaces, YIIS. The provided YIIS-θ data
points for each crack-load system were best-fitted in the
present work to a six-degree polynomial, see the example
presented in Appendix A. The resulting fitted YIIS-θ
curves are the lines, which are plotted in Figures 5 and 6
with μ = 0.
For each crack-load system, Equation (9) used the
corresponding fitted YIIS-θ curve to calculate the normalized mode II SIF, YIIF, of inclined central cracks (θ = 0–
90 ) in finite plates (ϕ = 0.3 and 0.5) for different biaxial
stresses (λy = 1 and 1, and λx ranging from 1 to 1 in
steps of 0.2) as shown in Figures 5 and 6. Now, the surfaces of the cracks are frictional, with μ ranging from 0 to
1 in steps of 0.2. Figures 5B–F and 6B–F show data
F I G U R E 4 Normalized mode I SIF, YI, of inclined central cracks (θ = 0–90 ) in finite plates (ϕ = 0.3 and 0.5) plotted against the
normalized tangential stress normal to the crack line, σσθθ (=λy s2 þ λx c2 ), for different biaxial stresses (λy = 1 and 1, and λx ranging from 1 to
1): s = sin θ and c = cos θ.
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HAMMOUDA
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F I G U R E 5 Normalized mode II SIF, YII, of inclined central cracks (θ = 0–90 ) in finite plates with ϕ = 0.3 for different biaxial stresses
(λy = 1 and 1, and λx ranging from 1 to 1); the crack surfaces are frictional (μ = 0–1).
corresponding to TCBL, CTBL, and CCBL with different
friction coefficients. Figures 5A and 6A show uniaxial
tension loading data with solid lines, where friction is
not a factor, and uniaxial compression loading data with
solid dotted lines that each correspond to a value of μ.
For TCBL, CTBL, and CCBL, respectively, the solid,
double solid, and square dotted lines in Figure 5 for
ϕ = 0.3 and 6 for ϕ = 0.5 plot YIIF against θ according
to Equation (9). For comparison, corresponding
numerically computed FE YIIF-θ values from Hammouda
et al.29 are shown in Figures 5 and 6 for each crack-load
system. The lines that correspond to Equation (9) agree
with the associated FE results that have already been
published.
General notes follow. (1) The sliding direction of the
two crack surfaces in TCBL is opposite to the sliding
direction in CTBL and CCBL, and thus, the YII values
in TCBL are positive, while those in CTBL and CCBL
are negative. (2) For a load pattern with a certain μ
value, (i) the relation YIIF-θ for ϕ = 0.3 is similar to that
for ϕ = 0.5, (ii) the value of YIIF for a certain θ at
ϕ = 0.5 is greater than that at ϕ = 0.3, and (iii) the magnitude of YIIF is less than or equal to the associated
magnitude of YIIS, depending on θ. (3) For a certain
crack-load pattern, the magnitude of YIIF decreases as μ
increases.
3.1 | Behavior of YII in TCBL
In the present work, the TCBL pattern is characterized
by λy = 1 and 0 ≥ λx ≥1. Here, the values of YII are
positive, see Figures 5 and 6. Irrespective of ϕ, μ, and λx,
at θ = 0 and θ = 90 , pure mode I CTD is operative.
Depending on the value of λx, at θ = 0 , YI < 0, see
Figure 3.
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F I G U R E 6 Normalized mode II SIF, YII, of inclined central cracks (θ = 0–90 ) in finite plates with ϕ = 0.5 for different biaxial stresses
(λy = 1 and 1, and λx ranging from 1 to 1); the crack surfaces are frictional (μ = 0–1).
For a certain value of λx, Equation (12) calculates the
crack angle θC that separates the two regimes (a), that is,
pure mode II CTD, and (b), that is, mixed mode I/II
CTD, see Equation (11), as shown in Figure 2 (A), that is,
tan θc ¼ λx
2
ð12Þ
The measure of θC increases from 0 to 45 as the
magnitude of λx increases from 0 to 1, independently of ϕ
and μ.
For certain values of ϕ, λx, and μ, the mixed mode I/II
CTD operates at θ, ranging between the associated θC
and 90 . In that range and irrespective of the value of μ,
YII starts with a zero value at θ = 90 , increases as θ
decreases, reaches a maximum value at θ ≈ 45 , and
then decreases till θ = θC, see Figures 5 and 6. For θ
≤ θC, YII decreases as θ decreases and becomes zero at
θ = θS. For θ ≤ θS, the two crack surfaces stick, resulting
in a dormant crack. The angle θS is calculated by setting
YIIF to 0, that is,
ð1 λx Þsc þ μ s2 þ λx c2 ¼ 0
ð13Þ
Thus, for TCBL pattern
1 λx
þ
tanθS ¼ 2μ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 λx 2
λx
2μ
ð14Þ
The measure of θS is independent of ϕ. Figure 7
(λy = 1) shows the variation of θS with λx and μ. The
measure of θS increases as μ increases and the magnitude
of λx increases; see also Figures 5 and 6.
3.2 | Behavior of YII in CTBL
The CTBL pattern is characterized by λy = 1 and
1 ≥ λx ≥ 0. Here, the values of YII are negative, see
Figures 5 and 6. Irrespective of ϕ, μ, and λx, at θ = 0 and
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HAMMOUDA
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F I G U R E 7 Variation of θS with λx and μ for the TCBL and
CTBL patterns.
θ = 90 , pure mode I CTD is operative. Depending on the
value of λx, at θ = 90 , YI < 0, see Figure 3.
For a certain value of λx, Equation (12) calculates the
crack angle θC that separates the two regimes (a), that is,
pure mode II CTD, and (b), that is, mixed mode I/II
CTD, as shown in Figure 2B. The measure of θC increases
from 0 to 45 as λx increases from 0 to 1, independently
of ϕ and μ.
For certain values of ϕ, λx, and μ, the mixed mode I/II
CTD operates at θ, ranging between 0 and the associated
θC. In that range and irrespective of the value of μ, YII
starts with a zero value at θ = 0, increases as θ increases,
till θ = θC, see Figures 5 and 6. For θ ≥ θC, the magnitude of YII decreases as θ increases and becomes zero at
θ = θS. For θ ≥ θS, the two crack surfaces stick, resulting
in a dormant crack. The angle θS is calculated by setting
YIIF to 0, that is,
ð1 λx Þsc μ s2 þ λx c2 ¼ 0
ð15Þ
Thus, for CTBL pattern
1 þ λx
þ
tan θS ¼
2μ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
1 þ λx 2
þ λx
2μ
F I G U R E 8 Regimes of active and dormant cracks in the case
of compression-compression biaxial loading, λy = 1 and 0 ≥ λx
≥ 1, in terms of crack angle θ, biaxial ratio λx, and friction
coefficient μ.
ð16Þ
The measure of θS is independent of ϕ. Figure 7
(λy = 1) shows the variation of θS with λx and μ. The
measure of θS increases as μ decreases and λx increases;
see also Figures 5 and 6.
3.3 | Behavior of YII in CCBL
In the present work, the CCBL pattern is characterized
by λy = 1 and 0 ≥ λx ≥ 1. Here, the values of YII are
negative, see Figures 5 and 6. Equation (12) and
Figure 2B demonstrate that a pure mode II operates at all
crack angles for all values of 0 ≥ λx ≥ 1. As illustrated
in Figures 5 and 6, YIIF for the CCBL is zero at two crack
angles, θS1 and θS2, which are given by making use of
Equation (9) or Equation (15). Hence,
1 þ λx
∓
tan θ S1 ¼
2μ
S2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
1 þ λx 2
þ λx
2μ
ð17Þ
Figure 8A shows θS1 (the continuous lines) and θS2
(the dashed lines) plotted versus λx for different values of
μ. For a certain μ value, as the magnitude of λx increases,
θS1 increases and θS2 decreases, that is, (θS2 θS1)
decreases till that difference becomes zero at a critical
value of λx = λx , at which θ = θ*. Both the measure of
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θ*and its associated magnitude of λx decrease as μ
increases. For a given value of μ, the corresponding value
of λx is obtained by setting the terms inside the square
root in Equation (17) to zero, that is,
1 þ λx
2μ
2
þ λx ¼ 0
1 þ λx
tan θ ¼
2μ
ð18Þ
ð19Þ
The variation of λx with μ is demonstrated in
Figure 8B, which displays the region of μ-λx in which all
cracks are dormant, irrespective of their angles. For a certain μ at λx with a magnitude less than that of the associated λx , a crack that has θ lying outside the range
between the associated angles θS1 and θS2 is dormant.
3.4 | General notes
For a certain crack-load pattern, (1) YI is an even function of θ, see Equation (11), that is, YI (θ) = YI (θ), and
(2) YII is an odd function of θ, see Equation (4), that is,
YII (θ) = YII (θ). Those two properties can be used to
determine YI and YII for a crack-load pattern at negative
measures of θ. Further, for a certain ϕ, YI (λy, λx, θ) = YI
(λx, λy, θ-90 ) and YII (λy, λx, θ) = YII (λx, λy, θ-90 ); see
the two-equivalent crack-load systems in Figure 9. Thus,
the present solution is comprehensive for it covers the
entire range of biaxiality, crack angle, and friction
coefficient.
3.5 | Manipulation of two-dimensional
compound stresses
The square plate in Figure 1 is assumed to be experiencing the normal stresses, σ1 and σ2, and the shear stress, τ,
FIGURE 9
load systems.
Two equivalent crack-
as shown in Figure 10A. A relatively short crack is
assumed to make an angle α in the clockwise direction
with the axis 1. This system of stresses may be reduced to
their equivalent principal stresses, σy and σx (see
Figure 10B), which are shown in Figure 10C and given
by
σ1 þ σ2
σy ¼
2
x
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ1 σ2 2
þ τ2
2
ð20Þ
The principal stress σy makes an angle ψ in the clockwise direction with the axis 1, see Figure 10B, such that
1
2τ
Ψ ¼ tan 1
2
σy σx
ð21Þ
Hence, the crack angle θ made in the clockwise direction with σy is
θ¼αΨ
ð22Þ
The cracked plate in Figure 10C may, now, be manipulated as described in the present work.
3.6 | Further note
The present solution is validated against FE numerical
data, which is artificial. Although a validation against
numerical data is useful, it may be incomplete. Therefore,
it would be recommended to include a comparison with
actual experimental data (measured on real specimens).
There are few experimental studies that examine the
impact of multiaxial loading on fatigue life, for example,
in the following tests: (a) plain cylindrical specimens
under cyclic tension-compression loading with cyclic
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HAMMOUDA
FIGURE 10
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Cracked plate experiencing the normal stresses, σ1 and σ2, and the shear stress, τ.
torsion39,40; (b) plain tubular specimens under cyclic axial
loading with inner pressure41; (c) notched round bars
under combined bending and torsion42; (d) thin-walled
hollow cylinders with a notch under proportional and
non-proportional cyclic tension-compression-torsion
loadings43; and (e) hollow cylinders with a central hole
under cyclic uniaxial and biaxial loading.44 The relevant
experiments found in the literature (a) concern mostly
the identification of the orientation of the growth of the
initiated surface crack and its subsequent growth rate,
and (b) lack knowledge with regard to the friction
between the surfaces of the initiated crack. Cruces et al.44
estimated the factors KI and KII of the cracks initiated in
their tests by making use of experimentally measured
crack tip openings and sliding displacements. Here, the
initiated cracks are small and imbedded in fully plastic
stress–strain fields. Thus, the measured SIFs are expected
to be much higher than the corresponding LEFM SIF
and are not suitable for comparison with the present
results.
4 | C ON C L U S I ON S
The present work successfully determines the analytic
mixed-mode SIF of a straight inclined frictional crack centrally located in a plate subject to biaxial loading in the
three patterns of tension-compression, compressiontension, and compression-compression for every combination of crack length and angle, biaxial ratio, and friction
coefficient. In general, the obtained results are in good
agreement with the available finite element results. In the
considered loading patterns, cracks are either open, exhibiting mixed-mode crack tip deformation, or closed, exhibiting a pure mode II loading with friction between its
crack surfaces. In cases of friction in action, the associated
normalized mode II SIF decreases with an increase in the
friction coefficient between the crack surfaces. Keeping
all other factors unchanged, the magnitude of the normalized mode II SIF increases as the magnitude of the biaxial
ratio (1) increases in the case of tension-compression and
compression-tension loading and (2) decreases in the case
of compression-compression loading.
The following detailed conclusions are related to a
crack angle in the range from 0 to 90 measured in the
clockwise direction along the vertical y-axis.
In tension-compression biaxial loading, for a certain
biaxial ratio, at 90 crack angle, the normalized mode I
SIF has a maximum value while the normalized mode II
SIF is zero. As the crack angle decreases, the following
events take place sequentially: (1) the normalized mode II
SIF increases and then decreases (following the behavior
of the crack with smooth surfaces) until a critical crack
angle is reached at which the mode I SIF reaches a zero
value while the normalized mode I SIF decreases until it
reaches that zero value; and (2) a pure mode II SIF operates with a decreasing value, due to friction between the
crack surfaces, at crack angles smaller than that critical
angle until it reaches a zero value to produce sticking surfaces. The critical angle at which mixed-mode CTD is
switched to a pure mode II CTD increases as the magnitude of the biaxial ratio increases independently of the friction coefficient. For a certain biaxial ratio, the sticking
crack angle increases with the friction coefficient.
In compression-tension biaxial loading, for a certain
biaxial ratio, at zero crack angle, the normalized mode I
SIF has a maximum value while the normalized mode II
SIF is zero. As the crack angle increases, the following
events take place sequentially: (1) the magnitude of the
normalized mode II SIF increases and then decreases
(following the behavior of the crack with smooth surfaces) until a critical crack angle is reached at which the
mode I SIF reaches a zero value while the normalized
mode I SIF decreases until it reaches that zero value; and
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10
(2) a pure mode II SIF operates with a decreasing magnitude, due to friction between the crack surfaces, at crack
angles greater than that critical angle until it reaches a
zero magnitude to produce sticking surfaces. The critical
angle at which mixed-mode CTD is switched to a pure
mode II CTD decreases as the magnitude of the biaxial
ratio increases independently of the friction coefficient.
For a certain biaxial ratio, the sticking crack angle
decreases with the friction coefficient.
In compression-compression biaxial loading, two
sticking crack angles exist for a certain biaxial ratio. For
crack angles between those two angles, the magnitude of
the normalized mode II SIF achieves a maximum magnitude. The difference between the two sticking angles (1) is
90 for smooth crack surfaces with a zero biaxial ratio,
and (2) decreases as either the magnitude of the biaxial
ratio or the friction coefficient increases. For a certain friction coefficient, there is a critical magnitude of the biaxial
ratio (which decreases with the friction coefficient)
greater than which a crack is dormant, whatever its angle.
NO MEN CLATU RE
2a
crack length
CCBL compression-compression biaxial loading
CTBL
compression-tension biaxial loading
CTD
crack tip deformation
c
= cos θ
FE
finite element
KI, KII mode I and mode II, respectively, SIF for a
crack in a finite plate
mode II SIF for a frictional crack
KIIF
mode II SIF for a frictionless crack
KIIS
s
= sin θ
SIF
stress intensity factor
TCBL
tension-compression biaxial loading
TTBL
tension-tension biaxial loading
T
= tan θ
2w
plate width
x, y
horizontal and vertical Cartesian coordinates,
respectively
normalized KIIF
YIIF
YIIS
normalized KIIS
α
crack angle measured from the 1-axis in twodimensional compound stresses
ϕ
= a/w
= σ x =σ
λx
= σ y =σ
λy
μ
coefficient of friction between the two surfaces
of a crack
θ
crack angle measured from the positive direction of the y-axis
critical crack angle for switching from the pure
θC
mode II CTD to a mixed mode I/II CTD
11
θS
σ
σθθ
σrr
σrθ
σx , σy
σ1, σ2
τ
Ψ
critical crack angle for sticking crack surfaces
positive normal reference stress
tangential stress normal to the crack line
normal radial stress along the crack line
shear stress acting on an infinitesimal material
element
normal stresses along the x-axis and y-axis,
respectively
normal stresses along the two axes 1 and
2, respectively
shear stress
=αθ
C O N F L I C T O F I N T E R E S T S T A TE M E N T
The author declares that there is no known competing
financial interests or personal relationships that could
have appeared to influence the work reported in this
paper.
DA TA AVAI LA BI LI TY S T ATE ME NT
All the data related to the present work are available with
the corresponding author who is ready to deliver upon
request.
ORCID
Mohammad M. I. Hammouda
0003-3599-3754
https://orcid.org/0000-
RE FER EN CES
1. Gagg CR, Lewis PR. In-service fatigue failure of engineered
products and structures – case study review. Eng Fail Anal.
2009;16(6):1775-1793.
2. Richard HA, Schramm B, Schirmeisen NH. Cracks on mixed
mode loading – theories, experiments, simulations. Int J
Fatigue. 2014;62:93-103.
3. Bonniot T, Doquet V, Mai SH. Mixed mode II and III fatigue
crack growth in a rail steel. Int J Fatigue. 2018;115:42-52.
4. Akama M, Kiuchi A. Fatigue crack growth under nonproportional mixed mode loading in rail and wheel steel. Part
2: sequential mode I and mode III loading. Appl Sci. 2019;
9(14):2866.
5. Hammouda MMI. Mixed-mode stress intensity factors
via centrally cracked Brazilian disc with wide-range controlling parameters. Fatigue Fract Eng Mater Struct. 2022;45(3):
713-724.
6. Hammouda MMI. Mixed-mode I/II stress intensity factors
for frictional centrally cracked Brazilian disk with confining
pressure. Fatigue Fract Eng Mater Struct. 2022;45(6):
1830-1841.
7. Fedelinski P. Boundary element analysis of cracks under compression. AIP Conference Proceedings 2078(1):Article
No. 020009, 2019; 1-7.
8. Li Z, Gong Y, Chen F. Simulation of mixed mode I-II crack
propagation in concrete using toughness-based crack
initiation-propagation criterion with modified fracture energy.
Theor Appl Fract Mech. 2023;123:103701.
14602695, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/ffe.14090 by Cochrane Portugal, Wiley Online Library on [04/07/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
HAMMOUDA
9. Shimamoto A, Ohkawara H, Nam JH, Hawong JS. Calculation
of stress intensity factor under mixed mode biaxial tensile load
by photoelastic hybrid method. Key Eng Mater. 2010;452-453:
837-840.
10. Young LJ. A further investigation of mixed mode loading center crack problem. Int J Solids Struct. 2001;38(42-43):7461-7471.
11. Fayed AS. Numerical analysis of mixed mode I/II stress intensity factors of edge slant cracked plates. Eng Solid Mech. 2017;
5(1):61-70.
12. Erdogan F, Sih GC. On the crack extension in plates under
loading and transverse shear. J Basic Eng. 1963;85(4):519-525.
13. Hammouda MMI, Pasha RA, Fayed AS. Modelling of cracking
sites/development in axial dovetail joints. Int J Fatigue. 2007;
29(1):30-48.
14. Hassan ZU, Hammouda MMI, Khushnood S, Aktar K,
Jahanzaib M, Malik AH. Experimental and numerical analysis
of fretting fatigue crack initiation in dovetail joint for aero
engine compressor. Nucl. 2012;49(4):255-264.
15. Rooke DA, Cartwright DJ. Compendium of stress intensity factors. Her Majesty's Stationery; 1976.
16. Tada H, Paris PC, Irwin GR. The Stress Analysis of Cracks.
Handbook. 3rd ed. American Society of Mechanical Engineers;
2000.
17. Gustavo VG, Jaime P, Manuel E. KI evaluation by the displacement extrapolation method. Eng Fract Mech. 2000;66(1):
243-255.
18. Kuang JH, Chen LS. A displacement extrapolation method for
two- dimensional mixed mode crack problems. Eng Fract Mech.
1993;46(5):735-741.
19. Albinmousa J, Merah N, Khan SMA. A model for calculating
geometrical factors for a mixed-mode I-II single edge notched
tension specimen. Eng Fract Mech. 2011;78(18):3300-3307.
20. Hedayati E, Vahedi M. Using extended finite element method
for computation of the stress intensity factor, crack growth simulation and predicting fatigue crack growth in a slant-cracked
plate of 6061-T651 aluminum. World J Mech. 2014;4(1):24-30.
21. Ismail AE, Abdul Rahman MQ, Ghazali MZ, et al. Combined
mode I stress intensity factors of slanted cracks. IOP Conf.
Series: Materials Science and Engineering 226: 2017; 012011.
22. Zakharova AP, Shlyannikova VN, Tartygashe AM. Couple
effects of mixed mode biaxial loading and crack tip configuration on plastic stress intensity factor behavior at small and
large scale yielding. Proc Struct Integr. 2019;18:749-756.
23. Shlyannikov V. Mixed-mode static and fatigue crack growth in
central notched and compact tension shear specimens. In:
Mixed-mode crack behavior. ASTM STP 1359. ASTM; 1999:
279-294.
24. Pook LP. An observation on mode II fatigue crack growth
threshold behavior. Int J Fracture. 1977;13(6):867-869.
25. Pook LP, Sharples JK. The mode III fatigue crack growth
threshold for mild steel. Int J Fracture. 1979;15(6):R223-R226.
26. Melin S. When does a crack grow under mode II conditions?
Int J Fracture. 1986;30(2):103-114.
27. Shlyannikov V, Fedotova D. Distinctive features of crack
growth rate for assumed pure mode II conditions. Int J Fatigue.
2021;147:106163.
28. Hammouda MMI, Fayed AS, Sallam HEM. Mode II stress
intensity factors for central slant cracks with frictional surfaces
in uniaxially compressed plates. Int J Fatigue. 2002;24(12):
1213-1222.
HAMMOUDA
29. Hammouda MMI, Fayed AS, Sallam HEM. Stress intensity factors of a central slant crack with frictional surfaces in plates
with biaxial loading. Int J Fract. 2004;129(2):141-148.
30. Hammouda MMI, Fayed AS, Sallam HEM. Stress intensity factors of a shortly kinked slant central crack with frictional surfaces in uniaxially loaded plates. Int J Fatigue. 2003;25(4):
283-298.
31. Murakami S, Hamada S. A new method for the measurement
of mode II fatigue threshold stress intensity factor range kth.
Fatigue Fract Eng Mater Struct. 1997;20(6):863-870.
32. Tuhkuri J. Dual boundary element analysis of closed cracks.
Int J Numer Methods Eng. 1997;40(16):2995-3014.
33. Kfouri AP, Brown MW. A fracture criterion for cracks under
mixed-mode loading. Fatigue Fract Eng Mater Struct. 1995;
18(9):959-969.
34. Candeias J, Baptista R, Claudio R, Reis L, Freitas M. On the
influence of different in-plane biaxial loading conditions over
FCG lives. Int J Fatigue. 2022;157:106714.
35. Mall S, Perel VY. Crack growth behavior under biaxial fatigue
with phase difference. Int J Fatigue. 2015;74:166-172.
36. Johnson KL. Contact Mechanics. Cambridge University Press;
1986.
37. Tong J, Yates JR, Brown MW. A model for sliding mode crack
closure part I: theory for pure mode II loading. Eng Fract Mech.
1995;52(4):599-611.
38. Hammouda MMI, El-Sehily BM. A two-dimensional elasticplastic finite element analysis of friction effects on sliding crack
surfaces in full or partial contact. Fatigue Fract Eng Mater
Struct. 1999;22(2):101-110.
39. Reis L, Li B, Leite M, de Freitas M. Effects of non-proportional
loading paths on the orientation of fatigue crack path. Fatigue
Fract Eng Mater Struct. 2005;28(5):445-454.
40. Reis L, Li B, de Freitas M. Crack initiation and growth path
under multiaxial fatigue loading in structural steels. Int J
Fatigue. 2009;31(11-12):1660-1668.
41. Cruces AS, Garcia-Gonzalez A, Moreno B, Itoh T, LopezCrespo P. Critical plane-based method for multiaxial fatigue
analysis of 316 stainless steel. Theor Appl Fract Mech. 2022;118:
103273.
42. Branco R, Costa JD, Antunes FV. Fatigue behaviour and life
prediction of lateral notched round bars under bending–torsion
loading. Eng Fract Mech. 2014;119:66-84.
43. Zerres P, Brüning J, Vormwald M. Fatigue crack growth behavior of fine-grained steel S460N under proportional and nonproportional loading. Eng Fract Mech. 2010;77(11):1822-1834.
44. Cruces AS, Mokhtarishirazabad M, Moreno B, Zanganeh M,
Lopez-Crespo P. Study of the biaxial fatigue behavior and overloads on S355 low carbon steel. Int J Fatigue. 2020;134:105466.
How to cite this article: Hammouda MMI.
Mixed-mode I/II stress intensity factors of a
biaxially loaded plate with a central slant frictional
crack: An analytical comprehensive solution.
Fatigue Fract Eng Mater Struct. 2023;1‐13. doi:10.
1111/ffe.14090
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12
13
A P P EN D I X A
The FE YIIS-θ data points for the crack-load system provided by Hammouda et al.29 were best-fit to a six-degree
polynomial. Figure A1 demonstrates an example for the
point data corresponding to ϕ = 0.3, λy = 1, and
λx = 0.6. The best-fit polynomial is displayed in
Figure A1, where β is the crack angle θ in radians.
F I G U R E A 1 An example to demonstrate FE YIIS-θ data
points29 best fit to a 6-degree polynomial.
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HAMMOUDA