Cent. Eur. J. Eng. • 3(2) • 2013 • 174-190
DOI: 10.2478/s13531-012-0063-8
Central European Journal of Engineering
Catalogue of maximum crack opening stress for
CC(T) specimen assuming large strain condition
Research Article
Marcin Graba∗
Kielce University of Technology,
Faculty of Mechatronics and Machine Design,
Chair of Fundamentals of Machine Design,
25-314 Kielce, Poland
Received 21 December 2011; accepted 29 October 2012
Abstract: In this paper, values for the maximum opening crack stress and its distance from crack tip are determined for
various elastic-plastic materials for centre cracked plate in tension (CC(T) specimen) are presented. Influences
of yield strength, the work-hardening exponent and the crack length on the maximum opening stress were tested.
The author has provided some comments and suggestions about modelling FEM assuming large strain formulation.
Keywords: Fracture mechanics • Cracks • Stress fields • FEM • J-integral • Large strain formulation • Maximum opening stress
• Local approach • CC(T) specimen
© Versita sp. z o.o.
1.
Introduction
The Hutchinson [1], Rice and Rosengren [2] (HRR) solution
for stress near the crack tip for non-linear (elastic-plastic)
materials concerns the small strains and consists of one
singular term only. The amplitude of the singular stress
field is the J-integral, which is path independent when
the strain energy is a unique function of strains. However the stress distribution for the plane strain model, can,
in most cases, be different from the HRR solution (see
Figure 1). This difference between the Finite Element
Method (FEM) stress distribution and the HRR solution
was named the “Q-parameter” by O’Dowd and Shih [3, 4].
In fact, the Q · σ0 -term (where σ0 is yield stress), when
∗
E-mail: mgraba@tu.kielce.pl
174
added to the HRR singular term, replaces all neglected
terms in the asymptotic expansions of the stress field in
front of the crack. Throughout literature there are many
papers referring to the influence of a materials properties
and specimen geometry on Q-stress [5].
In a real structural element with a crack, the stresses near
the crack tip are finite. The stress infinity is a result of
the assumption that the crack tip is perfectly sharp and
it remains sharp during the crack tip loading. When the
assumption of the small strains is relaxed, the crack tip
blunts and stresses in front of the crack become finite.
The opening stress reaches maximum at a distance equal
to r =(0.5 to 2.0)·Jσ0 , and is value dependent on material properties, specimen geometry and external loading.
This feature was first noticed by Rice and Johnson [6] and
McMeeking and Parks [7]. The Finite Element Method
(FEM) analysis of the stress and strain field in front of the
M. Graba
Figure 1.
The stress distribution near a crack tip for SEN(B) specimen – curves were obtained using the FEM (Finite Element Method) for small and finite strain and HRR formula
(based on [10, 11, 16])
Figure 2.
crack when the finite strains are used is not a trivial problem. The level of the stress maximum and its localization
depends on the FE mesh details when it is not properly
selected. For the small strain option such a problem is
not observed [3, 4, 8, 9].
2. Recommendations for FEM modelling assuming large strain condition
The numerical analysis of the stress distribution near
crack tip revealed, that results depend on the details of the
FE modeling, when the finite strain option is adopted [11–
14]. As shown in Figure2, one may notice that when the
number of the FE’s between the crack tip and the opening
stress maximum location is not large enough, the results
obtained don’t converge to a single curve. It is recommended to use at least 20 FE’s between the crack tip and
the stress maximum location [11]. Thus, the FE size in the
radial direction should be smaller than 0.1 · δT where δT
is the crack tip opening displacement (CTOD).
O’Dowd and Shih. [3, 4, 9] suggest, that the crack tip
radius should be smaller than the half of the crack tip
opening displacement δT c for critical moment:
rw = 0.5 · δT C = 0.5 ·
dn · JC
,
σ0
(1)
The influence of the number of the FE between the crack
tip and the opening stress maximum locationσ22 along θ =
0◦ direction on the stress distribution for SEN(B) specimen
for r ≤ 2J/σ0 (based on [11, 14, 16])
where JC is the critical value of J-integral (for material used in FEM analysis for preparing Figures, JC =
40 kN/m) and dn is the parameter introduced by Shih [15],
which connects J-integral, yield stress and crack tip opening displacement (for FEM analysis presented in this
paragraph, dn = 0.297).
The crack tip radius value calculated from equation (1)
is equal to rw = 1.89 · 10−5 m and is greater than the
crack tip radiivalues tested in the FEM analysis (in numerical analysis presented in this paragraph, the rw is
equal to rw =(0.5 to 2.0)·10−6 m and shown to be advisable. However O’Dowd et al. [3, 4, 9] was not interested
in the opening stress distribution at the distance r < J/σ0
which is of vital interest in large strain analysis. O’Dowd
et.al. used the small strain option during the computation [11, 14]. The FEM analysis presented by [11, 14, 16]
shows, that when the crack tip radius, denoted as rw decreases, the level of the maximum of the opening stress
in front of the crack increases and appears closer to the
crack tip. However, for sufficiently small values of rw , both
the opening stress maximum and its location in front of a
crack become independent of the crack tip radius. For
increasing external load, the saturation of the ξ0 = ξ0 (J)
and the ψ0 = ψ0 (J) curves was observed (see Figure 5),
175
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
Figure 3.
The influence of the number of the FE between the crack
tip and the opening stress maximum location σ22 along
θ = 0◦ direction, on normalized location of the opening
stress maximum (based on [11, 14, 16])
(a)
where:
σ22_ max
,
σ0
r22_ max · σ0
,
=
J
ξ0 =
(2)
ψ0
(3)
where ξ0 is normalized by yield stress maximum opening
stress value, ψ0 is the normalized distance of the maximum opening stress from crack tip, σ22_max is the maximum
opening stress value, r22_max is the distance of the maximum opening stress from crack tip, J is the J-integral, σ0
is yield stress.
Literature reveals two different methods for modeling the
crack tip for large strain assumptions as shown in Figure 5(a) and Figure 5(b). Brocks et al. [12, 13] suggest,
that the crack tip should be modeled in the way shown
in Figure 5(a). The computations confirm, that when this
model is used, the radius of the crack tip can be smaller
than for the model shown in Figure 4(b). If in the FEM
analysis the model of crack tip presented in Figure 4(a) is
used, the FEM analysis may be done for larger external
loads.
The curves level off and the level is independent of the
crack tip radius. The convergence of the FEM results is
observed, when rw is about 2.5 · 10−6 m (1/15 · δT C for
critical moment when J–integral value is equal to JC =
40 kN/m, respectively). Thus the crack tip model shown
in Figure 5(b) is recommended.
The preparation of numerical models assuming a large
deformation requires consideration of many facts and factors, for example: the shape and model of the crack tip,
176
(b)
Figure 4.
The influence of the size of crack tip radius on: (a) level
of the maximum opening stress; (b) normalized location of
the maximum opening stress (based on [11, 14, 16])
the size of the crack tip radius, the finite element size
and mesh density. These same problems are encountered
with the numerical designation of the J-integral, when the
assumption of large deformation was done. This problem
was discussed by [7, 10, 12, 13].
M. Graba
(a)
is considered independent of temperature, when within a
low temperature range.
O’Dowd and Shih [3, 4] adopted a modified small strain
HRR solution, called the OS model (O’Dowd-Shih model),
to derive simple, approximate formula in order to predict
the influence of in-plane constraint on fracture toughness
of a structural element. According to O’Dowd’s model the
critical conditions must be satisfied independently of the
level of constraint, which was quantified by the actual
value of the Q−stress, utilizing the OS theory:
σij = σ0 σ̃ij (n, θ)
(b)
Figure 5.
Two alternative crack tip models
J
αε0 σ0 In r
3. Fracture criteria using numerical
solution for large strain assumption
The maximum opening stress level and their position
against the tip of the cracks are extremely important because both parameters may be used to build a fracture
criteria, which will estimate the real fracture toughness of
the construction element. An example can there be local
fracture criteria is proposed in [4, 10, 17]. Several authors
adopted local fracture criterion to assess the constraint influence on fracture toughness (O’Dowd [18], Neimitz and
Gałkiewicz [19]). They assumed that for a cleavage fracture to happen it requires that the opening stress reaches
the critical length, σC at a certain distance from the crack
tip, rC or within a certain volume in front of the crack tip.
This critical value is characteristic for each material and
+ Qσ0 δij ,
(4)
where σ̃ij (n, θ) and In are well known functions, characteristic for the HRR stress field [1], ε0 = σ0 /E and Q is
computed according to the OS definition [3, 4].
Using Eq. (4) twice for the two different levels of constraint it was assumed that for the one state, called the
reference state, the value of Q = Qref = 0. The critical
distance rC was eliminated from the two equations and
the final formula to compute the actual value of fracture
toughness was derived in the form:
JC = JIC 1 −
Emergence of various problems in the FEM analysis when
large deformation assumptions were used have not discouraged researchers in using the maximum opening stress
and its distance from crack tip in solving engineering problems. Both parameters (maximum opening stress and its
distance from crack tip) were used in the proposals for
fracture criteria, what will be presented in the next paragraph.
1/1+n
Q
σC /σ0
n+1
,
(5)
where JIC is fracture toughness obtained for a specimen
with dominance with plane strain conditions [4], JC is
real fracture toughness, σC is critical stress according to
Ritchie-Knott-Rice model [17] and Q is the Q-stress (Qparameter) for real construction element.
Equation (5) follows from the RKR [17] hypothesis, provided, one assumes that the stress field in front of the
crack is singular (small strain assumption) and satisfies
Equation (4). In such a case the opening stress is greater
than critical one over the distance rC from the crack front.
To apply O’Dowd’s simple model, a very limited numerical analysis is necessary utilizing assumption of small
strains. However, the critical stress σ C is simply the parameter which adjusts theoretical prediction to experimental results. Almost all other theories concerning the local
approach to fracture use the stress distribution in front of
the crack characteristic for the finite strain (see Figure 1).
In 2007, Neimitz et al. [10] presented another form of the
fracture criterion (5), replacing the critical stress σC by
maximum opening stress σ22_max . They proposed that real
fracture toughness JC may be evaluated using following
expression:
JC = JIC 1 −
Q
σ22_ max /σ0
n+1
.
(6)
177
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
Formula (6) represents a two-parametric approach to determine fracture toughness. Its Equation is true until the
moment when the maximum opening stress value reaches
the saturation level (for example Figure 4(a)). It is a condition characteristic for the moment when a large plastic
zone covers almost the entire non-cracked section of the
specimen (structural component). Formula (6) is saddled
with a strong foundation, according to which, the maximum
opening stress level does not depend on crack length.
Expanded form of the criterion presented by (6) is the
relationship (7), presented in [10]:
"
JC = JIC
1
1
1+n
+
(7)
φQ=0
h
i
1+n
−1
max
max
)Q=0 − (σ22
)Q E 1+n
Q − (σ22
 φQ=0 ,
−
σ̃22
ασ0 In
max
max
)Q=0 and (σ22
)Q are maximum opening stress
where (σ22
level (normalized by yield stress σ0 ) for specimen dominated by plane strain (Q = 0) [18] and specimen characterized by Q stress value Q 6= 0 (for which the real
fracture toughness is desired) respectively; φQ=0 is normalized distance of the maximum opening stress from crack
tip for specimen dominated by plane strain conditions [20].
Using Equation (7) to calculate the real fracture toughness, engineers may consider the analysis of the flat dimension of the specimen (crack length, width), who have
effects on in-plane constraints, represented by Q-stress
and maximum opening stress level.
However, the use of local fracture criteria (Equations (6)
and (7)) next to the fracture toughness determined in the
laboratory for plane strain conditions [4] require knowledge of the Q parameter, which is determined by the assumption of small deformations and the maximum opening
stress level with the distance from crack tip which are determined by the assumption of large deformations. The
determination of these last two parameters by numerical calculations using the FEM is not easy and obvious,
because both values are sensitive to the quality of the numerical model, as mentioned in [11, 16]. Therefore, in this
paper, the FEM analysis with assumptions of the large
deformation will be presented. The measurable effect of
the paper is presented in the catalogue of numerical solutions obtained for centre cracked plate in tension (CC(T)),
obtained for domination of the plane strain assuming large
deformation. This catalogue includes the values for maximum opening stresses and their normalized distance from
crack tips, for different materials and geometric configurations of the CC(T) specimens.
178
4.
Details of the numerical analysis
Numerical analysis of the centre cracked plate in tension (CC(T)) was used (see Figure 6). Computations
were performed for plane strain using large strain option (finite strain option). The relative crack length was a
a/W = {0.20, 0.50, 0.70} where a is a crack length and
the width of specimens W was equal to 40 mm. The choice
of the CC(T) specimen was intentional, because the CC(T)
specimens are used in FITNET procedure [21] in order to
idealize the complex structural elements. This specimen is
used in laboratory tests in order to determine the critical
values of the J-integral, as shown in the papers prepared
by Sumpter and Forbes [22]. All geometrical dimensions
of the CC(T) specimen are presented in Table 1.
Computations were performed using ADINA SYSTEM
8.5 [23, 24]. Due to the symmetry, only a quarter of the
specimen was modeled. The finite element mesh was filled
with the 9-node plane strain elements. The size of the finite elements in the radial direction decreased towards the
crack tip, while in the angular direction the size of each
element was kept constant. It varied from ∆θ = π/13 to
∆θ = π/23 for various cases tested. The crack tip region was modeled using 36 to 72 semicircles. The first
of them, was between 20 to 100 times smaller than the
last. The first finite element behind to crack tip is smaller
(2000 to 10000) times than the width of the specimen.
The crack tip was modeled as half of the arc where the
radius was equal to rw =(1 to 5)·10−6 m. Selection of the
crack tip in the form of a semicircle, was dictated by the
ADINA SYSTEM instructions [23, 24], according to which
for structural elements with predominance with tension It
is recommended to use a semicircle model of a crack tip,
and for structural elements with a predominance of bending, itis better to use a model of the crack tip in the form
of a quarter arc. The CC(T) specimen was modeled using approximately 3500 finite elements and 12500 nodes.
The example finite element model for CC(T) specimen is
presented on Figure 7.
In the FEM simulation, the deformation theory of plasticity and the von Misses yield criterion were adopted. In
the model the stress–strain curve was approximated by the
relation:
ε
=
ε0
(
σ /σ0 for σ ≤ σ0
,
α (σ /σ0 )n for σ > σ0
(8)
where α = 1. The tensile properties for the materials
which were used in the numerical analysis are presented
in the Table 2. In the FEM analysis, calculations were
done for twelve materials, which differed in yield stress
and the work hardening exponent.
M. Graba
Table 1.
The geometrical dimension of the CC(T) specimen used in numerical analysis
2W [mm]
80
Figure 6.
Table 2.
4W [mm]
2L [mm]
160
176
The centre cracked plate in tension – CC(T) specimen
The mechanical properties for materials used in numerical
analysis
Young’s modulus, E [MPa]
206000
Poisson’s ratio, υ
0.3
yield stress, σ 0 [MPa]
315; 500; 1000
work-hardening exponent
in Ramberg-Osgood relationship, n
a/W
a [mm]
b = (W − a) [mm]
0.20
8
32
0.50
20
20
0.70
28
12
denotes the displacement vector and ds is the infinitesimal
segment of contour C .
The J-integral is path independent for small strain formulation only. However one may also calculate the J-integral
using large strain formulation, but the contour of integration should be sufficiently distant from the crack tip but
not too close to the specimen borders. In Figure 8 the
contours of integration used in this research project are
shown. The calculations and tests confirm, that the value
of the J-integral is almost independent of finite elements
used in FEM analysis. The most important parameter is
the way the contour of integration is drawn. It should
not lie too close either to crack tip or to the edge of the
specimen. The advised recommendation is to use a few
different integral contours in FEM analysis and compare
results [11, 16].
The external load of the specimen was carried out by applying a displacement to the upper edge. Selection of the
displacement resulted from a desire to obtain the maximum opening stress values and their location near crack
tip for an external load P, which satisfies the conditions
P/P0 = {0.2, 0.4, 0.6, 0.8, 1.0, 1.2}, where P0 is the limit
load [21, 25], which is set in accordance with the following
Equation:
4
P0 = √ · B (W − a) · σ0 ,
(10)
3
where B is the specimen thickness – for plane strain condition, B is equal to 1 m.
3; 5; 10; 20
5.
The J-integral were calculated using two methods. The
first method, called the “virtual shift method”, uses concept
of the virtual crack growth to compute the virtual energy
change. The second method is based on the J-integral
definition:
Z
J=
[wdx2 − t (∂u/∂x1 ) ds],
(9)
C
where w is the strain energy density, t is the stress vector
acting on the contour C drawn around the crack tip, u
Numerical results
In the analysis of the numerical results, the influence of the
yield stress, work-hardening exponent and crack length
on maximum of the opening stress and their location near
crack tip were tested. The influence of the yield stress
on maximum opening stress value and their location near
crack tip is presented in Figure 9.
Figure 9 shows, with increasing external load expressed
as a quotient of the force P and the load limit P0 , the value
of maximum opening stress is increasing. In some cases
(after an earlier analysis of the Tables in Annexes 6-6
179
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
ure 9(b)). For the same level of normalized external load
is observed that for higher yield stress values, the value of
normalized maximum stress position in front of the crack
is smaller. With the increase of external load, the value
of normalized position of maximum stress decreases. It
can be noted, that the actual physical location of maximum opening stress with increasing external load also
increases (see Figure 10).
The real argument is, therefore, that the maximum opening
stress value with increased external load initially increase
and then reach a saturation value. Figure 10 presents a
map of isolines of stress, which are presented, that maximum opening stress away from the crack tip. Their physical location (denoted as rmax ) from the crack tip grows.
(a)
(b)
(c)
Figure 7.
(a) The finite element model for CC(T) specimen used in
the FEM analysis assuming large strain option; (b) The
finite element mesh near crack tip; (c) Sample finite elements mesh around the crack tip
of this paper), it can be noted that the increase of external load, the value of maximum stress reaches a saturation point, and for the case of materials characterized
by weak (or very weak) hardening sometimes slightly decreases (within about 5% of maximum). Figure 9(a) shows
that for higher yield stress value of the maximum opening
stress is smaller (with including the same level of external
load). It can be noted, that for materials characterized
by smaller yield stress, σmax /σ0 = f(P/P0 ) curves are arranged above.
For materials characterized by lower yield stress, the
ψ = f(P/P0 ) curves which are presented the change of
normalized position of maximum opening stress as a function of normalized external load are arranged above (Fig180
For strong hardening materials, the higher values of the
maximum opening stress are observed. For weak hardening materials, the σmax /σ0 = f(P/P0 ) curves are arranged
below, and there are reached the saturation level for normalized external load equal to P/P0 = 0.6. For very weak
hardening materials and specimens characterized by long
crack (a/W = 0.70), increasing external load is caused a
decrease of the maximum opening stress value (see Figure 11(a)).
For strong hardening materials, the ψ = f(P/P0 ) curves
are arranged below (Figure 11(b)). For the same level of
normalized external load is observed that for larger values
of the work hardening exponent in the R-O law, the value
of position of the normalized maximum stress in front of the
crack is greater. Analysis of the ψ = f(P/P0 ) curves for
some cases, can lead to the conclusions that these curves
tend to reach saturation level. In the case of the CC(T)
specimens, which were used in the numerical analysis, it
can be noted, that normalized position of the maximum
opening stress from the crack tip is in the range ψ =(0.25
to 1.75), and this value depends on yield stress, work
hardening exponent and crack length.
Figure 12(a) demonstrates the fact that for longer cracks,
the greater values of maximum opening stress are observed. For the case of CC(T) specimens containing short
cracks, the σmax /σ0 = f(P/P0 ) curves are arranged below
(see Figure 12(a)). As shown in Figure 12(b), the crack
length hasn’t significant influence on the normalized maximum stress location in front of crack. It can be note, that
for longer crack, the ψ = f(P/P0 ) curves are arranged
above (see Figure 12(b)).
Larger values of normalized position of maximum stress
near crack tip are observed for specimens characterized
by long cracks, however, the difference between the values of ψ for the specimens described different values of
normalized crack length (a/W ) in many cases may not be
large, even the minimum (especially for weak and very
weak hardening materials).
M. Graba
(a)
Figure 8.
(b)
The three integration contours, which were used to calculation the J-integral
(a)
Figure 9.
(c)
(b)
The influence of the yield stress on maximum opening stress value σmax /σ0 (a) and their normalized location near crack tip ψ = rmax ·σ0 /J
(b) for CC(T) specimen – a/W = 0.20, n = 5
(a)
Figure 10.
(b)
(c)
The influence of the external load on physical location of the maximum opening stress near crack tip for CC(T) specimen – σ0 =
500 MPa, n = 10, a/W = 0.50, along with the selected location of maximum opening stress by triangle: (a) P/P0 = 0.4, σmax /σ0 = 3.07,
rmax = 0.025 mm, ψ = 3.09; (b) P/P0 = 0.8, σmax /σ0 = 3.15, rmax = 0.040 mm, ψ = 1.30; (c) P/P0 = 1.2, σmax /σ0 = 3.12,
rmax = 0.055 mm, ψ = 0.66
181
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
(a)
Figure 11.
a) The influence of the work hardening exponent on maximum opening stress value σmax /σ0 for CC(T) specimen – a/W = 0.70,
σ0 = 500 MPa; (b) The influence of the work hardening exponent on normalized location of the maximum opening stress near crack
tip ψ = rmax · σ0 /J for CC(T) specimen – a/W = 0.20, σ0 = 315 MPa
(a)
Figure 12.
6.
(b)
The influence of the normalized crack length on maximum opening stress value σmax /σ0 (a) and their normalized location near crack
tip ψ = rmax · σ0 /J (b) for CC(T) specimen – σ0 = 1000 MPa, n = 20
Summary and conclusions
In the paper the values of the maximum opening stress and
its distance from crack tip determined for various elasticplastic materials for centre cracked plate in tension (CC(T)
specimen) presented. The influence of the yield stress,
the work-hardening exponent and the crack length on the
maximum opening stress was tested. In the paper some
comments and suggestions about modeling FEM assuming
large strain formulation were given.
182
(b)
The maximum of the normal stresses in front of the crack
is observed for Ramberg-Osgood material and for blunted
crack. It must be computed numerically using the finite
strains option. The maximum of the opening stress is located at the normalized distance ψ · J/σ0 where ψ is usually equal to (0.25 to 1.75) – for CC(T) specimen for load
level P/P0 ≥ 0.8. When the external load acting on the
specimen is low (P/P0 ≤ 0.6), the maximum crack opening
stress is located at a normalized distance from the crack
tip ψ · J/σ 0 where ψ is equal to (3 to 27). The location of
M. Graba
maximum crack opening stress depends on the length of
the crack and the material characteristics.
For small scale yielding the maximum value of the opening stress component depends on the constraint level. The
maximum opening stress value σmax /σ0 and its normalized
distance from crack tip ψ depend on external load, work
hardening exponent, yield stress and crack length, what
was presented above. Presented in the paper catalogue
may be very useful for engineering problems, when, the
real fracture toughness for structural component is determined using local fracture criteria, presented in [10, 18].
Acknowledgements
The support of the Kielce University of Technology – Faculty of Mechatronics and Machine Design through grants
No 1.22/7.14 is acknowledged by the author of the paper.
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and Extending Cracks, Journal of the Mechanics and
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fields near the crack tip in the elastic-plastic materials. 3D problems., PhD dissertation, Kielce University of Technology - Faculty of Mechatronics and
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[19] Neimitz A., Gałkiewicz J., Fracture Toughness of
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183
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
[21]
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[23]
[24]
[25]
184
surement of Fracture Toughness. American Society for
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Palo Alto, CA, 1981
M. Graba
Appendix A: Numerical results for
CC(T) specimens characterized by
yield stress σ0 = 315 MPa
Table A1.
Table A2.
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 315 MPa and work hardening exponent
in R-O relationship n = 5
σ0 = 315 [MPa], n = 5, a/W = 0.20
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.29
2.86
13.85
0.4
1.50
3.52
3.58
0.6
3.19
3.88
1.38
σ0 = 315 [MPa], n = 3, a/W = 0.20
0.8
6.39
4.17
0.76
P/P0
J [kN/m]
σmax /σ0
ψ
1.0
10.29
4.41
0.60
0.2
0.34
3.34
6.33
1.2
23.96
4.67
0.43
0.4
1.38
4.61
1.53
σ0 = 315 [MPa], n = 5, a/W = 0.50
0.6
3.23
5.60
0.50
P/P0
J [kN/m]
σmax /σ0
ψ
0.8
5.84
6.41
0.37
0.2
0.40
3.35
6.32
1.0
9.95
7.23
0.30
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 315 MPa and work hardening exponent
in R-O relationship n = 3
0.4
1.59
4.03
2.19
σ0 = 315 [MPa], n = 3, a/W = 0.50
0.6
3.60
4.43
1.01
P/P0
J [kN/m]
σmax /σ0
ψ
0.8
6.48
4.68
0.58
0.2
0.35
3.96
2.04
1.0
10.34
4.87
0.48
0.4
1.50
5.51
0.66
1.2
15.42
4.99
0.41
0.6
3.32
6.52
0.28
σ0 = 315 [MPa], n = 5, a/W = 0.70
0.8
5.86
7.33
0.27
P/P0
J [kN/m]
σmax /σ0
ψ
σ0 = 315 [MPa], n = 3, a/W = 0.70
0.2
0.25
2.74
15.64
P/P0
J [kN/m]
σmax /σ0
ψ
0.4
1.00
3.43
5.20
0.2
0.25
3.65
5.13
0.6
2.25
3.81
2.63
0.4
1.00
5.04
0.70
0.8
4.02
4.09
1.42
0.6
2.25
6.00
0.29
1.0
6.33
4.29
1.00
0.8
4.00
6.76
0.28
1.2
9.24
4.47
0.65
185
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
Table A3.
186
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 315 MPa and work hardening exponent
in R-O relationship n = 10
Table A4.
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 315 MPa and work hardening exponent
in R-O relationship n = 20
σ0 = 315 [MPa], n = 10, a/W = 0.20
σ0 = 315 [MPa], n = 20, a/W = 0.20
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.37
2.69
15.88
0.2
0.41
2.63
25.90
0.4
1.49
2.97
5.36
0.4
1.46
2.57
7.22
0.6
3.18
3.03
2.42
0.6
3.40
2.54
3.00
0.8
6.03
3.06
1.18
0.8
6.44
2.54
1.48
1.0
11.29
3.08
0.78
1.0
12.31
2.42
0.67
1.2
48.88
3.14
0.38
1.2
175.91
2.36
0.34
σ0 = 315 [MPa], n = 10, a/W = 0.50
σ0 = 315 [MPa], n = 20, a/W = 0.50
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.40
2.74
14.76
0.2
0.40
2.64
20.60
0.4
1.59
3.11
5.00
0.4
1.60
2.83
6.72
0.6
3.61
3.18
2.11
0.6
3.62
2.82
2.84
0.8
6.49
3.22
1.52
0.8
6.50
2.75
1.47
1.0
10.36
3.20
0.88
1.0
10.41
2.65
1.15
1.2
16.62
3.21
0.66
1.2
17.96
2.58
0.74
σ0 = 315 [MPa], n = 10, a/W = 0.70
σ0 = 315 [MPa], n = 20, a/W = 0.70
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.23
2.53
25.60
0.2
0.25
2.53
27.58
0.4
0.92
3.00
8.77
0.4
1.00
2.83
9.90
0.6
2.37
3.16
3.30
0.6
2.26
2.87
4.26
0.8
4.12
3.18
1.83
0.8
4.22
2.82
2.17
1.0
6.89
3.22
1.42
1.0
6.86
2.69
1.25
1.2
10.56
3.20
0.85
1.2
10.93
2.65
1.01
M. Graba
Appendix B: Numerical results for
CC(T) specimens characterized by
yield stress σ0 = 500 MPa
Table B1.
Table B2.
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 500 MPa and work hardening exponent
in R-O relationship n = 5
σ0 = 500 [MPa], n = 5, a/W = 0.20
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.90
2.87
7.69
0.4
3.63
3.48
1.91
0.6
8.32
3.82
1.20
σ0 = 500 [MPa], n = 3, a/W = 0.20
0.8
15.38
4.08
0.66
P/P0
J [kN/m]
σmax /σ0
ψ
1.0
28.03
4.26
0.38
0.2
0.85
3.31
2.35
1.2
64.23
4.47
0.29
0.4
3.51
4.59
0.90
σ0 = 500 [MPa], n = 5, a/W = 0.50
0.6
8.02
5.49
0.37
P/P0
J [kN/m]
σmax /σ0
ψ
0.8
14.67
6.26
0.30
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 500 MPa and work hardening exponent
in R-O relationship n = 3
0.2
1.10
2.99
6.87
σ0 = 500 [MPa], n = 3, a/W = 0.50
0.4
3.59
3.53
2.03
P/P0
J [kN/m]
σmax /σ0
ψ
0.6
8.73
3.93
1.19
0.2
0.96
3.54
2.44
0.8
16.26
4.17
0.68
0.4
3.71
4.63
0.85
1.0
26.50
4.35
0.55
0.6
8.46
5.61
0.35
1.2
40.13
4.46
0.45
0.8
15.17
6.41
0.30
σ0 = 500 [MPa], n = 5, a/W = 0.70
1.0
24.40
7.13
0.27
P/P0
J [kN/m]
σmax /σ0
ψ
σ0 = 500 [MPa], n = 3, a/W = 0.70
0.2
0.56
2.64
10.98
P/P0
J [kN/m]
σmax /σ0
ψ
0.4
2.25
3.32
4.24
0.2
0.62
3.07
1.84
0.6
6.26
3.78
1.45
0.4
2.50
4.25
0.91
0.8
10.62
4.00
1.10
0.6
5.83
5.12
0.73
1.0
16.18
4.17
0.68
0.8
10.28
5.76
0.34
1.2
25.76
4.33
0.53
1.0
16.39
6.36
0.30
1.2
24.11
6.98
0.26
187
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
Table B3.
188
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 500 MPa and work hardening exponent
in R-O relationship n = 10
Table B4.
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 500 MPa and work hardening exponent
in R-O relationship n = 20
σ0 = 500 [MPa], n = 10, a/W = 0.20
σ0 = 500 [MPa], n = 20, a/W = 0.20
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.89
2.65
8.76
0.2
0.88
2.54
14.27
0.4
3.61
2.98
3.43
0.4
3.57
2.72
3.37
0.6
8.31
3.00
1.39
0.6
8.36
2.57
1.33
0.8
15.50
2.99
1.03
0.8
15.94
2.42
1.47
1.0
28.66
3.00
0.69
1.0
30.80
2.27
1.52
1.2
122.24
3.03
0.43
1.2
501.10
2.19
0.41
σ0 = 500 [MPa], n = 10, a/W = 0.50
σ0 = 500 [MPa], n = 20, a/W = 0.50
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.89
2.69
10.35
0.2
0.89
2.60
14.45
0.4
4.00
3.07
3.09
0.4
4.00
2.86
4.16
0.6
8.73
3.16
1.83
0.6
8.74
2.80
1.78
0.8
15.38
3.15
1.30
0.8
15.41
2.75
1.25
1.0
25.33
3.11
0.71
1.0
26.60
2.64
0.85
1.2
42.20
3.12
0.66
1.2
46.43
2.48
0.70
σ0 = 500 [MPa], n = 10, a/W = 0.70
σ0 = 500 [MPa], n = 20, a/W = 0.70
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.59
2.54
15.79
0.2
0.62
2.53
17.37
0.4
2.67
3.02
4.72
0.4
2.50
2.84
6.17
0.6
5.79
3.12
2.85
0.6
5.86
2.83
2.51
0.8
10.79
3.17
1.44
0.8
10.69
2.81
1.83
1.0
16.64
3.16
1.19
1.0
17.11
2.70
1.05
1.2
26.15
3.11
0.79
1.2
27.38
2.61
0.79
M. Graba
Appendix C: Numerical results for
CC(T) specimens characterized by
yield stress σ0 = 1000 MPa
Table C1.
Table C2.
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 1000 MPa and work hardening exponent
in R-O relationship n = 5
σ0 = 1000 [MPa], n = 5, a/W = 0.20
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
3.43
2.80
4.12
0.4
14.16
3.40
1.45
0.6
32.68
3.65
0.72
σ0 = 1000 [MPa], n = 3, a/W = 0.20
0.8
61.41
3.81
0.54
P/P0
J [kN/m]
σmax /σ0
ψ
1.0
109.15
3.93
0.33
0.2
3.40
3.27
1.15
1.2
256.45
4.06
0.28
0.4
14.00
4.45
0.42
σ0 = 1000 [MPa], n = 5, a/W = 0.50
0.6
32.54
5.26
0.40
P/P0
J [kN/m]
σmax /σ0
ψ
0.8
59.21
5.78
0.38
Numerical results for CC(T) specimens characterized by
yield stress σ0 = 1000 MPa and work hardening exponent
in R-O relationship n = 3
0.2
3.71
3.21
2.30
σ0 = 1000 [MPa], n = 3, a/W = 0.50
0.4
15.23
3.78
1.03
P/P0
J [kN/m]
σmax /σ0
ψ
0.6
34.19
4.02
0.69
0.2
2.55
3.08
2.55
0.8
62.57
4.14
0.46
0.4
10.20
4.18
0.42
1.0
101.66
4.21
0.39
0.6
22.97
4.92
0.41
1.2
161.34
4.23
0.35
0.8
40.84
5.57
0.39
σ0 = 1000 [MPa], n = 3, a/W = 0.70
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
3.84
3.38
1.19
0.4
14.94
4.51
0.39
0.6
33.94
5.29
0.38
0.8
60.74
5.79
0.36
σ0 = 1000 [MPa], n = 5, a/W = 0.70
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
2.55
2.68
5.60
0.4
10.21
3.31
1.79
0.6
23.05
3.62
1.15
0.8
41.72
3.80
0.79
1.0
66.98
3.92
0.52
1.2
101.43
4.00
0.42
189
Catalogue of maximum crack opening stress for CC(T) specimen assuming large strain condition
Table C3.
190
Numerical results for CC(T) specimens characterized
by yield stress σ0 = 1000 MPa and work hardening
exponent in R-O relationship n = 10
Table C4.
Numerical results for CC(T) specimens characterized
by yield stress σ0 = 1000 MPa and work hardening
exponent in R-O relationship n=20
σ0 = 1000 [MPa], n = 10, a/W = 0.20
σ0 = 1000 [MPa], n = 20, a/W = 0.20
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
0.89
2.65
8.76
0.2
3.41
2.54
7.26
0.4
3.61
2.98
3.43
0.4
14.49
2.61
2.50
0.6
8.31
3.00
1.39
0.6
34.06
2.63
1.34
0.8
15.50
2.99
1.03
0.8
64.88
2.38
1.05
1.0
28.66
3.00
0.69
1.0
124.18
2.16
1.22
1.2
122.24
3.03
0.43
1.2
1574.45
2.12
0.33
σ0 = 1000 [MPa], n = 10, a/W = 0.50
σ0 = 1000 [MPa], n = 20, a/W = 0.50
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
3.71
2.67
5.86
0.2
3.71
2.63
6.81
0.4
15.24
3.04
2.10
0.4
15.25
2.81
2.76
0.6
34.75
3.08
1.12
0.6
34.79
2.81
1.44
0.8
62.50
3.04
0.98
0.8
62.58
2.72
0.94
1.0
103.51
3.02
0.68
1.0
104.76
2.55
0.84
1.2
171.02
2.93
0.59
1.2
183.76
2.38
0.64
σ0 = 1000 [MPa], n = 10, a/W = 0.70
σ0 = 1000 [MPa], n = 20, a/W = 0.70
P/P0
J [kN/m]
σmax /σ0
ψ
P/P0
J [kN/m]
σmax /σ0
ψ
0.2
2.55
2.57
7.18
0.2
2.55
2.53
8.36
0.4
10.22
2.95
3.23
0.4
10.22
2.79
2.88
0.6
23.08
3.06
1.79
0.6
23.11
2.83
1.65
0.8
41.93
3.09
1.20
0.8
42.71
2.77
1.10
1.0
68.49
3.07
0.87
1.0
68.89
2.67
0.82
1.2
105.31
3.01
0.65
1.2
109.90
2.55
0.77