Failure by fracture in high pressure structure such as pipeline and pressure
vessel; can lead to greatly economic losses and, moreover, human lives impairment
(McHenry et al., 1986; NTSB,
2004; CEPPO, 1997). Accessing fracture based failure
criterion in design process of the high pressure structures is a mandatory.
One method to assess this criterion in design code and standard for pressurized
equipment (such as PD 6493) (BSI PD 6493 1980) is Failure
Assessment Diagram (FAD). It is based on comparison of the location of geometry-dependent
assessment point to the Failure Assessment Line (FAL). To construct the both
parameters, the elastic-plastic fracture parameter, like J-integral, need to
be calculated and predicted carefully. Mainly, these calculations are based
on Finite Element Analysis (FEA) and are particularly depend on modeling approach
of the crack geometry and material property.
Finite element modeling scheme for semi-elliptical crack problem is neither an instant nor a direct process with certain human intervention. In spite some effective schemes had been found (such as: Anderson with spider-web arrangement and singular tip element Shih and Shahani with sweep scheme Cao and Branco that combine above schemes) their systematic processes are still not openly published. Beside, almost all the schemes were using the complicated-external grid generator, like Nasgro, Wrap3d, Zencrack, Feacrack. This limitation led to the first part of this work, which is to explore and proposed in-house effective modeling scheme for particular semi-elliptical crack problem.
Many researchers currently found that fracture parameter is finely affected
by the constraint condition (the triaxiality stress state). In the low constraint
condition, the fracture toughness prediction based on single-parameter (such
as J-integral) is always underestimated. It implies that, in such condition,
overly conservative assessment results are obtained Ainsworth,
and ODowd, 1995. Addition of the second parameter that accounts the
affect of low constraint maximizes the prediction of load carrying capacity
of the cracked structure and at the end, provides engineers better information
to design structure (Yee and Kapper, 2006; Ferreno
et al., 2010; Gutierrezsolana and Cicero, 2009;
Ainsworth and Hooton, 2008; Flewitt,
2008; Ainsworth, 2000). Some theories have been
proposed to predict this fracture toughness dependence and to quantify the crack-front
constraint against the plastic flow. Among them are the J-T theory, J-Q theory
and the J-A2 three-term solution (Wang, 2009; Zhao
et al., 2008; Zhu and Leis, 2006; Kim
et al., 2001; Yuh-Jin and Poh-Sang, 1998).
The second subject of this study is to review briefly these current two-parameter
constraint-based fracture mechanics and their applications in construction of
In this study, as mention before, two subjects will be briefly discussed; the
modeling approach and the two-parameter fracture mechanics. First, a simple
modeling approach of the surface crack in plate is developed and reviewed.
|| Main algorithm of crack modelling
Singular element, spider-web pattern and sweep-mesh scheme is implemented cautiously,
along with several original strategies on free-mesh size control. The J-integral
using energy domain integral is calculated and compared to Raju closed-form
equation of fracture parameter. The convergence study and the numerical result
are reviewed. The proposed modeling approach is then summarized. Second, the
current theories of two-parameter fracture mechanics were reviewed shortly.
The characteristics of these two-parameters are discussed. The implementation
of these two-parameters on FAD is summarized.
MODELING APPROACH AND ITS RESULT
Careful attention is needed when building the crack model. The mesh has to
be light and simple to reduce the computation time, meanwhile the accuracy has
not to be much sacrificed. Simple development is performed, which is mainly
based on the following BS4 approach (Brick, Spider-web, Singular, Sweep and
Solid; Fig. 2).
||Brick (3D hexahedral) and the higher order 3D (20-node) element that exhibits
quadratic displacement behavior is prioritized
||Spider-web configuration (concentric rings of four sided elements that
are focused towards the crack tip) is utilized. The innermost elements are
degenerated to wedges. Since the crack tip region contains steep stress
and strain gradients, the mesh refinement should be greatest at the crack
tip. The spider-web approach facilitates a smooth transition from a fine
mesh at the tip to a coarse mesh remote to the tip
||The wedges element around the crack tip is transformed to (collapsed)
singular element by move the midside node to a quarter point. It compensates
the stress singularity (1/
) in elastic problem
||Sweep mesh scheme is employed to extend the two-dimension (2D) mesh to
3D mesh. It provides meshing that fit to semi-elliptical crack front geometry
||All above approach impose meshing that is based on solid (geometric) modeling
instead of free (automatic) meshing or direct modeling.
In the previous study Ariatedja (2009) employed BS4
modeling approach into the crack modeling algorithm Fig. 1
and developed the APDL (ANSYS Parametric Design Language) code. It presented
simple study on comparison between linear-elastic stress intensity factor, K
and elastic-plastic J-integral in order to divined their character and accuracy.
Using simple case semi-circular surface crack on flat plate under very low uniaxial
remote applied stress (<5% of yield stress), it showed that the J-integral
provides reliable accuracy than stress intensity factor, K. This parameter was
also more independent from the affect of element size and elastic singularity
||Development of crack modeling: (a) step of process; (b) comparison:
default and using BS4 approach
The BS4 modeling approach were succeeded to reduce the calculation time from
7 to half minutes (from 14200 to only 960 elements), meanwhile, it is also reducing
significantly the difference of J-integral (from 1.1 to 0.7%) and K (from 8
to 0.1%) to Raju closed-form equation (compared to default meshing from ANSYS
help, Fig. 2).
Current extend work Ariatedja (2010) had been done to
clarify the numerical convergence for the proposed model. The non-dimensional
J-integral of a semi-elliptical crack on flat plate under uniaxial loading and
linear-elastic material had been evaluated. All BS4 approach was obeyed. The
energy domain integral was used to obtain up to twenty contours of J-integral
of each crack front nodes. The crack baseline model is described in Table
1. Two different sweep schemes were employed, which were undistorted and
distorted sweep scheme. The ratio of crack depth to element size was considered
in the range between 15 and 80, division of crack tip area was in range between
3 and 12, division of sweep was in range between 4 and 15 and the ratio of plate
width to crack depth was in the range between 5 and 20. The non-dimensional
J-integral from Raju closed-form equation was used as reference.
|| Model parameters of semi-elliptical crack in plate
Almost all baseline parameters had been in fine arrangement (result I n minimum difference), except the J-integral dependence to the contour was found in distorted sweep scheme. The differences increased as the contour increase, especially for the midside nodes Fig. 3a. It implies that element distortion will produce significant error. Based on the results, the undistorted sweep scheme along with baseline parameters was recommended to be used as base mesh design along with BS4 modeling approach.
CONSTRAINT PARAMETER AND ITS IMPLEMENTATION ON FAD
Even though the single-parameter J-based fracture mechanics has long been regarded
as a material property into industry testing standards by American Society of
Testing and Materials (ASTM) (Rice, 1968; Hutchinson,
1968; Hutchinson, 1999), it has been found that fracture
toughness indeed depends on specimen size, thickness, crack depth, geometry
and loading condition, which are attributed to different crack-front constraints.
Fracture constraint at crack front means the resistance against the plastic
deformation. It is questionable to apply the fracture toughness value determined
from small laboratory specimens to integrity assessment of large defected structures.
The level of constraint at crack front play important role and can be revealed
by examining accurately the details of the crack-front stress and deformation
fields. Normally, the plain-strain state exhibits the highest constraint, generates
the highest triaxiality of stresses, whereas the plane-stress state yields the
Some approaches have been proposed to predict this fracture toughness dependence
and quantify the crack-front constraint against the plastic flow. Among them,
two representatives are the J-T theory proposed by Betegon
and Hancock (1991), J-Q theory developed by ODowd
and Shih (1991, 1992) and the J-A2 three-term solution
developed by Yang et al. (1993a, b)
and Chao et al. (1994).
||Convergence study of the non dimensional J-integral result
for two different sweep schemes: (a) distorted (b) undistorted
In the LEFM, it has been found that a second term, T-stress or A3
can represent the affect of specimen geometry and loading condition on crack-front
stress fields while the stress intensity factor K represent the applied loading.
Using Williams series solution for the crack-front fields, one has:
where, (r, θ) are polar coordinates with the origin located at the crack-front, KI is the Mode-I stress intensity factor, fij(θ) are non-dimensional angular functions, T is the second term and is a uniform stress parallel to the crack face commonly referred to as T-stress, δ1i and δ1j are the Kronecker delta. The indexes i, j have the range of 1-2.
In EPFM, J is used to represent the loading level. A second parameter Q is defined as hydrostatic stress, which is extension of parameter T for elastoplastic material. After simplification, the crack-front stress field is:
where, the first term is a standard ssy solution with T = 0 that replaces the HRR field, σ0 is reference stress that is generally equal to yield stress. The parameter Q is defined as hydrostatic stress by the difference of the HRR stress field and the full-field stress field:
parameter Q is engineering definition and depend of the location. The location of r/(J/ σ0) = 2 and θ = 0 is generally used for the determination of Q.
A more rigorous analysis of higher-order crack-front fields in power-law hardening materials is the J-A2 three-term asymptotic stress field:
where the angular functions ∂ij(k) with k = 1, 2, 3, the stress power exponents sk (s1<s2<s3) are only dependent of the hardening exponent n and independent of other material constants and applied loads. L is characteristic length parameter which can be chosen as the crack length a, the specimen width W, the thickness B, or unity. The parameter A1 and s1 are given by:
and s3 = 2s2 - s1 for n = 3. A2
is an undetermined parameter and may be related to the loading and geometry
of specimen. The angular function and s2 in (3a) are given in a report
written by Chao et al. (1994).
As the elastic T-stress requires only elastic calculations, it is recommended for initial evaluations. The hydrostatic Q-stress is expected to provide more accurate assessments, particularly when plasticity becomes widespread and should be used when more refined estimates of load margins are required or as part of sensitivity studies. On contrary, there are only a few works that study the J-A2 three-term asymptotic stress field. Further study is needed to explore particularly in its application on FAD.
To address the constraint affect on FAD, some design code and standard, such as SINTAP/FITNET, employ a special structural constraint factor β hat can be obtained from elastic T-stress and the hydrostatic Q-stress:
The β factor is then used to determine a constraint dependent fracture
toughness designated as Kcmat. The increase in fracture
toughness in both the brittle and ductile regimes may be represented by an expression
of the form:
Using finite element analysis, the J-integral (in elastic and in plastic property)
and the constraint parameter can be calculated. Both parameter is used to generate
the Kr and Fr as SINTAP/FITNET employs constraint parameter β to incorporate
the constraint effect on the FAD. This parameter can be obtained from elastic
T-stress and hydrostatic Q-stress.
The cost-effective meshing and modeling of crack under uniaxial load had been developed. The convergence study for single-parameter linear-elastic fracture mechanic had been done and cost-effective mesh design had been provided.
Critical review of fracture parameter that considered the constraint conditions had been done briefly. Works on implementing the two-parameter fracture mechanic and on developing failure assessment diagram are planned to be done further.
The gratitude feeling would like to be addressed to Universiti Teknologi PETRONAS, Malaysia for sponsoring this research.