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Identification of CTOA and fracture process parameters by drop weight test and finite element simulation

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Identiﬁcation of CTOA and fracture process parametersby drop weight test and ﬁnite element simulation
P. Salvini
a,1
, A. Fonzo
a,*
, G. Mannucci
b,2
a
Department of Mechanical Engineering, University of Rome ‘‘Tor Vergata’’, Via del Politecnico n. 1, 00133 Rome, Italy
b
Centra Sviluppo Materiali S.p.A. Via di Castel Romano n. 100/102, 00128 Rome, Italy
Received 30 July 2001; received in revised form 13 February 2002; accepted 6 March 2002
Abstract
This paper presents a new technique that is able to predict ductile fracture propagation occurrences in large metallicstructures, by means of an appropriate application of the ﬁnite element modelling. This technique takes account of acohesive zone in the vicinity of the crack tip, where a nodal release technique is implemented. Two parameters, gov-erning the process zone of the material under investigation, have to be determined: the process zone dimension (named‘‘
D
distance’’) and the critical value of crack tip opening angle (CTOA). CTOA
C
can be determined through an ex-perimental laboratory procedure two specimen CTOA test (TSCT) that is already known and used by researchers whostudy fracture propagation on pipelines [Demofonti G, et al. Step-by-step procedure for the two specimen CTOA test.In: Proceedings of the Second International Conference on Pipeline Technology, Ostend, vol. II. 1995]. The secondparameter required,
D
distance, is determined minimizing the diﬀerences of Finite Element results towards experimentaldata of an instrumented impact test (drop weight tear test). Some interesting improvements, concerning distinctionbetween the initiation energy and the propagation energy accounted in TSCT procedure, are also discussed, in order tosuccessfully extend its use to both high strength and high toughness steels.
2002 Elsevier Science Ltd. All rights reserved.
Keywords:
Ductile fracture; Cohesive zone model; Drop weight tear test; Crack tip opening angle; Essential work of fracture
1. Introduction
Ductile crack propagation in metals is accompanied with an extended plastic zone ahead the crack tip,particularly when thickness is small compared to other dimensions. If the driving force does not decreaseduring crack propagation, the fracture can widely extend, as it can eﬀectively happen in pressurizedpipelines suﬀering longitudinal cracks. Such events are characterised by a high dynamics, so that the centraldiﬀerence scheme for ﬁnite element solution is well suited. In the ﬁeld of the engineering structures here
Engineering Fracture Mechanics 70 (2003) 553–566www.elsevier.com/locate/engfracmech
*
Corresponding author. Fax: +39-6-2021351.
E-mail addresses:
salvini@uniroma2.it (P. Salvini), fonzo@ing.uniroma2.it (A. Fonzo), g.mannucci@c-s-m.it (G. Mannucci).
1
Fax: +39-6-2021351.
2
Fax: +39-6-5055452.0013-7944/03/$ - see front matter
2002 Elsevier Science Ltd. All rights reserved.PII: S0013-7944(02)00137-6
accounted, considering the wide plasticity induced during crack propagation, traditional crack modelscannot be easily managed. Furthermore, traditional crack models, applied to ﬁnite elements, need a ﬁnediscretisation which, for central diﬀerence scheme, causes an unacceptable decrease of the computationaltime step [1]. The alternative can be found by using cohesive fracture models which, if well suited on
material properties and geometry, allow the use of non-excessively reﬁned meshes, thus keeping a suﬃ-ciently high value of the time step. This is a mandatory requirement for practical use of central diﬀerenceintegration scheme. This approach is followed in the pipe crack propagation code, jointly developed by theUniversity of Rome ‘‘Tor Vergata’’ and the Centro Sviluppo Materiali S.p.A., where cohesive fracture zonemodelling is made possible together with other peculiar facilities (ﬂuid dynamic modelling of real gas ex-pansion, backﬁll constraint eﬀects,
. . .
) for the analysis of longitudinal cracks, running on pressurizedburied pipes [2].The cohesive zone model is one-dimensional, and ﬁnite element node release is carried out progressively;so that, the energy released during the steady propagation of a crack does not present discontinuities relatedto the mesh and keeps constant. As it is discussed within the paper, the use of the cohesive zone modelrequires the previous determination of two parameters: the critical crack tip opening angle (CTOA
C
) of thematerial [3], and the size of the cohesive zone involved in the local progressive softening.According to several researchers [4 – 6], far from transitory conditions, CTOA assumes a constant value
when stable propagation occurs; its value depends on material and geometry.Therefore, stable growth is achieved only if the applied CTOA reaches the critical one. CTOA is, on theother side, unable to establish crack initiation conditions; other criteria must be invoked in this case. Froma mathematical point of view, CTOA is the Lagrangian derivative of crack tip opening displacement. If CTOD is assumed to characterise the crack growth, CTOA is related to the changing rate of the crackspeed. Cohesive models also allow the monitoring of the essential work of fracture within the cohesive zone(fracture process zone, FPZ). This is the energy requested only for the generation of new crack surfaces.This quantity, if reliably estimated, could be a new key for the evaluation of high ductile crack propagation[7 – 10].
The aim of the present work is to describe the experimental set up and the numerical procedures that arerequired to tune the cohesive zone model for its use on engineering structures suﬀering long crack prop-agation. These conditions occur, as an example, in steel gas pipelines, when, due to incidental reasons(corrosion, impact with excavators, etc.), a wall thickness failure occurs and ductile fracture starts topropagate, driven by escaping gas force.
2. One-dimensional cohesive model
In the present paper a one-dimensional cohesive zone model for the simulation of the progressive for-mation of the crack ﬂanks is discussed. The progressive nodal release technique here discussed provides alayer (cohesive zone), located behind a virtual crack tip, where the ability of the material to resist to theseparation of the ﬂanks is gradually reduced to zero. According to the modelling technique proposed, theCTOA is the geometric angle that the crack ﬂanks form in correspondence of the virtual crack tip.One of the main advantages of this method is the possibility to take into account the energy locallydissipated during the creation of new fracture surfaces.The loads applied to the nodes of the mesh which lie within the cohesive zone (Fig. 1) are such as to beequivalent to the stress distribution which oﬀers resistance to ﬂanks
opening. The stress distribution isderived assuming a power law for the half-opening
V
of the ﬂanks in the cohesive zone versus an evolu-tionary parameter
a
. The parameter
a
is equal to the ratio between the actual stress and the higher stressthat keeps the ﬂanks closed immediately ahead the virtual crack tip:
554
P. Salvini et al. / Engineering Fracture Mechanics 70 (2003) 553–566
V
¼
V
0
ð
1
a
k
Þ
a
¼
r
C
r
C0
(
ð
1
Þ
The basic hypothesis of the fracture cohesive zone model is that a maximum of cohesive constraint isreached at the virtual crack tip, keeping the ﬂanks fully closed; the constraint stress reduces to zero at adistance which is equal to the FPZ size (
D
distance). At this position the two fracture ﬂanks are fully in-dependent each other. One of the main interesting points is that, moving from a continuous description to adiscrete model (FEM), the virtual crack tip location can still be described by means of a continuousvariable. In other words, the crack tip does not necessarily correspond to a node of the ﬁnite element mesh,so that crack evolution is independent of mesh reﬁnement.Stationary propagation implies that the speciﬁc energy released for formation of crack ﬂanks is inde-pendent of global amount of propagation. This obvious condition is the key that allows to correlate
a
to
d
considering the limit case of only one node present within the FPZ.The starting point is the deﬁnition of the essential work of fracture [11], computed as the energy dis-sipated in the softening zone behind the crack tip, for an unitary crack advance and unitary thickness.Being
D
the size of FPZ, the energy dissipated in the cohesive zone can be expressed asd
E
D
d
a
¼
2
B
Z
D
0
r
c
ð
x
Þ
d
V
ð
x
Þ
d
a
d
x
ð
2
Þ
where
B
is the specimen thickness,
V
is the half-opening of crack ﬂanks,
r
c
is the cohesive stress and
a
represents the crack size. Taking in mind the deﬁnition given by Cotterel, the essential work of fracture issimply the speciﬁc form of (2) with respect to specimen thickness:
w
f
¼
1
B
d
E
D
d
a
ð
3
Þ
Considering the case of only one node of the mesh falling inside the FPZ, cohesive stress distribution re-duces to a discrete
F
value. In this case the loads acting on this node, whatever its distance from the virtualcrack tip is, must be able to dissipate the same speciﬁc amount of energy, i.e.
w
f
ð
d
Þ ¼
w
f
ð
D
Þ
. In this discreterepresentation, the essential work of fracture is
Fig. 1. Simpliﬁed scheme of cohesive zone model.
P. Salvini et al. / Engineering Fracture Mechanics 70 (2003) 553–566
555
w
f
¼
2
B
d
Z
d
0
F
ð
x
Þ
d
V
d
a
d
x
ð
4
Þ
where
d
indicates the actual distance of the node from the virtual crack tip.The eﬀective value of
F
spans from a maximum value when the node falls exactly at the virtual crack tip,to a zero value when the node reaches the distance
D
from the virtual crack tip.
F
0
is, as a deﬁnition, theload necessary to keep the almost-opening nodes jointed together, just before the virtual crack tip reachesthe nodes themselves.If the mesh is geometrically regular, the evolution of the load
F
towards
a
presents the same trend as thestress (1).
F
¼
a
F
0
ð
5
Þ
Substituting (5) into (4) the essential work of fracture is now written as a function of the evolutionaryparameter
a
:
w
f
¼
2
F
0
V
0
B
d
k k
þ
1
ð
1
a
k
þ
1
Þ ð
6
Þ
Apparently
w
f
in (6) cannot be considered constant because of its dependency on
a
; however, it should bepointed out that each
a
corresponds to a particular value of
d
, so that the invariability of
w
f
can be forced.In particular, stationary propagation allows to equate the generic value of
w
f
with the value assumedimmediately before the unique ﬁnite element node, lying within the cohesive zone, is completely released(and a new node enters in FPZ):
w
f
ð
d
Þ¼
2
F
0
V
0
B
d
k k
þ
1
ð
1
a
k
þ
1
Þ¼
w
f
ð
D
Þ¼
2
F
0
V
0
b
D
k k
þ
1
ð
7
Þ
The previous equation can be solved for
a
giving its value as a function of the actual node position
d
, insideFPZ:
a
¼
F F
0
¼
1
dD
1
=
ð
1
þ
k
Þ
ð
8
Þ
Eq. (8) solves the question of deﬁning
a
according to the distance
d
.The value of the parameter
k
, in the exponent of (8), can be assumed equal to 1, as a formal attempt, inaccordance with a work of Rydholm et al. [12]. So that, ﬁnally, the value of nodal release force
F
, functionof actual position
d
, is simply
F
¼
F
0
1
dD
0
:
5
ð
9
Þ
Generalising the study to the case of a ﬁner mesh, where a generic number of nodes fall inside FPZ, it isreasonable to assume that position (7) is still valid. In this case
F
0
changes its value, being the stress ob-viously mesh independent. Thus it is more correct replacing
F
0
with
F
0
n
, whose value depends on the ef-fective ﬁnite element size, where
n
represents the number of nodes falling into FPZ.The value of
F
0
n
is not predeﬁned by the user, but is the result of numerical calculation in previous steps,when the crack tip reaches the node itself. Therefore,
F
0
n
is continuously updated and it can be modiﬁedwhether constraint or driving force changes during crack advance. The general expression of the EssentialWork of Fracture, valid for any reﬁnement of the mesh, is
w
f
¼
2
B
X
nodes
i
¼
1
F
0
n
V
0
d
i
k k
þ
1 1
h
ð
a
ð
d
i
ÞÞ
k
þ
1
i
ð
10
Þ
Here the sub ﬁx
i
applied to
d
indicates that
d
is taken into account diﬀerently for any node considered.
556
P. Salvini et al. / Engineering Fracture Mechanics 70 (2003) 553–566
However, if (8) is used into (10) it is easy to demonstrate that
d
i
disappears in the essential work of fracture:
w
f
¼
2
B
D
F
0
n
V
0
k k
þ
1
n
nodes
or
w
f
¼
2
B
D
F
0
V
0
k k
þ
1
ð
11
Þ
where
n
nodes
is the number of nodes falling into the FPZ. The second form of (11) shows the independence of
w
f
on the mesh since
F
0
n
is, for regular meshes, equal to
F
0
=
n
nodes
.This again conﬁrms the total independence of the cohesive zone model proposed on locally regular meshreﬁnement.In the next developments, it will be shown that a simple relationship can be highlighted between
w
f
andthe product
D
CTOA. CTOA gives a measure of the driving force applied to the crack; CTOA is deﬁned asthe angle emerging from the opening ﬂanks; from a mathematical point of view, introducing a variable
x
giving the distance from the crack tip, if
V
ð
x
Þ
(or half-COD
ð
x
Þ
) is known, the CTOA can be found by alimit applied to a derivative of COD:CTOA
a
¼
lim
x
!
0
2arctan 12dd
a
COD
ð
x
Þ
ð
12
Þ
x
is here oriented towards crack propagation.Considering
k
¼
1 in (11), a mean value of CTOA inside FPZ, and being
r
0
the eﬀective ﬂow stress of thematerial (see below for details) (11) becomes:
w
f
¼
2
B
D
F
0
V
0
k k
þ
1
¼
F
0
B
DD
tg CTOA2
r
0
D
CTOA
ð
13
Þ
This ﬁnal equation evidences that one-dimensional cohesive zone model is eﬀectively governed by a coupleof parameters, here CTOA and
D
are chosen, both necessary for energy release evaluation. Thus the mainobjective is to ﬁnd out an optimal procedure for a direct estimate of their values, elaborating test results in asimple manner.
3. First determination of CTOA parameter
Several approaches allow to determine the critical value of CTOA that is a characteristic of stationarycrack propagation in a material (CTOA
C
) as many authors [4 – 6] have highlighted. Within the present
paper, a particular method is explained, able to identify CTOA
C
on non-thin specimens, when opticalmethods lose eﬃciency, because of change in the emerging angle inside the bulk. This method is based onenergetic considerations, by means of an approach that is called two specimen CTOA test (TSCT).
3.1. Method 1: the two parameter approach for energy computation
According to Priest et al. [13], in a three point bending specimen, speciﬁc energy, required for a completecrack propagation, can be considered linearly dependent on the length of the specimen
s ligament:
E
T
A
¼
R
C
þ
S
C
ð
W
a
0
Þ ð
14
Þ
where
A
: product of initial ligament by thickness;
W
: total specimen width;
a
0
: initial crack extension;
E
T
: total energy absorbed during full propagation;
R
C
,
S
C
: material constants.
P. Salvini et al. / Engineering Fracture Mechanics 70 (2003) 553–566
557

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