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Benchmark for finite element analysis of rupture in
the ductile to brittle transition regime using the local
approach to fracture
C. Berdin , J. Besson , A. Dahl , P. Forget , A. Parrot
C. Poussard , C. Sainte–Catherine , B. Tanguy , N. Verdiere` .
Ecole Centrale de Paris, Laboratoire MSS–MAT, Grande voie des vignes, 92295
Chatenay–Malabryˆ , France
Ecole des Mines de Paris, Centre des Materiaux,´ UMR CNRS 7633, BP 87, 91003 Evry
Cedex, France
EdF les Renardieres Route de Sens - Ecuelles 77250 Moret-sur-Loing, France
CEA
Abstract
....
Key words: ductile to brittle transition, finite element analysis, local approach to fracture.
1 Introduction
[1]
2 Constitutive equations
2.1 Ductile rupture
Two models for ductile rupture are used in this study. The first one is the Gurson
model [2]. This model is based on a rigorous micromechanical analysis and has
been extended by Needleman and Tvergaard [? ] on a more phenomenological basis
(so called GTN model) to account for plastic hardening, nucleation, void interaction
and coalescence. The second model is the Rousselier model [3] which is based on a
thermodynamical approach of damage. Both models use a single damage variable:
the void volume fraction (porosity).
Preprint submitted to Frac. Fat. Engng Mater. Struct. 10 January 2004
...
Voir
the ductile to brittle transition regime using the local
approach to fracture
C. Berdin , J. Besson , A. Dahl , P. Forget , A. Parrot
C. Poussard , C. Sainte–Catherine , B. Tanguy , N. Verdiere` .
Ecole Centrale de Paris, Laboratoire MSS–MAT, Grande voie des vignes, 92295
Chatenay–Malabryˆ , France
Ecole des Mines de Paris, Centre des Materiaux,´ UMR CNRS 7633, BP 87, 91003 Evry
Cedex, France
EdF les Renardieres Route de Sens - Ecuelles 77250 Moret-sur-Loing, France
CEA
Abstract
....
Key words: ductile to brittle transition, finite element analysis, local approach to fracture.
1 Introduction
[1]
2 Constitutive equations
2.1 Ductile rupture
Two models for ductile rupture are used in this study. The first one is the Gurson
model [2]. This model is based on a rigorous micromechanical analysis and has
been extended by Needleman and Tvergaard [? ] on a more phenomenological basis
(so called GTN model) to account for plastic hardening, nucleation, void interaction
and coalescence. The second model is the Rousselier model [3] which is based on a
thermodynamical approach of damage. Both models use a single damage variable:
the void volume fraction (porosity).
Preprint submitted to Frac. Fat. Engng Mater. Struct. 10 January 2004
...
Benchmark for finite element analysis of rupture in
the ductile to brittle transition regime using the local
approach to fracture
C. Berdin
, J. Besson
, A. Dahl
, P. Forget
, A. Parrot
C. Poussard
, C. Sainte–Catherine
, B. Tanguy
, N. Verdi
`
ere
.
Ecole Centrale de Paris, Laboratoire MSS–MAT, Grande voie des vignes, 92295
Ch
ˆ
atenay–Malabry, France
Ecole des Mines de Paris, Centre des Mat
´
eriaux, UMR CNRS 7633, BP 87, 91003 Evry
Cedex, France
EdF les Renardieres Route de Sens - Ecuelles 77250 Moret-sur-Loing, France
CEA
Abstract
....
Key words:
ductile to brittle transition, finite element analysis, local approach to fracture.
1 Introduction
[1]
2 Constitutive equations
2.1 Ductile rupture
Two models for ductile rupture are used in this study. The first one is the Gurson
model [2]. This model is based on a rigorous micromechanical analysis and has
been extended by Needleman and Tvergaard [
?
] on a more phenomenological basis
(so called GTN model) to account for plastic hardening, nucleation, void interaction
and coalescence. The second model is the Rousselier model [3] which is based on a
thermodynamical approach of damage. Both models use a single damage variable:
the void volume fraction (porosity).
Preprint submitted to Frac. Fat. Engng Mater. Struct.
10 January 2004
2.1.1 Plastic yielding
In both cases, the model is defined by a plastic yield function
. In the case of the
GTN model it is given by:
(1)
In the case of the Rousselier model,
is given by:
(2)
is the yield stress of the undamaged material. It depends on the equivalent plastic
strain
.
,
,
and
are constant model parameter which need to be adjusted.
is the void volume fraction.
is a function of the porosity used to model the
material last stage of failure (coalescence). It is usually expressed as:
if
if
(3)
where
and
are adjustable material parameters.
The extension to rate dependent materials can be simply obtained writing
as a
function of both
and
[4]. This extension will only be used in the case of the
GTN model as it is not suitable for the orginal Rousselier model as shown in [5, 6].
Modifications of the Rousselier models have been proposed in [5, 6] to obtain a
satisfactory extension to rate dependent materials.
2.1.2 Plastic flow and damage rate
Plastic yielding occurs when
.The plastic deformation rate tensor is obtained
using the normality rule as:
(4)
where
is the stress tensor. In the case of the Gurson model
is defined
by expressing the equality between the microscopic and macroscopic plastic
dissipations:
. In the case of the Rousselier model,
coincides
with the usual von Mises equivalent strain. The evolution of the porosity is obtained
applying mass conservation so that:
. This equation can be
modified to account for void nucleation by particle debounding or cracking. In this
case, the evolution law for the porosity is given by:
(5)
2
where
is a function describing nucleation controlled by plasticity. In the
following, the function proposed by Chu and Needleman [7] will be used:
(6)
where
,
and
are parameters which describe nucleation.
2.1.3 Failure
At some point of the loading history, the material is considered as broken. In the
case of the Gurson model, failure occurs when
. In practice, the material
is considered as broken when
with
.
In the case of the Rousselier model, failure occurs for
. In pratice, one
assumes that the material fails when
reaches a critical value
which is usually
taken large enough to avoid numerical problems.
2.2 Brittle rupture
Brittle rupture will be modeled using the Beremin model [8] which accounts for
the random nature of brittle fracture. The model is based on the Weibull weakest
link theory. It can be applied as a post–processor of calculations including ductile
tearing. Care must be taken while computing the failure probability of the Charpy
specimen as the ductile crack advance leads to unloading of the material left behind
the crack front. Considering that each material point is subjected to a load history
(
time) the probability of survival of each point at time
is determined
by the maximum load level in the time interval
. An effective failure stress
is then defined as:
(7)
is the maximum principal stress of
. The condition
expresses
the fact than failure can only occur when plastic deformation occurs. The Weibull
stress is then defined as:
(8)
where the volume integral is taken over the whole volume of the specimen. The
rupture probability is then expressed as:
.
and
are
two model parameters.
is a reference volume which can be chosen arbitrarily.
3
2.3 Material coefficients used for the benchmark
2.3.1 Behavior of the undamaged material
The elastic properties are taken as constant : Young’s modulus
GPa,
Poisson ration
.
The plastic hardening behavior
is given as a table (see Table 1).
FE calculations were done using this table as an input data (i.e. fitting formulas
were not used). A linear interpolation is used between the different data points.
Note that for a total tensile strain larger than
the material behaves as a perfectly
plastic material. The yield stress of the material is referred to as
and is equal to
495 MPa.
In the case of a rate dependent material, the plastic behavior is described by a
Cowper–Symonds law so that:
(9)
with
and
.
2.3.2 Ductile rupture
The same initial porosity
was used for both models:
.
In the case of the GTN model the following material coefficients were used:
(10)
In the case of the Rousselier model the following material coefficients were used:
MPa
(11)
2.3.3 Brittle rupture
The evaluations of the Weibull stress were performed with a reference volume
and
.
4
2.3.4 Remarks
3 Finite element analysis
3.1 FE codes used
ABAQUS/UMAT
Castem3M
Code Aster
Z
´
ebulon
3.2 Numerical procedures
4 Benchmark tests
4.1 Volume element
Simple tests are performed on a single axisymmetric element with 4 nodes and
1 Gauss point. A first test is carried out under tension and two other tests under
prescribed displacement conditions as depicted on fig. 1 with
and
. The prescribed displacement was applied with steps equal to
.
Results show the evolution of the axial stress
and of the porosity
as a function
of the prescribed displacement
.
4.2 Notched axisymmetric bar
4.3 Cracked plane strain specimen
4.3.1 Calculation of the crack advance
The crack advance s determined as follows. The position of the last broken
Gauss point along the
axis is determined using post–processing calculation.
The calculation is performed in the deformed configuration. The crack length is
then obtained by substracting the position of the loading point (
l
on fig. 5) The
initial crack length is then substracted (i.e.
). Due to plastic deformation of the
geometry, this might result in a slight negative crack advance at the beginning of
propagation.
5
4.4 Charpy test
4.4.1 Calculation of the crack advance
The crack advance s determined as follows. The position of the last broken Gauss
point along the
axis is determined using post–processing calculation in the
deformed configuration. Due to the element removing technique, it is not possible
to compute the position of the node lying at the node root after the element it
belongs to has been removed. For this reason, the reference node used to determined
the crack advance is the one designed as
M
on fig. 8 and lying above the axis of
symmetry. The crack advance is then obtained as the difference of the position of
point
M
and the last broken Gauss point along the
axis. The original offset between
M
and the notch root (
) is then substracted.
5 Conclusion
6
References
[1] G. Bernauer and W. Brocks. Micro–mechanical modelling of ductile damage
and tearing—results of a european numerical round robin.
Fatigue and Fract.
of Engng Mat. Struct.
, 25(4):363–384, 2002.
[2] A.L. Gurson. Continuum theory of ductile rupture by void nucleation and
growth: Part I— Yield criteria and flow rules for porous ductile media.
J.
Engng Mater. Technology
, 99:2–15, 1977.
[3] G. Rousselier. Ductile fracture models and their potential in local approach of
fracture.
Nucl. Eng. Design
, 105:97–111, 1987.
[4] A. Needleman and V. Tvergaard. An analysis of dynamic, ductile crack growth
in a double edge cracked specimen.
Int. J. Frac.
, 49:41–67, 1991.
[5] B. Tanguy and J. Besson. An extension of the Rousselier model to viscoplastic
temperature dependent materials.
Int. J. Frac.
, 116(1):81–101, 2002.
[6] C. Sainte Catherine, C. Poussard, J. Vodinh, R. Schill, N. Hourdequin, P. Galon,
and P. Forget. Finite element simulations and empirical correlation for Charpy–
V and sub–size Charpy tests on an unirradiated low alloy RPV ferritic steel. In
Fourth symposium on small specimen test techniques, Reno, Nevada, ASTM
STP 1418
, 2002.
[7] C.C. Chu and A. Needleman. Void nucleation effects in biaxially stretched
sheets.
J. Engng Mater. Technology
, 102:249–256, 1980.
[8] F. M. Beremin. A local criterion for cleavage fracture of a nuclear pressure
vessel steel.
Met. Trans.
, 14A:2277–2287, 1983.
7
List of Tables
1
Plastic hardening data
10
2
Participants
11
8
List of Figures
1
Boundary conditions for volume element tests.
: prescribed
displacement. The initial volume of the element is equal to
(mm
).
11
2
Results of the one element tests
12
3
Axisymmetric notched bars: specimen geometry and meshes.
13
4
Axisymmetric notched bars: results
14
5
Simplified cracked specimen. The notation
means a length
equal to
(mm) and divided into
segments having the same
size. “l” indicates the loading point and “t” the crack tip. The area
labelled A
is elastic (
GPa,
). The area labelled
A
is modelled with a GTN of Rousselier material.
15
6
Results for the plane strain cracked specimens (GTN model)
16
7
Results for the plane strain cracked specimens (Rousselier model)
17
8
18
9
Results for Charpy specimens (GTN model)
19
10
Results for Charpy specimens (Rousselier model)
20
9
Table 1
Plastic hardening data
(MPa)
495.
0.0025
0.
530.
0.01
0.007323
570.
0.02
0.017121
604.
0.03
0.026949
634.
0.04
0.036798
660.
0.05
0.046666
681.
0.06
0.056560
700.
0.07
0.066464
716.
0.08
0.076383
730.
0.09
0.086313
740.
0.1
0.096262
749.
0.11
0.106217
755.
0.12
0.116187
765.
0.13
0.126136
770.
0.137
0.133111
771.
0.14
0.136106
778.
0.15
0.146071
809.
0.2
0.195914
855.
0.3
0.295682
889.
0.4
0.395510
917.
0.5
0.495369
940.
0.6
0.595253
960.
0.7
0.695152
977.
0.8
0.795066
993.
0.9
0.894985
1008.
1.0
0.9949
10
Table 2
Participants
Ecole des Mines Paris
Thick full lines
Ecole Centrale Paris
Thin full lines
EdF—MMC
Thick dashed lines
CEA—XXX
Thin dashed lines
Fig. 1. Boundary conditions for volume element tests.
: prescribed displacement. The
initial volume of the element is equal to
(mm
).
-1
0
1
2
3
4
-1
0
1
2
3
1
1
11
