Benchmark for finite element analysis of rupture in the ...

Description

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


































...

Sujets

• Displacement mapping
• Échantillon
• Rupture Tito-Staline
• Traction au point fixe
• Gforge
• Fracture temporelle

Informations

• Publié par: Phoem
• Nombre de lectures: 133
• Langue: English

Extrait

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