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Multi-scale analysis based study of effect of micro-level damage on

macro-level thermo-mechanical response of ductile material

Sriram Ganesan , Dr. C.S. UpadhyayDepartment of Aerospace Engineering, IIT Kanpur, Kanpur-208 016

Abstract Material models for porous materials

Discussion and Further Scope

Microstructure of 9 Cr-1Mo Steel

Acknowledgements

Micromechanically based models are based on the physical

understanding that in the course of plastic deformation, microvoids

nucleate and grow until a localised plastic necking or fracture of the

inter-void matrix occurs, which causes coalescing of neighboring voids.

The model was proposed by Gurson [3] and phenomenologically

extended by Tvergaard and Needleman[4] called the GTN model. An

approach based on continuum damage mechanics(CDM) and

thermodynamics has been proposed by Rousselier [5].

The plastic flow potential is written as

Where R is the yield stress of undamaged material(matrix) p is an

effective plastic strain representative of matrix hardening, σ* is an

effective scalar stress which is a function of both macroscopic stress

tensor and porosity. σ* is defined by the following equations:

Gurson Model:

Rousselier Model:

Where is the von Mises equivalent stress and σkk is the trace(tr)

of the stress tensor. q1, q2, q3,D and are material coefficients which are

assumed to be constant. f* is a function of the porosity f which was

introduced on a purely phenomenological basis to represent void

coalescence. ([4]).

Deformation Rule:

Obtained assuming normality rule

Where v_ is the normal to the flow potential. Total strain is expressed as

εe+ εp, where εp is the elastic strain rate.

In case of Gurson model

In case of Rousselier Model

Where e. is the deviator of the strain rate tensor. In this case p.

corresponds to the von Mises equivalent strain rate. p. can be

computed writing the consistency condition φ.=0 in case of plasticity.

Evolution of the porosity is given by mass conservation modified to

account for strain controlled void nucleation.[6]

•Porous Plasticity could not be incorporated into Plane Stress

model and so the material under elastic regime is shown in Fig 6.

•A separate extruded 3-d model having two interfering voids is

modelled and analysed as shown in Fig 7.

•The model can be expanded to include process inducedr

esidual stresses, vector damage potential and

micromechanics based evolution laws.

Undamaged Sample before Etching

The Sample was damaged by conducting a Tensile Testing in a 20

ton Universal Testing Machine

Testing Parameters:

Specimen Type: Flat

Test Mode : Stroke ( Displacement Controlled)

Gauge Length : 30mm

Displacement Rate : 1mm/min

Cross Sectional Area of Sample : 58.5239 sq-mm

Peak Stress : 404.1 kN/sq-mm

Yield Point Load: 23.1551kN

Constitutive descriptions for deterioration of material strength

capacity due to separation or rupture of material have been the

focus of numerous investigations in the field of continuum

damage mechanics. Presently, empirical scalar-based damage

descriptions are used in practical numerical simulations of impact

and failure. For ductile metals, the model introduces a damage

parameter—the cumulative scalar plastic strain at failure—

whose instantaneous value may depend upon the strain rate,

temperature, deviatoric stress, and/or hydrostatic pressure.

The broader aim of the investigation is to develop a framework

for describing the deformation and failure responses of ductile

materials from micromechanical considerations and volume

averaging techniques in a computationally efficient way and for

a broad range of load conditions. Two dimensional

micromechanical model based simulations were conducted to

study influence of interacting flows.

Also the microscopic structure of 9 Cr-1 Mo Steel, which is being

used in Nuclear applications was studied both in undamaged and

damaged state.

Cr-Mo ferritic steels are widely used as the structural material of

the steam pipes in fossil and nuclear power plants because of

their excellent strength as well as their good oxidation

resistance at a high temperature. The mechanical strength of

9Cr-1Mo steel at an elevated temperature can be improved by V

and Nb additions and it is called a modified 9Cr-1Mo steel.

Modified 9Cr-1Mo steel is being considered as a structural

material for the reactor pressure vessel and piping in advanced

reactors such as VHTR(Very High Temperature Reactor) and

SFR(Sodium Fast Reactor). The interest on the weld

characteristics of modified 9Cr-1Mo steel is increasing due to its

potentiality for application in all modern power plants and

advanced reactor system[1]

Properties of 9Cr- 1Mo Steel

Young’s modulus: 200GPaWork hardening rate 300MPa

Poisson’s Ratio 0.3 Specific heat 586 J/(kg°C)

Thermal expansion coefficient

1.2×10−5 per °C Conductivity 52 J/(m-s-°C)

Initial static yield stress 700MPa

Initial relative density 0.95 ( 0.05)

Size it will be on the PosterSize of original

.9 Comparison of Gurson and Rousselier Models

Fig compares both the yield surfaces in σeq-σkk plane in the

case of tensile stress (σkk >0). Under pure shear (σkk =0),

damage is still generated in the case of Rousselier model ( as the

normal to the yield surface does not coincide with the axis),

whereas in the absence of nucleation, the Gurson model does

not lead to damage growth.

Modelling porous plasticity using Simulia Abaqus 6.9

Elastic Model:

Porous Plasticity model:

Tensile Testing of the sample

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

Str

ess, kN

/sq

-mm

Strain

Damaged Sample Microstructure

Observations and Conclusions

•Since etching was not done, a secondary cleaning with an

ultrasonic cleaner and an appropriate surfactant as acetone

was done for removing inaccessible alumina located in

specimen cracks or pores (Fig. 1c).

•Since the specimen was small tensile testing could not be

carried out in accordance with ASTM specifications.

• The inclusions seen are the M23C6 precipitates.

•The crevices seen in the surface are due to a combination of

voids and polishing defects.

I would like to sincerely thank Prof. Sandeep Sangal, MME Department, for

the invaluable suggestions and inputs he had given me on Metallography. I

would also like to sincerely thank Mr. Jitender, Mr. Hari Om Bajpai, Mr.

Pravin Kumar Sharma, Mr. Nitin Awasthi, Mr. Shishir Pandya and Ms.

Aparna Mitra for their help during the entire duration of the project.

References

[1] 1. A. Barnes, ‘‘The Influence of Composition on Microstructural Development and Toughness of Modified

9%Cr-1%Mo Weld Metals,’’ Report 509/1995, TWI, Abington, UK, 1995

[2] B. Arivazhagan, Ranganath Prabhu, S.K. Albert, M. Kamaraj, and S. SundaresanMicrostructure and

Mechanical Properties of 9Cr-1Mo Steel Weld Fusion Zones as a Function of Weld Metal Composition

[3]Gurson.A ,1977. Continuum theory of ductile rupture by void nucleation and growth. Part-I Yield Criteria

and flow rules for porous ductile media. J. EnggMatter Tech. 99,2-15

[4]Tvergaard V, Needleman A, 1984. Analysis of cup cone fracture in a round tensile bar. Acta Metall157-69

[5]Rousselier G,1987, Ductile fracture models and their potential in local approach of fracture. Nuclaer engg

Des. 105, 97-111

[6]Chu C, Nedleman 1980 Void nucleation effects in biaxially stretched sheets, J. Engg Matter102,249-256

Chemical composition in weight %: 0.10% C, 0.49%

Si, 0.46% Mn, 0.085% P, 8.36% Cr, 0.97% Mo.

Table 1 Physical and Mechanical Properties of 9Cr-1Mo Steel

Fig 2 a,b) Undamaged unetched sample c) Alumina Impurities

Fig 2 a) Stress Strain Curve b) Universal Testing Machine

Fig 3 a,b,c) Damaged Unetched Sample

Fig 4 a,b) Damaged Etched sample c) 9Cr-1Mo Steel [2]

Fig 5 : Yield Surfaces for Gurson and Rousselier Model

Fig 6 a) Loading b) Stress Distribution c) Inclusion

Fig 7 a) Loading b) Stress Distribution c) Interference of Voids