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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