Elastoplastic Analysis of Polycrystalline Materials
-
Upload
aravind-kumar -
Category
Documents
-
view
223 -
download
0
Transcript of Elastoplastic Analysis of Polycrystalline Materials
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
1/17
Elastoplastic Analysis
of polycrystallinematerials
Under guidance of Prof. Biswanath Banerjee
By
Aravind Kumar Dammu
(09C!"0!#
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
2/17
Introduction
• Crysta$ %$asticity is the study of e$asto%$astic
&ehavior of meta$s at meso $ength sca$es.
• Understanding the microstructura$ features a$$ows
for &etter %rediction of macrosco%ic &ehavior.
• 'ode$ing at these $ength sca$es ena&$es us to
incor%orate anisotro%y eective$y &y s%ecifying the
distri&ution and orientation of grains.
•
)he deformations are com%ara&$e in magnitude tothe origina$ geometry.
• )he $inear stress*strain re$ationshi%s and strain
dis%$acement re$ationshi%s are not va$id.
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
3/17
Objectives
• Crysta$ %$asticity
• +ing$e crysta$ mode$
• ,ntegrate sing$e crysta$ mode$ to simu$ate
%o$ycrysta$ mode$
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
4/17
Large Deformation
• +ince undeformed and deformed geometries aredierent- the engineering strain cannot &e used for$arge deformation.
• +train is e%ressed using u$erian and /agrangian
strain measures.
• )hey are e%ressed in terms of deformationgradient.
• )he stresses are e%ressed in terms of cauchy
stress and %io$a*irchho stress.• ,n crysta$ %$asticity- it is assumed that the
deformation taes %$ace in two consecutive stages.
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
5/17
Large Deformation
• )he tota$ deformation gradient is given &y%roduct of %$astic deformation gradient ande$astic deformation gradient.
• )he ve$ocity gradient can &e e%ressed as thesum of %$astic ve$ocity gradient and e$asticve$ocity gradient.
1ig. "
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
6/17
Slip Representation
• P$astic s$i% occurs in a %$ane when the reso$ved shearstress reaches a critica$ va$ue.
• )he num&er of active s$i% systems de%ends on the stressstate- the crysta$ structure- the hardening mechanisms-
and the hardening history of the s$i% systems.
i. 1CC Crysta$ ii. BCC crysta$ iii.
2CP crysta$1ig. ii 3 1ig. iii
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
7/17
Slip Representation
• Dis$ocations which are initia$$y random distri&uted-%refer $ow energy %athways under $oading conditions.
• )hese distur&ances in their %ositions cause strainhardening.
• +$i% in a %$ane in4uences s$i% in every other %$ane. )his %henomenon is nown as $atent hardening.
• Under $ow strain rate and isotherma$ conditions- the%$astic 4ow can &e re%resented using %ower $aw.
5555555555...("#
•
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
8/17
Slip Representation
i. Dis$ocation +$i% ii. )winning iii. 7rain
Boundary +$iding
P$astic s$i% is main$y c$assi8ed into ! categories. ,n this %roject- on$ydis$ocation s$i% is considered.
1ig. iv
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
9/17
NumericalImplementation
• )he reso$ved shear stress is ca$cu$ated from thesecond %io$a*irchho stress using euation(:#.
5555555..(:#• After the reso$ved shear stress is ca$cu$ated- the
shear strain rate is ca$cu$ated on each s$i% %$aneusing %ower $aw shown in euation("#.
•
, have considered two s$i% %$anes to &e active andon$y se$f hardening is considered.
•
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
10/17
Evolution Equations
• )he evo$ution euations for deformationgradient in the intermediate con8gurationand the hardness are given &y euation
(!# and euation (;# res%ective$y. 5555555(!#
• uation(;# is nown as
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
11/17
Discretied Equations
• )he evo$ution euation for deformation gradientin the intermediate con8guration is discreti=edusing Bacward u$er method as shown inuation(>#.
……………...(5)
• )he evo$ution euation for hardness isdiscreti=ed using forward eu$er method as shownin euation (?#.
55.5..(?#
•
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
12/17
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
13/17
Algorit!m
1ig. v
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
14/17
Results
• )he deformation tensor was found to&e
•
• )he stretch tensor and the rotation matri arefound &y %o$ar decom%osition of deformation
gradient.• )he eu$er ang$es are ca$cu$ated from the
rotation matri and the %rinci%a$ stretches areca$cu$ated from the stretch tensor.
• )he deformed geometry was deve$o%ed using
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
15/17
Results
1ig. vi
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
16/17
"uture #or$
1ig. vii
)he features to &e considered in %o$ycrysta$mode$ing are shown in 8g vii.
-
8/19/2019 Elastoplastic Analysis of Polycrystalline Materials
17/17
%!an$ &ou