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