Metz HSM 2010 Lecture

8/8/2019 Metz HSM 2010 Lecture

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HSM 2010

Metz, December 8-10, 2010

Modelling of high strain rate behaviour of

metals accounting for micro-shear banding

Ryszard B. PCHERSKI

Institute of Fundamental Technological Research

Polish Academy of Sciences, Warsaw, Poland


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Outline

1. Shear banding:

- one of the mechanisms of plastic flow in metallic

materials (also in the case of HSM processes, e.g.:

A. Molinari, C. Musquar and G. Sutter, [2002], P. Chevrier et al.)

- the dominant mechanism of plastic deformation ofufg, nano-metals and glassy metals.

2. Constitutive relations.

3. Identification of the viscoplasticity model forufg and nano-crystalline Fe.

4. Effects of strength differential and Lode angle

5. Concluding remarks.


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Observations of shear banding - ufg Fe

d = 980 nm d = 268 nm

quasistatic high-strain rate deformations

Change in deformation mode of ultrafine grained consolidated iron under

uniaxial compression: (a) uniform low-rate deformation with d = 980 nm; (b)

non-uniform low-rate deformation with d = 268 nm and (c) non-uniform high-rate

deformation with d = 268 nm (Jia, Ramesh and Ma [2003]) .

d = 980 nm d = 268 nm

crystallographic slip shear banding

d = 268 nm


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Experimental results of Jia, Ramesh and Ma,Acta Materialia, 51 (2003)

Deformation of nano - and ufg metals

SBf - shear bandingcontribution

function

Z. Nowak, P. Perzyna

R.B. Pecherski,Archives of Metallurgy and

Materials 52, 2007 freepdf available on the Journal

website.

Typical stress-strain curves

obtained for the consolidated

iron under quasi-static and

high-strain-rate uni-axial

compression.


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Trace of

the cluster

of MSB

CSB

[Dziado, 1993]

(courtesy of profesor Andrzej Korbel)

Multiscale hierarchy of shear bands in

polycrystalline metals

Micro-shear band - longand very thin (ca. 0.1 )sheet-like region ofconcentrated andintensive plastic shearcrossing grain boundarieswithout deviation

Particular MSB operatesonly once and develops

fully in very short time

m


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Literature

1. R.B. Pcherski, Modelling of large plasticdeformations based on the mechanism of micro-shearbanding. Physical foundations and theoreticaldescription, Arch. Mech. (1992).

2. R.B. Pcherski, Macroscopic measure of the rate ofdeformation produced by micro-shear banding, Arch.Mech. (1997).

3. R.B. Pcherski, Macroscopic effects of micro-shear

banding in plasticity of metals, Acta Mechanica (1998)


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Multiscale hierarchy of shear bands

in polycrystalline metals

Schematic illustrationof multi-level hierarchy

of micro-shear bands:

a) polycrystalline RVE

with the increasingzone of shear banding,

b) cluster of active

micro-shear bands,

c) a single micro-shear

band (comp. of CSB)

R.B. Pcherski, Macroscopic measure of the rate of deformationproduced by micro-shear banding, Arch. Mech. [1997]


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Macroscopic measure of the rate of deformationaccounting for shear banding

R.B. Pcherski, Arch. Mech. 49, (1997)

, 2

p p p

S SB

p

S

SB

p

SB

p pSB SB

SB

rateof plastic deformation by slip

rate of plastic deformation by shear banding

d d d shear strain ra

d f instantaneous shear banding contribution

d

te

D D D

D

D

D D

(1 )

S SB

SB s

f

f

SBSBf

0


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Account for the change of deformation path (Lode angle )

K. Kowalczyk Gajewska, R.B. Pcherski (2005)

( ( ))1

oMS

ffa b

e

p

p

pp

D

Ddet

2

36)cos(

))cos(1()(


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V SBSB

Vf

V

RepresentativeVolume Element

traversed by shear

bands

V

VSB

VsViscoplastic flow law

accounting for shear

banding in application for

ufg metals(Z. Nowak, P. Perzyna, R.B.

Pecherski, Arch. Metall.

Materials, 2007)

S SB

s SBV V V

SBSBf

Volume fraction

of shear banding

Inst. contr.

of shear

banding


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,

, ( ),

(1 )

yield stren

(

gth at sh

(1 )(

ea

1 ) ,

r

) 01

s SB

s s s SB SB SB SB SB

V SB s SB V SB SB V SB

V

s S S SBB B

P P P

P k V P k V V P k V

Vk k f f k f f

k k

k

f

V

for kf

Balance of plastic deformation power in RVE

assumption - no hardening

0


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Viscoplastic flow law accounting for shear banding

12

0

2 2

0

2

(1 )(1 )

(1 ) , 0


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Viscoplasticity model accounting

for SDE

1

2 2 2

,

( )

1 3( 1) 9( 1) 4

2

p

G G

T

Y

m m e

G G

G F

F

D

=

G


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Identification procedure:

if d > 300 nm ( , , , )

if d < 300 nm ( , , , )

cal

eq

0SB

f 0 2n 3 1

3 0x10vp s

0 08D

0 1SBf 0B 5 0a 0 0 95SBf

0( ) ( )( ) 1 ( )

vpvp n

caleq A d B d d

D

0

0

0

121 exp( ( ) )

1 ( ) 1( )1 exp( ( ) )

SB vpvp

cal SB

vpeq

f

a b dfA d

da b d

D


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Grain diameter

d

A(d) [MPa]

A*

(d) [MPa]

b (d) B(d)[ MPa ]

80 nm 1205.0 98.28 0.0

138 nm 1120.0 94.95 0.0

268 nm 710.0 28.59 0.0

980 nm 106.78 ---- 602.25

20m 100.28 ---- 269.41

Identified parametersDynamic compression


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S ( )


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2 22 2

0 0

' '23

39 3 0

3

13

0 1

,

,

:

T C T C

C T T C Y Y Y Y

e m Y Y m Y Y

m etr

f m v cr + =

W. BURZYSKI (1900-1970) Study on Material Effort Hypotheses,1928 , PhD Thesis (in Pol.)

Ueber die Anstrengungshypothesen, Schweiz. Bauzeitung,1929.

Theoretical foundations of the hypotheses of material effort,

Engineering Transactions, 56, No. 3, 269305, 2008 the English translation of the paper published in Polish

in Czasopismo Techniczne, 47, 1929, 1-41, Lww.

A new, well founded physically, energy-based criterion was proposed:

f

v

- density of elastic energy of distortion

- density of elastic energy of volume change

2 23

Y Y

T C

Y Y Y

Huber Mises Hencky

condition

1

0

Beltrami

Huber


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3 T CY Y Y

The depiction of particular cases ofBurzyski yield condition inthe pressure - equivalent stress coordinates

(W. Burzyski [1928], M. yczkowski [1999]).

3 T CY Y Y

23 T CY Y Y

,e m ,e m

Paraboloid limit surfaces according to Burzyski criterion


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Paraboloid limit surfaces according to Burzyski criterion(T. Fr, Z. Kowalewski, R. B. Pcherski, A. Rusinek

Engineering Transactions [2010])

75%Cr - 25% Al2O3 composite 6061+2Zr+20Al2O3 composite

3


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Comparison of the formulations proposed by

F. Schleicher [1926] and W. Burzyski [1928]

According to W.B.

According to F.S. Poisson ratio


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Yield condition accounting for SDE and Lode angle(R.B. Pcherski, P. Szeptyski, M. Nowak [2010])

W. Burzyski [1928](cf. English translation in

Engineering Transactions, 2010

available on internet)

f

v

- density of elastic energy of distortion

- density of elastic energy of volume change


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Huber-Mises-Hencky cylinder

Burzynski-Drucker-Prager cone

Burzynski-Torreparaboloid

ellipse

m

e

YC

T

Y3

C

Y3

YT

The Burzyski yield condition and its particular cases:

ellipsoid, paraboloid, cone and cylinder.

SDE

SDE Strength Differential EffectC

Y

T

Y

k


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Examples of yield surfaces accounting to the

effect of Lode angle


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Examples how to account for Lode angle

Raniecki&Mrz [2009]


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Our identification for INCONEL 718

Empirical relaton:

Energy-based criterion


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Concluding remarks1. The plasticity flow law accounting for shear

banding was applied for modelling evolution oftexture and metal forming processes (K. Kowalczyk-Gajewska, Z. Mrz, R.B. Pcherski [2005], [2007], [2009]).

2. Viscoplasticity constitutive relations were

developed for modelling the behaviour of ufg andnc-metals (Z. Nowak, P. Perzyna, R.B. Pcherski [2007])

3. The effect ofLode angle was described in the

shear banding contribution in the rate ofdeformation accounting for the influence of the

change of deformation path and also in the

formulation of yield criterion.


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Thank you for your attention

Metz HSM 2010 Lecture

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