Presentation Workshop Jr Klepaczko 2009 Effect Of Strain Rate On Ductile Fracture

In memory of our Friend and Colleague J.R. Klepaczko WORKSHOP Dynamic Behaviour of Materials ENIM, Metz, May 13-15, 2009

EFFECT OF STRAIN RATE ON DUCTILE FRACTURE. A NEW METHODOLOGY

R.B. Pęcherski & W.K. Nowacki, Z. Nowak, P. Perzyna Institute of Fundamental Technological Research Polish Academy of Sciences, Warsaw

Introduction
Experimental tests – global response
Microstructural observations
Modelling and numerical calculations Cooperation with: Alexis Rusinek – ENIM Metz Ramon Zaera Jose A. Rodrigues-Martinez University Carlos III of Madrid

MOTIVATION - investigations of DH-36 steel leading to the new methodology of ductile fracture analysis

Microstructure in the strongly deformed shear zone volume 800

DH-36 steel compression

.

-2 -1

γ = 5.77*10 s

Evolution of temperature in the strongly deformed region

comp.

True stress (MPa)

eq. simple shear

600

400

200

0

The double shear testing set 0.0 0.2 0.4 True strain constructed in the Division The SD effect of Applied Plasticity IPPT Warsaw (W.K. Nowacki), σ YT ≠ σ YC G. Gary, W.K. Nowacki [1994]

0.6

W.K. Nowacki et al., Report [2006]

Examples of experimental investigations of DH-36 steel

∆Τ

200

2

100

1

0

0 0.0

0.4

0.8

1.2

Shear strain

600

3 -1

Shear stress (MPa)

γ = 6.3*10 s

400

11

5

2

.

-3 -1

.

0 -1

.

DH-36 steel

20

0 -1

γ = 10 s

σ

300

15 ∆Τ

200

10

100

5

0

0 0.0

0.5

1.0

1.5

Shear strain

Stress-strain and mean temperature curves at the strain rates: 10-1 and 1 s-1.

DH-36 steel .

Shear stress (MPa)

3

400

Temperature variation (K)

4

-1 -1

σ

300

Shear stress (MPa)

.

γ = 10 s

DH-36 steel

Temperature variation (K)

400

2 - γ. = 10 s -1 -1 5 - γ = 10 s

200

11 - γ = 10 s

Comparison of stress-strain curves obtained in quasi-static and dynamic tests

0 0.0

0.4

0.8

Shear strain

1.2

Outline of the methodology 1. Experimental tests under different loading rates with: - in-situ thermographic observations, - microstructural observations.

1. Numerical simulations of the process: - to identify the model, - to simulate the test until fracture initiation.

3. Evaluation of the local equivalent strain in the mostly strained region. 4. Analysis of the effects of strain rate and stress triaxiality.

Presentation of theoretical foundations In the viscoplasticity model the following effects should be accounted: • the strength differential effect, • the effects of shear banding.

SD effect - Burzyński yield condition • W. Burzyński, Study on Material Effort Hypotheses,1928 , PhD Thesis (in Polish)

• W. Burzyński, Ueber die Anstrengungshypothesen, Schweiz. Bauzeitung,1929. • W. Burzyński, Theoretical foundations of the hypotheses of material effort, Engineering Transactions, 56, No. 3, 269–305, 2008 – the English translation of the paper published in Polish, 1929, 1-41, Lwów.

An energy-based yield criterion:

η = 1 − B e ltr a m i

Φ f + η ( σ m ) Φ v = Φ cr ,

0 ≤η ≤1 η = 0 − H uber T C σY ≠ σY T C T C σYσY 2  3σ Y σ Y  2 C T T C σe + 9 − σ m − 3 (σ Y − σ Y ) σ m − σ Y σ Y = 0  2 2 τ0  3τ 0  2 2 3τ Y = σ Y , τ Y ≡ τ 0

σ m = 1 tr σ , σ e = 3

2 3

'

σ :σ

'

Φf - density of elastic energy of distortion

σ YT = σ YC = σ Y Huber − Mises − Hencky condition

Φ v - density of elastic energy of volume change R.M. Christensen [2007]

τY 3 = σ σ T Y

τY 3 = σ = σ T Y

C Y

C Y

(σ e , σ m )

3τ Y 2 > σ YT σ YC The depiction of particular cases of Burzyński yield condition in the plane with (pressure, equivalent stress) axes

(W. Burzyński [1928], M. śyczkowski [1999]).

Plastic potential with the effects of SD and micro-damage

{

}

1 ˆ e2 − σ YT G (σ ) = 3(κˆ − 1)σ m + 9(κˆ − 1) 2 σ m2 + 4κσ 2κˆ

σ YC A 1/3 = κˆ = κ ( λξ ) , λξ = α ξ , κˆ = + 1, λξ ∈ [λ0 , ∞) T σY λξ κˆ − the strength differential parameter λξ − a mean distance between voids ξ − void volume fraction (micro-damage parameter) A − constant to be calibrated

The depiction of Burzyński yield condition (T. Frąś, R.B. Pęcherski [2009]) The section of the Burzyński paraboloid yield condition

Coulomb-Mohr condition

C.A. Schuh, A.C. Lund (MIT), Atomistic basis for the plastic yield criterion of metallic glass, Nature Materials, 2, 449-452, 2003.

Multiscale hierarchy of shear banding in polycrystals CSB

msb – micro-shear bands CSB – coarse slip band

msb Trace of msb cluster

msb clusters shear bands [Dziadoń, 1993] (The micrograph provided by A. Korbel)

S. Nemat-Nasser, W.-G. Guo [2003]

Traces of micro-shear band clusters

Microstructure of a sample strained to γ = 74% at 800K and 0.001/s - DH-36 steel.

S. Nemat-Nasser, W.-G. Guo [2003]

Traces of micro-shear band clusters

Microstructure of a sample strained to γ = 59% at 77K and 3000/s - DH-36 steel.

Macroscopic measure of the rate of deformation accounting for shear banding R.B. Pęcherski, Arch. Mech. 49, (1997) p p p D = D S + D SB D Sp − rate of plastic deformation by slip p D SB − rate of plastic deformation by shear banding p d = D p , d SB = D SB

f SB

γɺ = 2d − shear strain ra rate te

d SB = − instantaneous shear banding contribution d

γɺ = γɺS + f SBγɺ γɺ (1 − f SB ) = γɺs

γɺSB f SB = γɺ

Instantaneous shear banding contribution function f SB identification with use of channel die test of ANAND et al. (1994).

f SB =

0 SB

1+ e

f (a − b ε 33 )

1. R.B. Pęcherski, „Continuum mechanics description of plastic flow produced by micro-shear bands”, Technische Mechanik (1998) 2. Z. Nowak, R.B. Pecherski, Plastic strain in metals ... II. Numerical identification and verification of plastic flow law, Arch. Mech. (2002)

Influence of the change of deformation path K. Kowalczyk – Gajewska et al., Archives of Metallurgy and Materials, 50,575-593, (2005) 0 SB

f fSB = (a −bχ(ε )) 1+ e

χ (ε p ) =

3 2

ε eqp (1 − α cos(3θ ) ), 0 ≤ α ≤ 1

 Dp cos(3θ ) = 3 6 det  p  D 

 p  , εɺeq =  

2 3

Dp : Dp

Vs V

VSB Representative Volume Element traversed by shear bands Inst. contr. of shear banding

f SB

γɺSB = γɺ

Viscoplastic flow law accounting for shear banding in application for ufg metals (Z. Nowak, P. Perzyna, R.B. Pecherski, Arch. Metall. Materials, 2007)

γɺ = γɺS + γɺSB V = Vs + VSB Volume fraction of shear banding

V SB f SB = V V

Balance of plastic deformation power in RVE assumption - no hardening

P = Ps + PSB , P = kγɺV , Ps = k sγɺ s (V − VSB ), PSB = k SBγɺ SBVSB kγɺ = k sγɺ (1 − f SB )(1 − fV ) + k SBγɺ SB fV , k = k s (1 − f SB )(1 − f ) V SB

instantaneous

volumetric

for k SB ≈ 0

f

V SB

VSB = V

Viscoplastic flow law accounting for shear banding Z.Nowak, P.Perzyna, R.B.Pęcherski, Arch. Metall. Mat.,[2007]

fSB - inst. shear banding

γɺSB γɺ (1 − f SB ) = γɺS , 0 ≤ f SB <1, f SB = γɺ

contribution function

shear strain rate γɺS - by dislocation

 F (σ)  − 1〉 γɺs = γɺo 〈Φ   k   F (σ)  γɺ0 〈Φ  − 1〉 γɺ = V (1 − f SB )  k s (1 − f SB )(1 − f SB )  t    0  

k s = K (∈p ,ϑ , ξ ) , ∈p = ∫

{

controlled slip

Φ - excess stress

1

2 p D :D 3

2 p   

dt ,

1 ˆ e2 F (σ ) = 3(κˆ − 1)σ m + 9(κˆ − 1)2 σ m2 + 4κσ 2κˆ

}

γɺ0

function in Perzyna viscoplasticity model [1971]

- viscosity parameter

Viscoplasticity model accounting for micro-damage

∂G ∂G µG = ∂σ ∂σ

D = γɺµG , p

µG

−1

G(σ) = F ( σ ) − σYT

ξɺ =

F ( σ)   g (ξ ,ϑ ) 〈Φ  − 1〉 V (1 − f SB )  ktr (1 − f SB )(1 − f SB ) 

γɺ0

ktr = Ktr (∈p ,ϑ,ξ ) − the void growht threshold stress

κˆ =

A

λξ

+ 1, λξ ∈[λ0 , ∞), λξ = αξ 1/3 , σYT = 3ks (1 − f SB )(1 − f SBV )

{

1 = ˆ e2 F ( σ) = 3(κˆ − 1)σ m + 9(κˆ − 1)2σ m2 + 4κσ 2κˆ

}

Conclusions The derived relations make a basis for the analysis of ductile fracture
Experimental tests – global response
Microstructural observations
Modelling and numerical calculations

Thank you for your attention

S. Nemat-Nasser, W.-G. Guo [2003]

pearlite ferrite

DH-36 plate; microstructure in the rolling direction of the plate.

S. Nemat-Nasser, W.-G. Guo [2003]

pearlite ferrite

DH-36 plate; microstructure transverse to the rolling direction of the plate.