Non-Associated Plastic Flow and Effects on Macroscopic Failure - - PowerPoint PPT Presentation

Vikranth Racherla

Mechanical Engineering IIT Kharagpur

Non-Associated Plastic Flow and Effects

• n Macroscopic Failure Mechanisms

by

Overview

• Background and motivation
• Basis for non-associated plastic flow
• Multi-scale nature of plastic deformation
• Generalized yield criteria for single crystals undergoing multiple slip
• Constitutive models for non-associated flow
• Uniqueness and stability of solutions to incremental BVPs
• Implication of second-order work (SOW)
• Classical rate-independent non-associated flow theory and negative

SOW

• Rate-dependent theory and effect of strain-rate sensitivity on SOW
• Effects on macroscopic deformation mechanisms
• Cavitation instabilities
• Localized necking
• Strain bursts – local instabilities
• Crack tip fields for associated and non-associated flow

BCC Metals and Intermetallic Compounds in High-Performance Applications

Nickel-Aluminum super-alloy in turbine blade applications Tungsten carbide drill bit High temperature furnace with molybdenum hot zone

• High strength and fracture toughness
• Creep resistance at elevated temperatures

nuclear technology (fusion reactors)

Single Crystal Crystal Plasticity Polycrystals Homogenization

identification of slip planes & non-glide stress components multi-slip models

Dislocation Core Atomistics

effective macroscopic behavior

Multiscale Analysis of Non- Associated Plastic Flow

• Atomistic studies of defect structures are the basis
• f models at progressively higher length-scales

which ultimately are used to study macroscopic response and, in particular, failure mechanisms.

• The strategy here is to pass only the most essential

information on to higher length scales.

Component Response Macroscopic Simulations

Single Crystal Yield Surfaces with Non-Glide Stresses

non-glide stress coefficients obtained from fits to atomistic simulations (a1, a2, a3) = (0.24, 0, 0.35) for each slip system (N=24 for BCC) specify 3 vectors, m, n, n1

1 1 2 2 3 3 cr

• +

+ + = = a a a

α α α α α α

τ τ τ τ τ τ τ

∗ ∗

≡

( ) ( )

1 1 2 3 1 1 1

= (Schmid stress) = = = = =

α α α α α α α α α α α α α α α α α

τ τ τ τ τ τ τ

⊥ ⊥

⋅ ⋅ ⋅ ⋅ × ⋅ ⋅ × ⋅ ⋅ mσ n = m σ n n m σ n n m σ n

Single crystal yield function

Ref: Qin and Bassani (1992)

p

α α

α α α α

γ γ ∂ τ ∂ = =

∑ ∑

D d σ   yield criteria: flow equation:

( ) ( )

( )

* * * * *

:

cr α α α α α α α α

τ τ γ τ τ γ γ − = − = − ≥ σ σ d      

Maximum Principle – Let σ denote the actual stress and an allowable stress:

σ 

Summing over all systems gives the convexity inequality:

* * * *

:

ij ij cr

a d

η

α α α α α α η

τ τ σ τ = + ≡ = = ∑ d σ

α η

τ

( )

( ) ( )

* * * *

: :

cr α α α α α α α

τ τ γ γ − = − ≡ − ≥

∑ ∑ σ

σ d σ σ ε      

( )

* *

;

cr α α α α

τ τ γ −

∑



p * * *

0 , ,

ij ij ij cr ij

d D d

α

α α α α α α

γ γ σ τ τ ≥ = = ≤

∑

 

QPP – minimize: subj to:

Single Crystal Plasticity with Non-Glide Stresses

Ref: Yin and Bassani (2006) Crystallographic tensors

Macroscopic Constitutive Theory: Non-Associated Flow

Kinematics – Constitutive Equation for Plastic Strain-rate

( )

p e

sym

ij ij ij ij

D L D D = = +

Velocity gradient plastic part elastic part Strain-rate

p p t

1

ij ij ij

G G D F E λ σ σ ∂ ∂ = = ∂ ∂  

flow potential yield function Plastic tangent modulus yield surface

( )

ij

F σ flow potential

( )

ij

G σ

F=G for classical associated flow behavior

Isotropic yield and flow surfaces predicted using a Taylor model of a random BCC polycrystal

σ ε

Ref: Yin and Bassani (2006)

e p e e-1 e p p-1 e-1

, = = + F F F L F F F F F F  

1

σ

2

σ b = -0.7, SD = 0.2

1

σ

2

σ

Yield and Flow Functions For Random Polycrystals

Yield function: isotropic – in terms of stress invariants

( )

1 2 2 tr

J

′ ′ ⋅

=

σ σ

( )

1 3 3 tr

J

′ ′ ′ ⋅ ⋅

=

σ σ σ

( )

1 3 tr ′ ′

= −

σ σ σ δ

A single parameter b characterizes the isotropic yield surface

( )

( )

1/3 3/ 2 2 3 1/3

3 3 3 2 F J b J b   = +   +

non-associated flow parameter

( ) ( )

t c t c

0.26

2 SD

b

σ σ σ σ

≈ −

− = +

Strength differential Yield stress in tension Yield stress in compression b is expressed in terms of strength-differential of a material

2

3 G J =

Flow Potential

Best fit to yield surface Best fit to flow surface

b 0, SD 0

~ ~ ~ ~

Ref: Racherla and Bassani (2007)

Effects of Non-Associated Flow on Cavitation Instabilities pcr

A void in an infinite matrix grows unbounded as the mean stress approaches the cavitation limit

( )

3

kk

σ

cr

P Constitutive Equations

p e t p p t e p p

• e
• e
• p

e p e

• e
• 1

, where and if if

ij ij N

G D F E E F σ σ ε σ ε ε ε ε σ ε σ ε ε ε ∂ = = ∂  ≤   = =    ≥             N is the strain hardening exponent

cr

• P

σ

( )

cr cr

P P

Local Sufficient Condition for Uniqueness and Stability in Incremental Boundary Value Problems

Governing Equations

( )

( )

σ u

in with , and

• n
• n

ij j t i m ij ijkl kl ijkl ijkl p q mn mn n i ij i t j j t

P u X v P L k X L L x N P t x x ρ ∂ = Ω ∂ ∂ = = = ∂ = Γ = Γ A A A         

Equilibrium equations Rate-independent constitutive equations Nominal traction-rates and velocity boundary conditions

Sufficient Condition for Uniqueness and Stability:

Second-order work (SOW) ij ji ij ij

L D P σ

∇

≅ > 

infinitesimal deformations:

At equilibrium, the sufficient condition ensures an increase in potential energy for any admissible perturbation in the displacement field

ij ji

P L > 

Non-Associated Flow and Negative SOW

p p t

1

ij ij ij

G G D F E λ σ σ ∂ ∂ = = ∂ ∂  

Classical Constitutive Equation:

p p p t t

1 1

ij ij ij ij

F G G D F E E σ σ σ

∇ ∇

∂ = = < ∂   

Plastic Part of Second Order Work In classical rate-independent non-associated flow second-order work can be negative even with positive hardening modulus so uniqueness and stability can be lost even at small strains

F G

ij

F σ ∂ ∂

ij

G σ ∂ ∂

1

σ

2

σ

ij

σ

∇

Stress rates in hashed region result in negative plastic part of second-order work and, therefore, can lead to negative SOW

Rate-Dependent Theory and Positive SOW

Hardening Relation

( ) ( )

p p eff e e m

g σ ε ε = 

Constitutive Equation

p G G F 2 e e

δ

ij ij kl ij k kl ij kl l

D qN t N N m m G q σ σ σ σ σ σ

∇ ∇

= +   +      ∂ ∂  ∂ 

m is the strain-rate sensitivity parameter

A corner-like term always makes a positive contribution to SOW and for a moderately large strain-rate sensitivity parameter m the SOW is always positive

F G

ij

G σ ∂ ∂

p ij

ε 

ij

σ 

2 π θ <

1

σ

2

σ

For a moderately large m, the angle between plastic strain rate and stress rate is less than

/ 2 π

Effect of Non-Associated Flow on Sheet Necking Bifurcations

A sheet is deformed under affine BCs. At bifurcation the IBVP admits two solutions: a uniform field and a field with localized deformation in a band

( )

det

i l ijkl

n n = A

If ni represents the normal to a localized band and Aijkl the incremental modulus ( ) the bifurcation condition is given by

ijkl ijkl kl

P F =   A

Critical strains at bifurcation for various loading strain ratios

The bifurcation strains are obtained using a corner theory

Localized band orientations Effect of corner coefficient

• n critical bifurcation strains

Corner theory:

2 p

• p

p t t kl kl ij ij kl ij kl

G F G D c F E E α α σ σ σ σ σ σ

∇ ∇

∂ ∂ ∂ = + ∂ ∂ ∂ ∂

Elastic-plastic transition function Corner coefficient

Effect of Non-Associated Flow on Sheet Necking Bifurcations

Critical necking strains for various strain ratios

Strain Localization: Growth of Inhomogenieties

Effect of non-associated flow

• n strain localization

Condition for sheet necking

m b

h h →   To obtain the forming limit diagram a uniform sheet with a thickness inhomogeneity in the form of groove (or band) is deformed at a fixed strain ratio until the sheet necks

Configuration

11

σ σ

11

ε

Load fluctuations N = 0.05, m = 0.0002 (nearly rate independent)

11 22

/ 0.2 ε ε = − Strain bursts from non- associated flow

m 11

ε

b 11

ε

( )

1/ e p

• e

p e

; ;

m N ij ij

G D F k σ φ φ φ σ σ ε   ∂   = = =   ∂       

Finite Element Analysis of Sheet Necking

Constitutive Equations

Specimen configuration in finite element analysis

F G

ij

σ

ij

σ  (

)

e

σ ≥ 

F ij

N

G ij

N

• 90

β ≥

p ij ij

σ ε ⇒ ≤  

bursts tend to

• ccur when

the angle between the flow surface normal and stress-rate exceeds 90o

11

ε

Second-order work

ij ij D

σ

∇

Correlation of Second-Order Work with Strain Bursts in the Band

b 11

ε

11

ε

22 11 b

σ σ   ′     ′  

11

ε

0.02 0.04 0.06 0.08 0.1 0.12 0.05 0.1 0.15 0.2 0.25

p b

ε

m = 0.0002

p m

ε

Non-associated flow Associated flow m = 0.002 m = 0.01 N = 0.1, ξ = 0.05, ρ = -0.2

Strain bursts are suppressed for large strain-rate sensitivity in agreement with M-K analyses

Effect of Strain-Rate Sensitivity m on Strain Bursts

Rate-Dependent Theory with Gradient Hardening

p e

;

ij ij

G D F φ σ σ ∂ = = ∂ 

( )

1/ e

• p

p y e e

; ,

m

σ φ φ σ ε ε     =       ∇   Constitutive relations with gradient hardening

( )

1 2 y p p 2

• e

e p e

d 1 1 d

N

k N l σ ε ε ε

− 

   = + + ∇        

strain gradient effect

Gradient effects are widely employed to capture size-scale dependent phenomenon in plastic flow (e.g., Fleck and Hutchinson, 1997; Bassani, 2001) and to regularize solutions in problems involving localization of deformation in the analysis to follow

2 l =

Mesh Sensitivity: Comparison of Sheet Necking Predictions for Associated and Non-Associated Flow

Evolution of the necking with strain gradient effects

0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25

p m

ε

p b

ε

1440 Elements 720 Elements 360 Elements 2880 Elements

0.02 0.04 0.06 0.08 0.1 0.05 0.1 0.15 0.2 0.25

1440 Elements 360 Elements 720 Elements 2880 Elements

Evolution of the necking without strain gradient effects

Non-associated flow Associated flow N = 0.1, ξ = 0.05,

11 22

/ 0.2 ε ε = −

p m

ε

p b

ε

5760 Elements

Ref: Morris Azrin and Walter A. Backofen, 1970 Experimental setup and test procedure

• Milling and abrasion treatment are used to get an

elongated patch at the center of a 6 square inch sheet with a thickness variation of less than 0.1 %

• Square grid of 100 lines per in., aligned with axis of

reduced section is photographically printed on the un- machined surface

• Polyethylene spacer is used to prevent the punch

from contacting the reduced section; the reduced patch remains flat as it deforms

• Strains are evaluated for each initial 0.01 guage

length and plotted at the initial location of the element

• Strain-ratio in the reduced section is controlled

through patch length to width ratio

• Errors in measurement of strains, are less than 0.5%

Experimental Analysis of Sheet Necking (Azrin and Backofen)

Ref: Morris Azrin and Walter A. Backofen, 1970

Results from Experiments on 70/30 Brass at a Strain- Ratio of 0.19 (Azrin and Backofen)

Important observations

• Fluctuations in are much greater

than the measurement error; for e.g. maximum fluctuation in for curve 2 is 3.5% and for curve 5 is 8.5%

• Strains are uniform along the width of

the patch

• Strain field has a wave like character
• The fluctuations in strain grow with

the magnitude of average strain

1

ε

1

ε

Effect of Non-Associated Flow

• n Crack Tip Fields

Elastic crack displacements for a semi-infinite crack loaded in Mode-I under plane strain conditions are imposed on the outer surface

( )

r R =

The J-Integral is nearly path independent even for non- associated flow

1 p e e p e

• ;

;

n m ij ij

G D F σ ε φ φ σ σ ε σ φ     ∂ = = =       ∂       

Constitutive Equations

( )

I

2

i i

K u f r θ π =

Effect of Non-Associated Flow on Crack Tip Fields n = 10

I

5 K K a σ = = 

SD = 0.2

Effect of strength differential on plastic zones which are defined as regions within which for

Displacement BCs

I 2

K cos 1 2sin , 1,2 2 2 2 2

i

r u k i θ θ µ π       = ± =             

p 3 e

10 ε

−

≥

5 K = 

Effect of Non-Associated Flow on Crack Tip Fields

The crack tip fields are still of HRR type, but the amplitudes of crack tip fields are slightly affected by non-associated flow

Angular variation of for associated and non-associated flow

θθ

σ

Pressure distribution ahead

• f the crack

• Positive second-order work guarantees unique and stable solutions to IBVPs

in the infinitesimal theory

• Classical rate-independent non-associated flow can lose uniqueness and

stability even at small strains

• Rate-dependent theory and a corner theory can guarantee unique and

stable solutions even for non-associated flow at small strains

• Non-associated flow significantly affects critical pressures at which

Cavitation instabilities occur

• Sheet necking is considerably affected by non-associated flow. Not only are

the critical strains affected but also the localized band orientations

• “Strain burst” instabilities occur in sheet necking with rate-independent non-

associated flow and the bursts can be correlated to the negative SOW

• The crack tip-fields under plane strain conditions are still of HRR-type but

the amplitude of the fields is slightly affected by non-associated flow

Conclusions