Extraction of Cohesive Properties of ElastoPlastic material using Inverse Analysis

07/18/2009 10th U.S. National Congress on Computational Mechanics, Columbus, Ohio©2009 Board of Trustees of the University of Illinois

Extraction of Cohesive Properties of Elasto-Plastic material using Inverse

Analysis

Arun Lal Gain, Jay Carroll, Glaucio H. Paulino, John Lambros

University of Illinois at Urbana Champaign

07/18/2009

10th U.S. National Congress on Computational Mechanics

07/18/2009 10th U.S. National Congress on Computational Mechanics, Columbus, Ohio

Contents

Introduction

– Cohesive Zone Modeling

– Elasto-Plastic Forward v/s Inverse Problem

Modeling Approaches

– Shape Regularization

– PPR model

Numerical Simulations

Summary & Conclusions

Future Work

2


Introduction: Cohesive Zone Model

Cohesive Zone Modeling - fracture seen as phenomenon

of gradual separation taking place across cohesive zone

(path of crack) and resisted by cohesive tractions

Four staged failure

– Stage 1: Material homogeneous

– Stage 2: Crack Initiation

• Criterion: Stress reaching tensile

strength (simplified)

– Stage 3: Crack propagation based on

traction v/s separation curve

– Stage 4: Complete failure

• Criterion: e.g. Crack width reaches

certain predefined value

3



4

Various approaches to obtain cohesive zone model are

available in literature

– Obtain through experiments

• Direct tension test

– van Mier, van Vliet, Uniaxial tension test for determination of

fracture parameters of concrete, Fracture of Concrete & Rock,

2002

– Assume the shape

• CZM shape significantly affects fracture analysis results –

should be chosen carefully

– Song S.H, Paulino G.H, Buttlar G.H, Influence of cohesive zone

model shape parameter on asphalt concrete fracture behavior,

American Institute of Physics, 2008



5

– Indirect method: Inverse analysis

• Development in experimental stress analysis techniques like

photo-elasticity, DIC have made Inverse Analysis attractive

– van Mier, Fracture processes of concrete : assessment of material

parameters for fracture models, CRC Press, 1997

– Hanson J. H. , An experimental - computational evaluation of the

accuracy of the fracture toughness tests on concrete, PhD Thesis,

Cornell university, 2000

– Hanson J.H., Ingraffea A.R. Using numerical simulations to compare

the fracture toughness values for concrete from the size-effect, two-

parameter and fictitious crack models, Engineering Fracture

Mechanics, 2003


Introduction: Forward v/s Inverse Problem

6

Forward Problem

Inverse Problem

P P

0 50 100 1500

2

4

6

CMOD(m)

Tn (

Mp

a)

0 0.1 0.2 0.30

200

400

600

CMOD (mm)

Lo

ad

(N

)

Global Response

P P

,x yu

P P

,x yu

DIC / Synthetic Data from forward problem

Optimization

Nelder-Mead Scheme0 50 100 150

0

2

4

6

CMOD(m)

Tn (

Mp

a)

Constitutive Response


Elasto-Plastic Forward Problem

7

1

1

12

2

element elementcoh coh=K u T

n xf t ld

d N N

1

coh coh+ ,j j j extb i i i i

K u K u u F

1

1

elementcoh

Tc nk t ld

K NN

1 2,

T

x x xd d d

Shen, B., 2009, “Functionally Graded Fiber – Reinforced Cementitious Composites : Manufacturing andExtraction of Cohesive Properties using Finite Elements and Digital Image Correlation”, PhD Thesis, UIUC

Plane Stress J2 Plasticityb K

s

l

2xd

1xd

21

34

Cohesive Element

FF

0 50 100 1500

2

4

6

n (m)

n (

Mp

a)

ck


Modeling Approaches: Shape Regularization

8

Elasto-Plastic Inverse Problem

1 21 1 2

ext int M

λmin , : R Rf fw w f w f λ F F Y X

1 2 1 2

Cohesive Parameters

X ,X , . .,X ,Y ,Y , . .,Y

coh

n n

λ

Yn c

n

Xn ncn

i

X i - 1 Xi Xi+ 1

Yi X , Yi i coh coh+ , extb K u K u u F

Nelder-Mead Optimization

1

1 1 iψ γ Y10 ,

i

f

Y 2 2

2iψ γ

10i

f

X

1 2

0

0

X X . . Xn

Constraints

1 1

1 1

2

2

i i

X X Xwhere,

X X

γ <<1, ψ >>1

i i i

i

i i


Modeling Approaches: PPR model

9

Unified potential based model: PPR (Park-Paulino-Roesler)

Park K., Paulino G.H., Roesler J.R., 2009, A unified potential-based cohesive model of mixed-mode fracture,Journal of the Mechanics and Physics of Solids, 57, 891

1

1

ψ , min ,

m

n nn t n t n n t

n n

n

t t

t t n

t t

m

n

ψ

,n n t

n

T

ψ

,t n t

t

T

0 100 200 3000

2

4

6

n (m)

Tn (

Mp

a)

= 1.3

= 2.0

= 4.0



10

Elasto-Plastic Forward Problem

1

1

12

2coh coh=K uelement element T

xf t ld

d N N

1

coh coh+ ,j j j extb i i i i

K u K u λ u F

1

1

elementcoh

Tc nk t ld

K NN

1 2,

T

x x xd d d

s

l

2xd

1xd

21

34

Cohesive Element

FF

0 50 100 1500

2

4

6

n (m)

n (

Mp

a)

ck

coh

max

Cohesive Parameters

, ,n

λ


Elasto-Plastic Inverse Problem

11

11 1

ext int 3

λmin w , : R Rfw f λ F F

max

Cohesive Parameters

, ,n

λ 1 Constraints:

1

ψ 110 ,f

Barrier Function:


coh coh+ , extb K u K u u F

Nelder-Mead Optimization

ψ >> 1


Problem Details

Numerical Simulations

12

• 5782 Nodes

• 5346 Q4 Elements

• 304 Cohesive Elements

• Displacement Ctrl: 100 Steps

70 0 25

0 14

20 100

GPa, . ,

.

Isotroplic Hardening

MPa, MPa

app

y

E

D mm

K

P P

38.4

150.4


Forward Problem

Numerical Simulations: Shape Regularization

13

0 0.05 0.1 0.15 0.2 0.250

100

200

300

400

500

600

CMOD (mm)

Lo

ad

(N

)

Elastic bulk material

Elasto-plastic bulk material

0 0.1 0.20

2.5

5

CMOD (mm)

(

MP

a)

Linear softening CZM


Inverse Problem: Different Loading Points


14

0 0.1 0.20

2.5

5

CMOD (mm)

(

MP

a)


• 6 Control Points

• Piecewise Cubic Hermite interpolation

• Synthetic data without any noiseForward Problem Plot

0 0.2 0.4 0.6 0.80

200

400

600

X: 0.3247Y: 170.5

CMOD (mm)L

oa

d (

N)

X: 0.3606Y: 143

X: 0.3964Y: 122.1

X: 0.4322Y: 105.7

X: 0.468Y: 92.52

X: 0.1808Y: 365.5

A

BC DE F

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

CMOD (mm)

(

MP

a)

Point APoint BPoint CPoint DPoint EPoint F

Initial Guess


Inverse Problem: Various Control Points


15

• Displacement field taken from loading point C = 122.1 N


• Synthetic data without any noiseForward Problem Plot

0 0.2 0.4 0.6 0.80

200

400

600

X: 0.3964Y: 122.1

CMOD (mm)L

oa

d (

N)

C

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

CMOD (mm)

(

MP

a)

3 - Control Points4 - Control Points5 - Control Points6 - Control Points

Initial Guess

0 0.1 0.20

2.5

5

CMOD (mm)

(

MP

a)



0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

CMOD (mm)

(

MP

a)

Inverse Problem: Different Initial Guess


16




• Synthetic data without any noise

Initial Guess 1

Initial Guess 2

0 0.1 0.20

2.5

5

CMOD (mm)

(

MP

a)


Forward Problem Plot

0 0.2 0.4 0.6 0.80

200

400

600

X: 0.3964Y: 122.1

CMOD (mm)L

oa

d (

N)

C


Inverse Problem: Noise in Synthetic Data


17




• Synthetic data with varying amount of noise

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

CMOD (mm)

(

MP

a)

Max Noise – 0.00 %

Max Noise – 0.02%



Initial Guess

0 0.1 0.20

2.5

5

CMOD (mm)

(

MP

a)



0 0.2 0.4 0.6 0.80

200

400

600

X: 0.3964Y: 122.1

CMOD (mm)L

oa

d (

N)

C


Forward Problem

Numerical Simulations: PPR model

18

500

5max

N / m

MPa

= 3

n

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

100

200

300

400

500

600

CMOD (mm)

Lo

ad

(N

)

Elastic bulk material

Elasto-plastic bulk material

0 0.1 0.2 0.30

2

4

6

CMOD (mm)

(

Mp

a)

= 3.0


Inverse Problem: Different Loading Points

19



0 0.1 0.2 0.3 0.40

200

400

600

X: 0.1245Y: 414.2

CMOD (mm)

Lo

ad

(N

)

X: 0.1549Y: 371.6X: 0.1824

Y: 334.9X: 0.2131Y: 296.4X: 0.2439

Y: 260.4X: 0.2749Y: 227.4

AB

CD

EF

0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

CMOD (mm)

(

MP

a)

Initial Guess

Point A

Point B

Point C

Point D

Point E

Point F

Initial Guess

0 0.1 0.2 0.30

2

4

6

CMOD (mm)

(

Mp

a)

= 3.0



Inverse Problem: Various Initial Guesses


20

• Displacement field taken from loading point D = 296.4 N


0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

CMOD (mm)

(

MP

a)

Initial Guess 2

Initial Guess 1

Initial Guess 3


0 0.1 0.2 0.3 0.40

200

400

600

X: 0.2131Y: 296.4

CMOD (mm)

Lo

ad

(N

)

D

0 0.1 0.2 0.30

2

4

6

CMOD (mm)

(

Mp

a)

= 3.0


Inverse Problem: Noise in Synthetic Data


21

• Displacement field taken from loading point D = 296.4 N

• Synthetic data with varying amount of noise

0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

CMOD (mm)

(

MP

a)

Initial Guess

Max Noise = 0.00 %

Max Noise = 0.05 %

Max Noise = 0.50 %

Max Noise = 5.00 %

Initial Guess


0 0.1 0.2 0.3 0.40

200

400

600

X: 0.2131Y: 296.4

CMOD (mm)

Lo

ad

(N

)

D

0 0.1 0.2 0.30

2

4

6

CMOD (mm)

(

Mp

a)

= 3.0


Load v/s CMOD from experiment

P P

Preliminary results using

PMMA

Inverse Analysis using DIC

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 200 400 600 800 1000 1200 1400 1600 1800

Load

(kN

)

Crack Opening Displacement (um)

Load vs. COD

Crack Propagation Observed in Images,Image 16

Image 17 DIC Data used for simulation

Load Line Displacement (mm)


Extracted Cohesive Relation using Inverse Analysis

23

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

2

4

6

8

10

12

14

16

18

n, mm

n, M

Pa

Shape of Cohesive Relation

• Results from different

simulation runs

• Shape similar to the

one used in reference

below

Elices M., Guinea G.V., Gomez J. & Planas J., 2002, The Cohesive Zone Model: Advantages, Limitations and Challenges, EFM, 69, 137

Preliminary results using

PMMA



Traction Separation relation for PMMA used by Elices et al.

24Elices M., Guinea G.V., Gomez J. & Planas J., 2002, The Cohesive Zone Model: Advantages, Limitations and Challenges, EFM, 69, 137



Summary & Conclusions

Developed inverse analysis techniques to extract

cohesive fracture properties of elasto-plastic

materials

– Shape regularization

– PPR model

Verified implementation for various conditions

Ongoing collaborative work: Hybrid technique

(Experimental DIC + Inverse analysis) for

polymers and metal/metal composites such as

Ti/Ti composites

25


Future Work

Inverse analysis for fatigue loading

Validation of elasto-plastic inverse analysis using

DIC experiments

Extension of elasto-plastic inverse analysis to

plates and shells

26


Thank You !

27

Acknowledgements:: Bin Shen, Jason Patric

Extraction of Cohesive Properties of ElastoPlastic material using Inverse Analysis

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