1

Modeling Fracture in Elastic-plastic Solids Using Cohesive Zones

CHANDRAKANTH SHETDepartment of Mechanical Engineering

FAMU-FSU College of EngineeringFlorida State UniversityTallahassee, Fl-32310

Sponsored byUS ARO, US Air Force

2

General formulation of continuum solids LEFM EPFM Introduction to CZM Concept of CZM Literature review Motivation Atomistic simulation to evaluate CZ properties Plastic dissipation and cohesive energy dissipation studies Conclusion

Outline

What is CZM and why is it important

In the study of solids and design of nano/micro/macro structures, thermomechanical behavior is modeled through constitutive equations. Typically is a continuous function of and their history. Design is limited by a maximum value of a given parameter ( ) at any local point. What happens beyond that condition is the realm of ‘fracture’, ‘damage’, and ‘failure’ mechanics. CZM offers an alternative way to view and failure in materials.

, , f( , , )

Formulation of a general boundary value problem

For a generic 3-D analysis the

equilibrium equation is given by

0

For a 2-D problem equilibrium equation reduces to

0; 0

where , and are the stresses

iji

j

xy y xyxx y

x y xy

fx

f fx y y x

1 1

2 2

2 3

within

the domain . , are the body forces.

Boundary conditions are given by

at

0

x y

x xy

f f

u u

u u at

l m t at

2

3

1

x

y

2 22

2 2

2 2

2 2

2

The strain compatibility conditions are given by

It can be shown that the all field equation reduces to

0

If is the Airy's stress function such that

y xyx

x y

x

x y x y

x y

2 2

2 2

4 2 4

4 2 2 4

, ,

Then the governing DE is

2 0

y xyy x xy

x x y y

Formulation of a general boundary value problem

For problems with crack tip Westergaard introduced Airy’s stress function as

Where Z is an analytic complex function

Re[ ] y Im[Z]Z

Z z z y z z x iybg Re[ ] Im[ ] ; = +

And are 2nd and 1st integrals of Z(z)Then the stresses are given by

Z,Z

2'

x 2

2'

y 2

2'

xy

'

Re[Z] y Im[Z ]y

Re[Z] y Im[Z ]x

y Im[Z ]xy

where Z =dZ dz

a

y

X

yy

Opening mode analysis or Mode I

Consider an infinite plate a crack of length 2a subjected to a biaxialState of stress. Defining:

Boundary Conditions :• At infinity • On crack faces

x y xy| z | , 0

x xya x a;y 0 0

2 2

zZ

z a

By replacing z by z+a , origin shifted to crack tip.

2

z aZ

z z a

s

sx

y

2a

And when |z|0 at the vicinity of the crack tip

KI must be real and a constant at the crack tip. This is due to a Singularity given by

The parameter KI is called the stress intensity factor for opening mode I.

Za

az

K

z

K a

I

I

2 2

1z

Since origin is shifted to crack tip, it is easier to use polar Coordinates, Using

z ei

Opening mode analysis or Mode I

Ix

Iy

Ixy

K 3cos 1 sin sin

2 2 22 r

K 3cos 1 sin sin

2 2 22 r

K 3sin cos cos

2 2 22 r

From Hooke’s law, displacement field can be obtained as

2I

2I

2(1 ) r 1u K cos sin

E 2 2 2 2

2(1 ) r 1v K sin cos

E 2 2 2 2

where u, v = displacements in x, y directions

(3 4 ) for plane stress problems

3 for plane strain problems

1

a

y

X

yy

u

Opening mode analysis or Mode I

10

.

Irwin estimates

Dugdale strip yield model:

rK

pI

ys

1

22

( )

rK

pI

ys

1

82( )

Small Scale plasticity

Singularity dominated region

EPFM•In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram.

sh a rp tip

Ideal elastic brittle behaviorcleavage fracture

P: Applied loadP : Yield loady Displacement, u

Loa

dra

tio,

P/P y

1.0

Fracture

Blunt tip

Limited plasticity at cracktip, still cleavage fracture

Displacement, u

Loa

dra

tio,

P/P y

1.0Fracture

Blunt tip

Void formation & coalescencefailure due to fibrous tearing

Displacement, u

Loa

dra

tio,

P/P y

1.0Fracture

large scaleblunting

Large scale plasticityfibrous rapture/ductilefailure Displacement, u

Loa

dra

tio,

P/P y

1.0 Fracture

EPFM•EPFM applies to elastic-plastic-rate-independent materials

•Crack opening displacement (COD) or crack tip opening displacement (CTOD).

• J-integral.

Sharp crack

Blunting crack

y

x

ds

2

2

0

4

( ),

ij

I

ys

ii

i

ij ij

K

E

uJ wdy T ds

x

w d

More on J Dominance

Limitations of J integral, (Hutchinson, 1993) (1) Deformation theory of plasticity should be valid with small strain behavior with monotonic loading(2) If finite strain effects dominate and microscopic failures occur, then this region should be much smaller compared to J dominated

region Again based on the HRR singularity

1

1

,n

Iijij y

y y n

Jn

I r

Based on the condition (2), inner radius ro of J dominance.

R the outer radius where the J solutions are satisfied within 10% of complete solution. R

or

3o CODr

15

HRR Singularity…1

0 0 0

0

Hutchinson, Rice and Rosenbren evaluated the character of crack tip

in power-law hardening materials.

Ramberg-Osgood model,

Reference value of stress=yield s

n

00

trength, strain-hardening exponent

, strain at yield, dimensionless constantE

n

1

Note if elastic strains are negligible, then

ˆ 3 3ˆ ;

2 2

n

y y

n

ij eq ijeq ij

y ij y

16

HRR Singularity…2

11

0 20

10

20

stress and strain fields are given by

,

,

Integration constant

,

n

ij ijn

nn

ij ijn

n

EJn

I r

EJn

E I r

I

Dimensionless functions of n and

HRR Integral, cont.

Note the singularity is of the strenth . For the specific case of n=1 (linearly elastic), we have singularity.

Note also that the HRR singularity still assumes that the strain is infinitesimal, i.e., , and not the finite strain . Near the tip where the strain is finite, (typically when ), one needs to use the strain measure . 

1

11 n

r

1r

1, ,2ij i j j iu u 1

, , , ,2ij i j j i k i k jE u u u u 0.1ij E

Some consequences of HRR singularity

In elastic-plastic materials, the singular field is given by

(with n=1 it is LEFM)

stress is still infinite at . the crack tip were to be blunt then since it is now a free surface. This is not the case in HRR field. HRR is based on small strain theory and is not thus applicable in a region very close to the crack tip.

1

1

1

1

1

2

n

ij

n

ij

Jk

r

Jk

r

0r 0 at 0xx r

18

HRR Integral, cont.

Large Strain Zone

HRR singularity still predicts infinite stresses near the crack tip. But when the crack blunts, the singularity reduces. In fact at for a blunt crack. The following is a comparison when you consider the finite strain and crack blunting. In the figure, FEM results are used as the basis for comparison.

0 at 0xx r

Large-strain crack tip finite element results of McMeeking and Parks.Blunting causes the stresses to deviate from the HRR solution close to the crack tip.

The peak occurs at and

decreases as . This corresponds to approximately twice the width of CTOD. Hence within this region, HRR singularity is not valid.

0x

J

1x

Fracture Mechanics - Linear solutions leads to singular fields-

difficult to evaluate Fracture criteria based on Non-linear domain- solutions are not

unique Additional criteria are required for crack

initiation and propagation

Basic breakdown of the principles of mechanics of continuous media

Damage mechanics- can effectively reduce the strength and

stiffness of the material in an average sense, but cannot create new surface

Fracture/Damage theories to model failure

IC IC ICK ,G ,J ,CTOD,...

ED 1 , Effective stress =

E 1 D

CZM can create new surfaces. Maintains continuity conditions mathematically, despite the physical separation.

CZM represent physics of fracture process at the atomic scale. It can also be perceived at the meso-scale as the effect of energy dissipation mechanisms,

energy dissipated both in the forward and the wake regions of the crack tip. Uses fracture energy(obtained from fracture tests) as a parameter and is devoid of any

ad-hoc criteria for fracture initiation and propagation. Eliminates singularity of stress and limits it to the cohesive strength of the the material. Ideal framework to model strength, stiffness and failure in an integrated manner. Applications: geomaterials, biomaterials, concrete, metallics, composites…

CZM is an Alternative method to Model Separation

Conceptual Framework of Cohesive Zone Models for interfacesConceptual Framework of Cohesive Zone Models for interfaces

S is an interface surface separating two domains 1, 2(identical/ separate constitutive behavior).After fracture the surface S comprise of unseparated surface and

completely separated surface (e.g. ); all modeled within the con-cept of CZM.Such an approach is not possible in conventional mechanics of con-tinuous media.

*2u

*1t

*1u

1

2

1ssP

N

1 1X , x

2 2X , x

3 3X , x

(a)

2s*

2t

*1t

*1u

1

Pn

*2u

2

P*

(b)

1S

2S

1n

2n

P

P

,Tn

t

1

2(d)

sepmax

maxnT

(c)x (X, t)

Interface in the undeformed configurationInterface in the undeformed configuration

1 2

1 1 2 2

1 1 2 2

and are separated by a common boundary S,

such that

and

and normals and

Hence in the initial configuration

S S

N N

S S

1 2

1 2

1 2

1 2

defines the interface between any two domains

is metal, is ceramic,

S = metal ceramic interface

, represent grains

S

N N N

S

1 2 1 2

in different orientation,

S = grain boundary

, represent same domain ( = ),

S = internal surface yet to separate

*2u

*1t

*1u

1

2

1ssP

N

1 1X , x

2 2X , x

3 3X , x

(a)

2s*

2t

1 2

After deformation a material point X

moves to a new location x, such that

(X,t)

if the interface S separates, then a pair of new

surface and are created bounding

a new do

x

S S*

*1 1 1 1 1 1

*2 2 2 2 2 2

*

main such that

ˆN moves to n

ˆ(S , N ) moves to ( , ) ( )

ˆ(S , N ) moves to ( , ) ( )

can be considered as 3-D domain made of

extremely soft glue, which can be shrunk to an

i

n

n

S S

S S

nfinitesimally thin surface but can be expanded

into a 3-D domain.

*1t

*1u

1

Pn

*2u

2

P*

(b)

1S

2S

1n

2n

P

P

,Tn

t

1

2(d)

Interface in the deformed configurationInterface in the deformed configuration

Constitutive Model for Bounding Domains 1,2Constitutive Model for Bounding Domains 1,2

After deformation, given by (X,t), if v is the velocity vector,

Then velocity gradient L is given by

Decomposing L into a symmetric part D and antisymmetri

x

vL

x

1 12 2

c part W

such that

Where, ( ) and W= ( )

D is the rate of deformation tensor, and W is the spin tensor

Extending hypo-elastic formulation

T T

L D W

D L L L L

to inelastic material by

additive decomposition of the rate of deformation tensor

where and are elastic and inelastic part of the rate of deformati

El In

El In

D D D

D D

1 2

on tensor

The constitutive model for the domains and can be written as

( )

where is elasticity tensor, and Jaumann rate of cauchy stress tensor.

InC D D

C

Constitutive Model for Cohesive Zone Constitutive Model for Cohesive Zone

*1t

*1u

1

Pn

*2u

2

P*

(b)

1S

2S

sepmax

maxnT

(c)

1n

2n

P

P

,Tn

t

1

2(d)

*

*ijkl

A typical constitutive relation for

is given by - relation such that

ˆ if ,

and

ˆif , 0

It can be construed that when

in the domain , the stiffness C 0.

sep

sep

sep

T

n T

n T

Molecular force of cohesion acting near the edge of the crack at its surface (region II ). The intensity of molecular force of cohesion ‘f ’ is found to vary as shown in Fig.a. The interatomic force is initially zero when the atomic planes are separated by normal

intermolecular distance and increases to high maximum after that it rapidly reduces to zero with increase in separation distance. E is Young’s modulus and is surface tension

  

oT(Barenblatt, G.I, (1959), PMM (23) p. 434)

m of ET / b E /10

Figure (a) Variation of Cohesive traction (b) I - inner region, II - edge region

Development of CZ Models-Historical Review

Barenblatt (1959) was first to propose the concept of Cohesive zone model to brittle fracture

The theory of CZM is based on sound principles. However implementation of model for practical problems grew exponentially for practical problems with use of FEM and advent of fast computing. Model has been recast as a phenomenological one for a number of systems and boundary value problems. The phenomenological models can model the separation process but not the effect of atomic discreteness.

Phenomenological Models

Hillerborg etal. 1976 Ficticious crack model; concrete

Bazant etal.1983 crack band theory; concrete

Morgan etal. 1997 earthquake rupture propagation; geomaterial

Planas etal,1991, concrete Eisenmenger,2001, stone fragm-

entation squeezing" by evanescent waves; brittle-bio materials

Amruthraj etal.,1995, composites

Grujicic, 1999, fracture beha-vior of polycrystalline; bicrystals

Costanzo etal;1998, dynamic fr.Ghosh 2000, Interfacial debo-

nding; compositesRahulkumar 2000 viscoelastic

fracture; polymersLiechti 2001Mixed-mode, time-

depend. rubber/metal debondingRavichander, 2001, fatigue

Tevergaard 1992 particle-matrix interface debonding

Tvergaard etal 1996 elastic-plastic solid :ductile frac.; metals

Brocks 2001crack growth in sheet metal

Camacho &ortiz;1996,impactDollar; 1993Interfacial

debonding ceramic-matrix compLokhandwalla 2000, urinary

stones; biomaterials

CZM essentially models fracture process zone by a line or a plane ahead of the crack tip subjected to cohesive traction.

The constitutive behavior is given by traction displacement relation, obtained by defining potential function of the type

n t1 t2, ,

n t1 t2, , where are normal and tangential displacement jump

The interface tractions are given by

n t1 t 2n t1 t 2

T , T , T

Fracture process zone and CZM

Material crack tip

Mathematical crack tip

x

y

29

Following the work of Xu and Needleman (1993), the interface potential is taken as

n nn t n n

n n

2tn

2n t

1 q, exp 1 r

r 1

r qq exp

r 1

where /tq

nnr /*

tn , are some characteristic distancen* Normal displacement after shear separation under the condition

Of zero normal tension

Normal and shear traction are given by

2 2n t tn n n

n 2 2n n n nt t

1 qT exp exp r 1 exp

r 1

2ttn n n n

t 2n t t n n t

r q2T q exp exp

r 1

30

C o n ta c t W ed g in g

C o n tac t S u rfa ce(fr ic tio n )

P l a s t i c W a k eP la s tic ity in d u ce d

c rack c lo su re

F ib ril (M M C b rid g in g

O x id e b rid g in g

P las ticz o n e

C le av ag efr ac tu re

W ak e o f c rac k t ip F o rw a rd o f c ra ck tip

E x trin s ic d is s ip a t io nIn trin s ic d is s ip a t io n

M eta llic

C e ram ic

C rac k M e a n d e rin g

T h ick n ess o fce ra m ic in ter fa ce

M ic ro v o idc o ale sc en c e

P la s tic w a k e

P rec ip ita te sC rac k D eflec tio n

C rac k M ean d e rin g

C y c lic lo ad in d u c edc rack c lo su re

M ic ro c rack in gin it ia tio n

M ic ro v o idg ro w th /co a le s cen ce

D e lam in a tio n

C o r n e r a to m s

B C C B o d y c e n te r e da to m s

F a c e c e n t er e da to m s

F C C

C o r n e r a to m s

P h asetran sfo rm a tio n

G rain b rid g in g

F ib ril(p o ly m e rs)b rid g in g

In te r /tran s g ran u la rfrac tu re

Active dissipation mechanisims participating at the cohesive process zone

Dissipative Micromechanisims Acting in the wake and forwardregion of the process zone at the Interfaces of

Monolithic and Heterogeneous Material

C

W A K E F O R W A R D

sep

max

D

C O H E S IV EC R A C K T IP

A C T IV E P L A S T IC Z O N E

IN A C T IV E P L A S T IC Z O N E(P la s tic w a k e )

E L A S T IC S IN G U L A R IT Y Z O N E

M A T H E M A T IC A LC R A C K T IP

M A T E R IA LC R A C K T IP

A

E D

x

y

D

max

sep

max

y

W A K EF O R W A R D

L O C A T IO N O F C O H E S I V EC R A C K T I P

A

B D

E

N O M A T E R I A LS E P A R A T IO N

l 1 l 2

C O M P L E T E M A T E R IA LS E P A R A T IO N

C

, X

Concept of wake and forward region in thecohesive process zone

31

32

33

CZM is an excellent tool with sound theoretical basis and computational ease. Lacks proper mechanics and physics based analysis and evaluation. Already widely used in fracture/fragmentation/failure

Importance of shape of CZM

Motivation for studying CZM

critical issues addressed here

m

Scales- What range of CZM parameters are valid?MPa or GPa for the tractionJ or KJ for cohesive energynm or for separation

displacement

What is the effect of plasticity in the bounding material on

the fracture processes

Energy- Energy characteristics during fracture process and how energy flows in to the cohesive zone.

34

Atomistic simulations to extract cohesive properties

Motivation

What is the approximate scale to examine fracture in a solid

Atomistic at nm scale or Grains at scale or Continuum at mm scale

Are the stress/strain and energy quantities computed at one scale be valid at other scales? (can we even define stress-strain at atomic scales?)

m

35

Embedded Atom Method Energy Functions(D.J.Oh and R.A.Johnson, 1989 ,Atomic Simulation of Materials,

Edts:V Vitek and D.J.Srolovitz,p 233Edts:V Vitek and D.J.Srolovitz,p 233)

Atomic Seperation (A)

Ene

rgy

(eV

)

2 4 6

-5

-4

-3

-2

-1

0

1

2

3

4

5AlMgCu

(5.44)

Cutoff Distances

(4.86) (6.10)

The total internal energy of the crystal

12

1

1

tot ii

i i ijj

i ijj

E E

E F r

f r

where

and

Contribution to electron density of ith atom and jth atom.Two body central potential between ith atom and jth atom.  

iF

ijf r

ij

iE Internal energy associated with atom i

Embedded Energy of atom i.

A small portion of CSL grain bounary before And after application of tangential force

9(221)

Curve in Shear directionT

Shet C, Li H, Chandra N ;Interface models for GB sliding and migrationMATER SCI FORUM 357-3: 577-585 2001

37

A small portion of CSL grain boundary before And after application of normal force

9(221)

Curve in Normal directionT

Summary complete debonding occurs when the

distance of separation reaches a value of 2 to 3 .

For 9 bicrystal tangential work of separation along the grain boundary is of the order 3 and normal work of separation is of the order 2.6 .

For 3 -bicrystal, the work of separation ranges from 1.5 to 3.7 .

Rose et al. (1983) have reported that the adhesive energy (work of separation) for aluminum is of the order 0.5 and the separation distance 2 to 3

Measured energy to fracture copper bicrystal with random grain boundary is of the order 54 and for 11 copper bicrystal the energy to fracture is more than 8000

A

2J / m

2J / m

2J / m

A

2J / m

2J / m

2J / m

Results and discussion on atomistic simulation

Implications

The numerical value of the cohesive energy is very low when compared to the observed experimental results

Atomistic simulation gives only surface energy ignoring the inelastic energies due to plasticity and other micro processes.

It should also be noted that the exper- imental value of fracture energy includes the plastic work in addition to work of separation (J.R Rice and J. S Wang, 1989)

p2 W

Material Nomenclature particle size

Aluminium alloys

2024-T351 35 14900 1.2

2024-T851 25.4 8000 1.2

Titanium alloys

T21 80 48970 2-4

T68 130 130000 2-4

Steel Medium Carbon

54 12636 2-4

High strength alloys

98 41617

18 Ni (300) maraging

76 25030

Alumina 4-8 34-240 10

SiC ceramics 6.1 0.11 to 1.28

Polymers PMMA 1.2-1.7 220

1/ 2ICK MPam 2

ICG J / m 2 J / m

2 3Al O mm

Table of surface and fracture energies of standard materials

40

Energy balance and effect of plasticity in the bounding material

Motivation

It is perceived that CZM represents the physical separation process. As seen from atomistics, fracture process comprises mostly of inelastic dissipative energies. There are many inelastic dissipative process specific to each material system; some occur within FPZ, and some in the bounding material. How the energy flow takes place under the external loading within the cohesive zone and neighboring bounding material near the crack tip?What is the spatial distribution of plastic energy?Is there a link between micromechanic processes of the material and curve.T

42

Al 2024-T3 alloy The input energy in the cohesive model

are related to the interfacial stress and characteristic displacement as

The input energy is equated to material parameter

Based on the measured fracture value

n

n max ne t max t

e

2

n

ICJ

mX

MPa

mJ

tn

ult

tn

6

max

2

105.4

642

/8000

Cohesive zone parameters of a ductile material

43

E=72 GPa, =0.33,

1/ 2ICK 25MPa m

Stress strain curve is given by1/ n

y

y

E

320MPa,

0.01347,

n 0.217173

where

and fracture parameter

Material model for the bounding material

Elasto-plastic model for Al 2024-T3

44

• The virtual work due to cohesive zone traction in a given cohesive element can be written as

3 45 6

7 8

Numerical Formulation• The numerical implementation of CZM for interface

modeling with in implicit FEM is accomplished developing cohesive elements

• Cohesive elements are developed either as line elements (2D) or planar elements (3D)abutting bulk elements on either side, with zero thickness

n n t tdS T T dS

1 2

Continuum elements

Cohesiveelement

The virtual displacement jump is written as Where [N]=nodal shape function matrix, {v}=nodal displacement vector

[N]{ v}

T T T 1n t Js

dS { v} [N] d{T } [N] d{T } dS

J = Jacobian of the transformation between the current deformed and original undeformed areas of cohesive surfaces

Note: is written as d{T}- the incremental traction, ignoring time which is a pseudo quantity for rate independent material

T

45

Numerical formulation contdThe incremental tractions are related to incremental displacement jumps

across a cohesive element face through a material Jacobian matrix as

For two and three dimensional analysis Jacobian matrix is given by

Finally substituting the incremental tractions in terms of incremental displacements jumps, and writing the displacement jumps by means of nodal displacement vector through shape function, the tangent stiffness matrix takes the form

czd{T} [C }d{ }

n n n t1 n t2

cz t1 n t1 t1 t1 t2

t2 n t2 t1 t2 t2

T T T

[C ] T T T

T T T

n n n tcz

t n t t

T T[C ]

T T

T 1T cz Js

[K ] [N] [C ][N] dS

46

Geometry and boundary/loading conditions

a = 0.025m, b = 0.1m, h = 0.1m

47

Finite element mesh

28189 nodes, 24340 plane strain 4 node elements, 7300 cohesive elements (width of element along the crack plan is ~ m77x10

48

Global energy distribution

are confined to bounding materialw e p cE E E E

e pE and E

cE is cohesive energy, a sum total of all dissipative process confined to FPZ and cannot be recovered during elastic unloading and reloading.

Purely elastic analysisThe conventional fracture mechanics uses the concept of strain energy release rate

Using CZM, this fracture energy is dissipated and no plastic dissipation occurs, such that

UG J

a

2G J 8000J / m

w e cE E E

49

Global energy distribution (continued)

IssuesFracture energy obtained from experi-mental results is sum total of all dissipative processes in the material for initiating and propagating fracture.

Should this energy be dissipated entirely in cohesive zone?Should be split into two identifiable dissipation processes?

Two dissipative process28000J / m

Plasticity withinBounding material

Micro-separation Process in FPZ

Analysis with elasto-plastic material model

where represents other factors arising from the shape of the traction-displacement relations

Implications

Leaves no energy for plastic work in the bounding material

In what ratio it should be divided?Division is non-trivial since plastic

dissipation depends on geometry, loading and other parameters as

maxp p i

y

E E ,n,S ,i 1,2,..

iS

What are the key CZM parameters that govern the energetics?

in cohesive zone dictates the stress level achievable in the bounding material. Yield in the bounding material depends on its yield strength and its post yield (hardening characteristics). Thus plays a crucial role in determining plasticity in the bounding material, shape of the fracture process zone and energy distribution. (other parameters like shape may also be important)

max

max y

y

51

Global energy distribution (continued)

u /E

nerg

y/(

1.0E

-2)

0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

4

1

2

3

4

Cumulative Plastic Work

Cumulative Cohesive Energy

yn

8 n

Variation of cohesive energy and plastic energy for various ratios

(1) (2) (3) (4)

max y max y 1 max y 1.5 max y 2.0 max y 2.5

Recoverable elastic work 95 to 98% of external work

Plastic dissipation depends on

Elastic behavior

plasticity occurs.

Plasticity increases with

eE

max y

max y 1 to 1.5 :

max y

max y 1.5 :

Relation between plastic work and cohesive work

Plastic Energy/( 1.0E-2)

Coh

esiv

eE

nerg

y/(

1.0E

-2)

0 1 2 3 40

0.5

1

1.5

2

2.5

3

max

max

max

y

y

y

= 2.0

= 2.5

y n

yn

(very small scale plasticity), plastic energy ~ 15% of total dissipation.

Plasticity induced at the initial stages of the crack growth

plasticity ceases during crack propagation.

Very small error is induced by ignoring plasticity.

plastic work increases considerably, ~100 to 200% as that of cohesive energy. For large scale plasticity problems the amount of total dissipation (plastic and cohesive) is much higher than 8000 Plastic dissipation very sensitive to ratio beyond 2 till 3 Crack cannot propagate beyond and completely elastic below

max y 1.5

max y 2.0

2J / m .

max y 1.5

max y

max y 3

53

Variation of Normal Traction along the interface

The length of cohesive zone is also affected by ratio.

There is a direct correlation between the shape of the traction-displacement curve and the normal traction distribution along the cohesive zone.

For lower ratios the traction-separation curve flattens, this tend to increase the overall cohesive zone length.

max y

max y

54

Local/spatial Energy Distribution

A set of patch of elements (each having app. 50 elements) were selected in the bounding material.

The patches are approximately squares (130 ). They are spaced equally from each other.

Adjoining these patches, patches of cohesive elements are considered to record the cohesive energies.

m

Variation of Cohesive Energy

The variation of Cohesive Energy in the Wake and Forward region as the crack propagates. The numbers indicate the Cohesive Element Patch numbers Falling Just Below the binding element patches

The cohesive energy in the patch increases up to point C (corresponding to in Figure ) after which the crack tip is presumed to advance.

The energy consumed by the cohesive elements at this stage is approximately 1/7 of the total cohesive energy for the present CZM.

Once the point C is crossed, the patch of elements fall into the wake region.

The rate of cohesive zone energy absorption depends on the slope of the curve and the rate at which elastic unloading and plastic dissipation takes place in the adjoining material.

The curves flatten out once the entire cohesive energy is dissipated within a given zone.

max

T max

maxnT

sep

Variation of Elastic Energy

Variation of Elastic Energy in Various Patch of Elements as a Function of Crack Extension. The

numbers indicate Patch numbers starting from Initial Crack Tip

Considerable elastic energy is built up till the peak of curve is reached after which the crack tip advances. After passing C, the cohesive elements near the crack tip are separated and the elements in this patch becomes a part of the wake.At this stage, the values of normal traction reduces following the downward slope of curve following which the stress in the patch reduces accompanied by reduction in elastic strain energy. The reduction in elastic strain energy is used up in dissipating cohesive energy to those cohesive elements adjoining this patch.The initial crack tip is inherently sharp leading to high levels of stress fields due to which higher energy for patch 1Crack tip blunts for advancing crack tip leading to a lower levels of stress, resulting in reduced energy level in other patches.

T

T

max

maxnT

sep

Variation of dissipated plastic energy in various patched as a function of crack extension. The number indicate patch numbers starting from initial crack tip.

Variation of Plastic Work ( )max y 2.0

max

maxnT

sep

yT

c eE and E

plastic energy accumulates considerably along with elastic energy, when the local stresses bounding material exceeds the yield After reaching peak point C on curve traction reduces and plastic deformation ceases. Accumulated plastic work is dissipative in nature, it remains constant after debonding. All the energy transfer in the wake region occurs from elastic strain energy to the cohesive zoneThe accumulated plastic work decreases up to patch 4 from that of 1 as a consequence of reduction of the initial sharpness of the crack. Mechanical work is increased to propagate the crack, during which the does not increase resulting in increased plastic work. That increase in plastic work causes the increase in the stored work in patches 4 and beyond

58

Variation of Plastic Work ( )max y 1.5

Variation of Plastic work and Elastic work in various patch of elements along the interface for the case of .

The numbers indicates the energy in various patch of

elements starting from the crack tip.

max y 1.5

, there is no plastic dissipation.

plastic work is induced only in the first patch of element

No plastic dissipation during crack growth place in the forward region

Initial sharp crack tip profile induces high levels of stress and hence plasticity in bounding material.

During crack propagation, tip blunts resulting reduced level of stresses leading to reduced elastic energies and no plasticity condition.

max y 1

max y 1.5

max

maxnT

sep

59

Contour plot of yield locus around the cohesive crack tip at the various stages of crack growth.

Schematic of crack initiation and propagation

process in a ductile material

Conclusion

CZM provides an effective methodology to study and simulate fracture in solids.

Cohesive Zone Theory and Model allow us to investigate in a much more fundamental manner the processes that take place as the crack propagates in a number of inelastic systems. Fracture or damage mechanics cannot be used in these cases.

Form and parameters of CZM are clearly linked to the micromechanics.

Our study aims to provide the modelers some guideline in choosing appropriate CZM for their specific material system.

ratio affects length of fracture process zone length. For smaller ratio the length of fracture process zone is longer when compared with that of higher ratio.

Amount of fracture energy dissipated in the wake region, depend on shape of the model. For example, in the present model approximately 6/7th of total dissipation takes place in the wake

Plastic work depends on the shape of the crack tip in addition to ratio.

max y

max y

max y

62

Conclusion(contd.)

ICJ

The CZM allows the energy to flow in to the fracture process zone, where a part of it is spent in the forward region and rest in the wake region.

The part of cohesive energy spent as extrinsic dissipation in the forward region is used up in advancing the crack tip.

The part of energy spent as intrinsic dissipation in the wake region is required to complete the gradual separation process.

In case of elastic material the entire fracture energy given by the of the material, and is dissipated in the fracture process zone by the cohesive elements, as cohesive energy.

In case of small scale yielding material, a small amount of plastic dissipation (of the order 15%) is incurred, mostly at the crack initiation stage.

During the crack growth stage, because of reduced stress field, plastic dissipation is negligible in the forward region.