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Theoretical and Computational Aspects of Cohesive Zone Modeling

NAMAS CHANDRADepartment of Mechanical Engineering

FAMU-FSU College of EngineeringFlorida State University

Tallahassee, Fl-32310

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( , , )

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 represents physics of the 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.

It is an 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

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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

sepd

maxd

Dd

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

Dd

maxd

sepd

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

d

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

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

is an interface surface separating two domains(identical/ separate constitutive behavior).After fracture the surface comprise of unseparated surface andcompletely 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.

11 2,

*1

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

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For Ductile metals (steel) Cohesive stress in the CZM is equated to yield stress Y Analyzed for plastic zone size for plates under tension Length of yielding zone ‘s’, theoretical crack length ‘a’,

and applied loading ‘T’ are related in

the form 2s Ta 4 Y2 sin ( )

(Dugdale, D.S. (1960), J. Mech.Phys.Solids,8,p.100)

Dugdale (1960) independently developed the concept of cohesive stress

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 relationship, 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

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• What is the relationship between the physics/mechanics of the separation process and shape of CZM? (There are as many shapes/equations as there are number of interface problems solved!)• What is the relationship between CZM and fracture mechanics of brittle, semi-brittle and ductile materials?• What is the role of scaling parameter in the fidelity of CZM to model interface behavior?• What is the physical significance of

- Shape of the curve C- tmax and interface strength

- Separation distance dsep and COD?- Area under the curve, work of fracture, fracture toughness G (local and global)

• What is the relationship between the physics/mechanics of the separation process and shape of CZM? (There are as many shapes/equations as there are number of interface problems solved!)• What is the relationship between CZM and fracture mechanics of brittle, semi-brittle and ductile materials?• What is the role of scaling parameter in the fidelity of CZM to model interface behavior?• What is the physical significance of

- Shape of the curve C- tmax and interface strength

- Separation distance dsep and COD?- Area under the curve, work of fracture, fracture toughness G (local and global)

Critical Issues in the application of CZM to interface modelsCritical Issues in the application of CZM to interface models

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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.

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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

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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.

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GRAIN STRUCTURE AND COMPUTATIONAL CRYSTAL

CONSTRUCTION OF COMPUTATIONAL CRYSTAL

CONSTRUCTION OF COMPUTATIONAL CRYSTAL

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Boundary Conditions for GB Sliding

Construct symmetric tilt boundaries (STDB) by rotating a single crystal (reflection)

Periodic boundary condition in X direction

Restrain few layers in lower crystal

Apply body force on top crystal

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

9(221)

Curve in Shear directionT d

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

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

9(221)

Curve in Normal directionT d

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

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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 micromechanics processes of the material and curve.T d

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Plasticity vs. other Dissipation Mechanisms

Since bounding material has its own inelastic constitutive equation, what is the proportion of energy dissipation within that domain and fracture region given by CZM.

Role of plasticity in the bounding material is clearly unique; and cannot be assigned to CZM.

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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

nd

n max ne dt max t

e

2 d

n

ICJ

mX

MPa

mJ

tn

ult

tn

6

max

2

105.4

642

/8000

dd

Cohesive zone parameters of a ductile material

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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

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Geometry and boundary/loading conditions

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

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Finite element mesh

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

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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

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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

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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

d

yn

8 dn

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

d y n

d

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

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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

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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 flattens out once the entire cohesive energy is dissipated within a given zone.

max

T d maxd

maxnT

sepd

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 d

T d

maxd

maxnT

sepd

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

maxd

maxnT

sepd

yT d

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

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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

maxd

maxnT

sepd

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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

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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.