Modeling Fracture in Elastic-plastic Solids Using Cohesive Zone

CHANDRAKANTH SHETDepartment of Mechanical Engineering

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

Sponsored byUS ARO, US Air Force

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

,Tdnd

td

1

2(d)

sepdmaxd

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

,Tdnd

td

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

sepdmaxd

maxnT

(c)

1n

2n

P

P

,Tdnd

td

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

d

d d

d d

d d

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

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

d d

d d where /tq

nnr d/*

tn dd , 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

d d d dd d

2ttn n n n

t 2n t t n n t

r q2T q exp exp

r 1

d d d d d d d

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

• 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 dSd d d

1 2

Continuum elements

Cohesiveelement

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

[N]{ v}d d

T T T 1n t Js

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

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

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