Transcript Modeling Fracture in Elastic-plastic Solids Using Cohesive Zones CHANDRAKANTH SHET

Modeling Fracture in Elastic-plastic
Solids Using Cohesive Zones
CHANDRAKANTH SHET
Department of Mechanical Engineering
FAMU-FSU College of Engineering
Florida State University
Tallahassee, Fl-32310
Sponsored by
US ARO, US Air Force
1
Outline
 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
2
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 , , f(, , ) 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.
Formulation of a general boundary value problem
For a generic 3-D analysis the
equilibrium equation is given by
ij
 fi  0
x j
1
3
For a 2-D problem equilibrium equation reduces to
 y  xy
 x  xy

 f x  0;

 fy  0
x
y
y
x
where  x ,  y and  xy are the stresses within
the domain . f x , f y are the body forces.
Boundary conditions are given by
u  u1 at 1
u  u2  0 at  2
l  x  m xy  t2 at 3
y
2
x
Formulation of a general boundary value problem
The strain compatibility conditions are given by
 2 y
2
 2  x   xy
 2 
2
x
y
xy
It can be shown that the all field equation reduces to
 2
2 
 2  2  x   y   0
 x y 
If  is the Airy's stress function such that
2 
2 
2
 x  2 ,  y  2 ,  xy  
y
x
xy
Then the governing DE is
 4
 2
 4
2 2 2  4 0
4
x
x y y
For problems with crack tip Westergaard introduced Airy’s stress
function as



  Re[Z]  y Im[Z]
Where Z is an analytic complex function
y
bg
Z z  Re[ z ]  y Im[ z ] ; z = x + iy

yy


And Z, Z are 2nd and 1st integrals of Z(z)
Then the stresses are given by
2
x  2  Re[Z]  y Im[Z' ]
y
2
y  2  Re[Z]  y Im[Z' ]
x
2
 xy 
  y Im[Z' ]
xy
where Z' = dZ dz
X
a

Opening mode analysis or Mode I
Consider an infinite plate a crack of length 2a subjected to a biaxial
s
State of stress. Defining:
Z
z
z
2
a
2

x
2a
Boundary Conditions :
• At infinity | z |   x  y  , xy  0
• On crack faces
 a  x  a;y  0 x  xy  0
By replacing z by z+a , origin shifted to crack tip.
Z
 z  a
z  z  2a 
s
y
Opening mode analysis or Mode I
And when |z|0 at the vicinity of the crack tip
a
KI
Z

2az
2 z
K I   a
KI must be real and a constant at the crack tip. This is due to a
Singularity given by 1
z
The parameter KI is called the
stress intensity factor for opening
mode I.
Since origin is shifted to crack
tip, it is easier to use polar
Coordinates, Using
z  ei
Opening mode analysis or Mode I

KI
 

3 
cos   1  sin   sin   
2 r
2
 2   2 
KI
 

3 
y 
cos   1  sin   sin   
2 r
2
 2   2 
KI
 

3 
 xy 
sin   cos   cos   
2 r
2 2
 2 
x 
From Hooke’s law, displacement field can be
obtained as
2(1  )
r
  1
 
u
KI
cos   
 sin 2   
E
2
2 2
 2 
2(1  )
r
  1
 
KI
sin   
 cos2   
E
2  2   2
 2 
where u, v = displacements in x, y directions
  (3  4) for plane stress problems
y
yy
u

X
a
v
3 
  
 for plane strain problems
 1  

Small Scale plasticity
.
K
Irwin estimates rp  1 ( I ) 2
2  ys
Singularity dominated region
Dugdale strip yield model:
1 KI 2
rp  (
)
8  ys
10
EPFM
In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram.
Ideal elastic brittle behavior
cleavage fracture
Load ratio, P/Py
sharp tip
P: Applied load
Py: Yield load
1.0
Fracture
Displacement, u
Blunt tip
Limited plasticity at crack
tip, still cleavage fracture
Load ratio, P/Py
•
1.0
Fracture
Displacement, u
Void formation & coalescence
failure due to fibrous tearing
Load ratio, P/Py
Blunt tip
Fracture
1.0
Displacement, u
Large scale plasticity
fibrous rapture/ductile
failure
Load ratio, P/Py
large scale
blunting
1.0
Fracture
Displacement, u
EPFM
• EPFM applies to elastic-plastic-rate-independent materials
• Crack opening displacement (COD) or
4 K I2
 
crack tip opening displacement (CTOD).
  2ys E
• J-integral.
J 
 ( wdy  Ti

w 

ij
0
ui
ds ),
xi
ij d ij
y
Sharp crack
x


Blunting crack
ds
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


J
 ij   y 



I
r
 y y n 
Based on the condition (2), inner
radius ro of J dominance. ro 3 COD
R the outer radius where the J
solutions are satisfied within 10% of
complete solution.
1
n 1
 I ij  , n 
ro
R
HRR Singularity…1
Hutchinson, Rice and Rosenbren evaluated the character of crack tip
in power-law hardening materials.
Ramberg-Osgood model,
 
 

  
0  0
0 
n
 0  Reference value of stress=yield strength, n  strain-hardening exponent
0
0 
, strain at yield,   dimensionless constant
E
Note if elastic strains are negligible, then
 

  
 
y
 y
n
 ij 3   eq 
 

y
2   ij 
n 1
ˆ ij
3
;  eq 
ˆ ij
y
2
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HRR Singularity…2
stress and strain fields are given by
1
 n 1
 EJ
 ij   0 

2


I
r
 0 n 
0 
n
 n 1
EJ
 ij  

E  0 2  I n r 
 ij  n, 
 ij  n, 
I n  Integration constant
 ,   Dimensionless functions of n and 
16
HRR Integral, cont.
Note the singularity is of the strenth  1r  n1 . For the specific case of n=1 (linearly
elastic), we have 1r singularity.
1
Note also that the HRR singularity still assumes that the strain is infinitesimal, i.e.,
 ij  12  ui , j  u j ,i  , and not the finite strain Eij  12  ui , j  u j ,i  uk ,iuk , j . Near the tip where
the strain is finite, (typically when ij  0.1), one needs to use the strain measure E.
Some consequences of HRR singularity
In elastic-plastic materials, the singular field is given by
J
 ij  k1  
r
J
 
 ij  k2  
r
1
n 1
1
n 1
(with n=1 it is LEFM)
 stress is still infinite at r  0.
 the crack tip were to be blunt then  xx  0 at r  0 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.
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  xx  0 at r  0 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.
The peak occurs at
x 0
and
J
decreases as x  1. This
corresponds to
approximately twice the width
of CTOD. Hence within this
region, HRR singularity is not
valid.
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.
18
Fracture/Damage theories to model failure
 Fracture Mechanics  Linear solutions leads to singular fieldsdifficult to evaluate 
 Fracture criteria based on K IC ,G IC ,J IC ,CTOD,...
 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
D  1
E

, Effective stress =  
E
1 D
CZM is an Alternative method to Model Separation
 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…
Conceptual Framework of Cohesive Zone Models for interfaces
t1
*
*
u
t1
*
1
u 1*
x  (X, t)
1
s
P
s1
N
P

s2
t
*
2
3
*
u2
3
X , x
1
X , x
2
1
(a)
(d)
,T  n

P


max
n̂
P
S2
t
P

n̂ 1
n̂ 2
*
max
(c)
S1
2
X , x
1
Tn
2
1
u *2
2
(b)
2
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 concept of CZM.
Such an approach is not possible in conventional mechanics of continuous media.
sep
Interface in the undeformed configuration
1 and  2 are separated by a common boundary S,
t1
u 1*
such that
1
S1  1 and S2   2
and normals
*
N1  1 and N 2   2
s
P
Hence in the initial configuration
S  S1  S2
s1
N
s2
t 2*
2
N  N1  N 2
S defines the interface between any two domains
X , x
1 is metal,  2 is ceramic,
S = metal ceramic interface
1 ,  2 represent grains in different orientation,
S = grain boundary
1 ,  2 represent same domain (1   2 =),
S = internal surface yet to separate
3
*
u2
3
X , x
1
X , x
2
1
(a)
2
Interface in the deformed configuration
After deformation a material point X
moves to a new location x, such that
*
t1
u 1*
x  (X,t)
1
if the interface S separates, then a pair of new
S1
surface S 1 and S 2 are created bounding
P

a new domain  such that
N moves to nˆ
*
*


n̂
P
S2
2
(S1 , N1 ) moves to (S 1 , nˆ1 ) (S 1  1  * )
t
(S2 , N 2 ) moves to (S 2 , nˆ2 ) (S 2  2   )
*
 can be considered as 3-D domain made of
*
extremely soft glue, which can be shrunk to an
infinitesimally thin surface but can be expanded
into a 3-D domain.
(d)
P

n̂ 1
n̂ 2
,T  n

P
u *2
1
2
(b)
Constitutive Model for Bounding Domains 1,2
After deformation, given by x  (X,t), if v is the velocity vector,
Then velocity gradient L is given by
v
L
x
Decomposing L into a symmetric part D and antisymmetric part W
such that
L  D W
Where,
D  12 ( L  LT ) and W= 12 ( L  LT )
D is the rate of deformation tensor, and W is the spin tensor
Extending hypo-elastic formulation to inelastic material by
additive decomposition of the rate of deformation tensor
D  D El  D In
where D El and D In are elastic and inelastic part of the rate of deformation tensor
The constitutive model for the domains 1 and  2 can be written as
  C ( D  D In )
where C is elasticity tensor, and  Jaumann rate of cauchy stress tensor.
Constitutive Model for Cohesive Zone 
*
t1
u 1*
A typical constitutive relation for *
Tn
1
is given by T -  relation such that

if   sep ,  nˆ  T
and
if   sep ,  nˆ  T  0
*
(d)
,T  n

P
It can be construed that when   sep
in the domain * , the stiffness Cijkl  0.


max
n̂
P
S2
t
P

n̂ 1
n̂ 2
(c)
S1
P
1
max
2
2
u *2
(b)
sep
Development of CZ Models-Historical Review
Figure (a) Variation of Cohesive
traction (b) I - inner region,
II - edge region
 Barenblatt (1959) was
first to propose the concept
of Cohesive zone model to
brittle fracture
 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 f m  ETo / b E /10 after that
it rapidly reduces to zero with increase in separation distance.
E is Young’s modulus and Tois surface tension
(Barenblatt, G.I, (1959), PMM (23) p. 434)
Phenomenological Models
 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.
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 fragmentation squeezing" by evanescent
waves; brittle-bio materials
Amruthraj etal.,1995, composites
Grujicic, 1999, fracture behavior of polycrystalline; bicrystals
Costanzo etal;1998, dynamic fr.
Ghosh 2000, Interfacial debonding; composites
Rahulkumar 2000 viscoelastic
fracture; polymers
Liechti 2001Mixed-mode, timedepend. rubber/metal debonding
Ravichander, 2001, fatigue
Tevergaard 1992 particle-matrix
interface debonding
Tvergaard etal 1996 elasticplastic solid :ductile frac.; metals
Brocks 2001crack growth in
sheet metal
Camacho &ortiz;1996,impact
Dollar; 1993Interfacial
debonding ceramic-matrix comp
Lokhandwalla 2000, urinary
stones; biomaterials
Fracture process zone and CZM
 CZM essentially models fracture process zone
by a line or a plane ahead of the crack tip
Material
crack tip
subjected to cohesive traction.
 The constitutive behavior is given by traction y
displacement relation, obtained by defining
potential function of the type
     n ,  t1 ,  t2 
x
where  n ,  t1 ,  t 2 are normal and tangential
displacement jump
The interface tractions are given by



Tn  
, Tt1  
, Tt 2  
 n
 t1
 t 2
Mathematical
crack tip
Following the work of Xu and Needleman (1993), the
interface potential is taken as
 
   n ,  t    n   n exp   n
 n
where q  t / 
r   *n /  n
 
 n   1  q  


  1  r 

 n    r  1 
 
 r  q  n 
 2t
q  
  exp   2
   r  1   n 
 t
 

 
 n ,  t are some characteristic distance
 *n Normal displacement after shear separation under the condition
Of zero normal tension
Normal and shear traction are given by
  
2
2


n





1

q
 n 










t
t  



n
n
Tn     exp 
   exp   2   
  r   1  exp   2   


 n 
  n     n 
  t    r  1    n  
  t   


  n    t   2 n     r  q    n 
 n
Tt       
q

exp
 
 


  n    t    t     r  1   n 
 n
 2t 

 exp   2 

 t 
29
Dissipative Micromechanisims Acting in the wake and forward
region of the process zone at the Interfaces of
Monolithic and Heterogeneous Material
Wake of crack tip
̂
Fibril (MMC bridging
 max
Microvoid
coalescence
C
Forward of crack tip
Plastic
zone
Metallic
Cleavage
fracture
Grain bridging
y
Oxide bridging
D
B
NO MATERIAL
SEPARATION
LOCATION OF COHESIVE
CRACK TIP
COMPLETE MATERIAL
SEPARATION
E
A
 max
l1
D
, X
 sep
l2
WAKE
FORWARD
Thickness of
ceramic interface
Crack Meandering
Plastic wake
Fibril(polymers)
bridging
Intrinsic dissipation
MATERIAL
CRACK TIP
MATHEMATICAL
CRACK TIP
COHESIVE
CRACK TIP
Precipitates
Crack Deflection
Crack Meandering
Ceramic
Extrinsic dissipation
Micro cracking
initiation
Micro void
growth/coalescence
Contact Wedging
INACTIVE PLASTIC ZONE
(Plastic wake)
 sep
E
 D  max
D
WAKE
A
C
Contact Surface
(friction)
Plasticity induced
crack closure
FORWARD
Delamination
Corner atoms
Plastic W ake
Face centered
atoms
FCC
Phase
transformation
y
Corner atoms
ACTIVE PLASTIC ZONE
Cyclic load induced
crack closure
x
ELASTIC SINGULARITY ZONE
Concept of wake and forward region in the
cohesive process zone
BCC
Body centered
atoms
Inter/trans granular
fracture
Active dissipation mechanisims participating at the cohesive process zone30
31
32
Motivation for studying CZM
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
critical issues addressed here
Scales- What range of CZM parameters
are valid?
MPa or GPa for the traction
J or KJ for cohesive energy
nm or m 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.
Importance of
shape of CZM
33
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 m 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?)
34
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 233)
The total internal
energy
E 
E of the crystal
tot
i
i
where
and
Ei  F   i  
1
2
  r 
j 1
5
ij
i   f  rij 
3
2
Internal energy associated with atom i
F  i  
ij 
Embedded Energy of atom i.
Contribution to electron density
of ith atom and jth atom.
f  rij   Two body central potential
between ith atom and jth atom.
Energy (eV)
j 1
Ei 
Al
Mg
Cu
4
Cutoff Distances
(4.86) (5.44) (6.10)
1
0
-1
2
4
6
Atomic Seperation (A)
-2
-3
-4
-5
35
T   Curve in Shear direction
A small portion of 9(221) CSL grain bounary before
And after application of tangential force
Shet C, Li H, Chandra N ;Interface models for GB sliding and migration
MATER SCI FORUM 357-3: 577-585 2001
T   Curve in Normal direction
A small portion of 9(221) CSL grain boundary before
And after application of normal force
37
Results and discussion on atomistic simulation
Summary
 complete debonding occurs when the
distance of separation reaches a value of 2
to 3 A .
 For 9 bicrystal tangential work of
separation along the grain boundary is of
the order 3 J / m2 and normal work of
2
separation is of the order 2.6 J / m .
 For 3 -bicrystal, the work of separation
ranges from 1.5 to 3.7 J / m2 .
 Rose et al. (1983) have reported that the
adhesive energy (work of separation) for
aluminum is of the order 0.5 J / m2 and the
separation distance 2 to 3 A
 Measured energy to fracture copper
bicrystal with random grain boundary is
of the order 54 J / m2 and for 11 copper
bicrystal the energy to fracture is more
than 8000 J / m2
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.
  2  Wp
It should also be noted that the experimental value of fracture energy
includes the plastic work in addition
to work of separation
(J.R Rice and J. S Wang, 1989)
Table of surface and fracture energies of standard materials
K IC MPam1/ 2
G IC J / m2
Material
Nomenclature
Aluminium
alloys
2024-T351
35
14900
1.2
2024-T851
25.4
8000
1.2
T21
80
48970
2-4
T68
130
130000
2-4
Medium
Carbon
54
12636
2-4
High strength
alloys
98
41617
18 Ni (300)
maraging
Al 2O3
76
25030
4-8
34-240
Titanium
alloys
Steel
Alumina
SiC ceramics
Polymers
6.1
PMMA
1.2-1.7
220
 J / m2
particle size
10 m
0.11 to 1.28 m
Energy balance and effect of plasticity in the
bounding material
40
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 T   curve.
Cohesive zone parameters of a ductile material
 Al 2024-T3 alloy
 The input energy in the cohesive model
are related to the interfacial stress and
characteristic displacement  n as
 n   max e n
e
 max  t
2
 The input energy n is equated to
material parameter
 Based on the measured fracture value J IC
t 
n  t  8000J / m 2
 max   ult  642MPa
 n   t  4.5 X 106 m
42
Material model for the bounding material
 Elasto-plastic model for Al 2024-T3
Stress strain curve is given by
1/ n

 

  
E
 y 
where  y  320MPa,
  0.01347,
n  0.217173
E=72 GPa, =0.33,
and fracture parameter
K IC  25MPa  m1/ 2
43
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
1
• The virtual work due to cohesive zone traction in a
given cohesive element can be written as
   dS    T  
n
n
Continuum
elements
3
5
2
4
6
7
8
Cohesive
element
 Tt   t dS
The virtual displacement jump is written as   [N]{v}
Where [N]=nodal shape function matrix, {v}=nodal displacement vector
   dS  {v}  [N] d{T }  [N] d{T }
T
T
s
T
n
t
1
J
dS
J = Jacobian of the transformation between the current deformed
and original undeformed areas of cohesive surfaces
Note: T is written as d{T}- the incremental traction, ignoring time
which is a pseudo quantity for rate independent material
44
Numerical formulation contd
The incremental tractions are related to incremental displacement jumps
across a cohesive element face through a material Jacobian matrix as
d{T}  [C cz }d{}
For two and three dimensional analysis Jacobian matrix is given by
T  n
[C cz ]   n
 Tt  n
Tn  t 
Tt  t 
 Tn  n
[C cz ]   Tt1  n

Tt 2  n
Tn  t1
Tt1  t1
Tt 2  t1
Tn  t 2 
Tt1  t 2 

Tt 2  t 2 
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
[K T ]   [N]T [C cz ][N] 1J dS
s
45
Geometry and boundary/loading conditions
a = 0.025m, b = 0.1m, h = 0.1m
46
Finite element mesh
28189 nodes, 24340 plane strain 4 node elements,
7300 cohesive elements (width of element along the crack plan is ~ 7x107 m
47
Global energy distribution
E w  Ee  E p  Ec
E e and E p are confined to bounding material
Ec is cohesive energy, a sum total of all dissipative
process confined to FPZ and cannot be recovered
during elastic unloading and reloading.
 Purely elastic analysis
The conventional fracture mechanics uses the concept
of strain energy release rate
U
GJ
a
Using CZM, this fracture energy
  G  J  8000J / m2 is dissipated and no plastic
dissipation occurs, such that
E w  Ee  Ec
48
Global energy distribution (continued)
Analysis with elasto-plastic material model
Two dissipative process
  8000J / m 2
Plasticity within
Bounding material

Issues
Fracture energy obtained from experimental 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?
Micro-separation
Process in FPZ
 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


E p  E p  max , n,Si  ,i  1,2,..
 y

where Si represents other factors arising from
the shape of the traction-displacement relations
What are the key CZM parameters that govern the energetics?
 max in cohesive zone dictates the stress level achievable in the bounding
material.
 Yield in the bounding material depends on its yield strength  y and its post
yield (hardening characteristics).
 Thus  max  y 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)
Global energy distribution (continued)
Recoverable elastic work Ee  95 to
98% of external work
  max  y  1.5 :  plasticity occurs.
 Plasticity increases with  max  y
Cumulative Cohesive Energy
3
2.5
4
2
3
1.5
2
1
1
0.5
0
0
20
40
u / n
60
80
8
 max  y  1 to 1.5 :  Elastic behavior
Cumulative Plastic Work
3.5
Energy/(y n  1.0E-2)
Plastic dissipation depends on  max  y
4
Variation of cohesive energy and plastic energy for
various max y ratios
(1) max y  1 (2) max y  1.5
(3) max y  2.0 (4) max y  2.5
51
Relation between plastic work and cohesive work
  max  y  1.5 (very small scale plasticity),
plastic energy ~ 15% of total dissipation.
  max  y  2.0 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 8000J / m2 .
 Plastic dissipation very sensitive to  max  y
ratio beyond 2 till 3
 Crack cannot propagate beyond  max  y  3
and completely elastic below max  y  1.5
3
Cohesive Energy/( y n 1.0E-2)
Plasticity induced at the initial stages
of the crack growth
plasticity ceases during crack
propagation.
Very small error is induced by ignoring
plasticity.
max  y  
max y = 2.0
2.5
2
 max y = 2.5
1.5
1
0.5
0
0
1
2
3
Plastic Energy/(  y  n 1.0E-2)
4
Variation of Normal Traction along the interface
The length of cohesive zone is also
affected by  max  y ratio.
For lower  max  y ratios the
traction-separation curve flattens, this
tend to increase the overall cohesive
zone length.
l2
Curve  max 
1
1
2
Tn /  y 
There is a direct correlation
between the shape of the tractiondisplacement curve and the normal
traction distribution along the
cohesive zone.
1.2
0.8
l2
3
y
l 2( m)
1
2.5
1100
2
2.0
2900
3
1.5
4800
4
1.0
11000
0.6
4
0.4
l2
0.2
l2
0
0
0.003
0.006
0.009
0.012
0.015
0.018
x(m)
53
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 m).
They are spaced equally from each other.
 Adjoining these patches, patches of cohesive
elements are considered to record the cohesive
energies.
54
Variation of Cohesive Energy
The cohesive energy in the patch increases
up to point C (corresponding to  max in
Figure ) after which the crack tip is
presumed to advance.
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 T  
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.
Cohesive Energy/(  y n 1.0E-7)
The energy consumed by the cohesive
elements at this stage is approximately 1/7
of the total cohesive energy for the present
CZM.
800
600
1
400
200
0
0.24
C
2
C C
0.25
3
4 5
C C C
0.26
0.27
6
C
7
C
0.28
a/b
8
C
9 10
Tn
max
max
C
0.29
0.3
0.31
sep
0.32
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
Variation of Elastic Energy
100
90
Elastic Energy/( y n  1.0E-8)
Considerable elastic energy is built up till
the peak of T   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
T   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 1
Crack tip blunts for advancing crack tip
leading to a lower levels of stress, resulting in
reduced energy level in other patches.
1
80
70
2 3
4 5
6
7
8
9
10
60
Tn
max
50
40
max
30
sep
20
10
0
0.24
0.25
0.26
0.27
0.28
0.29
0.3
0.31
0.32
0.33
a/b
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
Variation of Plastic Work ( max y  2.0)
plastic energy accumulates considerably
along with elastic energy, when the local
stresses bounding material exceeds the yield  y
After reaching peak point C on T   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 zone
The 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 Ecand Ee 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
Tn
max
max
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  1.5)
max y  1, there is no plastic dissipation.
 max y  1.5plastic work is induced only
in the first patch of element
125
 No plastic dissipation during crack
growth place in the forward region
Tn
 During crack propagation, tip blunts
resulting reduced level of stresses
leading to reduced elastic energies and
no plasticity condition.
Energy/(y n  1.0E-8)
 Initial sharp crack tip profile induces
high levels of stress and hence plasticity
in bounding material.
max
Plastic Work
100
max
75
sep
Elastic Energy
1
50
2
3
4
5 6
7
9
8
25
0
0.24
0.25
0.26
0.27
0.28
0.29
0.3
0.31
0.32
a/b
Variation of Plastic work and Elastic work in various patch
of elements along the interface for the case of  max  y  1.5 .
The numbers indicates the energy in various patch of
elements starting from the crack tip.
58
Contour plot of yield locus around the cohesive
crack tip at the various stages of crack growth.
59
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.
 max  yratio affects length of fracture process zone length. For smaller  max  y
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  max  y ratio.
Conclusion(contd.)
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 J IC 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.
62