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.