1. 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

2. 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

3. 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.

4. Formulation of a general boundary value problemx y

5. Formulation of a general boundary value problem

6. For problems with crack tip Westergaard introduced Airy’s stress
function as
Where Z is an analytic complex function
And are 2nd and 1st integrals of Z(z)
Then the stresses are given by

7. Opening mode analysis or Mode I
Consider an infinite plate a crack of length 2a subjected to a biaxial
State of stress. Defining:
Boundary Conditions :
At infinity
On crack faces
By replacing z by z+a , origin shifted to crack tip.

8. Opening mode analysis or Mode I
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.
Since origin is shifted to crack tip, it is easier to use polar Coordinates, Using

9. Opening mode analysis or Mode I
From Hooke’s law, displacement field can be obtained as

10. Small Scale plasticity
Singularity dominated region .
Irwin estimates
Dugdale strip yield model:

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

13. EPFM EPFM applies to elastic-plastic-rate-independent materials
Crack opening displacement (COD) or crack tip opening displacement (CTOD).
J-integral. y x
Sharp crack
Blunting crack ds

14. More on J Dominance
Limitations of J integral, (Hutchinson, 1993)
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
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.

15. HRR Singularity…1

16. HRR Singularity…2

17. Some consequences of HRR singularity
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 .
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.

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.
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.
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.

19. Fracture/Damage theories to model failure
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

20. 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…

21. Conceptual Framework of Cohesive Zone Models for interfaces

22. Interface in the undeformed configuration

23. Interface in the deformed configuration

24. Constitutive Model for Bounding Domains W1,W2

25. Constitutive Model for Cohesive Zone W*

26. 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 after that
it rapidly reduces to zero with increase in separation distance.
E is Young’s modulus and is surface tension
(Barenblatt, G.I, (1959), PMM (23) p. 434)

27. 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 Ficticious crack model; concrete
Bazant etal.1983 crack band theory; concrete
Morgan etal 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; composites
Rahulkumar 2000 viscoelastic fracture; polymers
Liechti 2001Mixed-mode, time-depend. rubber/metal debonding
Ravichander, 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,impact
Dollar; 1993Interfacial debonding ceramic-matrix comp
Lokhandwalla 2000, urinary stones; biomaterials

28. Fracture process zone and CZM
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
Material crack tip
Mathematical
crack tip x y
where
are normal and tangential displacement jump
The interface tractions are given by

29. Following the work of Xu and Needleman (1993), the interface potential is taken as
where
are some characteristic distance
Normal displacement after shear separation under the condition
Of zero normal tension
Normal and shear traction are given by

33. 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 for separation
displacement
What is the effect of plasticity in the bounding material on the fracture processes
Importance of shape of CZM
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?)

35. (D.J.Oh and R.A.Johnson, 1989 ,Atomic Simulation of Materials,
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 of the crystal
where
and
Contribution to electron density of ith atom and jth atom.
Two body central potential between ith atom and jth atom.   
Internal energy associated with atom i
Embedded Energy of atom i.

36. Curve in Shear direction
A small portion of 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:

37. Curve in Normal direction
A small portion of CSL grain boundary before
And after application of normal force

38. 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)
Summary
complete debonding occurs when the distance of separation reaches a value of 2 to
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
For 3 -bicrystal, the work of separation ranges from 1.5 to
Rose et al. (1983) have reported that the adhesive energy (work of separation) for aluminum is of the order and the separation distance 2 to 3
Measured energy to fracture copper bicrystal with random grain boundary is of the order and for 11 copper bicrystal the energy to fracture is more than 8000

39. Table of surface and fracture energies of standard materials
Nomenclature
particle size
Aluminium alloys
2024-T351 35
14900
1.2
2024-T851
25.4
8000
Titanium alloys
T21 80
48970
2-4
T68
130
130000
Steel
Medium Carbon 54
12636
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
220

40. Energy balance and effect of plasticity in the bounding material

41. 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.

42. 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 as
The input energy is equated to material parameter
Based on the measured fracture value

43. Material model for the bounding material
Elasto-plastic model for Al 2024-T3
Stress strain curve is given by
where
E=72 GPa, =0.33,
and fracture parameter

44. 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
Continuum
elements
The virtual work due to cohesive zone traction in a given cohesive element can be written as
Cohesive
element
The virtual displacement jump is written as
Where [N]=nodal shape function matrix, {v}=nodal displacement vector
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

45. Numerical formulation contd
The 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

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 ~ m

48. Global energy distribution
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
Using CZM, this fracture energy
is dissipated and no plastic dissipation occurs, such that
are confined to bounding material

49. Two dissipative process
Global energy distribution (continued)
Analysis with elasto-plastic material model
Two dissipative process
Plasticity within
Bounding material
Micro-separation
Process in FPZ
Issues
Fracture 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?
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

50. 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)

51. Global energy distribution (continued)
Recoverable elastic work to
98% of external work
Plastic dissipation depends on
Elastic behavior
plasticity occurs.
Plasticity increases with
Variation of cohesive energy and plastic energy for various ratios
(2)
(3) (4)

52. Relation between plastic work and cohesive work
(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

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.

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 ( ). They are spaced equally from each other.
Adjoining these patches, patches of cohesive elements are considered to record the cohesive energies.
To establish how the energies are absorbed within the cohesive zone along its length.
To provide good physics basis to postulate various types of micromechanisms active in the forward and wake regions of the crack.
To establish relationship between the spatial distribution of energy flow in the fracture process zone, dissipative mechanisms that absorb this energy, and an overall manifestation of traction-separation law that embodies those details

55. Variation of Cohesive Energy
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.
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

56. Variation of Elastic Energy
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 1
Crack tip blunts for advancing crack tip leading to a lower levels of stress, resulting in reduced energy level in other patches.
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

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

58. Variation of Plastic Work ( )
, 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.
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.

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

60. Schematic of crack initiation and propagation
process in a ductile material

61. 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.

62. 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 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.