Theoretical and Computational Aspects of Cohesive Zone Modeling

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Theoretical and Computational Aspects of Cohesive
Zone Modeling
NAMAS CHANDRA Department of Mechanical
Engineering FAMU-FSU College of
Engineering Florida State University Tallahassee,
Fl-32310
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• 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

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• CZM can create new surfaces.
• Maintains continuity conditions mathematically,
  despite the physical separation.
• CZM represents physics of the fracture process at
  the atomic scale.
• It can also be perceived at the meso- scale as
  the effect of energy dissipation mechanisms,
  energy dissipated both in the forward and the
  wake regions of the crack tip.
• Uses fracture energy(obtained from fracture
  tests) as a parameter and is devoid of any ad-hoc
  criteria for fracture initiation and propagation.
• Eliminates singularity of stress and limits it to
  the cohesive strength of the the material.
• It is an ideal framework to model strength,
  stiffness and failure in an integrated manner.
• Applications geomaterials, biomaterials,
  concrete, metallics, composites.

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Conceptual Framework of Cohesive Zone Models for
interfaces
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• 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 Youngs modulus
  and is surface tension
•  
•  

(Barenblatt, G.I, (1959), PMM (23) p. 434)
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• For Ductile metals (steel)
• Cohesive stress in the CZM is equated to yield
  stress Y
• Analyzed for plastic zone size for plates under
  tension
• Length of yielding zone s, theoretical crack
  length a, and applied loading T are related
  in
• the form

(Dugdale, D.S. (1960), J. Mech.Phys.Solids,8,p.100
)
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• 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 fragm-
• entation squeezing" by evanescent waves
  brittle-bio materials
• Amruthraj etal.,1995, composites

• Grujicic, 1999, fracture beha-vior of
  polycrystalline bicrystals
• Costanzo etal1998, 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 ortiz1996,impact
• Dollar 1993Interfacial debonding ceramic-matrix
  comp
• Lokhandwalla 2000, urinary stones biomaterials

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• CZM essentially models fracture process zone by
• a line or a plane ahead of the crack tip
  subjected
• to cohesive traction.
• The constitutive behavior is given by
  traction-displacement relationship, obtained by
  defining potential function of the type

y
where
are normal and tangential displacement jump
The interface tractions are given by
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Critical Issues in the application of CZM to
interface models

• What is the relationship between the
  physics/mechanics of the separation process and
  shape of CZM? (There are as many shapes/equations
  as there are number of interface problems
  solved!)
• What is the relationship between CZM and
  fracture mechanics of brittle, semi-brittle and
  ductile materials?
• What is the role of scaling parameter in the
  fidelity of CZM to model interface behavior?
• What is the physical significance of
• - Shape of the curve C
• - tmax and interface strength
• - Separation distance ?sep and COD?
• - Area under the curve, work of fracture,
  fracture toughness G (local and global)

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CZM is an excellent tool with sound theoretical
basis and computational ease. Lacks proper
mechanics and physics based analysis and
evaluation. Already widely used in
fracture/fragmentation/failure
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• Construct symmetric tilt boundaries (STDB) by
  rotating a
• single crystal (reflection)
• Periodic boundary condition in X direction
• Restrain few layers in lower crystal
• Apply body force on top crystal

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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 migrationMATER SCI FORUM 357-3
577-585 2001
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A small portion of CSL grain
boundary before And after application of normal
force
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• 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 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

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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
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Energy balance and effect of plasticity in the
bounding material
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Motivation

• It is perceived that CZM represents
• the physical separation process.
• As seen from atomistics, fracture
• process comprises mostly of inelastic
• dissipative energies.
• There are many inelastic dissipative
• process specific to each material
• system some occur within FPZ, and
• some in the bounding material.
• How the energy flow takes place
• under the external loading within the
• cohesive zone and neighboring
• bounding material near the crack tip?
• What is the spatial distribution of
• plastic energy?
• Is there a link between micromechanics
• processes of the material and curve.

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Plasticity vs. other Dissipation Mechanisms

• Since bounding material has its own
• inelastic constitutive equation, what
• is the proportion of energy dissipation
• within that domain and fracture region
• given by CZM.
• Role of plasticity in the bounding
• material is clearly unique and cannot
• be assigned to CZM.

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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

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Material model for the bounding material

• Elasto-plastic model for Al 2024-T3

Stress strain curve is given by
where
E72 GPa, ?0.33,
and fracture parameter
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Geometry and boundary/loading conditions
a 0.025m, b 0.1m, h 0.1m
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Finite element mesh
28189 nodes, 24340 plane strain 4 node elements,
7300 cohesive elements (width of element along
the crack plan is m
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Global energy distribution
are confined to bounding material
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Global energy distribution (continued)

• Analysis with elasto-plastic material model

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What are the key CZM parameters that govern the
energetics?
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Global energy distribution (continued)

• Variation of cohesive energy and plastic energy
  for various ratios
• (2)
• (3) (4)

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Relation between plastic work and cohesive work
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Variation of Normal Traction along the interface

• The length of cohesive zone is also
  affected by ratio.
• There is a direct correlation between the shape
  of the traction-displacement curve and the normal
  traction distribution along the cohesive zone.
• For lower ratios the
  traction-separation curve flattens, this tend to
  increase the overall cohesive zone length.

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Local/spatial Energy Distribution
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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
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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
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Variation of dissipated plastic energy in various
patched as a function of crack extension. The
number indicate patch numbers starting from
initial crack tip.
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Variation of Plastic Work ( )
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.
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Contour plot of yield locus around the cohesive
crack tip at the various stages of crack growth.
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Schematic of crack initiation and propagation
process in a ductile material
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Conclusion
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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.