Modeling Fracture in Elastic-plastic Solids Using Cohesive Zones

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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
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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

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Formulation of a general boundary value problem
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Formulation of a general boundary value problem
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For problems with crack tip Westergaard
introduced Airys stress function as
Where Z is an analytic complex function
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Opening mode analysis or Mode I
Consider an infinite plate a crack of length 2a
subjected to a biaxial State of stress. Defining
By replacing z by za , origin shifted to crack
tip.
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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
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Opening mode analysis or Mode I
From Hookes law, displacement field can be
obtained as
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Small Scale plasticity
. Irwin estimates Dugdale strip yield model
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• EPFM
• In EPFM, the crack tip undergoes significant
  plasticity as seen in the following diagram.

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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
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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.
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HRR Singularity1
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HRR Singularity2
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HRR Integral, cont.
Note the singularity is of the strenth .
For the specific case of n1 (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 n1 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.

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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.
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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 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

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Conceptual Framework of Cohesive Zone Models for
interfaces
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Interface in the undeformed configuration
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Interface in the deformed configuration
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Constitutive Model for Bounding Domains W1,W2
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Constitutive Model for Cohesive Zone W
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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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• 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 relation, 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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• 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

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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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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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• 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.

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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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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
1 2
3 4

• The virtual work due to cohesive zone traction in
  a given cohesive element can be written as

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Cohesive element
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The virtual displacement jump is written as
Where Nnodal shape
function matrix, vnodal displacement vector
J Jacobian of the transformation between the
current deformed and original undeformed areas
of cohesive surfaces Note is written as
dT- the incremental traction, ignoring time
which is a pseudo quantity for rate independent
material
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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

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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.