Presentation on theme: "Theoretical and Computational Aspects of Cohesive Zone Modeling"— Presentation transcript:
1 Theoretical and Computational Aspects of Cohesive Zone Modeling NAMAS CHANDRADepartment of Mechanical EngineeringFAMU-FSU College of EngineeringFlorida State UniversityTallahassee, Fl-32310AMML
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 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.
3 Fracture/Damage theories to model failure Fracture Mechanics -Linear solutions leads to singular fields-difficult to evaluateFracture criteria based onNon-linear domain- solutions are not uniqueAdditional criteria are required for crack initiation and propagationBasic breakdown of the principles of mechanics of continuous mediaDamage mechanics-can effectively reduce the strength and stiffness of the material in an average sense, but cannot create new surface
4 CZM is an Alternative method to Model Separation 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….
6 Conceptual Framework of Cohesive Zone Models for interfaces
7 Development of CZ Models-Historical Review Figure (a) Variation of Cohesive traction (b) I - inner region, II - edge regionBarenblatt (1959) wasfirst to propose the conceptof Cohesive zone model tobrittle fractureMolecular 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 thatit 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)
8 Dugdale (1960) independently developed the concept of cohesive stress For Ductile metals (steel)Cohesive stress in the CZM is equated to yield stress YAnalyzed for plastic zone size for plates under tensionLength of yielding zone ‘s’, theoretical crack length ‘a’, and applied loading ‘T’ are related inthe form(Dugdale, D.S. (1960), J. Mech.Phys.Solids,8,p.100)AMML
9 Phenomenological Models The theory of CZM is based on sound principles.However implementation of model for practical problems grew exponentially forpractical problems with use of FEM and advent of fast computing.Model has been recast as a phenomenological one for a number of systems andboundary value problems.The phenomenological models can model the separation process but not the effect ofatomic discreteness.Hillerborg etal Ficticious crack model; concreteBazant etal.1983 crack band theory; concreteMorgan etal earthquake rupture propagation; geomaterialPlanas etal,1991, concreteEisenmenger,2001, stone fragm-entation squeezing" by evanescent waves; brittle-bio materialsAmruthraj etal.,1995, compositesGrujicic, 1999, fracture beha-vior of polycrystalline; bicrystalsCostanzo etal;1998, dynamic fr.Ghosh 2000, Interfacial debo-nding; compositesRahulkumar 2000 viscoelastic fracture; polymersLiechti 2001Mixed-mode, time-depend. rubber/metal debondingRavichander, 2001, fatigueTevergaard 1992 particle-matrix interface debondingTvergaard etal 1996 elastic-plastic solid :ductile frac.; metalsBrocks 2001crack growth in sheet metalCamacho &ortiz;1996,impactDollar; 1993Interfacial debonding ceramic-matrix compLokhandwalla 2000, urinary stones; biomaterials
10 Fracture process zone and CZM CZM essentially models fracture process zone bya line or a plane ahead of the crack tip subjectedto cohesive traction.The constitutive behavior is given by traction-displacement relationship, obtained by defining potential function of the typeMaterial crack tipMathematicalcrack tipxywhereare normal and tangential displacement jumpThe interface tractions are given by
13 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)
14 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/failurecritical issues addressed hereScales- What range of CZM parameters are valid?MPa or GPa for the tractionJ or KJ for cohesive energynm or for separationdisplacementWhat is the effect of plasticity in the bounding material on the fracture processesImportance of shape of CZMEnergy- Energy characteristics duringfracture process and how energyflows in to the cohesive zone.AMML
15 Atomistic simulations to extract cohesive properties MotivationWhat is the approximate scale to examinefracture in a solidAtomistic at nm scale orGrains at scale orContinuum at mm scaleAre the stress/strain and energy quantities computed at one scale be valid at other scales? (can we even define stress-strain at atomic scales?)AMML
16 (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 crystalwhereandContribution to electron density of ith atom and jth atom.Two body central potential between ith atom and jth atom.Internal energy associated with atom iEmbedded Energy of atom i.AMML
18 Boundary Conditions for GB Sliding Construct symmetric tilt boundaries (STDB) by rotating asingle crystal (reflection)Periodic boundary condition in X directionRestrain few layers in lower crystalApply body force on top crystalA three dimensional bicrystal as shown above is constructed.To start with a single crystal is obtained for the given orientation. This crystal is rotated about the grain boundary plane to get STGBPeriodic BC in (X) direction. Few atomic layers in the lower crystal is fixed in direction. All the atoms in the upper crystal is subjected to body force. Top few layers in the upper crystal is restrained in Y-directionAMML
19 Curve in Shear direction A small portion of CSL grain bounary beforeAnd after application of tangential forceShet C, Li H, Chandra N ;Interface models for GB sliding and migration MATER SCI FORUM 357-3:
20 Curve in Normal direction A small portion of CSL grain boundary beforeAnd after application of normal forceAMML
21 Results and discussion on atomistic simulation ImplicationsThe numerical value of the cohesiveenergy is very low when comparedto the observed experimental resultsAtomistic simulation gives onlysurface energy ignoring the inelasticenergies due to plasticity and othermicro processes.It should also be noted that the exper-imental value of fracture energyincludes the plastic work in additionto work of separation(J.R Rice and J. S Wang, 1989)Summarycomplete debonding occurs when the distance of separation reaches a value of 2 toFor 9 bicrystal tangential work of separation along the grain boundary is of the order 3 and normal work of separation is of the orderFor 3 -bicrystal, the work of separation ranges from 1.5 toRose et al. (1983) have reported that the adhesive energy (work of separation) for aluminum is of the order and the separation distance 2 to 3Measured 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
22 Table of surface and fracture energies of standard materials Nomenclatureparticle sizeAluminium alloys2024-T35135149001.22024-T85125.48000Titanium alloysT2180489702-4T68130130000SteelMedium Carbon5412636High strength alloys984161718 Ni (300) maraging7625030Alumina4-834-24010SiC ceramics6.10.11 to 1.28PolymersPMMA220
23 Energy balance and effect of plasticity in the bounding material AMML
24 Motivation It is perceived that CZM represents the physical separation process.As seen from atomistics, fractureprocess comprises mostly of inelasticdissipative energies.There are many inelastic dissipativeprocess specific to each materialsystem; some occur within FPZ, andsome in the bounding material.How the energy flow takes placeunder the external loading within thecohesive zone and neighboringbounding material near the crack tip?What is the spatial distribution ofplastic energy?Is there a link between micromechanicsprocesses of the material and curve.
25 Plasticity vs. other Dissipation Mechanisms Since bounding material has its owninelastic constitutive equation, whatis the proportion of energy dissipationwithin that domain and fracture regiongiven by CZM.Role of plasticity in the boundingmaterial is clearly unique; and cannotbe assigned to CZM.AMML
26 Cohesive zone parameters of a ductile material Al 2024-T3 alloyThe input energy in the cohesive model are related to the interfacial stress and characteristic displacement asThe input energy is equated to material parameterBased on the measured fracture valueAMML
27 Material model for the bounding material Elasto-plastic model for Al 2024-T3Stress strain curve is given bywhereE=72 GPa, =0.33,and fracture parameterAMML
28 Geometry and boundary/loading conditions a = 0.025m, b = 0.1m, h = 0.1mAMML
29 Finite element mesh AMML 28189 nodes, plane strain 4 node elements,7300 cohesive elements (width of element along the crack plan is ~ mAMML
30 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 analysisThe conventional fracture mechanics uses the concept of strain energy release rateUsing CZM, this fracture energyis dissipated and no plastic dissipation occurs, such thatare confined to bounding materialAMML
31 Two dissipative process Global energy distribution (continued)Analysis with elasto-plastic material modelTwo dissipative processPlasticity withinBounding materialMicro-separationProcess in FPZIssuesFracture 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 relationsImplicationsLeaves no energy for plastic work in thebounding materialIn what ratio it should be divided?Division is non-trivial since plastic dissipation depends on geometry, loading and other parameters as
32 What are the key CZM parameters that govern the energetics? in cohesive zone dictates the stress level achievable in the boundingmaterial.Yield in the bounding material depends on its yield strength and its postyield (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)
33 Global energy distribution (continued) Recoverable elastic work to98% of external workPlastic dissipation depends onElastic behaviorplasticity occurs.Plasticity increases withVariation of cohesive energy and plastic energy for various ratios(2)(3) (4)AMML
34 Relation between plastic work and cohesive work (very small scale plasticity),plastic energy ~ 15% of total dissipation.Plasticity induced at the initial stagesof the crack growthplasticity ceases during crackpropagation.Very small error is induced by ignoringplasticity.plastic work increasesconsiderably, ~100 to 200% as that ofcohesive energy.For large scale plasticity problems theamount of total dissipation (plastic andcohesive) is much higher than 8000Plastic dissipation very sensitive toratio beyond 2 till 3Crack cannot propagate beyondand completely elastic below
35 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.AMML
36 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 detailsAMML
37 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 flattens 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
38 Variation of Elastic Energy Considerable elastic energy is built up tillthe peak of curve is reached afterwhich the crack tip advances.After passing C, the cohesive elements nearthe crack tip are separated and the elementsin this patch becomes a part of the wake.At this stage, the values of normal tractionreduces following the downward slope ofcurve following which the stress in thepatch reduces accompanied by reduction inelastic 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.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
39 Variation of Plastic Work ( ) plastic energy accumulates considerably along with elastic energy, when the local stresses bounding material exceeds the yieldAfter 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 beyondVariation of dissipated plastic energy in various patched as a function of crack extension. The number indicate patch numbers starting from initial crack tip.
40 Variation of Plastic Work ( ) , there is no plastic dissipation.plastic work is induced onlyin the first patch of elementNo plastic dissipation during crackgrowth place in the forward regionInitial sharp crack tip profile induceshigh levels of stress and hence plasticityin bounding material.During crack propagation, tip bluntsresulting reduced level of stressesleading to reduced elastic energies andno 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.AMML
41 Contour plot of yield locus around the cohesive crack tip at the various stages of crack growth.AMML
42 Schematic of crack initiation and propagation process in a ductile material
43 ConclusionCZM provides an effective methodology to study and simulate fracture in solids.Cohesive Zone Theory and Model allow us to investigate in a much morefundamental manner the processes that take place as the crack propagates in anumber of inelastic systems. Fracture or damage mechanics cannot be used inthese cases.Form and parameters of CZM are clearly linked to the micromechanics.Our study aims to provide the modelers some guideline in choosing appropriateCZM for their specific material system.ratio affects length of fracture process zone length. For smallerratio the length of fracture process zone is longer when compared with that ofhigher ratio.Amount of fracture energy dissipated in the wake region, depend on shape ofthe model. For example, in the present model approximately 6/7th of totaldissipation takes place in the wakePlastic work depends on the shape of the crack tip in addition to ratio.
44 Conclusion(contd.)The CZM allows the energy to flow in to the fracture process zone, where apart 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 regionis used up in advancing the crack tip.The part of energy spent as intrinsic dissipation in the wake region is requiredto complete the gradual separation process.In case of elastic material the entire fracture energy given by the of thematerial, and is dissipated in the fracture process zone by the cohesiveelements, 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, plasticdissipation is negligible in the forward region.AMML
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