A time-dependent cohesive zone model is employed to model adhesive failure and grain-boundary cracking. Through the incorporation of viscoelasticity and evolving damage, the viscoelastic cohesive zone model (VCZM) yields a time-dependent critical energy release rate. A double-cantilever beam configuration is investigated. Crack growth curves are presented for the Xu-Needleman and VCZM model. In addition, a fifty-five grain polycrystal is simulated in compression. Time-dependent grain boundary prying and sliding result in inter-granular separation parallel to the major axis of loading

Allen, D. H.

Foulk, J. W.

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Predicting Time-Dependent Interracial Behaviorwith a Viscoelastic Cohesive Zone ModelJ.W. FoulkSandia National Laboratories. The snbmitted mm~~ hag ~Livermore, CA, 94551, USAauthored by a contractor of&e UnitedStates Government onder contiact.Accordingly the United States Gov-D.H. Allenemment retains a non- exclnsi~royalty-bee license to publish or re-Department of Aerospace Engineering produce the published form of thieTexas A&M Universitycontribution, or allow others to do eo,for United Stat08 GOCollege Station, TX 77843-3141, USA ~.vernment~SummaryA time-dependent cohesive zone model is employed to model adhesive failure and grain-boundary cracking.Through the incorporation of viscoelasticity and evolving damage, the viscoleastic cohesive zone model(VCZM) yields a time-dependent critical energy release rate. A double-cantilever beam configuration isinvestigated. Crack growth curves are presented for the Xu-Needleman and VCZM model. IiI addition, afifty-five grain polycrystal is simulated in compression. Time-dependent grain bounday prying and slidingresult in inter-granular separation parallel to the major axis of loading.Introduction. Many material systems exhibit time-dependent fracture. Although much research has focused onpredicting the bulk behavior of organics and metals, fewer works have addressed rate-dependent crackgrowth. The situation is further complicated by the introduction of material interfaces on different lengthscales.The complications associated with material interfaces can be simplified through the introduction ofcohesive zone models. First proposed by Dugdale and Barenblatt, cohesive zone models provide an eleganttransition from crack initiation to traction-free boundmies. Since the works of Needleman [1], numerousinvestigators have employed cohesive zones to model de-adhesion in composite systems. These studies,however, have traditionally been performed with rate-independent cohesive zone laws. Knauss and Losi [2]proposed a rate-dependent cohesive zone constitutive law incorporating viscoelasticity and damage. Inaddition, Knauss [3] addressed crazing in thermoplastic polymers within the context of the model.Utilizing the Helmhokz free energy, Yoon and Allen [4] proposed a cohesive zone constitutiveequation in the form of a single hereditary integral for a linear viscoelastic material. The model relies on aninternal state variable which reflects the damage state of the cohesive zone. Allen and Searcy [5] have alsoarrived at a similar formulation through rnicromechanics. In addition, Zocher [6] and Allen [7] have proventhat the incrementalized equations. can be integrated analytically for a linear viscoelastic material. Recently,Rahulkamar [8] has also proposed a rate-dependent cohesive zone constitutive law.Given a time-dependent material system with multiple interfaces and the current state of theliterature, the engineer naturally gravitates towards a rate-dependent bulk material model and a rate-independent interface model. While this methodology may be sufficient for some boundary-value problems,research emphasizing rate-dependent interface models is needed to understand the source of dissipation., ,.T-’ - . . . . . -..<. ,DISCLAIMERThis repoti was prepared as an account of work sponsoredbyan agency of the United States Government. Neitherthe United States Government nor anyagencythereof, norany of their employees, make any warranty, express orimplied, or assumes any legal Inability or responsibility forthe accuracy, completeness or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, reco.mmendationt or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United States .Government or any agency thereof.IDISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.#4Theoretical FoundationThe cohesive zone constitutive law employed in this work was derived by Yoon and Allen [4].Equation (1) represents a local form of the traction relationship where p is indexed over the normal and[Qql-a(t)]. a’ +jE.(t-+ir“(A’’t)=a3P o 1 (1)tangential coordinate directions. APis the gap function and k, expressed in Equation (2), represents the norm(2)of the gap function where 8., ilir,and 5Sestablish the length scale and mode-mixity associated with L. Thedamage state is reflected by ct. One can observe that the crack faces become traction free as ct approaches 1.A simple damage evolution law has been proposed by Allen and Searcy [5]. Equation (3) reflects a power-law based on k where U1and m are fitting parameters. Note that damage does not accumulate on unloading.Finally, c? represents an initiation criterion and EC,constitutes the relation modulus of the cohesive zone. Inthe context of polymeric materials, one may view I as a measure of tibril stretching in a crazed zone. Voidgrowth sheds load to the extended fibrils. As the load bearing area decreases, a increases. The critical fiberdiameter coincides with a =1.Numerical ImplementationIn order to implement the cohesive zone constitutive law, one must develop an incremental form ofthe traction-displacement relation. A general procedure for a viscoleastic solid is presented by Zocher [6],while Allen and Sea.rcy [7] address Equation (1) specifically. First, one must assume that the relaxationEa(t+At–r) =E. +~Ejexp[1-(t+ At-r) ~j, pj=—j=l Pj Ej(4)modulus is well described by a prony series, Equation (4). Second, the rate of stretch, ~, must be constantover the chosen time step. Equation (5) is an acceptable condition provided small time steps are takenduring the simulation. Given these conditions, the convolution integrals can be integrated analytically and arecursive relationship can be established to capture the load history. The incremental traction-displacementequations were incorporated into the Sandia National Labs (SNL) code, JAS3D.— ,. . -=. , -- T-- : T-.; , . .:.4 ,,. . . . . . . t.- --, -.:,4 :.,-7.-;--..,,. . . .—— ...Example SimulationsThe following examples emphasize time-dependent model capability with minimal complexity. Thefirst study examines a weak adhesive joining two elastic, aluminum beams (aspect ratio = 20). Figure 1illustrates the double-cantilever beam (DCB) geometry and VCZM material parameters.behavior is represented by a standard-linear-solid for simplicity. The end displacement, h,8“ 0.001F c j~ :: :;:;T‘1’ 1“00.50~1 : .20Of 0.0Figure 1. Schematic of DCB geometry with VCZM material properties.The viscoelasticwas applied at aconstant rate of 0.01. Because the interface is loaded through bending, the normal opening rate for eachcohesive zone element along the interface will decreaseperformed simulations employing the Xu-Needleman [9]confirmed that the mesh composed of 3280 hex elementsthat the DCB geometry was not pre-cracked.—with increasing beam length. The authors alsoconstitutive law for comparison. A prior study(one-element thick) was converged. Please noteTraction-displacement relations for both models are shown in Figure 2. VCZM parameters werechosen such that fracture energy, @~, associated with the intermediate rate, VCZM(2), would roughlycorrespond to the Xu-NeedIeman input fracture energy. In the VCZM, the viscoelastic constitutive responseand the darnage evolution law combine to yield a rate-dependent fracture energy. The variance in the initialslope is indicative of the viscoelastic response while the general curvature of the traction-displacementrelation is a function of damage evolution. -211.60.40I 1[)& I 0Case -P, ~6* o .(x-w IVCZM(3) 0.025 0.63Xu-Needleman 0.100 1.00—VCZM(I)— VCZM(2)— VCZM(3)— Xu-Needlemano 1 2 3 4 5 6 7Anl&Figure 2. Traction-displacement relations and fracture energies as a function of loading rate., ,. .,Onecantrack thecrack tip as a function of opening displacement, 0.18.. Growth curves for bothmodels are presented in Figure 3. The Xu-Needleman model yields a curve consistent with linear elasticfracture mechanics. VCZM differs in that Glc, the critical energy release rate, is not a constant. Initially, theVCZM lags the Xu-Needleman prediction. Because the normal opening rate is a function beam length, GICdecreases with crack growth. Soon, the VCZM prediction surpasses the Xu-Needleman result. To further1.o-0.8-0.6-~0.40.21‘Xu-Needleman‘VCZM0.0 ! 1.0.0 20.0 40.0 60.0 80.0m“Figure 3. Rate-dependent crack growth for a DCB with a constant end-displacement rate.illustrate this point, we plot the peak traction in the cohesive zone. Notice that the peak tractions illustratedin Figure 2 span the behavior shown in Figure 4. In fact, the DCB end displacement rate and VCZM(l) areidentical, 0.01. The Xu-Needleman model remains unchanged.2.0-1.6-2*1.2 -g~0.8 -0.4-‘Xu-Needleman‘VCZM0.0-? , , , , i0.0 0.2 0.4 0.6 0.8 1.0CfiFigure 4. VCZM global peak traction decreases with increasing crack length..,, ,.,...,.- , ,.,.. . ...4... .... . .. . ..-—.,.. ~.-~,.:,.-.-..+.,,!?.,-..... \. .....:-,-..;-.-r >:..r.!i.%....?z~d?z~dn’ “:-- ..... :. — “-,’The second study investigates the evolution of microstructural interfaces through the incorporation ofcohesive zone elements along grain boundaries. Some polycrystalline material systems such as rocksaltexhibit rate dependence at room temperatures. Unable to measure bulk and grain-boundary behaviorseparately, the analyst must predict time-dependent grain-boundary fracture through the incorporation ofrate-dependent bulk and interface models. If we assume that the grains behave elastically, we maximize thegeometric effects of prying and sliding along grain boundaries. To further simpli~ matters, we assume thatall rate dependence stems from the darnage evolution law. Figure. 5 illustrates the trial RVE, correspondingZin 0.25 mmEN 137.9MPaal 0.2m 0.75(sf 0.0MPaE9nin 220.1GPaVglain 0.25Figure 5. Schematic of 55 grain RVE, corresponding rnicrograph, and chosen material parameters. .microstructure, and fictitious material parameters. The applied displacement and simulated reaction load areplotted in Figure 6. Although the ends are held fixed after 50 s, GICcontinues to decrement with time. Thestrain energy available in the bulk drives grain-boundary separation and eventual failure. While macroscopicunloading is illustrated in Figure 6, Figure 7 depicts load redistribution at the granular level. In addition, thedeformed geometry mirrors a frequent observation made in polycrystalline materials under compressiveloads - cracks form parallel to the major axis of loading.-8.00E-02 - r -1 .6E+04E~-6.00E-02 - --1 .2E+04$ gE vo ;&4.00E-02 - - -8.0E+03 ;m.-=sUYu mal= E&.2.00E42 . - -4.0E+03m—displacement— load0.00E+OO , O.OE+OO0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0time (s)Figure 6. Grain-boundary fracture causes trial RVE to unload under fixed-grip conditions.,, J, , ,. ,.~ .,, . . . ,, -!r!. i.zct-. J .’ ,,r,>fJ’ --? --- . . ..—, , 7:.”..”J;I>>. ..,, - ,:,.: .-,.. .. -~. ,...--- . —.- -A ,’7/, -.-! “. ...... I● ’P= -1.31C+04*WVO.oomal-1.oot%ol-MW+Ol-3.00M-014.W+O1time=50.00Figure 7. Stored energy drives crack growth, load redistribution, and eventual failure.ConclusionIn order-to predict time-dependent fracture in material systems with interfaces, one must not only relyon rate-dependent bulk models, but also rate-dependent interface models. The calculations presented hereinsupport a framework for incorporating time-dependence into a cohesive zone traction-displacement relation.Through material inelasticity and damage, one can account for interracial evolution and thus predict a time- .dependent critical energy release rate.References1.. Needleman, A. (1987): “A Continuum Model for Void Nucleation by hclusion Debonding~’ Journal2.3.4.5.6.7.8.9.1of Applied Mechanics, 54:525-531.Knauss, W.G., Losi, G.U. (1993): “Crack Propagation in a Nonlinearly Viscoelastk Solid withRelevance to Adhesive Bond Failure,” Journal of Applied Mechanics, 60:793-801.Knauss, W.G. (1993): “Time Dependent Fracture and Cohesive Zones;’ Journal of EngineeringMaterials and Technology, Transactions of the ASME, 115:262-267.Yoon, C. and Allen, D.H. (1999): “Damage Dependent Constitutive Behavior and Energy ReleaseRate for a Cohesive Zone in a Themoviscoelastic Solid;’ Inter. Journal of Fracture, 96:55-74.Allen, D.H. and Searcy, C.R. (1999); “A Micromechanical Model for a Viscoelastic Cohesive Zone,”to appear in International Journal of Fracture.Zocher, M.A., Allen, D.H., and Groves, S.E. (1997): “A Three Dimensional Finite ElementFormulation for Thermoviscoelastic Orthotropic Media:’ International Journal for NumericalMethods in Engineering, 40:2267-2288.Allen, D.H. and Searcy, C.R. (1999): “Numerical Aspects of a Micromechanical Model of aCohesive Zone:’ to appear in Journal of Reinforced Plastics and Composites.Rahulkamar, P., Jagota, A., Benison, S.J., Saigal, S. (2000): “Cohesive Modeling of ViscoelasticFracture: Application to Peel Testing of Polymers,” Int. J. of Solids and Struct., 37:1873-1897.Xu, X.-P. and Needleman, A. (1994): “Numerical Simulations of Fast Crack Growth in BrittleSolids,” Journal of the Mechanics and Physics of Solids, 42:1397-1434..—. ——. ,., .,,,.-, ,. ..,, .-,--r -

Predicting Time-Dependent Interfacial Behavior with a Viscoelastic Cohesive Zone Model

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