ebook img

NASA Technical Reports Server (NTRS) 20030067948: Micromechanics Modeling of Fracture in Nanocrystalline Metals PDF

8 Pages·0.58 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview NASA Technical Reports Server (NTRS) 20030067948: Micromechanics Modeling of Fracture in Nanocrystalline Metals

/rv; )+’ f AIAA 2002-1315 L7 _- ,y:, MICROMECHANICS MODELING OF FFLACTURE IN NANOCRYSTALLINE METALS E.IH. Glaessgen, R.S.Piascik, 1.S. Raju and C.E. Harris NASA Langley Research Center Hampton, VA 23681 4C3rd AIANASM WASC WAHSIASC !Structures, Structural Dynamics, and Materials Conference April 22-25, 2002/Denver, CO For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 22091 ~ ~~ AIAA-2002-1315 MICROMECHANICS MODELING OF FRACTURE IN NANOCRYSTALLINE METALS E.H. Glaessgen', R.S. Piascikt. 1.S. Raju' and C.E. Harris4 NASA Langley Research Center, Hampton, VA 23681, U.S.A. . Abstract Classical Approaches Fracture of standard engineering metals containing Nanocrystalline metals have very high theoretical micro-sized structure is thought of in terms of brittle strength, but suffer from a lack of ductility and and ductile fracture. Brittle metals usually have either toughness. Therefore, it is critical to understand the body-centered cubic (BCC) or hexagonal close packed mechanisms of deformation and fracture of these (HCP) atomic structure whereas ductile metals have materials before their full potential can be achieved. face-centered cubic (FCC) atomic structure. In BCC Because classical fracture mechanics is based on the and HCP metals, there are few planes for dislocation comparison of computed fracture parameters, such as movement allowing for little plastic deformation. In stress intlmsity factors, to their empirically FCC metals, there are many planes for dislocation determined critical values, it does not adequately movement and hence they can undergo significant describe the fundamental physics of fracture required plastic deformation. Figure 2 shows these three types to predict the behavior of nanocrystalline metals. of atomistic structures. Thus, micromechanics-based techniques must be considered to quanti@ the physical processes of deformation and fracture within nanocrystalline In general, high strength materials often exhibit metals. This paper discusses hndamental physics- brittle fracture and low toughness whereas low based modeling strategies that may be useful for the strength materials exhibit ductile fracture and high prediction Iof deformation, crack formation and crack toughness. Current fracture models employ classical growth within nanocrystalline metals. fracture mechanics. Classical fracture mechanics is based on the premise that brittle fracture (in plane Introduction strain) will occur when KI > KK-, i.e. when the computed value of the stress intensity factor, KI, is Fracture processes in materials such as greater than or equal to the fracture toughness, KK. nanocrystalline metals (see Figure 1 layered metals In classical fracture mechanics, fracture toughness is I), and powder metallurgy-formed materials cannot be considered to be a property of the material, and the modeled by traditional fracture mechanics-based plane strain fracture toughness, KIC, is the lowest concepts. While these materials often exhibit high value of material toughness. strength, they also tend to have low ductility and low fracture toughness. Low ductility and toughness is a result of the nano-scale structure; here, Hall-Petch behavior is no longer valid when the structural size approaches the size of dislocations, thus disabling the mechanisms that produce ductility and toughness.' Therefore, it is critical to understand the mechanisms of deformation and fmcture of these materials before their full potential can be achieved This requires a .3 new understanding at the micromechanics level. This paper discusses fundamental physics-based modeling strategies that may be useful for the prediction of deformation, crack formation and crack growth within nanocrystalline metals. Figure 1. A 3D View of Microstructure of Extruded A I-'Ti-Cu bulk Nanoaystalline Metal. (TEM Images)' Copynght 2002 b) the American Institute of Aeronautics and Astronautics, Inc No copyright is assened in the United States under Title 17. U S. Code The US Government has E. royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes All other rights are reserved by the copyright owner I American Institute of Aeronautics and Astronautics :a) BCC (b) FCC c) HCP Figure 2. Three Atomic Structures Found in Typical Metals. The relationship between the characteristic behavior tip opening angle (CTOA) are often used and tend to of standard engineering materials and the types of be associated with empirically determined length fracture criteria that are applicable for these materials scales resulting in two-parameter criteria for crack is shown in Figure 3.4 In this figure, LEFM denotes growth. Additionally, energy-based methods such as linear elastic fracture mechanics. The CTOD and the J-integral are unable to separate the relative CTOA are critical crack tip opening displacement and contributions of the energy driving plastic angle, respectively, and K-R and J-R curves are crack deformation and the energy spent on the creation of growth resistance curves based on K and the J- new crack surfaces to the perceived crack growth integral, respectively. The critical J-integral, Jlc., is resistance in ductile materials. yet another fracture criterion. In Figure 3, fracture behavior is captured in terms of empirical parameters Computational Strategies (K,(., J-R curves, CTOA, etc.) that are devoid of physics-based understanding. For example, plane strain fracture toughness (KK,) is an empirical Because classical fracture mechanics is based on the quantity that is only applicable to linear elastic comparison of computed fracture parameters (such as fracture in thick materials. the stress intensity factor) to their empirically determined critical values, the concept does not adequately describe the fundamental physics of Such empirical concepts used in classical fracture fracture required to predict the behavior of mechanics are not physics-based and do not capture nanocrystalline metals. In a large part, classically the fundamental mechanisms associated with crack formulated fracture mechanics (and classical growth. For example, in many medium and low continuum mechanics as a whole) is unsuitable for strength materials under plane stress, methods such the prediction of deformation and hcture of as crack tip opening displacement (CTOD) or crack nanocrystalline metals because it does not account for LEFM holds Brittle KI Stress Intensity Factor KIC Critical value at fracture (e g 7075 AI, Glass, - JIC KIC LEFM holds JIC Medium Strength (e.g. 2024 AI) - J-R or K-R curves LEFM invalid CTOD or CTOA / \ LEFM holds*- KIC Low Stength JIC ’ (e.g. Carbon Where plane strain conditions exist, Steel) e.g., for very thick sections J-R or K-R curves LEFM invalid- CTOD or CTOA Figure 3. Relationship Between Material Behavior and Suitable Fracture Mechanics Analysis. 2 American Institute of Aeronautics and Astronautics Table I. Characteristic Scaling Lengths and Associated Simulation Methods the length-scale dependencies that tend to dominate intractable as the number of atoms considered the behavilor of these materials. That is, the details increases, therefore, approximations to these of microstructural and nanostructural features are not interactions have been developed in the form of considered. Table 1, adapted from reference 5, empirical and semi-empirical potentials describing presents bounds on the domain of application for the potential energy of the interactions among the typical simulation methods over a broad range of atoms.6 Among the best known of the relationships length scales. for non-bonding potentials is the Lennard-Jones potential, Qi, This paper discusses three of the physics-based analysis approaches shown in Table 1 to predict deformation, crack formation and crack growth in metallic materials. These approaches are molecular dynamics analysis6, cohesive zone models' and L J variants of plasticity theory'. To varying degrees, these approaches have the ability to model aspects of where E is the depth of the energy well, o is the van der Waals radius and r,/ is the separation distance the material architecture and their effects on fracture behavior while classically formulated fixture between the i'" and jIh atoms in a pair. The r,;' term represents the attractive contribution to the van der mechanics cannot model these aspects. These three computational methods will be discussed next Waals forces between neutral molecules. The other component of the van der Waals interactions mimics followed by a brief discussion of their effects on the the Coloumb interaction of the nuclei and the Pauli emerging field of computational materials. repulsion between overlapping electron clouds and is modeled by the short ranged ri'* term.6 Molecular Dynamics Analysis The Lennard-Jones potential was originally developed Gaining accurate understanding of the mechanisms of and is most accurate for interactions among atoms of fracture at a crack tip requires modeling at the noble gasses.8 However, it has been used for a broad atomistic level. Quantum mechanical solutions for variety of gasses, liquids and solids. Great care is the interaction among atoms rapidly become needed in the selection of these potentials and hrther A (a) Before Deformation (b) After 10% Deformation Figure 4. Nanocrystalline Copper Sample Containing 100,000A toms.' 3 American Institute of Aeronautics and Astronautics majority of materials. One exception involves certain classes of nanocrystalline materials wherein the characteristic length scales are very small. Typical grain diameters in nanocrystalline metals range from 5 to 50 nanometers.' To resolve the issue of length scale in most other metals and larger domain problems in nanocrystalline materials, a multiscale modeling strategy has been proposed. The multiscale modeling strategy uses molecular dynamics models to provide constitutive input into cohesive zone relationships that can be embedded along regions of separation within finite element models of the grain structures. Figure 5. Fracture Along the Primary Cleavage { 1 IO} Planes of NiAI."' Cohesive Zone Models work is needed in developing accurate potential energy functions for specific types of atomic Cohesive zone models assume cohesive interactions interactions. Once developed, the interatomic among the grains of a material and permit the potentials can be used in molecular dynamics (MD) appearance of fracture surfaces in the continuum.6 simulations of deformation and fracture. One of the popular cohesive zone models is attributed to Tvergaard and Hutchinson," where the normal and shear components of the traction and displacement are Molecular dynamics analysis mimics elementary combined into single measures, f and A, respectively, atom istic path-dependent processes by solving the so that the responses are coupled.12 The coupled equations of motion for all atoms in the domain.6 cohesive zone model (CCZM) given in reference 12 MD simulations have yielded results such as the defines a traction potential, @ prediction of the deformation of nanocrystalline copper as shown in Figure 4. In this figure, stacking @(dn,dl)= s:,$r(A')dA' (2) faults left behind by partial dislocations that have run A through the grains during the deformation processes where A is a nondimensional measure of the relative are clearly seen.9 As another example, fracture along normal (&) and tangential (SI) displacements as the primary cleavage { 1 10) planes of NiAl is shown defined by in Figure 5." 112 [ ($ ( Atomistic simulations containing several hundred (3) million atoms are possible, however this number of A = + ;)2] atoms corresponds to a cube of less than IO00 atoms on each side. One thousand atoms of titanium span where S,, and 8,a re the critical values for the normal approximately 400 nm. Thus, it is readily apparent and tangential modes, respectively, and the initiation that an atomistic approach is not practical for of a crack is assumed to occur when A reaches a value determining deformation and fracture in the vast 4 Displacement, h I I1 Figure 6. Traction-DisplacementR elationship in CCZM. 4 American Institute of Aeronautics and Astronautics t t t t t .c 4 (a)G rain Structure and Far-Field Loading. (b) Local Decohesion at Location A. Figure 7. Decohesion Among Gmin Boundaries of a Crystalline Metal.” of unity. A graphical representation of the CCZM is heterogeneous character of crystalline micro~tructure.~ shown in Figure 6. For example, polycrystalline plasticity models have been used to examine strain localization at triple Grain boundaries in metals exist because of points and grain b0~ndaries.I~ Compared with incohereno: in the atomic lattice. Decohesion may standard continuum plasticity models, the predictions occur between the atoms within an individual grain of the polycrystalline plasticity-based analyses show (intragranular) or along the boundaries between the a more heterogeneous and physically realistic strain grains (intergranular). Currently, most analyses that field. Recently, crystalline plasticity has been implement cohesive zone models consider only the extended to a more general theory based on classical intergranular failure problem. Figure 7 shows results crystalline kinematics; classical macroforces; that are typical of a finite element calculation that microforces for each slip system consistent with a uses cohesive zone models to predict separation microforce balance; a mechanical version of the between the grains.I2 Figure 7(a) is a representation Second Law of Thermodynamics that includes, via of the grain structure, and Figure 7(b) is a detail of the microforces, work performed during slip; and a the decohesion among the grains at location A. rate-independent constitutive theory that includes While the results in reference 12 use heuristically dependencies on plastic strain gradient^.'^ derived relationships to define the coupled cohesive zone modl:l, the CCZM can also be developed from Based on the general framework of Cosserat couple results of molecular dynamics analyses allowing the stress theory, strain gradient plasticity allows for the physical insight of the MD analysis to be embedded inclusion of a single material length scale, I, and in the more computationally efficient finite element reduces to the conventional Jz plasticity theory when models. geometric length scales are large compared with 1.”. l6 The technique has been used to show strain, curvature Variants of Plasticity Theory and strength dependencies of materials containing rigid particles on the prescribed radius of the Because cd the complexities involved in developing particles. For materials with a strain hardening index meaningfill representations of atomistic or grain between 1 and 10, strength was shown to increase by configurations and the computational power required values of between approximately 6 and 8 when the to solve the resulting systems of equations, ratio of material length scale to particle size increased approaches allowing for homogenization of the from 0 to 1. domains of interest may facilitate study of the deformation and fracture problems. One such top- Another variant on P ,lasticity theory is discrete down approach is to use models based on variants of dislocation plasticity. In discrete dislocation plasticity theory. Among these approaches are plasticity, dislocations are treated as line singularities polycrystalline plasticity, strain gradient plasticity in an elastic solid. The formulation consists of two and discrete dislocation plasticity. parts: a many body interaction problem involving the discrete dislocations and a fairly conventional solid Polycrystalline plasticity replaces the microstructure mechanics boundary value problem. The long-range of the material by an appropriate constitutive interactions between dislocations are accounted for description and can, to some extent, account for the using the continuum elasticity fields, while the short- 5 American Institute of Aeronautics and Astronautics range interactions are incorporated through a set of systems exceeding a few hundred million atoms. In constitutive rules. The technique has been used to contrast, the cohesive zone models are suited for predict the local deformations inchdin5 localization larger domains including modeling decohesion along of plastic flow within small (48p.m ) blocks of individual grain boundaries, but do so with a loss of material under tension and bending and may be a detail near the crack tip. Plasticity-based models useful intermediary between atomistic and continuum tend to be even more homogenized, but lend form u~aiot ns. ” themselves to modeling larger domains. Computational Materials The real power of these techniques is not seen when the methods are implemented individually, but Over the centuries, the materials processing rather, when the methods are combined. For community has taken an Edisonian approach to example, molecular dynamics solutions can be used materials processing. This Edisonian paradigm is as the basis for discrete dislocation plasticity models archaic and needs to be replaced. Simultaneously, of deformation and as the foundation for cohesive the mechanics of materials community has developed zone relationships used in polycrystalline models of analytical approaches based on empirically crack initiation. determined parameters such as modulus of elasticity, strength and toughness. These empirical parameters Through these efforts, the mechanics of materials and offer limited insight into the internal mechanisms materials processing communities have embarked on from which they arise. Approaches such as the ones a bold paradigm shift that will lead a transition from outlined in this paper have the potential to the traditional separate, empirical and heuristic revolutionize mechanics of materials and couple it to qualitative methods that have existed for centuries to materials processing allowing material integrated physics-based quantitative methods that microstructures to be designed and processing will allow numerical experiments to drive materials parameters to be determined before physical processing. Many early steps have been taken to processing. develop useful techniques for modeling materials over a broad range of length scales. Some of them The work being undertaken to address the closely have been outlined in this paper. interwoven goals of developing techniques to predict the effects of processing parameters on microstructure References evolution and to determine the effects of material microstructure on properties will lead to true 1. Lee, Z., Rodriguez, R., Lavernia, E.J. and Nutt, computationally-designed materials. Materials that S., “Microstructural Evolution of Cryomilled can be computationally designed at the Nanocrystalline AI-Ti-Cu Alloy, Ultrafine Grained ” microstructural level, can sense microstructural Materials 11, Y.T. Zhu, T.G. Langdon, R.S. Mishra, damage and repair themselves to retard development S.L. Semiatin, M.J. Saran, and T.C. Lowe, Eds., of macroscopic cracks and have redundant internal TMS (The Minerals, Metals & Materials Society), load-paths that can circumvent damage once it does 2002. form appear to be possible. 2. Sanders, P.G., Youngdahl, C.J. and Weertman, J.R., “The Strength of Nanocrystalline Metals With Concluding Remarks and Without Flaws,” Materials Science and Engineering, A234-236, 1997, pp. 77-82. Several analyses that may contribute to the 3. Needleman, A. and Van der Giessen, E., understanding of deformation and fracture in “Micromechanics of Fracture: Connecting Physics to materials such as nanocrystalline metals, layered Engineering,” MRS Bulletin, March 2001, pp. 21 1- metals and powder metallurgy-formed materials have 214. been briefly described. These methods include 4. Broek, D., Elementary Engineering Fracture molecular dynamics analysis, cohesive zone models Mechanics, 4’h ed., Martinus Nijhoff, Dordrecht, and several variants on plasticity theory including 1986. polycrystalline plasticity and discrete dislocation 5. Raabe, D., Computational Materials Science: The plasticity. Simulation of Materials Microstructures and Properties, Wiley-VCH, Weinheim, 1998. Each of these methods has unique strengths and 6. Ortiz, M., Cuitino, A.M., Knap, J. and weaknesses. For example, the molecular dynamics Koslowski, M., “Mixed Atomistic-Continuum analysis provides considerable insight into the Models of Material Behavior: The Art of mechanisms of dislocation formation and the details Transcending Atomistics and Informing Continua,” of fracture at a crack tip but because of limitations in MRS Bulletin, March 200 I, pp. 2 16-221 . computing power, it is not suited for solutions of 6 American Institute of Aeronautics and Astronautics 7. Klein, ‘P. and Gao, H., “Crack Nucleation and Growth as Strain Localization in a Virtual-Bond Continuum,” Engineering Fracture Mechanics, vol. 6 I, 1998, IIP. 2 1-48. t 8. Moore. W. J., Physical Chemistry, 4’” ed. Prentice Hall Englewood Cliffs, New Jersey, 1972, pp. 913-915. 9. Schiotz. J., Di Tolla, F.D. and Jacobsen, K.W., “Softening of Nanocrystalline Metals at Very Small Grain Sizes,” Nature, Vol. 391, 1998, pp. 561-563. IO. lyer, V., Shastry, V. and Farkas, D., “Atomistic Simulations of Fracture in NiAI,” Processing and Fabrication of Advanced Materials V, T. S. Srivatsan and J. J. Moore, Eds., 1996, pp. 705-713. 1 I. Tvergaard, V. and Hutchinson, J.W., “The Relation Between Crack Growth Resistance and Fracture Process Parameters in Elastic-Plastic Solids,” J(wrna1 of the Mechanics and Physics of Solids, vol. 40, 1992, pp. 1377- 1397. 12. Iseulauro, E., Decohesion of Grain Boundaries in Statistical Representations of Aluminum Polycrystals, Cornell University Report No. 02-01, 2002. 13. Raabe., D., Sachtleber, M., Zhao, Z., Roters, F. and Zacfferer, S., “Micromechanical and Macromechanical Effects in Grain Scale Polycrystal Plasticity Experimentation and Simulation,” Acta Materialio, vol. 49, 200 1, pp. 3433-344 1. 14. Gurtin, M.E., “On the Plasticity of Single Crystals: Free Energy, Microforces, Plastic Strain Gradients,” Journal of the Mechanics and Physics of Solids, voll. 48, 2000, pp. 989-1036. 15. Fleck, N.A. and Hutchinson, J.W., “A Phenomenological Theory for Strain Gradient Effects in Plasticity,” Journal of the Mechanics and Physics of Solids, vol. 41, 1993, pp. 1825-1857. 16. Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinsoin, J.W., “Strain Gradient Plasticity: Theory and Experiment,” Acta Metallurgica et Materialia, vol. 42, 1994, pp. 475-487. 17. Needleman, A., “Computational Mechanics at the Mesoscale,” Acta Materialia, vol. 48, 2000, pp. 105- 124. 7 American Institute of Aeronautics and Astronautics

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.