Photonic crystals in biology - NanoTR-VI

Poster Session, Thursday, June 17Theme F686 - N1123Gradient Modeling of Strengthening and Softening in Inelastic Nanocrytalline Materials withReference to the Triple Junction and Grain BoundariesBabur Deliktas 1 * and George Z. Voyiadjis 21 Department of Civil Engineering, Mustafa Kemal University, Hatay, Iskenderun 31200, Turkey1 Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USAAbstract-The work presented here provides a generalized structure for modeling polycrystals from micro to nano size range. The polycrystalstructure is defined in terms of the grain core, the grain boundary and the triple junction regions with their corresponding volume fractions.Depending on the size of the crystal from micro to nano different types of analyses are used for the respective different regions of the polycrystal.The analyses encompass local and nonlocal continuum or crystal plasticity. Depending on the physics of the region dislocation based inelasticdeformation and/or slip/separation is used to characterize the behavior of the material. The analyses incorporate interfacial energy with grainboundary sliding and grain boundary separation. Certain state variables are appropriately decomposed to energetic and dissipative components toaccurately describe the size effects. Additional entropy production is introduced due to the internal subsurface and contacting surface.This new formulation does not only provide the internal interface energies but also introduces two additional internal state variables forthe internal surfaces (contact surfaces). One of these new state variables measures tangential sliding between the grain boundaries and the othermeasures the respective separation. A multilevel Mori-Tanaka averaging scheme is introduced in order to obtain the effective properties of theheterogeneous crystalline structure and to predict the inelastic response of a nanocrystalline material. The inverse Hall-Petch effect is alsodemonstrated. The formulation presented here is more general and it is not limited to either polycrystalline or nanocrystalline structured materialsHowever, for more elaborate solution of problems a finite element approach needs to be developed.The material modeling of nanocrystallines has beenemphasized recently by Gleiter (2000). He pointed out theoutstanding possibilities of the so called, microcrystallinematerials that are usually defined as the single or multi phasepolycrystalline metallic materials with grain sizes typicallyless than 100nm. It has been well recognized thatnanaocrystalline materials may exhibit increased strength andhardening, improved toughness, reduced elastic modulus andductility, enhanced diffusivity, higher specific heat, enhancedthermal expansion coefficient, and superior soft magneticproperties in comparison with conventional polycrystallinematerials (Ashby, 1970; Hall, 1951; Petch, 1953).The plastic deformation mechanisms of nanocrystallinestructure are much more complicated than those of thepolycrystalline material. Few of the controversial issues of theplastic behavior of the nano/polycrystalline materials are inregard to the work hardening and strengthening in suchmaterials. For example, experimental studies reported by Qingand Xingming (2006) showed that when the grain size ofnanocrystalline is greater than a critical value, the Hall-Petch(H-P) relation is satisfied for a wide range of nanocrystallinematerials. However, as the grain size of metals decreasebeyond the critical value, the H-P slope becomes negative.The so called inverse softening effect of the H-P relation isobserved for some nanocrystalline materials (Nieh and Wang,2005; Tjong and Chen, 2004; Zhao, et al., 2003). This inverseHall–Petch phenomenon was first explained in terms ofporosity in nanocrystalline materials. This explanation wasproved incorrect when high quality NC materials wereproduced and, they still exhibited a negative Hall–Petch slope(Khan, et al., 2000). To understand the inverse Hall–Petchphenomenon, numerous studies were conducted. Many modelsare based on the rule of mixtures and on the competition oftwo or more mechanisms (Carsley, et al., 1995). Meyers et al.(2006) and Qiang and Xingming (2006) presented veryapproximate models based on the rule of mixtures as incomposite materials in order to show the inverse softeningeffect of the H-P relation.The computational methods available for simulatingnanocrystalline materials are clearly imperfect but they maybe capable of providing important insights into the behaviorof nanoscale materials. Therefore, in this paper, thetheoretical bases for modeling the inelastic behavior of thematerial is based on the thermodynamic framework andconstitutive laws given in the works of Voyiadjis andDeliktas (Voyiadjis and Deliktas, 2009a; b) where thetheoretical concepts have been elaborated in detail. Thepresent treatment is different than that previously proposedby Voyiadjis and Deliktas (Voyiadjis and Deliktas, 2009a; b)in that this new formulation does not only provide theinternal interface energies but also introduces two additionalinternal state variables for the internal surfaces (contactsurfaces). By using these internal state variables togetherwith displacement and temperature, the constitutive model isformulated as usual by state laws utilizing free energies andcomplimentary laws based on the dissipation potentials. Oneof these new state variables measures tangential slidingbetween the grain boundaries and the other measures therespective separation. A homogenization technique isdeveloped to describe the local stress and strain in thematerial. The material is characterized as a composite withthree phases: the grain core, the grain boundaries and triplejunctions. The model presented for a general case is thenapplied to pure copper under uniaxial tensile load. The resultsare compared with the experimental data.The geometrical representation of the RVE proposed bydifferent authors (Mecking and Kocks, 1981; Pipard, et al.,2009) can be conceptually described by three regions such asthe grain core, the grain boundary, and triple junctions withtheir corresponding internal interfaces, respectively. In thiswork the simplified nanocrystalline structure shown in Figure1b is represented as a 2D triangle representative volumeelement (RVE) of a composite material with three phases;grain core, grain-boundary, and triple junction (see Figure 1).6th Nanoscience and Nanotechnology Conference, zmir, 2010 713