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PROLEGOMENON

J. Eng. Mater. Technol. 2002;124(3):289. doi:10.1115/1.1480411.
Abstract
Commentary by Dr. Valentin Fuster

TECHNICAL PAPERS

J. Eng. Mater. Technol. 2002;124(3):290-296. doi:10.1115/1.1479177.

An apparatus has been developed for performing compression deformation experiments on oriented metallic single crystals to provide data for validation of 3-D dislocation dynamics simulations. The experiment is performed under conditions that allow unconstrained motion of the upper and lower compression platen, and thus a relatively uniform state of axial stress is maintained during the deformation. Experiments have been performed on high-purity Mo single crystal and polycrystalline Cu. Various aspects of the experimental procedures and results are presented. Possible usages of the experimental data for the validation of 3-D dislocation dynamics simulations are discussed.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):297-301. doi:10.1115/1.1479178.

The configuration of dislocation wall structures and the interactions between dislocations and dislocation walls play a significant role in the understanding of deformation processes in metals. Samples of single-crystal aluminum deformed by tensile-straining (15%) were analyzed using TEM. In tensile-deformed (15%) single crystal aluminum, a cell structure is well developed and dislocations in the cell boundaries consist of either one set of Burgers vector or two sets of Burgers vector. The three-dimensional image of cell wall structure, misorientation angle across the cell boundaries and the Burgers vectors of dislocations in the cell boundaries are characterized.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):302-313. doi:10.1115/1.1479354.

High resolution experimental characterization of material stretch and rotation fields in relatively fine-grained polycrystals has been limited, inhibiting direct comparison with predictions of crystal plasticity theory. In this study, micron scale grids used more commonly in etching of substrates for microelectronic circuits were deposited on specimens of Oxygen Free High Conductivity Copper (OFHC Cu) subsequently subjected to uniaxial compressive deformations to effective strain levels up to unity. Material stretch and rotation fields were assessed for fields of view encompassing on the order of 20 grains. Some rather striking features emerge, including the apparent relative lack of deformation in regions sized on the order of large grains, and the apparent concentration of stretch and rotation in bands surrounding these relatively undeformed areas. Comparisons are drawn with results of 3D crystal plasticity calculations performed on digitized grain structures that conform to representative microstructures in terms of initial grain size and shape distributions. The crystal plasticity simulations predict regions of relatively large rotation and relatively localized stretch traversing multiple grains. The numerical solutions also exhibit slightly higher local stresses in the vicinity of grain boundaries and triple points than in grain interiors, a phenomenon attributed to local lattice misorientation among neighboring grains. However, the crystal plasticity calculations do not, in an average sense, predict larger-than-average maximum stretch or rotation in the grain boundary regions. The numerical solutions are also quite sensitive to initial lattice orientations assigned to the grains. Comments are made regarding the segmentation of slip within the grains and its implications for modeling, based upon direct comparison of results from experiments and simulations.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):314-321. doi:10.1115/1.1479355.

Many natural minerals and manmade materials are aggregates of crystals or polycrystalline solids with a nonrandom distribution of grain orientations. Macroscopic behavior in such textured polycrystals depends on directions and is thus anisotropic. In this paper we develop experimental and theoretical procedures for investigating grain orientation evolution and its effect on the tensile stress-strain curve. The micro-tensile experiments were executed in a self-developed micro-forcing-heating device together with a micro-recorder-image analyzer system. In the experiments the 0.1 mm thin foil specimens of pure nickel and copper were gradually loaded toward final failure and the evolution of grain boundaries and slip bands inside grains was observed and recorded digitally via microscope and CCD camera throughout the whole time history. The texture image data were then used in a theoretical micro-macro transformation procedure to simulate the orientation evolutions and the stress-strain curves. The procedure was based on a double-slip model of polycrystal plasticity and on averaging of polycrystalline behavior over all grain orientations weighted by an orientation distribution function. The comparisons made between the simulated and experimental data of orientation evolutions and between the simulated curves and the macro-curves concurrently obtained in the experiments confirm the proposed procedures capable of simulating the considered micro-macro relations.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):322-328. doi:10.1115/1.1480407.

We analyze simple shear and torsion of single crystal copper by employing experiments, molecular dynamics simulations, and finite element simulations in order to focus on the kinematic responses and the apparent yield strengths at different length scales of the specimens. In order to compare torsion with simple shear, the specimens were designed to be of similar size. To accomplish this, the ratio of the cylinder circumference to the axial gage length in torsion equaled the ratio of the length to height of the simple shear specimens (0.43). With the [110] crystallographic direction parallel to the rotational axis of the specimen, we observed a deformation wave of material that oscillated around the specimen in torsion and through the length of the specimen in simple shear. In torsion, the ratio of the wave amplitude divided by cylinder circumference ranged from 0.02–0.07 for the three different methods of analysis: experiments, molecular dynamics simulations, and finite element simulations. In simple shear, the ratio of the deformation wave amplitude divided by the specimen length and the corresponding values predicted by the molecular dynamics and finite element simulations (simple shear experiments were not performed) ranged from 0.23–0.26. Although each different analysis method gave similar results for each type boundary condition, the simple shear case gave approximately five times more amplitude than torsion. We attributed this observation to the plastic spin behaving differently as the simple shear case constrained the dislocation activity to planar double slip, but the torsion specimen experienced quadruple slip. The finite element simulations showed a clear relation with the plastic spin and the oscillation of the material wave. As for the yield stress in simple shear, a size scale dependence was found regarding two different size atomistic simulations for copper (332 atoms and 23628 atoms). We extrapolated the atomistic yield stresses to the order of a centimeter, and these comparisons were close to experimental data in the literature and the present study.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):329-334. doi:10.1115/1.1479692.

In copper and other face centered cubic metals, high-energy particle irradiation produces hardening and shear localization. Post-irradiation microstructural examination in Cu reveals that irradiation has produced a high number density of nanometer sized stacking fault tetrahedra. The resultant irradiation hardening and shear localization is commonly attributed to the interaction between stacking fault tetrahedra and mobile dislocations, although the mechanism of this interaction is unknown. In this work, we present results from a molecular dynamics simulation study to characterize the motion and velocity of edge dislocations at high strain rate and the interaction and fate of the moving edge dislocation with stacking fault tetrahedra in Cu using an EAM interatomic potential. The results show that a perfect SFT acts as a hard obstacle for dislocation motion and, although the SFT is sheared by the dislocation passage, it remains largely intact. However, our simulations show that an overlapping, truncated SFT is absorbed by the passage of an edge dislocation, resulting in dislocation climb and the formation of a pair of less mobile super-jogs on the dislocation.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):335-341. doi:10.1115/1.1479693.

A stochastic dislocation dynamics (SDD) model is developed to investigate dislocation glide through dispersed obstacles. The model accounts for: 1) the dynamics of the flight process between successive meta-stable dislocations under various drag mechanism using discrete dislocation dynamics, and 2) thermal activation processes for meta-stable pinned dislocations using a stochastic force. The integration of the two processes allows one to examine the transient regime of dislocation motion between obstacle-controlled motion and drag-controlled motion. Result pertaining to the stress-strain rate behavior in copper are obtained. The stress and temperature dependence of the average dislocation velocity show obstacle-controlled region below the critical resolved shear stress (CRSS) and drag controlled region above the CRSS, which is in good qualitative agreement with experimental data. In the transient region right below the CRSS, negative temperature sensitivity is observed due to the competition between the drag effects in dislocation flight process and thermal activation process.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):342-351. doi:10.1115/1.1479694.

Recent advances in 3-D dislocation dynamics include the proper treatment of free surfaces in the simulations. Dislocation interaction and slip is treated as a boundary-value problem for which a zero-traction condition is enforced at the external surfaces of the simulation box. Here, a new rigorous method is presented to handle such a treatment. The method is semi-analytical/numerical in nature in which we enforce a zero traction condition at select collocation points on a surface. The accuracy can be improved by increasing the number of collocation points. In this method, the image stress-field of a subsurface dislocation segment near a free surface is obtained by an image segment and by a distribution of prismatic rectangular dislocation loops padding the surface. The loop centers are chosen to be the collocation points of the problem. The image segment, with proper selection of its Burgers vector components, annuls the undesired shear stresses on the surface. The distributed loops annul the undesired normal stress component at the collocation points, and in the process create no undesirable shear stresses. The method derives from crack theory and falls under “generalized image stress analysis” whereby a distribution of dislocation geometries or entities (in this case closed rectangular loops), and not just simple mirror images, are used to satisfy the problem’s boundary conditions (BCs). Such BCs can, in a very general treatment, concern either stress traction or displacements.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):352-357. doi:10.1115/1.1479695.

The purpose of this two-part article, is first to give an update of recent developments of gradient plasticity as this was advanced by Aifantis and co-workers in the early eighties to address dislocation patterning and shear band problems, and then to elaborate on two specific issues of current interest: size effects and plastic heterogeneity. In Part I, a brief review of gradient dislocation dynamics as providing a direct motivation for the simplest form of gradient plasticity is given. Then, a more general phenomenological formulation of gradient plasticity is given and used to interpret size effects. In Part II, wavelet analysis is used as a potential tool to describe plastic heterogeneity at very fine scales for which experimental results are not available, as well as for providing another means to interpret size effects through the derivation of scale-dependent constitutive equations.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):358-364. doi:10.1115/1.1479696.

Wavelet analysis is used for describing heterogeneous deformation in different scales. Slip step height experimental measurements of monocrystalline alloy specimens subjected to compression are considered. The experimental data are subjected to discrete wavelet transform and the spatial distribution of deformation in different scales (resolutions) is calculated. At the finer scale the wavelet analyzed data are identical to the experimental measurements, while at the coarser scale the profile predicted by the wavelet analysis resembles the shear band solution profile provided by gradient theory in agreement with experimental observations. The different data sets provided by wavelet analysis are used to train a neural network in order to predict the spatial distribution of strain at resolutions higher than those possible by the available experimental probes. In addition, applications of wavelet analysis to interpret size effect data in torsion and bending at the micron scale are examined by deriving scale-dependent constitutive equations which are used for this purpose.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):365-370. doi:10.1115/1.1480408.

An analytical solution is presented for the borehole problem of an elasto-plastic plane strain body containing a traction-free circular hole and subjected to uniform far field stress. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. This gradient plasticity theory introduces a higher-order spatial gradient of the effective plastic strain into the yield condition to account for the nonlocal interactions among material points, while leaving other relations in classical plasticity unaltered. The solution gives explicit expressions for the stress, strain, and displacement components. The hole radius enters these expressions not only in nondimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect in a quantitative manner. The classical plasticity-based solution of the borehole problem is obtained as a special case of the present solution. Numerical results for the plastic region radius and the stress concentration factor are provided to illustrate the application and significance of the newly derived solution.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):371-379. doi:10.1115/1.1480409.

The micro-indentation experiments have shown that the indentation hardness depends not only on the indentation depth but also on the indenter tip radius. In fact, the indentation hardness displays opposite dependence on the indentation depth h for a sharp, conical indenter and for a spherical indenter, decreasing and increasing, respectively, with increasing h. We have developed an indentation model based on the theory of mechanism-based strain gradient plasticity to study the effect of indenter tip radius. The same indentation model captures this opposite depth dependence of indentation hardness, and shows the opposite depth dependence resulting from the different distributions of strain and strain gradient underneath a conical indenter and a spherical indenter. We have also used the finite element method to study the indentation hardness for a spherical indenter as well as for a conical indenter with a spherical tip. It is established that the effect of indenter tip radius disappears once the contact radius exceeds one half of the indenter tip radius.

Commentary by Dr. Valentin Fuster
J. Eng. Mater. Technol. 2002;124(3):380-387. doi:10.1115/1.1480410.

A phenomenological, polycrystalline version of a nonlocal crystal plasticity model is formulated. The presence of geometrically necessary dislocations (GNDs) at, or near, grain boundaries is modeled as elastic lattice curvature through a curl of the elastic part of the deformation gradient. This spatial gradient of an internal state variable introduces a length scale, turning the local form of the model, an ordinary differential equation (ODE), into a nonlocal form, a partial differential equation (PDE) requiring boundary conditions. Small lattice elastic stretching results from the presence of dislocations and from macroscopic external loading. Finite deformation results from large plastic slip and large rotations. The thermodynamics and constitutive assumptions are written in the intermediate configuration in order to place the plasticity equations in the proper configuration for finite deformation analysis.

Commentary by Dr. Valentin Fuster

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