George C. Kaschner

MECHANICAL RESPONSE OF ZIRCONIUM—I. DERIVATION OF A POLYCRYSTAL CONSTITUTIVE LAW AND FINITE ELEMENT ANALYSIS

Simulating the forming of anisotropic polycrystals, such as zirconium, requires a description of the anisotropy of the aggregate and the single crystal, and also of their evolution with deformation (texture development and hardening). Introducing the anisotropy of the single crystal requires the use of polycrystal models that account for inhomogeneous deformation depending on grain orientation. In particular, visco- plastic self-consistent models have been successfully used for describing strongly anisotropic aggregates. As a consequence, using a polycrystal constitutive law inside finite element (FE) codes represents a considerable improvement over using empirical constitutive laws, since the former provides a physically based description of anisotropy and its evolution. In this work we develop a polycrystal constitutive description for pure Zr deforming under quasi-static conditions at room and liquid nitrogen temperatures. We use tensile and compressive experimental data obtained from a clock-rolled Zr sheet to adjust the constitutive parameters of the polycrystal model. Twinning is accounted for in the description. The polycrystal model is implemented into an explicit FE code, assuming a full polycrystal at the position of each integration point. The orientation and hardening of the individual grains associated with each element is updated as deformation proceeds. We report preliminary results of this methodology applied to simulate the three-dimensional deformation of zirconium bars deforming under four-point bend conditions to maximum strains of about 20%. A critical comparison between experiments and predictions is done in a second paper (Kaschner et al., Acta mater. 2001, 49(15), 3097-3107). Published by Elsevier Science Ltd on behalf of Acta Materialia Inc.

Effects of texture, temperature and strain on the deformation modes of zirconium

Clock-rolled, high-purity, textured polycrystalline zirconium exhibits significant plastic anisotropy for compression along the through-thickness and in-plane directions and strong temperature dependence of flow stress for both orientations. Orientation imaging microscopy in a scanning electron microscope and defect analysis via transmission electron microscopy are used to characterize the defect microstructures as a function of initial texture, deformation temperature and plastic strain. The observed deformation mechanisms are correlated with the measured mechanical response.

Mechanical response of zirconium—II. Experimental and finite element analysis of bent beams

Abstract In a companion paper [Acta mater. 2001, 49(15), 3085–3096] we develop a polycrystal constitutive law that incorporates the deformation mechanisms operating in high purity zirconium (Zr) at liquid nitrogen (LN) and room temperature (RT). In this paper we present results of 4-point bending tests performed on beams of highly textured zirconium. These tests have been performed at LN and RT, in two orthogonal bending planes, and up to a strain of approximately 20% in the outermost fibers of the beams. A novel experimental technique, dot-matrix deposition and mapping (DMDM), has been developed and employed to analyze the distribution of local plastic strain and macroscopic deformation in the deformed beams. Automated electron backscatter diffraction (EBSD) pattern analysis has been used to evaluate the textures just below the outermost tensile and compressive surfaces and at the neutral plane. Experimental results compare very well with the predictions of finite element (FE) simulations obtained using the constitutive law developed in Part I. Specifically, we compare local deformation, macroscopic deformation and local texture in the beam. We show that the contribution of twinning to deformation results in different qualitative responses in the compressive and tensile fibers of the bent beam. Our results indicate the necessity of using a constitutive description that accounts for the anisotropy of the aggregate and for its evolution with deformation.

Advances in deformation twin characterization using electron backscattered diffraction data

This article reports on recent progress in the effort to develop an automated, crystallographically based twin identification and quantification routine using large sets of spatially correlated electron backscattered diffraction (EBSD) data. The proposed analysis scheme uses information about the most probably occurring twin types and the macroscopic stress state, taken together with the crystallographic theory of deformation twinning, to identify and classify twinned areas in a scanned cross section of a material. The key features of the analysis are identification of potential twin boundaries by their misorientation character, validation of these boundaries through comparison with the actual boundary position and twin-plane matching across the boundary, and calculation of the Schmid factors for the orientations on either side of the boundary. This scheme will allow researchers to quantify twin area fractions from statistically significant regions and, in turn, estimate twinned volume fractions with reasonable reliability.

Deformation twinning in polycrystalline Zr: Insights from electron backscattered diffraction characterization

The response of polycrystalline α-zirconium to various deformation conditions was investigated through electron backscattered diffraction (EBSD) characterization. The range of deformation conditions included quasi-static compression and tension at room and cryogenic temperatures, along with a Taylor cylinder impact experiment. The resultant data provided spatial resolution of individual with system activity as a function of the progression of deformation. Over 300 deformation twins were analyzed to identify the type of twin system and active variant, along with the Schmid factor in the parent orientation. These data supplied information on the distribution of Schmid factor and variant rank as a function of twin system and deformation condition. Results showed significant deviation from a maximum Schmid factor activation criterion and suggest deformation twinning is greatly affected by local internal stress heterogeneities and the sense of the applied stress.

Modeling Texture, Twinning and Hardening Evolution during Deformation of Hexagonal Materials

Hexagonal materials deform plastically by activating diverse slip and twinning modes. The activation of such modes depends on their relative critical stresses, function of temperature and strain rate, and the orientation of the crystals with respect to the loading direction. For a constitutive description of these materials to be reliable, it has to account for texture evolution associated with twin reorientation, and for the effect of the twin barriers on dislocation propagation and on the stress-strain response. In this work we introduce a model for twinning which accounts explicitly for the composite character of the grain, formed by a matrix with embedded twin lamellae which evolve with deformation. Texture evolution takes place through reorientation due to slip and twinning. The role of the twins as barriers to dislocations is explicitly incorporated into the hardening description via a directional Hall-Petch mechanism. We apply this model to the interpretation of compression experiments both, monotonic and changing the loading direction, done in rolled Zr at 76K.

Anisotropic Plasticity Modeling Incorporating EBSD Characterization of Tantalum and Zirconium

The application of automated EBSD techniques in the context of an overall predictive materials modeling effort incorporating anisotropic properties for tantalum and zirconium is covered in this chapter. The focus will be on the role of microtextural investigations as an integral tool supporting the development and validation of material models that incorporate anisotropic constitutive behavior. Continuum mechanics codes require accurate descriptions of materials behavior to adequately predict large-strain deformation response. The corresponding requirement of characterizing micro structure s after significant deformation places severe requirements on the EBSD system. In this work, a Philips XL30 SEM employing a warm Schottky FEG was used for all data collection; the combination of high resolution with adequate beam current was a necessity for analyzing fine detail amid heavily worked structures. The ability to spatially resolve orientation differences on the order of 100 nm is achievable. All EBSD data collection and analysis was performed with TSL’s OIM™ software, while the popLA code (Kallend et al., 1991) was used for x-ray texture analysis.

A study of the mechanisms of cyclic deformation in f.c.c. metals using strain rate change tests

Abstract Plastic strain rate change tests were performed during low cycle fatigue (LCF) of 7075-T6 aluminum and Type 304 stainless steel using plastic strain as the control variable. The evolution of dislocation interactions was observed by evaluating the activation area and true stress as a function of cumulative plastic strain. Activation areas for 7075-T6 aluminum at each of three plastic strain amplitudes, 0.2, 0.4, and 0.6%, have initial values of approximately 250–450 b2 which decrease to 70–115 b2, respectively, during cyclic loading to saturation. Activation areas for 304 stainless steel at both amplitudes tested, 0.4 and 0.6%, exhibit initial values of 90 b2 which increase slightly to 130 b2 at large cumulative strains. Both materials show a deviation from the Cottrell–Stokes law during cyclic hardening and softening. Tests performed at saturation reveal a mild dependence of activation area on plastic strain amplitude for aluminum but no such relationship for stainless steel. These results reflect a contrast between wavy slip for pure copper and 7075 aluminum versus planar slip for 304 stainless steel tested at room temperature. In addition, the Cottrell–Stokes law holds in both alloys at saturation. Dislocation motion in 7075 aluminum and 304 stainless steel is controlled by obstacles that are characteristically more thermal than forest dislocations; obstacles in 7075-T6 aluminum are identified as solutes from re-desolved particles; for 304 stainless steel, the obstacles are also in the form of solutes.

Evolution of dislocation glide kinetics during cyclic deformation of copper

Abstract Strain rate change tests were performed during low cycle fatigue of polycrystalline copper using plastic strain as the control variable. The evolution of dislocation interactions was observed by evaluating the activation area and true stress as a function of cumulative plastic strain. Activation areas at each of three plastic strain amplitudes, Δep/2=0.2, 0.4, and 0.6%, have initial values of approximately 2000b2 which decrease to 600b2 during cyclic loading to saturation. This observation suggests a transition from forest dislocation cutting to increasing contributions of cross-slip as the predominant rate-controlling mechanisms of dislocation motion. Haasen plots of normalized inverse operational activation area (b2/Δa) for specimens cycled to saturation exhibit a deviation from linearity similar to that observed for monotonic deformation. This nonlinearity corresponds to a failure of the Cottrell–Stokes law that correlates with the development of characteristic dislocation structures during cyclic deformation. Tests performed at various stresses at saturation reveal a linear dependence of b2/Δa on true stress. The athermal stress, σb=86.5 MPa, measured at saturation by extrapolating the activation area data compares favorably with the value determined from a Bauschinger analysis, σb=80 MPa, at a plastic strain amplitude of 0.6%. In addition, athermal stress values vary with plastic strain amplitude as expected, resulting in a constant value of approximately σb/σ=0.5.

Experiment-based validation and uncertainty quantification of coupled multi-scale plasticity models

Anisotropic materials, such as zirconium require the inclusion of evolution of the crystal structure in the finite element model representations of the mechanical behavior, naturally leading to coupled meso- and macro-scale plasticity models. For achieving such models, partitioned techniques where isolated models that resolve system behavior at different scales are coupled through iterative exchange of inputs and outputs are widely used. In this treatment, a finite element at the macro-scale provides strain information to a meso-scale visco-plastic self-consistent model to represent micro-scale properties. These properties are then returned to the macro-scale for new stress and strain calculations. During this iterative process, biases and uncertainties inherent within constituent model predictions propagate between constituents. This propagation creates a need for a multi-scale approach for experiment-based validation and uncertainty quantification, in which separate effect experiments conducted within each constituent’s domain test the validity of the independent constituents in their respective scales and integral-effect experiments executed within the coupled domain validate the entire coupled system. In this paper the authors implement a multi-scale experiment-based validation approach utilizing both separate and integral-effect experiments. Results demonstrate that training an independent error model for the bias of the constituent model utilizing separate-effect experiments as means of appropriately bias-correcting constituent model predictions during coupling iterations results in an improved predictive capability. This improvement is demonstrated through the use of integral-effect experiments.