David T. Fullwood

Microstructure sensitive design for performance optimization

Abstract The accelerating rate at which new materials are appearing, and transforming the engineering world, only serves to emphasize the vast potential for novel material structure, and related performance. Microstructure-sensitive design (MSD) aims at providing inverse design methodologies that facilitate design of material internal structure for performance optimization. Spectral methods are applied across the structure, property and processing design spaces in order to compress the computational requirements for linkages between the spaces and enable inverse design. Research has focused mainly on anisotropic, polycrystalline materials, where control of local crystal orientation can result in a broad range of property combinations. This review presents the MSD framework in the context of both the engineering advances that have led to its creation, and those that complement or provide alternative methods for design of materials (meaning ‘optimization of material structure’ in this context). A variety of definitions for the structure of materials are presented, with an emphasis on correlation functions; and spectral methods are introduced for compact descriptions and efficient computations. The microstructure hull is defined as the design space for structure in the spectral framework. Reconstruction methods provide invertible links between statistical descriptions of structure, and deterministic instantiations. Subsequently, structure–property relations are reviewed, and again subjected to representation via spectral methods. The concept of a property closure is introduced as the design space for performance optimization, and methods for moving between the closures and hulls are presented as the basis for the subsequent discussion on microstructure design. Finally, the spectral framework is applied to deformation processes, and methodologies that facilitate process design are reviewed.

Bragg's Law diffraction simulations for electron backscatter diffraction analysis

In 2006, Angus Wilkinson introduced a cross-correlation-based electron backscatter diffraction (EBSD) texture analysis system capable of measuring lattice rotations and elastic strains to high resolution. A variation of the cross-correlation method is introduced using Braggs Law-based simulated EBSD patterns as strain free reference patterns that facilitates the use of the cross-correlation method with polycrystalline materials. The lattice state is found by comparing simulated patterns to collected patterns at a number of regions on the pattern using the cross-correlation function and calculating the deformation from the measured shifts of each region. A new pattern can be simulated at the deformed state, and the process can be iterated a number of times to converge on the absolute lattice state. By analyzing an iteratively rotated single crystal silicon sample and recovering the rotation, this method is shown to have an angular resolution of approximately 0.04 degrees and an elastic strain resolution of approximately 7e-4. As an example of applications, elastic strain and curvature measurements are used to estimate the dislocation density in a single grain of a compressed polycrystalline Mg-based AZ91 alloy.

Estimations of bulk geometrically necessary dislocation density using high resolution EBSD

Characterizing the content of geometrically necessary dislocations (GNDs) in crystalline materials is crucial to understanding plasticity. Electron backscatter diffraction (EBSD) effectively recovers local crystal orientation, which is used to estimate the lattice distortion, components of the Nye dislocation density tensor (α), and subsequently the local bulk GND density of a material. This paper presents a complementary estimate of bulk GND density using measurements of local lattice curvature and strain gradients from more recent high resolution EBSD (HR-EBSD) methods. A continuum adaptation of classical equations for the distortion around a dislocation are developed and used to simulate random GND fields to validate the various available approximations of GND content.

Pattern Center Determination in Electron Backscatter Diffraction Microscopy

The pattern center of an electron backscatter diffraction (EBSD) image indicates the relative position of the image with reference to the interaction volume of the sample. As interest grows in high-resolution EBSD techniques, accurate knowledge of this position is essential for precise interpretation of the EBSD features. In a typical EBSD framework, Kikuchi bands are recorded on a phosphor screen. If the flat phosphor were instead shaped as a sphere, with its center at the specimens electron interaction volume, then the incident backscattered electrons would form Kikuchi bands on that sphere with parallel band edges centered on great circles. In this article, the authors present a method of pattern center (PC) refinement that maps bands from the planar phosphor onto a virtual spherical screen and measures the deviation of bands from a great circle and from possessing parallel edges. Potential sources of noise and error, as well as methods for reducing these, are discussed. Finally, results are presented on the application of the PC algorithm to two types of simulated EBSD patterns and two experimental setups, and the resolution of the method is discussed.

Material selection for elastic energy absorption in origami-inspired compliant corrugations

Elastic absorption of kinetic energy and distribution of impact forces are required in many applications. Recent attention to the potential for using origami in engineering may provide new methods for energy absorption and force distribution. A three-stage strategy is presented for selecting materials for such origami-inspired designs that can deform to achieve a desired motion without yielding, absorb elastic strain energy, and be lightweight or cost effective. Two material indices are derived to meet these requirements based on compliant mechanism theory. Finite element analysis is used to investigate the effects of the material stiffness in the Miura-ori tessellation on its energy absorption and force distribution characteristics compared with a triangular wave corrugation. An example is presented of how the method can be used to select a material for a general energy absorption application of the Miura-ori. Whereas the focus of this study is the Miura-ori tessellation, the methods developed can be applied to other tessellated patterns used in energy absorbing or force distribution applications.

The effect of length scale on the determination of geometrically necessary dislocations via EBSD continuum dislocation microscopy

Electron backscatter diffraction (EBSD) dislocation microscopy is an important, emerging field in metals characterization. Currently, calculation of geometrically necessary dislocation (GND) density is problematic because it has been shown to depend on the step size of the EBSD scan used to investigate the sample. This paper models the change in calculated GND density as a function of step size statistically. The model provides selection criteria for EBSD step size as well as an estimate of the total dislocation content. Evaluation of a heterogeneously deformed tantalum specimen is used to asses the method.

High Resolution EBSD-Based Dislocation Microscopy

Significant advances are reported in the application of HR-EBSD to the imaging of the dislocation structure of polycrystalline materials. The central assumption of the method is the compatibility of the total displacement field, which relates the (Nye) dislocation tensor to the (partially measurable) curl of the elastic displacement field. Two key challenges must be addressed, including: a) the fundamental limitation imposed by the electron-opacity of typical materials, which limits the measurement of gradients in the displacement field in the direction normal to the sample surface; and b) the inability of HR-EBSD to recover the spherical (elastic) distortions of the lattice. This second challenge can be overcome if a traction free boundary condition is applied. It is illustrated that consideration of the familiar stress equilibrium relations gives additional information, which may enable estimates of the missing components of the Nye tensor. An example of application of HR-EBSD to a Mg-Ce sample is presented.

Efficient Propagation of Error Through System Models for Functions Common in Engineering

System modeling can help designers make and verify design decisions early in the design process if the model’s accuracy can be determined. The formula typically used to analytically propagate error is based on a first-order Taylor series expansion. Consequently, this formula can be wrong by one or more orders of magnitude for nonlinear systems. Clearly, adding higher-order terms increases the accuracy of the approximation, but it also requires higher computational cost. This paper shows that truncation error can be reduced and accuracy increased without additional computational cost by applying a predictable correction factor to lower-order approximations. The efficiency of this method is demonstrated in the kinematic model of a flapping wing. While Taylor series error propagation is typically applicable only to closedform equations, the procedure followed in this paper may be used with other types of models, provided that model outputs can be determined from model inputs, derivatives can be calculated, and truncation error is predictable.

Analysis of traction‐free assumption in high‐resolution EBSD measurements

The effects of using a traction‐free (plane‐stress) assumption to obtain the full distortion tensor from high‐resolution EBSD measurements are analyzed. Equations are derived which bound the traction‐free error arising from angular misorientation of the sample surface; the error in recovered distortion is shown to be quadratic with respect to that misorientation, and the maximum ‘safe’ angular misorientation is shown to be 2.7 degrees. The effects of localized stress fields on the traction‐free assumption are then examined by a numerical case study, which uses the Boussinesq formalism to model stress fields near a free surface. Except in cases where localized stress field sources occur very close to sample points, the traction‐free assumption appears to be admirably robust.

Estimation of the Full Nye Tensor by EBSD-Based Dislocation Microscopy

An extension to a previously published, novel stereological method is reported which infers experimentally inaccessible components of the Nye GND tensor. Limitations imposed by electron-opacity of metals prevent direct measurement of four components of the Nye tensor, but it is possible to use additional experimentally-obtainable information in connection with underlying field equilibrium equations to estimate these additional components. This approach uses derivatives to the infinitesimal elastic distortion tensor to reduce error imposed by pattern center inaccuracy.