Brent L. Adams

Electron Backscatter Diffraction Microscopy and Basic Stereology

This chapter provides a brief introduction to the experimental techniques used to characterize the mesoscale structure of polycrystalline metals discussed in this book. At this scale (typically smaller than the grain size in the sample), we wish to capture the details of the spatial distribution of crystal lattice orientation and correlate it to properties or processing conditions. These measurements are based on electron backscatter diffraction on 2D surfaces in the sample. The recently developed high-resolution electron backscatter diffraction (HREBSD) is also introduced in this chapter. Stereology may be employed to extract 3D measures and is described in the closing section.

Higher-Order Microstructure Representation

This chapter develops the fundamental concepts needed for a higher-order description of microstructure using correlation functions. In particular, the theoretical framework needed for computing two-point correlation functions is developed rigorously. As an example of Fourier representation of these functions, the use of a simple primitive basis is illustrated. Rigorous relationships linking the variance in the local state distribution function to the two-point correlations are derived. The main concepts for quantitative representation of the interface microstructure are presented, along with the associated concepts of higher-order correlations and measures of variance. The connections between interface character distribution and the two-point correlation functions are developed.

Second-Order Hull, Property Closure, and Design

This chapter extends the concepts of the microstructure hull and properties closures introduced in previous chapters to include considerations of two-point statistics. Various approaches for building second-order properties closures are described and illustrated with examples. Microstructure design using two-point statistics is explored at the end of this chapter.

Structure–Property Relations: Continuum Mechanics

This chapter introduces the basic concepts of continuum mechanics and their application in the rigorous description of the single crystal behavior. The concepts of the stress tensor, the deformation gradient tensor, the strain tensor, the velocity gradient tensor, the stretching tensor, and the spin tensor are all introduced with illustrative examples. Important relationships among these different physical quantities are derived and presented. The field equations for satisfying stress equilibrium are derived. The continuum mechanics concepts are then used to formulate constitutive descriptions for single crystal regions, while accounting for the crystal symmetry. The basic concepts of the crystal plasticity theory are introduced.

Tensors and Rotations

This chapter provides a concise review of the important mathematical concepts needed for later chapters. Specifically, the concepts of tensors, the main rules followed in tensor operations, the conventions used for tensor notations, and the concepts of eigenvalues, eigenvectors, and invariants are concisely presented. The concept of Euler angles is introduced to describe relationships between two arbitrarily selected references frames in 3D space. The differences between coordinate transformations and rotations are clarified.

Higher-Order Homogenization

This chapter presents a highly sophisticated higher-order homogenization framework based on the statistical continuum theories developed by Kroner. This framework allows one to account for the details of the spatial distributions ( n -point correlations) in the microstructure in arriving at the homogenized values of properties for a selected microstructure. This chapter first presents a simpler form of the higher-order homogenization theory that is targeted for low-contrast composites. This is then followed by a formulation for high-contrast composites.

Description of the Microstructure

This chapter introduces the fundamental concepts of microstructure quantification used in MSDPO. We employ a framework for stochastic description of the microstructure using distribution functions defined over local state spaces of interest for the selected design problems. As examples, we introduce the grain size distribution function, the orientation distribution function, the grain boundary inclination distribution function, and the microstructure function. This chapter also introduces the concept of the fundamental zone in the orientation space.

Microstructure Hull and Closures

This chapter introduces the fundamental concepts of microstructure hull and property closure. These concepts are central to MSDPO. The importance of spectral representations in realizing these constructs is illustrated through simple examples. More complex case studies follow in Chapter 10 .

Chapter 8 – Homogenization Theories

This chapter introduces the basic theories of first-order bounds for effective properties of composites. Rigorous bounds are presented for the diagonal components of the elastic stiffness and compliance tensors, and degraded bounds are presented for the off-diagonal components. The first-order bounds for thermal expansion are presented for specific selected material systems.

Chapter 11 – Microstructure Evolution by Processing

This chapter introduces novel concepts for capturing the details of microstructure evolution within the framework of first-order microstructure quantification presented in previous chapters. Details of texture evolution are captured in an efficient spectral framework. The main points of the framework are illustrated through a simple process design example where the goal is to achieve a desired final texture using a combination of available deformation processing options.