Custom design, manufacture, and deployment of new high performance materials for advanced technologies is critically dependent on the availability of invertible, high fidelity, structure-property-processing (SPP) linkages. Establishing these linkages presents a major challenge because of the need to cover unimaginably large dimensional spaces. Hierarchical Materials Informatics addresses objective, computationally efficient, mining of large ensembles of experimental and modeling datasets to extract this core materials knowledge. Furthermore, it aims to organize and present this high value knowledge in highly accessible forms to end users engaged in product design and design for manufacturing efforts. As such, this emerging field has a pivotal role in realizing the goals outlined in current strategic national initiatives such as the Materials Genome Initiative (MGI) and the Advanced Manufacturing Partnership (AMP). This book presents the foundational elements of this new discipline as it relates to the design, development, and deployment of hierarchical materials critical to advanced technologies.
- Addresses a critical gap in new materials research and development by presenting a rigorous statistical framework for the quantification of microstructure
- Contains several case studies illustrating the use of modern data analytic tools on microstructure datasets (both experimental and modeling)
Surya R. Kalidindi earned a B.Tech. in Civil Engineering from the Indian Institute of Technology, Madras, an M.S. in Civil Engineering from Case Western Reserve University, and a Ph.D. in Mechanical Engineering from the Massachusetts Institute of Technology. After his graduation from MIT in 1992, Surya joined the Department of Materials Science and Engineering at Drexel University as an Assistant Professor, where he served as the Department Head during 2000-2008. In 2013, Surya accepted a new position as a Professor of Mechanical Engineering in the George W. Woodruff School at Georgia Institute of Technology, with joint appointments in the School of Computational Science and Engineering and in the School of Materials Science and Engineering. Surya's research efforts over the past two decades have made seminal contributions to the fields of crystal plasticity, microstructure design, spherical nanoindentation, and materials informatics. His work has already produced about 200 journal articles, four book chapters, and a new book on Microstructure Sensitive Design. His work is well cited by peer researchers as reflected by an h-index of 52 and current citation rate of about 1000 citations/year (Google Scholar). He has recently been awarded the Alexander von Humboldt award in recognition of his lifetime achievements in research. He has been elected a Fellow of ASME, ASM International, TMS, and Alpha Sigma Mu.
This chapter introduces the foundational concept of microstructure function as a spatially resolved probability distribution over the local state space. After introducing the central concept, this chapter explores key ideas on spectral representation of this function. The main advantages of discrete Fourier transform representations for the functional dependence on the spatial variable are discussed in great detail. Likewise, the advantages of employing suitable Fourier representations for the functional dependencies on the local state descriptors are discussed with several examples. It is also demonstrated that the concept is broadly applicable to a variety of material systems at different length/structure scales.
Microstructure function; local state; local state space; spectral representation; discrete Fourier transforms (DFTs); spectral interpolation
The previous chapter emphasized the importance of microstructure description and quantification, not only as a digital representation of the material itself but also as the main foundational block in establishing the high value PSP linkages needed to accelerate the materials development efforts. In this chapter, we now take a deep dive into a mathematically rigorous framework for the representation of the material microstructure. The reader is reminded that this framework has to necessarily address the hierarchy of material internal structure (spanning multiple length scales). As noted earlier, it is anticipated that most advanced materials used in emerging technologies will demand a tiered description of the material microstructure. In developing and presenting the fundamental concepts of such a framework, we will limit our attention in this chapter to two well separated length scales. It is assumed that the same overall philosophy can be applied repeatedly, as many times as needed, in describing materials whose microstructures exhibit salient features at multiple well separated length scales.
2.1 Length Scales
Let us now take a closer look at exactly what is meant by well separated length scales. This notion comes mainly from the concept of homogenization employed routinely in composite theories . In both the hierarchical description of the material microstructure as well as in efficient scale-bridging, we need to first understand and identify the pertinent length scales. Figure 2.1 depicts an example for a hypothetical composite material system, where the length scales of interest are the macroscale and the mesoscale. In this figure, L and l implicitly define the length scales of a material point at the two scales being investigated in some pertinent multiscale material phenomenon (e.g., the failure properties of the composite system subjected to a specified macroscale loading condition for which it is well known that the mesoscale distribution of stress plays a dominant role). Figure 2.1
Schematic description of the length scales involved in hierarchical description of microstructure and the use of homogenization theories.
In the example in Figure 2.1, L and l are selected such that they represent the smallest acceptable homogenization length scales for the macroscale and mesoscale, respectively. This essentially implies that it is possible (and reasonable) to formulate a sufficiently accurate homogenized material constitutive description at each of these length scales, by accounting for all of the salient (and often complex) features of the material structure that exist below that length scale. In other words, for the example shown in Figure 2.1, it is implicitly assumed that a volume of material of the order of L3 can be attributed with an effective value of a macroscale property of interest, denoted P* (generally a tensor, e.g., elastic stiffness, conductivity, coefficient of thermal expansion). It is deemed that this effective property allows a sufficiently accurate description of the material constitutive response for all considerations at the macroscale. For example, it is customary to describe the linear elastic response of a composite material as
*=C*e* (2.1) (2.1)
where s* and e* are suitably defined second-rank stress and strain tensors (these will be introduced formally in later chapters) at the macroscale, and C* is the effective macroscale material property of interest called the elastic stiffness (a fourth-rank tensor; this will also be formally introduced in later chapters). It is extremely important to identity and define L such that it facilitates a sufficiently accurate description of the material constitutive response for all design considerations at the macroscale. Implicitly, this requirement also means that L is significantly smaller than the length scales associated with the gradients of the fields of interest at the macroscale, i.e., it is assumed that the macroscale quantities such as stress and strain themselves do not vary much over the length scale L (obviously, if the macroscale measures of stress or strain vary significantly over lengths of the order of L, it precludes the description of the homogenized material response by Eq. (2.1)). It should be recognized that the actual value of L can vary dramatically depending on the specific application, from less than millimeters in small devices to centimeters or meters in larger structures.
The considerations at the mesoscale in Figure 2.1 are highly analogous to those described earlier for the macroscale. At the mesoscale, it is again assumed that there is a suitable length scale l, which allows us to describe the homogenized material constitutive response in a sufficiently accurate manner by accounting for all of the important material structure details at length scales much smaller than l. In other words, we expect the material structure to exhibit additional heterogeneity at length scales well below the length scale l (remember that we are dealing with hierarchical materials). In many ways, therefore, we are implicitly making the same assumptions in identifying length scale l, as we did earlier in identifying the length scale L (in a recursive manner).
In practice, the identification or selection of the length scales described above occurs quite naturally in most materials datasets generated by both experiments and models. In the experiments, the length scale is often set by the resolution limits of the specific machine used to acquire the datasets. For example, in any optical or electron micrograph, the length scale implicitly associated with a material point is the spatial resolution limit at which the image was acquired. Note the characterization machine implicitly provides information averaged at the spatial resolution limit. As another example, in any orientation map obtained by electron backscattered diffraction (EBSD), the length scale of each material point is typically associated with the measurement step size. Although the probe volumes in such measurements may be significantly smaller than the measurement step size, the lack of additional information forces us to adopt the step size as the appropriate length scale. In numerical simulations (e.g., using finite element methods or finite difference methods), each material point used in the computation is inherently associated with a specific length scale (e.g., the size of the element in the mesh or the computation grid) and the properties assigned to the material point are implicitly assumed to reflect averaged values over volumes defined by that length scale.
The concept of well separated length scales alluded to earlier implies that the hierarchical material being studied satisfies all of the requirements described above, while ensuring lL. This is very important for virtually all of the considerations described in this book because it is the main requirement for invoking the concept of a representative volume element (RVE). The RVE concept essentially allows us to identify volume element of size ?l at the lower length scale in such a manner that it effectively captures all of the salient features of the material internal structure at the lower length scale (Figure 2.1), with l<?lL. Without this simplification, scale-bridging for multiscale investigation of any materials phenomena of interest would become computationally impractical, as we would be forced to consider RVEs of size L.
The most commonly adopted definitions of RVE size in the current literature [2-9] focus largely on the convergence in the prediction of selected macroscale (effective) properties (e.g., C* in Eq. (2.1)) and do not explicitly consider whether or not the RVE has captured the desired microstructural details to sufficient accuracy. Incidentally, the classical definition of RVE provided by Hill  requires the RVEs to be large enough to capture both the representative microstructure as well as its homogenized effective properties. In this book, we will deviate from the current approaches in the literature and focus first on capturing the salient microstructure features in the RVE, and then subsequently address the convergence of the predictions of macroscale properties. The main motivation for this approach is the expectation that an RVE that already captures the salient microstructure features in the sample...