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D. Patel et al.
The broad implementation of data-driven approach requires validated and widely
adopted methods to describe and quantify hierarchical material structure(s) referred
to as microstructure(s). The performance behavior of materials is largely characterized and governed by the microstructure of the sample material; hence,
microstructure plays a significant role in the formulation of P-S-P linkages and
forms an input and/or output to the surrogate models. Accurate quantification of
microstructure is essential to determining the macroscopic constitutive response of
the materials subjected to various loading/processing conditions. It is imperative to
seek proper quantification of microstructures before addressing the formulation of
P-S-P linkages.
2.1 Microstructure Quantification
The success of materials design efforts hinges on how the salient microstructural
features of materials are quantified and tracked during various process/loading
conditions. Conventionally, the approach to microstructure quantification is motivated largely by our current partial understanding of physics, which includes
measures such as mean volume fraction, average grain size (and sometimes shape),
and rarely ensemble distributions. It is now increasingly recognized that there
are significant variations of all these parameters and they also miss important
topological information. Therefore, it is increasingly recognized that the statistical
information about the structure that includes topology is important to predict the
design allowables that depend on the variability of properties of interest.
Recently, various methods have been proposed and applied to quantify the
microstructure of heterogeneous materials at relevant length scales. For example,
correlation functions such as pair correlation functions [7], radial distribution functions [8], chord length distribution functions [9, 10], and n-point spatial correlation
functions [3, 9–12] at different length scales have been utilized to represent diverse
material systems ranging from solids to gaseous phases. A correlation function of
a variable of interest is a measure of the spatial order/disorder in a system with
respect to that variable. In addition, correlation functions quantify how different
variables covary with one another spatially. Studies have shown that these spatial
correlation functions are the microstructural features that naturally emerge in some
of the most sophisticated composite models/theories for effective, homogenized,
properties [13].
Although the quantification of microstructures using various correlation functions provides a means to capture the salient features and its interactions, the
statistical representation of the microstructure often span an unwieldy dimensional
space in comparison to the other variable(s) in the P-S-P linkages formulation [3, 5,
10–12]. From a practical perspective, the statistical representation of microstructure
needs to be cast in a compressed form for it to serve any useful purpose in order
to produce simple and easily applicable P-S-P linkages. Traditional approaches to
D. Patel et al.
The broad implementation of data-driven approach requires validated and widely
adopted methods to describe and quantify hierarchical material structure(s) referred
to as microstructure(s). The performance behavior of materials is largely characterized and governed by the microstructure of the sample material; hence,
microstructure plays a significant role in the formulation of P-S-P linkages and
forms an input and/or output to the surrogate models. Accurate quantification of
microstructure is essential to determining the macroscopic constitutive response of
the materials subjected to various loading/processing conditions. It is imperative to
seek proper quantification of microstructures before addressing the formulation of
P-S-P linkages.
2.1 Microstructure Quantification
The success of materials design efforts hinges on how the salient microstructural
features of materials are quantified and tracked during various process/loading
conditions. Conventionally, the approach to microstructure quantification is motivated largely by our current partial understanding of physics, which includes
measures such as mean volume fraction, average grain size (and sometimes shape),
and rarely ensemble distributions. It is now increasingly recognized that there
are significant variations of all these parameters and they also miss important
topological information. Therefore, it is increasingly recognized that the statistical
information about the structure that includes topology is important to predict the
design allowables that depend on the variability of properties of interest.
Recently, various methods have been proposed and applied to quantify the
microstructure of heterogeneous materials at relevant length scales. For example,
correlation functions such as pair correlation functions [7], radial distribution functions [8], chord length distribution functions [9, 10], and n-point spatial correlation
functions [3, 9–12] at different length scales have been utilized to represent diverse
material systems ranging from solids to gaseous phases. A correlation function of
a variable of interest is a measure of the spatial order/disorder in a system with
respect to that variable. In addition, correlation functions quantify how different
variables covary with one another spatially. Studies have shown that these spatial
correlation functions are the microstructural features that naturally emerge in some
of the most sophisticated composite models/theories for effective, homogenized,
properties [13].
Although the quantification of microstructures using various correlation functions provides a means to capture the salient features and its interactions, the
statistical representation of the microstructure often span an unwieldy dimensional
space in comparison to the other variable(s) in the P-S-P linkages formulation [3, 5,
10–12]. From a practical perspective, the statistical representation of microstructure
needs to be cast in a compressed form for it to serve any useful purpose in order
to produce simple and easily applicable P-S-P linkages. Traditional approaches to