- Ruhr-Universität Bochum

The characterization of three-dimensional (3D) microstructures that captures the essential features of a given material is oftentimes desirable for determining critical mechanisms of deformation and failure and for conducting computational modeling to predict the material’s behavior under thermo-mechanical loading conditions. However, acquiring 3D microstructure representations is costly and time-consuming because standard microscopic procedures can only produce 2D surface maps. Hence, current state-of-the-art methods for 3D microstructure characterization are serial sectioning techniques or X-ray tomography. Both methods produce a truthful characterization of the 3D structure of an individual specimen but require a rather high effort both in sample preparation and in software-based reconstruction of the 3D structure. In this work, an alternative path is suggested to generate synthetic 3D microstructures that resemble real ones in a statistical sense with a severely reduced effort, as only 2D surface maps from three orthogonal surfaces are used. The method is based on an inverse procedure that generates synthetic 3D microstructures with arbitrary parameters and then compares these artificial surface maps with the real ones. In an iterative procedure, the parameters of the microstructure generator are optimized until the best possible agreement between the corresponding surface maps of synthetic and real microstructures is achieved. In this way, the statistical descriptors of the real microstructure are gained as they are represented by the converged input parameters for the microstructure generator. Advantages of this method are that, after the converged microstructure parameters are obtained, different realizations of statistically equivalent 3D microstructures can be generated and also systematic parametric can be conducted by varying individual microstructure features. In this way, the influence of microstructure features on the material properties can be predicted by micromechanical simulations.

The work presented here focuses on microstructures of metastable austenitic steels where austenite and deformation-induced a-martensite co-exist at room temperature. The microstructure of this dual-phase steel is characterized by electron backscatter diffraction (EBSD) microscopy to produce three maps from orthogonal surfaces. The 3D microstructure reconstruction is performed in the way described above based on three EBSD maps from orthogonal surfaces of a dual-phase steel sample. As illustrated in the figure above, the workflow for such reconstructions incorporates a microstructure generator tool from the open-source python package pyMKS to produce synthetic 3D dual-phase microstructures based on physical parameters, such as volume fraction and phase shape. The input in the figure above is represented by three EBSD maps from orthogonal surfaces of a dual-phase steel (yellow: martensite, blue: austenite) from which a low-dimensional, yet representative vector of descriptors is extracted. The corresponding descriptors are generated from a synthetic 3D microstructure (red: austenite, blue: austenite) such that a scalar loss function can be evaluated to assess how similar the surface maps of real and synthetic microstructures are. In an iterative procedure, the parameters of the 3D microstructure generator are optimized such that the loss function becomes minimal. From the surfaces of the synthetic microstructure, 2D images are produced similarly to the surface maps of the real material. The primary challenge in minimizing the differences between real and synthetic surface maps lies in defining a proper loss function that quantifies the differences between surface maps in a physically sound yet numerically efficient way. In the present work, it is demonstrated that processing surface maps by spatial correlation functions, often referred to as 2-point statistics, and principal component analysis (PCA) results in a small set of unique descriptors that serve as a fingerprint of the 2D maps. These descriptors encode the topological information of 2D maps in a compact format and can be used to characterize both experimental and synthetic surface maps. In this way, the differences between the two surface maps can be quantified and iteratively minimized. After convergence, the synthetic 3D microstructure accurately describes the experimental system in terms of physical parameters such as volume fraction (here: 15% austenite) and phase shapes (here: aspect ratio of 20:4:3 for martensite regions). Hence, the presented approach ensures that the 3D reconstructed sample and the associated 2D surface maps are statistically equivalent.

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