From MMGWiki

Contents 1 Microstructure Sensitive Design for Performance Optimization 2 Design from the Property Closure 3 Design from the Microstructure Hull 3.1 Discretized Microstructure Hull 3.2 Microstructure Hull Delineated using Gram-Schmidt Orthonormalization 3.3 Exploiting the Convexity of the Microstructure Hull 4 References

Microstructure Sensitive Design for Performance Optimization

Current design practices neglect the constitutive material as a continuous design property. These methods focus on the optimizing the geometry of component while considering the material properties to isotropic and continuous throughout the part. The description of the material in this manner is incomplete; most commercially available materials used are polycrystalline and often possess a non-random distribution of crystal lattice orientations (as a consequence of complex thermo-mechanical loading history experienced in their manufacture). As a result, it is expected that the material should exhibit anisotropic properties.

Microstructure Sensitive Design offers a powerful mathematical framework that can describe the microstructure as a continuous design variable in the design process. MSD offers a method of statistically quantify the distribution of crystallographic orientations in a material while subsequently linking them to material properties.

The main focus of MSD has, thus far, been on the crystallographic texture (also called Orientation Distribution Function or the ODF) in polycrystalline metals and the macroscale elastic-plastic properties that are strongly influenced by this specific microstructural detail. Limited explorations have also been conducted into compositional variations in microstructures [1] and into fiber reinforced composites [2]. In cubic polycrystalline metals, the MSD framework has been successfully applied to a few design case studies. These have included maximizing the deflection in a compliant beam [3], maximizing the in-plane load carrying capacity of a thin plate with a central circular hole [4], and minimizing the elastic driving force for crack extension in rotating disks [5] and internally pressurized thin-walled vessels [6]. Property closures were also produced for a broad range of combinations of macroscale effective elastic and plastic properties in cubic metals [6-8] and for a limited number of hexagonal metals [9].

Design from the Property Closure

Design case studies were explored using the property closure as the design space [1]. This approach has the inherent advantage of greatly reducing the design space by mapping a higher dimensional Fourier space into a reduced dimensional property space. This method is particularly attractive when the property space (also called the property closure) is compact and convex. However, when the property space is not convex, defining the boundary of the property closure becomes extremely challenging, especially in design case studies where the overall performance is governed by a large number of macroscale properties. It should be noted that the property closures for a vast number of design problems are likely to be non-convex.

Design from the Microstructure Hull

Discretized Microstructure Hull

The first MSD design case studies were performed by discretizing the microstructure hull into bins. The performance at each bin was evaluated under the assumption that all the microstructures in each bin exhibited the same performance [4,5]. This method was very computationally inefficient which restricted the dimensionality of the microstructure hull to a low dimensionality.

Microstructure Hull Delineated using Gram-Schmidt Orthonormalization

The first case studies that fully explored the design space delineated the limits of the microstructure hull using Gram-Schmidt orthonormalization(GSO). This method was integrated into Finite Element Methods and optimized using a generalized reduced gradient method to optimize the performance of a composite plate [6] and turbine blade[5]. The GSO method is an efficient algorithm, however it is difficult to apply to case studies requiring a larger Fourier expansion.

Exploiting the Convexity of the Microstructure Hull

References

• [1]Saheli, G., H. Garmestani, et al. (2004). "Microstructure design of a two phase composite using two-point correlation functions." Journal of Computer-Aided Materials Design 11: 103–115.
• [2]Kalidindi, S. R. and J. R. Houskamp (2007). "Application of the Spectral Methods of Microstructure Design to Continuous Fiber Reinforced Composites." Journal of Composite Materials 41: 909-930.
• [3]Adams, B. L., A. Henrie, et al. (2001). "Microstructure-sensitive design of a compliant beam." Journal of the Mechanics and Physics of Solids 49(8): 1639-1663.
• [4]Kalidindi, S. R., J. R. Houskamp, et al. (2004). "Microstructure sensitive design of an orthotropic plate subjected to tensile load." International Journal of Plasticity 20(8-9): 1561-1575.
• [5]Sintay, D. S. and B. L. Adams (2005). Microstructure design for a rotating disk: with application to turbine engines. IDETC/CIE, 31st Design automation conference. Long Beach, California USA.
• [6]Houskamp, J. R., G. Proust, et al. (2007, Submitted). "Integration of Microstructure Sensitive Design with Finite Element Methods: Elastic-Plastic Case Studies in FCC Polycrystals." Int. J. Multiscale Computational Engineering.
• [7]Proust, G. and S. R. Kalidindi (2006). "Procedures for construction of anisotropic elastic-plastic property closures for face-centered cubic polycrystals using first-order bounding relations." Journal of the Mechanics and Physics of Solids 54(8): 1744-1762.
• [8]Knezevic, M. and S. R. Kalidindi (2007). "Fast computation of first-order elastic-plastic closures for polycrystalline cubic-orthorhombic microstructures." Computational Materials Science 39(3): 643-648.
• [9]Wu, X., G. Proust, et al. (2007). "Elastic-plastic property closures for hexagonal close-packed polycrystalline metals using first-order bounding theories." Acta Materialia 55(8): 2729-2737.