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The advance of additive manufacturing makes it possible to design spatially varying lattice structures with complex geometric configurations. The homogenized elastic properties of these periodic lattice structures are known to deviate significantly from isotropic behavior where orthotropic material symmetry is often assumed. This paper addresses the need for a robust homogenization method for evaluating anisotropy of periodic lattice structures including an understanding of how the elastic properties transform under rotation. Here, periodic boundary conditions are applied on two-material representative volume element (RVE) finite element models to evaluate the complete homogenized stiffness tensor. A constrained multi-output regression approach is proposed to evaluate the elasticity tensor components under any assumed material symmetry model. This approach is applied to various lattice structures including scaffold and surface-based Triply Periodic Minimal Surface (TPMS). Our approach is used to assess the accuracy of rotation for assumed anisotropic and orthotropic homogenized material models over a range of lattice structures.
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Numerical Homogenization of Orthotropic Functionally Graded Periodic Cellular Materials: Method Development and Implementation
This study advances the state of the art by computing the macroscopic elastic properties of 2D periodic functionally graded microcellular materials, incorporating both isotropic and orthotropic solid phases, as seen in additively manufactured components. This is achieved through numerical homogenization and several novel MATLAB implementations (known in this study as Cellular_Solid, Homogenize_test, homogenize_ortho, and Homogenize_test_ortho_principal). The developed codes in the current work treat each cell as a material point, compute the corresponding cell elasticity tensor using numerical homogenization, and assign it to that specific point. This is conducted based on the principle of scale separation, which is a fundamental concept in homogenization theory. Then, by deriving a fit function that maps the entire material domain, the homogenized material properties are predicted at any desired point. It is shown that this method is very capable of capturing the effects of orthotropy during the solid phase of the material and that it effectively accounts for the influence of void geometry on the macroscopic anisotropies, since the obtained elasticity tensor has different đ¸1 and đ¸2 values. Also, it is revealed that the complexity of the void patterns and the intensity of the void size changes from one cell to another can significantly affect the overall error in terms of the predicted material properties. As the stochasticity in the void sizes increases, the error also tends to increase, since it becomes more challenging to interpolate the data accurately. Therefore, utilizing advanced computational techniques, such as more sophisticated fitting methods like the Fourier series, and implementing machine learning algorithms can significantly improve the overall accuracy of the results. Furthermore, the developed codes can easily be extended to accommodate the homogenization of composite materials incorporating multiple orthotropic phases. This implementation is limited to periodic void distributions and currently supports circular, rectangular, square, and hexagonal void shapes.
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- Award ID(s):
- 2317406
- PAR ID:
- 10559741
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Materials
- Volume:
- 17
- Issue:
- 24
- ISSN:
- 1996-1944
- Page Range / eLocation ID:
- 6080
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/ma17246080
