Discrete Lattice Modeling of Wave Propagation in Materials with Heterogeneous Microstructures
Publication: Journal of Engineering Mechanics
Volume 147, Issue 10
Abstract
Many natural and human-made material systems (e.g., bone, shale, and cement-based composites) exhibit heterogeneous microstructures. Lattice models have reemerged to simulate such material systems because of their inherent simplicity while offering tremendous capabilities. However, current lattice models suffer from several deficiencies; for example, some lattice models cannot span the necessary range of Poisson’s ratio, some others are not isotropic (e.g., the standard square lattice), and others are not practical for complex domains (e.g., equilateral triangular lattice and hexagonal lattice). Thus, there is a need for a simple lattice that can handle Poisson’s ratio without any limitations, capture all the possible deformation modes of an isotropic elastic material, have minimal degrees-of-freedom, and provide positive definite stiffness and mass matrices. In this paper, we develop such a lattice model. Our approach hinges on equating the Lagrangians of the continuous (continuum) and discrete (lattice) systems, defining the strains consistently in terms of displacements in the lattice, adding a local interaction term to span the entire invariant space, and using an energy preserving time-stepping scheme. Using a Bloch wave analysis, we show that the lattice is, in fact, isotropic. Also, the lattice model does not suffer from volumetric locking, which is not the case with low-order finite elements. We verify the accuracy of the model using analytical solutions on benchmark problems. Finally, we demonstrate the application of the model on a practical example by performing propagation analysis in cement paste microstructure acquired from scanning electron microscopy (SEM).
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Data Availability Statement
The computer code is available from the corresponding author upon reasonable request.
Acknowledgments
This work was supported by the National Science Foundation (Grant No. 1825921). The authors acknowledge the use of the Sabine Computing Cluster at the University of Houston.
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Published In
Journal of Engineering Mechanics
Volume 147 • Issue 10 • October 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Dec 9, 2020
Accepted: May 3, 2021
Published online: Aug 9, 2021
Published in print: Oct 1, 2021
Discussion open until: Jan 9, 2022
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