Published 6 issues per year
ISSN Print: 1543-1649
ISSN Online: 1940-4352
Indexed in
Generalized Mathematical Homogenization of Atomistic Media at Finite Temperatures
ABSTRACT
In this manuscript, we derive thermomechanical continuum equations directly from molecular dynamics using the generalized mathematical homogenization (GMH) theory. GMH is a space-time multiple-scale asymptotic expansion method, which constructs the coupled atomistic unit-cell problem and the coarse-scale (continuum) problem. The fine-scale problem derived can be interpreted as a molecular dynamics problem on a unit cell, subjected to the coarse-scale fields including macroscopic deformation gradient and temperature. The coarse-scale problem derived is a constitutive law-free continuum thermomechanical equation, which calculates the overall stress and thermal flux vector directly from atomistics. Numerical experiments have been conducted to verify the formulation against the reference molecular dynamics solution. Attention is restricted to one-dimensional problems.
-
Yuan Zheng, Fish Jacob, Hierarchical model reduction at multiple scales, International Journal for Numerical Methods in Engineering, 79, 3, 2009. Crossref
-
Fish Jacob, Li Aiqin, Yavari Fazel, Adaptive generalized mathematical homogenization framework for nanostructured materials, International Journal for Numerical Methods in Engineering, 83, 8-9, 2010. Crossref
-
Cho Maenghyo, Yang Seunghwa, Chang Seongmin, Yu Suyoung, A study on the prediction of the mechanical properties of nanoparticulate composites using the homogenization method with the effective interface concept, International Journal for Numerical Methods in Engineering, 85, 12, 2011. Crossref
-
Rahul , De Suvranu, An efficient block preconditioner for Jacobian-free global-local multiscale methods, International Journal for Numerical Methods in Engineering, 87, 7, 2011. Crossref
-
Li Aiqin, Li Renge, Fish Jacob, Generalized Mathematical Homogenization: From theory to practice, Computer Methods in Applied Mechanics and Engineering, 197, 41-42, 2008. Crossref
-
Rahul , De Suvranu, Analysis of the Jacobian-free multiscale method (JFMM), Computational Mechanics, 56, 5, 2015. Crossref
-
Engelbrecht Jüri, Berezovski Arkadi, Reflections on mathematical models of deformation waves in elastic microstructured solids, Mathematics and Mechanics of Complex Systems, 3, 1, 2015. Crossref
-
Fish Jacob, Bridging the scales in nano engineering and science, Journal of Nanoparticle Research, 8, 5, 2006. Crossref
-
Murín Justín, Kutiš Vladimír, Masný Michal, Ďuriš Rastislav, Composite (FGM’s) Beam Finite Elements, in Composites with Micro- and Nano-Structure, 9, 2008. Crossref
-
Ma Hang, Qin Qing-Hua, Kompiš Vladimir, Computational Modal and Solution Procedure for Inhomogeneous Materials with Eigen-Strain Formulation of Boundary Integral Equations, in Composites with Micro- and Nano-Structure, 9, 2008. Crossref
-
Abdellatef Mohammed, Alnaggar Mohammed, Energy-Based Coarse Graining of the Lattice-Discrete Particle Model, Journal of Engineering Mechanics, 146, 5, 2020. Crossref
-
Berezovski A., Engelbrecht J., Salupere A., Tamm K., Peets T., Berezovski M., Dispersive waves in microstructured solids, International Journal of Solids and Structures, 50, 11-12, 2013. Crossref