Alternative multiscale material and structures modeling by the finite-element method
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ORIGINAL ARTICLE
Alternative multiscale material and structures modeling by the finite‑element method H. B. Coda1 · R. A. K. Sanches1 · R. R. Paccola1 Received: 13 February 2020 / Accepted: 14 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this study, we present an alternative finite element to multiscale analysis. In this strategy, strain energy comes only from semi-discrete or lattice elements immersed in a continuum without stiffness, enabling mechanical analysis from molecular scales to macroscopic scales. Some characteristics of the proposed element are: (1) geometrically non-linear exact description that allows the presence of large displacements and large strain, (2) general mapping that allows curved and distorted elements generation with automatic immersions, (3) total compatibility with standard finite elements, and (4) huge degrees of freedom reduction with a small loss of continuum mobility. Throughout the text, the proposed strategy is presented in detail and applied in the determination of suitable meshes for any scale of analysis, which is an important information for future applications. To be direct, a Lennard–Jones-like (LJL) potential is chosen to build different crystalline-like structures that, when immersed in finite elements without stiffness, results in the desired continuous behavior. In this sense, some space of the paper is used to determine the energy constant of the LJL potential for these different "crystalline" structures at any scale. Taking advantage of the total compatibility of the proposed element with continuum elements, the multiscale strategy is straightforward applied. Selected examples are used to demonstrate the good behavior of the proposed element and its applicability. Future developments to enhance applications are commented at the conclusion section. Keywords Finite elements · Multiscale mechanical problems · Geometrical and physical nonlinearity
1 Introduction The finite-element method (FEM) is a powerful technique for numerical analysis of engineering and physical problems including continuum and molecular mechanics. In the context of mechanical problems, regardless of the method that is applied, but maintaining our discussion within the FEM, in certain applications, one can imagine that the continuum mechanics strategies are better adapted than molecular mechanics. However, everything that occurs in the material occurs at the atomic level and the continuum mechanics is * R. R. Paccola [email protected] H. B. Coda [email protected] R. A. K. Sanches [email protected] 1
São Carlos School of Engineering, University of São Paulo, Av Trabalhador São Carlense, 400, São Carlos, SP 13560‑590, Brazil
able to solve problems only in an average way, losing details and, sometimes, the solid behaves more as an agglomerate of particles than a real continuum. This affirmation is supported, for example, by Ref. [1] that presents a multiresolution molecular mechanics trying to enable continuum applications from discrete modeling selecting re
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