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Article

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

1
Ishlinsky Institute for Problems in Mechanics RAS, 119526 Moscow, Russia
2
The Head Educational, Research and Methodological Center for Vocational Rehabilitation of Persons with Disabilities, Bauman Moscow State Technical University, 105005 Moscow, Russia
3
MEMSEC R&D Center, National Research University of Electronic Technology (MIET), 124498 Moscow, Russia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editors: David Steigmann and Francesco Dell’Isola
Symmetry 2021, 13(12), 2331; https://doi.org/10.3390/sym13122331
Received: 31 October 2021 / Revised: 29 November 2021 / Accepted: 1 December 2021 / Published: 5 December 2021
(This article belongs to the Special Issue Applications of Differential Geometry to Continuum Mechanics)
The present paper is intended to show the close interrelationship between non-linear models of solids, produced with additive manufacturing, and models of solids with distributed defects. The common feature of these models is the incompatibility of local deformations. Meanwhile, in contrast with the conventional statement of the problems for solids with defects, the distribution for incompatible local deformations in additively created deformable body is not known a priori, and can be found from the solution of the specific evolutionary problem. The statement of the problem is related to the mechanical and physical peculiarities of the additive process. The specific character of incompatible deformations, evolved in additive manufactured solids, could be completely characterized within a differential-geometric approach by specific affine connection. This approach results in a global definition of the unstressed reference shape in non-Euclidean space. The paper is focused on such a formalism. One more common factor is the dataset which yields a full description of the response of a hyperelastic solid with distributed defects and a similar dataset for the additively manufactured one. In both cases, one can define a triple: elastic potential, gauged at stress-free state, and reference shape, and some specific field of incompatible relaxing distortion, related to the given stressed shape. Optionally, the last element of the triple may be replaced by some geometrical characteristics of the non-Euclidean reference shape, such as torsion, curvature, or, equivalently, as the density of defects. All the mentioned conformities are illustrated in the paper with a non-linear problem for a hyperelastic hollow ball. View Full-Text
Keywords: incompatible deformations; additive technology; residual stresses; distortion of final shape; distributed defects; material connection; curvature and torsion; hyperelasticity incompatible deformations; additive technology; residual stresses; distortion of final shape; distributed defects; material connection; curvature and torsion; hyperelasticity
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MDPI and ACS Style

Lychev, S.; Koifman, K.; Djuzhev, N. Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches. Symmetry 2021, 13, 2331. https://doi.org/10.3390/sym13122331

AMA Style

Lychev S, Koifman K, Djuzhev N. Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches. Symmetry. 2021; 13(12):2331. https://doi.org/10.3390/sym13122331

Chicago/Turabian Style

Lychev, Sergey, Konstantin Koifman, and Nikolay Djuzhev. 2021. "Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches" Symmetry 13, no. 12: 2331. https://doi.org/10.3390/sym13122331

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